Florida State University Libraries
Electronic Theses, Treatises and Dissertations
The Graduate School
2005
Making Sense of Students' Understanding
of Fractions: An Exploratory Study of Sixth
Graders' Construction of Fraction Concepts
Through the Use of Physical Referents and
Real World Representations
Veon Stewart
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THE FLORIDA STATE UNIVERSITY
COLLEGE OF EDUCATION
MAKING SENSE OF STUDENTS’ UNDERSTANDING OF FRACTIONS:
AN EXPLORATORY STUDY OF SIXTH GRADERS’ CONSTRUCTION OF
FRACTION CONCEPTS THROUGH THE USE OF PHYSICAL REFERENTS
AND REAL WORLD REPRESENTATIONS
BY
VEON MURDOCK-STEWART
A Dissertation submitted to the
Department of Middle and Secondary Education
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Degree Awarded:
Fall Semester, 2005
Copyright © 2005
Veon Murdock-Stewart
All Rights Reserved
The members of the committee approved the dissertation of Veon MurdockStewart defended on October 12, 2005.
____________________________________
Elizabeth Jakubowski
Professor Directing Dissertation
____________________________________
Florentina Bunea
Outside Committee Member
____________________________________
Leslie Aspinwall
Committee Member
____________________________________
Maria L. Fernandez
Committee Member
Approved:
__________________________________________
Pamela Carroll, Chairperson, Middle and Secondary Education
The Office of Graduate Studies has verified and approved the above named committee
members.
ii
ACKNOWLEDGMENTS
I would like to express my profound gratitude to my Heavenly Father for His
constant watch care and provision throughout my entire life. All glory, praise and honor
to the One whom I adore. “How precious also are thy thoughts unto me, O God! How
great is the sum of them” (Psalm 139:17).
I could not have completed this work without the help of my committee members.
First, I would like to thank my major professor, Dr. Elizabeth Jakubowski for motivating
me to complete this degree. Thank you for all your advice. Thank you for being there
when I needed you the most. Special thanks to Dr. Leslie Aspinwall for your thoughtful
instructions; Dr. Maria Fernandez for your keen eye for perfection – I have really learned
a lot from you; and to Dr. Florentina Bunea who made Probability come alive for me.
Special thanks to the many individuals who have provided me guidance and
support throughout my doctoral journey. Without you this milestone would have been
more tedious. The Adamson’s family has been a tower of strength and support over the
years – THANK YOU. Richard and Daisy Cousins, thanks for being the brother and
sister that I needed and still need in Tallahassee. Words cannot express how grateful I
am for ALL that you did for me. Rose, and the rest of my friends in Tallahassee – thanks
a million.
To all my mathematics teachers from kindergarten to university, thanks for
instilling in me the love for this subject. Special thanks to my mentor, Avery Thompson.
My principal, Mr. Othniel Scott, thank you for believing in me. Thank you for giving me
the permission to work with twenty wonderful children. I would also like to express my
appreciation to those who volunteered to edit my paper: Mr. Edmund Harty, Ms. Martha
Morton and Mrs. Patty Hall. May God continue to bless you.
A BIG thank you to my parents who believe in education. Thank you for your
support throughout my many years of schooling. To the rest of my family members and
friends – your prayers meant a lot. Thank you.
Last, but definitely not least, the BIGGEST thanks goes to my wonderful
husband, Paul Stewart. This page is not enough to list all the things that you have done to
iii
make this moment possible. Thank you for all your care – in all forms. When I felt like
giving up – you were there. Without your love and support I would not have had the
strength to finish this project. I love you with all my heart. This project is dedicated to
you.
iv
TABLE OF CONTENTS
LIST OF TABLES ……………………………………………………………… viii
LIST OF FIGURES ……………………………………………………………..
ix
ABSTRACT ……………………………………………………………………
xi
CHAPTER 1
THE NATURE AND PURPOSE OF THE STUDY
Introduction …………………………………………………… 1
The Difficulty of Fractions …………………………………….. 2
Statement of the Problem ………………………………………. 5
Purpose and Significance of the Study ………………………… 8
Research Questions ……………………………………………. 9
Summary ………………………………………………………. 11
CHAPTER 2
THEORETICAL FRAMEWORK AND REVIEW OF RELATED LITERATURE
Introduction ……………………………………………………
The Teaching and Learning of Mathematics ………………….
Epistemology ………………………………………………….
The Fraction Concept ………………………………………….
Models of Understanding ………………………………………
Relational and instrumental understanding …………….
Kieren’s model of the growth of mathematical
understanding ……………………………………………
Herscovics and Bergeron’s model of understanding ……
Rationale for using Herscovics and Bergeron’s model ….
Criteria for the Components of Understanding in the Herscovics
and Bergeron’s (1988) Model of Understanding…………………
Understanding the Underlying Physical Concepts ………
Understanding the Emerging Mathematical Concept ……
Physical Referents and the Understanding of the Fraction
Concept ………………………………………………………….
Partitioning Strategies ……………………………………………
Partitive quotient construct strategies ……………………
Multiplicative strategies …………………………………
Iterative sharing strategies ………………………………
Summary ………………………………………………………..
v
12
13
14
18
21
22
23
26
30
31
32
33
34
38
40
42
42
44
CHAPTER 3
METHODOLOGY
Qualitative Interpretive Framework …………………………….
Participants ………………………………………………………
Research Design …………………………………………………
Pretest ……………………………………………………
Teaching Sequence ………………………………………
Interviews ………………………………………………..
Data collection ……………………………………………
Data Analysis …………………………………………………….
Assumptions and Limitations ……………………………………
Research Methodology …………………………………..
Teacher-Researcher ………………………………………
Sampling …………………………………………………
Summary …………………………………………………
46
46
48
48
49
55
56
57
57
58
59
61
62
CHAPTER 4
RESULTS: THE NATURE OF STUDENTS’ UNDERSTANDING OF
FRACTIONS
Introduction ………………………………………………………
Definition of the Fraction…………………………………………
Analysis ………………………………………………….
Intuitive Understanding ………………………………………….
Analysis …………………………………………………
Logico-Physical Understanding …………………………………
Analysis …………………………………………………
Logico-Physical Abstraction ……………………………………
Analysis …………………………………………………
Logico-Mathematical Procedural Understanding ………………
Analysis …………………………………………………
Logico-Mathematical Abstraction/Formulization ………………
Analysis …………………………………………………
Summary ………………………………………………………..
63
64
68
69
77
78
91
92
118
119
126
126
149
149
CHAPTER 5
RESULTS – STUDENTS’ PARTITIONING STRATEGIES
Introduction …………………………………………………….
Summary ……………………………………………………….
vi
152
162
CHAPTER 6
SUMMARY, CONCLUSION, DISCUSSION AND IMPLICATIONS
Introduction …………………………………………………….
The Nature of Students’ Understanding of Fractions ………….
The Students’ Partitioning Strategies …………………………..
Physical and Real World Representations ……………………..
Significance of the Study ………………………………………
Conclusions and Discussion ……………………………………
Implications for Teaching and Future Research ……………….
Concluding Remarks …………………………………………..
164
165
169
170
170
171
174
175
Pretest………………………………………………………….
177
Activities for Teaching Sequence …………………………….
185
Names, Categories and Group Assignments ………………….
238
Consent Forms ………………………………………...............
Human Subjects Approval Letter ……………………………..
239
242
REFERENCES .……………………………………………………………….
243
BIOGRAPHICAL SKETCH ………………………………………………….
255
APPENDIX A
APPENDIX B
APPENDIX C
APPENDIX D
vii
LIST OF TABLES
Table 1.
Fractions – Sample Item from NAEP ………………………….
3
Table 2.
Charles and Nason’s (2000) Partitioning Strategies ……………
41
Table 3.
Outline of Teaching Episodes ………………………………….
51
Table 4.
Research Questions and the Methods of Data Collection ………
56
Table 5.
Summary of Responses for Question 5 on Pretest ……………..
66
Table 6.
Summary of Responses for Question 9 on Pretest ……………..
70
Table 7.
Summary of Responses to Question 2 (Multiple Choice)
on the Pretest ……………………………………………………
72
Table 8.
Summary of Responses to Question 11 on the Pretest ………….
73
Table 9.
Summary of Responses to Question 1 (Multiple Choice)
on the Pretest……………………………………………………..
75
Table 10.
Summary of Responses for Assessment Task for Activity 1 ……
87
Table 11.
Configurations for Task 2a – Activity 4 ………………………… 113
Table 12.
Configurations for Three-Fourths – Task 2a Activity 7 ………..
133
Table 13.
Classification of Students’ Difficulties …………………………
172
viii
LIST OF FIGURES
Figure 1.
Steffe and D’Ambrosio’s (1995) Hypothetical
Learning Trajectory …………………………………………….
16
Figure 2.
Conceptual Scheme for Instruction on Rational Numbers ……… 19
Figure 3.
Kieren’s (1993) Model for the Recursive Theory of Mathematical
Understanding …………………………………………………… 24
Figure 4.
Potential Signing Process Appropriate for Fraction Readiness …. 29
Figure 5.
Herscovics and Bergeron Model of Understanding ……………… 30
Figure 6.
Lesh’s Translation Model ……………………………………….. 37
Figure 7.
Diagram for Question 15 – Pretest ……………………………… 67
Figure 8.
Students’ Configuration of the Parallelogram …………………… 82
Figure 9.
Mary’s Method of Equally Sharing 12 Chips …………………… 84
Figure 10.
Common Configurations for Exercise 2
Assessment Task - Activity 1……………………………………. 88
Figure 11.
Configurations for Task 2 – Individual Interview …….…………. 91
Figure 12.
Partitioning the Circle in Question 2 –
Activity 2 Assessment Task ……………………………………..
98
Figure 13.
Students’ Drawing of the Parallelogram – Item #4 ……………… 100
Figure 14.
Configuration for Item 4 (Second Part) …………………………. 101
Figure 15.
Squares Used for Activity 3 – Task 4 …………………………… 105
Figure 16.
Squares Used for Task 6 – Activity 3 …………………………… 109
Figure 17.
One of the Configurations Given in
Assessment Task – Activity 3 ………………………………......
110
Six-Part Configurations for Task 1b – Activity 4 ………………
112
Figure 18.
ix
Figure 19.
Common Configurations for Task 2b – Activity 4 ……………..
115
Figure 20.
Karla’s Configuration of Task 3d – Activity 4 …………………
116
Figure 21.
Configurations for Task 3d – Activity 4 ………………………..
117
Figure 22.
Richy’s Number Line Representation of One-Fifth …………….
123
Figure 23.
Alton’s Picture of Four
1
……………………………………….
5
129
Figure 24.
Item 3 – RNP Lesson 6 – Activity 7 ……………………………
135
Figure 25.
Item 8 – RNP – Lesson 6 – Activity 7 …………………………
136
Figure 26.
Students’ Configurations for Second Assessment Task (Item 3a) –
Activity 7 ………………………………………………………. 137
Figure 27.
Configurations for Second Assessment Task (Item 3b) –
Activity 7 ……………………………………………………….
138
Figure 28.
Item 11 – RNP Lesson 15 Student Page E – Level One ………..
147
Figure 29.
Example of Partitive Quotient Foundational Strategy ………….
154
Figure 30.
Karla’s Representation of the Sharing of the Pancakes …………. 157
Figure 31.
Example of the Preserved-Pieces Strategy ……………………… 157
Figure 32.
Item 5 – RNP Lesson 22 Student Page A - Level 2 ……………..
x
160
ABSTRACT
This study was an investigative, whole class descriptive research, on the
development of twenty sixth graders’ understanding of fractions as they interacted with
physical referents, hands-on task-based activities and activities that model real life
situations during eight weeks of a teaching sequence. The study was conducted in a
metropolitan school situated in southeast Florida. The teaching sequence consisted of 12
task-based activities that spanned 20 sessions with each session lasting for approximately
60 minutes. Data was collected through audio- and video-recording, in addition to the
numerous written tasks. The task-based activities that the students were involved with
during this study were analyzed to gain an insight into their understanding of fractions in
the context of subdividing, comparing and partitioning of continuous and discrete models
and the connections they made with the fraction ideas generated through these activities.
The study also examined how these students make sense of fractions and investigated
how their performance differed when fractions were presented using different models.
Herscovics and Bergeron’s (1988) extended model of understanding, and the partitioning
strategies identified by Charles and Nason (2000) and Lamon (1996) provided the
theoretical framework through which the investigation was explored.
Results from the study revealed that the participants exhibited an understanding of
unit and non-unit fraction based on the components of the above-mentioned model of
understanding. The students also displayed a number of different partitioning strategies.
The knowledge growth that was evident in the whole class confirms earlier studies as to
the significant role that partitioning plays in the basic development of the fraction
concept. Although discrete models were used by the students, a majority of the students
exhibited a preference for using continuous models as forms of reference for given
fractions. The students appreciated working with fractions that model real world
situations.
Preliminary findings from this study seem to indicate that students should be
introduced to fraction concepts via partitioning activities. The partitioning activities
should be introduced in grades earlier than sixth grade. Further research can be
xi
undertaken to investigate the role partitioning activities play in the development of
students’ ability to add, subtract, multiply and divide fractions.
xii
CHAPTER 1
THE NATURE AND PURPOSE OF THE STUDY
Introduction
There is a growing consensus among mathematics educators that rational number
concepts, in particular the fraction concept, are among the most ubiquitous, multifarious
and significant mathematical ideas that children encounter before reaching high school
(Behr, Lesh, Post & Silver, 1983; Mack, 1993) . The wide use of fractions in everyday
life makes information about fractions necessary as early as elementary grades. Smith
(2002) noted that students’ experience with fraction concepts begins even before formal
schooling and extends well into the high school years. However, even with this early
introduction, students still have trouble conceptualizing fractions, probably forming one
of the most critical barriers to the mathematical maturation of children (Aksu, 1997;
Behr, Harel, Post & Lesh, 1992; Bezuk & Cramer, 1989; Hope & Owens, 1987;
Schminke, Maertens, & Arnold, 1978).
Some researchers (e.g. Behr, Lesh, Post & Silver, 1983) went as far as attributing
many of the “trouble spots” in algebra to an incomplete understanding of earlier fraction
ideas. The students’ inability to perform basic operations on fractions has resulted in
error patterns in the successful completion of algebraic exercises and problems. Wu
(2001) underscored the idea by suggesting “that no matter how much algebraic thinking
is introduced in the early grades, and no matter how worthwhile this might be, the failure
rate in algebra will continue unless the teaching of fractions and decimals is radically
revamped. The proper study of fractions provides a ramp that leads students gently from
whole number arithmetic up to algebra” (p. 11).
1
The Difficulty of Fractions
Lamon (1999) said that “as one encounters fractions, the mathematics takes a
qualitative leap of sophistication. Suddenly, meanings, models, and symbols that worked
when adding, subtracting, multiplying and dividing whole numbers are not as useful” (p.
22). This “qualitative leap of sophistication” seems to cause such a level of confusion not
only in primary school students but has stimulated Riddle and Rodzwill’s (2000)
mathematical curiosity to ask, “Why is it that many adults, even after years of schooling,
still do not understand some mathematics topics, such as fractions” (p. 202)? Ohlsson
(1988) believed that the difficulty associated with fractions is semantic in nature: “The
complicated semantics of fractions is, in part, a consequence of the composite nature of
fractions. How is the meaning of 2 combined with the meaning of 3 to generate a
meaning of 2/3?” (p. 53). He further purported that the difficulty encountered in fractions
is partially due to the result of the “bewildering array of many related but only partially
overlapping ideas that surround fractions” (p. 53).
Results from state, national and international assessments have done very little in
convincing the mathematics education arena that the teaching and learning of fractions is
not a difficult and daunting task. Although the results from the 2003 National
Assessment of Educational Progress (NAEP) indicate a steady increase in fourth- and
eighth-graders’ average mathematics score, 51% of the fourth-graders tested scored
below the “Satisfactory” level on an extended constructed-response item dealing with
equivalent fractions and only 35% of eighth graders tested were able to accurately
identify the correct ordering of three fractions, all in reduced form. The twelfth-graders
fared even worst with only 25% of the test-takers able to write a fraction resulting from
dividing a fractional part of a unit into an integral number of parts (National Center for
Education Statistics [NCES], 2004, Wearne & Kouba, 2000). Although the eighthgraders were more successful at the fraction items than the fourth graders, the results
show that certain fractional concepts remain problematic to these middle school students.
These concepts include a) ordering a set of fractions, and b) using fractions to solve word
problems – the application questions. Students’ performance on the fraction test items
suggests that they have learned computational algorithms with little understanding of the
2
critical building-block concepts that are needed in the application of fractions to problemsolving tasks.
NAEP studies done on previous assessment years corroborated with the current
findings in highlighting the difficulty elementary students encountered with elementary
fraction concepts (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1981; Kouba,
Zawojewski, & Strutchens, 1997). “The 1996 NAEP data provide evidence that fractions
continue to be difficult for students, particularly for fourth-grade students” (Wearne &
Kouba, 2000, p. 164).
Table 1:
Fractions – Sample Item from NAEP (Wearne & Kouba, 2000)
Item
Percent Responding
Grade 4
How many fourths make a whole?
Answer: ___________________________
Correct response of 4
50
Any incorrect response
35
Omitted
16
Alongside the difficulty students faced with rational numbers, two observations of
the NAEP interpretive report done by Wearne and Kouba (2000) are worth mentioning.
Firstly, students at all three grade levels (4, 8 & 12) are more successful in solving
routine one-step tasks than nonroutine tasks including tasks that involved more than one
step. Wearne and Kouba (2000) noted that “fourth-grade students encountered fewer
problems representing a fraction on a region if the number of parts into which the region
3
was divided was equal to the denominator of the fraction than if the number of parts into
which the region was divided was a multiple of the denominator” (p. 166). Fundamental
to the notion of the fraction a/b is that the b represents the number of equal parts the
whole is divided into. An analysis of the test item related to this concept reveals that only
one-half of the test-takers in grade 4 were able to correctly state that 4 fourths are needed
to make a whole with one-sixth of the students not responding to the item (see Table 1 for
actual percentages). This becomes rather perturbing as one considers that this item
requires knowledge that should be acquired from first grade.
Secondly, students are able to construct more vigorous meaning for the unit when
fractions are represented as regions than the meaning attached to the unit when fractions
are represented as sets, as collections of objects. This coincides with the students’ use of
concrete and/or discrete objects to illustrate the unit fraction. Students should also be
able to connect written symbols of fractions with other representations such as physical
objects, pictorial representations and the spoken language.
The results of the Third International Mathematics and Science Study (TIMSS)
have shown that eight grade students in the United States performed below average in
fractions, proportionality and algebraic concepts compared to eight grade students
internationally (Jakwerth, 1996). The grim findings resulting from NAEP reports
coupled with those from TIMSS have aroused the interest of mathematics educators in
view of the fact that fractions occupy a great part of the middle school mathematics
curriculum and are also foundational to the mathematics encountered in high school and
beyond (Lesh, Post, & Behr, 1988).
A number of outstanding researchers in the field of mathematics education such
as Steffe and Olive (1991) and Lamon (1999) have documented various reasons primary
school students encounter difficulty with fractions. These include:
1)
The abstract way in which fractions are represented during classroom instruction.
It is a noted fact the students in the USA score significantly lower that their Asian
counterparts in the TIMSS. On observing the TIMSS videotape survey of 231 eighthgrade lessons in the United States, Japan and Germany, Geist (2000) noted that in a
Japanese classroom the teacher related the topic and concept to a real-world application
4
while in the United States classroom, the lesson was abstract, and no links to real life or
applications were made.
2)
Fractions do not form a normal part of the learners’ environment. The most
common fractions used are halves, quarters and thirds.
Other researchers in the quest to provide a solution to this dilemma, have
suggested that fraction is introduced too early in the school curriculum while some
notable researchers in the field, such as Behr, Wachsmuth, Post and Lesh (1984)
recommended that formal fraction instruction should begin in third grade instead of
fourth grade. Cramer and Henry (2002) added, “Students will be more successful if
teachers in elementary school invest their time building meaning for fractions using
concrete models and emphasizing concepts, informal ordering strategies, and estimation”
(p. 47). They suggested that much of the fraction symbolization done in fourth and fifth
grades should be taken up in middle grades. Watanabe (2001) suggested that this topic in
whatever form be eliminated from the primary school (K – 3) curriculum and encouraged
fraction instruction to begin when the students are developmentally ready.
Statement of the Problem
The importance of the understanding of fractions to mathematics learning,
coupled with the poor performance on fraction items on internal and external assessments
have prompted various researchers to take a keen interest in this particular area of
mathematics. Most of the prominent work on fractions date as far back as the early
1980’s with one of the most significant studies done on rational number - the Rational
Number Project (RNP) – starting in 1979 and officially ending in August 2000. Since
1980, the RNP has graced the research literature with reports on many investigations into
the teaching and learning of fractions among fourth and fifth graders (Bezuk & Cramer,
1989; Post, Wachsmuth, Lesh & Behr, 1985). Careful scrutiny of the research done on
fractions to date reveals an attempt by various researchers (e.g. Boulet, 1998; Charles &
Nason, 2000; Saenz-Ludlow, 1994; Tzur, 1999) to focus their study where fraction use
and/or instruction begins, whether formally or informally, that is, in fifth grade or lower
with sparse research on how middle school or junior high children make sense of this
concept.
5
Various studies on fractions have focused on middle school students’
misconception (e.g Behr, Lesh, Post, & Silver, 1983) or on the instructional approach
undertaken by the instructor to lead students to a functional understanding of the topic
(e.g. Mack, 1993). One such study was done by Erlwanger (1973) with an 11 year old
boy named Benny who was a student in a sixth grade class using Individually Prescribed
Instruction (IPI) Mathematics. Examining Benny’s concept of decimals and fractions,
Erlwanger (1973) noted “Benny’s view about rules and answers reveals how he learns
mathematics. Mathematics consists of different rules for different types of problems….
Therefore, mathematics is not a rational and logical subject in which one has to reason,
analyze, seek relationships, make generalizations, and verify answers” (p. 54). These
rote rules often result in various procedural misconceptions. Ashlock (2002) outlined a
number of the misconceptions plaguing the fraction concepts and stressed the importance
of students understanding the underlying concept of a fraction. These misconceptions
include:
1)
Students equating the number of shaded parts in a diagram to the numerator and
the number of unshaded parts to the denominator. For example, when Gretchen was
3
asked to name the fraction for the part that is shaded in the diagram below she wrote .
1
2)
Students associating the denominator of the fraction with the total number of parts
indicated in the diagram and the numerator with the number of shaded parts disregarding
the fact that the associations apply only when all the parts are equal in area. For example,
Carlos writes
3)
1
for
5
.
When asked to express a fraction in its lowest term the students will choose a
3
1
specific factor of the numerator and one for the denominator thus will be reduced to
8
4
by dividing the numerator by 3 and the denominator by 2. According to Ashlock (2002)
“This procedure is a very mechanical one, requiring no concept of a fraction; however, it
does produce correct answers part of the time” (p. 151).
6
A research review reveals that most studies done on fractions at the middle school
level have focused primarily on students’ misconception of computational procedures as
the ones described above or on other representations of rational numbers such as percent,
ratio, and proportional reasoning (Gay, 1997; Lamon, 1993; Singh, 2000 to name a few).
Consequently, only small portion of studies focused on the conceptual issues related to
the understanding of fractions (Peck & Jencks, 1981), how students make sense of the
concept and the role, if any, that physical referents including real life problem-solving
activities play in the students’ understanding of fractions.
Because of the complexity of the different subconstructs (part-whole relationship,
measure, operator, quotient, and ratio) that can be assigned to a fraction, more in-depth
and ongoing research need to be done in the quest to enhance conceptual and functional
understanding of this vital middle school topic. Boulet (1998) expounded on the
importance of students’ understanding of mathematical ideas and what would result from
the lack of it. She wrote:
Understanding is certainly the goal of learning, and teachers generally believe
that their pupils understand their lessons. Without understanding, the learning of
mathematics is reduced to the memorization of formulae and the rules governing
them. Mathematics thus learned cannot be meaningful, much less useful (p. 19).
Bezuk and Bieck (1992) purported that fraction concepts, order and equivalence
usually receive only shallow attention and are often taught in a meaningless manner.
They believed, and I agree, that “it is crucial for middle grades instruction to strengthen
students’ understandings before progressing to operations of fractions, rather than
assuming that students already understand these topics” (p. 119). As stated by Aksu
(1997), “A common error in teaching fractions is to have students begin computation
before they have adequate background to profit from such operation” (p. 375).
As a result of these findings, a number of recommendations have been gleaned
from these studies in the effort to improve students’ understanding of fractions. These
include (1) The use of manipulatives/physical referents and meaningful, engaging
activities in promoting higher order thinking skills and conceptual understanding; (2) The
need for students to develop the ‘correct’ concepts and relationships among fractions
necessary to fully undertake the tasks of performing and understanding operations on
7
fractions (Behr, Wachsmuth & Post, 1985; Bezuk & Cramer, 1989; Empson, 2003;
Morris, 1995; Yang, 2002); and (3) Students should have experience working with
different physical models - continuous and discrete (Behr & Post, 1992). In fact, a major
goal of mathematics education programs is to develop students’ mathematical reasoning
so that they become proficient in using the content knowledge and skills learned to solve
real-life problems (National Council of Teachers of Mathematics [NCTM], 2000).
Purpose and Significance of the Study
The results from a study on students’ use of part-whole and direct comparison
strategies on fractions with 4th, 6th and 8th grades students indicate that at the end of
middle school, students seldom recognize real-world situations in which they can apply
their knowledge of fractions other than halves and quarters (Armstrong & Larson, 1995).
Clearly, the mathematics education community and, in particular, practicing mathematics
teachers are bombarded with the demand for finding innovative ways to teach fractions
and to engage mathematics students in meaningful activities that will lead them to
conceptually understand fractions and to satisfy the students’ thirst for fun and relevance
of mathematics ideas.
This exploratory, whole class research study seeks to add to the knowledge base
about how students conceptualized fractions as they worked with manipulatives and
participated in process-oriented activities that model real-life events. These activities that
the students were involved with during this study were analyzed to gain an insight into
students’ understanding of fractions in the context of subdividing, comparing and
partitioning of continuous and discrete models and the connections they made with the
fraction ideas generated through these activities. The study also seeks to examine how
these students make sense of fractions and investigates how their performance differs
when fractions are presented using different models. The models selected for this study
include continuous quantity and discrete objects. Continuous quantity usually refers to
length, area or volume. The continuous whole is made up of one single object, which can
be circular, rectangular or other geometrical regions (e.g. triangles); the number line and
liquid measure that are not generally used in the classroom to illustrate a fraction.
Discrete objects refer to several separate objects, that is, the whole consists of more than
one object (Behr & Post, 1992).
8
Research Questions
This study is an investigative, whole class research, descriptive study on the
development of sixth graders’ understanding of fractions as they interact with physical
referents, hands-on task-based activities and activities that model real life applications
during eight weeks of a teaching sequence. The following questions served as the guide
in the investigation of these students’ understanding of this subset of rational numbers.
Each question is accompanied by a brief elaboration that serves the purpose of providing
explicit nuances relevant to the question.
1.
What is the nature of sixth grade students’ understanding of fraction?
Embedded in this question is the need to understand how middle school students;
•
define fractions
•
unitize and partition
•
make sense of fraction symbols
•
order fractions
•
generate equivalent fractions.
Behr, Wachsmuth, Post and Lesh (1984) concluded from the RNP conducted
from 1979 to 1983 that the performance of a considerable number of the
participants on fraction questions dealing with order and equivalence,
demonstrated a substantial lack of understanding. Research literature also
indicates that children’s knowledge of fractions is mainly algorithmic and is
flawed by the interference of the students’ whole number language (Hiebert &
Wearne, 1986; Moss & Case, 1999). This study utilizes the Herscovics and
Bergeron’s (1988) model of understanding to analyze the sixth graders’ responses
to the fraction tasks that are presented to them during the teaching sequence.
These tasks are designed to cover the various fraction concepts that are mentioned
above.
2.
What strategies do sixth grade students employ to ensure partitioning or equal
sharing as they engage in process-oriented activities?
“The action of partitioning an object or a set of objects is learned by a child in a
social setting” (Poither & Sawada, 1983, p. 311). Although the participants in
this study worked on numerous partitioning tasks, the answer to this research
9
question will be based largely on the process-oriented activities that the students
were involved with during the study. Process oriented activities are activities that
are designed to model real-life events with a purposeful twist to glean the
mathematical ideas deeply entrenched in the students’ solicited behavior needed
to complete the tasks. One such activity was the hosting of a Fraction Breakfast
where students were required to evenly share the group’s food among themselves.
For the purpose of this study, the partitioning activities that the students
performed were observed to determine their strategies for ensuring equal sharing.
Charles and Nason’s (2000) and Lamon (1996) partitioning strategies will be used
to analyze and interpret the partitioning activities. These categories will be
discussed with more elaboration in Chapter Two.
3.
How do physical and real world representations aid in the development of sixth
grade students’ understanding of fractions?
The task-based activities that the students were engaged in gave them the
opportunity to work with various physical referents. The terms physical referents,
physical models, concrete models, and manipulatives will be used synonymously
throughout this paper. For the purpose of this research, these are defined as the
concrete objects (discrete or continuous) and hands-on activities that students
work with as they solve fractional problems. They include objects and activities
that appeal to several senses, can be touched, handled, moved, or cut. The
participants in the study worked with fraction strips, fraction circles, fraction
squares, fraction triangles, fraction balance, measuring cups, chips, and regular
geometric polygons among other things. These will be identified throughout the
research. There is a need to understand how students’ performance differs when
modeling fractions using continuous and discrete objects.
Research has shown that the use of physical referents can, but not
necessarily, facilitate an improvement in student learning (Hiebert, et al., 1997).
Numerous studies (e.g. Behr, Wachsmuth, Post and Lesh, 1984; Cramer & Post,
1995; Lesh, Cramer, Doerr, Post, Zawojewski, 2003; Kolstad, Briggs and Hughes,
1993) tout the active use of physical referents as a means for students to develop
sound mathematical concepts. Behr, Wachsmuth, Post and Lesh (1984) have
10
also noted the negative effects of students’ dependence on the use of
manipulatives to aid in the solution of every mathematics problems they
encountered. Their dependence oftentimes dwarfed the need to make
generalizations. This type of dependence is not altogether inappropriate but the
learner who is able to connect the mental images of relations expressed via the
objects rather than on direct actions with the object is at a clear advantage.
Leinhardt (1988) defined real world representations as real life classroom
situations that are similar to events familiar to the students’ every day life. “Wellchosen tasks can pique students’ curiosity and draw them into mathematics. The
tasks may be connected to the real-world experiences of students, or they may
arise in contexts that are purely mathematical” (NCTM, 2000, pp. 18-19).
Streefland (1982), after perusing numerous articles on fractions up to the time of
his research, commented on the literature’s lack of “meaningful contexts both as
sources and as domains for the application of fractions” (p. 235). The
researcher’s review of current articles on the same subject reveals a similar
perspective.
Summary
The results of assessments done locally, nationally and internationally on
students’ fractional knowledge have displayed sufficient evidence that fraction concepts
are difficult to learn. This phenomenon is occurring despite the fact that children may
have informal knowledge of the concept before schooling and the fraction concept is
introduced in the early grades and continues to be taught or presented throughout the
school years. This has caused real concern in the mathematics education arena thus
whetting the research appetite of individuals seeking to understand, explain and/or offer
suggestions to alleviate this daunting problem. This chapter has outlined the necessity of
undertaking a study on students’ understanding of the fraction concepts thus adding to the
growing and varied collection of meaningful studies on this concept. The study seeks to
investigate and explore the sixth graders’ understanding of fractions as they work with
manipulatives and participate in various process-oriented activities during instruction.
The process-oriented activities include situations where students participate in real-life
events that model scenarios that are directly related and meaningful to them.
11
CHAPTER 2
THEORETICAL FRAMEWORK AND REVIEW OF RELATED LITERATURE
Introduction
“In qualitative work, theory is used at every step of the research process.
Theoretical frames influence the questions we ask, the design of the study, the
implementation of the study, and the way we interpret data” (Janesick, 1998, p. 5). Pirie
(1998) deemed the role of theory in qualitative research as very crucial and if not always,
the most dominant in the presentation of the data. This chapter serves the sole purpose of
providing the theoretical lens through which this present study can be viewed and
understood. In this study, the approach to understanding the dynamics of students’
meaning of fractions, partitioning strategies, and how they work with physical models
will be directly influenced by the teacher-researcher philosophy on the teaching and
learning of mathematics, the constructivist epistemological stance of Piaget (1970), von
Glasersfeld (1995), Steffe and D’Ambrosio (1995), and other literature relevant to the
study. Among the most relevant terminologies used in this study are constructivism,
conceptual and relational understanding, meaning of a fraction, unit fraction, discrete and
continuous quantities, sharing and partitioning, and logico-physical and logicomathematical understanding.
Good teaching sequences require both “a practical and a theoretical frame of
reference” (Higgins, 1973, v.). The extended model of understanding developed by
Herscovics and Bergeron (1988) was used as the framework to explore and interpret the
depth of the students’ understanding of a fraction as they performed a set of task-based
activities during an eight-week teaching sequence. The participants’ methods of
partitioning were observed, analyzed and categorized according Charles and Nason
(2000) and Lamon (1996) partitioning strategies.
12
The Teaching and Learning of Mathematics
Mathematics is a language that is used to describe the physical and non-physical
aspects of the world we are living in. This quotation from Everybody Counts summed it
up nicely:
Mathematics is a living subject which seeks to understand patterns that permeate
both the world around us and the mind within us. Although the language of
mathematics is based on rules that must be learned, it is important for motivation
that students move beyond rules to be able to express things in the language of
mathematics. … It involves renewed effort to focus on seeking solutions, not just
memorizing procedures; exploring patterns, not just memorizing formulas; and
formulating conjectures, not just doing exercises (National Research Council,
1989, p. 84).
Fueled by this definition, I have sought in my teaching experiences to explore and find
new ways to energize my mathematics classroom in the quest of understanding how my
students make sense of the mathematical ideas presented in the class.
Fundamental to my credo is the fact that all students can learn. This thinking
conforms to current ideologies of students’ learning. The National Council of Teachers of
Mathematics (NCTM) in its 2000 publication of Principles and Standards for School
Mathematics, outlined as one of the six principles for school mathematics: “Students
must learn mathematics with understanding, actively building new knowledge from
experience and prior knowledge. … Mathematics can and must be learnt by all students”
(pp. 11, 13). The National Research Council in its 1989 Nation Report stated:
Only in the United States do people believe that learning mathematics
depends on special ability. In other countries, students, parents, and teachers all
expect that most students can master mathematics if only they work hard enough.
The records of accomplishment in these countries--and in some intervention
programs in the United States--show that most students can learn much more
mathematics than is commonly assumed in this country (p. 10).
It is therefore the teacher’s responsibility to provide the opportunity for all
students to learn this subject. This includes mathematical concepts that are consistently
difficult for students to understand such as fractions, ratio and proportion. One aspect of
13
the teacher’s role is to provoke students’ reasoning about mathematics. Teachers do this
through the tasks they provide and the questions they ask. Teachers are responsible for
the quality of the mathematical tasks in which students engage. Teachers should choose
and develop tasks that are likely to promote the development of students’ understandings
of concepts and procedures in a way that also fosters their ability to solve problems and
to reason and communicate mathematically. There are documented evidences that direct
instruction may not provide an adequate base for students’ development and for students’
use of higher cognitive skills (Confrey, 1990).
Burns (2000) believed that for students to learn mathematics it becomes
imperative for them to create and re-create mathematical relationships in their own
minds. “Therefore, when providing appropriate instruction, teachers cannot be seduced
by the symbolism of mathematics. Children need direct and concrete interaction with
mathematical ideas …. Continuous interaction between a child’s mind and concrete
experiences with mathematics in the real world is necessary” (Burns, 2000, p.24). As a
practicing teacher, I believe that it is through interactions with children and deliberately
seeking ways to understand mathematics from their lenses that as teachers we begin to
make crucial distinctions between how children view mathematical situations and the
best way to teach them.
Epistemology
“Epistemology is the branch of philosophy that deals with the underpinnings of
how we know what we know, and in particular the logical (and sometimes the
psychological) bases for ascribing validity or ‘truth’ to what we know” (Goldin, 1990, p.
32). On this premise, the epistemological basis for this study is operated from a
constructivist view of knowledge. Constructivism focuses on how knowledge is acquired.
It emphasizes knowledge construction rather than knowledge transmission.
Constructivists believe that all humans have the ability to construct knowledge in their
own minds through a process of discovery and problem solving.
Piaget, a proponent of constructivism, believed that children construct knowledge
through the process of internalizing the physical operations on objects. As they move
sets of objects about by putting them together, arranging them and/or separating them,
they internalize properties of mathematical operations rather than the objects themselves
14
(Noddings, 1990). In the process of constructing meaning, the learner must actively
strive to make sense of new experiences. This epistemological view is based on an
assumption that knowledge is not passively received but is actively constructed by the
learner. This ideology is supported by numerous studies on constructivism (Confrey,
1990; Maher & Davis, 1990; Romberg & Carpenter, 1986).
Ideas and thoughts cannot be communicated in the sense that meaning is
packaged into words and “sent” to another who unpacks the meaning from the
sentences. That is, as much as we would like to, we cannot put ideas in students’
heads, they will and must construct their own meanings. Our attempts at
communication do not result in conveying meaning but rather our expressions
evoke meaning in another, different meanings for each person (Wheatley, 1991, p.
10).
Simon (1995) purported that constructivism does not stipulate a particular model
to be used in the teaching and learning of mathematics, in other words, it does not tell us
how to teach mathematics. He believed that the sole purpose of constructivism is to
provide a useful framework for thinking about mathematics learning in the classroom.
Steffe and D’ambrosio (1995) rebutted with the conjecture that “there is a kind of
teaching that can be legitimately be called ‘constructivist teaching’” (p. 146). From their
point of view, “a teacher regards the students’ mathematical language and action to
constitute a living mathematics and interacts with students in a learning space whose
design is in part based on that language and action” (p. 152). Simon’s (1995) view of
constructivism is implicitly embedded in their stance on constructivism. This view
becomes particularly interesting to the teacher-researcher as the fraction teaching
sequence conducted in the present study was designed to conform to Steffe and
D’Ambrosio’s (1995) working model of constructivist teaching. In conducting teaching
sequences modeled from a constructive standpoint, the teacher’s description of the
students’ schemes of action and operation would be at the forefront. This ideology forces
the mathematics teacher to make a concerted effort in reforming the classroom
environment to provide optimal learning. The underlying assumption pertaining to this
study is that mathematics learning can be facilitated by having the students participating
in preplanned, specific task-based activities. Figure 1 shows the working model
15
representing Steffe and D’Ambrosio’s (1995) hypothetical learning trajectory (HLT) that
formed the threshold for the learning environment that the sixth graders worked in. The
phrase “zone of potential construction” is used to refer to a “teacher’s working
hypotheses of what the student can learn, given her model of the student’s mathematics.
