USING A NEO-PLAGETIAN FRAMEWORK FOR LEARNING AND TEACHING
MATHEMATICAL FUNCTIONS
Mindy Kalchman
-4 thesis submitted in conformity with the requirements
for the degree of Doctor of Philosophy
Department of Human Development and Applied Psychology
Ontario Mtute for Studies in Education
University of Toronto
Cl Copyright by Mindy Kaichman 200 t
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Abstract
Using a neo-Piagetian fi;unework for Iearning
and teaching mathematical hctions
Doctor of Philosophy, 2001
Mindy Kalchman
Department of Human Devebprnent and Applied Psychology, University of Toronto
Using the framework of Case's theory of cognitive growth, an age-relatai developmental
sequence for students' understanding of mathematical functions is ptoposed. An experimental
cumculurn designed to foster that development is aiso presented. Sîudies at Grades 6,8, and 10
investigated (1) whether the experimental instruction better fosters students' leaniing of
functions than textbook approaches and (2) whether experimental students' understandings
devclop according to the predicted sequence.
Results indicate that the experimentai groups' understanding of functions continually
improved across Grades 6,8, and 10. Moreover, students in each successive grade had a mean
respective gain score approximately quidistant h m that of the grade bebw. Analyses of the
experirnental students' correct and partially correct stratedes showed few qualitative differences
in the ways these students approached problems. However, students at different levels of
developrnent exhibited difficdties with the mathematical intricacies of operating with rational
nurnbers, integers, and with aigebraic notation, Thus, there appeared to be some grade-related
lirnits to students' learning that resuited h m a combination of their experiences with these
mathematical intncacies and wiih their developrnental Ievel.
Results also showed that the Grade 8 and 10 experimentd p u p s performed significantly
better on the functions test tban did control groups at these same grades. Furthmore, the Grade
6 experimental group outperformed the Grade 8 control group, and perfonned equally to the
Grade 10 control students. Analyses of students' solution strategies on each test item Uidicated
that students in the experimental groups demonstrated a better understanding of how the graphic,
tabular, and algebraic representations of a function are connected.
Results show that a mode1 of development and instructional approach that share an
emphasis on developing children's understandings of how and why diffefent representations of a
function are connected, foster a deeper conceptual understanding of functions than do traditional
approaches to the topic. The experimental cuniculum's use of a familiar context, technology, and
theory-driven activities heiped participants leam concepts important for fiuictions. Furthemore,
only the experimental group demonstrated that they had integrated these functional concepts into
a rich conceptual network for their current and future learning of fiinctions.
This work is dedicated to the memory of Robbie Case, whose inspiration and wisdom are
on every page. There are no words to express the gratefuiness 1 feel for having had the honor and
pnvi lege of working with Robbie. Equally, there are no words to express the sadness I feel in
having lost him.
1 am especially grateful to Karen Fuson for welcoming me to Northwestern as a student,
colleague, collaborator and friend. Karen's generous professiond and personal support and
attention over the past year have been sources of intellectual and exnotional nourishment that
guide and hold me in so many ways.
1 am also very grateful for the comrnents, feedback, and suggestions made
by members of my cornmittee at OISE: Earl Woodruff; Doug McDougalI; and David OIson.
Special thanks to Ken Koediiger.
1 would also like to thank George Rutherford and Susan Dickinson for welcoming me
into their school and would like to thank Brenda Cmwder, Sue Jackson, Brad Stevens, Rob
Kennedy, Delfina Traxler, John Doma, and al1 their students for participating in the research.
This research was supported by the James S. Mchnnell Fouudation, h u g h a grant to
l
for conîhuing to fhd this
Robbie Case. 1would like to thank the M c D o ~ e lFoundation
research and for believing in its importance and contniutioas to human cognition and
mathematics education.
To Carlyle, Justine, Jeny, Jeunifer, Joe, TammyTNalini, and others who were fountains
of support over the past several yearsT
Table of Contents
Chapter
1
2
Page
Generai Introduction
1
Introduction
1
Related Paradigms for U n d d i n g Functions
4
Learning Mathematicai Fnnctions: A Devebpmentai Model and i
New Curriculum
Theoretical Framework
Central Conceptual Structure for Functions
Designing instruction for Functions Using the Framework
of Centra1 Conceptual Stnictures
The Study
Methods
Participants
Designs of the Experimental and Control instruction
Discussion
Discussion of Curricular Cornparison
Discussion of the Model of Development
3
Understandhg Mathenuthai Functions: Students' Solution
Strategies
Coding of Solutions
7
Page
Chapter
4
Conceptud Mappings and Requirements for Each Item
60
Empirical Results and Discussion of Individual Items
60
Level2 Items
66
Transitional Items 4 and 5
83
Level3 Items
94
Level4 Items
1O7
Discussion and Conclusions
118
E ffects of Curriculum
119
Effects of Experience
132
General Surnmary
133
General Summary and Conclusions
134
Discussion and Conchsions
140
Implications
t 47
Future Work
148
References
150
Appendices
162
List of Tables
Table
Page
1
Summary of Experirnental Cuiricular Sequence
24
7
Sumrnary of Grade 8 Control Curriculum
25
3
Summary of Grade 10 Control Curriculum
26
4
Means and Standard Dwiations for ail Groups at Pretest and Posttest
31
5
Percent Correct for Each Item on the Pretest
45
6
Percent Correct for Eac h Item on the Posttest
46
7
Conceptual Mappings and Requirernents for Each Item
61
S
Percentage of Students Who Gave Correct and Partiaily Correct
Solutions to item 1,2, and 3 (Level2)
67
9
Percentage of Students Who Gave Correct and Partially Correct
Solutions to item 4 and 5 (Transition h m Level2 to Leve13)
10
Percentage of Students Who Gave Correct and Partially Correct
Solutions to items 6 and 7 (Level3)
1I
Percentage of Students Who Gave Correct and Partially Correct
Solutions to items 8,9, 10, II, 12 &d4)
List of Figures
Page
Figure
Sample Computer Screen
28
Pretest and Posttest Mean Scores for Grades 8 and 10 Experfmental and
Conbol Groups
32
Pretest and Posttest Means for the Grade 6 Experimentai Group and the
Grade 8 and Grade 10 Control Groups
Pretest and Posttest Means for Grade 6,8, and 10 Experimental Groups
Pretest and Posttest Means for Grade 6 High-Achieving Experimental
Group and Grades 8 and 10 Contml Gmups
Pretest and Posttest Means for Low-AchieMng Grade 6 Experimental
Group and Grades 8 and 10 Control Gmups
Pretest and Posttest Means for High-Achieving Grade 6 Expimentai
Group and Grades 8 and 10 Experimental Groups
Pretest and Posttest Means for Low-Achieving Grade 6 Experimental
Group and Grades 8 and 10 Experïrnental Groups
Sample Correct, Partiaily Correct and Incorrect Solutions for Item 1
Sample Correct, Partiaily Correct and Incorrect Solutions for Item 2
Sample Correct, Partially Correct and Incorrect Solutions for Item 3
Sample Correct, Partially Correct and Incorrect Solutions for Item 4
SarnpIe Correct, Partialiy Correct and Incorrect Solutions for Item 5
Sample Correct, Partidy Correct and Incorrect Solutions for Item 6
Sample Correct, PartiaUy Correct and Iacorrect Solutions for Item 7
Sample Correct, Partialiy Correct and Incorrect Solutions for Item 8
SarnpIe Correct Responses for Items 9 and 10
Sarnple Correct Responses for Items 11 and 12
List of Appendices
h3e
Xppendix
.4
Functions Test
163
B
Tables of Strategy Categones for Each Item on the Functions Test
175
CBAPTER 1
General Introduction
The three pnmary purposes of this work are (1) to preseut an age-related learning
sequence that uses the general principles and ftamework of Case's theory of cognitive
development for describing and interpreting the way in which chikiren aged 10 to I8 come to
understand mathematical hnctions, (2) to pment an experimental curriculum for teaching
functions to students across that age range that was specificaIly designeci to facilitate the sort of
~rowthin understanding characterized in the proposeci model of development, and (3) to test the
ripplicability of the model of development and the powet of the experimental curriculum in the
context of classroom studies.
Introduction
Functions are widely considered to be a pivotal and central topic in chiIdren's leaming of
mathematics. They are the synthesis of many ofthe topics students traditionally lleam in isolation
in elementary schoo1 (e.g., coordinate graphing, solving for unknowns in arithmetic expressions,
and iïnding patterns within strings of numbers and rdationships beween pain of numbm), and
they are the gateway to more advanced topics in mathematics such as those found in cdculus.
Literamre has shown, however, that students of al1 ages (including pst-secondacy) have
di fficuity understanding the concept of fiinction and are thus ultimately limite-in rnakuigcareer
choices that include engineering, economics, and physics, al1 of which require an understanding
of func tions as a mathematical foundation.
Three themes with respect to the teaching and learning of functions were considered for
the present work: (1) the tirne at which childrea are first introduced to functions as a unified
mathematical concept; (2) the way in which subsequent leaming and instruction unfold over
tirne; and (3) the way in which the topic is introduced and taught in tems of appropriate
c timcular materials
In traditional sequences of school rnathematics programs, students are formally
introduced to functions in about Grade 9 (Leinhardt, Zaslavsky & Stein, 1990). Students then are
cxpected to take their elementary l e h g experiences with the individual and previously
disconnected extemal representations of a fùnction (prixnarily tables, graph, and equations) and
integrate them for understanding functions as a conceptual system. Furthemore, because of the
importance and the breadth and depth of the concepts and applications inherent to functions, it is
ri
topic that is revisited fiom year to year in children's school mathematics careers, continually
cxtending the topics taught. Consequently, consecutive year's instruction and curriculum depend
on students having learned the key concepts and having made the relevant mental connections
from the year before. However, the research literatwe indicates that students are generaily not
cven making the initial connections as they transition h m elementary to secondaq school
rnathematics, and thus the whole intended program of study is generally unsuccesstùl in helping
students understand functions as a conceptual system.
One explanation for why students have such difficulty with understanding functions is
that they are not making the interna1 cognitive comections that relate relevant concepts. Not
rnaking these connections may have two roots: (1) students' instructional and curricular
esperiences are not providing sufficient contexts and opportunities for students to make those
connections; or (2) the cognitive expectations placed on students in particula. grades are beyond
their capabiiities because of their position dong a dwelopment trajectory.
fn earIier work, Case and his colleagues constnicted theoreticai modets that describe agerelated Ieaming trajectories for children's understanding of two other centrat and pivotal topics in
mathematics: whoie numbers ( G r i f h & Case, 1996, 1997; Griffin, Case, & Siegfer, 1994) and
rational numbers (Muss,2000; Moss & Case, 1999). For both these mathematical domains,
curricular programs were developed for the express purpose of facilitating the sort of
development described in the leming models. Empirical studies were then cmied out to test the
accuracy of the modeis and the efncacy of the curricula. A related investigation ernbedded in
tlicse studies was the anempt to understand what concepts iaherent to these pivotal domains were
"
teachable" at which age levels, and what concepts w m resistant to instruction until a particular
dwelo prnental level of understanding had been achieved.
In both the whole number and rational number studies. it was found ( 1 ) that the
dcveloprnental trajectories were valid descriptions of at least one way that students build a nch
conceptual h e w o r k for understanding in the respective domains, and (2) that the
corresponding curricula better fostered deep conceptual understandingof the relevant topics and
concepts than did traditional instruction.
The present work buiIds on that eariier research and seeks to investigate the ways in
which children construct, over time, a rich conceptual framework for understanding
rnattiematical functions, and whether cmicular innovation intendeci to foster that development is
effective both with respect to heiping students move throue the hypbthesized leamhg sequence,
and with respect ro heIping students understand fûnctions more deeply and flexibly than do
traditional approaches to teaching the topic.
Relatai Paradiems far Understandine Funcaons
As it applies to the development of children's understanding of fimctions, Case's tbeory
is complemented by other accounts of how the concept of function deveIops as a mathematicai
consrruct. The most centrai of these accounts are a h complementary and include the process-toabject paradigm (Dubinsky & Harei, 1992; Sfard, 199f,1 WS), the CO-variationversus
correspondence approach (Borba & Confrey, 1996; C o n h y & Smith, 1994, 1995), and the
construction of the " Cartesian Comection" (Schoenfeld, Smith, & Arcavi, 1990). These models
for iinderstanding have aiso been used to orgaaize and build cocresponding instructional
programs, most of which have been technology-based.
In the process-to-object paradigm, students devetop an understanding of function as a
mathematical constnict by first working with functions in a procedual or operational way (e.g.,
g enerating tables and graphs h m equations). Students then further develop their understandings
to a point where a function can be s m and used "as sotnethhg to which actions and processes
may be applied" (Haret & Dubinsky, 1992, p. 19).
This conceptual shift aiIows the leamer Co
operate on the resulting object (mathematicai and conceptuai) to leam more sophisticaied
mathematics that require such reifiedknowledge. The curricular supports for the process-toobject paradigm are almost exclusively technology-based activities (Breidenbach, Dubinsky,
Hawks & Nichols, 1992; Goldenkg, 1988; Gddenberg, Lewis, & O'Keefe, 1992).
A second central approach to the teacbing and iearning h c t i o n s has been most widely
promoted and applied by Confiey and her colleagues-They use a CO-variationapproach to the
topic, in which students coordinate the patterns of values found in two different columns of
numbers (Confrey & Borba, 1996;Co*
& Smith, 1995). This approach kiigtilights variation
in a function over correspondence. The conespondencedefinition of function is the current
standard definition used in most textbooks. in this definition of a fimction, a fimction is descnbed
via a mle and the elements in one set correspond to elements in another set so that each element
in the first corresponds to exactly one element in the second (Conûey & Smith, 1995;
Thompson, 1994). Confrey and Smith (1995) wncluded that the conventional conespondence
approach to the teaching of functions neglecl the d e that tables play in students' constructing
an understanding of functions and "le& to an ovedance on algebraic representations" (p.
67). In their work, they promote the tabdar representation of a function as a valid mathematical
object from which students are encouraged to make connections that integrate the graph with the
operational (or procedural) basis of a function (Confrey & Smith, 1995). The software program
Function Probe (Confrey, 1992) is designed to facilitate many of the same sorts of integrations
benveen tables and graphs and equations described in the present mode1 for learning bction.
A third fiamework of analysis that has been used for discussing how students develop a
conceptual framework for functions is that of Schoenfeld et al's (1990) "Cartesian Comection."
