Building
Foundations
f o r Pre-Service
Mathematics
A Thesis Presented to
The Faculty of the Mathematics Program
California State University Channel Islands
In (Partial) Fulfillment
of the Requirements for the Degree
Masters of Science
by
Jaimee Morrison
May 2012
Teachers
in
© 2012
Jaimee Morrison
ALL RIGHTS RESERVED
Approved for t h e M a t h e m a t i c s Program
Doctor Ivona Grzegorczyk
Doctor Brian Sittinger
DateMay15,2012
DateMay15,2012
Approved for t h e University
Doctor
Gary A. Berg
DateMay15,2012
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Title of ItemBuildingFoundationsforPre-ServiceTeachersinMathematics
3 to 5 keywords or phrases to describe the itemPre-ServiceTeachersFractionsMathematics
Authors NamePrintJaimeeErinMorrison
Authors
Signature
DateMay24,2014
This is a permitted, modified version of the Non-exclusive Distribution
license from M I T Libraries and the University of Kansas.
Table of Contents
Abstract
Page 1
Introduction
Page 1
Motivation
Page 3
Hypothesis
Page 3
Methodologies and Tools
Page 5
Participants
Page 5
Variables
Page 5
Overview of Study Design
Page 6
Description of Surveys
Page 6
Description of Tests
Page 6
Description of Traditional Method Instruction of group C G
Page 7
Description of Experimental Method Activities for groups S G 1 and S G 2
Page 8
Data and Analysis
Page 16
Results
Page 25
Conclusions
Page 26
Future Plans
Page 26
References
Page 27
Appendix
Page 28
1
Abstract
Recent studies suggest t h a t many pre-service and in-service e l e m e n t a r y school t e a c h e r s are not
equipped with t h e a d e q u a t e c o n t e n t knowledge required t o t e a c h e l e m e n t a r y m a t h e m a t i c s .
This research study c o m p a r e d 3 sections of M a t h e m a t i c s for Elementary School Teachers
courses and investigated s t u d e n t understandings of fraction multiplication and division. One of
t h e sections was instructed using a traditional m e t h o d of teaching, while t h e o t h e r t w o sections
used c o n c r e t e hands-on models of real-life applications. The analysis shows t h a t t h e c o n c r e t e
models of real life applications improved pre-service t e a c h e r s ' conceptual understanding of
o p e r a t i o n s of fractions and improved their ability t o c r e a t e fraction story problems.
Introduction
Since t h e 1960's, t h e relationship b e t w e e n t e a c h e r characteristics, t e a c h e r conduct, and
s t u d e n t a c h i e v e m e n t has been investigated in both qualitative and quantitative studies. In
1986, Shulman introduced t h e idea of pedagogical c o n t e n t knowledge and described t h e
required knowledge for teaching as t h e intersection of c o n t e n t specific knowledge and
pedagogy (Shulman, 1986). Ma identified t h e d e e p understandings required by t e a c h e r s using
t h e phrase, "profound understandings of f u n d a m e n t a l m a t h e m a t i c s , " (Ma, 1999) and Hill, Ball,
and Shilling defined a f r a m e w o r k describing t h e required teaching knowledge, distinguishing
b e t w e e n subject m a t t e r knowledge and pedagogical c o n t e n t knowledge (2008). As explained
by Ball and Hill, in o r d e r t o effectively t e a c h m a t h e m a t i c s a c o m p e t e n t t e a c h e r n e e d s t o know
t h e correct knowledge of c o n c e p t s and procedures, an understanding of t h e underlying
principles and meanings, and an appreciation and understanding of t h e connection a m o n g
mathematical ideas; this is known as t h e knowledge of m a t h e m a t i c s for teaching, (Ball, 1990;
(HillandSchillingandBall, 2004).
It is essential t h a t t e a c h e r s have t h e required knowledge of teaching m a t h e m a t i c s , as it has
b e e n d e m o n s t r a t e d t h a t t h e r e is a positive correlation b e t w e e n a t e a c h e r ' s mathematical
knowledge and s t u d e n t a c h i e v e m e n t , (HillandRowanandBall, 2005). However, various studies
s h o w t h a t many pre-service and in-service e l e m e n t a r y school t e a c h e r s do not m e e t this
standard w h e n teaching fraction operations. At Tel-Aviv University, Tirosh found in her study of
enhancing prospective t e a c h e s ' knowledge t h a t most of t h e m knew how t o divide fractions but
could not explain t h e procedure, (Tirosh, 2000). She found t h a t t h e y had not developed a
conceptual understanding of fraction division. In research d o n e at t h e University of
Wollongong, Forrester and Chinnappan f o u n d t h a t pre-service t e a c h e r s ' understanding of
fraction division is predominately procedural, (ChinnappanandForrester, 2010). They did not
d e m o n s t r a t e a d e e p understanding of fraction c o n c e p t s and can t h e r e f o r e not pass conceptual
2
knowledge o n t o their s t u d e n t s . These studies are examples t h a t d e m o n s t r a t e t h a t overall, preservice t e a c h e r s have technical c o m p e t e n c e in fraction division and multiplication but are
unable t o conceptually explain t h e procedures, and t h e r e f o r e cannot t e a c h t h e s e concepts.
Recent m a t h e m a t i c s education literature identifies w o r d - p r o b l e m writing as an a v e n u e for
assessing s t u d e n t s ' conceptual understanding of various mathematical concepts, (Barlow
and
Cates, 2007; BarlowandDrake, 2008). As proposed by Luo, w o r d - p r o b l e m writing can also serve
as a tool for evaluating t h e mathematical c o n t e n t knowledge of pre-service t e a c h e r s , (2009).
Various studies have d e m o n s t r a t e d t h a t pre-service t e a c h e r s have significant difficulty in
writing word problems. At Washington University, Azim found t h a t only 28% of pre-service
t e a c h e r s w e r e able t o describe a situation modeled by multiplication with fractions, (1995). As
Ball investigated t h e mathematical understandings of prospective t e a c h e r s , she found t h a t very
f e w secondary c a n d i d a t e s and no e l e m e n t a r y c a n d i d a t e s w e r e able t o g e n e r a t e a
mathematically a p p r o p r i a t e r e p r e s e n t a t i o n of fraction division, (Ball, 1990). Also, Simon found
t h a t 70% of pre-service t e a c h e r s w e r e unable to symbolically r e p r e s e n t fraction division
problem, (1993).
A pre-service t e a c h e r ' s ability t o write word problems reflects their m a t h e m a t i c s literacy and
c o n t e n t knowledge, (Luo, 2009). According t o Kaiser and Willander, m a t h e m a t i c s literacy can
range f r o m illiteracy, "the inability t o cope with information regarded as culturally relevant," t o
multidimensional literacy, "Science literacy e x t e n d s beyond vocabulary, conceptual s c h e m e s ,
and procedural m e t h o d s to include o t h e r understandings a b o u t science," (2005). A pre-service
t e a c h e r ' s m a t h e m a t i c s c o n t e n t knowledge can be evaluated in t e r m s of their narrative and
paradigmatic knowledge, (Bruner, 1985; Chapman, 2006). In relation t o word problems,
narrative knowing requires a focus on t h e social context of t h e problem and paradigmatic
knowledge requires focus on mathematical models or mathematical structures t h a t are
universal and context-free, (Chapman, 2006).
In o r d e r t o improve instruction of m a t h e m a t i c s , researchers have investigated how t o improve
pre-service t e a c h e r s ' conceptual understanding. Schram, Wilcox, Lanier, and Lappan f o u n d t h a t
pre-service t e a c h e r s developed a conceptual understanding of many facts and f o r m u l a s they
had previously only memorized as a result of taking a class t h a t emphasized problem solving,
reasoning, discourse, group work, and t h e use of multiple r e p r e s e n t a t i o n s , (1988).
To e n s u r e e l e m e n t a r y s t u d e n t s have t h e mathematical knowledge for conceptually
understanding fraction multiplication and division, it is essential t h a t pre-service t e a c h e r s are
equipped with a well-developed understanding of t h e knowledge of m a t h e m a t i c s for teaching
including t h e ability t o c r e a t e fraction multiplication and division word problems, (Luo, 2009).
3
Motivation
This research w a s motivated by t h e a c u t e need to improve instruction in secondary school
m a t h e m a t i c s courses and t h e need for s t u d e n t s to develop d e e p understandings of basic
concepts. Overall, w e h o p e t o develop m e t h o d s to improve t h e conceptual understanding of
f u t u r e t e a c h e r s and e n c o u r a g e their appreciation of m a t h e m a t i c s connecting it t o real world
situations. As s t u d e n t s continue on in their education and careers their conceptual
understanding of and their disposition toward mathematical concepts will greatly influence
their ability t o t e a c h m a t h e m a t i c s . Since e l e m e n t a r y s t u d e n t s can only learn as much as their
t e a c h e r s know, pre-service t e a c h e r s should have t h e best opportunities t o build their
conceptual understanding in preparation for transferring their knowledge t o their pupils. One
of t h e most c o m m o n misconceptions concerning m a t h e m a t i c s is its abstract n a t u r e and lack of
applicability toward real world situations, so this research study examined pre-service t e a c h e r s '
understanding of how m a t h e m a t i c s c o n c e p t s are utilized in context. We believe t h a t concrete
models of real life applications will provide a d e e p e r conceptual understanding of c o n c e p t s and
allow t e a c h e r s t o c r e a t e engaging story problems in their classrooms.