The zone of potential construction is determined by the teacher as she interpreted the
schemes and operations available to the student and anticipates the student’s actions
when solving different tasks in the context of interactive mathematical communication”
(p.154). Though not a quantitative study, the working hypothesis for this study is that the
sixth graders participating in this study will display an understanding of fraction concepts
as outlined by Herscovics and Bergeron’s (1988) model of understanding as they work
through a set of task-based activities. Of major significance was the way in which the
information was presented, the scenarios were situated and how learners were supported
in the process of constructing knowledge. Von Glasersfeld (1995) was convinced that
within a constructivist learning environment students will be more motivated to learn
something.
Mathematics
of students
Student actions
and modifications
of actions
Figure 1.
Steffe and D’Abrosio’s (1995) Hypothetical Learning Trajectory
16
This learning environment contains:
•
The learner,
•
the setting or space where the learner acts, using tools and devices, collecting and
interpreting information and interacting with each other (Wilson, 1995).
According to Piaget (1970), knowledge arises from progressive social interactions
that actively take place between the subject and the outside world. This social interaction
is important and must be negotiated (Cobb, Yackel & Wood, 1992; Lo & Wheatley,
1992; Yackel, Cobb, Wood, Wheatley & Merkel, 1990). “Social interaction plays a
significant role in the development of mathematical knowledge in the individual, and of
the corpus of mathematics available to the human race as a whole” (Hunting & Davis,
1991). These researchers purport that a major if not determining role of constructing the
fraction one-half and other fractions, is played by the social activity of sharing.
Dialogues, which were encouraged through the act of sharing in the task-based activities
done in the study, provided the catalyst for knowledge acquisition. Oral exchanges
facilitated mathematical understanding that occurred through social interaction, through
questioning and explaining, challenging and offering timely support and feedback. The
classroom was eventually transformed into a community of learners, a knowledgebuilding community where students had firsthand knowledge of their peers’ thinking
processes. Besides the learning communities, concepts such as scaffolding, cognition and
cooperative learning lend support to the constructivist’s learning environment (Brown,
1994; Rogoff, 1990).
Burns (2000) and Mack (1990) have documented the need for teachers to organize
and plan classroom instruction that will build on children’s previous experiences with
fractions and help these children clarify the ideas they have encountered while grappling
with the fraction concept. The teacher in a constructivist environment will draw on the
culture (ethnomathematics) and resources available to the students. In the realm of the
study that was undertaken, it was crucial for the teacher to recognize that children’s
introduction to fractions occurred primarily outside of the mathematics classroom and
that active knowledge construction was taking place as the students interacted with each
during the activities.
17
The Fraction Concept
Fractions can be assigned various meanings according to the context in which the
concept is used. According to Ohlsson (1988) in order to understand the meaning of
fraction one has to “pay attention to the mathematical theory in which fractions are
embedded, to which fractions are applied, and to the referential mapping between the
theory and those situations” (p. 54). Fractions form a part of a subset of rational numbers
which form a subset of a larger set of real numbers. Freudenthal (1983) viewed fractions
as the phenomenological source of rational numbers. Kieren (1976) divided the concept
of fraction into five major subconstructs: part-whole relationship, measure, operator,
3
quotient, and ratio. Each subconstruct will be briefly described and the fraction , where
4
3 is called the numerator and 4 the denominator, is used to give a clearer idea of the
description.
•
Part-whole relationship: This “depends directly on the ability to partition either a
continuous quantity or a set of discrete objects into equal sized subparts or sets”
(Behr & Post, 1992). In this study the part-whole relationship is considered to be
the ratio of the part to the whole. The fraction
3
representing three equal slices
4
out of a cake cut into four equal pieces (continuous model) or three eggs from a
carton with 4 eggs (discrete model) can be shown diagrammatically as
•
Measure: This refers to the position of a number on the number line or a ruler
and for the purpose of this research will also be applied to the measure on
measuring cups or cylinders.
18
•
Operator: In this context, the fraction is considered as a transformation (Behr,
Lesh, Post, & Silver, 1983). The fraction
3
as an operator may be perceived as
4
finding three-quarters of some quantity. For example,
•
3
of 12 will result in nine.
4
Quotient: The quotient subconstruct of a fraction focuses on the operation. The
fraction
3
can be interpreted as 3 divided by 4 or the result of sharing 3 cookies
4
among four people.
•
Ratio: This subconstruct expresses a relationship between two quantities, for
example the relationship between the number of nickels and the total number of
coins in a purse, that is, there are three nickels out of the four coins in the purse.
A fraction is only a ratio if it is related to the ratio between the part and the whole
and not between part and the part such as the relationship between the number of
boys and the number of girls in the sixth grade classroom. This means that not all
ratios are fractions.
Ratio
Equivalence
Figure 2.
Operator
Quotient
Problem
Solving
Multiplication
Measure
Addition
Conceptual Scheme for Instruction on Rational Numbers
19
Behr, Lesh, Post and Silver (1983) deemed it “plausible that the part-whole
subconstruct, based both on continuous and discrete quantities, represents a fundamental
construct for rational number development. It is, in addition, a point of departure for
instruction involving other subconstructs” (p. 100). Figure 2 depicts the preliminary
conceptualization of the interrelationships among the various subconstructs. This
diagram suggests that “partitioning and the part-whole subconstruct of rational numbers
are basic to learning other subconstructs of rational number” (p. 100). The solid arrows
represent established relationships while the dashed arrows represent the hypothesized
relationships among the subconstructs and operations on the subconstructs.
Studies (Behr & Post, 1992; Charles & Nason, 2000; Lamon, 1996; Poither &
Sawada, 1983) of children’s partitioning and fraction knowledge suggested that children
use different mental processes according to whether the physical expressions of fractions
are discrete or continuous quantities. Behr, Wachsmuth and Post (1988) gave the
cognitive distinctions between continuous and discrete models:
There are significant similarities and differences between a continuous
model and a discrete model for showing rational number concepts. To represent
rational number concepts, each requires: (a) the identification of a unit and (b)
partitioning of the unit into parts of equal size (i.e. equal measure); for continuous
model, each part is again a single continuous piece and contiguous to other
part(s); for the discrete model, the parts may be a single discrete object, or several
discrete objects, and, in general, are not contiguous with other parts. Thus, there
seem to be distinctly different cognitive demands for representing fractions with
discrete and continuous models (p. 65).
Students partition discrete and continuous quantities in real-life situations, for
example when they have to share a chocolate bar (continuous model) or a set of cookies
(discrete model). Behr and Post (1992) claimed that geometric regions, sets of discrete
objects and the number line are the most frequently used models in representing fractions
in middle school.
Based on the subtle differences with the fraction subconstructs as mentioned
above, deliberate effort must be made by the mathematics teacher in the presentation of
this vital middle school topic. This study focused on the unit fraction, other proper
20
fractions, and the order and equivalence of fractions since these concept areas play a
critical role in the understanding of other topics relating to fractions. For the purpose of
this study, unit fractions are considered to be fractions of the type
any natural number (1, 2, 3, 4 and so on).
1
{
2
1
where n represents
n
}is an example of a unit fraction.
3
4
Non-unit fractions represent fractions whose numerator is greater than 1. For example
{
} is considered a non-unit fraction. Proper fractions refer to those
fractions that are less than the whole (the numerator is smaller than the denominator).
3
4
is also a proper fraction. Improper fractions are fractions that are greater than the whole.
Improper fractions can be expressed as mixed numbers. Thus
9
{
4
}can be
1
expressed as 2 .
4
Models of Understanding
The major goal for every practicing mathematics teacher is that mathematics be
learnt with understanding. “Teaching for understanding is not a new goal of instruction:
School reform efforts since the turn of the 20th century have focused on ways to create
learning environments so that students learn with understanding” (Carpenter & Lehrer,
1999, p. 19). But, what is understanding and how is it developed? Carpenter and Lehrer
(1999) suggested five forms of mental activity that contributed to the development of
mathematical understanding. These are a) constructing relationships, b) extending and
applying mathematical knowledge, c) reflecting about experiences, d) articulating what
one knows, and e) making mathematical knowledge one’s own. There exists, however, a
number of seemingly differing and overlapping models of understanding presented in the
mathematics education arena in the pursuit of answering the first part of the question
posed above. Among these are Skemp’s (1987) relational and instrumental
understanding, Kieren’s (1993) levels in the growth of understanding and Herscovics and
Bergeron’s (1988) two-tiered model of understanding. This section includes a detailed
description of the models of understanding mentioned above and a rationale for using the
21
Herscovics and Bergeron’s (1988) model for analyzing the students’ understanding of
fractions as they worked with physical referents. The tasks designed for the study aimed
at generating the students’ understanding and thinking on unit and non-unit fractions, and
the criteria needed for each component of the two-tier model of understanding.
Skemp’s Relational and Instrumental Understanding
Skemp (1987) grouped the varied views of mathematics understanding under two
broad and overlapping headings, namely relational and instrumental understanding.
Relational understanding includes both understanding of concepts and procedures. The
core of conceptual knowledge is the understanding of relationships or connections among
mathematical ideas and concepts (Hiebert & Lefevre, 1986). This type of understanding
can be perceived as a connected web of knowledge. A unit of conceptual knowledge
cannot exist as an isolated piece of knowledge but should have some relationship or
connection to other pieces of information. Procedural knowledge in its purest form
focuses on symbolism, skills, rules and step-by-step algorithms used in accomplishing a
mathematical task (Aksu, 1997; Ashlock, 2002). Conceptual and procedural knowledge
are considered crucial aspects of mathematical understanding. Students should learn
concepts concurrently with learning procedures so they can see the connections and
relationships (Carpenter, 1986).
Instrumental understanding or learning technique without reason focuses on
calculation, computation and problem solving without sense making. Students are unable
to make connections to a conceptual rationale. The goal of the student, who believes that
mathematics is just a set of formulas to be followed, is to understand a mathematics
problem instrumentally. Skemp (1987) cited three advantages for this type of
understanding.
1.
Instrumental mathematics is usually easier to understand.
2.
The rewards are more immediate and more apparent.
3.
One can often get the right answer more quickly.
In contrast, relational understanding provides opportunities for the students to adapt
methods learnt to accomplishing new tasks which students who exhibit only instrumental
understanding rarely can do. Since the students who possess relational understanding are
constructing knowledge from their own experiences, social interaction and peer
22
negotiation, it becomes easier for them to remember methods and strategies employed in
solving the problem.
Skemp (1987) also believed that more mathematical content might be involved
when teaching for relational understanding. For this type of mathematical understanding,
the learner is able to “build up conceptual structures (or schema) from which the
possessor can (in principle) produce an unlimited number of plans for getting from any
starting point within his schema to any finishing point” (p. 163). To achieve relational
understanding, it becomes pertinent for the mathematics teacher to help the students to
develop a conceptual and procedural knowledge of mathematics and the connections and
differences between them. Both Skemp (1987), and Hiebert and Lefevre (1986)
advocated for linking both types of understanding which will aid in the storage, retrieval
and effective use of mathematical procedures.
Teachers are urged to shift their classroom practices away from the focus on
computational accuracy and toward a focus on deeper understandings of mathematical
ideas, relations, and concepts (NCTM, 2000). It therefore becomes imperative for
teachers who want to support conceptual mathematics understanding in their students to
have knowledge about mathematics teaching and children’s mathematical thinking and
the pedagogical skills needed to put the knowledge to practice.
Kieren’s Model of the Growth of Mathematical Understanding
Pirie (1988) and Kieren (1988) viewed understanding as a dynamic, nonlinear
process and not as a single acquisition, or linear combination as suggested by Skemp’s
(1987) model. This view on understanding led them to collaborate on a model of
understanding which characterizes understanding as an organized and recursive process
which responds to the constructivist description of the understanding process (Kieren,
1993; Pirie and Kieren, 1989). This model describes mathematical understanding as
“recursive in that it involves the use of the same sequence of processes, but at a new
level, with new elements of action… a dynamic process that involves folding back on
itself for growth, extension, and re-creation (Kieren, 1993, pp. 72, 73).
In this model, mathematical understanding is represented using eight “levels" in
concentric circles (see Figure 3). The movement between the levels is not necessarily
from the smaller circle to the largest one but learners increase their knowledge base by
23
returning recursively to previous levels of understanding. A brief description of each
level is given to provide a theoretical comparison and contrast to other models of
understanding that are mentioned.
Inventing
Structuring
Observing
Formalizing
Property
Noticing
Image
Having
Image
Making
Doing
Figure 3.
Kieren’s (1993) Model for the Recursive Theory of Mathematical
Understanding
Primitive Doing. This is the core of action and involves the initial knowledge that
the learner possess. This forms the basis or starting point for the development of a
24
particular mathematical understanding. In reference to fractions, students’ ability to
partition an object (e.g. sharing a pizza) forms the primitive acts of doing that will
eventually form the core for understanding this daunting mathematical topic.
Image Making. At this level, images begin to form through specific actions where
the learner is able to apply the informal knowledge to new situations. “With respect to
fractions, this may mean working on various “sharing” problems and making a record of
such actions, using fractional language in a way that is very closely tied to results”
(Kieren, 1993, p. 73).
Image Having. From the activities included in the image making level, the
learners form mental objects and pictures – “an idea they have, rather than simply an
action they can take in response to a call to action” (p.73). Fractions become a mental
object for the learner.
Property Noticing. This is a recursion of the previous levels. The learner is able
to develop patterns, properties, conjectures and connection from and between the mental
objects formed before. For example, the learner may figure that a string of equivalent
fractions can be generated from a given fraction. Some of these fractions may be
fractions, which the learner cannot mentally “see.”
Formalizing. At this level, the learner engages in self-conscious thinking.
“Knowing” moves beyond specific actions and mental images – mathematical definitions
can be formulated. For instance, the learner is able to “see things in terms of ‘for all
fractional numbers,” and know that
a
can be any fractional number where b ≠ 0 ”
b
(Kieren, 1993, p. 73). This is considered to be the level where mathematics is no longer a
“metaphor for events in a physical or image world. It is abstract, although not necessarily
expressed in generalized mathematical terms or symbols (Pirie & Kieren, 1994, p. 40).
Observing. The learner, at this level, is able to observe his or her conscious
thought processes and have the ability to consistently structure and organize them. For
example, the learner will observe that “there is no least positive fractional number” (p.
73).
Structuring. The learner can now structure his or her own observations made at
the previous level into a set of systematic assumptions. Formal proofs can be formulated
25
at this level. With respect to fractions, the learner will not view the addition of fractions
as an action of combining quantities, or a property of equivalence, or as a relationship
between the numerator and denominator but as a logical consequence of the field
properties and the nature of formal equivalence.
Inventing. At this level, the learner possess the ability to change the entire way of
thinking about a mathematical topic without destroying and eliminating the structure and
understanding of previous ways of knowing. New questions can be generated which
might grow into a totally new concept.
Kieren (1993) described the levels in this model as “knowing organized by
understanding” (p.74). Learners are able to “fold back” to the inner level of knowing
thus resulting in an interweaving of intuitive (informal) and formal understanding. This
model suggests four bases for mathematical knowing, namely, action, image, form, and
structure. These bases are not the same as concrete, pictorial and symbolic modes.
Under this model, an understanding of rational numbers is characterized as a recursive,
dynamic whole and of knowing rational numbers at many levels simultaneously (Kieren,
1993).
Herscovics and Bergeron’s Model of Understanding
Herscovics and Bergeron (1981) suggested a model which identified four levels
of understanding, namely, intuitive understanding, initial conceptualization, abstraction
and formalization. This model, however, provided little help in analyzing the formation
of mathematical concepts since in some cases it was not clear how to characterize and
distinguish between the different levels. This led Herscovics and Bergeron (1988) to
refine and clarify this model of understanding and forced them to extend their model to
include the physical aspects of the concept under consideration – the number scheme.
This extended model of understanding (Herscovics & Bergeron, 1988) divides the
construction of mathematical concepts that stem from actions performed on objects in the
real world into a two-tiered model. The first tier identifies three different levels of
understanding of the preliminary physical concepts. The second tier identifies three
distinct constituent parts of the comprehension of mathematical concepts. “An
understanding that results from observing and thinking about situations involving
physical objects (logico-physical understanding) is crucially different from an
26
understanding resulting from observing and thinking about situations involving
mathematical objects (logico-mathematical understanding); the first kind is natural and
real, the latter is artificial and abstract” (Boulet, 1998, pp. 20, 21).
The three components or levels of the first tier are intuitive understanding, logicophysical procedural understanding, and logico-physical abstraction.
Intuitive understanding. This refers to a global perception of the notion at hand; it
results from a type of thinking based essentially on visual perception; it provides rough
non-numerical approximations.
Logico-physical procedural understanding. This is the acquisition of logicophysical procedures which the learners can relate to their intuitive knowledge and use
appropriately.
Logico-physical abstraction. This type of understanding refers to the construction
of logico-physical invariants or the reversibility and composition of logico-physical
transformations, or as generalization.
Since the teaching and learning of mathematics is usually characterized by
symbolic and numerical manipulation and not the handling of physical objects, the
second tier of Herscovics and Bergeron’s model (1988) deals with the understanding of
the mathematical concept itself (Boulet, 1998). It is worthy to note that the levels of
understanding in the first tier are essential to the constituent parts of understanding in the
second tier. Boulet (1993) put it this way: “without the understanding of such concepts
as those describe in the first tier of the model, understanding at this level can be both
superficial and fleeting…. In fact, it is this very connection with the real, physical world
which gives meaning to elementary mathematical notions” (p. 33).
The second tier consists of three constituent parts: logico-mathematical
procedural understanding, logico-mathematical abstraction and formalization. At this
point in the model Hersovics and Bergeron (1988) replaced the word ‘level’ with
‘constituent part’ with the purpose of preventing “an overly hierarchical interpretation”
(Herscovics & Bergeron, 1988, p.20).
Logico-mathematical procedural understanding. At this constituent part of
understanding, the learner possesses explicit logico-mathematical procedures that the
learner can relate to the underlying preliminary physical concepts and use appropriately.
27
Logico-mathematical abstraction. This refers to the construction of logicomathematical invariants together with the relevant logico-physical invariants, or the
reversibility and composition of logico-mathematical transformations and operations, or
generalization.
Formalization. This component of understanding involves a gradual development
of various mathematical notations. This constituent part is defined as the “axiomatization
and formal mathematical proof” (Boulet, 1998, p. 23). At the middle school level, it can
be viewed as the discovery of axioms and the ability to justify these axioms logically.
Formalization also includes the enclosure of a mathematical notion into a formal
definition and the use of mathematical symbolization for notions for which prior logicomathematical procedural understanding or abstraction already exist to some degree. The
research literature (e.g. Saxe, Taylor, McIntosh & Gearhart, 2005) does indicate,
however, that the students’ knowledge of the standard notation of fractions and the partwhole relationships can develop to a certain extent independently.
Chaffe-Stengel and Noddings (1982) seeking to clarify the symbolic process
involved in the treatment of fractions made mentioned of the connection that Charles
Pierce (Eisele, 1962) made with the visual, written and spoken representation of a
mathematical concept that form an integral part of the formalization constituent part of
the Herscovics and Bergeron’s (1998) model of understanding. Pierce “characterized the
signing process as the triadic interrelation between the concrete objects involved in the
problem, the sign which stands for those objects, and the signification, or meaning, of the
sign. The signification serves as a mediator, making sense of the object-sign relation.
The function of the sign is to stand in place of the object, to allude to the object but
remain more easily available than the object” (Chaffe-Stengel & Noddings, 1982, p. 43).
Figure 4 illustrates Chaffe-Stengel and Noddings (1982) process of signing which
corroborates with the connections that exist between the two tiers of Herscovics and
Bergeron’s (1988) model of understanding.
28
Verbal mediator
Sign
1 1 1
, ( ), , etc
2 3 4
Part of a
thing
Concrete representation
Figure 4.
Potential Signing Process Appropriate for Fraction Readiness (ChaffeStengel & Noddings, 1982)
Although the understanding of a mathematical concept relies heavily on the
understanding of the preliminary physical concepts, this does not suggest that the
“understanding of a mathematical concept needs to await the prior three levels of
understanding of the preliminary physical concepts” (Herscovics & Bergeron, 1988, p.
20). This means that the construction of a mathematical concept does not necessarily
progress in a linear fashion but are interconnected as Figure 5 shows.
This model has several pedagogical implications as suggested by Herscovics and
Bergeron (1988).
1.
It explicitly links up the children’s mathematics to their physical world.
2.
Thus, the physical world should be used as a starting point in the construction
of the children’s mathematical concept.
3.
Teachers can develop tasks related to every aspect of understanding of a given
concept.
The broad range of activities that the students are engaged in will complement the rich
cognitive environment provided in a constructivist learning environment.
29
UNDERSTANDING THE UNDERLYING PHYSICAL CONCEPTS
Intuitive
understanding
Logico-physical
procedural
understanding
Logico-mathematical
procedural
understanding
Logico-physical
abstraction
Logico-mathematical
abstraction
Formalization
UNDERSTANDING THE EMERGING MATHEMATICAL CONCEPT
Figure 5.
Herscovics and Bergeron’s (1988) Model of Understanding
Rationale for using Herscovics and Bergeron’s (1988) Model of Understanding
Close examination of Kieren’s (1993) recursive theory of mathematical
understanding reveals the possible steps a child encounters in the learning process of a
mathematics concept. It does not explicitly provide an instrument for analyzing the
development of a mathematical concept. This model is mainly geared towards problem
solving and does not adequately provide the tools needed to describe the comprehension
involved in concept formation. In particular reference to the fraction concept, Kieren’s
(1993) model does not specify the content of the fractional understanding needed at each
level. Skemp’s (1987) categorization of understanding is too broad and general to use for
the analyzing of the students’ concept of the unit and non-unit fractions.
30
Although Herscovics and Bergeron’s (1988) do not claim that their model of
understanding is suitable to describe the understanding of all mathematical concepts,
there have been successful attempts to use this model to analyze the addition of small
numbers, measurements, surface areas, algebraic concepts and coordinate geometry.
Boulet (1993) used this model and applied it to the unit fraction as she worked with a set
of fourth-graders. In this study, this model was used to interpret the sixth-graders’
construction of the unit and non-unit fraction from a physical and mathematical
standpoint. From my review of the different discussions on models of understanding
Herscovics and Bergeron’s (1988) model of understanding that was partially influenced
by Skemp’s (1987) instrumental and relational understanding, provides a more refined,
detailed and systematic model for interpreting the students’ actions on the tasks provided
to analyze their mathematical behavior.
Though this model does not provide an explicit didactical approach to the
handling of the fraction it does provide for the leeway for the building of one.
Herscovics and Bergeron’s (1988) model is not concerned with the pedagogical aspects
of the fraction but it can be used to “supply concrete cues for the formation of an
instructional program in that activities can be designed specifically for each of the
components” (Boulet, 1993, p. 45). This model of understanding does not provide any
inkling of how the fraction concept is learned, however, it is able to clarify the
components needed to interpret the understanding of the unit and non-unit fraction as the
students work with the physical referents. It can also be used as a template or frame of
reference for the development of tasks and activities that are designed to expunge some
of the cognitive obstacles students encountered while working with the fraction.
Criteria for the Components of Understanding in the Herscovics and Bergeron’s
(1988) Model of Understanding
Having established the rationale for the use of the Herscovics and Bergeron’s
(1988) model of understanding, an outline of the criteria necessary for each level of
understanding is included below. These criteria are adapted from Boulet (1993) study of
the unit fractions with two sets of fourth graders.
31
Understanding the Underlying Physical Concepts
The physical concepts underlying the emerging mathematical concepts are the
focal points of the first tier of the Herscovics and Bergeron (1988) model. Based on the
assumption that partitioning and the part-whole subconstruct are fundamental to learning
the other subconstructs of rational numbers, a teaching sequence was designed to develop
an understanding of fractions based on partitioning and the part-whole relationship.
These concepts were expanded to the order and equivalence of fractions, and the
quotient, measure and operator subconstructs. Similar sequencing of the concept was
done by the Rational Number Project (RNP).
Intuitive understanding of the unit fraction. The sole criterion necessary for
intuitive knowledge is the awareness of the role of partitioning in part-whole
relationships. There are four conditions that are necessary for this awareness. Students
should recognize:
1.
the equality of the parts in a partitioned whole despite the position of the parts in
the whole.
2.
the inequality of the parts in similar wholes regardless of the equal number of
parts (shaded or unshaded).
3.
the inequality of the parts of similar wholes that are partitioned differently.
4.
the similarity in the part-whole relationships in spite of the inequality of the size
of the parts.
Logico-physical procedural understanding. The most fundamental criterion for
this level of understanding is the ability to partition a whole into a given number of equal
shares. The methods to ensure equality of the parts can vary with the students’ freedom
to use whatever materials (e.g. pipe cleaners, rulers, cutouts) present at hand to aid in
their estimation.
Logico-physical abstraction. According to Boulet (1993): “It is within the
context of this component in the model of understanding that a first glimpse of the
breadth and depth of a child’s understanding of the unit fraction is given” (p. 39). At this
level the fraction concept is distinguished from the simple action of dividing a whole into
equal parts but rests upon the understanding of the relationship between the part and
whole. There are five criteria related to this component.
32
1)
The child’s ability to accurately understand the equivalence of the part-whole
relationship in spite of a variation in the physical attributes of the whole, that is,
the wholes can be of different shapes and/or sizes.
2)
The child’s ability to reconstitute the whole from its parts.
3)
The child’s ability to recognize the equivalence of the part-whole relationship
regardless of the physical transformations of the whole. For example, will the
student recognize the part-whole relationship in these identical whole
to be the same?
4)
The child’s ability to recognize the relationship between the size of the parts and
number of equal shares.
5)
The child’s ability to repartition already partitioned whole.
Understanding the Emerging Mathematical Concept
The quantification of the fraction is the main consideration in the second tier
instead of the part-whole relationship.
Logico-mathematical procedure. The procedures that the child uses correctly in
association with the underlying physical concepts gained in the first tier become the focus
of analysis at this level of Herscovics and Bergeron (1988) model of understanding. The
criterion set for this component include the child’s ability to “quantify part-whole
relationships depicted in all sorts of situations and to produce illustrations of orally given
unit fractions” (Boulet, 1993, p. 41).
Logico-mathematical abstraction. The criteria for this level is similar to those of
the logico-physical abstraction level in that the main objective is to determine the scope
of the child’s ability to comprehend and understand the invariance, reversibility and
generalization aspects of unit and non-unit fractions. The main difference between both
levels is the fact that at the logic-mathematical abstraction level, the child is able to
quantify the part-whole relationships connected to the physical attribute of the fraction.
Mainly, the student is now able to verbally associate a numerical value to the physical
referent illustrating the fraction. For example, in examining the child’s apprehension of
the invariance (i.e. the part-whole relationship remains unchanged regardless of the
physical operations and/or transformations) of the fraction as demonstrated using discrete
33
objects, the child is shown a set of 10 chips – 4 white and 6 black. The chips are firstly
arranged similar to this representation.
Using the chips to represent cookies, the student will be asked what fraction of the set of
cookies is eaten if only the white ones are eaten. The chips will then be rearranged to
look like this.
At this point the child can be asked to decide if the same amount of the set of cookies
would be eaten if only the white ones are eaten (logico-physical abstraction invariance)
and what fraction of the set of cookies was eaten if only the white ones are eaten (logicomathematical abstraction invariance). Boulet (1993) noted that children may be able to
recognize a logico-mathematical invariant while not recognizing the underlying logicophysical one. Consequently, at this constituent part both the logico-physical and logicomathematical abstractions need to be verified.
Formalization. At this final constituent part of the model of understanding, the
formal aspect the fraction is considered. Embedded in this component of Herscovics and
Bergeron’s (1988) model is the need to symbolize the fraction that has been internalized
and spoken. Boulet (1993) defines formalization (at the elementary level) to mean the
discovery of axioms which will lead to the discovery of logical mathematical
justifications. For example,
2
is two-thirds as well as what it represents. Included in the
3
set of criteria for formalization is the students’ ability to symbolically order a set of
fractions, and symbolically reconstitute wholes as well as parts of given fractions.
Physical Referents and the Understanding of the Fraction Concept
The issue of the use of manipulative materials in the mathematics classroom is
hotly debated. Some research articles reported on the ineffectiveness of physical models
in helping the students understand the underlying mathematical concept involved with a
fraction, and the difficulty children have in moving from using physical models to using
34
symbolic notation (Bezuk & Bieck, 1992; Thompson & Lambdin, 1994). Majority of the
research conducted on physical referents in the mathematics classroom, however, (e.g.
Hall, 1998; Hasemann, 1981; Hinzman, 1997; Post, Cramer, Behr, Lesh & Harel, 1993)
have shown that the active use of physical referents may aid in the conceptual
development of the fraction concept. “These concrete analogs are used by teachers with
the conviction that they will facilitate the construction, understanding and retrieval of
mathematical concepts, reduce learning effort, serve as memory aids, provides a means of
verifying the truth, … mediate transfer between tasks and situation, and indirectly
facilitate transition to higher levels of abstraction” (Boulton-Lewis, 1998, p. 220).
Physical referents play a pivotal role in the acquisition and retention of vital
mathematical concepts. Wearne (1990) from a study on decimal fractions with a set of
students, argued that the participants were able to use their reasoning abilities about
decimal fractions at the same level or better as they appeared to be using the meanings
derived and developed from the firsthand experiences the students had with the physical
referents they used as aids to justify processes and solve problems. NCTM (2000) in its
Representation Standard for Grades 6-8 stated that “representation is central to the study
of mathematics. Students can develop and deepen their understanding of mathematical
concepts and relationships as they create, compare, and use various representations.
Representations – such as physical objects, drawings, charts, graphs, and symbols – also
help students to communicate their thinking” (p. 280). From the students’ perspective the
representation could be their reconstruction or drawing of a contextualized problem
situation or their interpretation or use of a representation that was structured or designed
by the teacher-researcher.
From the very onset of fraction instruction students are introduced to this concept
using circular and/or rectangular objects such as a pizza, a cake or sheet of rectangular
paper used in paper folding. Thus, when asked to represent a fraction pictorially or
diagrammatically, students most frequently resort to drawing and partitioning these
geometric regions. The task-based activities that the students’ worked with during this
study gave them the opportunity to manipulate physical referents of different shapes,
sizes and quantity besides the ones mentioned above. Their behavior were observed as
they labored with continuous wholes that are not confined only to circular or rectangular
35
regions but triangular regions and solid/liquid quantities. Discrete models consisted of
objects of varying shapes - circular, triangular and rectangular in appearance. Behr,
Wachsmuth and Post (1988) purported that “discrete-embodiment tasks are more difficult
for children than continuous-embodiment tasks” (p.73).
One the major work done on rational numbers, of which fraction is a subset, is the
Rational Number Project (RNP) pioneered by Merlyn Behr (now deceased), Richard
Lesh and Thomas Post and funded by the National Science Foundation (NSF). From this
extensive work a RNP curriculum was developed and used which reflected the following
beliefs: “(a) Children learn best through active involvement with multiple concrete
models, (b) physical aids are just one component in the acquisition of concepts – verbal,
pictorial, symbolic and realistic representations also are important, (c) children should
have opportunities to talk together and with their teacher about mathematical ideas, and
(d) curriculum must focus on the development of conceptual knowledge prior to formal
work with symbols and algorithms”
The NCTM Principles and Standards (2000) recommends that physical models
be used in the classroom to aid in the development of students’ understanding of
fractions. As a matter of fact, “representations are ubiquitous in the middle-grades
mathematics curriculum” (p. 280) that is proposed by NCTM. Most often than not,
young children are presented with physical referents in mathematics learning that are
expected to facilitate the learning process. These include everyday materials, structure or
semi-structured concrete embodiments of mathematical concepts such as Cuisenaire Rods
bundles of tens. However, as the students get into middle grades teachers seem to
eliminate these concrete representations for one reason or the other. NCTM 2000 and
other researchers (e.g. Boulton-Lewis, 1998; Cramer, 2001; Kamii, Lewis, Kirkland,
2001) encourage the use of manipulatives at all levels of education suffice these physical
referents aid the students to make relationships through constructive abstraction in
problem solving.
Building on the theories of Piaget, Bruner and Dienes, the Lesh’s translation
model (first cited in Lesh, Landau, & Hamilton, 1983) suggested “that a deep
understanding of mathematical ideas can be developed by involving students in activities
that embed the mathematical ideas to be learned in five different modes of representation
36
with an emphasis on translations within and between modes” (Cramer, 2003, chap. 24).
These five modes - manipulatives, pictures, real-life contexts, verbal symbols, and
written symbols - stress that understanding is reflected in the ability to represent
mathematical ideas in multiple ways, plus the ability to make connections among the
different embodiments (Lesh, Cramer, Doerr, Post & Zawojewski, 2003) . Translations
within and between various modes of representation make ideas meaningful for students.
Lesh’s translational model (Lesh, Post & Behr, 1987) shows the connections among
different external representations including real world situations and manipulative aids.
This instructional model (see Figure 6) necessitates the active involvement of students as
well as the extensive use of manipulatives and other learning aids such as pictures,
diagrams, student verbal interactions, and real life situations simulations modeled in the
classroom. Each mode is connected to every other mode forming a translation such as
picture-to-symbolic. Each connecting link is bidirectional. This signifies the
interconnection with the different modes evidenced during problem solving.
Pictures
Written
Symbols
Spoken
Symbols
Manipulative
Aids
Real
World
Figure 6.
Lesh’s Translation Model (Lesh, Post & Behr, 1987)
37
Among the pedagogical beliefs that emerged from the literature, the use of multiple
physical models such as fraction strips, fraction circles, Cuisenaire rods, and chips over
an extended period of time is important in optimizing children’s learning of fractions.
The art of paper folding is also useful in developing and extending students’
understanding of the unit fraction (Cramer and Henry, 2002). In Piaget’s stages of
cognitive development he proposed that children’s learning is enhanced and optimized
when they are able to manipulate concrete objects in the development of the concept then
further extending his knowledge to include abstract ideas (Higgins, 1973).
Partitioning Strategies
Partitioning is defined as the “act of dividing a quantity into a given number of
parts which are themselves qualitatively equal” (Kieren & Nelson, 1981, p. 39). This
process produces equal parts that are not overlapping (Lamon, 1999). An understanding
of partitioning is fundamental in the development of students’ understanding of fractions
(Poither and Sawada, 1990). It forms the basis for equivalence that in turn forms the
foundation for operations on fractions. Partitioning also plays an active role in the
development of the fraction concept as it relates to the part-whole, measure and quotient
subconstructs.
Fair sharing has been part of most students’ everyday experiences, thus,
partitioning activities should build on what the students already know and provide the
opportunity for them to extend their knowledge to include more complex tasks (Mack,
1995). Findings from several studies on fraction understanding show that students who
fail to extend their knowledge usually exhibit a lack of understanding about partitioning
(Behr, Lesh, Post & Silver, 1983; Lesh, Landau & Hamilton, 1983). Behr and Post
(1992) recommended that “children be given the opportunity to partition various types of
objects” (p. 211). This act of partitioning should go beyond the visual separation of a
regular polygonal region into congruent parts. The focus should be on the appropriate
interpretations of the children’s actions as they operate on the partitioning activities
(Steffe, 2002). The students partitioning experiences should include partitioning of
objects in even and odd number of parts.
The process of unitizing and the recognition of the unit play a crucial role in
partitioning. Children’s understanding of fraction, for the most part, is delayed due to the
38
failure to recognize the importance of the unit. The unit is different in every context. For
example in equally sharing 12 quiches among 4 people (Activity 12 – see Appendix B),
the unit is the twelve quiches. Each person would get 3 quiches, that is
3
1
or of the
12
4
quiches. However, in dividing a circular birthday cake among 4 friends, the cake is the
unit. The process of operating on the unit is called unitizing (Lamon, 1996). In the case
of the 12 quiches, a child might consider the unit as one unit consisting of 12 quiches
continuous while another child may think of the unit as 12 separate and distinct quiches.
This study will not delve into the process of unitizing but will make mention of it as is
necessary in explaining the students’ partitioning strategies.
Poither and Sawada (1983) and Lamon (1996) suggested that children use an
assortment of intuitive strategies when given partitioning activities. A number of these
partitioning strategies are identified in the literature such as Armstrong and Larson (1995)
part-whole and direct comparison strategies and Poither and Sawada’s (1990) algorithmic
halving and evenness.
However, as these strategies proved to be too general in their
scope, the more explicit partitioning strategies that Charles and Nason (2000) and Lamon
(1996) had generated from their study on young children’s partitioning strategies were
used in this study as a tool to analyze and interpret the participants’ sharing behavior as
they participated in the task-based activities. The strategies provide an explicit
framework or template based on the observation of partitioning strategies in children and
is therefore suitable for use in this research.
Charles and Nason (2000) identified twelve partitioning strategies, six that
emerged from the literature and the remaining six emerged from the strategies used by
the students in their study that were not reported in the literature. They categorized the
twelve strategies into three areas:
•
Partitive Quotient Construct Strategies
•
Multiplicative Strategies
•
Iterative Sharing Strategies.
They identified the steps that are involved in the utilization of each strategy. A deliberate
attempt is made to include the steps for each strategy similar to ones identified in Charles
39
and Nason (2000) with the aim of providing the readers of this study with the explicit tool
used as the frame of reference for the interpretation of the students’ partitioning behavior.
Table 2 shows how the partitioning strategies identified by Charles and Nason
(2000) are categorized and whether they are gleaned from the research literature on
fractions or not. The authors referred to these as a “taxonomy for classifying young
children’s partitioning strategies in terms of their ability to facilitate the abstraction of the
partitive quotient fraction construct from the concrete activity of partitioning objects
and/or sets of objects” (p. 192).
Partitive Quotient Construct Strategies
Strategies in this category utilize the “relationship between number of people
sharing and the fractional name to generate the denomination for each share” (Charles &
Nason, 2000, p. 199).
Partitive quotient foundational strategy. This strategy involves the application of
the following steps:
Step 1:
Recognition of the number of people (n)
Step 2:
Generation of fraction name from the number of people
Step 3:
Recognition of relationship between fraction name and number of equal
pieces in each whole object (n equal pieces)
Step 4:
Partitioning of each whole object into equal pieces (n equal pieces)
Step 5:
Sharing the pieces ( 1 n of each object to each person)
Step 6:
Quantifying each share (addition of 1 n pieces)
Proceduralised partitive quotient strategy. This is a reduced version of the strategy
mentioned above. The utilization of this strategy to formulate a solution to the task
results in three fewer steps than the foundational strategy. They are:
Step 1:
Recognition of number of people (n)
Step 2:
Recognition of number of analog objects (m)
Step 3:
Quantification of each person’s share ( m n )
Partitive and quantify by part-whole notion strategy. This strategy is one of the
six strategies that was not reported in the literature but emerged from the researchers’
observation of some of the students’ partitioning behavior. It is similar to the
foundational strategy without Steps 2 and 3.