These researchers looked specificdly at how one student understood the connection between
gaphs and equations in linear fimctions. in their notion of the Cartesian Comection as "the
underlying structure of the Cartesian plane and the mathematical conventions for interpreting the
properties of linear graphs" (Schoenfeld et al, 1990, p. los),they identified the importance of
students understanding that "specific algebraic expressions have graphical identities. For
example, O.r -y,) is a directed üne segment with both direction and magnitude specified by
mathematical convention" (p. lm), S c h d e i d and his coUeagues also use tecbnology m tbeu
work. They have developed the program Grapher (Schoenfeld, 1990) for examining how students
understand the way in which siope, intercept, and cwrdinate pairs are related in graphs and
equations. This software is used mainiy as a diagnostic leaniing tool. Students*problems and
misconceptions c m be identified through the use of the program and then remediated in a
technology-based environment that provides instant and automatic ieedback on ideas and
predictions.
The overall structure of this dissertation is as iollows: In Chapter 2, the mode1 of
development, the present curricular approach and general resuiîs fiom its implementation are
described and discussed. The results presented and discussed in Cbapter 2 focus on and compare
pretest and posttest scores achieved by both experimental and control groups on a bctions test.
Specifically, cornparisons are made between experimental grogs and control groups within and
between grades 6 , 8 , and 10, and comparkons are made among three experimental groups'
results (Grades 6,8, and 10). in Chapter 3, common strategies students in each group used for
and discussed. The focus in this chapter is
solving each item on the functions test are de~~n'bed
on comparing the experimental students*solution strategies for each item on the bctions test to
those of the control groups and discussing any ciifferences in the context oistudents' respective
instructional experiences. in Chapter 4, a general summary of the hdings is given dong w i t .
general conclusions and implications for the study as a whole. Particular ways in which the
present work relates to these other perspectives wiii be discussed in the general conclusion.
CIIAPTER 2
Learning Mathemahi Functions:
A DevefopmentalModel and a New Cumculum
The topic of functions is widely considered to be central to mathematics as a discipline.
Fiirthemore, it has been suggested that little is possible in the way of higher mathematicai
lcaming until the concepts and constructs in this area have k n understood (Artigue, 1992;
Confrey Sr Doerr, 1996; Eisenberg, 1992; Selden & Selden, 1992). Traditionally, functions have
becn introduced to students in about Grade 9 (Leinhardt et ai., 1990). Ongoing and recent
initiatives, however, by The National Council of Teachers of Mathematics (NCTM) have called
tbr the leaming of this pivotal topic to begin in the elementary grades (NCTM, 1989,2000).
Two of potentially many positive eatécts ofintroducing the topic to elementary school
students are key for the present work. F'mt, it may stave off some of the more common
difficulties students experience with fumions. A considerable arnount of attention has been
givcn to invesrigating students' understanding of and difficulties with the concept of function
(CS., Artigue,
1992; Confiey & Smith, 1995; Davis, 1987; Dubinsky & Harel, 1992;
Moschovich, 1996; Schoenfeld et al., f 990; Sfd1992;Zaslavsky, 1997). Results of this
rcsearch suggest that, with tnditionai instructiond appmaches, students generally have limited
understanding of and flexibilitywith (1) the relevant foundationai ideas such as the concept of
\,ariable (Freudenthal, 1982; Usiskin, 1988; Wagner, 1981) ônd rate of change (Confrey &
Smith, 1994; Saldanha & Thompson, 1998), (2) the common representatious of a h c t i o n
(tables, graphs, equations, and verbal descriptions) (e-g., Goldenberg, 1988; 1995; Leinhardt et
al, 1990; Markovitz, Eylon & Bnrckheimer, 1986; Moschovich, Schoenfeld & Arcavi, 1993;
Sierpinska, 1992), and (3) possible applications of the function concept to other domains such as
crilculus (Confrey & Smith, 1995)' and to other disciplines such as physics and economics.
Second, it rnay allow for a conceptuai foundation for understanding fiuictions to be laid
earlier in children's school mathematics careers. Researchers' and educators' efforts to help
students better understand the concepts inhefent to hinctions have resulted in many innovative
and motivating approaches to instruction (e.g., Confrey & Smith, 1995; Davis, 1987; Carraher.
Schliemann Sr Brimela, 2000). The most cornmon of these approaches are technology-based
(Confrey, L992; Dugdole, 1982; Koedinger, Anderson, Hadley, & Mark, 1997; Magidson, 1992;
Schoenfeld, 1990).Technology allows students ta go between the representations of a function
qiiickly and easily and to get irnmediate Feedback on their ideas and actions (Goldenberg, 1988,
1395; Hillel, Lee, Laborde, & Linchevski,l992; Kalchman, Case, Kelly, & Cassidy, 2000;
Schoenfeld et al, 1990). Earlier experiences using appropriate cumculum would aIso allow
rnough time for the key concepts to build across grade levels as new ideas are introduced and for
students continually to consolidate and make connections among the central ideas.
The contributions of researchers and educators to the design of more effective instruction
based on fine-gained analyses of functions as a mathematicai consmct and of students'
difficulties with the relevant concepts (e.g., Schoenfeld et al, 1990; Sierpinska, 1992: Zaslavsky,
1997), have led to considerable progress being made in addressing students' leamhg needs in
the domain in general. However, in order to address the question of how to help students build
these ideas across grade tevets, acomprehensive and systematic ûamework for leamhg and
instruction is needed. Such a ~ e w o d may
c be derived fiom the traditions of developmental
cognitive psychology. in the past, little attention has b e n paid to the influences that the general
principles of developrnental cognitive psychobgy may have on students' Iearning of Eiuictions,
and to the ways in which these prïnciples may be used as a guiding h e w o r k for the design of
effective instruction (Kalchman, Moss, & Case, 2001). These principles include age-related
information-processing capacities, working memory, and children's natural tendencies to
construct knowledge from local experiences.
For the present work, an age-related stage model of children's conceptuai development in
the domain of functions was constructed and used as an heuristic for designhg a supporthg
cumculum. Preliminary versions of this model and descriptions of the eariy curriculum may be
found in Kalchman and Case (1998,1999) and in Kalchman, Moss, and Case (2001). Prior to the
present work, however, a cross-sectional anpiricd andysis of the pmpsed made1 had not been
done. Furthemore, results fiom comparing the experimental cumcuium to more traditional
textbook instructional approaches to tünctions had o d y been reporteci for single grade levels
(Kalchman & Case, 1998,1999). in ihis paper, revised versions ofboth the model of
development and the curriculum are given, and results h m cross-sectional empirical studies
reported. Specifically, instnictional studies were carried out at each of sixth, eighth, and tenth
gades (1) to compare the effects of the experimental instnictional approach to more traditional
textbook approaches both w i t b and accoss grade levels, and (2) to detennine if the postinstructional understanding of studenîs who participated in the experimental curriculum at each
grade level could be predicted by the pmposed model of development. In other words, does the
sort of instruction proposed hm - one designed using principles h m developmentai cognitive
psychology - foster students' understanding in the domain of fimctions more readily and
comprehensively than do traditional textbook approaches to the topic?
Theoretical Framework
The theoretical position taken for the present work is the neo-Piagetian one developed by
Case ( 1985, 1992, 1996). C a ' s gened mode1 of intellectuai dwelopment combines g e n d
age-related stages of cognitive growth with the development of domain-specific knowledge
structures, or central conceptual structures. Accotding to Case (1996), central conceptual
structures are complex networks of semantic nodes, relations, and operaton that (1) represent the
core content in a domain of knowledge, (2) help cbiidren think about the problems that the
domain presents, and (3) serve as a tool for the acquisition of highersrder insights into the
domain in question. Case and his colleagues argue that the development of a central conceptual
structure may be faciiitated by thougtiüüi instructional design and training (Case & Sandieson,
1992; Griffin & Case, 1996; McKeough, 1992). However, commoa age-related ceihgs linked to
chiidren's information-processing capacity and wotiOng memory Iimit the complexity of these
stnictures as students progress through hypothesized stages of intellecnial growth.
Generally speaking, central conceptuai structures within a domain are constmcted by
integrating two "primitive" schemas. The k t of these schemas is primady digital, verbal, and
sequential, and the second is primarily spatial. in the f h t pbase of chiidren's leamhg (Level 1).
t hese
two core schemas are elaborated in isolation- in the second phase, (Level2), they becorne
more complex as independent structuresyand they are dso mapped on to each other. The r d t
is that students' understanding in a domain is ûansformed, and a new psychoIogical unit is
constructed. During the next phase (Leve13), students elaborate theirnewly forrned integrated
structure, and they begin to discriminate among diffetent cantexts in which the new unit can be
applied. They do this by creating different representations of the new stnrcttne, each with its own
distinctive properties. In the final phase (Level4), students build explicit representations of how
these different variants of the core structure are related to each other, and leam to move among
them fieely and fluently.
Central Concebtuai Structures for Functions
Level 1 (9 - 11 years of aee)
Level 1 is the entry level into the domain of functional thinking. Here, the primitive
schemas are still separate. The initial digital schema is one where children can iteratively
compute within a single string of whok numbers. That is, given a string of positive whote
numbers such as 0,2,4,6,8 ..., studenîs are able to add 2 to each successive number and
consequently extend the pattern. The initial spatial schema is one where children use two
orthogonal reference axes (Case, Marra, Bleiker & Okamoto, 1996; Okamoto & Case, 1996) for
representing quantities as bars on a graph. The bars on the graph are read as discrete quantities
froin the vertical axis (y - axis), and as qualitative and discrete categories h m the horizontal
mis (x - mis). Students perceive patterns of qualitative changes in amount by a left-to-right
visual scan of the graph (for example, each bar is longer than the previous bar), but they do not
quanti@those changes. These two primary schemas are hypothesized to be in place by the t h e
c hildren are 9- 10 years of age.
Level2 ( 11 - 13 vears of ape)
In the second level, the two initial schemas are elaborated and mapped on to each other,
h m i n g the central conceptuai structufe for fiuictions that is hypothesized to underpin much of
children's further learning in this domain, Two kinds of elaboration are hypothesized to take
place at this level for the digitai scbema. In one, students iteratively apply a single operation on,
rather than within, a string of positively ascending whole numbers to generate a second swing of
numbers. For example, if children multiply each number iu the sequeace O, 1,2,3,
rcsult is a set of pairs of values: 0-0, 1-2,2-4,3-6,
... by 1, their
....The second kind of elaboration is one
where students constnict an aigebraic expression for this repeated operation by generalizing the
pattern to y = Ir.
An elaboration p d l e l to the creation of numerical x,y pairs is presumed to take place in
the spatial schema. The categories dong the horizontal axis become fintquantitative intervals
(Lehrer & Schauble, in press) and then continuous quantities and thus can be used to represent
quantitative rather than qualitative data. Any pair of values is now understaod to be representable
in this (Cartesian) space, and the pattern that these pairs yield is representable by joining the
points and looking at the shape of the line that results, Also in this elaboration, students see the
pattern of change represented in the graph, and they are able to compare that pattern (for
example its steepness or starting place) to another by assigning relative values to salient
c haracteristics. For example, students can tell the ciifference between a linear graph that
represents a steep pattern of change and one that shows zero change. The latter may then be
assigned a value of zero and the former a value greater than zero.
As these two initial schemas are elaborated, they are aiso mapped on to each other so that
the base set of whole numbers on which a computation is executed become points that can be
-
represented fiom left to right dong the g axis. The resuits of this computation (for example,
-
multiply by 2) become points that can be represented using the y axis. The overall pattern can
be seen in the size and the shape of the step that is taken h m one pokt to the next when the
points are joined. That is, there is a constant U&J
2 in the theght-hand column of a table
J
I& 2 on a graph, which generates
representing pairs of numeric values, and a constant nmneric ~
a linear pattern (spatial) with a rise of 2 (numeric). The d
t of this elaboration and integration
is an integrative representation (a coordinate graph) that corresponds to a central conceptual
structure for functions. When this new structure has been fomed, an algebraic representation
such as y = 2r cm begin to have both numaic and spatial meanings for children.
This same sort of structural elaboration is presumed also to take place when students
employ any other operators in the positive domain. For example, students can add a certain
arnount to each value in one set of numbers (in the table and on the graph) to generate the other
set o f numbers which are represented in the fonn y = x + b (rather than the earliery = mr form).
These functions cross the y - axis at a place ùther than zero (in both the graph and the generated
number sequence). Students can also multiply each value in the first set by itself'(at lest once) to
generate y = .r2. This latter operation characterizes non-constant increasing differencs between y
- values and results in a non-linear graph.
Recall that the general theoreticai h e w o r k guiding this model posits age-related
developrnental ceilings, which are Iinked to limitations of information-processingcapacity
and/or working memory.Therefore, the model for leamhg suggested thus far is concerned with
students' constructing a basic integrated schema for functions. Complicating factors such as
operating with negative and rational numbes and including multiple operators that present
challenges to students' fluency with computation are not included at this level. Limiting these
comp Iicating factors was intended to minimize processing and working memory loads. This
allows students to focus on the essence of the inkgration of the initiai schernas.
Level3 ( 13 - 15.5 vears of aeel
As children progress to the thid level in their learning, they can begin to elaborate the
above integration. In doing so, they begin to combine the operations they execute on the domin
of a function to the point where theù conceptual structure includes fluency with fuuctions in the
fom (i) y = mr + b, where m and b are positive or negative rationai numbers or integers, and (u)
y = .r" + b, where n is a positive whole number, and b is any rational number or integer. For a
full elaboration to occur, however. it is necessary for children to understand integers and rational
numbers fùlly and to have facility computing with both of these number systems. They also must
have weI1-differentiated models for different meanings of the minus symbol. Students m u t
understand when the symbol is a subtraction operator and when it is a negative number. This is
necessary for differentiating between hctions such as y = x - 10 and y = 10 - x. Furthemore,
students must elaborate their integrative structure to the point where they can differentiate the
four quadrants of the Cartesian plane, and understand the telatioaship of these quadrants to each
other and to positive and negative numbers. This elaboration is necessary for students to extend
their understanding of the behaviors and characteristics of various functions to include the lefihand and lower-right quadrants.
Finally, students diffefentiate families of hctions. Such differentiations facilitate
students distinguishing between the spatial and numeric qualities of, for example, linear versus
quadratic functions, and quadratic versus cubic fimctions. Once students elaborate that schema to
include negative values in the domain, they can constnict independent models for each of these
sorts of functions based on their Level2 stnicture.