Hypothesis
In this p a p e r we study t h e following hypothesis: If pre-service t e a c h e r s learn fraction division
and multiplication using c o n c r e t e models and visual r e p r e s e n t a t i o n s of real life applications,
t h e n t h e y will develop a d e e p conceptual understanding of t h e concept and be able t o c r e a t e
fraction story problems to use with their s t u d e n t s . The statistical hypothesis of this thesis is
t h a t s t u d e n t s ' ability t o c r e a t e story problems will not change during t h e course of this
experiment, and t h e alternative hypothesis is t h a t t h e s t u d e n t s ' ability will significantly
improve.
During this research, various additional statistical h y p o t h e s e s w e r e t e s t e d at various stages.
Paired t-tests w e r e d o n e with a 95% confidence level b e t w e e n t h e m e a n scores of t h e pre-tests
and post-tests of each individual class section in accordance with t h e following hypotheses:
Control Group:
Study Group1:
Study Group2:
Hsub0:mupre-test
C Gequalsmupost-test
CG
HsubO:mupre-test
S G1equalsmupost-test
SG 1
HsubO:mupre-test
S G2equalsmupost-test
SG2
Hsuba:mupre-test
C Glessthanmupost-test
CG
Hsuba:mupre-test
S G1lessthanmupost-test
SG 1
Hsuba:mupre-test
S G 2lessthanmupost-test
SG2
4
W h e r e HsubOr e p r e s e n t s t h e statistical hypothesis, Hsubar e p r e s e n t s t h e alternative hypothesis, and
Mu pre-test
andmupost-test
r e p r e s e n t t h e m e a n p r e t e s t scores post-test scores, for each section,
respectively.
Also, unpaired t - t e s t s w e r e d o n e with a 99% confidence level b e t w e e n t h e post-tests and pret e s t s results of each study g r o u p with t h e control g r o u p with t h e following hypotheses:
Control Group and Study Group 1:
Control Group and Study Group 2:
HsubO:mupre-test
C Gequalsmupre-test
SG 1
HsubO:mupre-test
C Gequalsmupre-test
SG2
Hsuba:mupre-test
C Glessthanmupre-test
SG 1
Hsuba:mupre-test
C Glessthanmupre-test
SG2
Control Group and Study Group 1:
Control Group and Study Group 2:
HsubO:mupost-test
C Gequalsmupost-test
SG 1
HsubO:mupost-test
C Gequalsmupost-test
SG2
Hsuba:mupost-test
C Glessthanmupost-test
SG 1
Hsuba:mupost-test
C Glessthanmupost-test
SG2
Pearson correlations w e r e d o n e b e t w e e n survey q u e s t i o n s and post-test results with values
based on t h e scale depicted in Figure 1.
Figure 1
5
M e t h o d o l o g y and Tools
Participants:
S t u d e n t s at C S U Channel Islands w h o are working toward obtaining their Bachelor of Arts in
Liberal Studies: Teaching and Learning Option, and s t u d e n t s working toward obtaining a minor
in Foundational Mathematics, take t h e course M a t h e m a t i c s for Elementary School Teachers, as
a pre-requisite for M a t h e m a t i c s for Secondary School Teachers. The course is designed t o
discuss K-8 m a t h e m a t i c s curriculum, including abstract thinking and problem solving
a p p r o a c h e s t o teaching.
The participants of this study consist of s t u d e n t s enrolled in t h r e e different sections of Math for
Elementary School Teachers, at CSUCI in Fall 2 0 1 1 and o n e section of M a t h e m a t i c s for
Elementary School Teachers. The s t u d e n t s enrolled range f r o m s o p h o m o r e s t o seniors and have
all completed or satisfied t h e r e q u i r e m e n t s for college algebra.
The control group, later d e n o t e d C G, consists of 20 s t u d e n t s , 18 f e m a l e s and 2 males; study
g r o u p 1, ST 1, consists of 20 s t u d e n t s , 19 girls and 1 boy, enrolled in Math for Elementary School
Teachers; study g r o u p 2, S T 2, consists of 18 s t u d e n t s . Each section met twice a week for 75
m i n u t e s and w a s in t h e middle of their m a t h e m a t i c s education course w h e n participating in this
research and had received instruction in whole n u m b e r o p e r a t i o n s and fraction concepts.
Variables:
We considered f o u r main variables for t h e design of this study. As this area of research relates
t o specific teaching strategies, we tried t o assess t h e different learning styles of t h e participants
by administering a learning styles survey. Since w e are also interested in s t u d e n t s ' general
opinions and feelings toward m a t h e m a t i c s and fractions, we administered a Revised
M a t h e m a t i c s Anxiety Rating Scales, R MARS, Survey. Participants c o m p l e t e d a d e m o g r a p h i c
survey so w e could analyze any relationships b e t w e e n a s t u d e n t ' s background and previous
education with their pre- and post-test p e r f o r m a n c e . Finally, t o evaluate participants' prior
knowledge of fraction division and multiplication, (including their conceptual and procedural
understanding of t h e concepts), s t u d e n t s completed a pre-test.
6
O v e r v i e w of s t u d y d e s i g n :
Each groups' participation consists of f o u r consecutive sessions.
C G- Control Group: During t h e first day of t h e study, C G took t h e fraction pre-test. During t h e
second session, C G received traditional lecture-based instruction on fraction multiplication and
during third session they received traditional lecture-based instruction on fraction division.
During t h e final session, C G c o m p l e t e d t h e fraction post-test.
S G 1/ S G 2- Study G r o u p 1 and Study Group 2: During t h e first day of t h e study, S G 1 and S G 2 took
t h e fraction pre-test and filled o u t t h e Learning Styles Survey, Demographic Survey, and R MARS
Survey. During t h e second session, S G 1 and S G 2 participated in t h e designed fraction
multiplication activities. During t h e third session, S G 1 and S G 2 participated in t h e designed
fraction division activities. During t h e final session, S G 1 and S G 2 took t h e fraction post-test.
Description of Surveys:
Groups SG1 and S G 2 completed t h r e e surveys on t h e first day of their participation, t h e
Learning Styles Survey (Appendix- A 1), t h e Demographic Survey (Appendix- A 2), and t h e R MARS
Survey (Appendix- A 3) .The Learning Styles Survey consisted of q u e s t i o n s used t o identify
s t u d e n t s as an auditory language, auditory numerical, kinesthetic, visual language, visual
numerical, individual, group, expressiveness oral, a n d / o r expressiveness written learner. The
results of this survey are t h e n totaled and c o m p a r e d on a scale measuring learning styles as
major, minor, or not applicable for each participant. The Demographic Survey consisted of 10
q u e s t i o n s identifying each participants' background including, but not limited to, ethnicity, age,
m a t h courses completed, work schedule, and f u t u r e goals. The participants also completed t h e
R MARS Survey w h e r e t h e y rated how anxious different mathematical situations m a d e t h e m
feel on a scale f r o m 1 t o 5, 1 meaning "not at all anxious" and 5 meaning "very anxious." These
surveys are used t o d e t e r m i n e if t h e r e is a significant correlation b e t w e e n participants' learning
style, d e m o g r a p h i c information, and level of math anxiety and t h e post-test results.
Description of Tests:
Groups C G, S G 1, and S G 2 all c o m p l e t e d t h e fraction pre-test (Appendix- B 1) on t h e first day of
their participation and a fraction post-test (Appendix- B 2) on t h e last day of their participation.
The q u e s t i o n s on t h e s e t e s t s w e r e designed t o d e t e r m i n e t h e progress s t u d e n t s have m a d e
during their participation in this study. The pre-test and post-test differed in t h e n u m b e r
7
s e n t e n c e s with which t h e pre-service t e a c h e r s worked, but w e r e of equal difficulty so
i m p r o v e m e n t could be m e a s u r e d .
The word problems on t h e pre- and post-test w e r e scored using a rubric (Appendix-B 3)
developed by Luo, based on Kaiser and Willander's discussion of mathematical literacy and
Bruner's description of pragmatic and narrative knowledge, (Luo, 2009). Each word problem
w a s assigned o n e of five p e r f o r m a n c e levels ranging f r o m 0, a failing level, t o a perfect score of
4, w h e r e each level described t h e d e m o n s t r a t e d conceptual understanding of t h e participant.
In addition, t h e computational problems on t h e pre- and post-test w e r e scored using a rubric
(Appendix-B 4) developed to m e a s u r e t h e procedural c o m p e t e n c y of t h e participants. Each
c o m p u t a t i o n problem was also assigned o n e of five p e r f o r m a n c e levels ranging f r o m 0, a failing
level, to a perfect score of 4.
Description of Traditional M e t h o d Instruction of C G- Control Group
During session 1, C G participants completed t h e fraction multiplication and division pre-test.
During session 2, participants w e r e t a u g h t multiplication of fractions as described in t h e
t e x t b o o k Math for Elementary
School Teachers,
(MusserandBurgerandPeterson, 2006).
Participants w e r e p r e s e n t e d with t h r e e m e t h o d s / models of fraction multiplication: t h e
r e p e a t e d addition approach, t h e n u m b e r line approach, and t h e area model. Participants w e r e
provided with t h e following definition of fraction multiplication:
Letadividedbybandcdividedbydbeany fractions.
by
d
equals
a
c
divided
by
Participants
two-fifths.
w e r e
t h e n
p r e s e n t e d
w i t h
v a r i o u s e x a m p l e s :
Thenadividedbybtimescdivided
b
c.