40
Step 1:
Recognition of the number of people (n)
Step 2:
Partitioning of each whole object into equal pieces (n equal pieces)
Step 5:
Sharing one piece from each object to each person
Step 6:
Quantification of each person’s share by application of part-whole system
mapping to determine fraction name.
Table 2: Charles and Nason’s (2000) Partitioning Strategies
Partitive Quotient
Construct
Partitive quotient
foundational
Proceduralised partitive
quotient
Partition and quantify by
part-whole notion
Multiplicative
People by objects
Regrouping
Horizontal partitioning
Iterative Sharing
Halving object then halving
again and again
Half the objects between
half the people
Whole to each person then
half remaining objects
between half the people
Half to each person then a
quarter to each person
Repeated sizing strategy
Repeated halving/repeated
strategy
Regrouping strategy. The steps for this strategy are:
Step 1:
Recognition of the number of people (n)
Step 2:
Recognition that the number of people gives the fraction name (nths)
Step 3:
A realization that the fraction name gives the number of pieces in the
whole (n)
Step 4:
A realization that the total number of nths to be shared can be generated
by multiplying the objects (m) by the number of nths in each whole.
41
Step 5:
Quantification of each share through a recognition that the number of nths
for each person can be calculated by dividing the total number of nths by
the number of people (m)
Horizontal partitioning strategy. The steps for this strategy are:
Step 1:
Recognition of the number of people (n)
Step 2:
Generation of fraction name from number of people (nths)
Step 3:
Recognition of relationship between fraction name and number of pieces
in each whole object (n)
Step 4:
Horizontal partitioning of each whole circular object into pieces (n pieces)
Step 5:
Quantification of each share ( 1 n piece)
Step 6:
Recognition that shares are equal.
Multiplicative Strategies
For this type of strategy, a multiplication algorithm is used to find the number of
parts in each whole.
People by objects strategy. This strategy involves the following steps:
Step 1:
Recognition of the number of people (n)
Step 2:
Recognition of the number of objects (m)
Step 3:
The number of pieces in each whole is generated by multiplying (n) by
(m)
Step 4:
Partitioning each whole into (nm) pieces
Step 5:
Sharing the pieces between people
Step 6:
Quantification of each share
Iterative Sharing Strategies
Halving object then halving again and again strategy. The steps for this strategy
are:
Step 1:
Iterative halving of each whole object into eight equal pieces
Step 2:
Recognition that each piece is 1 8
Step 3:
Sharing out the pieces
Step 4:
Quantification of each share
42
Half the object between half the people strategy. This strategy is slightly different
from the halving strategy identified in the literature. For the purpose of their study,
Charles and Nason (2000) noted that this strategy was employed for the sharing of two
objects between four people. These are the steps for this strategy.
Step 1:
Recognition of the number of people (4)
Step 2:
Recognition that the number of objects (2)
Step 3:
Realization that halving the objects will generate enough pieces to share
between all the people.
Step 4:
Partition of the objects into halves
Step 5:
Sharing the halves between all the people
Step 6:
Quantification of each share
Whole to each person then half remaining objects between half people strategy.
This strategy is an extension of the above strategy and involves the following steps:
Step 1:
Recognition of the number of people (n)
Step 2:
Recognition of the number of objects (m)
Step 3:
Realization that the number of objects (m) is greater than the number of
people (n)
Step 4:
Sharing of one whole object to each person
[Step 3 and 4 are repeated until m < n]
Step 5:
Application of ‘half objects between half the people strategy’
Step 6:
Quantification of each share
Half to each person then a quarter to each person strategy. This strategy is also
an extension of the ‘half the objects between half the people strategy.’ The steps utilized
in the strategy are:
Step 1:
Application of the ‘half objects between half the people strategy’
Step 2:
Realization that there are not enough remaining objects to reapply the
‘half the objects between half the people strategy’
Step 3:
Application of the ‘partitive quotient foundational strategy’
Step 4:
Quantification of each share
Charles and Nason (2000) study was conducted in Eastern Australia. They reported that
pizzas were hardly ever cut into equal slices in that country. This facilitated the
43
observation of the next two partitioning strategies. The strategies were not used in this
study but are included for the complete citation of Charles and Nason (2000) twelve
partitioning strategies.
Repeated sizing strategy. This steps involved in this strategy are:
Step 1:
Partitioning of each whole object into an even number of unequal pieces
Step 2:
Sharing of pieces using attribute of area rather than attribute of number in
an attempt to achieve equal shares.
Repeated halving/repeated sizing strategy. The steps for this strategy are as
follows.
Step 1:
Application of the ‘halving the object again and again strategy’
Step 2:
Application of the ‘repeated sizing strategy’
The next three partitioning strategies were used in Lamon (1996) study of 346
children from grades four through eight. These strategies were used alongside Charles
and Nason (2000) strategies to categorize the sixth graders partitioning acts.
Preserved-pieces strategy. In sharing situations where a person should receive
more than one whole of the total quantity, the students will leave the required wholes
intact and then mark and partition the whole that requires cutting.
Mark-all strategy. As the name of this strategy suggests, the students will mark
all the pieces but at the time of sharing only the pieces that require cutting will be cut.
Distribution strategy. All the pieces of the whole are marked and cut. Each piece
is then distributed.
Summary
This study was set within a constructivist environment where notions
about the teacher and students’ role in the learning environment, how they construct
fraction knowledge as they participate in the activities and the role of the physical
referents and real world representations in this construction of knowledge come together.
Kieren (1976) divided the concept of a fraction into five subconstructs: the part-whole,
quotient, measure, operator and ratio subconstruct. Herscovics and Bergeron’s (1988)
model of understanding and the partitioning strategies identified by Charles and Nason
(2000) and Lamon (1996) were explicitly discussed as they provided the theoretical
frame of reference in investigating the students’ method of determining fraction size,
44
partitioning and repartitioning the whole, and generating equivalent fractions. These
partitioning strategies are definitive and are well suited for the purpose of the present
study.
45
CHAPTER 3
METHODOLOGY
Qualitative Interpretive Framework
Current trends in research in mathematics education reflect a paradigm shift from
an emphasis on scientific or quantitative studies to the use of qualitative, interpretative
methodologies (Teppo, 1998). Qualitative research is primarily concerned with human
understanding, interpretations and intersubjectivity and “is a field of inquiry in its own
right” (Denzin & Lincoln, 2003, p. 3). This type of research necessitates the mingling of
the researcher with the everyday life of the setting for the study where the researcher
would enter the participants’ world. Through the ongoing interactions with the
participants, the researcher sought to obtain a clear understanding of the participants’
perspectives on a particular phenomenon. The research questions posed in this study are
based largely on students’ understanding of fractions as they engage in certain processoriented activities, thus necessitating a qualitative study. The methods used in this
qualitative study are better able to provide a “deeper understanding of social phenomena
than could be obtained from purely quantitative data” (Silverman, 2001). One such
social phenomenon is the act of fair sharing. Besides highlighting the technical aspects
of this study, this chapter also includes the limitations and assumptions associated in
conducting the type of research design mentioned above.
Participants
The participants involved in this study come from a private middle school situated
in a metropolitan area of southeast Florida where the teacher-researcher currently works.
The school serves a predominantly Hispanic community with a number of students from
West Indian origin. The participants for this study were members of the sixth grade class
during the school year the study was conducted. The study took place between the
46
months of February and April, 2005. The researcher focused on sixth grade students for
various reasons. Many students at this level exhibit insufficient understanding of
fractions despite prior instruction in lower grades. This fact is based on the teacherresearcher’s years of experience in teaching sixth grade, comments from other sixthgrade mathematics teachers, and reports from literature (e.g. Lamon, 1993; Mack, 1995)
dealing with this vital middle school topic. Another reason for choosing this group was
the minimal attention and instruction that are given to fractions in grades higher than
sixth, thus making it a priority for teachers in sixth grade to seek an in-depth
understanding of how students at this level make sense of the fraction concept with the
aim of providing meaningful engaging activities that will promote relational
understanding of the concept.
Two colleagues who have experiences in teaching middle school mathematics,
served in the role of teacher-observer. These teacher-observers scored the pretest and
made the different group assignment. One of them regularly attended and observed the
activity sessions. Informal conversations were held with the teacher-observers that help
the teacher-researcher in planning review sessions and interpreting the findings of the
study.
The sixth grade class involved in the study consisted of 10 males and 10 females
ranging from ages 10 to 12. The results of the pretest that was administered prior to the
teaching sequence were used to categorize the students on a scale from 1 to 3 with 1
representing very little understanding of the fraction concept and 3 representing a basic
understanding of fractions. For the purpose of the cooperative activities, the students
were divided into five heterogeneous groups with at least four students in each group.
The members of the group were selected randomly by the teacher-observers based on the
categories defined by the pretest results. At least one member from each category was in
each group. Appendix C shows the students by category the different group assignments.
The teacher-researcher intentionally chose to involve the whole class in this
undertaking since studies (e.g. Boulet, 1998; Mack, 1990) have shown that research done
in a natural classroom setting yields more accurate data which are of more educational
value. Written permission was granted to the teacher-researcher to pursue this study from
the administration of the school. Permission forms and letters were sent to the sixth-
47
grade parents explaining the purpose and nature of the study and were returned prior to
the beginning of the study. The teacher-researcher met with individual parents who had
questions concerning their child’s participation in the research. Their main concern was
what they needed to do to aid in the study. The tasks and process-oriented activities that
were used in this study did not yield a grade for the students. They were used for the sole
purpose of seeking to understand how students make sense of fractions. A pseudonym
was given to each student to aid in the non-recognition of the participants involved in the
study.
Research Design
The research design involved three phases that lasted for a period of eight weeks.
The phases comprised the pretest, teaching sequence packed with task-based activities,
and selected interviews. Classroom–based research has become more accepted in the
mathematics education arena due to criticisms of the limited applicability of the findings
of studies conducted in out-of-school laboratory controlled environments (Confrey &
Lachance, 2000). Teachers fear that these findings have very little use in the classroom
and lack the vigor and practicality of the everyday classroom, that is, they are theoreticalbased and not classroom-based. As Steffe, Thompson and von Glasersfeld (2000) wrote
“a large chasm existed between the practice of research and the practice of teaching” (p.
271). Conducting a whole-class research study within the confines of the researcher’s
class allows the researcher to experience, firsthand, students’ mathematical learning and
reasoning about fractions. In this section, each phase will be described and the rationale
for including these phases in the study will be established. No instruction on fractions
was given in sixth grade prior to the study.
Pretest
At the beginning of the study a pretest consisting of seven multiple choice and 18
free-response items was administered to the 20 participants. A copy of the pretest is
included as Appendix A. One of the main purposes of this test was to obtain a detailed
diagnostic skill profile for each learner as it related to the students’ current understanding
of fractions. Some of the items from this test were also used to verify the intuitive
understanding of fractions as detailed in the Herscovics and Bergeron’s (1988) model of
understanding and to obtain the students’ definition of fractions prior to the
48
administration of the teaching sequence. The first seven items (multiple-choice) of the
test are NAEP and TIMSS sample test questions. The Free-Response items are gleaned
from research related to fractions (Boulet, 1993; Mack, 1990) with the last seven
questions coming from the written test databank of the Rational Number Project [RNP]
(Cramer, Behr, Post & Lesh, 1997). To ensure the feasibility and fairness of the test
items for the sixth grade level, the pretest was administered to two sets of sixth-graders
from two schools – two private institutions with the same affiliating body as the school in
which the study was conducted and with similar demographics, with the exception that
one of the schools had a predominantly West Indian population. The results of the tests
and suggestions from the volunteer teachers aided the teacher-researcher in the final draft
of the pretest items.
Teaching Sequence
The teaching sequence consisted of 12 task-based activities (see Table 2). The
students performed two types of tasks during the course of the study. Some tasks were
individually done while others were completed in cooperative groups. The activities with
the relevant tasks were designed and arranged particularly for this study. The fraction
tasks were adapted from Boulet’s (1993) study on the unit fraction with the researcher
modeling them to suit the sixth grade audience. An assessment task was given at the end
of most of the activities and served the purpose of verifying or determining whether the
students comprehended the concepts that were highlighted in the tasks for that particular
activity. Some of the assessment and activity tasks were from RNP, while others,
including the interview protocols, emerged from other fraction-related literature. The
tasks and interview protocols are found in Appendix B.
The final activity of the teaching sequence was centered on a “Fraction
Breakfast.” The idea for conducting an actual breakfast emerged from a similar task –
“Cake Problem” used in Poither and Sawada’s (1983) study. Unlike the “Cake Problem,”
the “Fraction Breakfast” activity used real food that the students were allowed to eat at
the end of the partitioning tasks. The questions given on the activity sheet were
fashioned from similar questions used by Behr and Post (1992), Lamon (1993), Mack
(1995) and Moss and Case (1999) in their quest to obtain students’ meaning of fractions
and its operations. Each breakfast table was set with the utensils for eating and sharing
49
including gloves to be used while partitioning the food. Items for the breakfast were
distributed to each group. Each group had the same food items in the same quantity.
They were required to share the food equally among the group members and then answer
the questions on the activity sheet that accompanied the breakfast. One member of each
group was chosen to initiate the discussion, but each person was required to write on the
individual task sheet. The “Fraction Breakfast” activity had been field tested and had
proven to be successful in facilitating children’s partitioning capabilities and techniques.
This task also provided key insights into the partitioning process.
Table 3 gives an outline of the actual timeline for the study. The order of the
sessions was determined from the field tests that the researcher conducted with the
previous set of sixth graders at the same school as the present study. It should be noted
that there was a review session of the previous class topic at the beginning of every
session. This lasted for approximately 10 minutes and was in the form of an oral exercise
on the previous tasks.
The middle school where this study was conducted operated on a
block schedule with mathematics occurring for sixth-grade three times for a Block B
week and two times for a Block A week. The eight week period resulted in 20 sessions.
Each session lasted for 60 minutes except the “Fraction Breakfast” which lasted for one
and a half hours.
In addition to the task-based activities that formed the major component of the
teaching sequence employed in this whole class exploratory study, other aspects of the
sequence need to be considered. These include the teaching agent (the researcher), the
group of participants (sixth grade) that were dealt with earlier in the chapter, the learning
space (classroom setting), witnesses of the teaching sequence (peer mathematics
teachers), and the various methods of data collection that were used to answer the
research questions posed in Chapter 1. Initial conjectures about the teaching sequence
were based on the research literature, however, modifications were made daily based on
the ongoing analysis of the students’ activity (Ellerbruch & Payne, 1978; McClain,
2002). The activities done during the teaching sequence were adapted and designed by
the researcher in an attempt to provide opportunities for the child to learn by judiciously
selecting tasks, offering suggestions, and posing questions” (Cobb, Wood & Yackel,
50
1990). The teaching sessions were flexible and were susceptible to the students’ grasp of
the concept in the subsequent sessions.
Table 3: Outline of the Teaching Sequence
Types of
Understanding/Session
Tasks
Logico-Physical
a. Intuitive
Pretest
b. Logico-physical procedural
i.
ACTIVITY 1
(Partitioning the whole)
1, 2 & 3
ii.
iii.
iv.
v.
c. Logico-physical Abstraction
ACTIVITY 2
(Reconstitution of the whole from
its parts)
4
Partitioning the continuous whole in odd and even
parts: circle into 3 parts; triangle into 6 parts;
rectangle into 8;parts; parallelogram into 4 parts; L
into 5 parts
Partitioning discrete whole (12 chips per child)
4, 6 3 and 5 equal shares
Partitioning a 4” line (measure) using a ruler
2, 5, 8 equal parts
Liquid partitioning (measuring 1 cup) 1 per child
2, 3 and 5 equal shares
Verification Task
i.
Making a whole out of given parts (12 circular
parts)
ii. Given 1-fourth of circular whole to predict the
missing parts to complete the whole (One part of
the whole)
iii. Given 5-eighths of a triangle to predict the missing
parts to complete the whole (Many parts of the
whole)
iv. Given 1-fifth of a circle to predict the missing
parts to complete the whole (One part of the
whole)
v. Given 8-twelfths of a square to predict the missing
parts to complete the whole (One part of the
whole)
vi. Give each child 1-eighth (2 oz) of a cup and then
predict the missing parts to complete the whole (1
cup)
vii . Verification Task
51
Table 3: Continued
i.
ACTIVITY 3
(Equivalence of the part-whole
relationship in spite of variation in
physical attributes transformation
of the whole)
5&6
Given four different sizes of rectangles with a third
shaded in each
ii. Given two different sizes of circle with a half
shaded in each
iii. Given 1-fourth shaded in different shapes and
sizes: circle, triangle, square, rectangle, L, star
iv. Given four identical squares with differently
partitioned halves
v. Finding equivalent part-whole relationship in
discrete whole (circles) with differing amount in
each set (10 chips with 4 shaded and five with 2
shaded) and different shapes (diamonds – 10 with
4 shaded)
vi. Transforming a quarter of a square (continuous)
vii. Transforming shaded parts in discrete whole
8 chips with 2 shaded - transform the set
viii. Verification Task
i.
ii.
iii.
ACTIVITY 4
(Repartitioning already partitioned
whole)
7
iv.
Partition a circle divided into half into 4,8,16,32
equal parts and a rectangle already partitioned in
3 into 6, 12 equal parts
Partition a rectangle and a circle already
partitioned in 3 into 4 equal parts
Repartition a circle divided into 8 equal shares
for 4 persons, a circle divided into 5 to repartition
into 3, a rectangle divided into 7 parts for 3
persons, a rectangle divided into 8 to share with
two and the letter L divided into 7 to share with
4.
Verification Task
Logico-Mathematical
a. Logico-Mathematical
Procedural
ACTIVITY 5
(Quantification of part-whole
relationships in unit and non-unit
fractions)
8&9
i.
ii.
iii.
iv.
vi.
Naming unit fractions in continuous wholes of
various shapes: circles, squares, rectangles,
parallelograms; star and others and unit fractions
in discrete wholes
Orally quantify part-whole relationships other
than unit fractions.
Naming unit fractions in continuous and discrete
wholes from part-whole relationships such as 2fourths 3-sixths
Give concrete illustrations of given unit fraction
Verification Task
52
Table 3: Continued
b. Logico-Mathematical
Abstraction & Formalization
i.
ACTIVITY 6
(Writing conventional symbols for
unit & non-unit fractionsFormalization)
10
ii.
iii.
iv.
Use paper folding to model and name unit and
unit fractions
Writing symbols for unit and non-unit fractions
in continuous wholes
Writing symbols for unit and non-unit fractions
in discrete wholes
Verification Task
[Spring Break]
i.
ACTIVITY 7
(Mathematical reconstitution of
wholes from unit fractions – the
whole is made up of its parts)
11 & 12
ii.
iii.
iv.
v.
i.
ACTIVITY 8
(Relative size of unit and nonfractions)
13 & 14
ii.
iii.
i.
ii.
ACTIVITY 9
(Quantifying equivalent fractions)
15 & 16
iii.
iv.
Orally determine how many of the unit fractions
are needed to make a continuous whole – use
different shapes and different sizes
Orally determine how many of the unit fractions
are needed to a make a discrete whole
Determine how many of the symbolic unit
fractions are needed to make a whole
Reconstitution of a part from symbolically given
unit fractions
Verification Task
Orally determine by comparison the size of unit
and non-unit fractions using models and stories
Determine size of unit and non-unit fractions
symbolically
Verification Task
Finding equivalent fractions of one-half using
models and symbols
Symbolically represent equivalent fractions RNP
– Lesson 10 Student Page A, and Lesson 15
Student Page B.
Orally and symbolically find equivalent
fractions of unfamiliar unit and non-unit
fractions
Verification Task
53
Table 3: Continued
i.
ACTIVITY 10
(Ordering proper fractions)
17 & 18
ACTIVITY 11
(Problem Solving TasksExamining partitioning strategies)
19
ACTIVITY 12
Fraction Breakfast
20
ii.
iii.
i.
ii.
Ordering fractions by comparing them to half
[Rational Number Project level 2 lesson 8]
Ordering fractions by comparing them to 0, ½ , 1
Verification Task
The “Cake Problem”
Problem solving tasks from the Rational Number
Project
“Fraction Breakfast” where the students will use the
knowledge gleaned throughout the study to share food
items equally amongst themselves.
A range of instructional approaches were used during the study. Small group
discussion, whole class discussion, teacher interviewing and the use of hands-on
activities formed an active part of the daily classroom experience. The learning space set
up for this study did not only include the actual physical setting of the classroom. It also
encompassed the setting where students were active in the fraction environments. In this
environment they engaged in tasks, talked and wrote about what they know as they
attempted to reason about their actions and their thinking. The learning space was open
in nature thus resulting in a number of responses that provided variety in the
understanding of the students’ actions. To foster a healthy creation of fraction learning
space, it was important for the teacher-researcher to understand the different
subconstructs of fractions as mentioned in Chapter 2. With this understanding, the
teacher was able to design fraction tasks and activities that allow the students to build and
effectively use fractions in a variety of settings. Consequently, the students were able to
reflect on the various uses to which fractional numbers can be applied (Kieren, 1995).
In conjunction with the constructivist epistemological perspective for this study,
“the space for learning fractions should deliberately foster the students’ construction of
54
their own meaning for fractional numbers and operations on them” (Kieren, 1995, p. 45).
It should provide ways for the students, if necessary, to change from a more to less
sophisticated activity, to extend their knowledge from working with the physical models
to symbolic notation and for students at different levels of understanding to function
together (Kieren, 1995). The teacher-researcher, therefore, made deliberate effort to
model the learning environment to be compatible to the suggestions given by Kieren
(1995). In this learning space the participants were allowed to express their fraction
number ideas using physical referents which included models (continuous and discrete),
diagrams, pictures and charts.
The role of the researcher (teaching agent) in this study was twofold. As the
researcher, the students’ mathematical behavior was observed documented, analyzed and
interpreted as they participated in the different activities. The students’ critical thoughts
on the fraction tasks presented to them were probed and recorded. As the teacher,
fraction concepts were presented using the theoretical foundations set by Steffe and
D’Ambrosio (1995) hypothetical learning trajectory and Lesh’s model of translation
(Lesh, Post & Behr, 1987). Students’ interactions among themselves and with the teacher
were facilitated and encouraged. The focus of this study was not on the method of
instruction but on the students’ as they performed the tasks given them. With that in
mind, no formal lecturing was given. The teacher, instead, moved around, visiting
groups and individuals conducting spontaneous on-the-spot interviews and asking
questions that encouraged the students to think critically.
Interviews
The initial plan was to conduct two sets of interviews during the study. The first
set of interviews was aimed at the tasks that were done individually. The second set of
interviews would be geared towards the group activities including Activity 12. As the
study unfolded, it became possible for the researcher to visit the groups and question the
students while they were on task, thus gleaning valuable information on their thought
processes. Two sets of individual interviews were done with students who experienced
difficulty while working on particular tasks. These interviews lasted for approximately
35 minutes and were conducted during the student’s afternoon elective sessions. The
interview sessions were held in their classroom with the researcher serving as the
55
interviewer. Information gathered from the interviews provided further inferences about
the students’ cognitive processes while sharing and constructing fractions. De Groot
(2002) believed that, “Students talk about their experiences of learning in unstructured,
in-depth individual interviews” (p. 42). Due to time restraints no individual interviews
were conducted at the end of the study.
Data Collection
Various methods were used for collecting data for the duration of this study.
Besides the observation of the students’ mathematical behavior, each session and
interview were video- and audio-taped as the students’ verbal and nonverbal behavior
constituted a major part of the database. Students’ written work from the classroom
activities and tasks were collected, reviewed and analyzed continuously by the researcher
after each session. Audio-tapes and transcripts of the teacher-researcher’s reflection were
recorded and transcribed after every session and played a pivotal role in the planning of
the next activity. The conversations between the teacher-observer who attended the
sessions and the teacher-researcher were also documented and used as an aid in
validating the interpretations of the findings of the study. Table 4 shows the method(s) of
data collection used to gather information that was pertinent in answering each of the
research questions. The audio column represents the taped individual interviews, and
whole class and group discussions done during the study. The test represents the pretest
that was administered at the beginning of the study.
Table 4:
Research Questions and the Methods of Data Collection
Observations/Journals
Methods
Audio
Research
Questions
1
2
3
♦
♦
♦
Video
♦
♦
♦
Test
♦
56
Tasks
Sheets
Researcher
Teacher-Observer
♦
♦
♦
♦
♦
♦
♦
♦
♦
Data Analysis
“Data analysis involves organizing what you have seen, heard, and read so that
you can make sense of what you have learned” (Glesne, 1999, p. 130). During this
process you discover themes and concepts embedded throughout the interviews and
discussions (Rubin & Rubin, 1995). As mentioned earlier, data for the study consisted of
the students’ verbal and written responses to the task-based activities and the participants’
responses to the semi-structured interviews and any other relatable responses from the
researcher’s reflection and teacher-observers. The analyses of the information gathered
during the teaching sequence were done in three phases.
(1)
The students’ (individual or as a group) responses whether verbal or written were
continuously analyzed during and immediately after each session. The teacherobserver’s observation played an important role at the end of sessions that she attended.
(2)
There were daily analyses of the entire class responses coupled with the teacher-
researcher’s observation.
(3)
At the conclusion of the study the analyses that were done in (1) and (2) were
combined with the final analysis to create a rich, descriptive narrative of authentic insight
of the sixth graders’ understanding of fractions. Pertinent solution behavior patterns were
noted and responses to interview questions were transcribed. Careful notes were taken of
the mathematical language the students employed during the discussions of the taskbased activities. The following steps were used in the analysis of the data: (a) coding of
each group’s discussion, interviews and task sheet; (b) grouping excerpts per categories
revealed from the coding; and (c) organizing the categories in a coherent manner,
following an interpretive, narrative approach. The major aim of the data analysis was to
seek an in-depth understanding of the participants’ understanding of the fraction concept
as they worked with the task-based activities individually and within groups.
Assumptions and Limitations
One of the characteristics of qualitative research is that the researcher is the
primary instrument for data collection and analysis. This can result in certain limitations
and biases that will impact on the study. Instead of trying to eliminate these
shortcomings and biases, this study seeks to “identify them and monitor them as to how
they may be shaping the collection and interpretation of the data” (Merriam, 2002). The
57
limitations will be discussed under the following sections: research methodology, the
active involvement of the researcher as the main teacher, and the sample. The
assumptions are embedded in the discussion for each section.
Research Methodology
Due to the nature of the research questions, a qualitative approach was used as the
lens through which the data acquired from this study were viewed and interpreted. A
summative description of this research methodology is given at the beginning of this
chapter. In this section, the major issues concerning this type of research methodology
are highlighted. This includes the steps that were taken to check for accuracy and
credibility of the results of the study.
One of the major setbacks for using this design is the need to validate the findings
of the study. Creswell (2003) explained: “Validating does not carry the same
connotations as it does in quantitative research, nor is it a companion of reliability … or
generalizability” (p. 195). Instead, he saw validity as one of the strengths of qualitative
research that should be used to determine whether the results harvested from the data are
accurate from the viewpoint of the researcher, the participant or the readers to which the
study is targeted. Hammersley (1990) defined validity as truth “interpreted as the extent
to which an account accurately represents the social phenomena to which it refers” (p.
57). This issue of validity in qualitative research is a hotly debated topic (Lincoln &
Guba, 2000; Silverman, 2001). Suffice it to say, Creswell (2003) offered a number of
synonyms for validity. These include trustworthiness, authenticity and credibility.
Addressing the issue of generalizability in qualitative studies, Stake (2003) wrote,
“Damage occurs when the commitment to generalize or to theorize runs so strong that the
researcher’s attention is drawn away from features important for understanding the case
itself” (p. 141). A major strategy to ensure for generalizability in this qualitative research
is the provision of narrative with rich, thick description. Enough information including
excerpts from students’ interaction with the researcher and each other are included in the
research so that the readers are able to determine how closely their situations match, and
therefore determine whether the interpretations of the study can be transferred to their
own situation. As the researcher, I aspired to use the data collected in this study to
produce qualitatively valid and reliable information in an ethical manner that provided
58
the groundwork for the development of further study in the understanding of students’
fractional scheme.
One the most significant limitations of this study is that it involves inferences
about the learners’ observed behavior. Although the twelve activities done during the
teaching sequence were audio- and video-taped, with such a large number of students
working at the same time, there existed moments when clear cut audible discussions were
not possible and reasonable inferences had to be made from the visual data. Since the
inferences drawn were based on the teacher-researcher’s background and experience with
these students, the study is limited in objectivity. Being a mathematics teacher for over
eighteen years has possibly shaped the way the teacher-researcher viewed and interpreted
the data thus devaluing the validity of the study. Every effort was made, however, to be
objective and a number of strategies were employed to ensure the validity of the study.
Besides using rich, thick description to convey the findings, there was a triangulation of
the different sources of data with the aim of building a coherent and justifiable
interpretation of the results. The different methods of data collection were compared to
see whether they corroborate. The interpretation of the findings of this study was also
subjected to the reviews of the teacher-observers who helped to determine whether the
findings are plausible based on the data (Merriam, 2002). Related literature also played a
significant role in supporting the interpretation of the data.
Teacher-Researcher
In defending researchers acting as teachers, Cobb and Steffe (1983) believed that
the “activity of exploring children’s construction of mathematical knowledge must
involve teaching” (p. 83). Among the reasons given to substantiate the position of
teacher-researcher is the idea that teachers are able to form close relationships with the
students thereby helping them to reconstruct the contexts within which they learn
mathematics. Other advantages include a) the teacher has access to the detail of the class
that others would not; b) she may understand the students’ talk and mannerisms in ways
that an outside researcher would not, and c) she knows the history of shared examples,
problems, and discussions that she can use to probe the children’s ideas, be better able to
frame the tasks and interpret the students’ problem solving activities. This study did not
attempt to examine the students’ construction of fractions due to the instructional
59
processes. Instead, the teaching sequence provided the setting or learning space for
fractions where the participants could construct their knowledge of fraction as they
interacted with their peers and teacher while performing the assigned task-based
activities.
This first-person perspective, often refered to as “backyard” research, has its
downside. The data collected can be biased, incomplete, or compromised (Creswell,
2003). The teacher-researcher commitment to helping her students can impede the
capacity to see and hear the students’ problems and difficulties. Aptly stated by Ball
(2000, pp. 389, 390) the teacher may not be “able to notice the subtle ways in which her
manner affects particular children and, hence, may not be able to probe their responses.”
To minimize these discrepancies, Ball (2000) included three crucial questions that a
person who wishes to undertake the role as both teacher and researcher should ponder.
The researcher will provide the answers to these questions, thus providing the rationale
for conducting this study and interpret the data collected during the study.
Question 1(a). Does the researcher have an image of a kind of teaching, an
approach to curriculum, or a type of classroom that is not out there to be studied? Given
the complexity of the teaching process and the unique learning environment of every
classroom, this qualitative study does not claim to be projecting or highlighting any
teaching approach or pedagogical skill on the part of the teacher-researcher. Instead, the
researcher was simply interested in what goes on in the mind of her students as they
engaged in meaningful activities that are relevant to their everyday lives and as they
made sense of fractions while working with physical models. Teachers are always
encouraged to understand “what students know and need to learn and then challenging
and supporting them to learn it well” (NCTM, 2000, p. 11).
Question 1(b). Given that there is a need for a first-person research, is the
researcher well-equipped to be the designer and developer for this study? The researcher
has taken several courses that dealt with conducting qualitative studies. She has also
conducted small-scale studies that have been reviewed by professors who have
experiences in qualitative research. In addition, the teacher-researcher has been engaged
in an extensive review of related literature to cull tasks, activities and designs that will
prove beneficial and significant for this study.
60
Question 2.
Is the phenomenon to be studied only uniquely accessible from the
inside? Seeking to understand how students make sense of fractions is not uniquely
accessible from the inside. However, the researcher believes that a natural setting
environment will add a fresh view on this cannot-be-exhausted topic.
Question 3(a). Is the phenomenon one in which other scholars have an interest?
Numerous studies have focused on the fraction concept. This subset of rational numbers,
along with its counterparts such as ratio, has been hotly discussed and has also been the
main focus of the 2002 NCTM yearbook.
Question 3(b). Will conducting the study within the teacher’s natural environment
offer perspectives crucial to a larger discourse? Besides adding to the reservoir of
fraction studies done over the decades, this study can possibly trigger a discussion of the
advantages of doing similar research within the teacher-researcher’s natural environment.
Sampling
The participants in this study do not represent a randomly selected group as the
teacher-researcher is working with her own group of students. This research is not
testing a well-defined theory, but seeks to add to the knowledge base about how students
conceptualize fractions as they work with manipulatives and participate in processoriented activities that represent real life events, thus the nature of the sample is deemed
appropriate. One of the underlying assumptions was that the students would perform to
the best of their ability as no grades were assigned to the tasks and activities used in the
study and that the tasks the students did elicited information that aided in answering the
research questions. With the exception of one student (Ben) who was sometimes lowkeyed, the participants were enthusiastic about the activities and possessed the same
energy level throughout the eight weeks. The students could express themselves verbally
through written and non-written responses using the English language. A simple survey
of the students involved in this study showed that they were all born in the United States
of America and spoke English fluently. Only a few misspellings prevailed throughout the
written tasks that were clearly understood by the researcher. The written tasks always
corroborated with the video- and audio-taped recordings.
61
Summary
The chapter began with a brief description of the type of study that will be
pursued by the researcher. All research design that includes the qualitative design, calls
for a clear outline of the sample, methods of data collection and how the data will be
analyzed. This whole-class research project included a teaching sequence, a pretest, and
task-based activities. All these formed the network through which the students’
understandings of fractions were scrutinized and interpreted. Despite the limitations
mentioned above regarding this study, there is sufficient reason for the use of this type of
study and setting to investigate students’ understanding of fractions. The teacher acting
as the researcher had first-hand knowledge of the students’ ability to do the work thus
providing the opportunity for her to provide tasks that were suitable for the different
levels of understanding in the classroom.
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CHAPTER 4
RESULTS - THE NATURE OF STUDENTS’ UNDERSTANDING OF
FRACTIONS
Introduction
The discussion of the results of this study is organized around the research
questions. Chapter 4 is solely dedicated to the answering of research question number
one and the remaining two research questions will be dealt with in Chapter 5. Being a
whole-class study, thousands of statements generated during the teaching sequence, and
tens of written tasks were examined to distill patterns that were relevant to the nature of
the study. The participants’ written and oral behaviors were observed for commonalities
in the way they approached, worked with, understood and talked about fractions.
The first ten task-based activities done during the teaching sequence formed the
core of the coding for the levels and constituent parts of the Herscovics and Bergeron’s
(1988) model of understanding and were designed to deduce the nature of the students’
understanding of fractions thus serving the purpose of answering the first research
question. The remaining activities were designed to investigate the participants’
partitioning strategies and the way they solved problems with real world applications.
At the end of each activity, an individual and/or group assessment task was given
to determine the level of each student’s understanding or misunderstanding of the
concepts introduced during the activity. An outline of the activities done during the
teaching sequence is shown in Table 3 in the previous chapter. An analysis of the
activity and its accompanied assessment task was done instantaneously with the aim to
inform the subsequent session. As Merriam (1998) aptly put it: “Data collection and
analysis is a simultaneous activity in qualitative research. Analysis begins with the first
63
interview, the first observation, the first document read. Emerging insights, hunches, and
tentative hypotheses direct the next phase of data collection …” (p. 151).
This study was done with a set of six-graders who attended a private metropolitan
school in southeast Florida. The class was comprised of 10 males and 10 females. The
sequence of activities was done during the last two quarters of the 2005 – 2006 school
year. At the beginning of the study each student was given a transparent bag containing a
ruler, flexible wires, eraser, two sharpened pencils and a set of circular red and yellow
chips The students were at liberty to use any of manipulatives in their bag and those
assigned during the specific activities to help them with the assigned tasks.
Research Question #1:
What is the nature of sixth grade students’ understanding
of fractions?
The various task-based activities were designed and adopted from the research
literature to glean an in-depth view of the nature of the participants’ understanding of
fractions. As mentioned in Chapter 1, Herscovics and Bergeron’s (1988) model of
understanding was used to analyze the sixth graders’ responses to the fraction tasks that
were presented to them during the teaching sequence. These tasks were designed to
solicit the participants’ definition of a fraction, their process of unitizing and partitioning,
their sense-making of fraction symbols, and the order and equivalence of fractions. Each
level and constituent part of Herscovics and Bergeron’s (1988) model will be highlighted
in the effort to answer the first research question. Items from the pretest (Appendix A)
and the “Fraction Breakfast” task sheet (Appendix B) were used to verify the
participants’ definition of a fraction and the first level of Herscovics and Bergeron’s
(1988) model of understanding – intuitive understanding. An interpretative analysis of
the data will conclude each section.
Definition of the Fraction
Prior to scrutinizing some of the pretest items for verification of the participants’
intuitive knowledge of fractions, an analysis of the students’ definition of a fraction was
done with the purpose of revealing their perception of what constituted this mathematical
concept. These students had previously completed two years of formal fraction
instruction and thus it was the researcher’s intent to gain insight into the students’ sense
64
making of the fraction before they began formal work with the concept in sixth grade.
This sense making is deeply entrenched in the meaning they assigned to the fraction.
Four questions (1, 5, 6 & 15 – Free Response Section) from the pretest were used
as the lens through which the students’ definition of a fraction was verified. These four
questions were chosen on the merit that they could solicit from the students any of the
major subconstructs of a fraction as outlined by Kieren (1976). Question 1 directly asked
the participants to give their verbal definition of a fraction and to make different
representations of a fraction of their choice. Question 5 gave a fraction and then similar
to Question 1, asked the students to generate two different representations of this fraction.
Question 6 hinged on the fact that the sum of the equal parts make up the whole which is
an essential point in the understanding of the fraction, and Question 15 investigated the
students’ knowledge of the role equal parts play in the naming of a fraction. A detailed
summary of the responses to each question will be done, followed by an analysis in
reference to the students’ meaning of a fraction.
The first question required the students to tell a friend, who did not know, what a
fraction was and to illustrate a fraction in more than one ways. 45% (or 9 out of 20) of
the participants made no attempt to answer any part of the question. Here are some of the
responses. The number beside the name represents the category that the student was
placed in by the teacher-observers after the completion of the pretest. The results of the
pre-test that was administered prior to the teaching sequence were used to categorize the
students on a scale from 1 to 3 with 1 representing very little understanding of the
fraction concept and 3 representing a basic understanding of the fraction. Each student
was assigned a pseudonym. Carol was the only student placed in Category 1, seven
students were placed in Category 3 while the remaining twelve students were placed in
Category 2. Appendix C shows the students by category.
Ashley (2):
I would tell them sorry ’cause I don’t know it either.
Clauia (1):
Let’s say we bought 4 slices of pizzas, if I took 2 we would have a
fraction of two pieces left.