LeveI 4 (15.5 - 19 vears of age)
FinalIy, at Level4, children leam how h e a r and non-hear terms of a polynomial can be
reIated and understand the properties and behaviors of the d t i n g entities by analyzing these
relations. To achieve this, students must have well-constnicted and diff'erentiated models of
different soirsolfiinctions suchas quadratics inthe f o m y = d + b x + c o r y = a ( ~ - ~ ) ~ + q ,
polynornials, reciprocal, exponential, and even growth functions. They must also have the
necessary computational, algebraic, and graphing facility to inter-relate the representations of
these complex Eunctions. Moreover, students who have achieved this level of understanding wiil
be cible CO generalize their understanding in this domain to a broader system of conceptuai
structures that have developed in related domains such as physics and caiculus.
Desimine tnstruction for Functions Usine the Framework of Central Conceatual Structures
When using Case's theory as a guiding h e w o r k for instructional design, the primary
goal is to facilirate children's movement through, and understanding of, the proposed
dcveloprnental sequence. This is done by (1) considering the initiai numeric and spatial schemas
that children are presumed to bring to the domain, and (2) coming up with a core context that
serves ris a conceptual bridge between the initial schemas and the integrated conceptual structure
thrit
is the ultimate target. The particular "bridging context" used for the functions cmiculum
\vas
Y
walkathon, in which both the cornputational and the p p h i c aspect of a function c m be
simultaneously represented and understood as tables of numbers or a series of bar pphs,
respectively. In this context, the money eamed depends on the distance waiked according to
sorne specified rule of sponsorship.
In order to muimize the gains fiom these multi-modal representations. situations were
also created in which quantities were represented as objects having Iocations chat are kxed in
physical space. Students' introduction to siope is an example of this notion, where children
corne to understand that the slope of a line occupies a fixeci relative position, or steepness, in
Canesian space. Such situations may foster an appreciation of the properties of ntmbers that
children's spatial cognition naturally disposes them to notice (e-g., steepness, doseness,
srraightness, etc.), and which they might otherwise miss (Kalchrnan et al., 2001). Situations were
also designed in which children move back and forth among the different representations to
ivhich they were exposed, thus getting practice fiom the start in the sorts of problems that are
O ften
considered critical for demoastrating a "sense" of my mathematicai domain (Case, 1998;
Eisenberg, 1992; Greeno, 1991; Sowder, 1992). Finally, children were encouraged to taik about
ihe integrative representations using natural and mathematid language. The intent of this is to
fostcr a direct and simple mapping between the naturai language that children use spontaneously
and the more formal linguistic and symbolic mathematical ternis of the domain.
The Study
Three instructionai studies were carried out, one at each of Grades 6,8, and 10. in the
Grades S and Grade 10 studies, there was one experimentai and one control condition. At the
sisth-grade level there were two experimental classes. No control condition was used at this
grade because there is no locai curriculum whose mathematical content is a viable comparison
for the experimental condition. That is, functions is not a topic in the curriculum perse, although
swdents do work with tables, gnphs, and algebraic notation as independent matùematicai
representations.
The first assessrnent measure was a specially designed 12-item functions used to evduate
al1 students' learning (see Appendix A). The Grade 8 and Grade 10 teachers agteed that items 1
to 9 were tasks they felt their respective students should be able to do following standard
instruction on functions. They also felt that items 10 - 12 were items their Grade 12 and 13
mathematics students should be able to do following standard instruction. AU students wrote the
test before and after their respective instructionai units, in a maximum of one hour during a
regular mathematics period. The test inchdes four items at each of the hypothesized leveIs of
development described earlier. A detaiied analysis ofeach test item will be presented in Chapter
3. However, an example of an item h m each level will be described below.
The first four items represent tasks that require a basic integration of the primitive digital
and spatial schemas. For example, the 6rst task was for students to draw the shape of the
function y
= x'
+ 1 on a set of unmarked axes. The requùed integration is one where students
must recognize the generated numeric pattern of tbis particular computation as non-constant and
increasing beginning with 1 (when r = O), and integrate that with an understandhg that such a
numeric pattern is spatially manifest as an increasing curve that meets the y - axis at 1.
The next four items were designed such ihat students must integrate one more dimension
(either a nurnenc or spatial) to the established core conceptuai structure constructed by the end of
Level2. An example of a Level3 ta& is one where students were asked to compare the g q h of
j- =
10 - .r to the equation y = x - 10 and to decide if the former could be represented by the
latter. To determine if they are equivaknt expressions, students needed to recognize the numeric
patterns for, and the spatial implications of, both representatioas, while simuitaneously
understanding and cornparhg the more abstract algebraic (numeric) generalizationof bok
The final set of items represented Level4 tasks, in which students were expected to have
achieved significant flexibility with abstract pmcessing in the domain as a whole. An exampie of
a Level4 task is the following: What shape would the fimctiony = lOax - 252 + 2 likely have?
For this item, it was expected that students would have constructeci a mental mode1 for
polynomial functions such that algebraic expressioas where the leading term has an even versus
an odd exponent would be differentiated by their conjoint nurneric and spatial patterns. Thus,
students wouId know that the h c t i o n wouid have the base shape of a cubic because of the
spatial and numeric effects of cubing positive and negative x values, and they would then
integrate the effects of the linear and quadratic ternis into their solutions.
The second assessment measure was the Canadian Achievement Tests, Second Edition
(CAT12) (Canadian Test Center [CTC],1992), which is a standaràized ".*.test series designed to
measure achievement in the basic skills commoniy taught in schools across the country
[Canadar (CTC, 1992, p. 1). Students' "Total Math" scores, obtaiaed by c o m b i i g the scores
from two subtests: Mafhem~iicsConcepts and Applications and Mathematics Cornpuration, and
their "Grade Equivalency" scores, were used for anaiyses. Al1 students in the school write a
complete version of the CATI2 in Grades 7,9, and 11. Specifically for tbis study, the sixth-grade
students wrote the test approximately three montbs prior to the beghnhg of theu instructional
unit. Results fiom the CATI2 assessment were used for two purposes: (1) to detemine
equivalency between the experimental and contml groups at Grades 8 and 10; and (2) to compare
ihc academic profiles of students in the experimental grogs for the developmental analysis.
Background of the Srudents.
Ali studies took place at an independent school north of a major urtran center in Canada.
Tliere is a substantial tuition fee for attending the school, and thus, students generally corne fiom
middle- to upper-income homes. This particular school was selected because it housed the
primary, middle school, and secondary grades. Thus, the socio-ecommic status (SES) and the
ticademic protiles of students in different grades would be relatively similar.
Euuivalencv Between Ex~enmentaland Control Gmu~s.
Grade 6 students. Two intact classes of Grade 6 students participateci as experimentai
classes. CATf:! scores showed that bath were very high-achieving classes. Class A had 24
studcnts ( 13 girls and 11 boys), with a mean age of 11.5 (1 1 y e m and 6 months). Their mean
grade equivalency score from the CATI2 was 9.04 (SD of 1-42)?and their mean '"rotai Math"
pcrcentile rank was the 92" (10). Class B aiso had 24 students (13 girls and 11 boys), with a
mean age O f 1 1.6. Their mean grade equivalency was 9.1 (1.3), and theu mean 'Total Math"
percentile r d was ais0 the 92* (10).
Grade 8 students. The Grade 8 students also were composed oftwo high-achieving
-zroups of students. The experirnental group (II= 33) was comprised of 12 girls and 11 boys, with
3 merin
age of 13.8 yem. Twenty of these students were at the schooI in seventh grade, when
their most recent CAT/2 scores were obtained. They had a CATIS mean grade equivalency score
of 9-17 (.99), and a mean 'Total Math" percentile cauk of 88" (13). The Grade 8 conttol p u p (o
= 33) was
made up of 14 $ 1 ~ and 10 boys, with a mean age of 13.7, Nineteen students in this
goup were at the school in seventh-grade. At that time they had a CATI2 mean grade
equivalency of 9.04 (1.46), and abTotaiMath" mean percentile rank of 82" ((21).
To examine the equivaiency of these groups, -tests for independent samples were
conducted on their grade equivalency and percentile rank scores. No differences were found, 1=
.461,Q = .164and t = -953,
= .347,respectively.
Grade 10 students. A specid political circumstance of the Grade 10 sample is that the
199912000 provincial cohort is the last one required to complete Grade 13 in order to graduate
tiom high school. Students who enterai ninth grade in
1999 comprised the first cohon for whom
Grade 13 will not exist. They will graduate h m high school after complethg Grade 12. This
means that province-wide, two cohorts of students will be graduating high school in the year
2003. To avoid this "double-cohort effect" on college entrance applications, the Grade 10
stiidents in the present sample were enrolled in a specially-offered mathematics class in which
ihc tenth-gracie mathematics curriculum is completed in the fmt half of the school year and the
eleventh-grade curriculum in the second half. The school's administration explained that these
particular students. who comprise 47% of the school's tenth-grade population, wanted to "fast[rack" through high school. However, they were not considered advanced enough to be working
ri
full grade level ahead of themselves. Both groups of Grade IO students were in the second half
of their year and thus at the beginning of the Grade 11 mathematics curriculum.
The Grade 10 experimental group (n = 16) consisteci of 7 girls and 9 boys, with a mean
Lige of 15.7 years. Fourteen of the students were at the school in Grade 9, when theumost ment
C.L\T/2 scores were obtained. They had a CATI2 mean grade equivaiency of 10.78 (1-191, and a
'Total MaW mean percentile rank of70" (18). The control group (n = 17)was made up of 6
-eirls and 1 1 boys, with a mean age of 15.5. Sixteen students in this group wrote the CATI2 in
ninth grade, with a rnean grade equivalency of 11.47 (1.29), and a "Total Math" mean percentile
rank of 8 1" (14). Results fiom t-tests for independent samples showed no difference betwew
either the groups' grade equivalency or ''Total Math" scores, t = -1.4,
e = .I74 and 1= -1.841. p =
.072, respectively.
Deveio~rnentalDifferences Amow Grade Levels
The second purpose for obtaining students' CATj2 results was to examine the absolute
di ffcrences between the experirnental groups' grade equivalency performances'. The hope was
tlirtt
the same absolute difference in grade equivalency would result between the Grade 6 and
Grade S srimpies as between the Grade 8 and the Grade LO samples. (Only the experimental
groups were considered here because it was uot expected that the control cunicuIa would foster
hc sort of learning necessary for considering optimal leaming in the domain.) As it tumed out,
Iiowever, ris the grade level increased, students' comparative performances on the CATf?
dccrerised. That is, the gap between the Grade 6 and the Grade 8 sample was srnaIler thm the gap
between the Grade 8 and Grade 10 sample. Specifically, the Grade 8 and the Grade 10
cxperimental groups' wrote their latest respective CATI2 tests at the same tirne, and the
di fference between their mean grade equivalency scores was 1 S6 grades. This is a Iarger
di ffcrence than the di fference betwtxa ihe Grade 6 and the Grade 8 samples. Calculating an
mual dit'ference was not possible because the Grade 6 students wrote their test in sixth grde
and the Grade 8 students wrote their test in swenth grade, a full calendar year earIier than the
sisth graders. Nevertheless, it is clear from th& respective grade equivalency scores that the
' Students write specific grade-associated teveis of the CATR.For the pment study, the sixth gndcrs -te
LcvcI
is designed for students in grades 6.0 - 72;the eight graders m t e Level 17 when in scvmth pde
i drsigncd for students in gndes 7.0 - 8.2); and the tenth p d e n wmte Level 18 when in ninth grade (desiged for
jtudents in grades 8.0-10.2).
16. which
sixth graders were more advanced than the eighth graders (means of 9-07 and 9.22, respectively),
The implications for generalizability of forthcoming deveiopment analyses given this
stratification wiIl be addresseci in the generai discussion of the findings.
Desimis of the Ex~enmentaland Control Instruction
InstmctionaI Time
Each of the experimental classes received approximately 600 minutes of teaching of the
c..uperimentalcurriculum. Grade 6 srudents had approximately 50 minutes of daily class t h e
over 12 school days (600 minutes). Because the content of the cunicuiwn was not fmiIiar to
cither of the sixth-grade cIassroom teachers, the author taught the unit. Eighth-Me students in
the sxperimental group had
13 consecutive days of teaching. The first mathematics period of
cvery tour-day cycle lasted 75 minutes, and the rernaining three days of the cycle were 35-
minute periods for a total of 615 minutes. nie regdar classroom teacher and the author cri-taught
the cumcuium. This CO-teachingstructure was requested by the classroom teacher in order to
cnsure continuity for his students.The tenth-grade experimental goup's instruction tvas spread
out over eight consecutive 80-minute pends for a totaI of640 minutes. Again, the regular
classroorn teacher and che author co-taught the unit.
Students participating in a control condition had the same weekiy schedule as their gradeIcvet experimental counterpart. However, the Grade 8 contml class had 825 minutes (comparai
10
6 5 0) in 19 (versus 13) iessons. The Grade 10 control dass had 1200 minutes (compareci to 640
minutes) in 15 versus 8 lessons.
Content of Instruction
The experimental cumcular sequence remained constant from grade to grade, although
the amount of material per lesson varied with each group's schedule. A s u m n i a ~of~the
instructional sequence of the experimental curriculum is given in Table 1.
The Grade 8 control curriculum was a teacher-designed compilation of tacher-prepared
activities ruid of activities fiom textbook chapters on functioas and relations h m two different
~sxtbooks.The textbooks used were written pnmarily for ninth-grade students. They were used
~ i tliis
t
school for eighth graders because the Grade 8 mathematics p r o g m at this schoot is
comparable to what ninth-grade students have in the advanced Stream of the public school
s y st em. This class's regular classroom teacher was responsible for al1 planning and instruction
Cor this goup. Table 2 shows the outline of curricular topics, with brief descriptions of the
relevant instruction and activities. The Grade 10 control class worked pnmarily from a single
tsxtbook with some teacher-prepared worksheets. This class's regular classroom teacher was
responsible for al1 planning and instruction for this group. Table 3 shows the outline of curcicdar
topics for this group, with brief descriptions of the relevant instruction and activities.