C o m p u t e
t h e
f o l l o w i n g
p r o d u c t stwo-thirdstimesfive-thirteenths,th
Finally, participants w e r e p r e s e n t e d with t h e properties of fraction multiplication: closure,
associativity, distributivity, commutativity, identity, and
the
existence of t h e multiplicative
During session 3, participants w e r e t a u g h t t h e t h r e e different t y p e s of fraction division
t e x t b o o k Math for Elementary
School Teachers,
(MusserandBurger
of fractions with c o m m o n d e n o m i n a t o r s :
Let
b
adividedbybandcdividedbybbefractions
equals
a
with
cdoesnotequal0, thenadividedbybdividedbycdividedby
divided
by
c.
8
Participants w e r e also t a u g h t t h e missing factor approach for dividing fractions:
Let
a divided by b and c divided by d be fractions with cdoesnotequal0.
If a divided by b divided by c divided by d equals e divided by f, then a divided by b.
S t u d e n t s w e r e then t a u g h t t h e m e t h o d of dividing fractions with u n c o m m o n d e n o m i n a t o r s as
follows:
Let a divided by b and c divided by d be fractions with c does not equal 0, then a divided by b
divided by c divided by d equals a divided by b times d divided by c.
Finally, participants did various examples using each definition and m e t h o d of fraction division.
During t h e fourth session, participants took t h e fraction multiplication and division post-test.
Description of Experimental M e t h o d Activities for S G 1 a n d S G 2
During t h e first session, groups S G 1 and S G 2 participants took t h e fraction multiplication and
division pre-test, and filled out t h e Demographic, Learning Styles, and R MARS Survey.
During t h e second session, groups S G 1 and S G 2 participated in t h e designed fraction
multiplication activities. When s t u d e n t s arrived t o class t h e r e w e r e six stations set up around
t h e classroom, with the following materials:
Pumpkins and gourds- pumpkins and go
plastic bags, bowl, scissors, glue, q u e s t i o n s s h e e t , a n d manipulative c u t - o u t s
9
C o f f e e G r o u n d s - 8ounceso f s a n d ( c o f f e e ) , m e a s u r i n g s c o o p sone-fourth,...,1 cup,
bags, scissors, glue, q u e s t i o n s s h e e t , and manipulative
zip lock
cut-outs
Ribbons and Tiaras- Tiara, ribbon, measuring tape, ruler, scissors, glue,
questions sheet, and manipulative cut-outs
Hershey's Chocolate- Hershey's Chocolate bars, dominos, glue, scissors,
questions sheet, and manipulative cut-outs
10
Police A c a d e m y - p o l i c e m e n , scissors, glue, q u e s t i o n s s h e e t , a n d
manipulative
cut-outs
C h e x P a r t y B r a i n M i x - B o x o f C h e x m i x c e r e a l , MandM s , p r e t z e l s , a n i m a l c r a c k e r s ,
m e a s u r i n g s c o o p sone-fourth,..,1 c u p , z i p l o c k b a g s , s c i s s o r s , g l u e , q u e s t i o n s s h e e t , a n d
manipulative
cut-outs
T h e f o l l o w i n g c o n s i s t e d of t h e p r o b l e m s r e l a t e d t o e a c h a c t i v i t y ( A p p e n d i x - C 1):
Pumpkins and Gourds Activity
Y o u n e e dnine-fourthsp o u n d s o f p u m p k i n s a n d g o u r d s t o m a k e a d e l i c i o u s p u m p k i n ( a n d g o u r d s )
p i e . If t h e p u m p k i n s a n d g o u r d s c o s t $2 p e r p o u n d ; h o w m u c h m o n e y w i l l y o u s p e n d
buying the pumpkins and gourds?
Answer the following questions:
1. W h a t n u m b e r s e n t e n c e c a n b e u s e d t o s o l v e t h i s p r o b l e m ?
2 . S o l v e t h e p r o b l e m u s i n g t h e p u m p k i n a n d g o u r d s a n d d o l l a r bill m a n i p u l a t i v e s .
y o u r p r o c e s s u s i n g t h e p u m p k i n s a n d d o l l a r bill p a p e r c u t - o u t s , g l u e , a n d s c i s s o r s .
3 . S o l v e t h e n u m b e r s e n t e n c e in p a r t 1 u s i n g t h e r e p e a t e d a d d i t i o n
approach.
Demonstrate
11
Coffee Grounds Activity
You purchase at t h e store a 6 cup bag of coffee. You w a n t t o portion o u tone-thirdof t h e c o f f e e
you bought into individual servings. How much coffee will you be portioning o u t ?
Answer t h e following questions:
1. What n u m b e r s e n t e n c e can be used t o solve this p r o b l e m ?
2. Solve t h e problem using t h e "coffee grounds" and measuring cup manipulatives.
D e m o n s t r a t e your process using t h e zip lock bag cut out, glue, and scissors.
3. Solve t h e n u m b e r s e n t e n c e in part 1 using t h e r e p e a t e d addition a p p r o a c h .
Ribbons and Tiaras Activity
You are creating princess tiaras for a princess t h e m e d birthday party and each tiara
requires 10 f e e t of ribbon. This ribbon is t h e n cut into strips and tied o n t o t h e tiara. If
each strip isone-fifthof t h e total required ribbon length, how long is each strip?
Answer t h e following questions:
1. What n u m b e r s e n t e n c e can be used t o solve this p r o b l e m ?
2. Solve t h e problem using t h e ribbon and measuring t a p e manipulatives. D e m o n s t r a t e your
process using ribbon paper cut-outs, glue, and scissors.
3. Solve t h e n u m b e r s e n t e n c e in part 1 using t h e m e a s u r e m e n t a p p r o a c h .
Hershey Bars Activity
You havethree-fourthsof a candy bar left over f r o m last night's trip t o t h e movies. If you w a n t to e a t
one-thirdofyour leftovers, how much of t h e candy bar will you e a t ?
Answer t h e following questions:
1. What n u m b e r s e n t e n c e can be used t o solve this p r o b l e m ?
2. Solve t h e problem using t h e Hershey Bar (or dominos) manipulatives. D e m o n s t r a t e your
process using t h e Hershey Bar p a p e r cut-outs, glue, and scissors.
3. Solve t h e n u m b e r s e n t e n c e in part 1 using an area model.
Police Academy Activity
One-third of a class of police a c a d e m y s t u d e n t s will be out of class and participating in physical
fitness course. If t h e r e are 30 s t u d e n t s in t h e class, how many s t u d e n t s are participating
in t h e course?
Answer t h e following questions:
1. What n u m b e r s e n t e n c e can be used t o solve this p r o b l e m ?
2. Solve t h e problem using t h e police officer figure manipulatives. D e m o n s t r a t e your process
using t h e police officer figure p a p e r cut-outs, glue, and scissors.
12
3 . S o l v e t h e n u m b e r s e n t e n c e in p a r t 1 u s i n g t h e m e a s u r e m e n t
approach.
C h e x Brain P o w e r Party M i x Activity
T h e f o l l o w i n g c o n s i s t s of i n g r e d i e n t s f o r C h e x Brain P a r t y Mix.
S e r v i n g Size: 9 H u n g r y S t u d e n t s
Ingredients:
4andone-halfc u p s C h e x c e r e a l ( a n y v a r i e t y )
1
cup Animal Crackers
2andone-fourthc u p s p r e t z e l s
three-fourths c u p
p e a n u t MandM ' s
Answer the following questions:
1.
C r e a t e y o u r C h e x p a r t y mix f o r 3 s t u d e n t s i n s t e a d of 9. S o l v e t h e M & M
fraction multiplication problem using an area
2.
portioning
model.
D e v e l o p y o u r o w n f r a c t i o n m u l t i p l i c a t i o n s t o r y p r o b l e m . W h a t n u m b e r s e n t e n c e is u s e d
t o s o l v e it?
At e a c h s t a t i o n t h e r e w e r e v a r i o u s c u t - o u t m a n i p u l a t i v e s f o r p a r t i c i p a n t s t o u s e a s t h e y
c o m p l e t e d a c t i v i t i e s i n c l u d i n g : p u m p k i n s , d o l l a r bills, zip l o c k b a g s , r i b b o n s , c h o c o l a t e b a r ,
n u m b e r lines, a n d police m a n n u m b e r lines. (For activity m a n i p u l a t i v e c u t - o u t s s e e A p p e n d i x C 2)
P a r t i c i p a n t s w o r k e d in s m a l l t e a m s o f 3 o r 4 f o r 1 5 m i n u t e s a t e a c h s t a t i o n o n t h e a b o v e
f r a c t i o n multiplication real w o r l d
activities.
D u r i n g s e s s i o n t h r e e , S G 1 a n d S G 2 p a r t i c i p a t e d in t h e d e s i g n e d f r a c t i o n d i v i s i o n a c t i v i t i e s .