Polly (3):
A fraction shows what part of a whole has been taken or shaded.
Mary (3):
A way to divide things.
65
Marla (3):
A fraction is when something is divided into 1 or more parts and
then a certain amount is taken away from it to show what is left.
Dave (3):
Fraction is when you divide things into even pieces.
Bob (3):
A fraction is a piece of something.
Two students – Richy (2) and Paul (3) – were able to produce a diagram and a
symbolic notation of a fraction but did not give a word definition for a fraction. All the
participants who attempted to illustrate the fraction using diagrams used a continuous
model (square or rectangle). Jay (3) used the fractions
1
2
and as two ways of
2
4
representing a fraction.
Question 5 – “What does the fraction five-eighths look like? Using a diagram,
show this fraction in two different ways.” produced some rather noteworthy responses. A
summary of the responses is shown in Table 5.
Table 5: Summary of Responses for Question 5 on Pretest
RESPONSE
5
only
8
Diagram and Fraction Symbol
Circle and Rectangle
Circle and Triangle
Rectangle and Discrete Chips
2 Sets of Rectangles
No Response
Fraction Symbol
Number of Students
(Total = 20)
2
8
5
1
1
2
1
Ashley showed three representations of five-eighths. She used the unshaded parts
to indicate the numerator.
SYMBOL
CONTINUOUS MODEL
5
8
66
DISCRETE MODEL
Question 6 asked the students “how many fourths are in a whole?” They were
required to produce a diagram to illustrate this. 35% of the participants did not respond
to the question. The remaining participants correctly answered “4” and used either a
circle or a rectangle divided into four parts as the illustration.
In Question 15, the students were asked to determine the fraction that was
represented by c in Figure 7 below. 40% of the students did not respond to this question.
Two students (Bob and Chrissy [3]) gave the correct answer
Alton [2], Jay, Dave and Kisha [2]) named the fraction as
1
. Five students (Carol [2],
6
1
with three students (Josh [3],
5
1
1
1
Paul and Richy) writing . Mary wrote 2 while Marla wrote .
3
5
7
1
Figure 7.
Diagram for Question 15 – Pretest
As the students participated in 12 task-based fraction activities their formal
understanding of unit and non-unit fractions should have strengthened. No post-test or
follow-up activity was given but Item 7 on the “Fraction Breakfast” task-sheet asked a
similar question to Question 1 on the pretest free response section. It should be noted
that none of the previous activities solicited or gave a formal definition of a fraction.
Here are some examples of the students’ emerging definitions by the end of the study:
Ashley:
A piece of a whole.
Josh:
Dividing things into equal parts.
Dahlia:
A piece or pieces that come from dividing into partitions.
67
Alton:
My definition of a fraction is the equal shaded parts of a whole.
Chrissy:
A fraction is a part of an equal partitioned square (doesn’t have to
be a square) the part that is shaded.
Carol:
A fraction is a group of numbers that helps you to divide and
multiply equally.
Claudia:
A fraction is something that we use everyday.
All the students had a definition with the majority of them hinting towards the
part-whole subconstruct with one student (Carol) connecting the concept of equivalence
in her definition.
Analysis
Fundamental to the meaning of a fraction is the fact that the parts that constitute
the whole must be equal in area. This is also an important component in intuitive
understanding of fractions. From the responses for Question 15, it became apparent that
the majority of the students (with at least two years of formal fraction instruction) did not
consider that the parts represented by a, b, c, d, and e were not all equal. Bob, Richy,
Paul, Chrissy, and Josh recognized the need for equality of the parts but unlike Bob and
Chrissy who ignored the previous divisions and partitioned into six equal parts, Richy,
Paul and Josh chose to ignore half of the whole and named their fractions using the three
equal parts. Marla’s problem was simply that of over counting while Mary had no idea
how she came up with that answer – she just “guessed.” Dave alluded to the equality of
parts in his written definition of a fraction when he referred to “even pieces.”
Careful scrutiny of the results for the four questions showed that the participants
in this study had very little or no understanding of the meaning of a fraction except for
the symbolic representation of a fraction. Mary, Marla and Dave’s definitions hinted
towards the quotient subconstruct. Interestingly, with the exception of Polly, none of the
participants used the “part out of a whole” definition that is prevalently used in textbooks
and often used by mathematics teachers (Kieren, 1993). Deliberate efforts were made not
to influence the students towards any definition of a fraction with the hope that their
meaning of the fraction would continue to emerge as they participated in the meaningful,
task-based activities.
As previously mentioned, by the end of the study all the students
68
had a personal definition of a fraction, which, incidentally made reference to the partwhole subconstruct.
In making reference to the numbers constructing the fractions, the students mostly
refered to the top number instead of numerator and bottom number instead of
denominator. When illustrating a fraction, the participants mostly used the common
continuous geometric regions (circles, squares or rectangles) with little use of other
shapes or discrete models. One participant (Ashley) used a discrete model to illustrate
the fraction five-eighths.
Intuitive Understanding
Items from the pretest were used to verify the students’ intuitive understanding of
the fraction. As mentioned in Chapter 2, the sole criterion for this type of understanding
is the awareness of the role of partitioning in part-whole relationships. There are four
conditions necessary for this awareness. Each condition will be examined with the
relevant question(s) followed by a table summary of the responses to the question(s) and
a summative narrative of the results.
Condition 1: Students should recognize the equality of the parts in a partitioned
whole despite the position of the parts in the whole.
Question 9:
A.
Here are some cookies:
Suppose you eat only the dark ones. How much of the cookies would you
eat? Why do think so?
If the cookies are rearranged as follows:
69
B.
How much of the cookies would you eat now if you eat the dark ones?
Why do you think so? (adapted from Boulet, 1998).
Table 6: Summary of Responses for Question 9 on Pretest
RESPONSE
Number of Students
(Total = 20)
17
2
(Part A)
8
2
(Part B)
8
Other (Part A)
Other (Part B)
17
3
3
The students had very little difficulty in recognizing the equality of the parts in
the discrete whole despite the rearrangement of the cookies. Jonah (2) correctly put
2
8
for both parts of the question giving the same reason “because there are 8 dots and only 2
are shaded.”
Alton (2):
Chrissy:
(Part A) 2 out of 8. Because you ate 2 and there are 8 cookies.
(Part B) 2 out of 8. Because it’s the same thing but you ate the
pizzas in different ways.
2
(Part A) because 2 cookies were eaten out of 8.
8
(Part B)
2
because you are still eating 2 cookies out of 8.
8
None of the students wrote the equivalent unit fraction
1
to represent the amount
4
of cookies eaten. Carol answered the first part of the question correctly but wrote the
70
answer to Part B as
1
“because there is one cookie eaten out of four and when you add
8
all the cookies together they add up to 8 cookies.” Richy wrote
1
for Part A counting
8
the two shaded dots as one since they are lined up under each other but then failed to
count how many groups of twos were present to arrive at the denominator of 4. He wrote
2
for Part B. When questioned about his answers, he remarked that he remembered
8
learning something similar in a lower grade but had forgotten how to do it. Brian (2) and
Ashley wrote
6
for both parts of the question misinterpreting the question thus giving the
8
answer for the fraction of non-dark cookies that were not eaten. Brian initially wrote
adjacent to the diagram but scratched it out and then wrote
2
8
6
below Parts A and B of the
8
question. Ashley’s misinterpretation was consistent with her response to Question 5 on
the pretest. No interventions were planned for the students who did not arrive at the
“correct” solution of
2
since the researcher was simply examining the presence of the
8
students’ intuitive knowledge of fractions based on the test items.
The responses to this item indicated that the sixth graders exhibited the ability to
recognize the equality of the parts in a partitioned discrete whole despite the position of
the parts in the whole. Tasks involving continuous wholes were used during the teaching
sequence where all the students also displayed much ease at recognizing the equal parts
of the whole even though the parts were transformed (Activity 3 – Appendix B).
Condition 2: Students should recognize the inequality of the parts in similar wholes
regardless of the equal number of parts shaded or unshaded.
Two items were used to verify this condition.
71
Question 1:
Which shows
3
of the picture shaded? (NAEP Sample Item)
4
A.
B.
C.
D.
Table 7:
Summary of Responses to Question 2 (Multiple Choice) on the Pretest
RESPONSE
Number of Students
(Total = 20)
17
3
C
C and D
This item was chosen to examine the second condition despite the fact that choice A is
not a similar whole to the other three choices and choice B is not divided into four parts.
None of the students attempted to chose this option (see Table 7). Polly originally chose
C and D as her response with words “main answer” pointing to C. She eventually
eliminated D leaving C as her only choice. The three students (Carol, Dahlia and Jonah)
who chose C and D were placed in category 2 indicating that they had difficulty with
some of the basic concepts embedded in the understanding of fractions. If you recall,
Carol also displayed some form of confusion in her response to Question 9.
72
Question 2:
The circles below represent two pies of the same size – one for you and
one for your friend.
you eat this much
your friend eats this much
Did you eat as much pie as your friend? Why do you think so? (Boulet, 1993)
Table 8:
Summary of Responses to Question 11 on the Pretest
RESPONSE
Number of Students
(Total = 20)
12
6
2
Yes
No
No Response
The responses to this question suggested that at least 70% of the participants
experienced difficulty in recognizing the inequality of the parts in similar wholes
regardless of the equal number of parts. Here are some of the reasons for answering
“Yes” to the question.
Kisha (2):
As you can see that my friend’s part is cut in different shapes and
you can also see that mine is cut differently but we still ate the
same amount.
Chrissy:
There are three pieces in each pie and both of us ate one piece.
73
Dahlia (2):
We both ate the same amount of pie because both of the pies are
half of 3 pieces of pie which means we both ate the same amount of
pie.
Dave:
Because even though the shapes and sizes are different we both ate
one piece.
Ashley:
Because there are 3 slices, I ate one and she ate 1.
1
We both ate .
3
Polly:
Josh (3), Brian and Claudia gave similar responses to Polly.
Two of the students who answered “No” offered no suggestion as to why they
arrived at that conclusion with the other four students making some form of implicit
reference to the shape and size of the slices.
Paul:
No, because the one I ate is longer and fatter than the one my
friend ate.
Jay:
The thirds were not equal.
Bob:
They are unequal pieces.
Mary:
No, more. It’s like my friend ate maybe 2 pizzas and I look like I
1
ate 3 pizzas.
2
Although Mary’s response seemed to indicate a lack of understanding of the unit fraction
in reference to the whole, it was obvious that she recognized that the slices are different
with one slice bigger than the other.
A review of the combined results of these two items reveals that at least 50% of
the students did not recognize the role that equal parts played in the construction of the
fraction. Polly, for example, assigned one-third to the shaded parts in both circles
disregarding the fact that the parts were not equal and was initially confused with choice
D as a possible option for the fraction three-quarters. Dahlia, Ashley, Carol, Brian, Jonah
and Claudia were not sure of the necessity of the equality of the parts.
Condition 3: Students should recognize the inequality of the parts of similar wholes
that are partitioned differently.
74
Question 1:
Which rectangle is not divided into four equal parts? (NAEP Sample Test
Item)
Table 9:
Summary of Responses to Question 1 (Multiple Choice) on the Pretest
RESPONSE
Number of Students
(Total = 20)
17
3
D
Other Choices
Most of the students recognized the inequality of the parts of similar wholes that
were partitioned differently. It is noteworthy to mention that the question never used the
word “fraction” or had any fraction symbol. It is therefore possible that the students were
able to make their choices without any recognition of the role equality of parts plays in
the fraction concept. Based on the evidence shown above, however, the participants were
able to recognize the inequality of the parts of similar wholes that were partitioned
differently. Two students chose option B (Carol and Dahlia) while Kisha chose option A.
Carol and Dahlia chose C and D as responses to Question 2 (Multiple Choice) on the
pretest. As mentioned before, this could be an indication that they had little
understanding of the role that the equality of parts played in the generation of a fraction.
75
Condition 4: Students should recognize the similarity in the part-whole relationships
in spite of the inequality of the size of the parts.
Question:
Think carefully about the following question. Write a complete answer.
Be sure to show all your work.
1
1
of a pizza. Ella ate of another pizza. Jose said that he ate
2
2
more pizza than Ella, but Ella said they both ate the same amount. Use
words and pictures to show that Jose could be right.
Jose ate
Karla, Ben (2), Ashley, Alton, Claudia and Brian did not respond to this question.
Below are some of the varied responses.
Polly:
1
is equivalent to saying half. The
2
smaller amount of slices of pizza there are, the bigger the slices.
Jose could be right because
Jose
Carol:
Ella
1
Jose could have gotten a huge of the pizza, while Ella could have
2
1
gotten a medium of the pizza.
2
Ella’s Pizza
Kisha:
Jose’s Pizza
I think that they both ate the same because the pizza can be the
same size.
1
2
1
2
76
Bob:
Jose could be right. One pizza could be smaller.
Paul:
It depends on the size of the pizza.
Jose’s pizza
Ella’s pizza
Dave:
They both bought an 8 slice pizza. If Jose eats half and Ella eats
half, they both eat 4 slices of pizza.
Jay:
Ella could have eaten a smaller pizza.
Josh:
Jose’s pizza might have been bigger than Ella’s pizza.
Dahlia:
To me, I think Jose is wrong. Ella is right. They both ate the same
amount.
At least 50% of the students who responded to this question made reference to the
number of slices of pizzas eaten instead of referencing the size of slices. Although the
exercise asked them to defend Jose’s statement twelve of the respondents supported
Ella’s position. Chrissy wrote: “Jose is wrong!”
Analysis
The combined results shown for the four conditions indicated that at least 85% of
the students (e.g. Bob, Jay, Dave and Paul) possessed an intuitive understanding of
fractions. All of the students satisfied at least two or more of the conditions which
indicated some form of intuitive knowledge. This intuitive knowledge could possibly be
a result of previous lessons done on fractions in the lower grades prior to Grade 6. No
instructions/lessons on fractions were done within the sixth grade curriculum prior to the
study.
77
Logico-Physical Procedural Understanding
Logico-physical procedural understanding is exhibited when the student can relate
and use intuitive knowledge suitably. As partitioning plays a critical role in fraction
understanding (Behr, Lesh, Post, & Silver, 1983), the first four activities of the teaching
sequence were centered on this fundamental building block. The main criterion at this
level of understanding is the learner’s ability to partition a whole into a given number of
equal shares. Activity 1 (see Appendix B) formed the central activity for this level of
understanding and was done on an individual basis. A whole class discussion followed at
the end of the task. This activity occupied three sessions where each session was geared
mainly towards the students practicing their partitioning skills. The activity consisted of
four tasks with each task having at least 3 subtasks. Some of the tasks were adapted from
Boulet’s (1993) work on the unit fraction. The researcher developed the tasks involving
the number line and the cup of water. At the beginning of the activity each student was
handed his/her personal bag, a folder containing the task sheets, drawn figures, a cutout
L, and a cup of water with five empty plastic containers.
Tasks 1, 3 and 4 allowed the students to work with continuous models (geometric
regions, lines, and liquid measure) while Task 2 gave the students the opportunity to
work with a discrete model. The tasks were designed to engage the students in
partitioning in odd and even shares. It is noted in the literature that students seemed to
experience more difficulty partitioning into odd shares (depending on the shape of the
region) than even shares (Poither & Sawada, 1983). This was very evident as the sixth
graders attempt to partition the circle into three equal parts (Task 1 – Activity 1). At the
onset, Ashley, Claudia and Alton partitioned the circle using vertical lines (
).
Claudia and Alton eventually erased their configuration and divided their circle similar to
most of the students (
) but Ashley was adamant that her way of partitioning the circle
produced equal shares.
I (Interviewer):
Are they equal parts?
Ashley:
I think so.
I:
Why do you think so?
Ashley:
I measure them.
I:
What did you measure?
78
Ashley:
Two inches for each part. [She measured along the diameter and
divided the diameter into three equal parts.]
I:
What about the curved sides?
Mary, who was sitting at the same table, butted in. Claudia also chimed in.
Mary:
The two sides are fatter than the one in the middle so I don’t think
they are equal. (She pointed to the curved arcs of the circle.)
Claudia:
Also because the middle part is not even equal. The middle is not
like the two sides.
I:
What if your pizza represents a real pizza [pointing to Ashley’s
circle]. I cut the pizza just the way you have it. I give you the
middle slice; I give Karla one of the outer slices and give Ben the
other outer slice, who would get the most?
Ashley:
Ben and Karla.
I:
Would each of you get the same amount?
Ashley:
No
I:
So, are the 3 parts equal [pointing to Ashley’s circle]?
Ashley:
No.
In subsequent activities, when given the task to partition a circle in 3 parts, Ashley
proceeded to the conventional configuration (
).
85% of the students divided their circle in the conventional manner but more than
three-quarters of them had no idea how to check for equality of the parts. When it was
stressed that the three parts needed to be equal the students began to use items from their
bag such as the ruler, flexible wires and even the chips to verify equality.
I:
[Speaking to Jay] Does your circle have 3 equal parts?
Jay:
I got three parts but I not sure they are 3 equal parts.
I:
Why aren’t you sure?
Jay:
I don’t have a protractor.
I:
What would you use the protractor to do?
Jay:
I would have to measure each angle to see if they are all equal.
79
The sixth grade class had previously finished a unit on geometry where the students
worked with angles and protractors. They also worked with different types of polygons.
I:
Is there anyone else not sure of the equal parts.
Alton:
I’m not sure. It doesn’t look equal.
I:
What can you do to let your parts be equal?
Alton:
I could use a protractor.
I:
Did you think of protractor because of what Jay said.
Alton:
I was thinking about the circle thing. Not a protractor, a compass!
Marla intervened:
Marla:
I measured the lines from one point to the next.
On the teacher-researcher’s invitation she proceeded to illustrate her strategy on the circle
drawn on the transparency. She used the ruler to draw three radii and then draw the
chords connecting the outer endpoints of the radii (
). Chrissy used the same method
to confirm equality of parts.
Chrissy:
I got my ruler and I measure it and put it on each line to see if it is
the same and one of them was different.
I:
What did you do?
Chrissy:
I fix it.
The remaining students proceeded to check their circle using Marla’s strategy and made
the necessary corrections.
The students were then asked to divide the given equilateral triangle into six equal
parts. At first Polly, Brian, Bob and Josh had difficulty in perceiving how this could be
done. At the first attempt, Polly’s triangle looked similar to this
. She then
realized that she had different shapes: “two small triangles, two large triangles and two
trapezoids.” She sat at her desk in frustration, then started to measure each side of the
triangle and realized that they had the same measurement. Marla, sitting next to her, had
already begun to join the midpoints of each side of the triangle to the opposite vertex of
the triangle to produce six equal parts:
80
Polly then copied Marla’s strategy.
I:
How do you know that the six parts are equal?
Polly:
They look alike, uhm, they have the same shape. At least I know
that they have one side in common [referring to half line on each
side of the original triangle].
To partition her triangle in 6 parts, Carol put a point in the center (her perception) of the
triangle. She then drew lines from the point to touch the sides of the triangle. She
estimated where the lines should end without any idea of finding and using the midpoint
of the sides of the triangle. She contended that her parts were equal because they all
looked alike. She had no suggestion as to how to check for equality of parts. Josh joined
the midpoints of the sides of the triangle but discovered that it only produced four equal
parts:
. Bob, on the other hand, sought to do his partitioning a little different from
the other resulting in a diagram looking similar to this:
. He recognized that his
division produced seven parts instead of the required six and then proceeded to partition a
new triangle similar to Marla’s. Brian, Mary and Jonah did not complete their
partitioning considering it an impossible task.
Partitioning the rectangle into eight parts and the parallelogram into four parts
proved to be an easy task for majority of the students. Various configurations for each
shape were observed. The given rectangle measured 4” by 1” thus making it an easy task
for the students who measured the length of the rectangle. They marked points ½” apart
on both sides of the triangle and then continued to draw vertical lines to construct 8 small
rectangles (
). They were pretty sure that these rectangles were equal
considering the measurements they used. Dave partitioned his rectangle into 4 equal
squares and then drew the diagonals to create eight equivalent triangles (
Dave:
).
I know the squares are equal so this line [pointing to the diagonal]
will make two equal triangles.
Ashley decided she was going to think outside the box in partitioning her triangle. She
divided the rectangle lengthwise and widthwise to produce four equal rectangles. She
then drew the diagonals for each rectangle to produce a rectangle looking like this
. She argued that the parts were equal because she drew only drew the half
lines at each turn.
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Ben and Karla partitioned their rectangles in a similar fashion. To verify the
equality of the parts, they each looked at the small rectangles and decided that each
triangle divided the rectangle into two equal parts. They were very confident with their
decision. The class had previously worked with rectangles and triangles in a geometry
unit. Alton divided his rectangle lengthwise into four rectangles and then used one line in
the center of the rectangle widthwise to create his eight rectangles. He did not use his
ruler for measurement and thus his rectangles were not equal. He, nonetheless, insisted
that if he spent a little more time with it he would be able to produce eight equal
rectangles with this configuration:
. Marla’s rectangle produced
a replica of this configuration.
Partitioning the parallelogram (4” x 4”) into four shares generated various
configurations. 70% of the participants partitioned their parallelogram similar to Figure
8(a). Jay and Polly were the only students to divide their parallelogram resembling
Figure 8(b). Neither of them used their ruler to mark off 1” points on top and bottom
sides of the parallelogram and then joined them to form parallel lines thus forming four
equal parallelograms. Instead, they used the estimate-erase-readjust (Boulet, 1993)
method of partitioning the shape. Kisha drew one diagonal of the parallelogram but
failed to recognize that drawing the other diagonal would produce four equal triangles.
She did not complete her partition but continued to the next task. Karla, although
admitting that her division did not result in equal parts, left her diagram looking like
Figure 8(c).
(a)
(b)
Figure 8.
(c)
Students’ Configuration of the Parallelogram
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The last shape to be partitioned in Task 1 of Activity 1 is the letter L. They were
required to partition the L into five equal parts
. 17 out of 20 students did the partition in a similar fashion by closing off the bottom of
the L. They found that the dimensions of the square was 2” x 2” and then proceeded to
mark off four 2” x 2” squares to complete five equal squares (
). Jonah closed the
bottom of his L but continued to divide the long side without using any measurement
derived from the closed square. His diagram, therefore, showed five unequal parts.
Kisha and Claudia never completed their partition.
The second task of Activity 1 required the participants to partition twelve chips
into four, six, three and five equal shares respectively. The first three shares never posed
any difficulty for the students who displayed a variety of ways to do their equal
groupings. After using the chips to form the required partition they then made a model
on the task sheet. Jay, Josh, Polly, Bob, Paul and Chrissy employed the algorithmic
approach of division to decide how many chips would be in each group.
Polly: 12 divided by 4 gives three, therefore I will have 3 chips in a group.
Others such as Kisha, Ben, Mary, Richy, Brian, Carol and Dahlia simply placed four
chips on the table to represent the four groups and then continued to place a chip in each
group until there were no more chips. This strategy is called “dealing” (Davis &
Hunting, 1990; Davis & Pitkethly, 1990). Similar methods were used for partitioning in
six and three shares.
The last subtask proved rather tricky for the students. Proceeding as they did in
the previous subtasks, those who were using division to aid in forming their groups found
out that 5 is not a factor of 12. Polly said: “You cannot do the problem. 5 times nothing
equals 12.” All students made some attempt to form the five groups.
I:
How many chips do you have there?
Jay:
10
I:
Where do the other two chips go?
Jay:
I’m trying to find out where to put them. It has to be five equal
shares.
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Marla and Paul resolutely stated that only 10 chips could be partitioned. Chrissy
gave this configuration
choosing to assign four chips to one group.
Mary, Kisha, Richy, Bob, Ben and Paul decided to make the groups of five with two
chips in each group with remaining two chips set aside as remainders (see Figure 9 for
Mary’s distribution). Ashley considered this task to be “impossible” and 25% of the
students agreed with her making no suggestion as to how this task could be
accomplished.
S. 1
S. 4
S. 2.
S. 5.
S. 3
Rem. 2/
Figure 9.
Mary’s Method of Equally Sharing 12 Chips
The subtask was then placed in the context of dividing 12 cookies with five
persons. Jay then offered this suggestion for doing the partition:
Jay:
Take each cookie and divide each cookie into 5 pieces and then
give each person one-fifth.
Carol agreed with him. Claudia decided she would break each of the two remaining
cookies in halves but quickly retracted when she recognized that each of the five persons
would not get a half. Dave had a different idea.
Dave:
You only need to divide the leftover 2 cookies in five equal parts
each and give one piece to each person. Each person would get,
uhm, 2 wholes and 2 small pieces.
He made this sketch on the task sheet:
.
represented one of the pieces from the leftover cookies.
Task 3 required the sixth graders to partition the number line into two, five and
eight equal parts. The given line measured 4”. Dividing the line in halves posed little
84
problem although the students employed different tactics to accomplish the task. After
measuring the number line, which was labeled 0 and 1 respectively at the endpoints, most
students simply used their ruler to guide them to the half of 4. Others used the estimateerase-readjust method to find the center of the line. Partitioning the line in eight parts
never posed much problem as the subtask before it – partition into five parts. Still using
the 4” measure, Marla, Karla, Jonah, Alton, Jay, Mary, Carol and Brian recognized that
they could not easily partition into five parts therefore they resorted to estimating the
parts to be less than an inch.
Claudia, Ben, Dave, Josh, Bob, Mary and Paul decided to use the centimeter (cm)
side of the ruler showing 10cm in close proximity to 4”. Using the division property,
they assigned approximately 2cm to each part to produce the five equal parts on the
number line. Polly and Dahlia completed this task outside of the regular session due to
their absence at the last part of the session.
The last task in this activity proved to be a rather interesting one. Some students
were a little baffled as to the meaning of a “cup of water.” Does it mean that the measure
of the water was one cup, or the container was referred to as a cup? They were told to
use the former meaning. At each of the three subtasks, they assumed that there was no
water loss. They were asked to equally share the cup of water into two, three and five
shares.
I:
Can anyone share how they partition their water into three equal
parts?
Karla:
I count the stripes around the cup.
Dave:
I kinda did what Amanda did, except this time I just pour the water
into the cup, look at each one until only two stripes are left.
Josh:
I put water to four stripes, another one to four stripes and another
to four stripes.
I:
Does it work exactly?
Josh:
Yes
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Chrissy, Kisha and others decided to use some innovative ways to ensure their equal
partition. This included using the chips, the erasers, the rulers and the graduated
measurement on the cup.
I:
Does anyone want to share their method of making sure the parts
are equal?
Kisha:
My water is up to 250 ml. I divide it by 5 and get 25 and then give
everyone 25 ml.
Ashley drew my attention to what she was doing by exclaiming that she had water left
over.
I:
What should you do with your extra water?
Ashley:
I should drink it [laughing]. I will use all of it. [She poured and
observed until all her water was used up contending that she
partitioned her water into five equal parts.]
The students found this task an interesting one as they worked with non-conventional
ways to verify the equal sharing of the cup of water.
At the end of the three sessions for Activity 1, students were given the
opportunity to share their work with the whole class and a review timeslot was scheduled
at the beginning of the next session. As mentioned earlier, each student participated in
individual assessment tasks at the end of some of the activities. These tasks were
designed to verify the students’ learning gains that were captured during their
participation with the meaningful, engaging task-based activities coupled with their
interaction with the physical referents at their disposal.
Keep in mind that the main objective for this level in the model of understanding
is the discernment of equal and unequal parts. For each of the tasks in Activity 1 some
students displayed signs of miscomprehension and uncertainties as to the way in which
the partitions should be done. The assessment task required the students to recognize and
partition discrete and continuous wholes. There were three exercises in the task sheet
(see Appendix B – Assessment Task Activity 1). On the first exercise, the participants
were presented with four circles divided into three parts. The students were asked to put
a check mark beside the ones that were partitioned. Choices b and c were pre-designed
86
to be partitioned but with different orientation while a and d were not partitioned. Table
10 gives a summary of the students’ choices.
Table 10.
Summary of Responses for Assessment Task for Activity 1
RESPONSES
Number of Students
N = 20
1
2
17
b only
c only
b and c
Some of the reasons for choosing b and/or c were:
Dahlia:
I know that d and a are not equal because I observe both of them
and I measure b and c in my head.
Claudia:
c seems to be equal and a looks really messy.
Richy:
I know that circle c and b are partitioned because I measured from
point to point using centimeters and if the space is the same size.
Circle b looks like its partitioned because the angles are the same
size and circle c looks partitioned because the edges are all the
same size.
Mary:
Kisha:
Well, I saw that letter b was equal because I measure it with my
ruler. Also letter c. Both letter b and c has 4 cm from point to
point.
Marla:
You can solve this by drawing an equilateral triangle in to the
middle to measure each side.
Polly:
For each circle I would measure the three lines with the ruler. If
all three lines were the same length, they were partitioned.
Josh:
I used the ruler for circle b and c. I measured from one point to
the other.
Jay:
c is just an upside-down b.
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None of the students chose a indicating that they were not confused with the conventional
orientation of a circle dividing in three equal parts. Ashley, who had earlier in the
activity divided her circle similar to d, chose b and c without offering any explanation for
her choices.
On the second exercise the students were asked to partition a rectangular-shaped
cornbread among six people. The three common configurations were:
(a)
Figure 10.
(b)
(c)
Common Configurations for Exercise 2 Assessment Task for Activity 1
Bob and Jonah were the only students who differed from one of these configurations. For
Bob’s second configuration he partitioned his rectangle by first dividing the figure into
three squares and then drew one of the diagonals for each square resulting in a
configuration looking similar to this
. His first partition looked
similar to Figure 10(b). Jonah chose to draw the two diagonals of the rectangle and then
a line cutting vertically through the center of the rectangle to form 6 triangles. He had no
idea of how to check if the triangles were equal. He was satisfied with the fact that he
could divide his rectangle into six parts.
The final exercise asked the students to equally divide eight cookies among three
persons. Here are some of the students’ responses:
Dave:
I gave each person two cookies and later broke the two extra
cookies into three parts.
Karla:
I give two cookies to all of the three people then two cookies are
left. I break one cookie in half and give one piece to the first and
the other half to the 2nd kid. I break the last cookie into four pieces
and give one to the 3rd kid and one to the 1st kid another piece to
the 2nd kid and the last piece to the 3rd kid.
88
Alton:
I get two cookies for each person and then divide the two by three
slices.
Jay:
I counted the circles and there were two left and I divided them
into thirds. So each person gets two and two-thirds cookies.
Claudia:
I give everyone one-half of a cookie.
Ben:
I put two cookies each then split up the other two into three and
give each person two pieces.
Kisha:
Well each person had two cookies so I divided the other two
cookies into three halves so everyone could get two cookies and
two thirds.
Bob:
I divide the cookies into thirds so each one could get equal amount
of cookies.
All the students, except Jonah, Karla, Claudia and Kisha figured that they would have to
split the remaining two cookies in three equal parts. They distributed the thirds in two
different ways, either by giving each person one-third of each cookie or putting the thirds
together to form six thirds and then gave each person two thirds. Jonah did not make any
attempt to share the cookies with three persons instead decided to share the cookies with
four persons thus distributing two cookies to each person. Karla focused mainly on
sharing all the cookies making sure that each person had the same number of pieces of
cookies instead of ensuring that each person got the same amount of cookie. So for her,
each person got four pieces of cookie disregarding the inequality of the pieces. While
referring to “halves” in her explanation of how she performed the task, Kisha partitioned
the circle into three equal parts using the conventional method (
). This was evident in
the diagram she drew to accompany her explanation.
Based on the less than satisfactory results on the individual assessments tasks,
individual interview sessions were planned with Claudia, Ashley, Carol, Kisha, and
Dahlia. These sessions were held during their elective session and lasted for
approximately 30 minutes. They were designed to provide re-verification tasks that
allowed the students to talk aloud their strategy for completing them. The interview
protocol and tasks are shown in Appendix B. The first task invited the interviewees to
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divide 11 cookies among three friends. Using whatever physical referents they needed,
they were encouraged to talk aloud as they solved the problem.
Dahlia:
Uhm… Yea. I am going to give three cookies to each person then I
have two remaining cookies, and from those two cookies I break
them up in three equal parts and two parts go to each person. So
it’s three cookies and two parts of a cookie to each person.
At first Dahlia divided the two remaining cookies using vertical lines but when
questioned about the equality of the parts, she figured that the parts were not equal and
then proceeded to partition the circle the conventional way. She was able to verify the
equality of the parts using the strategy that Marla shared in a previous session.
Kisha:
I found out that each person will get three cookies but two are left
and so I broke them in three parts and gave each person one part
from each cookie.
Kisha proceeded to do the partition in a similar fashion as Dahlia but without any
intervention she employed the regular strategy of dividing the circle into three equal
parts. Carol, Ashley and Claudia distributed the cookies in the same manner giving three
whole cookies to each person and then dividing the last two cookies into three equal parts
and then gave two parts to each child. Claudia used the chips to aid in her distribution.
She even attempted to partition the last two remaining chips. Carol made a table with the
three names as a header then put a stroke (|) to represent a whole cookie and a dot (•) to
represent the third of the circle. Carol used the word “half” to mean the third as she did
in a previous task.
I:
So each of the part of this cookie is a half.
Carol:
Yes.
I:
So this cookie has how many halves?
Carol:
Three. Hmm! But something can’t have more than 2 halves.
I:
How do you define a half?
Carol:
Two equal pieces. I think I am using the wrong word.
90
To verify that Carol knew what she was doing, she was asked to share seven cookies
among three friends. She was able to do so this time using the word one-third to
represent one equal part.
The second task required the interviewees to partition a square cake among eight
friends. They were expected to show the partition in two different ways. None of the
students experienced any difficulty in partitioning and verifying the equal parts of the
square. They suggested measuring the sides of the parts, cutting out and placing the part
on top of each other (superimposing) and visual observation as means through which they
could verify that the square is partitioned into eight parts. A summary of their
configurations is shown in Figure 11.
Figure 11.
Configurations for Task 2 – Individual Interview
Analysis
As you may recall, the most fundamental criterion for the logico-procedural
understanding is the ability to partition a whole into a given number of equal shares. The
twenty students in this study, during the various Activity 1 tasks, assessments tasks and
interviews demonstrated that they possessed this level of understanding. At the end of
the three sessions dedicated to the partitioning activities, the students were eager to share
their strategies for partitioning the continuous and discrete wholes. The word “partition”
became a standard word in their mathematics vocabulary. By the end of Activity 1, the
participants exhibited much ease with working with continuous and discrete models.
Dividing the circle into three equal parts and sharing discrete objects among
individuals where the number of objects was not a multiple of the number of equal parts
were the two most trying tasks for the students. As the participants of the study
91
interacted with each other they were able to agree on acceptable methods of ensuring
equal parts when dividing the whole. Those who had difficulty in authenticating the
equality of the three parts eventually abandoned their methods and adopted the method
that Chrissy and Marla used – measuring the chords formed at the end of the partitioned
lines. They remarked that the chords should form an equilateral triangle.
The students employed various methods in verifying equality of parts in
continuous models. These methods ranged from measuring with whatever physical
referents (e.g. rulers, flexible wires, fraction circles, fraction squares, fraction triangles,
graduated marks on the objects, their fingers) made sense to them at the time. They also
relied on visual observation (especially when working with the parallelogram) and little
facts that they gleaned from the unit previously done in geometry.
The participants experienced little to no difficulty when partitioning discrete
objects where the number of objects was a multiple of the number of partitions. The two
most common methods that these students used were the “dealing” strategy and the use of
their multiplication/division facts. The main manipulative used to aid them in their
strategy was the circular chips that were provided in their bags. Some students also used
the circle cutouts that were given during the tasks. Partitioning a number of discrete
objects where the situation was different than what is cited above posed considerable
struggle for the participants. Some of the students, initially, regarded the task as
impossible but were later able, with the help of their classmates, to accomplish this feat.
There were open class reviews of partitioning ideas the students developed in the
previous sessions. These review sessions were student-based and student-centered where
the students shared what they learned. The teacher played the part of a facilitator at these
review sessions by asking leading questions and opening the floor for discussion using
excerpts from the previous tasks. This practice became a standard feature throughout the
teaching sequence.
Logico-Physical Abstraction
This level of understanding is the last component in the first tier of Herscovics
and Bergeron’s (1988) model of understanding. At this level the fraction concept is
distinguished from the simple action of dividing a whole into equal parts (logico-physical
procedural) but rests upon the understanding of the relationship between the part and the
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whole. The five criteria related to this component (see Chapter 2, page 42) will be
evaluated through three activities (see Appendix B). Each activity consisted of several
subtasks which covered the participants’ perception of the relationship between the part
and the whole in spite of the transformation, reversibility and repartitioning of the part or
whole. Activity 2 majored on the reconstitution of the whole from its part, Activity 3
focused on the equivalence of the part-whole relationship regardless of variation in
physical attributes and/or transformations of the part or the whole, while Activity 4
centered on repartitioning an already partitioned whole. Activity 3 spanned two sessions.
The three activities were group-based. An assessment task accompanied each activity.
At the beginning of each activity the students were handed their personalized
manipulative bag which included all the materials they needed for the tasks at hand.
They were given twelve one-twelfth pieces of a circle (green or black) and one-quarter of
circle part (yellow). Each group was also given a bag containing five one-eighth parts of
a triangle, one one-fifth part of a circle, eight one-twelfth part of a square. There were
also provided with two ounces of water in a small plastic container and a measuring cup.
They sat in their group at the beginning of each group session.
The first two tasks in Activity 2 were done individually. For Task 1 each student
was expected to use the twelve equal parts of a circle to reconstruct a whole circle. The
sixth graders had no difficulty in accomplishing this task. Task 2 required the
participants to determine how many of the given part (one-fourth) of the circle would be
needed to reconstitute the whole circle. Again, this task posed no problem for all the
students who recognized that four parts were needed in total. To verify this, some of the
students (e.g. Alton, Paul, Bob, Cindy, Karla, Ashley, and Carol) placed the given part on
a blank paper and traced it. They repeated this action three more times laying the given
part adjacent to the drawn part until the circle was complete. Others such as Ben, Jay,
Dave, Mary, Chrissy, Marla, and Jonah immediately recognized that the piece
represented one-fourth or one-quarter (Jay’s terminology) and stated that it would take
four of the pieces to complete the circle. A number of students linked Task 1 with Task 2
using the circle formed in Task 1 to recognize that three additional pieces are needed to
complete the circle.
93
The remaining six tasks required the students to work in their groups to discuss
the solution to each problem. The members of each group were previously assigned by
the teacher-observers. Each task was accompanied with the required props necessary to
arrive at the solution. Without any prompting from the researcher, the students assigned
roles to each member. They took turns in reading the questions and in monitoring the
tape recorder placed at the center of the table. Because of the relative ease at which the
groups completed the tasks for this activity only the results from two of the groups will
be displayed. These two groups represented the overall work done by the members of the
different groups. The excerpt below is pulled from the conversation among Bob, Kisha
and Jay. Dahlia was absent that day.