In addition to the cunicular differences found in the tables, there were five other
significant contrasts between the experimental instructional approach and the textbook
approaches. The first contras was the use of contexts or situations as vehicles for. or bridges to.
iinderstanding. The experimental approach used the uniQing context of a walkathon for
introducing the key concepts and formai notation structure. This particular context was chosen
because (1) children are famiiiar with money and distance as variables quantities, (2) chiIdren
understand the contingency relationship between the variables, (3) chiidren are interesteci in and
rnotivated by the rate at which money is eamed, and (4) for this context, the relationship between
Topic
Introduction
Description
The walk-a-thon context was inaoduced.
a d students recordcd in tables the money
earned for each km waiked, and plottcd each
pair of values for a variety of des- Ushg
km and S. an equatioa was consmcted
based on the nile of s p o n ~ ~ ~ ~ h i p .
Activities
Student parmers invent at least two of theh
own spomorsbip anangemcnts for which
their parmcr consmicts tables, graphs. and
equations.
Slope
Slope was introduced as the const~nt
numeric uJ,by (or d o m bv) amount
between successive $ values in a table or a
gnph. It is a niativc masure of the
steeuness of a bction. If is the amount by
which each kilometer (x - value) is
multiplied.
Studcnts arc asked to f d thc slope of
x v d diffcrcnt functions expressed in
tables, grapk, and equations.
-v - intercept
y
-
intercept was introduced as the "starter
offer", that is, a fmed startirig bonus shidents
receive before the walk-a-thon bcgins. It
affects onIy the vertical starting place of the
numeric sequence and graph. It does not
affect steepncss or the gencnl shape of the
line.
Each student invented two lineu functions
that wouid allow himto taraexactly
$153.00 after walkhg 10 kilometen.
Studenu recordeci the slope and y - inmcept
of each h a i o n and explaincd how the y
interccpt of each tunction could be found in
its table, graph, and equation.
Curving
functions
Non-linear functions wcn maoduccd as
those having uu by amounts that incrcase (or
decrease) afier each kilometer walked. They
are derived by multiplying the kilometers (x)
by itself at least once. n i e grrater the
number of cimes x is multiplicd by itself, the
bigger the diffaencc betweeri douar values,
and thus, the steeper the c w e .
Students wcn asked to decide which of four
huictions cxpressed m tables w m aonlinear and to explain their reasoning. They
w m also askcd to &te an cquation for, and
to sketch and label the graph of each
huiction. Shidents were asked to corne up
with cwed-line fiuiction for camirig
3 153.00 over 1O kilometers.
Computer
aciivities
Students used spreadsheettcchmlogy and
prepared files and activity sheets to
consolidate and extend the undcrstandings
they had constnicted about slope, y intercept, and linearity in the nnt pan of the
curriculum.
Shidents changed the steqmcss, y
intercep, and direction of y = x and y = i m
order to maLe the function go through prcplottcd pomts. Smdents rccorded the
numeric, algebiaic, and graphic &mof
their changes. Stridents aiso mvented
functious with spccific amiutes such as
paralie1to y = 4x and a y - mtercept below
thex-axis.
Groups of students mvcstigatcd, and then
prepared a presentation abow a particular
type of hction. This srirmilatcddiscussion
and summarizing of key concepts and as a
partial teacher assessrnent for cvaluatbg
students' post-insuuction understandmg
about tunctions.
Groups of studem uscd cornputer-gcncr;ued
output of graphs, equatious, and tabks to
illustrate a
type of funCaon's
gend
and behavim. Students
then gave presentations about thcü fMction
and sharcd their expertise with ciassrnates,
-
-
Table 1
Summarv of Grade S Connol Curriculum
Topic
Relations as
Ordered Pain
Inrerpolation
2nd
Exirapolation
Dcsnipti~a
Shideatr did activitics m which they
Students wne told tbat 70find the 5intercept, put y = O [in the quation], and
to €id y - interccpt, put x = 0"-
Students were given graphs of hcar
Interpolation and extrapolation w m
inmduced in the coniat of Whc h of
best fit" for a scatter plot Thus. tbe
panerns in data cm bc capnïred by
interpolahg andlot cxtrapolating a p p h .
Siudcnts practisedintetpoiating and
txtrapoiating pairs of values fmm thc
grapbs of givcn fimctions.
Srridenu were shown b w to find the
intersectjon of two h a r fuactions by
çnphing. 'Tbe ublcs for a h function
were not used IO fiad the point of
intersection.
Siudents wcre givcn pairs of cquations.
which thcy graphcd on the same set of
axes. ïbcy locatcd the point of intersection
fiom the grid
Slope was defuicd as the ratio of rise to
nrn. Studenis w m shown b w to calculate
the slopc of a funcuou givcn its rige and its
Studeats caldtcd tfic slopcs of (a) the
hypotenuscs of ri@t aijngics (b) line
stgmcats givcn on unit ppcr and (c)
bctwced two points wing risdrun.
nrn.
Direct and
Partiai
Variation
Activitics
A relation may bt a set of oiderod pairs, a
hctions and asked to turb algebraidy,
dit points whnc the Iine crosses each mis.
Direct variaiion was definicd as a lhcar
relation that goes thugh the origh
Partid variation was defincd as a I i
relation hat bas a î k â amount wbich
causes the graph to mcet ihty - axis at a
place olhcr than (0,O).
Using direct and partial variation snidtnts
soIvad wwd pmblcm~by dcrrmining the
mlc ofvariation (the funcrian) rind solving
fm a pdcular vahie of y for a givm x.
y = mr + b was inaoduccd as another way
Stuclents graphcd quations, CalcuIatcd the
slow ami gave the y - inlcrcept Thcn,
of writiag,
y=nunibet~x+aiiothanimtbct.
&cy fouad tht slope and ihc y - inmept
h m quadons and gnphs with ~ i points
o
ÏdenntTcd (mcluding the y - intercept). and
putthcfunctionintoy=mr+bformat
Students wcrc told that the grapb of a
reIation with ''2"in it is caltcd a paabola,
SnidtDts made tables and graphedy =.?,y
=-$-2,v=x'+ I , ~ = Z ~ , ~ = . $ , ~
Table 3
Surnmarv of Grade 10 Control Curriculum
equation of a straight Le: the point-slope form;
the two-point f o m the dope-x-mtercept forni;
and the intercept form.
in which they found (a) one of the speciiïc
forms of the equation for a liae mentioned
above, and (b) the dope and interccpt of a hue
given the quation of a üne in the form a;r i by
+c=O.
Parallel and
Pcrpendicular
Lines
Use of geomeuic proofs to show snidenI rhat
(a) two non-vemcal pmiicl lines bave the samc
slope. and (b) the p m i mof ihe slopes of
perpendicuiar iines equais -1.
Saidents did exwhere they detcmimcd if
two qlations reprcscntedparalie[ or
perpmdicular lines and wrote equations for
lines parallel or pcrpendidar to given
cgnntiom.
Linear Systems
Linear systems were iniroduced as methods for
determining coordinates that a pair of linear
equations has in cornmou. ibis c d d k done
by graphing both hcs and finding the pomt on
the gnph, or algebnically by eitha addmg or
subuacting the equations or by substitution.
From the tcxtboolr. studcnts salved hcat
systmis in two and thcn m three variables using
each of the methods dcscribcd
Function
Sotation and
thc Definition
(if a Function
Students were shown how to "translate" fonnal
function notation ( h i is,flx) is said "fofx")
and how the notation is used O! W A X ) .ï h e
defmition of a funcrion and descriptions for the
domain and range of a funciionw m given.
Students worked h m pages in the rexibook and
substitutcd in values to solve for equations such
asAx)=801(;~3)=80*3andf(t)=r4-?+ 1;
8-1)
=(-1)'
++')l-(
1. Siudcnts did a
workshcet in which they decided if certain
relations were functions.
Quadratic
Functions
Students were inwduced to quadratic fiuictiom
as those whose gtaphs are parabolas and whose
equations are in the f o m y = c
d + bx + c
Studem did a workshcet in which they dccided
if given quations rrrpnsmtcd quahtic
functions. They gnphcd quadntic fimctioas,
and gave the domainand range of eacb
Studcnts also solvcd relevant word problems.
Cornputer
:ktivities
Using compter graphing soffwarr, snidcnts did
a series of activities in which thcy made vertid
and laterd s h i h on y =x', and they metchcd
and compresseci the same base functio~
Snidents drew graphs using the software for
recordhg the vcrta. axis of symuicay,
direction of opcning, and mtcrcepts of each
funcbon in a table, Studem skeicbed ami
d e s c r i i graphs incorporating two or more
transfomlations.
Cornpleting the
Square
Students were showabw to converty = u? +
b.r + c into y = a (x - p ) + q by cornpichg the
square.
.Lla.xMin
Problems
'
Students were given aramples of how ta h d
the tna..um or minmnmipomt of a fimction
by determining its vcncx tiom the fonn y = a (x
- p ) + q, and consequentiy, k i d i n g w h e k
the fùnction opens up or d o m A texthmk
'
Students answered two sets of questions h m
the textbook in which they sketched quamatic
functions and gave the cange for each thaction.
Students solvcd 21 madmh word problems
fkit h m the taabook and then h m a tacherprocniced woritsheet
the variables cm be equaiiy recorded in words, tables, graphs, and quatiom. This is in conuast
to
two aspects of' the control curricula in ihese*different contexts wete used for introducing
di fferent constructs. For example, in the Grade 8 cwiculwn, the gradient of a hi11 was used for
introducing slope, and fixed costs in production was used for introducing partial variation.
Second, some of the topics in the curricula, especialIy for the tenth graders, were iniroduced as
context-less procedures for manipulating algebraic expressions m order to find for example, the
solution to a Iinerir system, or to determine if two lines are parallel or perpendicular.
A second major distinction between the two cwicula was the role of multiple
representations. In the experimental appmach, tables, graphs, equations, and verbal d e s were
p m e n ted as di fferent, yet equivalent, representations of the same mathematical relationship.
Thus. a strong emphasis was placed on students moving among these representaiions in both the
\valkathon activities and in technology-based exercises. ln the former curriculum, children built a
variety of representations for a number of spomrship rules and expressed any one of those
representations in terms of the others. In the latter*students were able to effm change in the
gaphic and tabular representation of a function by changin? W f i c parameters in a fimction's
squation (for example, the slope, y - intercept, or degree of exponent.) Spreadsheet technolog
\vas chosen because it uniquely and sïmultaneously displays the graphic, tabular, and dgebraic
representations of a function. A sample of the screen ifiat students worked with during the
computer exercises is found in Figure 1. By contrast, the control curricula focused on algebraic
procedures and manipulations in a way that suggested that the tabies and graphs were the product
of. rather than equivaient to, a function's equation.
Third, but related to the second, was the focus of the control curricula on procedures for
example for finding the slope of a Iinear function or the vertex of a quadratic function h m
Fieure 1. Sarnule cornouter screen. With this spreadsheet screen configuration, -dents do
nctivities in which they alter the dope andior the y - intercept of the funchon y =x by changing
the value of m and/or b, respectively.
forma1 notation. In the experimental appniach, students cagaged in the construction of notation
that
represented a functional relationship, and thus built constnicts such as dope and y - intercept
into their use of notation. Over the course of the experimental mstniction, students progressively
formalized (National Research CounciI, 1999) their own notation until it corresponded with
conventional general equations such asy = mx + b.
A fourth difference was the isoIation and greater number of topics addressed in the
control approaches. In the experimental approach, students worked with a limited number of
concepts in an integrating fashion. Each of the control curricula addressed more topics, 0 t h in
isolation. For example, in the Grade 8 control cwriculum, the topic of dope and direct variation
tvere introduced as separate topics, when in fact they address very similar mathematical issues.
Krcping topics isolated also meant that the Grade 10 control students manipulated equations for
quadratic functions (for example, hding the vaiue ofy wben .r equals -1, in fiinctions such asf
( s ) = X' + 4.r + 1) before they even had any experience with or exposure to situations, tables, or
lrraphs of qiiadratic functions.
"
.4 final difference between approaches was the son of activities students did following
instmction and the s h i h over the unit for the experimentai groups. Activities for students in the
csperimental groups remained in the bridging context initidly (that is, related to a walkathon),
and thcn focused on consolidating and ùitegrating these concepts into the core conceptual
structure in the computer environment. Fuially, each group of -dents demonstratecl their
understanding by giving a presentaîion on a particular kind of fimction to their classrnates (for
example, linear, quadratic, recipnical, cubic), and included their understanding of and expertise
on certain key characteristics and behaviors of functions (such as the degree of steepness
reflected in the steepness of a line or c w e or the tabular patterns found in quadratic functions.)
The control groups, however, worked primady on textbook exercises generaiiy dixonnecteci
frorn students' own experiences, and completed many exetcises in a relatively short period of
time (for exarnple, the Grade 10 contrai group's assignment fotiowing instmctiort on Max/Min
problems was to complete 2 1 word problems by the next day.)
Results
Two general approaches to anaiysis were use& (1) ANOVA models using the mean
pretest and posttest scores foilowed by pst-hoc tests and calculations for effect size, and (2)
csarninations of the percentage ofstudents in each class who answered each item correctly on
dit. pretest
and posttest.
A11 items on the functions test were dichotomously scored as correct or incorrect.
Bccause most items required students to explain the reasoning that led to their solution,and
scverril factors went into a correct solution, it was not always immediately apparent whether or
not ri response was correct or incorrect. Thus, a general sconng rubric was designed, and hvo
iiidcpendent raters were trained in scoring the test using this scoring system. They achieved 91%
Ligrcernent on a randomly selected sub-sample of 12% of al1 the tests (4% €rom each _-de
Ievel).
Statistical Results
Means and standard deviations for al[ groups at pretest and posttest are given in Table 4.
Paircd-samples
<:
- tests
showed that all groups improved significantly h m pretest 10 posttest @
.O1 1 on the functions test regardless of the program of instruction.
To mesure the effects of the experimentd cturiculum as compared to the control
ciirricula at Grades 8 and 10, a 2 x 2 x 2 (tirne [pretdposrtestl x instructional g m u ~
[rsperimentaYcontrol]x g
p
&
[8/10]) ANOVA with repeated measures was carried out. Results
sliowed (a) a siynificant within-subject main effect of tirne.IF [I, 701 = 143.120,e= .CHIO, with
prctcst and posttest means of 1.61 and 4.29, respectively), and (b) sigüficant between-subject
main effects for & in favor oftenth grade @ [l, 701 = 11.305,2= -001, with means of 2-45
for Grade 8,and 3.45 [ 1.881 for Grade IO), and instructionai m u u in favor of the experimental
groups (F [ l , 701 = 12.871,~=.00t, withtneansof3.47 fortheexperimmtal p u p and2.41 for
the control group). A significant within-subject tirne x instructional mu^ interaction favoring
the esperimental group at posttest was also found (F [ 1,701 = 25.422, E = -000).This interaction
is s h o w in Figure 2.