W h e n participants arrived t o class, t h e r e w e r e five s t a t i o n s set u p a r o u n d t h e c l a s s r o o m with
the following materials:
Pumpkins and gourds- pumpkins and gourds labeled with associated
weights,
plastic bags, bowl, scissors, glue, q u e s t i o n s s h e e t , a n d manipulative c u t - o u t s
13
C o f f e e G r o u n d s - 8ounceso f s a n d ( c o f f e e ) , m e a s u r i n gs c o o p sone-fourth,...1cup,
bags, scissors, glue, q u e s t i o n s s h e e t , and manipulative
zip lock
cut-outs
Ribbons and Tiaras- Tiara, ribbon, measuring t a p e , ruler, scissors, glue,
q u e s t i o n s s h e e t , and manipulative cut-outs
Hershey's Chocolate- Hershey's Chocolate bars, dominos, glue, scissors,
q u e s t i o n s s h e e t , and manipulative cut-outs
14
Police Academy- brownie pan, plastic knife, scissors, glue, q u e s t i o n s sheet, and
manipulative cut-outs
Chex Party Brain Mix- Box of Chex mix cereal, MandMs, pretzels, animal crackers,
measuringscoopsone-fourth...,1cup, zip lock bags, scissors, glue, q u e s t i o n s sheet, and
manipulative cut-outs
The following consisted of t h e problems related t o each activity (Appendix-C 1):
Pumpkins and gourds
You are decorating your living room table for Thanksgiving dinner and you go to t h e
store t o buy s o m e pumpkins and gourds. You have 9 g u e s t s coming for dinner and you
w a n t aone-fourthpound pumpkin or gourd for each guest, so you need t o purchasenine-fourthsp o u n d s of
pumpkins and gourds. The grocery bags used to purchase t h e pumpkins and gourds can
15
each holdthree-fourthsof a p o u n d . How many bags will you need t o purchase your nine pumpkins
and gourds?
Answer t h e following questions:
1. What n u m b e r s e n t e n c e can be used t o solve this p r o b l e m ?
2. Solve t h e problem using t h e pumpkin and gourds and pumpkin grocery bag manipulatives.
D e m o n s t r a t e your process using t h e pumpkins and bag p a p e r cut-outs, glue, and scissors.
3. Solve t h e n u m b e r s e n t e n c e in part 1 using t h e c o m m o n d e n o m i n a t o r a p p r o a c h .
Coffee Grounds
From t h e last class, we havesix-thirdscups of c o f f e e t o portion out. If w e w a n t to make daily
servings in zip lock bags oftwo-thirdscup portions, how many bags will w e fill?
Answer t h e following questions:
1. What n u m b e r s e n t e n c e can be used t o solve this p r o b l e m ?
2. Solve t h e problem using t h e coffee g r o u n d s and measuring scoop manipulatives.
D e m o n s t r a t e your process using t h e zip lock bag cut-outs, glue, and scissors.
3. Solve t h e n u m b e r s e n t e n c e in part 1 using t h e c o m m o n d e n o m i n a t o r a p p r o a c h .
Ribbons and Tiaras
Suppose you have 2andfour-twelfthsf e e t of ribbon, and you w a n t to cut it into stripsfour-thirdsf e e t long. How
many strips will you have?
Answer t h e following questions:
1. What n u m b e r s e n t e n c e can be used to solve this p r o b l e m ?
2. Solve t h e problem using t h e ribbon and measuring t a p e manipulatives. D e m o n s t r a t e your
process using t h e ribbon cut-outs, glue, and scissors.
3. Solve t h e n u m b e r s e n t e n c e in part 1 using t h e n u m e r a t o r / d e n o m i n a t o r multiples a p p r o a c h .
Hershey's Chocolate
If you have 1andone-halfHershey chocolate bars, and you w a n t t o give sandone-fourthof a chocolate bar t o
each of your friends, how many friends can you give chocolate t o ?
Answer t h e following questions:
1. What n u m b e r s e n t e n c e can be used to solve this p r o b l e m ?
2. Solve t h e problem using t h e Hershey bar (or dominos) manipulatives. D e m o n s t r a t e your
process using t h e Hershey Bar cut-outs, glue, and scissors.
3. Solve t h e n u m b e r s e n t e n c e in part 1 using t h e invert and multiply a p p r o a c h .
16
Police Academy
To celebrate t h e end of t h e police a c a d e m y program, o n e of t h e s t u d e n t s brings in a pan
of brownies. As t h e f u t u r e officer drove t o t h e a c a d e m y he got hungry at a red light and
a t etwo-eighthsof t h e brownie pan he m a d e . If he w a n t s t o give each personthree-sixteenthsof t h e left over
brownies, how many s t u d e n t s can have a brownie?
Answer t h e following questions:
1. What n u m b e r s e n t e n c e can be used t o solve this p r o b l e m ?
2. Solve t h e problem using t h e brownies and plastic knife manipulatives. D e m o n s t r a t e your
process using t h e brownie cut-outs, glue, and scissors.
3. Solve t h e n u m b e r s e n t e n c e in part 1 using t h e invert and multiple a p p r o a c h .
Chex Party Brain Power Mix
M e a s u r e o u tfive-thirdscups of t h e Chex Party Brain Mix prepared t h e previous day. Suppose you
w a n t t o makeone-fourthcup portions of your party mix in your small zip lock bags. How many
portions can you make?
Answer t h e following questions:
1. What n u m b e r s e n t e n c e can be used t o solve this p r o b l e m ? Solve this n u m b e r s e n t e n c e in
using t h e invert and multiple a p p r o a c h .
3. Develop your own fraction division story problem. W h a t n u m b e r s e n t e n c e is used t o solve it?
At each station t h e r e w e r e various cut-out manipulatives for participants to use as t h e y
c o m p l e t e d each activity including pumpkins, plastic bags, zip lock bags, ribbons, chocolate bars,
and brownies. (For activity manipulative cut-outs see Appendix- C 2)
Participants worked in small t e a m s of 3 or 4 for 15 minutes at each station on t h e above
fraction division real world activities.
During session 4, participants took t h e fraction multiplication and division post-test.
Data and Analysis
We start with descriptive statistical analysis of data collected f r o m t h e pre- t e s t s of groups C G,
S G 1, and S G 2. Recall t h a t t h e pre-test consisted of t w o word problems and t w o c o m p u t a t i o n a l
problems worth f o u r points each. The following graphs s h o w t h a t all s t u d e n t s scored low,
typically 1 or 0 out of 4, on each question of t h e pre-tests, d e m o n s t r a t i n g t h a t t h e y n e e d e d
additional instruction t o m a s t e r fractions. The following Figures 2.1-2.12 s h o w s t h e individual
scores on each question.
17
Figure 2.1
Figure 2.3
Figure 2.5
Figure 2.2
Figure 2.4
Figure 2.6
18
Figure 2.7
Figure 2.8
Figure 2.9
Figure 2.10
Figure 2.11
Figure 2.12
19
The following Figures 2.13, 2.14, and 2.15 s h o w total scores for each s t u d e n t on t h e pre-test.
(Score o u t of 16 points)
Figure 2.14
Figure 2.13
Figure 2.15
The statistics below confirms that all three sections were initially very similar. Figures 3.1- 3.3 show
descriptive parameters.
SG 1
Mean
Median
Mode
Standard Dev.
Variance
Range
Minimum
Maximum
Sum
Count
Figure 3.1
SG 2
1
1
0
1.12
1.26
4
0
4
20
20
Mean
Median
Mode
Standard Dev.
Variance
Range
Minimum
Maximum
Sum
Count
CG
1.06
1
1
.64
.41
2
0
2
19
18
Figure 3.2
Mean
Median
Mode
Standard Dev.
Variance
Range
Minimum
Maximum
Sum
Count
1.05
1
0
1.09
1.21
4
0
4
21
20
Figure 3.3
20
We performed un-paired t-tests to compare the three sections. The results displayed in Figure 3.4 show
no significant difference between the three groups
Un-Paired T-test Results (1-tail)
S G1, S G 2
pequals.425
S G1, C G
pequals.444
S G2, C G
pequals.492
Figure 3.4
Since p-values are close to .5, T-test initial comparisons, with significance level of 100% , show that there
were no initial differences between the three sections and they were therefore comparable.
The following consists of descriptive statistical analysis of data collected from the post- tests of C G, S G 1,
and S G 2. Recall that the post-test also consisted of two word problems and two computation problems
worth four points each. From the graphs below we can see that the study groups S G 1 and S G 2
performed much better than the control group. Figures 4.1-4.12 show scores on individual questions
(out of 4).
Figure 4.1
Figure 4.3
Figure 4.2
Figure 4.4
21
Figure 4.5
Figure 4.6
Figure 4.7
Figure 4.8
Figure 4.9
Figure 4.10
22
Figure 4.11
Figure 4.12
Figures 4.13, 4.14, and 4.15 show the post-test results for each section, out of 16.
Figure 4.13
Figure 4.14
Figure 4.15
23
The following descriptive statistics s h o w s t h a t group C G's post-test scores w e r e much lower
t h a n groups S G 1 and S G 2, shown in Figures 5.1-5.3.
SG 1
Mean
12.2
Median
13.5
Mode
14
Standard Dev. 3.41
Variance
11.6
Range
13
Minimum
3
Maximum
16
Sum
244
Count
20
SG 2
Mean
Median
Mode
Standard Dev.
Variance
Range
Minimum
Maximum
Sum
Count
Figure 5.1
CG
13.3
13.5
13
2.7
7.03
12
4
16
239
18
Mean
Median
Mode
Standard Dev.