Task 3:
You are given five parts of a triangle. How many more pieces of the same
size are needed to complete the whole triangle?
Jay:
We need about four more.
Kisha:
We could measure the sides to help us. We could make it equal
sides with our ruler.
The others ignored what she was saying.
Jay:
We need about two more, no, three more.
Kisha:
That’s true, we need three more triangles.
Bob agreed with them.
Task 4:
You are given one part (green) of circle. How many pieces of the same
size are needed to complete the whole circle?
Bob:
I think we need about (pause) 3 or 4 more circle parts to complete
the circle.
Kisha:
I don’t agree that we need 4 more. I think we only need 3 more
because we already have one.
Jay:
We need only 3 more parts to complete the circle.
Bob:
Actually, the answer can’t be four or three. It is five.
Jay:
Why?
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Bob:
If you take this part of the circle and trace it (he traces the given
part of the circle on the paper) you will see that you need about five
more.
Jay:
So each piece is a sixth?
Bob:
Yes.
Kisha:
Bob:
Four more pieces?
Yes, four more pieces.
Kisha:
I guessed it right.
Jay:
So each piece is a fifth.
Bob:
I made the mistake ‘cause I count the one we already had.
They all agreed that four more pieces are needed.
Task 5:
You are given eight parts of a whole square. How many more pieces of
the same size are needed to complete the whole square?
Kisha put the pieces together. The group decided that they needed four more little
rectangles to complete the square.
Task 6:
You are given one part of a cup of water. Predict how many more similar
parts will give you a whole cup of water.
They poured the water in the measuring cup.
Kisha:
We have one-fourth of a cup.
Bob and Jay agreed with Kisha that three more fourths of the part of the cup was needed
based on their observation.
Alton’s group had quite an interesting twist to Task 6. Here is the excerpt of their
discussion from Task 3 to Task 6.
Task 3:
Paul:
We need 3 more pieces.
Chrissy:
That’s impossible.
Ashley:
We need six more pieces.
Chrissy:
We need seven more pieces to do this triangle.
They reassembled the parts until they all agree that three more pieces are needed.
95
Task 4:
Chrissy:
I think we are missing five pieces from this one. There are six
pieces in all.
They all agreed and moved on.
Task 6:
Paul:
How much is the whole cup?
Chrissy:
8 oz. or 250 ml and the part is75 ml.
Ashley:
What is 75 times 3?
Chrissy:
75 times 4.
They struggled with the multiplication realizing that they were not getting the 250 they
desired. Chrissy reread the question.
Alton:
I’m lost.
Paul:
Wait, why don’t we do 250 take away 75?
Ashley:
That’s what I did and get 175.
The group decided the answer was 175 ml.
Dave had quite a task in convincing his group members that Task 6 required only three
more parts to complete the whole cup.
Dave:
I think 4 more parts of the cup because on the metric side of the
cup, 1 cup is 250 ml and we have 50 ml already so we need 4 more
50 ml to get the whole cup.
Carol:
I also get 4 more parts by observing it.
Karla:
It could be 5 more parts or even 4 ½.
Claudia:
I agree.
Dave:
It doesn’t need so much. It only needs 4 more parts. (He said the
last part with much emphasis).
Claudia and Karla were satisfied with Carol’s observation method but needed
more tangible proof that four more parts were needed. They decided to listen as Michael
repeated his method of determining that four more parts were needed to obtain a cup of
water. The group then agreed.
From the discussions and conclusions reached by the students concerning Activity
2 it became evident that the students experienced little difficulty. They employed
96
various strategies to ensure the completion of the whole whether they are given many
parts of the whole or one part of the whole. Some of the strategies included observation,
drawing pictures, working with the manipulatives and in the case of the liquid measure,
graduated marks on the container. The students were adequately able to reconstitute the
whole from its parts. To further investigate this ability, however, an assessment task was
given before the start of Activity 3.
The assessment task for Activity 2 was a paper-pencil task-sheet that required the
participants to determine how many of the shaded parts were need to complete the whole
object. The group members were at liberty to use any physical referents to aid in the
derivation of a solution. Only continuous models were used. The first exercise proved
fairly simple to all the students. The dotted frame used to complete the whole provided
an easy scaffold thereby enabling all the participants to recognize that one more part of
the shaded region was needed.
After agreeing that the first exercise was simple, Chrissy’s group grappled with
the circle. Here is an excerpt from her group’s discussion on the entire task-sheet.
Chrissy:
That was easy. [Speaking of the first task.]
For number 2, the circle, I started with the three lines, put the
ruler on each line and then make a line half way through the circle
and I did that for each three lines. Therefore 4 more pieces are
needed.
Ashley:
That’s what all of us did.
Paul:
What I did, I draw the other three lines and made them longer so
the other half [meaning the other half of the circle] could have the
same amount of lines. That gives me four pieces.
Alton:
What I did was I split them in half to make each a triangle and
them make them equal ‘cause I saw that each of them was a
triangle. [He used the word ‘triangle’ to refer to the sector of the
circle.]
Alton, Ashley, Chrissy and Paul drew lines in the diagram to partition the circle
into eight pieces (see Figure 12).
97
Figure 12.
Partitioning the Circle in Question 2 – Activity 2 Assessment Task
This would mean that they needed seven more of the shaded part to complete the circle.
The group focused, however, on the half circle that did not have any lines thus obtaining
four as the needed number of parts. The four remaining groups concluded that seven
more of the shaded part was needed. They verified their decision by completing the
circle as shown in Figure 12. Dahlia stated:
1
The circle needs 7 more parts because the shaded part is only .
8
The circles in Task 3 posed some challenge to some of the students as they recognized
that they could partition the shaded part into any number of desired pieces and then use
one piece of the shaded part to determine how many of that piece was needed to complete
the circle.
Josh:
This circle needs sixteen pieces.
Richy:
That’s not true.
Mary:
Or a thousand million little pieces.
Richy:
This little circle only needs one more piece.
Josh:
Richy, it doesn’t need one more piece. It needs sixteen more pieces.
Richy:
Okay, it doesn’t matter. We could still divide it into sixteen pieces.
You put that and I will put the one piece of fourth.
Josh:
As a matter of fact, it can take any amount.
Mary:
But the pieces have to be equal pieces.
98
Richy:
It just means that the shaded parts would have to be divided into
smaller equal parts.
Josh:
It can be done but it is not going to be easy to draw the parts. You
could put even 16 billion equal pieces in your one part, 16 billion
there, 16 billion there and 16 billion there [making reference to the
three parts of shaded fourths.]
Josh and Alton’s groups chose to partition the whole differently. Each member in both
groups divided his or her circle into different equal parts. Josh decided to divide the first
circle into twelve parts instead of the sixteen parts he argued about. He first divided his
circle into four equal parts, and then partitioned each fourth into three equal parts thus
resulting in a circle that was partitioned into twelfths. From this Josh concluded that “If
you put three equal parts in the blank spot you will get 12 equal pieces all around. Threetwelve more pieces are needed.” Alton did a similar partition. Paul partitioned the circle
into sixteen equal pieces thus determining that four more pieces of sixteenths were
needed.
Paul:
I divided each four into four little pieces which comes out to be 16
pieces.
Chrissy and Mary divided their circle into eight equal parts and deduced that two-eighths
were missing. Karla, Kisha and Brian’s groups (see Appendix C for group members)
partitioned their circle in four thus stating that one more fourth (¼ or quarter) was
missing.
Dave:
This one is made up of four parts [referring to the first circle in
item 3.] We’re guessing that they cut it up into four parts. So we
are guessing that we need one fourth more.
The other group members chimed in with their confirmation of Dave’s guess. The
groups mentioned above expressed similar reasoning. Ashley also opted for one missing
part of fourth.
Ashley:
For the first circle, I close it and then I put it for four equal pieces
so one piece of four is missing.
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Jonah misunderstood what his group members were discussing concerning the number of
equal parts that could be drawn in the missing part and chose to divide his circle into five
parts. He drew three lines in the shaded region resulting in four parts plus the blank part
completing the five parts.
Armed with some insight from the first circle of item 3, most of the participants
moved ahead to divide their circles into parts with no apparent method of ensuring the
parts were really equal. There was little group discussion on this exercise. 45% of the
class partitioned the circle into eight equal pieces by completing the fourths and then
drew center lines through the quarters resulting in three more fourths needed to complete
the entire circle. Josh, Polly and Brian made six parts and figured that two more sixths
were needed while Kisha and Dahlia simply drew a line to divide the missing part into
two equal parts stating that two more fourths were missing. Paul did the same as Kisha
and Dahlia but concluded that the shaded part consisted of three parts.
Paul:
I divided the circle into five and so two pieces are needed.
Bob is the only student who divided his circle into three parts while Richy divided his
circle into 10 pieces.
Bob:
One more third is missing because I noticed that 2/3 were shaded.
Richy:
Four more pieces of ten are missing.
The responses to the tasks revealed that the students were aware that the parts must be
equal but were still unable to apply suitable methods of checking that the parts were
actually equal. Most of the students relied on the method of observation making little use
of the manipulatives that they worked with throughout the previous activities.
Two parallelograms were used as the whole in item 4. All the students stated that
two more fourths (quarters) were needed to complete the first parallelogram. They drew
the two lines needed to form the parallelogram (see Figure 13a) as a form of verification
with Jay adding that he used his “common sense” to know the answer.
Dave:
I think in the parallelogram we need two more pieces ‘cause all we
have to do is to complete it to make slanted lines.
Carol:
And it’s obvious too.
100
For the second parallelogram, all the students drew the necessary lines to complete the
figure with 60% of the entire class dividing the parallelogram into six parts as shown in
Figure 13b.
(a)
(b)
Figure 13.
Students’ Drawing of the Parallelogram – Item #4
Dahlia and Kisha who were working together concluded that only one more third was
needed giving no explanation of how they arrived at their solution. Richy drew lines to
make eights parts in his parallelogram thus needing “One more piece of eight.” Mary
and Jonah simply closed their figure without making any suggestion to how many parts
were missing. Josh and Paul made two different configurations after completing the
parallelogram (see Figure 14).
Paul’s
Figure 14.
Paul:
Josh’s
Configuration for Item 4 (Second Part)
For the box, I just close it in. I divided it into twelve equal parts. 2
pieces of 12 are missing.
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Josh:
I divide the parallelogram into 24 equal pieces so I need four more
pieces of twenty-four to complete the parallelogram.
80% (four groups) of the participants completed the existing lines in the first
circle of item 5 to form six parts thereby determining that four more sixths were needed.
6
Brian wrote that “ more pieces are needed” indicating his confusion with the
4
positioning of the shaded parts with the total number of equal parts in the whole. No
intervention was planned at this point as formal fraction language and symbolization
would not be considered until further activities in the logico-mathematical tier of the
model. Dave and his group members divided the circle into five parts and unanimously
agreed that three more fifths were needed. Most of them did not offer a method of
ensuring equal parts. Bob argued with Jay that his parts looked larger than the ones that
were shaded and then concluded “we need a protractor to be exactly sure.” Kisha used
the words “equal” and “half” synonymously. All the students except Mary responded
that two more thirds were needed for the second circle in item 5. Mary, possibly trying to
be different from the others, said that 12 more pieces of twelfths were needed. She drew
lines to divide the blank portion of the circles into twelve parts.
The last item on the assessment sheet for Activity 2 never posed much challenge
to the students who were able to use their ruler to measure one of the shaded parts of the
rectangle and then used that measurement to mark off the remaining parts. They all
concluded that seven more pieces of tenths were needed in the first rectangle and three
more pieces of fourths were needed for the second rectangle.
Chrissy:
For the rectangle, I measured how big the shaded piece is and then
I made the same size on the rectangle and for the last rectangle I
did the same thing.
Ashley:
For the first one of the rectangles, I measure how wide and then I
put my ruler to the length and I put it equal, then lay it straight
down. For the second one, I measure it again, I made a line with
10 equal parts.
Based on the responses to the tasks in Activity 2 and the assessment sheet, the
participants displayed the ability to reconstitute the whole from its part. This includes
102
making the whole given many of the parts and making the whole given only one part.
Although majority of the students were not very clear in their oral discourse about their
method of ensuring that the parts were equal, on observing their behavior on the video
clip they could be seen using the chips, ruler, and even the flexible wires to aid in their
decision. Others just simply used the method of observation or “common sense”
especially on the circular diagrams, which proved more of a challenge to them than the
parallelograms. There was evidence that the students made effort to show that the parts
were equal. There was very little teacher intervention as future activities would clear up
some of the discrepancies.
At this level the students were expected to employ the whole number language in
their responses as the use of the “fraction language would elevate the task into the logicomathematical tier of the model” (Boulet, 1993 p. 121). However, as sixth graders who
had been introduced to fractions at least two years earlier, it was anticipated that the
fraction language would be used widely in some of their responses. This does not outlaw
the activity as logico-physical as emphasis was placed only on the numerical aspect of
their solutions and not on the language itself. Careful scrutiny of most of the responses
revealed that the students mostly used the whole-number language to express their idea
both in their written and oral work. These include “three pieces of ten” and “four pieces
of 24.”
Activity 3 was the second activity at the logico-physical abstraction level. The
main objective of the seven tasks comprising this activity was to foster awareness and
understanding in the students of the equivalence of the part-whole relationship in spite of
variation in physical attributes and transformation of the whole. The first three tasks
were aimed at determining the students’ understanding of the preservation of the partwhole relationship regardless of the physical attributes of the whole while the remaining
four tasks focused on gleaning the students’ comprehension of the preservation of the
part-whole relationship despite the transformation of parts in the whole. In five of the
seven tasks the students worked with continuous models (circular and rectangular
regions). At this point, the students were clear about the term “part-whole relationship”
since it was discussed during one of the review sessions.
103
For the first two tasks, the students were presented with two stories accompanied
with the necessary materials (see Appendix B) to aid in their solution. As there were four
persons in a group, in solving task 1, all the students within the groups assigned to
him/her one of the names in the task. In Claudia’s group consisting of two boys and two
girls, they changed one of the male names to a female name. All the members of each
group easily recognized that one part out of three parts was eaten. Polly and Ashley’s
groups, after acknowledging that all four persons had the same amount of his/her
chocolate bar, made note that each person had a different size.
Paul:
Each student didn’t eat the same size because they have different
sizes but they are all going to eat one-third of their chocolate. So
they didn’t really eat the same because there are different sizes but
they each ate one-third. It still doesn’t mean that they ate the same
size.
Chrissy:
What Paul said is true but if you’re saying that they are eating the
same amount as in fractions then they did eat the same amount.
Ashley:
They eat the same amount in fractions but they have different sizes
so it isn’t the same amount as for each.
Alton:
If you put it into regular pieces it wouldn’t be the same amount but
when you put it into fractions they each have the same amount of
pizza to eat.
The students continued to use fraction terminologies though the teacher did not use the
terms and the word “fraction” was not mentioned in the tasks.
Bob:
All of us ate the same amount of chocolate – one third.
Polly:
Everybody ate 1/3 of their chocolate bar, but each ate different
size.
Claudia:
We still ate the same 1/3. Even though they are different sizes its
still the same fraction.
The second task was similar to the first task except that the continuous wholes were
circles. Similar responses were given where students made note of the part-whole
relationship of the pizza that was eaten (one-half) but mentioned that Mitch’s portion was
bigger than Kris’. Here is an excerpt from Carol’s group.
Claudia:
Mitch had a lot more to eat.
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Dave:
But if you put it fraction-wise it’s the same fraction.
Carol:
Mitch had to lose weight. He had to do exercising.
Claudia:
Our conclusion was that even though they are different sizes it is
still the same fractions.
Task 3 required the students to look at a set of figures (circle, square, rectangle,
equilateral triangle, L and a star) and to determine if the part-whole relationship was the
same for all the figures. Polly was confused with the term “part-whole relationship” but
Marla using two of the figures on her desk, explained the term to her. The part-whole
relationship was the same for the circle, star, rectangle, square and triangle with the Lshape being the odd one out. The students quickly recognized the difference and readily
determined that the part-whole relationship was not the same in all the figures. Carol,
along with most of her classmates, made note that the circle, triangle, rectangle, square
and the star had the same part-whole relationship.
Task 4 focused on the part-whole relationships in wholes where the parts were
transformed. The students were presented with four squares (see Figure 15) where halves
were shaded using varying configurations. They all figured that the part-whole
relationship was the same after some discussion and negotiation.
Polly:
Look at these two squares [referring to the first two squares in
Figure 15], do you think the part-whole relationship is the same?
Marla:
I think so because they look equal, the only thing that is different is
that they are on different side of the square but they are still half of
the square.
(a)
(b)
Figure 15.
(c)
Squares Used for Activity 3 – Task 4
105
(d)
The group agreed that the part-whole relationship was the same. They examined the
remaining squares:
Marla:
The other squares are still the same, this one is divided into four
but it is still missing two parts (same as shaded part for her) which
is the same as a half and this one is divided into two but it is still
the half of the square.
This discussion was similar to the ones shared in the other groups. Ashley read the
question in her group and laid the squares on the table so all the group members could see
them.
Alton:
No, the part-whole relationship is not the same.
Paul:
No, they are not the same because they all have different shaded
parts.
Chrissy:
Yes, it is ‘cause each one is half of a whole. The question asks if
the part-whole relationship is the same. It doesn’t matter if this is
two and this is one. It is still half of the whole.
Ashley:
It is the same. If we put it together it is still a half.
Alton:
Yea, right, it doesn’t matter how it looks it’s still going to be a half
(nodding in agreement).
Josh and his group members agreed that the part-whole relationship was the same
in Task 4 because each shaded part represented half of the square no matter the size or
how the shaded parts were placed. They were able to arrive at a consensus with the
transformation parts by mentally flipping the parts to match and compare with the other
parts.
Based on the responses mentioned above, the students exhibited some knowledge
of the preservation of the part-whole relationships in wholes that vary in shape, size and
transformation of the parts within the whole. Although they consistently mentioned the
size of the part, for example, “Mitch gets a bigger slice,” they were able to differentiate
between continuous wholes that possessed the same part-whole relationship and those
that did not.
106
Tasks 5 and 7 were designed to examine the participant’s understanding of the
equivalence of the part-whole relationship in the case of discrete wholes despite physical
attributes and transformation of the parts. There were two subtasks in Task 5. Task 5a
had the students working with sets of circles and diamonds. Each set of 10 had four of
them shaded. The students agreed within their groups that the part-whole relationship
was the same “four out of 10.” A third set of circles was added with two of five of the
circles shaded. Initially some of the students refuted the others who claimed that the partwhole remained the same. With much discussion and negotiation the groups said ‘Yes’ to
the equivalence of the part-whole relationship. Here are some of the strategies used to
verify this equivalence.
Richy:
Yes, they are all equal because you can divide the black circles
and diamonds into twos.
Kisha’s group:
Bob:
They both can be simplified to two-fifths.
Jay:
(Covering one set) Take out this part. That’s two-fifths.
Dahlia:
They are not equal.
Jay and Bob: They are equal.
Jay:
Okay, just divide by two.
Dahlia:
What?
Jay:
You divide both numbers by two and you get two-fifths.
Dahlia:
Okay. So two-fifths is a half of four-tenths.
Marla’s group:
Polly:
Is it the same part-whole relationship?
Marla:
I don’t think so.
Brian:
I don’t think so either.
Polly:
Why?
Brain:
Because they are not… they are not all equal amount.
Marla looked again at the set of 10 circles with four shaded and the set of five circles
with two shaded.
Marla:
Oh yes I do. I think they are the same because they are all missing
one fifth even though there are ten here, if you make one row you
can see that’s like one fifth. There’s two rows of five and it looks
107
the same as this one which has one row of fives – you see. This
one is like four tenths, two fifths is half of this. This is the same
thing. If you move these here and these here [pointing to the set of
10 circles] it will look the same as this ones.
Polly:
I disagree with you. These two are divided into ten pieces and they
are all missing four while this is divided into 5 pieces and only
missing two. I know it’s half of this. The 10 diamonds and 10
circles with the four parts shaded are the same but the ones that
have the five circles and only two are shaded is not the same as
this one. This is five circles and missing two and this is ten.
Marla:
Yea but when I was in fifth grade they taught me that if you have
two fourths it is the same as one half? [She makes the illustration
2 1
on her paper – = .]
4 2
Polly:
Yea because it’s half.
Marla:
When you have 4-tenths, it is the same as two-fifths because you
can divide the four in half and get two and divide the ten in half
4 2
and get five. [She wrote on her paper = .]
10 5
Polly:
That’s right.
Polly’s acceptance of Marla’s explanation could be based on the fact that Marla
used the word “half” in simplifying four-tenths. Previous activities revealed that the
word “half” for some students did not necessarily mean two equal parts but could be used
for different connotation such as “even parts” and in Polly’s case may mean the only
fraction that another fraction can be reduced to or reduced by.
The last two tasks focused on the preservation of the part-whole relationships in
spite of physical transformation of the parts. In Task 6 the students worked with
continuous models. The part-whole relationship was the same in all the squares though
some of the parts were transformed (see Appendix B). Again the students employed
different strategies to determine this fact. Each person easily identified Figure 16(a) as
having one part shaded out of four making note that the parts were equal. At this point
Polly expressed that she now understood the term “part-whole relationship.” Referring to
Figure 16(b):
108
Polly:
They are the same. If you put this part [pointing to the half of the
quarter that is shaded] with the other part it creates one square. Oh
I know that the part-whole relationship is now one-fourth yeah!
Alton’s group was talking about Figure 16(d):
Paul:
This one is not the same.
Alton:
Yes, it is.
Chrissy:
No, it’s not.
Ashley:
It is Chrissy.
Alton:
Yes it is, ‘cause if you take out these two lines and put this piece
right here and like this then it will be just like this one.
Chrissy:
They are not the same because you can’t just move things around
like that.
Ashley:
But we move the others [referring to the square shown in Figure
16(b).]
Chrissy:
Okay, I guess they have the same part-whole relationship.
Using similar strategies the concluded that all five squares possessed the same part-whole
relationship.
(a)
(b)
Figure 16.
(c)
(d)
Squares Used for Task 6 – Activity 3
Task 7 (discrete model) posed no problem to any of the students despite the
transformation of the parts as evident in some of the students’ dialogue given below. In
this task the students determined if the same share of cookies was eaten in the two
instances where the shaded cookies changed relative positions in the whole. A similar
109
task was included in the pretest and was used to test the students’ intuitive knowledge of
the equality of the parts in a partitioned discrete whole despite the position of the parts in
the whole.
Karla:
It’s equal cause in each one there is 6 cookies and I ate 2 cookies.
Dave:
There are 2 cookies eaten and in each set you have 4 left.
Josh:
It’s the same. There’s two out of six in each one.
Jonah:
Yes, because there is
Mary:
Yes, because there’s two out of six in each set just not in the same
positions.
Polly:
The part-whole relationship is the same because they are all
2
in each set of cookies.
6
missing two. That was an easy task – one of the easiest ones.
The group assessment task for Activity 3 required the students to use all the
figures they worked with including some new ones and sort them according to their partwhole relationships. They found this task rather exciting as they negotiated where each
representation of the part-whole should go. An onsite interview with each group revealed
that they were able to identify diagrams showing the same part-whole relationships and
those that did not. They were also able to discard those figures that show unequal parts
though these figures possessed the same number of divided parts as other diagrams.
Much discussion took place as they collaborated on the part-whole relationships depicted
on the diagram shown in Figure 17.
Figure 17.
One of the Configurations Given in Assessment Task – Activity 3
Brian:
That’s one-half.
I:
Why one-half?
110
Brian:
It is one part shaded out of two.
Carol:
It’s one-fourth
Carol used the transparency copy of the diagram and shared with the class how she got
one-fourth as her solution. She connected the center line to make four equal parts. Brian
ignored the unshaded half and looked at the whole as the two equal parts.
Activity 4 - the final activity of the logico-physical abstraction level of
understanding - was used to investigate the students’ understanding of the relationship
between the part and the whole at the logico-physical abstraction level. The main
objective of this activity was to examine the students’ ability to repartition wholes that
were already partitioned. This activity formed the prerequisite physical basis for
understanding equivalent fractions. Partitioning continuous wholes were only considered
in this activity as Boulet (1993) noted that in repartitioning discrete wholes that were
already partitioned the original partitioning would have been lost. There were three main
tasks with each task comprising of at least two subtasks. Task 1(a) required the students
to repartition a circle that was already partitioned in half (see Appendix B). It aimed to
evaluate how the participants would partition the circle into an even number of parts. The
students found this task rather easy as they drew the perpendicular bisector (‘cutting the
circle in the middle’) of the diameter of the circle. The remaining three subtasks then
asked the students to share the assumed circular pie among eight, sixteen and thirty-two
people. They were to show the shares for eight and sixteen while they were expected to
extrapolate the shares to thirty-two people. The students utilized the halving algorithm to
obtain the required number of equal shares. To share the pie for 32 people Carol
remarked that she “would draw one line through all the sixteen lines to get thirty-two.”
3
Richy started to use the division algorithm ( 8 32 ) to aid him in acquiring the
number of thirty-two equal shares. He recognized that his solution was incorrect, erased
it, and then confirmed that he could cut each eighth in four parts to get the desired shares.
There was clear evidence from the whole class that they knew that cutting the parts in
half would double the number of parts.
Karla:
I split all of the parts into halves.
Brian:
I would divide each piece in half.
111
To verify that they understood what they doing, the teacher extended the exercise to
sharing the pie already partitioned into thirty-two shares among sixty-four people. They
all used the doubling algorithm to get the required sixty-four shares.
The rectangle for Task 1(b) was pre-partitioned in three parts and the participants
had to further partition this rectangle into six and twelve parts respectively. This task
proved to be of little challenge to them with each student partitioning the rectangle using
one of the configurations shown in Figure 18. Bob was the only student who did the
partition shown in Figure 18(c). The dark lines represent the original partitioned lines.
They applied the halving algorithm recognizing that “if you double the 3 you will get 6”
(Josh). The students used their rulers to measure the sides of the rectangle and then
proceeded to do the required partitions. They applied the halving algorithm in
partitioning into twelve parts by drawing the vertical center lines for the three original
given parts in Figure 18(a) and for each created part in Figure 18(b). To complete his
twelve equal parts, Bob simply drew the diagonal for each of the three given parts.
(a)
(b)
Figure 18.
(c)
Six-Part Configurations for Task 1b – Activity 4
For the first part of the second task, the students were asked to share equally a
rectangular chocolate bar already partitioned into three parts for four persons. At first
this task caused a lot of perturbation as the students negotiated among themselves how
should this be done.
Jonah:
This is impossible.
Josh:
I don’t think it is impossible, but it is going to be hard to do.
Ben:
Can’t we just add another chocolate bar to it?
112
Dave:
That’s what I was going to do it [referring to Ben’s method.]
Kisha:
I find a way. I give everybody one piece of the larger part and
then break up one piece into four equal parts and give everybody
one piece of that.
Table 11.
Configurations for Task 2a – Activity 4
CONFIGURATONS
NAMES OF PARTICIPANTS
Richy
Jonah
Kisha
Josh, Brian
A
A
B
C
C
D
A
B
B
C
D
D
Ashley, Polly, Jay, Alton, Dahlia,
Mary, Bob, Marla, Claudia,
Chrissy, Paul, Ben
Dave
Carol
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Kisha’s verbal explanation of her solution started a chain reaction of solutions.
Richy chose to “divide the rectangle into 24 equal pieces. Each person gets four equal
parts but I have eight left and I give each person two more pieces.” Ashley partitioned
each of the three equal parts into four equal shares. She then assigned the shares as
shown in Table 11. A, B, C, and D represented the four people she shared the chocolate
bars with. Karla partitioned the rectangle similar to Ashley but chose to distribute the
parts in a different manner. She gave two pieces to each person than gave one piece each
of the last four parts to the four persons. Dave “divided each third into eight. It was 24
so 24 divided by four is equal to six pieces for each person.” Table 11 gives a summary
of the different configurations accompanied with the names of the participants who drew
that particular configuration. Of the other configurations not previously described, one
share of the larger piece along with one of the smaller piece was distributed to each
person.
Task 2(b) also requested the students to divide a circular whole already
partitioned in three to be divided into four equal shares. With the insight gained from the
previous subtask, this assignment posed little difficulties to the sixth-graders. 50% of the
participants divided the circle into twelve pieces by using the halving algorithm and
distributed three parts to each person [see Figure 19(a)] while 35% of them partitioned
the thirds in half and then divided the last third into four pieces distributing “one large
piece of the third and one of the small piece of the third to each person” [see Figure
19(b)]. They made much effort in ensuring that the parts are equal. Each person got a
half of the third and a small piece of the third. Richy partitioned his circle into twentyfour shares halving each part continuously until he got 24 parts and then distributed six
shares to each person. Polly chose a unique way to share her circular pie. She halved
each third and then halved one of the sixths in two of the original third. Each person
would get “half of a third and a little piece” as shown in Figure 19(c). Dave struggled
with the exercise as he “divided it into eight slices and each person gets two slices.” He
did not produce a diagram to match his statement.
114
(a)
(b)
Figure 19.
(c)
Common Configurations for Task 2b – Activity 4
The four subtasks of Task 3 sought to evaluate the sixth-graders’ ability to
repartition a continuous whole already partitioned in larger parts to smaller parts. For the
first task, the children would equally share a circular birthday cake already divided into
eight equal slices among four persons. This task proved to be a very simple one as they
easily recognized that each person would get two slices. Most of them applied the
division algorithm (eight divided by four equals two) while others used the counting
principle to arrive at their solution. Task 3b presented the birthday cake now cut in five
equal slices to be shared equally with 3 persons. Four different configurations emerged
from the students’ work.
1.
Partitioned each of the five parts into three parts producing fifteen parts. Each
person would get five slices.
2.
Partitioned two of the original parts into three parts each. Each person would get
one of the original parts with two of the smaller pieces.
3.
Partitioned two of the original parts into six parts each. Each person would get
one of the original parts with four of the smaller pieces.
4.
Partitioned four of the original parts into two pieces. The last original part would
be partitioned into three pieces. Each person would get three of the larger slices
and one piece of the smaller slice.
A rectangular whole previously divided in seven equal parts was used for Task 3c.
The students were to share this rectangular pizza among three persons. This task is
similar to the previous one with the exception that the continuous whole was now
rectangular. The participants were now comfortable in doing the partitions with some of
them looking for innovative ways to do their equal sharing. One of the most prominent
115
configuration was the one in which each person got two of the original slices plus one of
the slices from the last original slice which was divided into three equal slices. Ashley
chose to divide four of the original slices into twelve equal pieces and then gave each
person one of the large slices plus four pieces from the twelve slices:
Jay divided “the seven parts into 21 parts and gave each person seven pieces”:
Alton divided “the first piece into three pieces and make a long line across”:
Paul had a slightly different configuration than the ones above:
In Task 3(d) the students were asked to shade one person’s portion of a
rectangular cake already sliced in eight pieces if the cake should be shared with only two
persons.
C
C
C
D
D
D
D
C
Figure 20.
Karla’s Configuration of Task 3d – Activity 4
116
Three different methods of assigning the shares were observed from the students’ work.
Karla noted: “eight divided by two is four so each person would get four.” She then
proceeded to assign the shares as shown in Figure 20.The rest of the students had
configurations similar to
or
.
The final task for Activity 4 gave the participants the opportunity to partition an
L-shaped candy that was already striped into seven equal parts. The candy should be
shared equally among four people. This task was given to aid the students’ in their
practice of reconstituting larger shares from smaller shares. With the skills they acquired
in the previous tasks, the sixth-graders got creative in their partitioning. The most
popular strategy of sharing, however, had the students assigning one of the original
pieces of the candy to each of the four individuals and then dividing the remaining three
pieces into four equal parts each with every one getting three of those parts. The
partitioning of the seventh was done one of three ways, (a) using intersecting
perpendicular bisectors, (b) four vertical lines, or (c) drawing the two diagonals. The
distribution of three small parts varied with the students. Some distributed the three
smaller pieces from one of the original part with the remaining quarter counting with two
pieces from the next partitioned part to complete the three parts for the next person as
shown in Figure 21. Others assigned one of the quarters from each partitioned seventh to
each person. Below are some of the configurations that were employed in completing the
task.
(a)
(b)
(e)
(f)
Figure 21.
(c)
(d)
(g)
Configurations for Task 3d – Activity 4
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The individual assessment task for Activity 4 modeled the previous tasks
requiring the students to partition already partitioned continuous wholes. The first
exercise asked the students to repartition a parallelogram, already divided in halves, into
twelve pieces. The source of challenge for the students was the need to make sure the
lines were parallel and the same distance apart to produce the required equal parts. Some
of the sixth-graders drew five lines to cut the given horizontal to arrive at their solution.
Others did similar to Jonah who “put two lines going down so that made 6 pieces then on
the top I put a line across then I did the same thing on the bottom one.” He was referring
to the top of the given line.
The second exercise sought to assess their ability to repartition a smaller share
from a larger using a circular continuous whole already partitioned in halves. Dave chose
to divide his circle into “twelve equal pieces then gave each person four,” while Alton
“made it equal pieces by making an X and the two sides will have three pieces and each
person will get two.” Chrissy divided the cookie in four equal pieces by drawing the
perpendicular diameter and dividing one of the quarters into three equal pieces. She gave
each person “one of the quarter and a small piece.” It was evident from the peer
observer and teacher’s observation including the students’ drawings that care was taken
to ensure that the parts were equal. Bob made note of his method: “I measure the circle
to find the exact center then drew lines through it.” Erased marks could be seen all over
their diagrams.
For the final exercise the students shaded one child’s share of a rectangular
chocolate bar that was previously partitioned into six pieces which should now be shared
for four people. The sixth graders completed this task with relative ease with the
majority splitting two of the original slices into halves each and then assigned one of the
larger slices plus one of the smaller slices to each child. The children apparently recalled
what they did while performing similar tasks.
Analysis
The three activities that comprised the logico-physical abstraction level of
Herscovics and Bergeron’s (1988) model of understanding covered the breadth of the five
criteria related to this component. The responses given on the tasks coupled with the
results of the assessment tasks indicated that the sixth-graders were capable of
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reconstituting the whole from its parts, were aware of the equivalence of the part-whole
relationship in spite of a variation in the physical attributes of the whole, recognized the
equivalence of the part-whole relationship regardless of the physical transformations of
the whole, recognized the relationship between the size of the parts and the number of
equal shares, and were capable of repartitioning already partitioned whole.
The students readily figured whether the part-whole relationships were the same
or not but consistently noted that the sizes were not the same in relevant tasks. This
awareness fulfilled one of the criteria for this component of understanding. Physical
transformations of the part or differently shaped continuous wholes with the same partwhole relationship did very little in deterring their newfound concept as they negotiated
and agreed on their solutions. The students frequently and freely use fraction language
on most of the tasks. At this physical tier of the model of understanding the fraction
language was not required and was not encouraged. The students, possibly because of
the previous instructions they had in fractions, had no difficulty in expressing their
solutions using the fraction language. Besides the usual stopping at the group tables
and/or individual desks to make sure the students were on task or clarifying any
misunderstanding of the tasks, teacher interventions at this point were seldom.
Partitioning the circle and the parallelogram gave more challenge to the students’
than the rectangular whole. They were aware, however, of the importance of the equality
of the parts and made much effort to ensure that this was done using whatever they had in
their manipulative bag or just their imagination. Jay remarked that if he had a protractor
he would be able to make sure the parts in the circle are equal. When repartitioning an
already partitioned whole, the sixth graders displayed much excitement in finding
creative ways of doing the sharing as shown in some of the configurations above.
Logico-Mathematical Procedural Understanding
This is the first constituent part in the second tier of Herscovics and Bergeron’s
(1988) model of understanding. The quantification of the fraction is the main
consideration in the second tier instead of the part-whole relationship. One highlight of
this model is that it “extracts the mathematical from the physical in the analysis of a
particular mathematical concept” (Boulet, 1993, p. 165). As a reminder, at the logicomathematical procedural understanding the learner possesses explicit logico-
119
mathematical procedures that the learner can relate to the underlying preliminary physical
concepts and use appropriately. This takes into the account the ability to orally quantify
concretely presented part-whole relationships and to also generate part-whole
relationships from concrete representations of the fraction.
Activity 5 was designed to assess the sixth graders’ ability to orally quantify partwhole relationships depicted in all sorts of situations and to produce illustrations of orally
given unit fractions. This day’s session started off with the usual review session where
the students shared what they learned from the previous activities, followed by an
introduction to Activity 5. Based on previous misspellings and wordings used by the
participants in an effort to express part-whole relationships using the fraction language, it
became necessary for the teacher at this point in the study, to lead the students in a
discussion of the naming of unit and non-unit fractions, for example, one-twelfth and
two-fifths. This was followed by the tasks for Activity 5.
For the first task, the students were shown a set of diagrams on transparencies
depicting unit fractions. These included new diagrams as well as some that they had
worked with on previous tasks. Both discrete and continuous representations were
shown. They should determine what the part-whole relationships were, orally quantified
them and then wrote the words on their task sheet. The following is an excerpt of the
discussion that ensued after the first diagram [a parallelogram partitioned into sixths with
one-sixth shaded] was shown:
Mary:
One sixth.
Paul:
One-third [not agreeing with Mary].
I:
Give us a reason why you think it is one third, Paul.
Paul:
The top part has 3 with 1 shaded and the bottom part also has
three parts.
Dave and Mary partially agreed with Paul.
Dave:
I think it could be one-third. It is wrong but it does look like onethird.
Mary:
[She repeated Jeffrey’s explanation.]
All of us could be right - one sixth and one third because there are
six pieces that can be divided into threes.
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The teacher asked the class to identify the number of equal parts in the whole and the
number of shaded parts. Jay and Richy then quantified the part-whole relationship. To
show that they now understood how to orally quantify part-whole relationships, Paul and
Dave chose to give the answer “one-seventh” for the second diagram:
and then noted that they could not tell the part whole relationship for the fifth diagram
because the parts were not equal:
. The students never had any difficulty in naming
the unit fractions in the first task and the non-unit fractions presented in the next task.
This second task was very similar to the first one in that the students had to orally
quantify non-unit fractions. The class easily recognized that the diagram showing a circle
divided into 12 equal pieces with two of the pieces shaded represented the non-unit
fraction, two-twelfths. Marla said it could be “one-sixth.” If you recall from the excerpts
included from Activity 3, Marla had previous knowledge of reducing fractions.
The students were expected to say what the unit fraction was for each of the
diagram shown to them in the next task. The first diagram shown was a set of eight
diamonds with two of them shaded. Claudia said it could not be expressed as a unit
fraction. Others disagreed. Chrissy shared her reason for believing that two-eighths can
be expressed as the unit fraction – one-fourth.
Chrissy:
It’s eight and if you cut that in half it will be four and then the two
if you cut that in half it will be one [the two shaded regions]. So it
will be one-fourth.
Richy:
I divided by two.
To verify if the students got the concept, a diagram with ten discrete circles with two of
the circles shaded, was shown. The class agreed that the diagram represented the unit
fraction one-fifth. Jonah made up his own scenario to validate his understanding.