Means and Standard Deviations for dl Grou~sal Pretest and Posttest
Group
Grridc 6 class A
Grtitic 6 class B
C'ornbGd Grade 6
Grade 8 expenmental
Grade 8 controI
Gradc 10 elcpcrimental
Grade 10 control
M
Posttest
Pretest
SD
-78
1-00
-17
-48
.39
.8 1
I.i8
1.27
2-00
1.10
2.00
1.07
1.52
1.71
-n
M
SD
-n
23
23
46
4.00
4.29
4-15
4.87
2.43
5.88
3.94
1.91
1.8 1
1.84
1.36
1.59
2.03
1.98
24
24
22
22
16
15
48
23
23
16
17
These results support ehe hypothesis chat studenis who engaged in the experimentd
curriculum would significantly outperform those who had more traditionai textbook-bttsed
instnicrion on functions. The tinding of no signifiant within-abjects interaction f o r a x
m d s (F( 1.70) = 1.O72,g = -340)or tirne x g
&
-
bctween-subjects gr&
xpuu
(!, 70) = 245, Q = .622), and no
x soua interaction (F (I,70) = .038, g = 347) support the notion that
sach yroup's respective curricuIum pmmoted comparable p w t h
for students at each grade
l e i d That is, the experimental curriculum engendered comparabIe gains in both the eigbth- and
tmth-grade students, and likewise for the more traditionai cmicula used (see the roughly parallei
lines for the experimental groups in Figure 2).
Figure 2. Pretest and Posttest Meaas for Grades 8 and 10 Experimental and Control Groups.
Delta (A) effect sizes were calculateci in order to determine the magnitude of the
diffcrence behveen the Grade 8 groups and between the Grade 10 groups. Effect sizes were
c;ilculated by subtracting the mean gain score of the control group (that is, posnest - pretest
rcsult)
from the mean gain score of the experimental group and then dividing the result by the
standard deviation of the gain score distribution of the control group (Fraenkef & Wallen, 1996).
For the Grade 8 study, an effect of 1.28 was found, meaning that, at posttest, the experimental
and control goups were more than one standard deviation apart (favoring the experimental
~ O L I P ) .In
educational research, an effect size of -50 or Iarger is considered an important finding
Fracnkel (Y: Wallen, 1996) with practical significance (Borg & Gall, 1989). For the Grade 10
(
srtidy. an effect size of 1.O0 was found, again favoring the experimentai group.
Before comparing the Grade 6 leaming to that of the experimental and control p u p s ai
hoih Grades 8 and 10, a two-way ANOVA with repeated measures was done using the pretest
;incl
posrtcst means of the two Grade 6 classes. Results showed only a significant within-subjects
c t'ficct o C time, F (1,44) =
192.301, p = .000,with pretest and posttest means of .48and 4.24,
rcspectively. Because the effect of(
:i
and the time x class interaction were not significant (F
1 , 44) = .123, g = -728 and F (1,44) = 3.399,
e = .072), al1 other analyses were conducted using
cornbined Grade 6 experimental group.
The next anaiysis compared the Grade 6 learning using the experimental curriculum to
the lèrirning of older students (Grades 8 and 10) who had traditional textbook approaches. A 3 x
2 ANOV.4 (gr&
[6/8/10] x time [pretest/posttest]) with repeated measures was conducted.
Rcsults showed a significant within-subject effect of tirne,(F (1,79) = 93.307, g = -000, with
pretest and posttest rneans of 1.255 and 3.527, respectively), and a significant within-subject
time .u rrade
-interaction F (1,79)
= 14.987,
p = -000. The interaction, seen in Figure 3, shows
F i a r e 3. Pretest and Posttest Means for the Grade 6 Experimental Group and
the Grade 8 and Grade 10 Conml Groups.
how the sixth graders rnoved fkom below the Grade 8 and 10 conîrol gtoups on the pretest to
bcing above them on the posttest. There was also a sipificant between-subjects main efkct of
m d e (F ( 1 , 79) = 3.720, g = .O291) with means of 2.359,1.88l, and 2.933 for Grades 6,8, and
10. respectively.
Post hoc tests were done to locate the significant differences in this interaction. At
pretest, the Grade 6 scores were significantly lower than those of the Grade 10 control group, g
=.O0 1 , but not significantly lower than the Grade 8 control group,E = .098. No difference was
found between the Grade 8 and Grade 10 control groups' pretest scores. (Tamtiane's T2 test was
IISC~
because the variance mong groups was not equivalent). At posttest, the Grade 6 scores
w r e significantly higher than those of the Grade 8 control group's, e = .O01 (Fisher's LSD test
\vas
used bccause equal variance was found). This effect size was 1.34. A difference was also
foiind between the Grade 8 and the Grade 10 control groups' posttest means @ = -025) in ravm
of the tenth graders. Strikingly, no difference was found between the Grade 6 experimental group
rind the Grade 10 control groups' posnest means, p = -551-
To examine the developmental questions, a 3 x 2 ANOVA ( d e 16/3/10] x time
[prctcstlposttest]) with repeated measures comparing the sixth-grade experimentd group with the
~ i v oot her
experimental groups was done. Results showed a significant within-abject effect of
t i nie (F [ 1. S 1 ] = 286.270, I,= -000, with respective pretest and posttest means of I .220 and 4.98
-
[ 1 .%Il). rind a significant between-subjects effect ofgrade (F (1,811 = 11.782. g = -000).with
per tomance increasing with grade level (withmeans of 236 for Grade 6,3.00 for Grade 8, and
3.93 7 for Grade 10). The absence of a significant within-subjecttirne x d
e interaction (F (1,
S 1) = .073, p = .924)supported the notion that the curriculum woked equally well for each of
Fieure 4. Pretest and Posnest Means for Grades 6-8, and 10 Experimental Groups.
rhe grade levels. The near parallel iines found inFigure 4 display this result visually. Tbus,
regrirdless of the students' starting points in terms of their domain-relevant howledge, students
in cach group were able to benefit fiom the experimental curriculum and to advance their
itnderstandings at comparable rates.
Post hoc tests were done to locate any significant dierences. At pretest, the Grade 6
scores were significantly lower than those of the Grade 8 and Grade 10 experimental groups', E
= ,034 aiid
p = .009, respectively (Tamhane's T2 test). At posttest, the only difference found was
that the Grade 10 experirnental group's scores were significantly higher than those of the Grade
0 p ~ l p ' s p, = .O02 (Fisher's
LSD).No differences were found between the Gnde 6 and Grade 8
cspcrimcntal groups (Q= 201) or between the Grade 8 and Gnde 10 experimental goups (p=
.070).
Follwv-UDAnalvses with Higher- and Lower-Achievin~Sixth Graders.
The relatively high scores reported on ihe CAT/2 tests for the Grade 6 students compared
to
the Grade 8 and 10 students raises issues of the generalizability of the results. To address this
question, each sixth-grade student was put into either a higher- or iower-achieving group,
dcpending on the individual's CATJ2 grade equivalency score. These two groups' grade
cquivalency means were 10.19 (.84)and 7.95 (-62).A two-way ANOVA (tirne [pretest/posttest]
\;
goup [Grade 6 higher versus lower achievers] with repeated rneasures was cmied out on the
tiincrions test. Pretest and posttest means were -63 1-97) and 5.04 ( f -68)for the higher CATI2
p u p , and .32 (-57) and 3.25 (1.57) forthe lowerpup. A significant between-subjects main
efft'ct of croup, F (1,44) = 13.255, g = .OOt as well as a signifïcantwithin-subjects tirne x m u u
interaction, F (1,44) = 8.105, g = .007,were found. No differences were found fiom comparing
the Iiigher and lower achievers in the other two experimental groups.
To identifi the effects of the higher general niath achievement score, each of the
stat ist ical tests reported above was repeated with the bigher- and lower-achieving group of sixth
graders. The results when comparing the higher-achieving Grade 6 group with the two controi
-m i i p s were similar to those reported earlier. A significant within-subjects time x made
interaction was found, F (1,57) = 17.756, g = -000; and a significant posttest difference was also
hund benveen the Grade 8 control group and the higher-achieving sixth graders (e = -000).
However. a significant posttest difference was found between the Grade 10 control group and
thesc higher-achieving sixth ,mders
(e = .046) in favor o f the sixth graders. Figure 5 shows ttiese
rcsults. When cornparing the lower-achieving Grade 6 group to the two control goups, a
s i-n i ficant time x
interaction wirs still found, F (I,55) = 5.473.2 = .007, but they
pcrfonned cornparably to, rather than better han, the Grade 8 control group on the posttest (E=
.094). There was no significant difference on the posttest between the Grade 10 control group
;ind the lower-achieving sixth p d e r s @ = 382). F i p 6 shows these results.
When cornparin; the higher-achieving Grade 6 group with the other two experirnentat
-croups. the only difference with respect to sigiuficant resuits between this analysis and the one
rcported earlier was that no difference was found at posttest between the higher-achieving sixthgraders and the Grade 10 experimental gcoup, (g = -129). In the previous analysis, the Grade 10
group outperformed the sixth-grade group as a whole. When comparing the lower-achieving
Grade 6 group with the other two experimental p u p s , the only difference fiom the initial
anal ysis was behveen the lower-achieving sixth-graders and the Grade 8 experimental p u p on
-4- Grade 0 CON
n
, d66HI
+Grade
10 CON
F i a r e 5 . Pretest and Posttest Means for Grade 6 High-Achieving Experimeatal
Group and Grades 8 and 10 Control Groupsi
+Gdc
8 CON
G n d e 6 LO
-C-Gndc
10 CON
Figure 6. Pretest and Posttest Means for Low-Achieving Grade 6 Experimental Group
and Grades 8 and 10 Control Groups.
the posttest in favor of the latter, @ = .005). In the initiai analysis, no dserence was found
bctween the Grade 6 and Grade 8 groups at posttest. Generally, the findings with respect to the
effects of the curriculum remained unaffected by splitting the Grade 6 group into higtier- and
lo\vrr-achievers. These results are shown in Figures 7 and 8, respectively.
Cornnaring Percent Correct on Individuai ltems
Tables 5 and 6 show the percentage of students in erich class who answered each item
correctly on the pretest and the posttest, respectively. Limited success with the pretest in general
n as
predictcd for the Grade 6 and Grade 8 levels because the present experiment constinited
thcir first formal experience with fùnctions. However, the Grade 10 students had studied
liinctions for two years prior to this study. Nevertheless, as shown in Figure 5, they onIy passed
one item at the pre-designated 50% passing rate. Students' specific item-by-item solution
straic-ies are beyond the scope of this chapter, but are discussed in Chapter 3.
Posttest results found in Tabie 6 show differences in performance within and b e ~ e e n
-~ r a d clevels. Results conceming control groups wiil be discussed first. Then results concerninr:
tlic tliree proposed levels of development will be discussed for the experimental groups.
Between-Grade Results. The Grade 6 group performed better than the Grade 8 control
sroup and equal to the Grade 10 control group (see Table 6).The Grade 6 group outperfonned
tlic Grade S control students on five of the first six items and tied them on the Lust item. They
also outperformed the Grade 10 control group on five of the h
t six items.
Within-Grade Results. The Grade 8 experimental group was considerably more
successful than the Grade 8 control p u p on items 3 t b u g h 6 and a bit more successfiii on the
tirst two items. The lowest Level2 item for the experimental group was item 2. On this item, the
most common incorrect response for both groups was to draw a straight rather than a curved Iine
Fi rure 7. Pretest .and Posttest Means for High-Achieving Grade 6 Expenmentd
Croup and Grades 8 and 10 Experirnental Groups.
Figure 8. Pretest and Posttest Means for Law-Achieving Grade 6 Exp-mtal
Group and Grades 8 and 10 Experimental Gmups.
io represent y = x2 + 1. This problem seemed to stem h m shidents' weak understanding ofthe
meaning of algebraic statements as illustrateci by a group of Grade 8 students who explained that
".Y'
is the same as 2r since r is being doubled, and thus the dope of the function is 2." Both
I ns truc tional
programs promoted students' ability to express a v&aI description of a functional
rel ationship as an equation, which led to mcreases on item 1. Stuàents in both were groups
generalIy unsuccessful with items 7 - 12.
The Grade 10 experimental group outperfionned the Grade 10 control group on items 1 7, w i t h large di fferences on five of these items. Snidents who received the expximental training
wcre more competent with a range of numeric underpinnings (tabular and algebraic) of
functions. Even on those items on which both groups exceeded the 50% passing rate, the
cxperimental group had qualitatively better strategies and solutions (see Ciiapter 3).
On three of four Level 4 items, the Grade 10 control gmup somewhaî exceeded ihe experimental
group. Item 9, however, shows the most outstanding Merence. Part ofthe Grade 10 control
curriculum dealt expIicitly the transformations of quadratic hncbons necessary for correctly
answering item 9. Nevertheless, the majority of these students were still unable to successfuily
answer this question.
Cross-Sec tional Results
The experimental groups' posttest performance was predicted as foiiows: at least 50%of
the Grade 6 students wouId be successful with items 1-4 (LmeI 2) but unsuccesshi with Level3
and Level4 items (5-8 and 9- 12, respectively); at least 500/0 of the Grade 8 students wodd be
successful with the LeveI 2 items and some Level3 items, but unsnccessfbl with LeveI 4 items;
and at least 50% of the Grade 10 students wodd be successfùl with ail LeveI2 and 3 items, and
Perceiii Correci for Eacli lteiii on the Prctest
Grade 6
Exp
hem Description
L,evel 2 items
1
Draw the sliape of y = x2+1
2 Give an equaion f& a function ihat crossesy=x+7
3 Write a function that shows earning $ 3 0 h
Make a table for increasing linear function with a
4
with a positive y - int&cep!
Canthegraphofy=lO-xbethesameasy=x-IO
7 Give a function that generates 2,5,8, 11, 14.,.
8 --- Sketch y = x3
graph with y-= x4
- onlo- already
- - --- there
Cevel4 items
9
Give equation for inverted parabola shified to riglit
find x - intercepis k r y = lair - x2
10
II
Whnt shape wowld graph o f y =Iûûx - 25x2+ r3have?
12
I
Whirt happens iii the function y = " a s g increases?