Variance
Range
Minimum
Maximum
Sum
Count
Figure 5.2
4.7
4
3
34
11.5
12
1
13
94
20
Figure 5.3
We p e r f o r m e d paired t - t e s t s t o evaluate t h e i m p r o v e m e n t for each section f r o m pre-test to
post-test. The following chart, Figure 5.4, displays t h e results:
Paired t-test Results
SG 1
1.06052 E-11
SG 2
1.37597 E-13
CG
3.38205 E-06
Figure 5.4
Note t h a t p-values are lower t h a n .01, hence our s t a t e m e n t s are valid at 99% c o n f i d e n c e level.
Recall t h e following h y p o t h e s e s :
Study Group l:
Control Group:
Study Group 2:
HsubO:mupre-test
S G1equalsmupost-test
SG 1
Hsuba:mupre-test
S G1lessthanmupost-test
SG 1
HsubO:mupre-test
Hsuba:mupre-test
S G 2equalsmupost-test
S G 2lessthanmupost-test
SG2
SG2
HsubO:mupre-test
C Gequalsmupost-test
CG
Hsuba:mupre-test
C Glessthanmupost-test
CG
For group S G 1, with a p — v a l u eequals1 . 0 6 0 5 2 Eminus11lessthan.01, at t h e 99% confidence level we
reject t h e null hypothesis (Ho) and w e accept t h e alternative hypothesis (Ha), and t h e m e a n
scores for t h e post-test is g r e a t e r t h a n t h e m e a n scores for t h e pre-test. Therefore,
participants' scores in this section improved significantly t h r o u g h t h e designed activity.
For group S G 2, with a p — v a l u eequals1 . 3 7 5 9 7 Eminus1 3lessthan.01, at t h e 99% confidence level we
reject t h e null hypothesis (Ho) and w e accept t h e alternative hypothesis (Ha), and t h e m e a n
24
scores for t h e post-test is g r e a t e r t h a n t h e m e a n scores for t h e pre-test. Therefore,
participants' scores in this section improved significantly t h r o u g h t h e designed activity.
For group C G, with a p —valueequals3.38205Eminus0 6lessthan.01, at t h e 99% confidence level w e reject
t h e null hypothesis (Ho) and we accept t h e alternative hypothesis (Ha), and t h e m e a n scores for
t h e post-test is g r e a t e r t h a n t h e m e a n scores for t h e pre-test. Therefore, participants' scores in
this section improved significantly through t h e traditional m e t h o d .
In addition, unpaired t-tests w e r e p e r f o r m e d t o evaluate t h e level of i m p r o v e m e n t of
participants' post-test scores c o m p a r e d with each section. The following chart, Figure 5.5,
displays t h e results:
Unpaired t-test Results
S G 1 and C G
1.33067 E-08
S G 2 and C G
1.40724 E-10
Figure 5.5
Recall our hypotheses:
Control Group and Study Group 1:
HsubO:mupost-test
C Gequalsmupost-test
H sub a: mu p o s t - t e s t C Glessthanmupost-test
SG 1
SG 1
Control Group and Study Group 2:
HsubO:mupost-test
C Gequalsmupost-test
SG2
Hsuba:mupost-test
C Glessthanmupost-test
SG2
For S G 1 and C G, with a p — valueequals1 . 3 3 0 6 7 Eminus0 8lessthan.01, at t h e 99% confidence level w e
reject t h e null hypothesis (Ho) and w e accept t h e alternative hypothesis (Ha). Therefore, S G 1's
post-test scores w e r e significantly g r e a t e r t h a n C G's post-test scores.
For S G 2 and C G, with a p — v a l u eequals1 . 4 0 7 2 4 Eminus10lessthan.01, at t h e 99% confidence level w e
reject t h e null hypothesis (Ho) and w e accept t h e alternative hypothesis (Ha). Therefore, S G 2's
post-test scores w e r e significantly g r e a t e r t h a n C G's post-test scores.
Now we will study t h e influences of qualitative variables collected on t h e Demographic, R MARS,
and Learning Styles Survey and t h e post-test results. The following chart, Figure 6, consists of
t h e results f r o m comparing t h e strength and direction of a linear relationship b e t w e e n t h e
r a n d o m variables using a Pearson correlation. The assigned significance is based on Figure 1:
25
Correlations Between Survey Questions and Post Test Results
Survey Question
Correlation Value
Correlation Significance
Learning Styles: Visual
.2329
Weak Positive Correlation
Learning Styles: Auditory
-.1798
Weak Negative Correlation
Learning Styles: Kinesthetic
.5509
Moderate Positive Correlation
R MARS
.3352
Weak Positive Correlation
Demographic: Gender
-.1105
Weak Negative Correlation
Demographic: Age
.09941
Weak Positive Correlation
Demographic: Ethnicity
.00030
Weak Positive Correlation
Demographic: Employment
-.1796
Weak Negative Correlation
Demographic: Semesters on campus
.1127
Weak Positive Correlation
Demographic: Level of education
-.2132
Weak Negative Correlation
Demographic:numberof units
.0089
Weak Positive Correlation
Demographic:numberof math classes
.1224
Weak Positive Correlation
Demographic: First education class
-.1647
Weak Negative Correlation
Demographic: Goal
.0436
Weak Positive Correlation
Figure 6
T h e r e f o r e t h e r e is a m o d e r a t e positive correlation b e t w e e n s t u d e n t s w h o are identified as
Kinesthetic Learners and post-test results. Since o t h e r correlations are weak, we conclude t h a t
o t h e r variables w e r e irrelevant in this study.
Results
In s u m m a r y , pre-test results d e m o n s t r a t e t h a t s t u d e n t s in all sections w e r e c o m p a r a b l e in
t e r m s of their ability t o c r e a t e fraction multiplication and division story problems and
performing fraction c o m p u t a t i o n s . Post-test results d e m o n s t r a t e t h a t s t u d e n t s in all sections
improved significantly following t h e instruction t h e y received on fraction multiplication and
division. However, f u r t h e r statistical analysis d e m o n s t r a t e s t h a t pre-service t e a c h e r s t h a t learn
fraction division and multiplication using concrete models of real life applications improved
significantly m o r e t h a n s t u d e n t s t a u g h t with a traditional m e t h o d . Hence, we claim t h a t t h e r e is
sufficient evidence t h a t t h e designed activities produce b e t t e r results than t h e traditional
method.
Note t h a t this included a specific group of individuals, s t u d e n t s working toward becoming
e l e m e n t a r y school t e a c h e r s . Although participants w e r e placed in sections randomly, t h e r e
w e r e various factors t h a t could have created bias in t h e results. The following factors t h a t may
have influenced t h e validity of our results include:
Previous math courses taken, students major
Students confidence level in math
Students English Language skills
26
Physical state on testing days
Previous experience with fraction division and multiplication
In addition, t h e study groups w e r e t a u g h t by a different instructor t h e n t h e control group so
teaching style and personality might have influenced t h e results. There is a f r e e tutoring c e n t e r
on c a m p u s at C S U Channel Islands, and s o m e s t u d e n t s might have used private tutors; hence
t h e r e is also t h e possibility t h a t t h e results w e r e affected by this and o t h e r outside resources.
Finally, each section studies w a s small; so it is unclear w h e t h e r real world application stations
would work in large sections.
Conclusions
We studied specific hands-on real world applications as a special teaching t e c h n i q u e to
evaluate their influence on pre-service t e a c h e r s ' ability to c r e a t e fraction division and
multiplication story problems. The data shows t h a t pre-service t e a c h e r s ' p e r f o r m a n c e
improved significantly a f t e r participating in t h e designed activities. This research highlights t h e
need for implementing word problem writing into t h e pre-service t e a c h e r curriculum, as well as
d e m o n s t r a t e s t h e need to improve pre-service t e a c h e r s ' conceptual understanding of fraction
multiplication and division. The results f r o m this study suggest t h a t if pre-service t e a c h e r s have
t h e o p p o r t u n i t y t o explore math c o n c e p t s using hands-on explorations their ability to c r e a t e
story problems will improve.
Since t h e r e are various factors t h a t may have influenced our results, this e x p e r i m e n t should be
r e p e a t e d under different circumstances with different instructors and possibly a different
venue.
Future Plans
There are various possible extensions of this study. O t h e r concepts can be t a u g h t using handson real world activities and pre-service t e a c h e r s ' ability t o c r e a t e story problems of t h e s e
c o n c e p t s can be m e a s u r e d . It would be also interesting t o conduct a study w h e r e s t u d e n t s '
residual learning of creating fraction story problems is m e a s u r e d at t h e end of t h e s e m e s t e r as
o p p o s e d t o directly a f t e r instruction and activities. It would be valuable t o investigate w h e t h e r
s t u d e n t s participating in a class designed around word problem writing would increase their
conceptual understanding of concepts including fraction operations. It is also worth knowing
w h e t h e r an e m p h a s i s on t h e transition b e t w e e n w h o l e - n u m b e r and non-whole n u m b e r
o p e r a t i o n s would improve pre-service t e a c h e r s ' ability t o c r e a t e word problems. Finally, a study
should be conducted with a larger group of s t u d e n t s t o evaluate t h e effectiveness of this
m e t h o d in bigger classrooms.
27
References
Azim, D. S. (1995). Preservice elementary teachers' understanding of multiplication involving fractions. In D. T.
Owens, M. K. Reed, & G. M. Millsaps (Eds.), Proceedings of the seventeenth annual meeting of the North American Chapter of the
International Group for the Psychology of Mathematics Education (pp. 226-232). Columbus, OH .