Jonah:
So, if you have six circles and two of them are shaded, that would
be one-third, right?
The class members gave their approval.
Jonah:
Oh, I get it.
Because the first two diagrams involved the familiar halving algorithm the next
diagram depicted twelve arrows with three of them shaded. More than half of the
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students shouted one-fourth. Ashley’s reasoned that “there are three rows of four that’s
twelve and each row has one shaded. So it is one-fourth.” The next diagram showed
fifteen heart-shaped discrete objects arranged horizontally with three of the hearts shaded.
Initially some of the students called the non-unit fraction name – three-fifteenths – and
then renamed the fraction as a unit fraction. Some students exhibited signs of confusion,
as they were not able to group in sets of twos or divide by two. This was an indication
that they never fully understood the solution given by some on the previous diagrams.
Polly:
Why is it one fifth?
Josh explained how he got one-fifth.
Josh:
Three times five equals fifteen. Then I give one little heart to each
little five.
Polly:
Oh I get that. [She proceeded to explain to her tablemates who
were puzzled at Josh’s solution.] I rearranged the hearts to make
it three rows of five.
Claudia:
O I get it.
I:
Explain what you got.
Claudia:
You group them in fives and give each group one.
Dave shared his method with the class.
Dave:
I divided it. I divided fifteen by three and it comes out to be five.
The remaining discrete model diagrams were no challenge for the students so diagrams
depicting continuous wholes were introduced. The first continuous whole was a circle
partitioned into sixteen parts with four shaded.
Jay:
One-eighth
Chrissy:
How did you get one-eighth?
Jay:
I mean one-fourth.
I:
How did you get one-fourth?
Jay:
If you divide 4-sixteenths by 2 you get 2-eighths and then you
simplify again and it comes out as one-fourth
Brian:
[Interrupting Jay] There are sixteen equal parts and if you divide it
by four you get four and the four shaded ones, if you divide it by
four it equals one.
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The last diagram showed a rectangular whole partitioned into seven parts with two
shaded parts. The sixth graders readily expressed that they could not say the unit fraction
for the shaded part as they could not find any common number that two and seven are
divisible by.
The final task of Activity 5 sought to examine how the students would produce
concrete illustrations of orally given fractions. They were advised to use any of the
manipulative in their bag or draw pictures to represent the fractions. The three fractions
were one-half, one-sixth and one-fifth. 5 out of 20 students used the discrete model to
illustrate the three fractions. Each of these students formed the fractions using the chips
and then reproduced the model using paper and pencil. Bob portrayed his fractions as an
equivalent fraction of the particular unit fraction, that is, for one-half he drew six circles
including three shaded ones. Two of the students (Claudia and Ashley) represented at
least one of the unit fractions as a discrete model. The remaining thirteen students
utilized the continuous model (circle, rectangle, L, number line) to represent the
fractions. The students used their rulers to ensure equality of the parts. Richy and Jay
were the only students who used the number line to illustrate at least one of the fractions.
Figure 22 shows one of Richy’s representations.
Figure 22.
Richy’s Number Line Representation of One-fifth
As some confusion was evident at the start of Activity 5, the assessment task was
designed to determine whether or not the intended objectives of the activity were
accomplished. Were the sixth-graders capable of orally quantifying unit and non-unit
fractions? Can they reproduce the fraction given an oral representation of the fraction?
The paper and pencil assessment task was made up of four exercises. The students were
given the opportunity to work with their groups as they orally quantified the part-whole
relationships shown. The first exercise was the simple task of determining whether or not
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the diagram shown represented one-half. The students readily recognized that the parts
were not equal and unanimously said so. They were then required to show one-half
whichever way they chose to. This was done with relative ease. Some students (such as
Karla, Dahlia, Jay, and Richy) used two red and yellow chips with a chip showing one
color and the other chip showing the next color. Josh, Dave, Paul, and Mary used more
than two chips to successfully represent one-half. Carol, Ashley and Marla used other
manipulatives such as the fraction circle, fraction squares or fraction triangles to illustrate
the fraction. The remaining students drew continuous models splitting them in halves to
depict the required fraction.
Exercise 2 had three subparts where the students had to identify parts of the whole
and verbalized the solutions. The students immediately recognized the part-whole
relationships in the first two. The third subpart proved a bit tricky for the students. The
scenario presented a patio in the form of a half circle. The students were expected to
partition the patio in two and then determine whether the patio was one-half or a whole.
A number of students drew a semi-circle to complete the circle, thus determining that
Mary was right in saying that the patio was really one-half and not a whole. After Ben
read the question, the group members unanimously said ‘one-half’ for the first part of
question. Brian reread the last part of the task. Here is an excerpt of the conversation
within the group and across groups as they negotiated whether Mary was right or wrong.
Polly:
You’re supposed to cut this half into half again.
Ben:
She is right.
Brian:
Me, too.
Polly:
I think she is right also… because a whole [completing the circle]
would be the whole circle.”
Marla:
I don’t think she is right because it is a whole. That’s the shape of
his patio.
Polly:
You maybe right.
Marla:
It’s not half of the patio, that’s the size, that’s the shape.
Polly:
Okay, I think Marla is right.
Marla:
It may be shaped like this but that doesn’t mean it is a half.
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Polly:
So I would tell her that she is wrong ‘cause that’s the shape, even
though it looks like half a circle.
Dave:
It’s half of a circle.
Chrissy:
But the picture shows half of a whole.
Paul:
That could be the whole.
Chrissy:
How can half of a circle be a whole?
I:
Did the question say the patio is a circle?
Marla:
No, they said the patio looks like that. It looks like half of a circle.
It did not say it is a whole circle.
Mary:
It can’t be a whole because obviously they don’t show a whole
there.
Polly & Marla:
Brian:
Yes they do.
It is really weird to see a patio looking like a full circle anyway.
Marla repeated her strategy to the class in an effort to convince them that the whole was
not a circle but was the shape that was shown on the paper. To confirm that she
understood what Marla was explaining Dahlia used a model square, cut off a part of it,
resulting in a rectangle, which she said, was a “whole in itself.” She mentioned that any
object whatever the shape or size could be considered a whole. Consequently, majority of
the class agreed with Marla. Chrissy, Alton and Ashley reluctantly went along with the
majority thus confirming the assumption that most students got stereotyped into believing
the whole must either be a circle or a rectangle.
The third exercise required the students to make concrete representations of onehalf, two-thirds and one-fifth. The students had no difficulty in completing this exercise
as they used both continuous and discrete models to illustrate the fractions. Mary used
circles to represent discrete wholes as well as continuous wholes. The frequency of the
usage of discrete representations increased in comparison to the related tasks in Activity
5. Careful consideration of the equality of parts was given while showing the part-whole
relationships using continuous models. The final task asked the students to identify and
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say the unit fraction related to a set of discrete objects – ten cookies with two shaded.
The ease at which the students did this exercise bore evidence to the fact that they were
comfortable in identifying unit and non-fractions in discrete wholes. Chrissy expressed
the answer shared by all the students: “one-fifth or two tenths.”
Analysis
The main purpose of Activity 5 and the accompanying assessment task was to
reveal and provide opportunities where the students can orally quantify part-whole
relationships and also make concrete representations of part-whole relationships from
orally given fractions. It is evident from the responses shared above that the sixthgraders demonstrated logico-mathematical procedural understanding of unit and non-unit
fractions. The major hurdles encountered in this activity were a) renaming a non-unit
fraction as a unit fraction where possible and b) the assumption by some students that a
semicircle could not be a whole. The students were able to discuss, negotiate and agree
on solutions. They seemed to work comfortably with discrete and continuous models.
Based on the responses to the different tasks, the students were capable of extracting the
mathematical from the physical in the quantification of the unit and non-unit fraction.
The responses corroborate with the possession of logico-mathematical procedural
understanding where the learner possesses explicit logico-mathematical procedures that
he/she can relate to the underlying preliminary physical concepts and use appropriately.
Logico-Mathematical Abstraction/Formalization
The logico-mathematical abstraction constituent part of the second tier of
Herscovics and Bergeron’s (1988) model of understanding is analogous to the logicophysical abstraction level in the first tier. The criteria for this component is similar to
those of the logico-physical abstraction level in that the main objective is to determine the
scope of the child’s ability to comprehend and understand the invariance, reversibility
and generalization aspects of unit and non-unit fractions. The main difference between
both components was the fact that at the logic-mathematical abstraction level, the child
was able to quantify the part-whole relationships connected to the physical attribute of
the fraction. The use of the fraction language was expected and encouraged as students
displayed the ability to obtain unit fractions from already partitioned wholes,
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mathematically reconstitute wholes from unit fractions, determine the size of unit and
non-unit fractions, quantify equivalent fractions and order fractions that are less than one.
Formalization is the final constituent part of the Herscovics and Bergeron (1988)
model of understanding. Besides the formal meaning of formalization (discovery of
axioms and using these axioms to justify mathematical arguments), this component
embodies the use of mathematical symbolization for concepts for which the previous
levels and/or constituent parts hane already established. As noted in Chapter 2, this
model of understanding does not necessarily function in a linear fashion yet as Boulet
(1993) noted “the connections among the components are by no means arbitrary. Some
components are necessary for other components” (p. 232). For the purpose of this study,
the first four levels and constituent part were used as the prerequisites for formalization.
Logico-mathematical abstraction and formalization will be considered simultaneously in
this section of the paper. This decision was based on the pervading presence of the
fraction symbol in the responses of the participants ever since the start of the study and
the relative ease at which they were able to orally quantify unit and non-unit fractions
thus confirming logico-mathematical procedural understanding.
With this in mind, Activity 6 was designed to aid the students in quantifying oral
and written part-whole relationships in symbolic form. Paper folding was introduced as a
way to create fractions from continuous wholes. The students were to orally quantify the
fractions modeled and generate convention symbols for these fractions. Activities 7 -10
included tasks relevant to both logico-mathematical abstraction and formalization
constituent parts. For example, in tasks that related to the size of fractions, the students
were expected to orally compare and order given fractions and then order unit and nonunit fractions given in symbolic form. Numerous tasks from RNP were used in the effort
to assess the students for the last two components of understanding in Herscovics and
Bergeron’s (1988) model. References to the relevant RNP tasks are noted in Appendix
B.
Prior to the students working on Activity 6 the students were asked to give their
7
understanding of the fraction . Activity 6 formally introduced the students to fraction
8
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symbols so it was pertinent for the teacher to know what understanding the students had
of these symbols.
Ashley:
Seven is how much is shaded and the eight is the number of equal
parts in the whole.
The other sixth-graders agreed with her. After working with two more fractions the
group then went through the exercise of creating fractions via paper folding. This was
done with the entire class as a means of providing one more strategy that they could use
to decipher fraction problem-tasks. Besides, the art of paper folding provides an
innovative way of generating equivalent fractions. The students used triangular,
rectangular and circular shaped papers to form fractions such as
1 1 3
2
, , , and . The
2 4 4
5
rectangular-shaped papers were among the favorite for the students who were free to
choose which of the shapes they wanted to work with.
2
gave them the most challenge
5
as they could not successfully apply the halving algorithm that they used in creating the
half and the fourths. The teacher modeled how to create sevenths and then allowed the
students to create the desired fifths. There were required to write the fraction name and
the fraction symbol for the fraction part that was shaded. Karla noted that she had to
make her half before she could make the one-fourth.
Kisha said that “it is easier to
make the fractions on the paper because it’s easier to make equal parts.”
The task sheet used in Activity 6 was copied from RNP’s level one activities
designed for middle school students. Lesson 5 – Student Pages A – E were assigned with
the main purpose of investigating the students’ understanding of three different
representations of a fraction, namely, fraction name, fraction symbol and a pictorial
representation. For the first task on Page A the sixth-graders had to express given
fraction symbols in words. The next two tasks reversed the instruction with each of the
tasks expressing the fraction name in two different formats. For example, item 2a had “3
of 5 equal-size parts are shaded,” while 3a used the format “9-tenths.” All the students
were able to complete the tasks with no apparent difficulty. On items 4 and 5, the
students were expected to name the fraction that was depicted in the narrative and then
draw a picture to illustrate the fraction. Instead of writing a fraction to tell how much is
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shaded in all, Alton simply wrote the number of parts that were shaded, for example, “3
are shaded” in response to item 4 which asked them to tell what fraction was represented
if “three
1
parts are shaded.” His pictorial representation showed three separate
4
rectangular wholes with one-fourth shaded in each whole. He did similarly on item 5.
1
Figure 23 shows Alton’s picture of four parts. He used his ruler to make equal parts.
5
He read two-fifths instead of one-fifth.
Figure 23.
1
Alton’s Picture of Four Parts
5
Dahlia ignored the words ‘three’ and ‘four’ in items 4 and 5 respectively thus drawing
pictures depicting
1
1
and . She repeatedly used the phrase ‘the top number’ in her
4
5
response to the fraction representing the total shaded parts in each item. Alton and
Dahlia blamed misreading and misunderstanding the questions for the responses they
initially gave for items 4 and 5.
1 1
Ben used mixed number fraction symbols ( 3 ,4 ) to name the fractions that
4 5
represented the shaded parts. He drew a blank circular whole partitioned into four for
item 4 and a blank rectangular whole partitioned into fifths for item 5. He was unsure
what the whole numbers (three and five) meant in front of the fraction and how they
related to the number of shaded parts. When questioned, he said he was confused. He
could not remember what he did in earlier grades and cannot recall the difference
1
3
between 3 and . No intervention was done as mixed numbers and improper fractions
4
4
were outside the scope of the study. Instead, the teacher-researcher relied on the
129
remaining task-based activities in the study to aid in the clearance of this misconception.
Ben defined
3
as “three parts shaded out of four equal parts.” The responses to the
4
remaining tasks indicated that the students including Ben were successful in representing
fractions in three different ways. On Lesson 5 Student Page C the students matched
pictorial representation of a fraction with its symbol or word name. Page D asked them
to shade each continuous circle to show the designated fractional amount while on Page E
they were required to write the word name and symbols for the shaded part of shown in
the rectangular whole.
Four simple assessment tasks were used to assess the students’ use of the symbols
associated with the fraction notation. These tasks were done with relative ease
indicating that the sixth-graders were capable of linking pictorial representations of a
fraction with its fraction name and fraction symbol. Task 2 depicted eight discrete circles
as the whole with four shaded ones. Marla gave both the general fraction name (4eighths) accompanied with its symbol (
symbol (
4
) and the unit fraction name (1-half) and
8
1
) associated with the diagram. The responses to these tasks including the ones
2
done during Activity 6 indicated that the students possessed a level of competence in
working with fraction symbols coupled with the meaning attached to these symbols. The
remaining activities will continue to assess the students’ formal understanding as they
worked to generate equivalent fractions and order symbolic unit and non-unit fractions.
Activity 7 was done over a period of two sessions. The main objectives of this
activity were to have the participants determine the number of unit fractions needed to
reconstitute the whole (continuous or discrete) and to construct given unit and non-unit
fractions from preset discrete wholes. In Task 1 the sixth-graders were given four unit
fractions to determine how much of each would compose the whole. If you recall,
similar tasks were done at the logico-physical abstraction level. At the logicomathematical abstraction constituent part, however, the students were expected to use the
underlying physical concept garnered at the parallel level to quantify (orally and written)
the situations under consideration. Some of the sixth-graders (e.g. Jay, Marla, Bob,
Dave, Josh and Polly) were already exhibiting logico-mathematical understanding and
130
these tasks posed no problem to them. Others such as Ashley, Carol, Claudia, Brian and
Mary showed signs of growth in the understanding of basic fraction concepts. For
instance, Brian who earlier had written
6
to indicate four more pieces of sixth was able to
4
complete Activity 6 and its accompanied assessment task and Activity 7 with little
difficulty.
All the students were able to determine that “four fourths are needed to make the
whole” (Task1a). They used the quarter pieces of the fraction circle to make the whole.
They easily recognized that seven pieces of one-seventh (Task 1b) and six pieces of onesixth (Task 1d) were needed to make the whole. When asked how many
1
were needed
5
to complete the whole (Task 1c), Chrissy and Dahlia answered confidently that “seven
more pieces are needed.” A pictorial representation of
1
(
5
) was given. This
apparently posed a possible distracter for a number of the students whose first initial step
to solve the problem was to construct the complete circle. Chrissy and Dahlia drew
similar pieces of the given one-fifth to make a complete circle thus resulting in their
answer. Others erased their drawing when the number of pieces they drew conflicted
with their desired answer ‘five.’ The misunderstanding that the whole must be a circle if
the given part was a portion of a circle was evident in Task 2c of the Assessment Task for
Activity 5. Dave reminded the class that the whole could be of any form. Jay, Josh,
Jonah and Karla were the only students who did not attempt to complete the circle or
draw any form of diagram to answer the question. Karla seemed to have discovered a
pattern to find out how many of the unit fractions would make the whole: “one-fifth
would need five, one-sixth would need six and one-twelfth would need twelve.” An
obvious highlight of this task was the students’ awareness that the parts had to be equal.
Task 2a - 2c ended the first session for Activity 7. For the first two parts of the
task, the students were to use their chips to model the given wholes and then used the red
side of the chips to indicate the given fraction. The teacher with Claudia’s help, modeled
three-fifths on transparency at the beginning of the session. The students modeled the
fraction using their chips and moved on to Task 2a, which asked them to show threefourths using twelve chips. Alton, Brian and Ashley initially formed four rows and
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shaded three chips in each row to represent three-fourths. Paul explained to Alton that he
was only showing one-fourth. He argued that he was showing three-fourths because he
had four rows and shaded three in each row. Brian and Ashley gave similar reasons for
their configuration.
Brian:
I have four rows and three shaded ones.
These students were able to identify the unit fraction (one-fourth) from their
configurations but argued that there were three one-fourths thus resulting in three-fourths.
Bob showed Ashley his model; she looked at hers and then exclaimed: “Oh, the three
means I should shade three groups out of the four groups and that would be threefourths. I get it.” She changed her configuration (see Table 12).
Carol was unable to use her chips to display the solution she had in mind. She
decided to draw two rows of six and then shaded the first three circles in each row. She
then split the next circle in each row into halves as shown below.
Karla tried to ‘correct’ her but she was adamant that she had three-fourths. When
interviewed she pointed to the three shaded ones to represent the ‘three’ in the fraction
and the three unshaded ones to represent the ‘fourth’. The halves were counted as two to
complete the ‘fourths’. The second row was a duplication of the first row. Carol
displayed a common error noted by Ashlock (2002) where students equated the number
of shaded parts to the numerator and the number of unshaded parts to the denominator.
This is usually the case in continuous wholes but Carol used discrete whole to display a
similar misconception. She was able to do a three-fourths’ configuration after working
with her in a one-on-one session. In this session the teacher used a modified version of
one of Ashlock’s (2002) strategies to aid Carol in clearing up the misconception. In this
strategy – “start with the whole” (Ashlock, 2002, pg 153), Carol was asked to identify the
whole. She quickly grouped all 12 chips as the whole. After explaining her
understanding of the meaning of three-fourths – four chips with three shaded, she was
prompted to make the three-fourths. Reminding herself that the ‘fourths’ mean four
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equal parts she took four chips and created three-fourths; she did the same with the next
set of four chips and the next set until there were no more chips to work with (see Table
12). Kisha and Karla did a similar configuration.
Table 12.
Configurations for Three-Fourths – Task 2a Activity 7
CONFIGURATIONS
NAMES
Dahlia
Carol, Karla, Kisha
Jonah, Jay, Richy, Polly, Mary, Alton
Marla, Bob, Claudia,
Chrissy, Paul, Dave, Josh, Brian
Ashley
The remaining set of students focused on the four equal parts by creating four
groups first and then shaded three parts in each of the group to successfully representing
three-fourths using twelve chips. A summary of the different configurations
accompanied with the names of students are shown in Table 12. Ben never completed
Task 2. He was not in the mood to work that day.
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The students experienced no difficulty in completing Task 2b and 2c. However,
for Task 2b which required the students to model two-sixths with six chips some of the
students including Alton, Ashley and Dahlia were not sure how to group the chips –
should all six chips be laid side by side or should they be grouped in two rows? They
were not able to resolve those issues but decided on shading only two of the chips no
matter the configuration.
Alton:
I have three rows with two chips in each row or six chips with two
rows of chip shaded.
Some students resorted to drawing discrete or continuous models to ‘describe’
the steps they would take to show
3
(Task 2c) while others used words “I’ll get 7 chips
7
and shade in 3 chips,” (Alton), “I would draw seven equal parts and shade in three”
(Paul). They were all successful in completing this task.
Before the students completed the final part of Activity 7 in the next session they
had to do an assessment task based on first two tasks done in the previous session. The
students were able to complete the paper and pencil work within minutes of getting the
work. Majority of them were able to arrive at a solution without the aid of the chips.
The second part of Activity 7 contained problems that were used to verify the
students’ understanding of making concrete representation of discrete wholes given parts
of wholes. The students worked in groups and were encouraged to verbally discuss the
problems before writing their solution on paper. As all ten questions (RNP – Lesson 6 Student Pages A and B – Level 2) were based on the same principle only the responses to
two of the questions (Items 3 and 8) will be considered. These questions were chosen
based on the level of difficulty the students experienced while working on them.
One of the criteria necessary to complete the unit is the understanding wrapped up
in the meaning of the numerator and denominator of a fraction. Based on the previous
assessment task, the students seemed to have a basic understanding of the meaning of
both – the top number stands for the number of shaded parts while the bottom number
represents the total number of equal parts. Their first step in solving the different tasks
134
was to put the circles in groups of twos. This worked fine for the first two tasks. Item 3
(see Figure 24) caused some perturbation. Brian, Richy, Alton and Carol exhibited signs
of confusion while working on this item revealing the need for understanding the
underlying principles involved in the meaning of both the denominator and numerator of
a fraction other than the meaning mentioned above.
Figure 24.
Item 3 – RNP Lesson 6 – Activity 7
Alton:
That’s impossible. There’s one left over [after grouping in twos].
I:
How many groups should you have in all?
Alton:
3, no 5.
I:
How many groups do you have there?
Alton:
4 with one left over.
I:
What does the three mean?
Alton:
How many circles are shaded.
I:
If you shade 3 circles here, will you have three-fifths?
Alton:
No.
I:
You said you should have 5 groups in all because of the five at the
bottom. Since shading 3 circles did not give you your answer, do
you have any other idea for the three/
Alton:
Uhm, no.
I:
Can you get three equal groups here?
Alton:
Yes. [He regrouped the circles to get three groups of threes.]
135
I:
How many groups do you need in all?
Alton:
5, so I will make two more groups of three. [He completed the
whole.]
I:
Very good. I’ll come back to you later.
Similar conversations were held with Brian, Richy and Carol.
Figure 25.
Item 8 – RNP – Lesson 6 – Activity 7
Alton was still a little unsure of what do with item 8 (see Figure 25). He
interpreted the numerator two to mean groups of twos instead of two equal groups of
fours. Seven students (Richy, Carol, Jonah, Ben, Claudia, Brian and Kisha) had
difficulty with item 8. Two features about the problem caused the difficulty the students
faced:
1.
There was an even number of x’s so the students could easily group in twos.
2.
The two in the numerator enforced their misconception.
Dave shared his method of knowing how many discrete objects to group together: “I
count how many x’s there are and then divide by the top number.” Jay and Paul agreed
with him. The students regrouped using Dave’s strategy.
Due to the recurring misconception a second assessment task for Activity 7 was
administered. The students (except Kisha) readily determined that nine one-ninths and
13 one-thirteenths were needed to complete the wholes. The teacher worked with Kisha
in a follow-up interview that took place after the verification task. Item 3 was designed to
136
evaluate the students’ understanding of making the whole (continuous and discrete) given
parts of the whole. For the first subpart of Item 3 the students were required to fold a
square sheet of paper into eight equal parts and shade three-fourths while for the second
part the students folded a rectangular sheet of paper into nine equal parts and then
1
shade of the parts. A deliberate attempt was made to use the fraction word name and
3
the fraction symbol in Items 3a and 3b respectively. This was to verify whether the
students had a working understanding of both forms of fraction representation as these
were necessary conditions for the last two constituent parts of the Hercovics and
Bergeron (1988) model of understanding.
All the students were able to accurately partition the square paper using one of
two sets of configurations as shown in Figure 26(a) and 26(b). There were some students
who folded their paper similar to Figure 26(b) but chose to shade the parts differently
[see Figure 26(c)]. The shaded display shown in Figure 26(c) verified that the students
were able to identify and use the equivalence of the part-whole relationship despite the
physical transformation of the parts in the whole.
(a)
(b)
Figure 26.
(c)
Students’ Configurations for Second Assessment Task (Item 3a) – Activity 7
Item 3b produced similar results as the previous one with the exception of Brian
who shaded one of the parts (one-ninth) to represent one-third. Initially, Brian exhibited
lack of interest in the tasks but recognizing that his classmates were deeply involved in
137
their work he decided to spend more time on his tasks. He was able to accurately show
the required one-third. Two different configurations (shown in Figure 27) were observed
as the children folded the rectangle. Figure 27(b) and 27(c) show the different patterns of
shading using the same configuration. Figure 27(b) depicts three equal groups of threes
with one of the groups shaded while Figure 27(c) shows one-third shaded in each group
of threes.
(a)
Figure 27.
(b)
(c)
Configurations for Second Assessment Task (Item 3b) – Activity 7
The remaining three tasks of Item 3 required the students to reconstitute discrete wholes
given parts of the wholes. Item 3c required the students to draw the whole if ☺☺ is
1
of
4
whole. Interestedly, Chrissy and Kisha drew a rectangle and circle respectively to
represent the whole ignoring the smiling circles but possibly revealing that they knew
that four fourths were needed to complete the whole. The remaining students easily did
this item possibly because the part of the whole was already in a pair. Although item 3d
was a duplicate of Item 8 in Activity 7 similar misconceptions prevailed necessitating a
follow-up interview with Dahlia, Karla, Kisha and Carol. All the students were able to
give an ‘accurate’ diagram of item 3e.
Dahlia, Karla, Kisha and Carol were interviewed an hour after they completed the
second assessment task for Activity 7. Three tasks were designed as follow-up tasks.
1
The first task required them to draw the whole if ☺☺☺☺ represented a of whole.
4
I:
What is the meaning of the four in the fraction, anyone can answer.
Carol:
It means how many groups.
138
I:
What if I make four groups like this [I made four groups with each
group having different number of chips], could that be the four
groups in the fraction.
Dahlia:
No.
I:
Why not?
Dahlia:
The groups are not equal. The others agreed with her.
I:
You are all saying the four groups must be equal.
They nodded in agreement.
I:
What does the 1 mean?
Karla:
How many groups are supposed to be shaded.
I:
The faces are not shaded so tell me how many of them should be in
the group.
Kisha:
Four.
I:
Why?
Kisha:
‘Cause it says one group.
I:
So the one means that all of this is one group [reiterating what
Kisha said.] Go ahead and complete the task and then we will
continue to talk about it.
They worked on the task individually. Carol and Karla started off by grouping in twos
thus creating two groups of twos.
I:
How many groups have you formed?
Carol and Karla:
Two.
I:
How many groups should you have?
Carol:
Uhm, one.
Karla:
[Ignoring the question] We don’t need three more groups only two
more.
Dahlia:
The happy faces in the problem are counted as one group.
Carol and Karla corrected their work to reflect the four equal groups of four. They
moved onto the second task where they were expected to create the whole if 10 x’s
139
represented
2
of the whole. They were advised to share their response orally before they
7
began to draw the whole. This time it was Dahlia who grouped in twos and added two
additional groups of twos to complete the whole.
I:
How many groups did you form?
Dahlia:
Five.
I:
So what does the 2 in the numerator mean?
Dahlia:
Two equal groups.
I:
Did you form two equal groups?
Dahlia:
No. I get it. I will group by fives to form two equal groups.
She redrew her groups of chips and then successfully completed the last task. The other
three students also completed the last two tasks with relative ease. They were then given
the second verification task of Activity 7 to redo Items 3c, 3d and 3e, which they did
successfully. Kisha worked on Items 1 and 2 which she had previously done incorrectly.
Using commercial plastic circular wholes partitioned into fixed number of parts such as
tenths, fourths and sevenths, Kisha was asked to count the number of tenths that were in
the whole. She did the same for the next two wholes. She discovered the pattern and
then made the changes to her activity sheet.
Activity 8 was designed to determine whether the sixth-graders could order unit
fractions according to their size. The first task of this activity required the students to
paper fold and shade one-half and one-quarter on separate sheets of rectangular wholes
(same size). Using the fractions they created they were able to determine that one-half is
bigger than one-fourth. Jay and Josh said they knew that because they were taught that in
fourth grade.
Bob:
1
1
1
1
is bigger than because is half of a .
2
4
4
2
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The students were then asked to determine if one-third was larger than one-fifth.
Dave:
One-third is larger than one-fifth because dividing a circle into 3
parts gives a bigger piece than dividing it into 5 parts.
Brian:
I don’t get it. One-third is smaller than one-fifth.
I:
Why do you think so?
Brian:
Because 5 is a higher number than 3.
Jonah:
[Butting in] But you said half is bigger than one-fourth and the
four is bigger than two.
Brian:
I know that one-half is bigger than one-fourth because one-fourth
is a half of a half [repeating what Bob said earlier.]
I:
Look at the one- half and the one-fourth you made. Which one is
bigger?
Brian:
One-half is bigger. It has a bigger piece shaded.
I:
Use the pieces on the Fraction Stack Bar and try to find out which
fraction is bigger, one-third or one-fifth.
Brian looked at the two pieces on the Fraction Stack Bar and then decided that one-third
is larger than one-fifth because “a third has a bigger piece than one-fifth.”
The entire group of students was able to determine that one-seventh is larger than
one-eleventh. As Josh put it:
I would prefer to share my pizza with seven people than 11 because I
would get a bigger piece.
The students were able to order the fraction according to the instructions with very little
problem. Most of them applied Josh’s method in determining the relative size of the
fraction in comparison to other fractions.
In Task 2 the students were to help Tyra decide which table she should sit if she
wanted to get the most pizza to eat – the table with two friends or the table with three
friends. The entire group decided for one reason or the other that Tyra should sit at the
table with the two friends. Here are some excerpts of the conversations that accompanied
the task:
Marla:
With the group of two because since there are only two people
that’s less people to share with.
141
Richy:
2 because 2 is bigger than 3.
Karla:
At table 2 with the two friends ‘cause she would get a bigger piece.
Jay:
1
1
The table with 2 friends because is greater than .
3
4
Polly:
The group with 2 people because her slice is larger.
They drew pictures of two circles divided into thirds and fourths to verify their decision.
Marla was the only one who showed a method of verifying that the thirds are equal.
The last two tasks (RNP – Lesson 6 Student Pages B and C – Level One) of this
activity had the students identifying larger unit and non-unit fractions from given pairs of
fractions. The students were able to use Fraction Stack Bar, Fraction circles, squares, and
triangles in their comparison. They reasoned aloud in their groups. The excerpts below
represent some of the rationale the students used in determining their solutions to Student
Page B.
Ben:
2/3 is larger than 1/3 because there are two pieces of 1/3. So…
2/3 has to be larger than 1/3.
His group members agreed.
Dave:
4/5 is bigger is bigger than 3/5, the parts are equal shapes, there is
just one more in 4/5 therefore 4/5 is bigger.
Jay:
Ben:
I say 6/7 is bigger than 2/7. Once again the parts have the same
shape and size but 6 is more than 2.
11/12 is bigger. You have more twelfths in this fraction.
Paul:
2/9 is bigger than 2/7.
Jay:
I don’t think so: 2/7 is bigger. It’s like sharing a pizza. You get
more to a slice when you have to share for 7 people instead of 9.
You’re going to have 2 bigger sevenths than 2 ninths.
Dave:
I’m guessing 2/7 ‘cause for 2/9 you are breaking up into more
pieces which are smaller.
They proceeded to identify the others using the methods they discussed above with the
aid of fraction circles. The last two problems posed no challenge to them. The other
students used one or more of the strategies shared in Ben’s group to decide which of the
142
fraction was larger in each question. Claudia initially said that “two-ninths is a larger
fraction than two-sevenths because the ninths give more pieces,” but Bob reminded her
that the “sevenths are larger than the ninths.” Chrissy explained to Ashley why
larger than
9
is
10
9
:
100
If you divide something in one hundred pieces you get very small pieces
than if you divide into ten pieces. So if you shade 9 of the ten pieces and 9
of the hundred pieces you will get more with the ten pieces.
There were four items on Task 4 - Student Page C. Because of the relative ease at which
the sixth-graders completed this page, only the responses to items 4 and 5 will be
considered. The students all agreed that Matthew ate more candy than Cassandra because
“two-thirds is larger than two-fifths” (item 4). In item 5 the students had to determine
whether it is possible that Ellen spent more money than Andrew if she spent one-third of
her allowance on a movie and Andrew spent one-half of his allowance on candy. Dave’s
group assumed an allowance amount.
Dave:
I think Andrew spends more. Let’s say the allowance is $6.00, she
gets $2 and he gets $3.00. It’s impossible.
Jay:
What if the allowance is $45.00. Then Ellen gets $15.00. And then
he gets $23.50 (excited). He spent $23.50 on candies. He got a lot
of cavities.
Dave:
She spent $15 on movie. She’s smart. I would prefer to go to the
movie than spend my money on candies.
Paul:
It’s impossible. Andrew will spend more.
Josh:
Andrew spent more money.
Bob, Jonah and Claudia reasoned that Ellen “may have more money because she may
have a bigger allowance.” Carol believed that the possibility that Ellen spent more than
Andrew depended on the amount of money she had in comparison to Andrew. With the
143
exception of Dave’s group, the students believed that Ellen could have spent more
money.
RNP’s Lesson 7 Student Pages A and B were used as the individual assessment
task for Activity 8. The students (except Brian) experienced little difficulty in
completing Pages A and B which required the students to determine the larger of two
fractions. Brian continued to choose the larger number in the denominators to be the
reference point thus he chose one-third to be greater than one-half and one-twelfth to be
larger than one-half. The teacher used paper strips, Fraction Stack bar and fraction
circles to help in clearing up his misconception. At this point Brian clearly displayed a
dependency on the physical aspects of the fraction to aid him in making crucial decision
relating to the tasks at hand.
Both the tasks on Activity 8 and the relevant assessment task revealed that the
students possessed the ability to order symbolic and non-symbolic fractions. The
students used various manipulatives and mental image to order the given fraction (orally
and/or written). Their success at completing the tasks showed that they possessed one of
the key criteria for logico-mathematical abstraction understanding. Activity 10 contained
more challenging tasks on ordering symbolic fractions with the aim of assessing students’
formal understanding in a more specific way.
Tasks 1, 2 and 3 of Activity 9 had the students discovering equivalent fractions
via paper folding. Equivalent fractions of one-half (Task 1) were created by folding the
paper in two equal parts at each turn. The students wrote the equivalent fractions that
they generated and were advised to write three more equivalent fractions without folding
the paper. The students easily recognized the pattern:
Mary:
1 2 4 8 16 32 64
.
= = =
=
=
=
2 4 8 16 32 64 128
We just need to multiply the top number and the bottom number by
two.
The next two tasks allowed the students to generate two equivalent families of onefourth. The sixth-graders never encountered any difficulty when determining the first
three equivalent fractions from their folded paper. They easily recognized that they can
obtain the required fractions by folding the quarter in half and third respectively. It took
quite a time before Claudia recognized that she could “multiply both numbers in the
144
fraction by two to get the next one” (Task 2). The students’ responses revealed that they
were capable of forming equivalent fractions without the use of manipulatives. As
verification, the students were asked to generate five equivalent fractions for two-thirds.
All the students generated equivalents fractions derived from multiplying two-thirds by
two. When asked if six-ninths was an equivalent fraction of two-thirds, most of the
student hesitated to given an oral answer.
Polly:
No, because you cannot divide nine by two.
Mary:
I don’t think so. If you multiply the numbers by two you will not
get six-ninths.
Dave:
It is. It’s like you take the paper with two-thirds and fold into
thirds.
Chrissy:
If you multiply the numbers by three you will get six-ninths so I
guess they are the same.
The teacher drew nine circles with three in each row on the chalkboard. Two circles in
each row were shaded.
I:
What fraction is represented here?
Alton:
Six-ninths.
Most of the students agreed with him.
I:
Very good. Is there another fraction that could be used to describe
this same diagram?
Marla:
Yes. Two-thirds
I:
Does anyone agree with her?
Dave, Jay, Paul, and Chrissy agreed with her. Marla reminded them of the previous tasks
(Activity 5) they did. Dave chimed in: “take one row at a time and you will see one twothirds in each row.” Karla, Carol and Kisha identified with Dave’s explanation and
nodded in agreement.
Karla:
So I can multiply by any number to get equivalent fractions?
I:
Let’s see. If you multiply one-fourth by four what will you get?
Brian and Josh:
I:
Four-sixteenths.
Use your chips to form four-sixteenths. Make four equal groups.
How many will be shaded in each group?
145
Class:
One.
The students formed four-sixteenths with the chips as they were instructed to do.
I:
Based on what you know and what you can see from your model
are one-fourth and four-sixteenths the same.
Class:
Yes.
I:
So, Karla, do you think if I multiply both the numerator and
denominator by five, I’ll get an equivalent fraction?
Karla:
Yes.
Polly:
We can divide by the same number too, right? ‘Cause if you divide
four-sixteenths by four you will get one-fourth.
Jay:
I am going to multiply by one million.
The students started suggesting numbers that they could multiply one-fourth by to get an
equivalent fraction.
RNP Level One Lesson 10 Student Page A and Lesson 15 Student Page B
completed the set of tasks for Activity 9. The first task sheet was used to form a link
between the students’ logico-physical abstraction understanding and their logicomathematical abstraction and formal understanding. The students’ had already
encountered partitioning already partitioned wholes. After symbolically and orally
identify the original fraction shown in the diagram the students were expected to create
equivalent fractions by repartitioning the given diagram to reproduce the required
equivalent fraction. The students found this task interesting and completed it with ease.
Carol used the whole number to represent the fraction shaded. When questioned, she said
that she knew it should be a fraction but she could not be bothered to write so much.
Lesson 15 Student Page B questions were modeled similarly to the second part of
Activity 7. Given the number of chips to use, the students were expected to show the
given fraction and then give an equivalent fraction. These tasks proved challenging to
some of the students. With no teacher intervention, the students were able to negotiate
among themselves and then arrived at agreed solutions. For example, on item 2 which
asked them to show two-thirds using 15 chips and then give another name for two-thirds,
the students wondered if they should make groups of threes and shade two in each group
146
or make three groups of five and shade two of the groups. After deciding on the
configuration it became a simple task to name the equivalent fraction. Most of the
students chose to make three groups of five and then shade two of the groups. Initially,
Alton displayed similar confusion as experienced in Activity 7 but as he worked with his
classmates he was able to arrive at the ‘correct’ solution.