6
2
O
7
O
O
O
O
O
A
Note. Darkly shaded cells indicate a passing rate of over 50% for the clitss,
Grade 10
Grade 8
Exp
Coiitrol
EXP
Conlrol
9
5
13
5
14
25
6
40
l'able 6
Percent Correct for Each Item on the Posltesl
Gratlc 6
EXP
Item Description
Exp
Grade 8
Control
Grade 10
EXP
Conlrol
Level 2 ilenis
I
Write a function that shows eamiiig $30/lir
2
Draw the shape of y = ?+1
3 Give an equation for a function that crossesy=x+7
4 Make a table for increasing linear function with a
negative
-y - intercept
.-- --.-.
-.- - ---- ..
Level 3 items
5 Give a function that generates 2,5,8, 1 1, 14., .
6 Give 2 equations that produce increasing lincar funciion
with iipositive y - intercept
7 C a n t h e g r a p h o f y = 1 0 - x b e t h e s a n i e a s y = x - 10
Sketch y -=--x3
onto
grapli
already tliere
.----8
--..- - ..--- wiih
-----y-=- x4
.- - - - -- -- .
--Level4 items
9
Cive equalion for inverted parabola shiRed 10 right
10
find x - iniercepts for y = I ûx - x2
1I
What shape would graph o f y =10k - 25x2-k x3 have'?
12
1
What Iiappoiis in the funclion y = " as g iticreases?
..
-.
.
..
. .
. .
.. ..-.
.. .
..
il
Note.Darkly shnded cells indicate a passing rate ofover 50% for the class. Liglitly sliaded cells indicate a passing rate for only the top
hnlf of the class, Iierns wiibin level2 have been rearrniiged Io Iiighlighi the paiieni of respoiises foiind aiiiorig the experiniental groups.
significant progress would be made by some students on the Level4 items (9 - 12). The shaded
regions in Table 6 are rhose for which 504/0or more of the students gave correct solutions.
Results show that over 50% of the Grade 6 students got items 1 - 5 correct; over 50% of
the Grade 8 experimental group got items 1 - 6 correct; and over 50% of the Grade 10
experimental group got items 1 - 7 correct but not item 8 and few made progres on Level4
items. These results siightly exceed the predicted pattern for the Grade 6 students and are
somewhat less than predicted for the Grade 10 students. Thus, results particular to the Grade 6
and Grade 10 experimental groups will be discussed.
Grade 6. The percentage of students who got each item correct was caiculated for the
CATI:! higher- and lower- achieving Grade 6 groups. More than haifof both groups passed items
1 - 3, but only the higher group passed items 4 and 5 (67% and 75% versus 42% and 38%).
Item 4 proved to be more difficult than predicted. For this item, students were required to
write a table of values for the graph of an increasing linear function with a negative y - intercept
(al1 four quadrants were drawn, but the line began at the y - axis). To do this successfülly,
students needed to recognize the general numetic pattern that generates an increasing straight
linc with a coordinate pair forx = O l e s than zero. They then had to integrate two main
fundamental concepts inherent to the domah, namely dope and y - intetcept. That is, the
function must have a constant increasing slope
it must have a uegative y - uitercept. The
unsuccesshl sixth graders seemed able to represent either a positive sIope or a negative y intcrcept, but had difficultycoorduiating the two. This suggests that the intepuion of the two
concepts, at Ieast when a negative value is uivolved, was d
ibeyond the capabilities of these
younger students.
Grade 6 students were not expected to be successful with item 5. Snidents were given a
sequence of numbers (2,5,8,11,14 ...) and asked to write an equation for a function that would
generate that pattern of values, Students needed to understand that there are necessarily two
variables at play, and it is by operating on one unseen set of quantities (the domain) that the
second set is generated. The successive difference of 3, or " +3", in the dependent variable
corresponds to a "multiply by 3" operation on the independent variable and thus y = 3.r.
Furthemore, since the first value in the given sequence is not O, a y - intercept must be included
in the response to account for the 2. Thus y = 3x + 2 and y = 3x - 1 were the two correct
responses given by students. It was expected that younger students could likely elaborate their
numeric schemas and absîract a correct solution h m a table that shows both the dependent and
independent variables. However, it was not expected that these younger students' still
unelaborated integrated structure for function, which rnay or rnay not support the notion of an
implicit and unseen domain, would supersede their models of addition, and thus, a solution ofy =
s + 3 was expected.
One explanation for the success that was achieved by the higher-fùnctioning sixth graders
on items 4 and 5 is that they may be transitional items and c h t e r i z e students' capabilities as
they transition Erom LeveI 2 to Level3. That is, simple elaborations of the integrated schema
may have already begun to occur.
Grade 10. The majority of Grade 10 students were able to answer al1 items on which both
the Grade 6 and Grade 8 students were successfûl, plus item 7. Still, they fell short of the
predicted results. They were unsuccessflll with item 8, the last item in Level3, and showed little
progress with the Level4 items.
For item 8, students were given the gnph ofy =f on unmarked axep and asked to sketch
in the graph of y = .y3. This item had the misleading feature of 4 being larger than 3.
Consequently, many students concluded that the former is "biggef and therefore wider. Having
to compute with negative x - values and graph the resulting coordinates made this item even
more difficult. The main error made by d l experimentd groups at posttest was to draw a
parabola entirely outside the given parabola rather than a cubic shape that comes up tfom the
lower-left quadrant and passes through the origin as it continues to rise "outside" of y = .r4in the
upper-right quadrant. This cornmon incorrect representation indicated both a general
understanding that the higher the exponent in a fiinction, the steeper or more rapid the nse of the
curve. and an inability to integrate that understanding with the numeric and graphic implications
of cubing a negative number.
With respect to the Level4 items, it was hoped that as students in each successive grade
level moved through the experimentd curriculum, they would build on understandings and
would be better able to apply their leaniing to more sophisticated and novel circumstances. The
limitations shown by the Grade 10 experimentd group may have been the result of these students
having had minimal initial competence with the sort of tasks that require the basic Level2 spatial
and numeric integration. Such integrations have been pmposed here as necessary for developing
competency with the Level3 and Level4 tasks. Particular approaches to and strategies for items
9- 13 were not expressly addresseci in the experimentai unit, but many of the core elements of
them were featured throughout the cuniculum. More time was spent than anticipated with the
Grade 10 students on concepts t'bat they were presumed to have leamed in previous years of
schooling such as siope, y - intercept, cwrdinate graphing, and algebra. Cousequently, t h e ran
short. Perhaps if the Grade 10 experimentd group had the same amount of class tirne the Grade
10 control class did, they rnight have been able to elaborate the Level2 and 3 integrations they
did construct, and make greater progress with the Level4 tasks.
Discussion
The central assumption of this wock was that in order for students to develop flexibility
and Huency with the concept of fiuiction, instruction must be designed to elaborate and integrate
preexisting spatial and numeric understandings that support learning in the domain. Using Case's
general theory O lintellectual development as a guiding h e w o r k , a theoretical model detailing
the age-related developmentril sequence through which students progress as they elaborate and
in tegrate these SChemas
was proposed, and a cuniculum was designed to foster that
development. Instructional studies were canied out to investigate (1) whether this sort of
instruction fosters students' understanding more readily and comprehensively than traditional
tex tbook
approaches to the topic, and (2) whether the understandings of students in each grade
who expenenced the expenmental curriculum develop as predicted by the model.
Discussion of Cunicular Com~arison
Results showed that the Grade 8 and Grade 10 experimental groups performed
significantly bener on the functions test after instruction than did the coatrol groups.
Furthemore, the Grade 6 experimental group outperformed the Grade 8 controt group, and
performed comparably to the Grade 10 control students, who not only had twice as much
instructional time using a more traditionai approach to the leamhg of functions, but also hadhad
two years of prior leaming on the topic- This latter finding in particular raises questions having
to do with the approach and content of more traditional mathematics instruction, at l e s t in the
domain of functions. Analyses of the individual items indicated that, in generai, the experimental
group demonstrated superior fluency and enriched competency with moving among the
representations of a function and with identifying the numerical patterns that characterize the
spatial entailments of functions.
The posttest differences between instructional groups may be considered in the context of
the di fferent curricular emphases, Teachers ofboth control classes believed that students at their
school who complete the standard tocd secondary mathmatics curriculum should develop the
sort of flexibility and competency necessary for success with the items that appeared on the
functions test. Students in both these classes, however, had limiteci success with the tasks, which
required fluency and flexibility when wotking w i t h or among the various representations of a
function. The hypothesis here is that such fluency and flexibility is developed through the
merging of students' initial numeric and spatial undemandings that are relevant to the domain.
In both control curricda, each of the digital and spatial schemas was elaborated in
isolation, but no contexts or instructional approaches that fostered the integration of the SChemas
were uscd. For exarnplc, the Grade 8 snidents codd Iook at a table of values and identiQ a
relationship such as "multiply by 2" and reptesent that relationship as "y = W .These students
could also make tables and plot graphs given equations. Students did not, however, engage in the
sort of meta-analyses necessary for coordinathg th& numeric uoderstanding of a tùnction
rcpresented in verbal descriptions, tables, and equations with the particular spatial characteristics
that are representable in the graph of that same k t i o n .
For the Grade 10 coatrol students, the two schemas were also elaborated independently,
without sufficient opportunity for integration. For example, students worked on dgebraic
manipulations and complex computations found in polynomial expressions. They solved a
number of problems where they substituted a certain value for into given fiinctions of the form
Ax). They also spent tirne on the computer transforming the graphs of fiinctions by aitering the
parameters of the equation y = a(x -p)2 + p. However, midents were not required to examine the
numenc patterns that resulted h m these manipulations and parametric alterations. Thus, the
connections between the numeric and spatiai behaviors of this fiinction were not explored,
In contrast, the design of the eqwirnentd c ~ c u l u r nwas based on the assumption that
students require ample opportunity to integrate their numeric and spatial schemas in otder to
develop flexibility and competency in the domain. in the experimental approach, the walkathon
situatim acted as the bndging context h m which each of the various representationsof a
function could be naturalIy and intuitively constnicted and C O M ~ & ~ . The relative emphasis on
the tabular form of a hnction seemed to be a distinNshing feature of the expenrnental unit, and
one that tumed out to be a significant bridging representation with respect to the two schemas.
Although a table of values is primarily numeric, it also has the quaiities of a spatial analog if one
considers the pattern found in successivey - values to be going
or going d o m by a constant
or increasing arnount. These changes were related to the spatially visible changes in the graph.
There were also other distinctive features of the curriculum that may have fostered
çreater flexibility and cornpetence among the experimental students: (1) In the experimental
cumculum, a single context of the walkahon for introducing key concepts in the domain was
used rather than different contexts for diffetent concepts. (2) The various representations of a
function were introduced simultaaeouslyas equivalent forms of the same mathematical
relationship rather than as isolated representations of a fimction that may be derïved h m one
another. (3) The sorts of activities students did throughout the curricula were also diflierent-
Students in the control groups mostly completed pmblerns h m the textbook or h m
n-orksheets. The experimental group first did exercises that were related to the walkathon
situation. They then consolidated and extended their learning tbrough computer activities.
Finally, students gave presentations to their classmates on a particular h c t i o n for which they
lixi becorne "experts."
Comparing the use of computer activities between the experimental groups and the Grade
10 control group
is particularly interesting given the emphasis on software development and
irnplernentation for the learning of functions throughout a11 levels of education, including poststcondary (Hillel et al., 1992). The experimental groups worked on computer activities that
reviewed, consolidated, and applied concepts that had been presented during ctassroom-baseci
instruction. The Grade10 control group, on the other hand, did one day of exercises on the
cornputer that served as instruction for, rather than consolidation and application of, transforming
tiinctions. It is not possible to measure the impact of the experirnentd students' use of the
cornputer as a tool for consolidating and applying concepts separate h m the impact of their
otlier experiences. It is possible, however, to look at the Grade10 control group's posttest results
for item 9, which instantiated the instruction and activities of these students with respect to the
transformation of quadratic functions. The majority of these students did not pass item 9. This
rcsult suggests the need for further investigation into questions about what may be the most
beneficial use of the cornputer as a tooi for Ieaming in this domain. That is, are technology-based
leriming activities (including those that use graphing calculators) better suited for consolidation
and application of concepts fïrst inintroduced in the regular classroom setting than as instructional
vehicles unto themsehes?
In sum, results suggest that the experimentai approach to teaching fùnctions fosters better
than the textbook approaches the sort of Iearning about functions that mearchers and
inathematics educators deern crucial for supporthg more-advancd mathematical thinking and
for app l ying the central concepts to other discipluies such as physics and econornics. The present
q u m e n t is thai such l e d n g , which indudes ffexibility and ffuency w i t h and among the
rcpresentations of a function, is promoted through instruction that both elaborates and integrates
studcnts' numetic and spatial understandings that underpin the key concepts in the domain.
Discussion of the Model of Deveio~rnent
The pattern of item-by-item results was Iargely consistent with the one hypohesized. It
wris predicted that the majority of sixth graders would be succeshl with items 1 - 4 (LeveI 2);
rhc rnajority of eighth graders would be successful wirh items i - 4 and at least two of items 4 S ILcvcl 3 ): and thai the majority of tenth graders wouid be successfid with items [ - 8 and rnake
significant progress towards understanding items 9 - 12. Resuits showed that the majority of the
sisth sraders was successful not only with items 1- 4, but a
h with item 5; the majority of
cighth graders was successfd with al1 Level2 items and with items 5 and 6 at Leve13; and the
mrijority of tenth graders was successfiil with the level2 items plus items 5 - 7 at IeveI 3, but not
\i.ith item 8 at Level 3 or with any Level4 items. Thus, overalI, grade-reiated Levels of
~tndrirstandingwere found,
Items on the boundrines of the leveis, however, were miscategorized. Items 4 and 5 were
onIy passed by a majority ofthe higher-achieving Grade 6 students, dut is those Grade 6
students whose CATI2 grade equivalency scores were in the upper half of the group. Thus, these
niay be transitional items that characterize students' moving h m a Level2 to a Level3
understanding. Item 8 proved to be more difficdt for Grade 10 students than anticipated. Success
with this item required students to make many more complex connections among the level2
idcas tlian were foreseen. Thus, this item will be reçategorized as a level4 item for fiiture work.