Ball, D. L. (1990a). The mathematical understandings that prospective teachers bring to teacher education. The
Elementary School Journal, 90, 449-466. Fenqjen Luo 95
Barlow, A. T., & Cates, J. (2007). The answer is 20 cookies. What is the question? Teaching Children Mathematics,
13, 326-332.
Barlow, A. T., & Drake, J. M. (2008). Division by a fraction: Assessing understanding through problem
writing. Teaching Children Mathematics, 13, 326-332.
Bruner, J. (1985). Narrative and paradigmatic modes of thought. In E. Eisner (Ed.), Learning and teaching
the ways of knowing (p. 97-115). Chicago, University Press.
Chapman, O. (2006). Classroom practices for context of mathematics word problems. Educational
Studies in Mathematics, 62, 211-230.
Chinnappan, M., Forrester, T. (2010). The Predominance of Procedural Knowledge in Fractions.
MERGA, 33rd, Freemantle, Western Australia.
Hill, H.C., Rowan, B., & Ball, D. L. (2005). Effects of teachers' mathematical knowledge for teaching on student
achievement. American Educational Research Journal, 42(2), 371-406.
Hill, H.C., Schilling, S.G., & Ball, D.L. (2004). Developing measures of teachers' mathematics knowledge for
teaching. The Elementary School Journal, 105(1), 11-30.
Kaiser, G., & Willander, T. (2005). Development of mathematical literacy: 96 Evaluating the Effectiveness
and Insights Results of an empirical study. Teaching Mathematics and its Applications, 24, 48-60.
Luo, Fenqjen. (2009) Evaluating the Effectiveness and Insights of Pre-Service Elementary Teachers'
Abilities to Construct Word Problems for Fraction Multiplication. Journal of Mathematics Education, Vol. 2, No. 1, 83-98
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental
mathematics in China and the United States. Hillsdale, NJ: Erlbaum.
Musser, G. & Burger, W. & Peterson, B. (2006). Mathematics for Elementary School Teachers, A
Contemporary Approach, 9th edition. Courier Kendallville Inc.
Schram, P., Wilcox, S., Lanier, P., & Lappan, G. (1988). Changing mathematics conceptions of preservice teachers: A
content and pedagogical intervention . (Research Report No 88-4). East Lansing, MI: NCRTL, Michigan State University.
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational
Researcher, 15, 4-14.
Simon, M. A. (1993). Prospective elementary teachers' knowledge of division. Journal for Research in
Mathematics Education, 24, 233-254.
Tirosh, D. (2000). Enhancing prospective teachers' knowledge of children's conceptions:
The case of division of fractions. Journal for Research in Mathematics Education, 31(1),
5-25.
28
Appendix A 1
Learning Styles Survey
Question
Most Like Me
1. Making things for my studies helps me to remember what I have learned.
4
3
2
1
2. I can write about most of the things I know better than I can tell about them.
4
3
2
1
3. When I really want to understand what I have read, I read it softly to myself.
4
3
2
1
4. I get more done when I work alone.
4
3
2
1
5. I remember what I have read better than what I have heard.
4
3
2
1
6. When I answer questions, I can say the answer better than I can write it.
4
3
2
1
7. When I do math problems in my head, I say the numbers to myself.
4
3
2
1
8. I enjoy joining in on class discussions.
4
3
2
1
9. I understand a math problem that is written down better than one that I hear.
4
3
2
1
10. I do better when I can write the answer instead of having to say it.
4
3
2
1
11. I understand spoken directions better than written ones.
4
3
2
1
12. I like to work by myself.
4
3
2
1
13. I would rather read a story than listen to it read.
4
3
2
1
14. I would rather show and explain how something works than write about how it works.
4
3
2
1
15. If someone tells me three numbers to add, I can usually get the right answer without
writing them down.
4
3
2
1
16. I prefer to work with a group when there is work to be done.
4
3
2
1
17. A graph or chart of numbers is easier for me to understand than hearing the numbers
said.
4
3
2
1
18. Writing a spelling word several times helps me to remember it better.
4
3
2
1
Least Like Me
19. I learn better if someone reads a book to me than if I read it silently to myself.
4
3
2
1
20. I learn best when I study alone.
4
3
2
1
21. When I have a choice between reading and listening, I usually read.
4
3
2
1
22. I would rather tell a story than write it.
4
3
2
1
23. Saying the multiplication tables over and over helps me remember them better than
writing them over and over.
4
3
2
1
24. I do my best work in a group.
4
3
2
1
25. I understand a math problem that is written down better than one I hear.
4
3
2
1
26. In a group project, I would rather make a chart or poster than gather the information
to put on it.
4
3
2
1
27. Written assignments are easy for me to follow.
4
3
2
1
28. I remember more of what I learn if I learn it alone.
4
3
2
1
29. I do well in classes where most of the information has to be read.
4
3
2
1
30. I would enjoy giving an oral report to the class.
4
3
2
1
31. I learn math better from spoken explanations than written ones.
4
3
2
1
32. If I have to decide something, I ask other people for their opinions.
4
3
2
1
33. Written math problems are easier for me to do than oral ones.
4
3
2
1
34. I like to make things with my hands.
4
3
2
1
35. I don't mind doing written assignments.
4
3
2
1
36. I remember things I hear better than things I read.
4
3
2
1
37. I learn better by reading than by listening.
4
3
2
1
38. It is easy for me to tell about the things I know.
4
3
2
1
39. I make it easier when I say the numbers of a problem to myself as I work it out.
4
3
2
1
40. If I understand a problem, I like to help someone else understand it , too.
4
3
2
1
41. Seeing a number makes more sense to me than hearing a number.
4
3
2
1
42. I understand what I have learned better when I am involved in making something for
the subject.
4
3
2
1
43. The things I write on paper sound better than when I say them.
4
3
2
1
44. I find it easier to remember what I have heard than what I have read.
4
3
2
1
45. It is fun to learn with classmates, but it is hard to study with them.
4
3
2
1
29
Appendix A 1
Learning
Styles Scoring
Rubric
30
Appendix A 2
31
Appendix A 3
RMARS S u r v e y
Rate how anxious the following situations make you feel on a scale from 1 t o 5, where 1 represents "not at all" anxious, and 5 represents "very much" anxious
Rating of 1to5
Question
1.
Studying for a math test.
2.
Taking the mathematics section of college entrance exam
3.
Taking an exam (quiz) in a math course
4.
Taking an exam (final) in a math course
5.
Picking up math textbook to working on a homework assignment
6.
Being given homework assignments of many difficult problems that are due the next class meeting
7.
Thinking about an upcoming math 1 week before
8.
Thinking about an upcoming math test 1 day before
9.
Thinking about an upcoming math test 1 hour before
10. Realizing you have to take a certain number of math classes to fulfill requirements in your major
11. Picking up math textbook to begin a difficult reading assignment
12. Receiving your final math grade in the mail
13. Opening a math or stat book and seeing a page full of problems
14. Getting ready to study for a math test
15. Being give a "pop" quiz in a math class
16. Reading a cash register receipt after your purchase
17. Being given a set of numerical problems involving addition to solve on paper
18. Being given a set of subtraction problems to solve
19. Being given a set of multiplication problems to solve
20. Being given a set of division problems to solve
21. Buying a math textbook
22. Watching a teacher work on an algebraic equation on the blackboard
23. Signing up for a math course
24. Listening to another student explain a math formula
25. Walking into a math course
Total
32
Appendix A 3
R M A R S S c o r i n g Rubric
Range of Total Score
Level of math Anxiety
25to45
Not anxious
45to65
Slightly Anxious
65to85
Moderately Anxious
85to105
Anxious
105to125
Very Anxious
33
Appendix B 1
Pre-Test
1.
Demonstrate these division problems a, b, and c using three different methods of fraction division.
a. twelve-thirteenths divided by six-thirteenths equals
b.twenty-onefortiethsdividedbyseven-eighthsequals
c. six-fifteenths divided by four-sevenths equals
d. Explain why and how the invert and multiply method works.
2.
Demonstrate these multiplication problems a, b, and c using three different models/ methods of
fraction multiplication. Identify the method/ model you chose.
a.
3timesone-fourthequals
b.one-halftimessixequals
c.one-thirdtimesfive-seventhsequals
d.
3.
U s e p a t t e r n b l o c k s t o s h o w h o w t o t h i s m u l t i p l i c a t i o n p r o b l e m c a n b e c o m p u t e d :one-thirdtimestwoandone-half
Imagine that you are teaching multiplication with fractions. Create a story problem that can be
solved
w i t h t h e n u m b e r s e n t e n c eoneandthree-fourthstimestwo-thirdsequals
4.
I m a g i n e that you are teaching division with fractions. C r e a t e a story p r o b l e m that can be solved
the n u m b e r
s e n t e n c eoneandone-thirddividedbyone-fourthequals
with
34
A p p e n d i x B2
Post Test
1.
D e m o n s t r a t e t h e s e division p r o b l e m s a, b, a n d c u s i n g t h r e e d i f f e r e n t m e t h o d s of f r a c t i o n division.
a.fifteen-seventeenthsdividedbyfive-seventeenthsequals
b.twenty-seven-forty-fifthsdividedbynine-fifthsequals
c.seven-thirteenthsdividedbythree-eighthsequals
a.
2.