RNP’s Lesson 15 Student Pages D and E – Level One were used as the
assessment task for Activity 9. These lessons were chosen as they reflected the goal of
Activity 9, that is, to assess the students’ ability to determine equivalent fractions of
fractions given through varying media – pictures, orally and symbolically. An example
of one of the items on the task sheet is shown in Figure 28. The students regrouped and
quantified fractions as they completed the number sentences and write equivalent
fractions from the discrete wholes shown in each item (11items in all).
Figure 28.
Item 11 – RNP Lesson 15 Student Page E – Level One
The final activity used to complete the levels and constituent parts of Herscovics
and Bergeron’s (1988) model required the students to use the underlying physical
properties of unit and non-unit fractions to order fractions mentally by comparing them to
one-half and the whole. This activity was designed to specifically assess the students’
formal understanding. RNP’s Lesson 18 Student Page A – Level One and Lesson 8
Student Pages A and B – Level 2 were used to achieve this goal. The first three items on
Lesson 18 presented the problems using real-life situations. In item 1 the students had to
5
6
determine if Margo who ate of a large pizza had more to eat than Jose who ate of
8
16
147
another large pizza. The excerpt below represents the typical responses given by the
sixth-graders.
Josh:
Margo because with 8 pieces the pieces are bigger and because
she ate more than one-half.
Dahlia:
Jose ate less than one-half while Margo ate more than one-half.
6
5
would be smaller than .
16
8
6
3
None of the students made note of the fact that
is equivalent to which is smaller
16
8
Carol:
Margo ate more pieces because
5
than . This was probably due to the title of the lesson: “Comparing to 1-half.” Some of
8
the students (e.g. Brian, Dave and Jay) drew diagrams to illustrate their decisions. They
were able to accurately complete the other items of Lesson 18 using similar strategies as
employed on the first item.
RNP Lesson 8 Student Pages A and B – Level Two required the students to
determine larger fractions by picturing the fractions in their minds. Page A asked them to
1
decide whether a given fraction is greater than or less than . The students found this
2
task relatively simple. Most of them found the half of the denominator of the fraction
that is compared to one-half while others picture the slices of a pizza or a rectangular bar.
Page B made no reference to the one-half but the students had to order six sets of familiar
and unfamiliar fractions. At the end of ordering the fractions, they were to determine
which fractions were close to 0, ½ or 1. Here are some of the strategies they used. All
the students used at least one of these strategies.
Polly:
I picture them being pies, and I tried to find the bigger slice.
Richy:
I just compared all the fractions to ½ or 2/4.
Jonah:
I thought of them as rectangle.
Chrissy:
I picture them as a fraction that was divided into whatever the
fraction was.
Alton:
I tried to find which fraction is closer to 0, ½ or 1.
148
Karla:
I looked at the top numbers to see which one was bigger than I
looked the bottom numbers and compared to see how big it was
and how much would be shaded.
Analysis
By the end of the last four activities cited above, the participants of this study
demonstrated logico-mathematical abstraction and formal understanding as they
performed tasks specifically designed to examine their manipulation of unit and non-unit
fractions smaller than the whole. They were able to write conventional symbols for unit
and non-unit fractions, mathematically reconstitute wholes from unit fractions by
recognizing and using the fact that the whole is made up of its parts, use the relative size
of unit and non-unit fractions to order fractions and generate equivalent fractions.
The most challenging task that the students encountered as they were assessed for
these two types of understanding was the reconstitution of discrete wholes given parts of
the whole. The major deterrent was the students’ insistency to inadvertently group in
twos despite the number of parts mentioned in the numerator. After a succession of
assessment tasks and follow-up interviews the students who experienced the difficulty
were able to perform similar tasks accurately. They were able to represent fractions in
three different formats: fraction name, pictorial representation and fraction symbol. They
were capable of orally quantifying the part-whole relationships depicted in any of the
formats as they generated equivalent fractions and unit fractions from non-unit fractions.
SUMMARY
This chapter was dedicated to the answering of research question number one,
which sought to investigate and examine the nature of the participants’ understanding of
the fraction concept. Specific aspects of this vital middle school topic, such as the
students’ definition and understanding of the term ‘fraction’, their method of unitizing
and partitioning, and the strategies they employed when ordering fractions and generating
equivalent fractions, were spotlighted through the medium of ten task-based activities
some of which were adapted from Boulet (1993) work on the epistemological analysis of
the unit fraction with a set of fourth-graders. Herscovics and Bergeron’s (1988) model of
understanding and the pretest provided the crucial lens through which the teacherresearcher scrutinized and gained a structured insight to these students’ sense making of
fractions.
149
At the beginning of the study, most of the students showed signs of very little
understanding of this concept as became evident from the results of the pre-test. Their
definitions were superficial. As they participated in the activities, there appeared more
structured definitions of fractions in one form or the other. Brian, for example, used
mixed numbers to name fractions smaller than the whole at the early part of the study but
showed signs of fraction ‘maturity’ as he worked with the task-based activities. This
‘maturity’ was not only seen in Brian but also in Carol, Claudia, Dahlia, Alton, Karla,
Kisha , Ben and Richy. The other students were able to perform most of the tasks with
relative ease with just some misconceptions here and there. These misconceptions were
addressed through either one-to-one follow-up interviews, group sessions, and/or review
sessions at the beginning of the previous activity. Based on the items on the pretest, the
participants possessed level of intuitive knowledge, as they were able to satisfy the
criterion for this level of understanding – the recognition of equal parts.
An analysis of ten task-based activities was carried out in accordance with the
Herscovics and Bergeron (1988) model of understanding to explicate what encompassed
the understanding of the fraction concept among the participants of the study. These
students were able to partition discrete and continuous wholes of different shapes and
sizes with very little difficulty thus fulfilling the only criterion for the logico-physical
procedural understanding. The main difficulty encountered by the students occurred
when they had to share a discrete set of objects if the number of objects is not a multiple
of the number of equal shares. This difficulty was alleviated as the students worked
through the various tasks that involved partitioning. In a number of instances the
students existing network of fraction knowledge conflicted with the new knowledge
gained through their construction of the fraction.
Three activities (2, 3 and 4) were used to assess for logico-physical abstraction
understanding – the last task in the logico-physical tier of Herscovics and Bergeron’s
(1988) model. The task-based activities up to this point dealt exclusively with the nonquantified part-whole relationship. Some students, however, exhibited aspects of the
logico-mathematical tier due to fraction instructions prior to grade six. Based on the
responses (verbally and written) to the tasks for these activities the students had
satisfactorily fulfilled the five criteria related to this component of understanding. The
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students had no difficulty reconstituting the whole from its part given one of the parts or
many of the parts. One misconception became dominant when part of the whole
resembled a sector of a circle. Some of the students claim that the whole must be a circle.
Group discussion was used to circumvent this misconception. The students were also
able to recognize the part-whole relationships in discrete and continuous wholes despite
the transformations and physical attributes of the different parts. There was much
perturbation as the students repartitioned already partitioned wholes. This formed the
concrete foundation for finding a common denominator a necessary concept in the
addition and subtraction of fraction. They eventually developed several partitioning
strategies to divide the varying figures.
The next six activities allowed the students to perform tasks geared at assessing
their logico-mathematical understanding. It was obvious from the results of Activity 5
that the majority of the students were able to quantify part-whole relationships in unit and
non-unit fraction. The major hurdle encountered at the logico-physical mathematical
constituent part was the renaming of unit fraction from non-unit fractions. The
participants’ logico-mathematical and formal understandings were assessed
simultaneously due to the pervading presence of the fraction symbol since the beginning
of the study. There were clear evidences of these types of understandings as the students
(orally and written) quantified equivalent fractions and ordered unit and non-unit
fractions. There were signs of formal understanding as the students were able to generate
equivalent fractions and order fractions without the aid of physical referents.
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CHAPTER 5
RESULTS – STUDENTS’ PARTITIONING STRATEGIES
Introduction
The remaining two research questions will be addressed in this chapter. Based on
the students’ responses on the ten activities that were discussed in the previous chapters,
the sixth graders seemed to exhibit the components of understanding as outlined by
Herscovics and Bergeron (1988). As the students engaged in these task-based activities,
plus two additional activities not discussed before, they displayed varying partitioning
strategies that will be scrutinized and discussed in an effort to answer the second research
question. These task-based activities also gave them the opportunity to work with
various physical referents. These include continuous and discrete objects. The last two
activities depicted problem-solving sessions that allowed the students to apply their
fraction knowledge in solving problems that model real life situations. The participants’
behaviors and discussions as they manipulated the objects were carefully observed and
recorded and were paramount to the answering of the final question.
Research Question #2:
What strategies do sixth grade students employ to ensure
partitioning or fair sharing as they engage in the processoriented activities?
By the end of Activity 3, a majority of the students were aware of the significant
role equality of the parts played in determining the part-whole relationship. As the
students divided the given wholes, they employed varying strategies to guarantee that the
parts were congruent. The act of partitioning or fair sharing is essential in the
development of students’ understanding of fractions and has been part of most students’
everyday experiences. Lamon (1996) purported that students’ partitioning strategies
were most times influenced by social practices and the commodity being shared.
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It forms the basis for equivalence that in turn forms the foundation for operating on
fractions.
Partitioning also plays an active role in the development of the fraction
concept as it relates to the part-whole, measure and quotient subconstructs.
As mentioned in Chapter 1, the students’ partitioning habit will be explored using
the partitioning strategies outlined by Charles and Nason (2000) and Lamon (1996) as the
frame of reference. Charles and Nason’s (2000) strategies involved the quantification of
the part-whole relationship which was not required for the first tier of Herscovics and
Bergeron’s model (1988) where much of the partitioning tasks were done. Consequently,
a purposeful partitioning activity dubbed ‘Fraction Breakfast’ (see Appendix B) was
planned. During this final activity of the study – Activity 12 – the students participated
in an actual breakfast where they had to share food items representing continuous and
discrete objects among their group members. There were five groups with four members
to a group. Despite the lack of the necessary step of quantifying the part-whole
relationship, partitioning tasks from Activities 1 and 4 will be referenced with the aim of
confirming the students partitioning habit. For the purpose of this study each strategy
will be listed followed by the findings generated from the students’ observed behaviors.
The first nine strategies are the ones mentioned by Charles and Nason (2000) while the
last three are gleaned from Lamon’s (1996) study on young children’s process of
unitizing. Three of the strategies reported by Charles and Nason (2000) were not
observed during the study and were, therefore, omitted from this chapter. A complete
listing of the strategies is included in Chapter 2.
1.
Partitive quotient foundational strategy: This strategy involves recognizing the
number of people the object(s) must be shared for, generating the fraction name from the
number of people, partitioning of the object into that number of equal pieces, and sharing
and quantifying each partition. This strategy was used by the members of three groups
(Bob’s, Mary’s and Polly’s) in accomplishing Task 3 of Activity 12. This task required
them to share three rectangular tostados [toasted bread with cheese] among the four
group members. Each tostado was considered to be a whole.
Ben (Polly’s group): We divided each tostado into fourths to get twelve pieces of
fourths. Each person would get one-fourth from each
tostado and three-fourths in all. (See Figure 29).
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Figure 29.
2.
Example of Partitive Quotient Foundational Strategy
Proceduralised partitive quotient strategy: This strategy is a condensed version
of the one above. The students exhibiting this strategy would take fewer steps in
quantifying the fraction. A person demonstrating knowledge of this strategy would be
able to quantify the fraction without actually partitioning the whole. Task 2 in Activity
12 required the students to share a 64 fluid ounces container of orange juice equally
among the four group members. Each student had a nine fluid ounces plastic drinking
container. Due to the liquid nature of the whole, the students did not attempt to make any
drawing to help them decide what fraction of the orange juice each person would get at
the first serving assuming that each person would be served the capacity of the container.
All five groups readily demonstrated knowledge of this strategy by quantifying each
person’s share as
3.
9
before partitioning and sharing the orange juice.
64
Partitive and quantify by part-whole notion strategy: The steps for this strategy
involves the partitioning of each whole object to match the number of parts needed,
sharing one part from each object to each person before quantifying the part-whole
relationship. This strategy is different from the first strategy in that for the partitive
quotient foundational strategy the learner would start the quantification process from the
onset of the partitioning procedure whereas for the strategy mentioned here the
“quantification of each share could not be achieved until after the sharing had been
completed” (Charles & Nason, 2000, p. 201). This strategy was evident in the
completion of Task 3 in Activity 12. Two groups (Dave’s and Chrissy’s) assigned each
3
person their share of the tostado before concluding that each person got .
4
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Carol[ Dave’s group]:
4.
We cut all the wholes into 4 pieces each to get 12
pieces in all and gave each person 3. [She paused
while the tostado was shared.]
3
So each person gets .
4
Regrouping strategy: After recognizing the number of parts and that the number
of parts gives the fraction name, the students using this strategy will realize that the
number of pieces to be shared can be obtained by multiplying the number of discrete
objects by the number of parts in each whole. This is done prior to the quantification of
each share. This strategy was utilized by four of the five groups in determining the
fraction of the set of quiches each person should get if 12 quiches should be divided
equally among four persons (Task 1 – Activity 12). The following two excerpts represent
the responses received from the students who used this strategy.
5.
Mary:
I multiply by 3. And since there are four people and there are 12
quiches so you also divide 12 ÷ 4.
Richy:
The way I got 3 parts is that I divided 12 by 3, also because 4
people multiply by 3 quiches equal 12.
Horizontal partitioning strategy: On recognizing the number of parts needed to
partition the whole, the students employing this strategy will horizontally divide the
circular whole into the desired number of parts; quantify each part before recognizing
that the parts are unequal, for example using vertical lines to divide a circle into four
equal parts. Considering earlier misconceptions displayed by Ashley, Kisha and Dahlia
concerning partitioning a circle in three shares, it is worthy mentioning that none of the
students used this strategy when sharing the pancakes in Task 4 of Activity 12.
6.
People by objects strategy: In sharing circular objects among a number of people,
the students using this strategy will recognize the number of people and the number of
objects. The number of pieces that each whole will be divided into is generated from the
product of the number of people and the number of objects. This is followed by the
quantification of each share. Although this partitioning strategy was not evident in any of
the tasks that involved circular food items (quiches and pancakes), Mary’s group
employed this strategy in determining each group member’s share of tostados for Task 3
– Activity 12. They figured that there were three tostados and four group members so
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each whole needed to be partitioned into twelfths. Each person would get one twelfth of
each tostado. They quantified each person’s total share as
7.
3
.
12
Halving object then halving again and again strategy: This strategy involves an
iterative halving of the object until the desired number of parts is obtained. This strategy
is referred to as algorithmic halving by Poither and Sawada (1983). Although this
strategy was not evident in the Fraction Breakfast, some students repeatedly used this
strategy in partitioning tasks for Activities 1 and 4. For example in Task 2b – Activity 4
where the students had to repartition a circular pie previously partitioned in thirds to
share for four people, some of the sixth-graders halved the thirds, then halved each of the
halves of the thirds resulting in four equal parts to the third [12 parts to the whole.]
During the paper-folding exercises done during the study, it was observed that the
initial procedure to obtain a fraction for a majority of the students was to fold the paper in
halves. For instance, in one session the students were asked to fold one-half followed by
a quarter. They were all successful in completing this task. They were then asked to fold
the same paper, already folded in fourths, to obtain twelve equal parts. Most of the
students proceeded to fold the already folded paper in half. This was done three times.
They recognized that 3 × 4 = 12 but could not understand the reason for obtaining thirtytwo equal parts on the paper instead of the desired twelve parts. They were encouraged
to think aloud about what they did. Polly spoke up: “Every time we fold, we are
doubling the parts.” They discovered Polly’s pattern after much practice. Jay
suggested and then demonstrated how to make a tri-fold.
8.
Whole to each person then half remaining objects between half people strategy:
This strategy and the one immediately below are extensions of the halving strategy.
Karla and Claudia deviated from their group members who displayed the mark-all
strategy mentioned below. During the group discussion on the strategy that could be
used to share the pancakes in Task 4 – Activity 12, Karla reasoned that “each person can
get a whole. Take two of the pancakes that are left over and cut them in half, give each
person one piece and then cut the last pancake in four and give each person one of those.
She made a pictorial representation of her strategy (see Figure 30). Claudia nodded in
agreement.
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Claudia:
“Each person gets one whole, one half and one fourth.”
Figure 30.
9.
Karla’s Representation of the Sharing of the Pancakes
Half to each person then a quarter to each person strategy: This strategy is self-
explanatory. Dahlia’s group utilized this strategy in assigning the amount of pancakes
each member should receive in Task 4 of Activity 12. They halved each pancake, further
divided the last pancake into fourths. Each person in the group received three halves and
a fourth as shown here:
10.
.
Preserved-pieces strategy: Whenever the sharing involves more than one of the
discrete objects of the total number of objects to be shared, the student exhibiting
knowledge of this strategy would partition only the object(s) that needed dividing leaving
the other objects intact. This strategy was widely used in completing Task 4 of Activity
12 where the students had to equally share seven pancakes among the four of them.
Marla’s group left four of the pancakes intact and then partitioning each of the remaining
three pancakes into fourths with each person getting one-fourth of each. By the end of
the process, each group member would receive one whole pancake and three-fourths (see
Figure 31). Similar sharing was done by Jonah’s group.
Figure 31.
Example of the Preserved-Pieces Strategy
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11.
Mark-all strategy: In this strategy all the discrete objects will be partitioned to
obtain the desired number of pieces. However, at the time of sharing, only the objects
that needed cutting will be cut. The pictorial representation that Dave and Carol gave to
accompany Task 4 on Activity 12 indicated that they visually partitioned each pancake
into four pieces but in assigning the actual share they gave each person one whole plus
three-fourths of the remaining three pancakes. Each person would get
12.
.
Distribution strategy: For this strategy, all the discrete objects will be partitioned
and cut. The pieces will then be distributed to each person until finished. Alton’s group
applied this strategy in sharing the pancakes in Task 4 Activity 12. Each pancake was cut
in fourths where each person was given one fourth of each pancake eventually resulting
in each person getting the equivalent to seventh-fourths. The members of this group
recognized that that they could use the preserved-pieces strategy but opted to use this
method instead “to make it interesting.”
Research Question #3:
How do physical and real world representations aid in the
development of students’ understanding of fractions?
Paramount to the first tier of Herscovics and Bergeron’s (1988) model of
understanding is the physical manipulation of objects that aid in laying the foundation for
the constituent parts of the model. Throughout the study, these concrete objects formed
the basic building blocks for the conceptual understanding of fractions. As mentioned in
the previous chapters, varying manipulatives were integrated in the task-based activities.
These included two-sided colored chips or counters, fraction circles, fraction triangles,
fraction squares, Fraction Stack Bar, Fraction Balance and fraction stencils. From the
onset of the study, the students were presented with a personal “manipulative bag”
containing a ruler, erasers, flexible wires (similar to pipe cleaners) and two pencils. The
students were free to use any of manipulatives in their bag, any of the ones mentioned
above and those assigned during the specific activities to aid them with the assigned
tasks. These include numerous task-based cut-outs, measuring cups, plastic containers for
liquid and the liquid itself.
As Steffe and D’Ambrosio’s (1995) hypothetical learning trajectory indicated,
major significance was placed on the presentation of information, the situation of the
scenarios and the support given to the learners during the process of constructing
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knowledge. Lesh’s translational model (Lesh, Post & Behr, 1987) depicts the
connections among different external representations including real world situations and
manipulative aids. This translational model reflects the ideology that the understanding
of fraction is reflected in the students’ ability to represent fraction ideas in multiple ways,
including the ability to make connections among the different representations. As the
students worked with several manipulative models, they were advised to consider how
these models were alike and different. They worked individually, in small groups as well
as interacted with the teacher-researcher in a whole class setting as they discussed
fraction ideas and worked with the various physical referents. Except for a number of the
RNP activities which specified which manipulative to use, the students were not just
limited to one type of manipulatives but were encouraged to use the ones best suited to
aid in the completion of the task(s) at hand. Even though an attempt was made by the
teacher-researcher to use the fraction circles, fraction triangles and fraction
squares/rectangles to illustrate fractions, the students often resorted to using only the
fraction circles and rectangles to model their fractions. This was also noted during the
paper folding exercises. The students had the opportunity to work with paper circles,
paper triangles or paper rectangles. Only Josh, Dave and Jay took up the challenge of
folding the paper triangles to yield the given fractions. The paper rectangle became the
most popular as it was “easier to bend in shape” (Ben). Karla remarked:
It is easy to use the fraction triangles but it is hard to draw them and show equal
parts for the fifths and sixths.
For a number of the activities, the students worked with both discrete and
continuous analogs. Even after working with both continuous and discrete model for the
duration of the study, it was noted that 85% of the participants chose to draw a
continuous diagram to illustrate a fraction in response to Task 7b in Activity 12. The
remaining 15% opted for a symbolic representation. Left on their own, most of the
students displayed an affinity to working with continuous rather than discrete objects. As
Carol put it: For this one [showing the rectangular-shaped paper] there is only one thing
for the whole but more than one thing here [showing the set of counters] stand for the
whole.”
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In Activities 5, 6 and 7, the students worked extensively with discrete models
mainly because they were required to. They exhibited relative ease when working with
these models. Completing discrete wholes in Activity 7 posed quite a challenge to a
number of the students. The main problem, however, was not in the type of analogs
(continuous or discrete) used but in the students’ interpretation and understanding of the
numerator and denominator. Despite the difficulty that they faced to ensure equality of
the parts when dividing a circle, the circle seemed to be the most popular of all the
continuous wholes introduced during the study.
Although no specific attempt was made to eliminate the use of physical referents
on problem-solving tasks, it was observed that for the most part, the sixth graders treated
problems written in symbolic form differently from those connected to physical referents,
partly because the first set of activities never stressed the quantification of fractions or
used fraction symbols or words. Activity 11 presented a number of problem-solving
tasks including Poither and Sawada (1983) “Cake problem”, Lesson 6 – Student Page C
and Lesson 22 – Student Page A (Level 2) from RNP. These activities asked the students
to use chips or draw pictures to help them solve the problem. Majority of the students
ignored the instructions and proceeded to solve the problems without the use of any
physical referents. For example, on item 5 from Lesson 22 (shown in Figure 32), Marla
applied the division algorithm to arrive at the solution for the problem. She divided 18
by nine to get two and then used the two to multiply the five to yield the desired answer –
10. She continued to use the same algorithm for the remaining problems on the task
sheet. Her method typified the method used by most of the students. Carol, Karla and
Claudia were the only students who attempted to use some form of diagram to illustrate
their solution to that problem.
Todd’s mother cut a cake into 18 pieces. She is going to take 5/9 of
the cake to a party. How many pieces of cake will she take to the
party?
Figure 32.
Item 5 – RNP Lesson 22 Student Page A - Level 2
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The ease at which these students solved these problem solving tasks on fractions
surprised the researcher. In my teaching experiences over the years I have seen eightand nine-graders baffled over problem solving questions such as the one shown in Figure
34. The participants in this study were able to transfer knowledge gained from previous
activities to the problem solving tasks. For example, Richy reasoned that “if 1/3 of the
box of gumdrops has seven gumdrops and there are three thirds in a whole then the box
has 21 gumdrops, ‘cause 3 ×7 = 21” (see Item 2 – Lesson 6 Student Page C – RNP
Level 2).
Karla:
21 gumdrops are needed because you have to get 2/3 more so you
add 1 more group of 7 then in order to get 3/3 you add 1 more
group of 7 to equal 21. [She used x’s to illustrate her solution
when asked to verify.]
Chrissy, one of the few who did some form of pictorial representation, arrived at
the solution by drawing a circle, cut the circle in three equal parts, assigned seven to each
part then add the three seven’s to get 21. Alton drew seven small shaded circles to
represent one group of one-third and then drew two more sets of seven circles to obtain
21. No form of teacher intervention was necessary as the students were simply working
individually and verifying their results among themselves. During the problem-solving
sessions, the teacher-researcher main task was to ensure that the students were on-task.
When asked about the impact of the real world representation tasks, the students
had positive things to say. Here are a few of the comments.
Polly:
They helped me to understand fraction better because they are
used in examples and stories.
Chrissy:
I am able to look at fractions in different ways.
Marla:
I am better able to understand fractions. They seem to let working
with fractions become easier.
Alton:
This shows that we use fractions everyday without realizing it.
When we grow up we will be having fractions everywhere we go.
Brian:
I love the sharing activities. Fractions seem to be fun.
Bob:
The way the fractions lessons are done are really cool. I know I
will always remember how to work with fractions.
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Mary:
They help to show that fractions are interesting.
Paul:
They teach me how to use fractions.
Carol:
I can use what I learn to divide a cake or pie equally if someone
asks me.
In ordering fractions, a number of the students conjured pictures of slices of a
pizza or a rectangular chocolate bar. Polly and Jay placed the fractions within a real
world setting before arriving at a solution. The class was ordering fractions.
Polly:
I picture them being pies, and I tried to find the bigger slice.
Jay:
I don’t think so: 2/7 is bigger. It’s like sharing a pizza. You get
more to a slice when you have to share for seven people instead of
nine. You’re going to have two bigger sevenths than two ninths.
The “Fraction Breakfast” gave the students the opportunity to practice their
sharing abilities in real ways. Some of the students credited this activity as one of their
reasons for understanding fractions a whole lot more.
Jay:
The Fraction Breakfast shows that learning fraction can be fun.
Josh:
It shows how fractions are important in our everyday lives without
realizing it
.
Mary:
I will never look at fractions the same way again. They will have
more meaning to me.
SUMMARY
The equality of the parts in a whole plays a critical role in the conceptual
understanding of fractions. Besides being cognizant of that fact, a learner should possess
the capability to divide given wholes (continuous or discrete) into equal parts. To
achieve this end, various partitioning strategies were observed and documented in the
literature. Poither and Sawada (1983), Charles and Nason (2000) and Lamon (1996)
have shared a number of partitioning strategies they observed while working with young
children. Some of the strategies are common among the prominent literature mentioned
above, with slight variation. This study used the partitioning strategies reported by
Charles and Nason (2000) and Lamon (1996). Activities 11and 12 were purposefully
designed to examine the students’ partitioning behavior and their strategy for solving
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problems related to the real world. The students’ responses to these activities revealed
that they were able to partition the given objects in a variety of ways. Their partitioning
skills were honed during the tasks for Activities 1 and 4. The students expressed the need
to make other mathematics topic relevant by designing the tasks similar to the problem
solving tasks used during the study. They enjoyed interacting with each other.
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CHAPTER 6
SUMMARY, CONCLUSION, DISCUSSION AND IMPLICATIONS
Introduction
This study that was done with the set of sixth graders from a private school in
southeast Florida was a whole class exploratory study designed to examine these
students’ understanding of fractions as they participated in task-based activities during an
eight week teaching sequence. These activities included tasks depicting real world
representations and those requiring the use of physical referents. The students’
understanding of fraction was analyzed and assessed using the Herscovics and
Bergeron’s (1988) model of understanding. Mathematical understanding is achieved
when the learner is capable of making connections between ideas, facts, or procedures
(Hiebert & Carpenter, 1992). In reference to the current study, an understanding of the
part-whole and quotient subconstructs of a fraction does not necessarily imply that the
students are able to make the connection between each.
Throughout this study, a careful analysis was undertaken to reveal the
connections, if any, the students made while engaging in the process-oriented activities.
The analyses and interpretations of the behaviors of the students were made from the
written task sheets and the video- and audio-tapes collected during the study. These were
augmented with the teacher-researcher’s journal and input from a peer reviewer. The
function of this chapter is primarily to discuss and summarize the findings unraveled
during the teaching sequence, and to outline the implications of the study. The summary
deriving from the study will be discussed under the major themes: the nature of the
student understanding of fractions, the students’ partitioning strategies and physical and
real world representations. The limitations of the study were already discussed in
Chapter 3.
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The Nature of the Students Understanding of Fractions
It was noted during the study that very few students initially had a formal
definition of a fraction although they were able to give a symbolic representation. The
few who attempted a definition hinted towards the part-whole concept of the fraction. By
the end of the study, the majority of the students viewed the fraction from a part-whole
perspective. The other subconstructs of the fraction were possibly ignored due to the
emphasis that was placed on the part-whole relationship for the first five activities done
during the study. With the exception of the ratio subconstruct, the other three
subconstructs (measure, operator and quotient) as outlined by Kieren (1976) were
included in some of the activities but were not accentuated in a verbal or written form as
much as the part-whole relationship. The students worked with the number line and
measuring cups (measure subconstruct) in at least three activities. The quotient
subconstruct was referred at least two activities. For example, in Activity 11 the students
were asked to share nine cookies equally among four persons. Another problem solving
task required them to find how many children went on a trip if
3
of the children from a
5
class of 25 students went on the trip (Activity 11). Findings related to the study also
indicated that the students possessed a clearer idea of the distinction between the top
number (numerator) and the bottom number (denominator) of the fraction.
Herscovics and Bergeron’s (1988) model was used as the lens through which the
students’ understanding of fractions was observed. This model was developed by
Herscovics and Bergeron in an attempt to “provide an epistemological analysis of the
various conceptual schemata taught at the elementary level” (Herscovics and Bergeron,
1988, p. 15). It is a two-tiered model; the first tier describes the understanding of the
preliminary physical concepts while the second tier describes the understanding of the
mathematical concept itself. Each tier has three levels or constituent parts which do not
necessarily occur in a linear fashion. Due to the physical connotation as well as the
mathematical phenomenon that is attached to the fraction concept, this model of
understanding is well suited to be used as the analysis tool through which this subset of
the rational number concept was observed.
165
Although there were evidences that these students had previous formal
instructions in fraction, none of the students were operating at an advanced stage of
formalization. This was evident from the findings related to the pretest where most of the
students were placed in category 2 revealing the idea that the students mostly had a
limited idea of the fraction concept. Jay, Polly and Marla frequently quantified the partwhole relationship before it was required but still exhibited signs of misconception in the
understanding of the physical concepts associated with the fraction. The students
satisfied the criterion for intuitive understanding during the pretest and later activities by
recognizing parts that were partitioned. This included the recognition of equality of parts
even though the parts were transformed.
During the teaching sequence, the students participated in twelve activities
designed and developed to investigate the remaining five levels or constituent parts of the
Herscovics and Bergeron’s (1988) model of understanding. The students exhibited signs
of possessing logico-physical procedural understanding as they partitioned continuous
and discrete wholes – the main criterion for this level of understanding. The main hitch
came when the students were dividing the circle into equal parts of three. Ashley drew
two vertical lines to complete her partition but was later challenged by the students as to
the equality of the parts. Marla shared with the students her method of ensuring that the
parts were equal. This method became the standard strategy used by the students when
partitioning a circle into three parts throughout the study. Partitioning the equilateral
triangle did prove to be challenging to some of the students but they were able to
negotiate and practice until they were satisfied the parts were equal. When partitioning
discrete wholes, the students experienced little difficulty when the number of objects to
be partitioned is a multiple of the number of partitions. They used the division algorithm
or ‘dealing’ strategy (Davis & Hunting, 1990; Davis & Pitkethly, 1990). When asked to
partition 12 circular chips into 5 parts, all of the students were momentarily stumped as
they were not sure of what to do with the remaining two chips. Jay offered a solution that
made sense to them so the students proceeded to divide the remaining two cookies into
five parts each then dealt one part of each circle to the five groups. They also used
varying unique and familiar techniques to ensure equality of the parts. It was noted that
information gained from a previous unit in geometry was referenced while checking for
166
equal parts. The geometry unit was done before the fraction unit solely because of the
reorganization of the time to facilitate the study.
The logico-physical abstraction level of understanding is the last component in
the first tier of Herscovics and Bergeron’s (1988) model of understanding. At this level
the relationship between the part and whole is emphasized. As soon as the students were
clear on the meaning of the term ‘part-whole relationship’ it became evident that the sixth
graders possessed logico-physical abstraction understanding as they successfully satisfied
the five criteria related to this level. The findings of the first four activities of the
teaching sequence indicate that the students understood the physical concepts that
underlie the fraction. These activities aimed at looking at the unquantified part-whole
relationship via partitioning, repartitioning and determining part-whole relationships.
The use of fraction language at this level was ignored and not encouraged. The first two
activities showed that the students were capable of reconstituting the whole from its parts,
were aware of the equivalence of the part-whole relationship in spite of a variation in the
physical attributes of the whole, they recognized the equivalence of the part-whole
relationship regardless of the physical transformations of the whole making note of the
relationship between the size of the parts and the number of equal shares. The most
challenging criterion to be satisfied was encountered when the students were given the
opportunity to repartition already partitioned continuous whole. Repartitioning discrete
whole would result in the simple task of regrouping where the original partition would be
lost. The art of regrouping was already evident while assessing for logico-physical
procedural understanding and consequently tasks involving discrete wholes were
eliminated from the tasks comprising this activity. The participants were challenged by
the tasks of repartitioning already partitioned wholes where they could not apply the
‘halving algorithm’ thus the first of the three tasks were accomplished with ease. When
asked to repartition a three-part chocolate bar so four persons could get an equal share,
the students considered the task to be impossible. After much discussion and negotiation,
seven different configurations were evident. The most popular configuration that
emerged yielded twelve equivalent parts – the number of parts analogous to the lowest
common multiple of 3 and 4. The remaining tasks in this activity were completed with
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relative ease as the students used the knowledge gleaned from this task to do their
repartitioning.
Five activities were used to assess the students’ understanding of the emerging
mathematical concept as it related to the fraction concept. The quantification of the
fraction, verbally and written, formed the core of understanding for the second tier of
Herscovics and Bergeron’s (1988) model of understanding. At the logico-mathematical
constituent part, the students were able to orally express the fractions associated with a
number of concretely depicted part-whole relationships and to make concrete
representations of part-whole relationships from orally given fractions. This was done
with relative ease with the greatest challenge occurring when the students had to rename
non-unit fractions to unit fractions. It was evident from the responses that the sixth
graders demonstrated logico-mathematical procedural understanding of unit and non-unit
fractions. One of the misconceptions that emerged from the activities had to do with the
identification of the whole if the whole was part of a circle. The initial inclination of the
students was to complete the circle figuring that the whole must be a circle. It took much
discussion before some of the students became convinced that John’s patio (see
Assessment Task – Activity 5 in Appendix B) represented a whole although it had a
semi-circular shape. By and large, the students possessed explicit logico-mathematical
procedures that they could use to relate to the underlying preliminary physical concepts
and were capable of using these procedures appropriately. The use of physical referents
at this point was downplayed although the students had the opportunity to reference them
if necessary. The art of paper folding was introduced and used in the second tier of the
model of understanding.
Due to the prevailing use of the fraction symbol in previous activities by the
students, the logico-mathematical abstraction and formal understanding were dealt with
simultaneously. One of the prerequisites for formal understanding is the mastery of the
conventional fraction symbol. With the aid of a RNP activity, the students had the
opportunity to view the relationship that existed among three different representations of
a fraction - fraction name, fraction symbol and a pictorial representation. There were
able to complete this task with very little challenge. The tasks at the logico-mathematical
abstraction level paralleled those encountered at the logico-physical abstraction level with
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the exception that at the second tier constituent part, the use of the fraction language was
permissible and evaluated. The possession of logico-mathematical abstraction
understanding as it relates to the fraction concept indicates that the learner is capable of
reconstituting the whole from its parts, ordering fractions according to their size and
generating and quantifying equivalent fractions. It was obvious that some of the students
possessed this type of understanding even before they did the tasks designed to assess
such understanding. Activities 7 – 10 solidified this notion and also gave proof to the
fact that the participants of this study, despite the hurdles they encountered in completing
some of the tasks, did possess this type of understanding. The students worked to
reconstitute continuous as well as discrete wholes. Activity 7 provided the most
challenge as their understanding of the numerator and the denominator was highlighted.
Two sessions of Activity 7, two assessment tasks and individual interviews were held
with the aim of clarifying and verifying the students’ ability to reconstitute discrete
wholes given parts of the whole. The major deterrent was the students’ insistency to
inadvertently group in twos despite the number of parts mentioned in the numerator.
Formal understanding was evident as the students were able to generate equivalent
fractions and order fractions without the aid of physical referents. On tasks that required
the use of the fraction as an operator, the students displayed a level of formal
understanding that was not expected by the researcher. The students solved real world
representation problems without any form of teacher intervention and without little use of
manipulatives even when instructed to do so.
The Students’ Partitioning Strategies
One of the aims of this study was to reveal the partitioning strategies that the sixth
graders employed while completing various tasks throughout the duration of the study.
These strategies were categorized according to Charles and Nason (2000) and Lamon
(1996) classification of partitioning strategies. Depending on the task at hand, each child
opted to use one or more of the partitioning strategies that were outlined by the
researchers above. Throughout the study the students were given the opportunity to
partition continuous and discrete wholes. They partitioned regular geometric shapes such
as the circle, parallelogram, triangle and rectangle. They partitioned cutout shapes of
stars, the letter L, heart shapes and other irregular geometric shapes. They partitioned
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lines and liquids and at the end of the study had the opportunity to share items of food
equally among themselves. Twelve distinct partitioning strategies (see Chapter 5) were
observed. The three most popular strategies were the regrouping strategy identified by
Charles and Nason (2000), preserved-pieces strategy and mark-all strategy identified by
Lamon (1996).
Physical and Real World Representations
The manipulatives or physical referents used during this study were carefully
selected and ranged from cutouts of figures to graduated containers for assessing
students’ partitioning strategies. Each manipulative was categorized as either a
continuous or discrete whole depending on the context in which it was placed. For
example, one circle could be used as a continuous whole and a set of circles used as a set
of discrete objects. Based on the findings of the study the students showed a preference
toward working with the continuous whole than the discrete whole although some
research literature purported that children develop an understanding of discrete quantities
before a comparable understanding of continuous quantities (Hiebert & Tonnessen, 1978;
Poither & Sawada, 1983). Although discrete models were used by the students, majority
of the students exhibited a preference in using continuous models as forms of reference
for given fractions. The participants of the study have expressed in verbal and written
form their appreciation for working with fraction problems that represent real world
situations.
Significance of the Study
A study of students’ construction of fractions as they participate in problemsolving activities and their subsequent understanding of this very important middle
school topic is significant for a number of reasons:
1.
It adds to the growing scholarly documents on students’ sense-making of
fractions and how this affects further operations on fractions.
2.
It provides real-time activities and application where students work with
discrete and continuous quantities interchangeably.
3.
It provides support for several instructional and curricular strategies.
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4.
It provides teachers with an alternative curriculum that addresses not only
prerequisite knowledge and skills which support the acquisition of fraction
concepts but also the learners’ intuitive knowledge base.
5.
The findings and interpretation of the study can inform classroom practice
and pedagogy.
6.
The inquiry method that was used during the tasks and interviews will
foster an important transformation in the students’ thinking of fractions.
CONCLUSIONS AND DISCUSSION
This research project was designed to probe a set of sixth graders sense-making or
understanding of fractions as they worked with physical referents and real
representations. At the beginning of the study, students such as Brian, Alton, Kisha,
Dahlia, Claudia, Ben and Carol’s understanding of fraction appeared to be a bit unstable.