With the re-categorization of items 4 and 5 as transition items and item 8 as a Level4
itm.
the numbcr of tasks remaining at Level2 and Level3 was reduced to three and two.
respectively. In addition, a new transition stage was inhvduced. Thus, new tasks need to be
cic~dopedto explore students' understandings M e r at LeveI 2 m d Level3 and as they
transition from Level 2 to Level3. Furthemore, opportunities to evaiuate the suitability of the
csisting level3 tasks must be createti. Given the low level of performance of the Grade 10
control students on Level2 and Level3 tasks, the need for such opportunities seems to be in the
t'om of an extended snidy that heips students build Level2 and Level3 understandings before
advoncing to Level4 topics.
What has been shown here is an attempt to apply the principles of developmentai
coçni tive psychology to the leaming of mathematical hctions. The study reported is one
crnpirical step toward validating the notion that a defined developmental sequence may provide a
coinprehensive and systematic framework for domain-relevant leaming and instruction.
Furthemore, results support the hypothesis that amodel of deveIopment and instructionai
qproach that share an emphasis on the integration of chiidren's spatial and numeric schemas
faster a deeper conceptual understanding of the domain than do traditional approaches.
It is not possible to say with certainty that these grade-related IeveIs of understanding are
3
rcsult of the son of development ûajectory suggested in the learning sequence. This is because
i t is no t for certain what the leaming profiles would look like ha& for instance, the Grade 6
students been able to continue with their study of functions and to reach a place where their
school-based expenence with h c t i o n s matched that of the older students. What is supposable is
that students' experience with mathematics as a school-taught subject in combination with
ii-iatliernatics as a day-to-day activity outside of the classrnom influenced performance. In this
~ i c wthe
. older students indeed had had more relevant experiences going into this study. This
notion may also be illuminated in the context of schwl learning being culturally systematized in
such a way that students' Iearning on any particular topic in school mathematics is ''unitized"
riricl
thus their instructional experiences are limiting.
A next step dong this path of inquiry is to extend this work to students who are less
advantaged rnathematically in order to ascertain what their developrnental limits are. Careful
rinalyses of the mathernatical prerequisites for functions knowledge are d s o important so that
they can be addressed within instnrction at any grade IeveI if necessary. Such an anaiysis couid
dso inform cumculum design for earlier grades so that a11 students could enter sixth grade with
the rcquisite understandings for hnctions - a topic that wiU underpin so much oftheir higherniathematical learning.
In conclusion, the relationship between cognitive structurai modeling and curricular
design is a circular one. Once the central conceptuai structure that underpins children's
understanding has been described, the task of designing a curricuIar sequence and piloting
ixious bridging contexts can begin. However, as the original model is impiemented via the
Jesignated cumculum, ways in which the model and curriculum must be expanded or modified
\vi ll be
illuminated by children's performance in the leaming context. tt is hoped that as the
developmental mode1 and instructional techniques are refined, a deeper understanding of the
psychologicai processes that are involved in the successfùl ieaming and teaching of fùnctions
\vil! be discovered and thus contribute to a more detailed mode1 and program of instruction.
cxAPTER3
Understanding MathematicaiFiinctions:
Stridents' Solution Stntegies
The results reported in Chapter 2 support the hypothesis that the experimental cuniculum
better facilitates students' leaming of functions. Two questions follow up these fidngs in order
to attempt to understand them more deeply.
First, are there general qualitative differences in the sorts of reasoning about each item on
the functions test between the experimentai and control groups? More specificaily, is there
evidence that students in the experimental groups had indeed integrated their spatial and numeric
understandings? And if so, is it the case that the control students did not demonstrate such
integration? Included in this inquiry is the question of how each group's approach to each item
changed from pretest to posttest as a result of the respective instructional experiences.
The second follow-up question concems the cross-sectionai anaiysis. Even where there
were no notable differences in the percentage of students who got an item c o m t among the
experimental groups, are there qualitative differences in the experimentai students' approaches to
that item? That is, were correct strategies more common among experirnental groups even
though the strategies were not compieted niccessfully? And, where differences in passing rates
were found, was there evidence that the differences in the correct and incorrect strategies were
attributable to the students' level of development as predicted by the model?
Questions about the experimental students' reasoning are important to ask because they
pertain to the validity and practicality of the general theoreticai c l a b that students progres
through a series of elaborations and htegrations of their numeric and spatial uuderstanduigsof
functions as they deveIop a central conceptual structure for fiinctions. Specificaiiy, the questions
scek to explore whether the theoretid position has explanatory value for how students develop a
conceptual Eramework for functions, and whetber the theoreticai stance is a pra~ticalheuristic for
designing curriculum that fosters such developrneat.
Codingof Soiutions
Tù answer these two questions, students' pretest and posttest solution strategies were
examined and categorized. Each sïrategy category was then characterized as correct, partially
c m e c t, or incorrect. Correct solutions were those that were clear and correct. PartiaiIy correct
soltitions were those for which students showed that they had used a strategy requiring at least a
partial understanding of how to solve the probIem. Dificdties that stemmeci fiam incornpetence
or carelessness with complex computation, algorihs, or algebraic constructions were the
p r i m q cause for responses being categorized as partially correct. Incorrect respoases were
those for which students did not demonstrate even partially c o m t steps of any correct strategy.
To establish the strategy categorïes, each tesponse to each item was characterized and
sirnilar responses and related approaches were identifieci. Prior to categories being fonned, it was
decided that at l e s t 20% of responses would have to be a particular strategy for it to be
considered an independent category. Item groupings that did not reach this threshotd and
responses that did not fit into any particular grouping were considered anomalous. Anomaious
categories exist for both paitiaiiy correct and mcorrect sdutions. Moreoveri ifa student made no
discernable effort to answer the question, the response was categorized as "Blauk." Mer the
initial coding and categorkation of each respanse, an independent rater was trained in the
scoring and coding procedures and was givm a randomly selected 12% of the sample tests to
score (a total of 327 responses). A 98% inter-rater agreement was reached.
Conceptual Mappuigs and Requirements for Each Item
Each item on the functions test required t h students had made some progress through
the model of development described for the learning of hctions. The conceptual elaborations
and integrations hypothesized for the leaniing are operationalized in items 1 through 12 in an
order that mirrcrs the pattern of results achieved by the expenmental students on the posttest.
The conceptual requirements for each item as it relates to the model of learning are described
below. Items are grouped accordhg to revis& b e l s of development that resulted h m the first
set of analyses. Table 7 summarizes this pattem of deveiopment in an item by item format. A
cornplete set of the items is found is Appenix
as weU as in individual Figures that support
the discussion of each item in the upcoming strategy anaiysis.
Level 2 items
Items 1,2, and 3 constitute tasks that represent Level2 learning: an elaboration of the
nurnenc schema, an elaboration of the spatid schema, and an htegration of the two. In Iteml,
students were given the following situation and task: "!kppose 1agree to pay you $30.00 for
every hour you work. Give a function h t we could use to caIcuIate the total amount of money
you have eamed &er you have finishedx hours of work" For success with this item, it was
' The order the items appear on the fimaions is diîX~feathmIhc order a which thcy are laid out here due to the
post-instructional analysis of dficuIty and tht p a nofd t s €4
among the cxpcrimmtai groaps. In tbe
original functions test, item 3 is discusd h m as k m 1, aiad item 7 Ïs discussed as item 5. The rcmaining items
remam in the order in which they appcar on ihc test.
2
Draw ihc shape o f y -x2.t I
I~lühoriiiiorio f initial spiitiiil scherno
3
Give an cqutttion for a funciion ihar crossçsy-x+7
Firiil iiitcgraiion of clabornird niinieric and spatial sclreiiitrs
.Transiiion IO 1,evel
4
5
.--
6
7
....-.-..---.------.-.AL
3 Itcnis
Makc a table for increasing linear function wiih a iiegalive y iniercept
Inteyaiion o f y = r~wand y = x + b --+y
Give a hnction ~ h agcnrtratcs
t
2, 5, 8, I 1, 14,..
*- ."- .
Lave1 3 i i e ~
Giva 2 equatiotis thai produce iricreasing lincar hnclion wiih a
positive y - intercepi
fggure
..
... . " . --
.. . - -
-
-
~
-
. -. .
8
9
Give cquaiion for invened parabla shiAed io riglii
10
find x - iniarccpts for y = 10x - x'
II
What skape would graph o f y = IOOX - ZSX'
Whut Iisppeiis i n the fiinctiony =
+ x'
Irtivc'?
--1 as x increases?
..
. .
-
.
---
- -
= riw
+ b plus misleading "+Y
--
-
"-A
Iniegrated y = IILX + b siructurc plus geiieralizing and abstruciing frorii
individual funciions i o furnilies o f bnctions
Ceii the graph ufy = IO - x be the saine as y = x - IO
ve14 items
%eh y x' onio graph with y a r' already thcrr
12
Integruted structure plus elaboraiions of numeric and spatial understandings
include negaiive constants and coefficients
IO
,
Iniegraiing siruciures that includa rlabarations fur spatial and numeric
iinplicaiions o f negativc consianis and caeftlcienis plus misleading "-10"
. ka%
- - ---- .- - .-.- --.. -.... . -,
- - .-A. .. . ..--..-- . . .
.-.-m.
-
lnicgratiiig varialits o f structures consiructed for curving fiunciions plus
relating four quadranis to each oiher
Iiitegrating variaras o f structures consiructrd for translations o f quadratic
fiinci ions
Kelating intryratrd striiciiires for lineur and non-linear funciioiis
Fully iniegrated systeiii for funciions including polynoniinls
I:ully inicgruted systrin for functions including iioa-intcr x volucs
.A*
expected that students would have an elaborated numeric scherna such that they could represent
the relationship between a function's dependent and independent variables when both variables
were given.
For item 2, students were asked to draw the shape ofy = 2 + 1. They were given axes
that represent the upper-right quadrant of a Cartesian grid. Tu get this item correct, it was
expected that students would have an elaborated spatial understanding such that they understood
that a function in which the x value is multiplieci by itself at least once creates a curvilinear
pattern when graphed because the y-values increase 'By more and more" for each unit change in
s. It was not essential that students were able to calculate$ -t 1.
In the third item, students were given the graph ofy = x + 7 in the upper-right quadrant.
The scale was unitized fiom 1 - 10 on both axes. Students were asked to think of a f'unction that
would cross the function seen in the graph and to give the equation of the function they thought
of. (Students were explicitly asked to give a function that would pass through, not overlap, the
given function within the space provideci on the grid.) To achieve a correct solution, it was
expected that students would have integrated theu numenc and spatial undetstandings such that
they showed understanding of, and facility with, the way in which a graph, table, and equation of
a particular Function are numerically and spatially connected. Students were expected to
demonstrate this understanding by applying an equation (an elaborated numeric understanding of
a function) to a spatial representation of that same fûuction (a graph).
Transitionine fiom Level2 to LeveI3
From the eariier analysis of the experimentai groups' ov-di pretest to posttest
performance, it was decided that items 4 and 5 were items that represent students'
understandings as they transition h m LeveI2 to 3. In item 4, students were asked to make a
table of values that would produce an increasing Iinear fimction with a negative y-intercept. This
item required that students had fulIy integrated their numeric and spatial understandings of
functions, and had gone on and elaborated their numeric and spatial schemas to include negative
-.-values. The integrated structure is found in students' ability to show that both the graph and the
table for an increasing linear function have a constant dope. This is shown by students
generating a table that represents a pattern of covarying quantities (Confiey & Smith, 1994,
1995) with a constant increase iny for every unit change i n x Students dernonstrate the required
elaborations by representing negativepvalues in that pattem of change in both the table and the
graph.
In item 5, students were given a sequence of values (2,5,8, 11, 14, 17...) and were asked
to write an equation for a function that would generate that particular pattern of values. To get
this item correct, students had to understand h t a b c t i o n (at least of the sort they had been
studying) was an expression of the reiationship between two variables (elaborated numetic
scherna), where the dependent variable is generated by carrying out actions on the independent
variable. Thus, students had to recognize that there was an impiicit independent variable. in
terms of the connections students had to make to answer this question correctly, they had to
integrate their integrated structures for dope (embodied iny = mx) with their integrated structure
for y-intercept 0, = x + 6).That is, they bad to integrate the basic pattern of linear change found
in the sequence (a constant ùicrease of 3), with the recognition that the k
tvalue in the sequence
is not zero and thus, the whole funetion must have a y-intmept other than O. Together, these
two structures generate a single structure for uaderstandingy = mx + b.
Level 3 items
Items 6 and 7 were Level3 items because they required that the integrated conceptual
structure for functions developed throughout Level2 can be abstracted and generalized to a
variety of novel circumstances, thus adapting the conceptuai structure to meet the demands of a
probiem.
For item 6 , students were aslced to give two equations that would produce a graph of an
increasing linear function with a positive y-intercept. They were given a graph with unmarked
axes in the upper-right quadrant. ibis item tested students' understandings about what sorts of
algebraic expressions make an inmasing linear fuaction with a positive y-intercept. Although
this item required rnany of the same understandings and connections needed for items 4 and 5,
the underpinning connections were more abstract and cornplex. That is, students needed not only
an integrated mental structure fory = mx + b, but they also needed to understand how that
structure may be instantiated in the context of the given graph (in contrast to k i n g given the
instantiation as in item 5). The task was summed up by a student who wrote on her posttest paper
"any equation which is a positive siope with a positive intercept."
In item 7, students were asked if y = x - 10 can represent a given graph (of y = 10 - x).
Students were also ~ t e tod sketch the graph ofy =x - 10 if they thought it could not be
represented by the graph. This item required that d e n t s had elaborated their integrated
conceptual structure such that they understood the spatial and numeric impact of negative
nurnbers. The -IO in the equation was a distracter for students either who had o d y tenuously
made the many connections necessary for conectly anmering this question or who had mt made
them at all. Eariy in primary school, students have a well-esîabiished schema for subtraction as
reducing or lessening a quantity. It is couter-intuitive, therefore, for many students to imagine
that y = .r - 10 has a graph that inmases.
Level4 items
Level4 items required that studeats had integrated the variants of the conceptual
structure developed in Level3 as a result of applying that structure to novel situations. These
fi na1 integrations combine with a clear understanding of how the four quadrants of the Cartesian
system relate to each other to fonn compkte and flexible understandings that facilitate solving
sophisticated and cornpiex probkms in the domain of fiinctions in general. Each item at tbis level
progressively requires the application of a conceptual structure ihat resulted h m the Uitegration
of variants of less elaborated structures deveIoped in Levels 2 and 3. Students had to be able to
apply these sophisticated understandings to aü four quadrants of a Cartesian plane and use
complex computation and algebraic structures involvùig polynomials and non-integer values of
.r. Because pretest and posttest performance on these items was so Iow for aü groups, the Levek 4
items wilI not be discussed here in m e r detail.