E x p l a i n w h y a n d h o w t h e invert a n d multiply m e t h o d w o r k s .
D e m o n s t r a t e t h e s e m u l t i p l i c a t i o n p r o b l e m s a, b, a n d c u s i n g t h r e e d i f f e r e n t m o d e l s / m e t h o d s of
f r a c t i o n multiplication. I d e n t i f y t h e m e t h o d / m o d e l y o u chose.
d.fourandone-halfequals
e.one-thirdtimessixequals
f.one-fourthtimestwo-fifthsequals
a.
3.
U s e p a t t e r n b l o c k s to s h o w h o w to this m u l t i p l i c a t i o n p r o b l e m c a n b e c o m p u t e d :two-thirdstimetwoandone-thirds
I m a g i n e t h a t y o u a r e t e a c h i n g m u l t i p l i c a t i o n w i t h fractions. C r e a t e a story p r o b l e m t h a t c a n b e solved
w i t h t h e n u m b e r s e n t e n c eoneandone-fourthstimesone-halfequals
4.
I m a g i n e t h a t y o u a r e t e a c h i n g division w i t h f r a c t i o n s . C r e a t e a story p r o b l e m t h a t c a n b e solved w i t h
t h e n u m b e r s e n t e n c etwoandone-halfdividedbythree-fourthsequals
35
A p p e n d i x B3
Multiplication W o r d Problem
Failing
(Scoreequals0)
( a ) A n individual d o e s not write a w o r d
Sample:oneandtwo- p r o b l e m ; o r
( b ) A n individual i g n o r e s b a s i c
thirds times
four equals question mark
m a t h e m a t i c a l c o n c e p t s a n d m e t h o d s ; his
C a t h e r i n eoneandtwo-thirdso r a n g e s . S h e
o r h e r w o r d p r o b l e m is not solvable b y
w a n t s t o s h a r e it a m o n g h e r
u s i n g the g i v e n multiplication sentence.
friends. H o w much oranges
each friend
does
get?
Poor
( a ) A n individual d e m o n s t r a t e s a b a s i c
u
n d e r s t a n d i n g of m a t h e m a t i c a l c o n c e p t s
(Scoreequals1)
a
n
d methods, but does not demonstrate
Sample:one-halftimes
the
ability to u n i f y a n d relate central
one-third equals question mark
J o e m a d e b r o w n i e s a n d s p l i t t h epaninone-half
mathematical concepts and methods.
f o r h i s s i s t e r . S h e a t e one-third (b)The w o r d p r o b l e m is solvable by using
a multiplication s e n t e n c e , b u t d o e s n o t
o f t h eone-halfo f h e r p a r t . If y o u t o o k
m a t c h with t h e given n u m e r i c a l value.
thetotalpan,howmuchdidshee a t ?
Weak
(Scoreequals2)
Sample:1andtwo-thirdstimes4
equals
question mark
Four children have each
of a s h e e t c a k e . H o w
( a ) A n individual d e m o n s t r a t e s a n ability
to u n i f y a n d relate central m a t h e m a t i c a l
concepts and methods, but does not
demonstrate a contextual understanding
a n d a p p l i c a t i o n of m a t h e m a t i c a l c o n c e p t s
g o t1andtwo-thirds
and methods.
many
c a k e s d o t h e y h a v e all
together?
Fair
(Scoreequals3)
Sample:one-halftimesone-third
equals question mark
Mom
Division W o r d Problem
( a ) A n individual d o e s not write a w o r d
problem; or
( b ) A n individual i g n o r e s b a s i c
mathematical concepts and methods;
his o r h e r w o r d p r o b l e m is n o t solvable
b y u s i n g the g i v e n d i v i s i o n sentence.
m a d e a cookie.
Sally
f r o s t e done-thirdso f t h e c o o k i e . O n
frosted part, she put
o none-half.H o w m u c h h a s
the
sprinkles
sprinkles?
( a ) A n individual d e m o n s t r a t e s a b a s i c
u n d e r s t a n d i n g of m a t h e m a t i c a l
concepts and methods, but does not
d e m o n s t r a t e t h e ability to u n i f y a n d
relate central m a t h e m a t i c a l c o n c e p t s
and methods.
(b)The w o r d p r o b l e m is solvable by
using a division s e n t e n c e , b u t d o e s n o t
m a t c h with t h e given n u m e r i c a l value.
( a ) A n individual d e m o n s t r a t e s a n
ability to u n i f y a n d relate central
mathematical concepts and methods,
but does not demonstrate a contextual
u n d e r s t a n d i n g a n d a p p l i c a t i o n of
mathematical concepts and methods.
(b)The w o r d p r o b l e m m a t c h e s t h e given
(b)The w o r d p r o b l e m m a t c h e s t h e
multiplication s e n t e n c e , b u t includes a
given division s e n t e n c e , b u t includes a
logical e r r o r or misleading c o n t e x t .
logical e r r o r or misleading c o n t e x t .
( a ) A n individual d e m o n s t r a t e s a
contextual u n d e r s t a n d i n g a n d a p p l i c a t i o n
of m a t h e m a t i c a l c o n c e p t s a n d m e t h o d s ,
b u t d o e s n o t d e m o n s t r a t e it in a clear o r
c o h e r e n t m a n n e r sufficiently.
( a ) A n individual d e m o n s t r a t e s a
contextual u n d e r s t a n d i n g a n d
a p p l i c a t i o n of m a t h e m a t i c a l c o n c e p t s
and methods, but does not demonstrate
it i n a clear o r c o h e r e n t m a n n e r
sufficiently.
(b)The w o r d p r o b l e m is logically a n d
contextually correct, but does not show
(b)The w o r d p r o b l e m is logically a n d
sufficient clarity or c o h e r e n c e .
contextually correct, but does not
s h o w sufficient clarity or c o h e r e n c e .
Good
(Score=1.00)
Sample:oneandtwo-thirdstimesfourequals
M a r y h a d landtwo-thirdsp i e c e s o f p i e .
had 4 times as m a n y
H o w m a n y pieces of pie
Joe
have?
Joe
pieces.
does
( a ) A n individual d e m o n s t r a t e s a
contextual u n d e r s t a n d i n g a n d a p p l i c a t i o n
of m a t h e m a t i c a l c o n c e p t s a n d m e t h o d s in
a clear a n d c o h e r e n t m a n n e r .
(b)The w o r d p r o b l e m is logically a n d
c o n t e x t u a l l y c o r r e c t , a n d clearly a n d
coherently described.
( a ) A n individual d e m o n s t r a t e s a
contextual u n d e r s t a n d i n g a n d
a p p l i c a t i o n of m a t h e m a t i c a l c o n c e p t s
a n d m e t h o d s in a clear a n d c o h e r e n t
manner.
(b)The w o r d p r o b l e m is logically a n d
c o n t e x t u a l l y c o r r e c t , a n d clearly a n d
coherently described.
36
A p p e n d i x B3
Points
Multiplication C o m p u t a t i o n
Division C o m p u t a t i o n
0
S t u d e n t is u n a b l e t o c o r r e c t l y
S t u d e n t is u n a b l e t o c o m p u t e
c o m p u t e fraction multiplication
f r a c t i o n division c o m p u t a t i o n o r
computation and cannot
explain w h y t h e invert and
d e m o n s t r a t e s t h e multiplication
multiply m e t h o d works
p r o b l e m using p a t t e r blocks
1
Student correctly c o m p u t e s
Student correctly c o m p u t e s
fraction multiplication
f r a c t i o n division c o m p u t a t i o n s
c o m p u t a t i o n s using 1 a p p r o a c h
using o n e a p p r o a c h or explains
or correctly d e m o n s t r a t e s t h e
w h y t h e invert and multiply
multiplication p r o b l e m using
method works
p a t t e r blocks.
2
Student correctly c o m p u t e s
Student correctly c o m p u t e s
fraction multiplication
f r a c t i o n division c o m p u t a t i o n s
c o m p u t a t i o n s using 2
using t w o different a p p r o a c h e s
a p p r o a c h e s or c o m p u t e s using 1
or 1 approach and explains w h y
model correctly and s t u d e n t
t h e invert and multiply m e t h o d
correctly d e m o n s t r a t e s t h e
works
multiplication p r o b l e m using
p a t t e r blocks.
3
Student correctly c o m p u t e s
Student correctly c o m p u t e s
fraction multiplication
f r a c t i o n division c o m p u t a t i o n s
c o m p u t a t i o n s using t h e n u m b e r
using t h e c o m m o n d e n o m i n a t o r
line a p p r o a c h , m e a s u r e m e n t
approach, multiplies approach,
approach, and t h e area model or
and invert and multiply a p p r o a c h
computes 2 models and student
or c o m p u t e s 2 approaches and
correctly d e m o n s t r a t e s t h e
explains w h y t h e invert and
multiplication p r o b l e m using
multiply m e t h o d works
p a t t e r blocks.
4
Student correctly c o m p u t e s
Student correctly c o m p u t e s
fraction multiplication
f r a c t i o n division c o m p u t a t i o n s
c o m p u t a t i o n s using t h e n u m b e r
using t h e c o m m o n d e n o m i n a t o r
line a p p r o a c h , m e a s u r e m e n t
approach, multiplies approach,
approach, and the area model
and invert and multiply a p p r o a c h
and s t u d e n t correctly
and s t u d e n t correctly explains
d e m o n s t r a t e s t h e multiplication
w h y t h e invert and multiply
p r o b l e m using p a t t e r blocks.
m e t h o d works.