However, by the end of study, these students along with their classmates seemed to make
the “qualitative leap of sophistication” (Lamon, 1999) towards a formal understanding of
fractions. Kamii, Lewis and Kirkland (2001) made the distinction between physical and
logicomathematical knowledge which coincides with Herscovics and Bergeron’s (1988)
model of understanding. Physical knowledge refers to the knowledge of objects that exist
in the external world while logicomathematical knowledge consists of the mental
relationships, which each learner creates from within. The responses to the tasks done in
this study revealed that the participants’ logicomathematical knowledge was constructed
through their own thinking. The physical referents provided the launching pad for such
critical thinking. According to Behr, Harel, Post and Lesh (1992) semantic and
mathematical analysis of the part-whole, quotient and operator subconstructs of rational
number concepts there exists a one-for-one matching between a sequence of
manipulations of physical objects and a sequence of manipulations of mathematical
symbols. This matching reveals that children’s understanding occurs from the
manipulation of objects.
In traditional classrooms, fractions are taught algorithmically with little attempt to
ground them in a meaningful contextual basis. More often than not, they do not have a
deep understanding of the mathematics behind them. Consequently, when the given
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problems do not fit neatly into the structure within which the algorithm was taught, even
competent students can have difficulty (Bulgar, 2003; Davis & Maher, 1990).
Table 13.
Classification of Students’ Difficulties
CLASSES
PARTITIONING
•
RECONSTITUTION
•
•
•
ORDER
•
QUANTIFICATION
•
DIFFICULTIES
Inability to produce odd partitioning especially in
circular wholes.
Unreliable strategies to ensure equality of the parts
Repartitioning already partitioned whole
Assuming that the whole must be a circle if the
part is a sector of the circle
Reliance on the whole number language (e.g since
4 > 2 then ¼ > ½
Identifying unit fractions from its equivalent
fraction
Table 13 highlights the major challenges that one or more students encountered
during the study. Research (Behr, Wachsmuth, Post & Lesh, 1984; Lamon, 1999, Mack,
1995) has shown that students’ whole number language often interfered in their ability to
quantify a fraction appropriately. This is especially noted when students are comparing
unit fractions such as 1/5 and 1/7. Some students figured that 1/7 is larger than 1/5
because seven is bigger than five. Biddlecomb (2002) commented that this
overgeneralization was not only evident in young children but also in college algebra
students who made similar errors “in claiming that 1/(x + 1) is bigger than 1/x because (x
+ 1) is larger than x” (p. 167).
It is important for the teacher to recognize the fraction knowledge that the
students bring into the classroom (Mack, 1995). Children at an early age are familiar
with sharing their cookie in half with someone else. However, as they get older more
sophisticated partitioning methods become more evident. Poither and Sawada (1983)
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noted the difficulties students faced when dividing into odd number of parts. As students
are encouraged to discuss the part-whole relationship using different shapes and sizes
they realize that it is easier to represent certain fraction using a particular shape. This
was evident during this study where the participants frequently used the rectangle to
represent fractions with odd number denominators rather than the well-used circle. Very
few students in the middle grades (grades 5 – 9) are able to partition into any number of
given parts whether mentally or physically (Smith, 2002). Consequently, significant
attention should be placed on partitioning tasks at this level or an earlier grade level as
partitioning forms the basic building block for the fraction concept.
One danger in emphasizing the part-whole relationship in the construction of a
fraction is the possible reinforcement of fraction as two numbers or not a number at all
instead of the fraction as a single number (Kieren, 1988). It is also possible that over
dependence on the part-whole subconstruct can thwart the development of the fraction as
other subconstructs. In addition, sometimes the learners’ conception of the part-whole
meaning of fractions interferes with their ability to show fractions on a number line
(Bright, Behr, Post, & Wachsmuth, 1988). Children often consider the given segment of
number line as the whole as was evident in the pretest.
Critical to the students’ sense-making or understanding of fractions is the
knowledge that the same fraction always represents the same relative amount regardless
of the variation in physical attributes and/or transformation of the parts in the whole as
highlighted in Activity 3. Different sizes and shapes of objects should be used to aid
students to come to this realization. Repartitioning already partitioned whole provides it
own set of challenges to students. Repartitioning tasks should be aimed at allowing the
students to ‘discover’ the lowest common multiple or a common multiple of the
denominator of original fraction and the denominator of the new required fraction part.
For example, when repartitioning a rectangle already partitioned in thirds, into fourths,
the strategies used should lead to common multiples such as 12 and 24. The concepts of
equivalence and addition and subtraction of fractions arise in repartitioning activities.
Manipulatives have been at the forefront of recent educational reforms of
mathematics teaching and learning. Yeatts (1991, p. 7) suggested that “manipulatives
assist students in bridging the gap from their own concrete sensory environment to the
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more abstract levels of mathematics.” However, as Kamii, Lewis, and Kirland (2001, p.
31) noted: “the mathematics we want children to learn does not exist in manipulatives. It
develops as children think, and manipulatives are useful or useless depending on the
quality of thinking they stimulate.” Ball (1992) pointed out that working with
manipulatives do not automatically create mathematical knowledge. In the context of the
teaching and learning of fractions, Smith (2002) placed the burden of the proper use of
fraction manipulatives on the shoulders of mathematics teachers. He noted that fraction
manipulatives that are skillfully used by teachers can support children’s developing
knowledge of fractions, but only when these materials are seen as representations of
many, many examples of divided quantities.
Implications for Teaching and Future Research
From their constructions of students’ fraction schemes, D’Ambrosio and
Mewborn (1994) concluded from their findings that “developing a conceptual
understanding of fractions takes time” (p. 160). Even after eight weeks of intense
investigations and observations of students’ performance on fraction activities, the
surface of this vital middle school topic is barely scratched. Considering the number of
topics to be covered in the middle school mathematics curriculum, hardly any
mathematics teacher can realistically spend so much time on only the basic aspects of the
topic. Operations on fractions must be done before seventh grade. I am suggesting that
fewer topics be taught at the sixth grade level so that ample time can be spent in the
teaching and learning of fractions. This will minimize the time spent on fraction
instruction in seventh and eighth grade giving way for the introduction of the topics
eliminated from the sixth grade curriculum. The preliminary partitioning activities
should be done in earlier grades (3 – 5) thus setting the pace for the operations and
applications of fractions in sixth grade.
Based on her studies on young children’s informal knowledge of rational
numbers, Mack (1990, 2000) recommended that mathematics teachers seek to develop a
strand of rational number knowledge based on partitioning, as a way to build upon
students’ informal knowledge. As previously mentioned, the research literature is replete
with studies that suggest that the knowledge of partitioning is necessary for having a
conceptual understanding of fraction (Armstrong & Bezuk, 1995; Behr,
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Harel, Post, & Lesh, 1992; Mack, 2000). This study has shown how the partitioning
activities that the students were engaged in provided the base for understanding
equivalent fractions and the concept of the fraction as an operator. Fraction tasks should
be designed to elicit a variety of strategies and representations from the learners. The
results of the study indicate that with adequate instruction and meaningful engaging
activities, students are capable of solving problems relating to fair sharing and the order
and equivalence of fractions. Fraction words can be used instead of fraction symbols in
the initial conceptualization of the fraction (Sáenz-Ludlow, 1994). As a matter of fact,
research (Saxe, Taylor, McIntosh & Gearhart, 2005) shows that students’ knowledge of
conventional fractional symbolization can develop somewhat independently from partwhole relations.
Students are capable of developing a deeper understanding of mathematical
symbols by relating symbolic representations to physical referents that are meaningful to
them (Ball, 1993; Mack, 1995). Implications from this study support the pedagogy of
using physical referents to aid in building middle school students understanding and
foundations of fraction concepts. Much emphasis should be placed on the unit fraction
prior to the introduction to non-unit fractions. Continuous and discrete wholes should be
used with similar intensity. Students should be provided with numerous experiences with
manipulative materials.
Further research is needed to elaborate how the activities undertaken during the
study will impact the students’ ability to add, subtract, multiply and divide fractions. A
more detailed study is necessary to investigate the measure and operator subscontructs.
Also, considering the prevalence of the acquired knowledge from the unit of geometry
that was taught before the fraction unit, an investigation could be launched as to the
effects, if any, that a geometry unit would have on the teaching and learning of fractions.
CONCLUDING REMARKS
This research does not, by any means, provide an exhaustive study on the
understanding of basic fraction concepts. The findings may confirm the role that
partitioning and problem solving tasks play in the development of students’
understanding of fractions but there may be other mathematical ideas that play a similar
critical role. Ongoing research and classroom practice will aid in identifying some of
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these ideas. In the meantime, mathematics teachers are continuously faced with the
responsibility of creating tasks that meaningful and engaging – tasks that will foster
higher order thinking skills in students. This responsibility becomes especially crucial
and challenging as US students’ mathematics scores on standardized and international
assessment tests are relatively low when compared with scores from other countries.
Certain mathematics topics have contributed to this demise, the fraction concept being at
the forefront. For the past 25 years there has been concern for the nature of students’
understanding of fractional numbers. This study attempts to investigate this phenomenon
with the hope that the findings will provide a framework from which teachers can begin
to create and model task-based activities that will prod the children towards a formal
understanding of the fraction concept. There is much to be done. This study is just one
of the springboards to achieve this goal.
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APPENDIX A
PreTest
Fraction
Name:
_______________________________
Multiple-Choice Questions
1.
Which rectangle is not divided into four equal parts?
2.
Which shows
3
of the picture shaded?
4
A.
B.
C.
D.
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3.
4.
The figure below shows that part of a pizza has been eaten. What part of the
pizza is still there?
A.
3
8
B.
3
5
C.
5
8
D.
5
3
What fraction of the circle is shaded?
A.
B.
C.
D.
1
4
1
1
Between and
4
2
1
3
Between and
2
4
3
Between and 1
4
Between 0 and
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2
4
is equivalent to ?
5
10
5.
Which picture shows that
6.
Which of these fractions is smallest?
7.
A.
1
6
B.
2
3
C.
1
3
D.
1
2
4
2
Students in Mrs. Johnson’s class were asked to tell why is greater than .
5
3
Whose reason is the best?
A.
Kelly said, “Because 4 is greater than 2.”
B.
Keri said, “Because 5 is larger than 3.”
C.
Kim said, “Because
D.
Kevin said, “Because 4 + 5 is more than 2 + 3.”
4
2
is closer than to 1.”
5
3
Free-Response Questions
General Information
1.
(a)
What is your definition of a fraction?
(b)
How would you show a friend what a fraction is? How would you show
this same fraction in another way?
179
2.
Do you like learning about fractions? Why? Why not?
3.
Why do you think we need to learn about fractions in school?
Fractions
4.
Where should you put
0
1
3
on the number line? Why?
5
2
3
4
5
5.
What does the fraction five-eighths look like? Using a diagram, show this
fraction in two different ways?
6.
How many fourths are in a whole? Draw a diagram to show this.
7.
Think carefully about the following question. Write a complete answer. You
may use drawings, words, and numbers to explain your answer. Be sure to show
all your work.
1
1
of a pizza. Ella ate of another pizza. Jose said that he ate more
2
2
pizza than Ella, but Ella said they both ate the same amount. Use words and
pictures to show that Jose could be right.
Jose ate
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8.
6 persons shared 3 pizzas. How much was each person’s share? Why do you
think so?
9.
Here are some cookies:
Suppose you eat only the dark ones. How much of the cookies would you eat?
Why do think so?
If the cookies are rearranged as follows:
How much of the cookies would you eat now if you eat the dark ones? Why do
you think so?
10.
The shaded part of each strip below shows a fraction.
A.
This fraction strip shows
3
.
6
B.
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What fraction does this fraction strip show? ______________
C.
What fraction does this fraction strip show?
What do the fractions shown in A, B, and C have in common?
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
Fill in the fraction strips below to show two different fractions that are equivalent
to the ones shown in A, B, and C.
11.
The circles below represent two pies of the same size – one for you and one for
your friend.
you eat this much
your friend eat this much
Did you eat as much pie as your friend? Why do you think so?
182
12.
Bert’s father cuts a cake into 8 pieces. He is going to take 3-fourths of the cake to
the party. How many pieces of cake will he take with him?
A. Draw a picture below to solve the problem.
B. The number of pieces of cake taken to the party is: ______________________
13.
is the unit. You have two pieces this size. What fraction is
this?
_____________________
14.
15.
Draw in the box a set of circles 3-fourths as many circles as you see here:
What fraction of the circle is part c? _________________
183
16.
How many fifths are shaded?
17.
Circle
18.
is 3-fourths of some length. Draw the whole
length below. Explain why it is the whole.
2
.
3
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APPENDIX B
ACTIVITIES FOR TEACHING SEQUENCE
ACTIVITY 1
Investigating Fractions:
Task 1:
a.
b.
c.
d.
e.
Task 2:
a.
b.
c.
d.
Task 3:
a.
b.
c.
Task 4:
a.
b.
c.
You are given a set of shapes – a circle, rectangle, triangle, a
parallelogram, and the letter L.
Partition the circle into three parts.
Partition the triangle into six parts.
Partition the rectangle into eight parts.
Partition the parallelogram into four parts.
Partition the letter L into five parts.
You are given twelve (12) chips.
Partition the set of chips into four parts.
Partition the set of chips into six parts.
Partition the set of chips into three parts.
Partition the set of chips into five parts.
You are given three (3) lines.
Partition the first line into two parts.
Partition the second line into five parts.
Partition the third line into eight parts.
You are given one cup of water.
Partition the cup of water into two shares.
Partition the cup of water into three shares.
Partition the cup of water into five shares.
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Individual Assessment Task
Activity 1
1.
Here are four circles each divided into three parts. Decide which circles are
partitioned and which are not. Put a check mark (√) beside the ones that you
believe are partitioned.
a.
b.
c.
d.
Describe your method of verifying that the circles are partitioned. You may use the letter
written beside the circle to identify the circle.
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
186
2.
The rectangle below represents a yummy cornbread. You want to share the
cornbread equally with five of your friends and yourself. Show two different
ways in which in you could divide the cornbread equally among the six of you.
a.
b.
3.
Romy has given you 8 cookies all of the same size to share equally for three
persons. In the space provided below, describe and model how you would share
the cookies.
187
Activity 2
Individual Tasks
Task 1.
Assemble the parts of the circle to form a whole circle. How many parts
make up the whole? Make a model of the whole with its parts in the space
below.
Task 2.
You are each given one part (yellow) of a circle. How many pieces the
same size do you believe are needed to complete the whole circle?
Group Tasks
Discuss and find the solution to these tasks.
Task 3.
You are given 5 parts of a triangle. How many more pieces of the same
size are needed to complete the whole triangle?
Task 4.
You are given 1 part (green) of circle. How many pieces of the same size
are needed to complete the whole circle?
Task 5.
You are given 8 parts of a whole square. How many more pieces of the
same size are needed to complete the whole square?
Task 6.
You are given 1 part of a cup of water. Predict how many more similar
parts will give you a whole cup of water.
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Group Assessment Task
Activity 2
Instructions: How many more pieces of the shaded parts are needed to complete the
whole object? For each one, share the method or strategy that you use to complete the
whole. You may write your solution beside the figure.
1.
Rectangle
2.
Circle
189
Instructions: Look carefully at the diagram below. How many more parts are needed to
complete the whole object? For each one, share the method or strategy that you use to
complete the whole. You may write your solution beside the figure.
3.
Circle
4.
Parallelogram
5.
Circle
6.
Rectangle
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Activity 3
(Shapes for the Activity are found at the end of Appendix B)
1.
Phil, John, Jane and Tom each have a rectangular bar of chocolate of different
sizes. The rectangles represent the size of the chocolate bars and the shaded part
represents how much of each chocolate was eaten. Did each child eat as much of
his/her chocolate bar as the other? Discuss in your group.
2.
Kris bought a small pizza from Pizza Hut while Mitch bought a small pizza from
Papa John. When the pizzas came, they found out that Kris’ pizza is smaller than
Mitch’s is. The circles below represent the pizza. The shaded part represents
how much of each pizza was eaten. Did Kris eat as much of his pizza as Mitch?
Discuss in your group?
3.
Look carefully at the shaded parts of the wholes in the following figures. Is the
part-whole relationship the same? If so, what is it? If not, why? Discuss in your
group.
4.
Is the part-whole relationship the same in these squares? If so, what is it? If not,
why? Discuss in your group.
5.
a)
Look at the set of circles and the set of diamonds. Is the part-whole
relationship the same in both set? If so, what is it? If not, why? Discuss
in your group?
b)
Compare the larger set of circles with the smaller set of circles. Is the partwhole relationship the same in both set? If so, what is it? If not, why?
Discuss in your group?
6.
Are the shaded parts in the following squares the same? What is the part-whole
relationship? Discuss in your group?
7.
The circles on the given sheet of paper represent a set of cookies. The dark ones
represent the ones that are eaten. Is the same share of cookie eaten in each set?
Discuss in your group? Explain your reasoning.
Group Assessment Task
Activity 3
Shapes are shown at the end Appendix B
Using all the figures you have been working with including some new ones, sort them
according to their part-whole relationship.
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Activity 4
Shapes are at the end of Appendix B
Task 1 (a)
1.
Partition the circle into four parts.
2.
Let’s assume that the circle represents a pie. Share this pie evenly among eight
people. Describe in words what you did.
3.
You had already divided the pie into eight equal pieces and then realize that you
have to share the pie for eight more people that is, for sixteen people? Show the
partition on your circle and describe what you did.
4.
What if you have to share the same pie with 32 people? Describe what you would
do.
Task 1 (b)
1.
Divide the rectangle into 6 equal parts. Describe what you did to arrive at your
solution.
2.
Divide this same rectangle into 12 equal parts. What did you do?
Task 2(a)
Partition this rectangle into four parts. [You could let the rectangle represent a 3 bar
chocolate and you have to share it evenly with four people.] Describe what you would do
to obtain four equal shares.
Task 2(b)
Romy ordered a pizza from Pizza Hut for you. When the pizza came it was divided into
3 equal slices. There are four of you. How would you divide this pizza into 4 equal
shares? Use the circle to show.
Task 3 (a)
This circle represents a circular birthday cake that is already divided into equal slices.
Show by shading the parts, how much one person would get if only four persons share
the cake. Explain your reasoning.
Task 3 (b)
This birthday cake is already divided into five equal pieces. Show how you would
partition this cake so that 3 people will get an equal share. Explain what you have done.
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Task 3 (c)
Divide this rectangular pizza from Dominoes (already sliced in 7 equal parts) into 3 equal
shares. Show by shading one person’s share. Explain your reasoning.
Task 3 (d)
This rectangular cake is sliced into 8 pieces. How would you share this cake with 2
persons so that each one will get the same amount? Shade the portion for 1 person.
Task 3 (e)
This L represents a flat candy cane already striped into equal shares. Repartition this
candy to share with four people.
193
Assessment Task
Activity 4
Task 1.
Here is a parallelogram that needs to be divided into 12 equal pieces. List
the steps that you would use to do this. Follow your steps on the drawing.
STEPS:
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________
Task 2:
John and his two friends have just won a giant cookie from the Cookie
Factory for their art work. When the cookie arrived they found out that
194
the cookie is partitioned only for two persons and there are three of them.
Using the circle below, how can John divide this cookie so each of the
three persons get an equal amount of the cookie? Shade the amount that
one person would get. Describe your steps.
STEPS:
___________________________________________________________
___________________________________________________________
____________________________________________________________
____________________________________________________________
195
Task 3:
This chocolate bar is divided into equal pieces. Four children have
decided to share the bar equally among them. On the rectangle below,
shade one child’s share.
196
Activity 5
TASK 1:
INSTRUCTION:
Look at the diagrams that I will be showing you. Tell what the
part-whole relationship is? Decide what it is; write it down on the paper.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
n.
o..
TASK 2:
INSTRUCTION:
Look at the diagrams that I will be showing you. Tell what the
part-whole relationship is? Decide what it is; write it down on the paper.
a.
b.
c.
d.
e.
f..
TASK 3:
INSTRUCTION:
Look at the diagrams that I will be showing you. Tell what the
unit fraction is for each of them? Decide what it is; write it down on the paper.
a.
b.
c.
d.
e.
f.
g.
h.
i.
TASK 4:
Listen to the unit fractions that I will be saying. Represent these fractions whichever way
you want to. You can do your drawing in the space below or on the blank paper you are
given.
197
Assessment Task
Activity 5
1.
Look on the diagram. Show a half.
2.
Work on these problems in your group.
a.
Here is picture of a pizza with one piece removed. The piece is
_______________________ of the whole pizza.
(say the word)
b.
Here is a picture of a candy bar which someone has started to cut into
pieces.
The small piece is ___________________ of the whole candy bar.
(say the word)
Draw lines to finish cutting the candy bar into equal parts.
c.
John has a patio that looks like this.
Draw on John’s patio to show it divided into two equal-sized parts. Each
part is ___________________ of John’s patio.
(say the word)
3.
Mary said “John’s patio is really one-half (not a whole.)” What would
you say to Mary?
Listen to the fractions that I will be saying. Draw diagrams to illustrate these
fractions.
198
4.
Gina is given these cookies. She ate some (shown by the shaded ones.)
Gina ate __________________ of her cookies.
(say the word)
Say and write the unit fraction for the cookies that Gina ate.
199
Activity 6
TASK 1:
Use your paper to fold, shade and name the following fractions.
1 1 3 1
, , ,
2 4 4 5
Task 2:
The students completed Lesson 5 (Student Pages A, B, C, D, E) of RNP Level one.
Discrete Tasks for Activity 6
Task 1:
Write the word name and the symbol name for the fraction shown below.
____________________ ________
Task 2:
Write the word name and the symbol name for the fraction shown below.
_____________________ ________
Task 3:
Write the word name and the symbol name for the fraction shown below.
____________________ _________
Task: 3:
2
Draw a picture of the fraction two-thirds ( ).
3
200
Activity 7
TASK 1:
Answer the following:
a.
How many one-fourths would be needed to make a whole? _______________
Draw a picture to show your answer.
b.
How many one-sevenths would be needed to make a whole?
_______________
c.
This shape represents
the whole.
d.
1
of whole. How many of them are needed to complete
5
_______________
Sydney divides his chocolate bar into
1
pieces. How many pieces did he get?
6
________________
DO NOT WORK ON TASK 2 UNTIL YOU ARE TOLD TO DO SO!!!
Task 2:
a.
Show three-fourths with chips. Use 12 chips in all. Draw a picture of your
display.
b.
Show two-sixths with chips. Use 6 chips in all. Draw a picture of your display.
c.
3
Describe the steps you would take to show .
7
d.
Work in your groups to complete the worksheet.
201
Assessment Task
Activity 7 (Task 1)
1.
How many fifths are in a whole?
_____________________________
2.
John ordered a large pizza from Papa John. He found out that each slice is
1
of
12
the whole pizza. How many slices of pizza did he get?
____________________________
3.
Show two-fifths with chips. Use 10 chips in all. Draw a picture of your display.
4.
Show two-sevenths with chips. Use 7 chips in all.
5.
Show
5
with chips. Use 16 chips in all.
8
202
Assessment Task
Activity 7 (Task 2)
1.
How many one-ninths are needed to make a whole?
2.
How many
3.
You may use the chips to help you to do these problems.
1
are needed to make whole?
13
__________________
__________________
a.
Fold the square sheet of paper into eight equal parts. Shade three-fourths.
b.
Fold the rectangle sheet of paper into nine equal parts. Shade
parts.
c.
This ☺☺ is
1
of whole. Draw the whole.
4
X X
d.
X X X X is
2
of a unit. Draw the unit.
9
X X
e.
♦♦♦♦
♦♦♦♦♦
is
3
of a whole. Draw the unit.
8
203
1
of the
3
Interview Tasks
Activity 7
a.
This ☺☺☺☺ is
1
of whole. Draw the whole.
4
X X X
b.
X X X X is
2
of a unit. Draw the unit.
7
X X X
c.
♦ ♦ ♦ ♦ ♦ ♦ is
♦♦♦♦♦♦
3
of a whole. Draw the unit.
4
204
Activity 8
TASK 1:
Listen carefully to the questions that I will be giving. Give your answers to these
questions.
Which is larger?
a.
One-third or one-fifth?
b.
One-seventh or one-eleventh?
Order these unit fractions from the smallest to the greatest.
1 1
1
, or
3 2
6
Order these fractions from the largest to the smallest.
1 1 1
1
, , and
.
15 7 4
10
TASK 2:
a.
Tyra entered Papa John Pizza Factory. She saw 2 friends at one table and 3
friends at another table. Both groups have just been served a large pizza. Which
group should she sit with so that she gets the most to eat? Explain your
reasoning.
b.
Using the circles below, show Tyra’s share at the table with the 2 friends and also
at the table with the 3 friends.
Table with 2 friends
Table with 3 friends
Tasks 3 & 4: Do the worksheets – Lesson 6 Student Pages B and C (RNP – Level One)
Verification Task for Activity 8: Lesson 7 – Student Pages A and B (RNP – Level
One)
205
Activity 9
Equivalent Fractions
TASK 1:
1
. Using the same strip, fold your paper into four
2
1
equal parts (1/4). How many fourths are shaded? What is the relationship between
2
2
and ? Fold the same strip into eighths, how many eighths are shaded? Write the
4
1
relationship for the three fractions and write two more equivalent fractions of .
2
Use your paper strips to fold and shade
TASK 2:
1
. Using the same strip, fold your paper into
4
eight equal parts (1/8). How many eighths are shaded? What is the relationship
1
2
between and ? Fold the same strip into sixteenths, how many sixteenths are shaded?
4
8
Write the relationship for the three fractions and write two more equivalent fractions
1
of .
4
Use your paper strips to fold and shade
TASK 3:
1
. Using the same strip, fold your paper into
4
twelve equal parts (1/12). How many twelfths are shaded? What is the relationship
1
3
between and ? Fold the same strip into twenty four equal pieces, how many twenty4
12
fourths are shaded? Write the relationship for the three fractions and write two more
1
equivalent fractions of .
4
Use your paper strips to fold and shade
206
Activity 10
Students completed Rational Number Project Lesson 18 Student Page A – Level One and
Lesson 8 Student Pages A and B – Level 2 .
Activity 11
TASK 1:
Work on Lesson 22 Task sheet.
TASK 2:
You are invited to a birthday party where 10 girls are invited. The circular
birthday cake is to be divided amongst the ten individuals. Using the given circle, show
how you would divide the cake so each person gets the same amount.
TASK 3:
You are expected to share 9 cookies equally for 4 persons. Use a model to
show how you would make your partition.
207
Activity 12
Task Sheet for Fraction Breakfast
ADDING TASTE TO FRACTIONS
GRADE 6
Name:
____________________________________________
Group #:
____________
Introduction
Today you will be taking a delicious journey into the world of fractions. As you
partake of the morning’s breakfast that is provided for you, be mindful of the
mathematics that is involved. Remember to share your insights with your group
members.
Expectations
1.
2.
3.
4.
5.
There should be no sharing of food across groups.
The food should be shared equally amongst each group member.
Answer all questions on the worksheet.
Remember that you are being videotaped and audio taped.
Please speak clearly in your discussions so that the recording will be clear.
Enjoy Your Meal!!!
208
FRACTION BREAKFAST WORKSHEET
1.
You have 12 quiches to share equally among your group members.
a.
How many quiches did each person get?
____________________________________________________________
b.
What fraction of the set of quiches did each person get?
____________________________________________________________
c.
Describe in detail the strategy or method you used in determining the
fraction of quiche each person will get. You may draw a picture to
demonstrate what you did.
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________
d.
Write two fractions that are equivalent to the fraction you wrote above.
____________________________________________________________
e.
If you give one of your quiches to one of your group members, how many
quiches will you now have? Write a fraction to represent the number of
quiches that person will have?
____________________________________________________________
f.
Write the fraction that represents the number of quiches for three group
members? Describe how you get your answer.
____________________________________________________________
____________________________________________________________
____________________________________________________________
209
2.
Share the orange juice equally among your group members. The bottle holds
64 fluid ounces. Each cup holds 9 fluid ounces.
a.
What fraction of the orange juice does each person get?
____________________________________________________________
b.
How much fluid ounces were served?
____________________________________________________________
c.
What fraction of the juice was served?
____________________________________________________________
d.
What fraction of the juice remains in the container?
____________________________________________________________
e.
If you divide the remaining portion of the juice equally with your group
members, how much fluid ounces will each of you get?
____________________________________________________________
f.
How many cups full will you get from the remaining portion of juice?
____________________________________________________________
3.
Share the three tostados (flat bread) equally among yourselves. One tostado is a
whole.
a.
What fraction of the tostados did you get?
____________________________________________________________
b.
Describe your sharing method.
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________
210
c.
Did you get less than a whole, a whole or more than a whole?
____________________________________________________________
d.
Write two fractions that are equivalent to the fraction you wrote above.
____________________________________________________________
4.
I have decided to give each group 7 pancakes (syrup is also provided) to share
equally among the group members.
a.
Describe the way you choose to share the pancakes. You may draw
pictures to show your partition.
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________
b.
With your group members, discuss then write another way you believe the
pancakes could be shared equally.
____________________________________________________________
____________________________________________________________
____________________________________________________________
c.
If you do not want pancakes, how many pancakes will each member get if
you divide the 7 pancakes equally with the other members of the group?
____________________________________________________________
211
Extended Activity [Individual – ON YOUR OWN]
5.
1 pint of milk is divided equally between 3 cups, all the same size. How much of
the pint of milk will each cup hold?
__________________________________________________________________
6.
A sausage has been sliced into 28 slices of the same size. How many slices
3
would equal of the sausages. Use picture to demonstrate your answer.
7
Thoughts and Comments
7.
8.
(a)
What is your definition of a fraction?
(b)
How would you show a friend what a fraction is? How would you show
this same fraction in another way?
Do you like learning about fractions? Why? Why not?
212
9.
Why do you think we need to learn about fractions in school?
10.
How have the exercises you have been doing for the past weeks helped you in
working with fractions?
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________
11.
You may add any comments pertaining what you have been doing for the past 6
weeks in fractions in the lines below.
____________________________________________________________
____________________________________________________________
_____________________________________________________________
______________________________________________________________
______________________________________________________________
213
INTERVIEW #1
(20 – 30 minutes)
I am going to ask you some questions about the tasks that we previously did in class. I
am very interested in how you come up with the answers, so it is important for you to tell
me what you are thinking about. The interview will not be graded, so you do not have to
worry about wrong answers. Are you ready?
Listen to this story:
Fay, Laura and Jim bought a bag of circular cookies. The bag contains 11 cookies. How
should they share the cookies so each person gets the same amount? Here are some paper
cookies that you can use to help you. Talk aloud as you solve the problem. Tell me what
you are thinking. You can also make a picture to help you.
Listen to this story:
March 5, 2005 is Paul’s birthday. He bought a square cake for his seven friends and him
to eat. He wants to divide the cake into eight equal parts. Can you show Paul two
different ways that he could share his cake? You can use the cut out squares to help you.
Talk aloud as you solve the problem.
214
SHAPES FOR ACTIVITIES 3 & 4
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
APPENDIX C
NAMES, CATEGORIES AND GROUP ASSIGNMENTS
PSUEDONYMS
Names
Category
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
1
Jay
Dave
Josh
Chrissy
Paul
Polly
Mary
Richy
Alton
Kisha
Claudia
Bob
Jonah
Ashley
Karla
Dahlia
Marla
Ben
Brian
Carol
The Five Groups with the Student’s Pretest Category
1
Jay -3
Bob - 2
Kisha - 2
Claudia - 2
2
Polly -3
Brian - 2
Marla - 2
Ben - 2
3
Mary - 3
Josh - 3
Jonah - 2
Richy - 2
238
4
Dave - 3
Karla - 2
Dahlia - 2
Carol - 1
5
Chrissy – 3
Paul – 3
Alton – 2
Ashley – 2
APPENDIX D
CONSENT FORMS
PARENTS’ LETTER
UNDERSTANDING STUDENTS’ CONSTRUCTION OF FRACTIONS
I am a graduate student under the direction of Dr. Elizabeth Jakubowski in the College of
Education at Florida State University. I am conducting a research study to get an indepth look into sixth-graders’ understanding of constructing fractions.
To get further understanding of this phenomenon, the students will participate in a group
event that will be video- and audio-recorded. Each student will complete a set of
questions directly related to the group activity. This will be used for the solely used for
document analysis. Some students will also be interviewed on an individual basis. Each
interview session will take approximately 30 – 50 minutes and will be audio-taped
I am inviting your child, _____________________________, to participate in this study.
Participation in this study is voluntary. If your child chooses not to participate or to
withdraw from the study at any time, there will be no penalty (it will not affect his/her
grade.) The results of the research may be published. A pseudonym will be used to
protect the subject’s identity.
If you have any questions concerning this research study, please call me at (305) 6211077 or you can email me at [email protected].
_____________________________________
Veon Murdock-Stewart
239
CHILD ASSENT FORM
UNDERSTANDING STUDENTS’ CONSTRUCTION OF FRACTIONS
I have been informed that my parents have given permission for to participate, if I want
to, in a study concerning my understanding of constructing fractions. I maybe
interviewed and I will also be asked to do some written work which is apart of a group
activity. Non-participation or withdrawal from the study will not affect my grade in any
way. If I choose to participate and the results of this research study be published, my
name or identity will not be revealed. All records of my participation will be kept strictly
confidential.
__________________________________
Child’s Name
__________________________________
Researcher's signature
__________________________________
Date
240
INFORMED CONSENT
UNDERSTANDING STUDENTS’ CONSTRUCTION OF FRACTIONS
I HAVE BEEN INFORMED THAT:
1.
Veon Murdock-Stewart, who is a student at Florida State University, has
requested the participation of my child in a research study done at his/her school.
2.
The purpose of the research is to get an in-depth look into the child’s
understanding of constructing fractions.
3.
My child’s participation will involve a group activity and possibly an interview
which will be audio-taped. Non-participation or withdrawal from the study will
not affect my child’s grade.
4.
Your participation, as well as that of your child, in this study is voluntary. There
will be no penalty for your child’s non-participation.
4.
The results of this research study may be published but the child’s name or
identity will not be revealed. All records of participation will be kept strictly
confidential. The results from this study will be reported in a written research
report and an oral report during a class presentation. Information about the project
will not be made public in any way that identifies individual participants.
5.
I have read the above form, understand the information read, understand that I can
ask questions or withdraw my child from the study at any time.
6.
I will not be paid for my child’s participation.
I give consent for my child, ______________________________________ to participate
in the above study.
__________________________________
Child’s Name
______________________________
Parent’s Name
__________________________________
Date
______________________________
Parent's signature
241
Florida State
UNIVERSITY
Office of the Vice President For
Research Human Subjects
Committee Tallahassee, Florida
32306-2763 (850) 644-8633 • FAX
(850) 644-4392
APPROVAL MEMORANDUM
Date: 1/10/2005
To: Veon Murdock-Stewart 18401 NW 56 st Ave Miami FL 33055
Dept.: MATHEMATICS EDUCATION
From: John Tomkowiak, Chair
Re: Use of Human Subjects in Research
Making sense of students' understanding of fractions: An action-research
study of sixth graders' construction of fraction concepts through the use of physical
referents and real world representations
The forms that you submitted to this office in regard to the use of human subjects in the
proposal referenced above have been reviewed by the Human Subjects Committee at its
meeting on 12/8/2004. Your project was approved by the Committee.
The Human Subjects Committee has not evaluated your proposal for scientific merit,
except to weigh the risk to the human participants and the aspects of the proposal related
to potential risk and benefit. This approval does not replace any departmental or other
approvals which may be required.
If the project has not been completed by 12/7/2005 you must request renewed
approval for continuation of the project.
You are advised that any change in protocol in this project must be approved by
resubmission of the project to the Committee for approval. Also, the principal
investigator must promptly report, in writing, any unexpected problems causing risks to
research subjects or others.
By copy of this memorandum, the chairman of your department and/or your major
professor is reminded that he/she is responsible for being informed concerning research
projects involving human subjects in the department, and should review protocols of such
investigations as often as needed to insure that the project is being conducted in
compliance with our institution and with DHHS regulations.
This institution has an Assurance on file with the Office for Protection from Research
Risks. The Assurance Number is IRB00000446.
cc: E.Jakubowski
HSC No. 2004.847
242
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BIOGRAPHICAL SKETCH
VEON MURDOCK-STEWART
PROFESSIONAL PROFILE
Experienced mathematics teacher seeking to complete a doctorate degree in
mathematics education.
OBJECTIVE
To be an effective and efficient tech-savvy mathematics teacher and educator.
ACADEMIC PREPARATION
M. Sc. in Education, Florida State University, Tallahassee, Florida, 2001
Concentration: mathematics education
A. Sc. in Information Science, West Indies College, Mandeville, Jamaica, W.I,
1993
Major Courses: COBOL, Pascal, C++, Systems Analysis
B. Sc. in Mathematics, West Indies College, Mandeville, Jamaica, W.I., 1986
Major Courses: Calculus, Finite Mathematics, Statistics, Differential
Equations, Abstract Algebra, Trigonometry, College
Geometry, Number Theory
Diploma in Primary Education, West Indies College, Mandeville, Jamaica,
W.I., 1981
PROFESSIONAL EXPERIENCE
Mathematics Teacher, 2001 – present
Greater Miami Academy, Miami, Florida
Grades 6 - 9
Tutor
American Reading and Tutoring Services, 2001 - 2002
Algebra I & II, Trigonometry, Analytic Geometry
255
Teaching Assistant
Florida State University, Department of Curriculum & Instruction, January
– August 2001
Teaching Assistant
Florida State University, Department of Mathematics, August December 2000
Mathematics Teacher
Head of Mathematics and Computer Science Department
Belair School, Mandeville, Jamaica, W.I.,1994-1999
Grades 7 – 12
Instructor
Northern Caribbean University (Formerly West Indies College),
Mandeville, Jamaica, W.I., 1990-1999 & Summer 2000
College Algebra, Pre-Calculus, COBOL & Pascal Programming
Instructor
Northern Caribbean University (Formerly West Indies College),
Summer 2000
College Algebra, COBOL & Pascal Programming
Mathematics Teacher
Savanna-la Mar High School, Westmoreland, Jamaica, 1981-1983 &
1986-1990
Grades 7-11
PROFESSIONAL MEMBERSHIPS AND AFFILIATIONS
- National Council of Teachers of Mathematics (NCTM)
- Association for Supervision and Curriculum Development
HONORS AND AWARDS
- Recipient of College of Education Teaching Fellowship, 1999-2000
- Teacher of the Year Award, Project Motivational Math, Inc., 2003
- Teacher of the Year Award, Greater Miami Academy, 2002
RESEARCH NTERESTS
Impact of technology on the teaching and learning of mathematics
Teaching fractions for understanding
Students’ understanding of fractions
256