37
Appendix C 1
Pumpkins and gourds
You n e e d-nine-fourthsp o u n d s of p u m p k i n s a n d g o u r d s t o m a k e a d e l i c i o u s p u m p k i n ( a n d g o u r d s ) p i e . If t h e
p u m p k i n s a n d g o u r d s c o s t $ 2 p e r p o u n d . H o w m u c h m o n e y will y o u s p e n d b u y i n g t h e p u m p k i n s
and gourds?
Answer t h e following questions:
1. W h a t n u m b e r s e n t e n c e c a n b e u s e d t o s o l v e t h i s p r o b l e m ?
2. S o l v e t h e p r o b l e m u s i n g t h e p u m p k i n a n d g o u r d s a n d d o l l a r bill m a n i p u l a t i v e s . D e m o n s t r a t e y o u r
p r o c e s s u s i n g t h e p u m p k i n s a n d d o l l a r bill p a p e r c u t - o u t s , g l u e , a n d s c i s s o r s .
3 . S o l v e t h e n u m b e r s e n t e n c e in p a r t 1 u s i n g t h e r e p e a t e d a d d i t i o n a p p r o a c h .
38
Appendix C 1
Coffee Grounds
Y o u p u r c h a s e a t t h e s t o r e a 6 c u p b a g o f c o f f e e . Y o u w a n t t o p o r t i o n o u tone-thirdso f t h e c o f f e e
y o u bought into individual servings. H o w m u c h coffee will you b e portioning out?
Answer t h e following questions:
1. W h a t n u m b e r s e n t e n c e c a n b e u s e d t o s o l v e t h i s p r o b l e m ?
2. S o l v e t h e p r o b l e m u s i n g t h e " c o f f e e g r o u n d s " a n d m e a s u r i n g c u p m a n i p u l a t i v e s . D e m o n s t r a t e y o u r
p r o c e s s u s i n g t h e zip lock b a g c u t o u t , g l u e , a n d s c i s s o r s .
3 . S o l v e t h e n u m b e r s e n t e n c e in p a r t 1 u s i n g t h e r e p e a t e d a d d i t i o n a p p r o a c h .
39
Appendix C 1
Ribbons a n d Tiaras
You a r e c r e a t i n g p r i n c e s s t i a r a s f o r a p r i n c e s s t h e m e d b i r t h d a y p a r t y a n d e a c h t i a r a r e q u i r e s 1 0
f e e t of r i b b o n . T h i s r i b b o n is t h e n c u t i n t o s t r i p s a n d t i e d o n t o t h e t i a r a . If e a c h s t r i p isone-fifthof t h e
t o t a l r e q u i r e d r i b b o n l e n g t h , h o w l o n g is e a c h s t r i p ?
Answer t h e following questions:
1. W h a t n u m b e r s e n t e n c e c a n b e u s e d t o s o l v e t h i s p r o b l e m ?
2. S o l v e t h e p r o b l e m u s i n g t h e r i b b o n a n d m e a s u r i n g t a p e m a n i p u l a t i v e s . D e m o n s t r a t e y o u r p r o c e s s
using ribbon p a p e r cut-outs, glue, a n d scissors.
3 . S o l v e t h e n u m b e r s e n t e n c e in p a r t 1 u s i n g t h e m e a s u r e m e n t a p p r o a c h .
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Appendix C 1
H e r s h e y Bars
Y o u h a v e - o f a c a n d y b a r l e f t o v e r f r o m last n i g h t ' s trip t o t h e m o v i e s . If y o u w a n t t o eat
one-third o f y o u r l e f t o v e r s , h o w m u c h o f t h e c a n d y b a r w i l l y o u e a t ?
Answer t h e following questions:
1. W h a t n u m b e r s e n t e n c e c a n b e u s e d t o s o l v e t h i s p r o b l e m ?
2. S o l v e t h e p r o b l e m u s i n g t h e H e r s h e y Bar ( o r d o m i n o s ) m a n i p u l a t i v e s . D e m o n s t r a t e y o u r p r o c e s s u s i n g
t h e H e r s h e y Bar p a p e r c u t - o u t s , g l u e , a n d s c i s s o r s .
3 . S o l v e t h e n u m b e r s e n t e n c e in p a r t 1 u s i n g a n a r e a m o d e l .
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Appendix C 1
Police A c a d e m y
One-third of a c l a s s of p o l i c e a c a d e m y s t u d e n t s will b e o u t of c l a s s a n d p a r t i c i p a t i n g in p h y s i c a l f i t n e s s
c o u r s e . If t h e r e a r e 3 0 s t u d e n t s in t h e c l a s s , h o w m a n y s t u d e n t s a r e p a r t i c i p a t i n g in t h e c o u r s e ?
Answer t h e following questions:
1. W h a t n u m b e r s e n t e n c e c a n b e u s e d t o s o l v e t h i s p r o b l e m ?
2. S o l v e t h e p r o b l e m u s i n g t h e p o l i c e o f f i c e r f i g u r e m a n i p u l a t i v e s . D e m o n s t r a t e y o u r p r o c e s s u s i n g t h e
police officer figure p a p e r cut-outs, glue, a n d scissors.
3 . S o l v e t h e n u m b e r s e n t e n c e in p a r t 1 u s i n g t h e m e a s u r e m e n t a p p r o a c h .
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Appendix C 1
Chex Brain Power Party Mix
The following consists of ingredients for Chex Brain Party Mix.
Serving Size: 9 Hungry Students
Ingredients:
4andone-halfcups Chex cereal (any variety)
1 cup Animal Crackers
2andone-fourthscups pretzels
three-fourths cup peanut M & M ' s
Answer the following questions:
3.
Create your Chex party mix for 3 students instead of 9. Solve the MandM portioning
fraction multiplication problem using an area model.
4. Develop your own fraction multiplication story problem. W h a t number sentence is used
to solve it?
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Appendix C 1
Pumpkins and gourds
You are decorating your living room table for Thanksgiving dinner and you go to the store to buy
some pumpkins and gourds. You have 9 guests coming for dinner and you want aone-fourthpound
pumpkin or gourd for each guest, so you need to purchasenine-fourthspounds of pumpkins and gourds.
The grocery bags used to purchase the pumpkins and gourds can each holdthre -fourthsof a pound. How
many bags will you need to purchase your nine pumpkins and gourds?
Answer the following questions:
1. What number sentence can be used to solve this problem?
2. Solve the problem using the pumpkin and gourds and pumpkin grocery bag manipulatives.
Demonstrate your process using the pumpkins and bag paper cut-outs, glue, and scissors.
3. Solve the number sentence in part 1 using the common denominator approach.
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Appendix C 1
Coffee Grounds
From the last class, we havesix-thirdscups of coffee to portion out. If we want to make daily servings in
zip lock bags oftwo-thirdscup portions, how many bags will we fill?
Answer the following questions:
1. What number sentence can be used to solve this problem?
2. Solve the problem using the coffee grounds and measuring scoop manipulatives. Demonstrate your
process using the zip lock bag cut-outs, glue, and scissors.
3. Solve the number sentence in part 1 using the common denominator approach.
45
Appendix C 1
Ribbons and Tiaras
Suppose you have 2andfour-twelfthsfeet of ribbon, and you want to cut it into stripsfour-thirdsfeet long.
How
many
Answer the following questions:
1. What number sentence can be used to solve this problem?
2. Solve the problem using the ribbon and measuring tape manipulatives. Demonstrate your process
using the ribbon cut-outs, glue, and scissors.
3. Solve the number sentence in part 1 using the numerator/denominator multiples approach.
46
Appendix C 1
Hershey's Chocolate
If you have 1andone-halfHershey chocolate bars, and you want to giveone-fourthof a chocolate bar to each
of your friends, h o w many friends can you give chocolate to?
Answer the following questions:
1. What number sentence can be used to solve this problem?
2. Solve the problem using the Hershey bar (or dominos) manipulatives. Demonstrate your process using
the Hershey Bar cut-outs, glue, and scissors.
3. Solve the number sentence in part 1 using the invert and multiply approach.
47
Appendix C 1
Police Academy
To celebrate the end of the police academy program, one of the students brings in a pan of
brownies. As the future officer drove to the academy he got hungry at a red light and atetwo-eighthsof
the brownie pan he made. If he wants to give each personthree-sixteenthsof the left over brownies, how
many students can have a brownie?
Answer the following questions:
1. What number sentence can be used to solve this problem?
2. Solve the problem using the brownies and plastic knife manipulatives. Demonstrate your process
using the brownie cut-outs, glue, and scissors.
3. Solve the number sentence in part 1 using the invert and multiple approach.
48
Appendix C 1
Chex Party Brain Power Mix
Measure outfive-thirdscups of the Chex Party Brain M i x prepared the previous day. Suppose you
want to m a k eone-fourthcup portions of your party mix in your small zip lock bags. H o w many
portions can you m a k e ?
Answer the following questions:
1. What number sentence can be used to solve this problem? Solve this number sentence in using the
invert and multiple approach.
3. Develop your own fraction division story problem. What number sentence is used to solve it?
49
Appendix C 2
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Appendix C 1
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Appendix C 2
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Appendix C 2
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Appendix C 2
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Appendix C 2
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Appendix C 2
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Appendix C 2