Empirical Resuits and Discussion ofIndividua1 items
The percentage of students in each cIass at pretest and posttest who used the three levels
O f strategy
(correct, partiaiiy c o m t , or incorrect) was recorded in item-specific tables for each
item on the fùnctions test. Figures presentmgthe item and sainple correct, partialIy correct, and
incorrect so1utions will be given for each of these items as they are discussed below. TabIes that
describe the actual strategies within the correct, partially correct, and incorrect classifications can
be found in Appendix B.
The mathematical specificity of each item requires that results be given and discussed in
an itern-by-item format. Generai cornparisons will be made between the ways in which the
csperimental students approached the problems and the ways in which the control students
approached the problems. Patterns of strategies for each instructional group will be related to the
conceptuai expectations for the item as described above and in Table 7.
Pretest scores were g e n e d y Iow across groups, and many solutions were categorïzed as
Incorrect Anomalous. That is, they were inconect solutions and did not fit into any particular
strategy grouping. Comparing experimental and control students' pretest responses did not yield
any outstanding patterns of differences between the groups. Thus, the pretest strategies will not
uenerally be discussed for each item. Exceptions to this will occur in cases where something was
2
rcrnarkable in the pretest strategies with respect to a robust preconception about hctions among
the sixth and eighih graders in general (neither of which had received any forma1 instruction on
functions pnor to this study) or to a common strategy used on the pretest by the tenth graden,
who had experienced three years of textbook instruction on the topic.
Level2 Items
Table 8 shows the percentage of students in each p u p who gave correct, partially
correct, and incorrect solutions for items 1,2, and 3.
Item 1
Item L and sarnple correct, partidly correct and incorrect solutions are found in Figure 9
Item 1 pretest strateaies. Item 1 was the only item on the pretest that students in any
class "passed" (that is, an overall rate of 50% correct was achieved). Al1 gmups except the Grade
6 group achieved a passing rate on the pretest (see Table 8)- with an overail mean of 53% for the
Grade 8's and an overail mean of 68%for the Grade 10's. mus, one-half to two-thirds of the
oldcr students had already begun to elaborate their initial numeric understandings to include the
ability to represent in an equation the relationship between two given variables. An analysis of
thc content of the Grade 8 and 10 conûol cunïcula suggested that traditional approaches io the
leaming of functions privileges the numeric aspect of a function.
One particulariy relevant sort of activity that students do in many traditional mathematics
programs is to practice interpreting and modeling word problems in two variables using
algebraic constructions. Students are taught a particularmethod for deriving eqmîons h m
situations, and this method was found repeatedly in studenrs' pretest solutions: "Let x represent
the nurnber of hours worked. Let y represent the amount of money earned." M y the Grade 6
students did not use this approach. Thus, the method was likely introduced to students in Grade 7
or exly Grade 8. Most of the Grade 8 and Grade 10 students had obviously had experience with
this sort of problem pnor to the test, aud this experience was reflected in the higher success
pretest rate with this item. This type of activity encourages students to elaborate theu prhary
numeric schema.
Although the sixth graders had not yet had experience with modehg quantitative
relationships in two variables, some students were able to give a correct representation of the
Suppose 1 agree to pay you $30.00 for every hour you work. Give a furiciion that we could use to
calculale lhc total arnount of nioney you have eamed afier you have finished x hours of work.
I
Correct Rcspansc
Part ially Correct Response
9. Samr>le PPartinll~Co-
Incomct Response
O
'O
for w
prob lem on the pretest and others were able to give a single instantiation of the pmbIem for a
partialIy correct soiution. Using instantiation to represent general situations bas been consideseci
a step aIong the path of children's
developing an understanding of the role of algebra in
representing mathematical relationships (Carraheri Schliemann & Brimela, 2000; English &
Halford. 1993; Swafford & Langrall, 2000).
Item 1 posttest results. Large pretest to posttest gains were made by al1 groups except the
Grade 10 control class (59% on the posttest for a decrease of 7% h m the pretest). Posttest
results in general indicated a post-instmctioaal
understaading of the probIem (means for the
other 4 groups were 86% correct and 6% pattidly correct).
There were differences in the ways the eXpenmental and conml p u p s appmached a
correct solution for this item. Conect responses were divided berween those that included one
general representation of the relationship between h o m worked and money eamed and those
that inciuded more than one representation. For example, the former was usual1y an equation
such as y = 30.r or M = 30h. The latter was d l y an equation as weIl as a tabk of values that
represented the relationship (see the first column ofFigure 9).Thirty-seven percent of the
experimental students and 10% of the control students used a multiple representation strategy,
and 52% of the experimental students and 59% of the control students used a single
representation strategy. This difference may be attniutable to the fact that the experimentai
groups had grown accustomed to using multiple representaiions in their class work. It is also
possible that a table or p p h was used by the experimentai students as a scaffold for determining
the equation of the h c t i o n or even for checking that their equation 'korked". This possibility is
supponed by the fact that the G d e 10 experimental students used the multiplerepresentation
strategy l e s than students in the two younger grades (25% versus 44% pooled, respectively), and
may reflect the older students' comfort with bypassing the table of values and cons~ictingthe
equation directly.
Both Grade IO groups began with about the same swcess rate for this item and no growth
or change in solution strategies was found for the Grade 10 control class. This lack ofchange
arnong the Grade 10 control students combined with the significant growth for the Grade 8
controI students suggests that traditionid instruction pnor to tenth grade is respom'ble for
building students' competence in generating algebraic expressions for situations in two variabIes.
In their functions unit, the Grade 10 control students did exercises h m the textbook in which
they were required to solve linear systems in two and three variables using a specified algebraic
procedure. However, they were given the equations with which to work and were not required to
construct the equations that went into those linear systems.
There was one change in the Grade 10 conûol group worth mentioning. Almost onequarter of the students in this group gave some rendition of the generai linear equationy = nur +
h as the final response (see third column of Figure 9). Indeed the situation presented a liear
relationship. However, without the slope of 30 substituted for m into the g e n d equation, it
cannot be used specifically for solving the probiem. Students may have been overgeneraiizing
the utility of the general linear equation, or it may have been that they did mt understand the
roles of the individual letters in the equation (Schoenfeld et ai, 1990).
Item 1 surnmarv. Some members of each of the Grade 8 and Grade tO groups at least
passed this item on the pretest thus showing some pre-instructional understanding of how to
soive the problem. Al1 groups showed relative competence on this item on the posttest, with
large pretest to posttest gains for al1 groups except the Grade IO control group, which a c M y
showed a decrease in correct responses. Successfd solutions required that sh~dentshad
elaborated their digital schema to the extent that they were able to represent the relationship
behveen a hnction's dependent and independent vaciables when both variables were given. Both
i nstructional approaches fostered an elabration of the primary numeric schema. Thus, fbr al1
groups except the Grade 10 controI p u p , the relevant instructional program resulted in an
elaborated numeric sciiema.
Item 2
Item 2 and sarnple correct, partially correct, and incorrect strategies are found in Figure
hem 2 ~reteststrategies. Pretest scores were low for al1 groups on this item with an
overai1 pretest mean of 1 1% correct. However, the mean pretest partially correct score was 23%,
with the Grade 8 groups at 41% and 42% parhaHy correct for the experimentai and control
croups, respectively. A sampIe partially correct solution for this item is found in the second
"
column of Figure 10. Although no one strategy stood out for these partiaily correct solutions, the
Grade 8 control group's responses did coUect amund one strategy. This strategy was to draw a
tUnction with the correct shape but beginning h m the origin. Many of these students even made
tables to represent y = x' rather than y = .2 + 1. It is not possiile to h o w h m the studenrs'
responses if they did not know how to include the constant (1) in their caIculations, or if they
sirnply forgot to include it.
There was one outstanding pretest problem particular to the Grade 10 p u p s . Thiry-six
percent of the Grade IO students attempted to use an aigorithmic procedure for 6nding a solution
to
item 2 on the pretest. This procedural generdy involved finding two cwrdinate points by
substi tuting in two randomly chosen d u e s forx to get y; plotting the two coordinate points; and
What shape would ihc graph of the function y =.?+ 1 have? Draw it bclow.
x-axis
Correct Response
Partially Comct Response
Incorrect Response
Curved Line from (O. 1)
Full Parirrtbola with Vertex
Lu
W r e 10%
Sample Correct, Partially Correct and hcotrect Solutions for Item 2.
then joining them with a straight üne (see the first example in the thirà column of Figure 10).
This strategy is in keeping with students' p s t instructional experiences with linear functions,
which emphasized the need to find only two points dong a straight line in order to graph that
line. Literature suggests that early mathematics instruction places an intense focus on linearity
(Howden, 1989; Leinhardt et al, 1990). Arnong students who used the algorithmic strategy
described above were some who prefaced theu caiculations by writing the equation y = mr + 6,
thus indicating an assumption that the fimction was linear. This assumption is an example of how
students constmct mainly linear graphs when considering a set of given points because they have
had lirnited expenences with non-hear mathematical relationships (Bell & Janvier, 1981;
Dreyfus & Eisenberg, 1982; Geer, 1992; Karplus, 1979; Lovell, 1971; Markovits et ai, 1986;
Swan, 1982; Zaslavky, 1997).
These students, who had spent years building a repertoire of formulas and methods, either
had too little experience with non-hear functions or an inadequate amount of experience to
for hctions. Such "number sense" ailows students to
enable them to develop "number senseTV
select the best method for approaching a problem and to judge the "reasonableness" of their
solutions in the context of the problem (Case, 1998; Greeno, 1991; Kalchman et al, 2001;
Sowder, 1992).
Item 2 oosttest results. Large pretest to posttest gains were achieved by al1 groups (mean
gain for al1 groups pooled was 57%). Even the Grade 8 contra1 group, which did not pass item 2
on the posttest, had a 38% ciifference h m pretest to posttest. The total posttest partially correct
rnem score dropped to 16%, but more significantly, the Grade 8 groups' partiaily correct posttest
scores dropped to 2 1% and 17%.
Patterns of pusttest respoIlSeS were afso reiatively untemarkable. Although the Grade 8
control group did not achieve a passing rate on this item, here was no single incorrect sirategy
that could point to a problem that may have resuIted h
m insiruciion. Stiil, the diffetences
between the Grade 8 groups' strategies for a correct solution were notable. Students in the
experirnental groups in general tended ta draw smooîh increasing curves h m a point on the y-
axis that could represent (O, 1) (see the k t exampIe in the first column of Figure IO). The
concentration o f correct responses for the Grade 8 control group (22%)was on plotted c w e s
beginning from the point (I,2). Most of these sîudenîs did not indicate an understanding of the
y-intercept as a spatial or numeric feature of the functiony =x2 + 1. The mIe ofy-intercept as a
special coordinate that can be used to identify any bction's g e n d position on graph was not
addressed in the Grade 8 control curriculum in the way that it was in both the experhentai
curriculum and in the transfomation exercises the Grade 10 contrd students did on the
cornputer. Thus, the "+I" did not have any special meaning for these students and could not be
irsed as a heuristic for their solutions.
The Grade IO control group's most popdar c o m t response was a fun parabola with a
vertex of (O, 1) (65% of their responses were of this sort) (see the second example in the 6rst
column of Figure 10). The instructional focus ofthe Grade 10 conid curriculum clearfy
iduenced students' responses. What is not clear is whetber or not the Grade 10 control students
who used this strategy understwd that the patabola could be "split up" and adequately
represented in just the upper-rîght quadrant.
Although al1 three experimentai groups passai this item, both the Grade 6 and the Grade
8 experirnental groups had passing rates considerably below the Grade 10 groups. Quaiitatively,
no differences were found among the experimeatal groups in the approaches they used for a
correct solution on the posttest. With respect to the Grade 6 students, no one sort of enor wu
found among their solutions. A common enor found for the Grade 8 experUnental students
(71%) was to make a table and graph for the fiinctiony = 2x + 1 (see the second example in the
third colurnn of Figure 10). In this strategy, students t m k 2 to be the same as 2r, calculated
points, plotted them, and rightly connected the points with a straight line. Students who used this
strategy clearly still had some unelaborated numeric understandings that would allow them to
recognize a pattern of co-variation between variables h m a power fiinction's equation. Without
this numeric elaboration, the spatial elaboration was impeded.
Summarv of item 2. Al1 groups except the Grade 8 control group passeci this item on the
posttest, with both the Grade 10 groups doing markedly better than the Grade 6 and Grade 8
experimenta1 groups. The pretest to posttest &op in the eighth-graders' partially correct scores
combined with the large pretest to posttest gains achieved by al1 groups, hdicated that both
instructional programs promoted some leaming for this item. However, the fact that the Grade 8
control group did not pass this item shows that in general, the curriculum for this group did not
foster an elaboration of these students' spatial understandiigs of a function.
The trouble the Grade 8 experixnental students in particular had with the exponentid
notation in the problem (confùsiq? and 2x) suggests that the current fom of the item may not
have been the best test of students' spatial elaboration independent of their numeric elaboration.
It
may be better to give an equation such as y =x*x + 1 rather than y =.$ + 1 next time. This
problem with exponential notation was also common on the pretest for tbis group. Thus, it is
unlikely that the experimental curriculum engendered this problem. Howwer, the curriculum
also did not elirninate the problem, and consideration shouid be given to friture work for building
into the curriculum more direct practice and explanarion having to do with the meaning and use
of conventional notation systems involving exponents.
Item 3
Item 3 and corresponding sample correct, partiaily correct and incorrect responses are
found in Figure 1 1.
Item 3 nretest straterries. ûverall, a mean of 13% of students got item 3 correct on the
pretest and 17% got it partially correct. Two pretest scores stand out despite their being below a
passing rate: the Grade 8 experimentai group's partially correct pretest score (45%), and the
Grade 10 control group's correct pretest score (40%). In addition to these salient results, two
particular incorrect strategies were identified for the Grade IO groups on the pretest, both of
which reflected past instructional experiences that led to an inappropriate use of the general
equation and an incorrect algorithmic appmach.
The Grade 8 experimental group's partially correct pretest score WU dominated by one
particular strategy. These students gave "y =x + TT'as h e u solution. Students who gave this
solution obviously gave the equation for tbe given fimction, despite king explicitly instmcted to
give a function that would cross and not overlap the one given. A fairly large proportion of the
Grade 8 control group also gave this answer on the pretest (27%). It is possible that these
students did not fùlly understand the ta& However, some students' explanations for their
mswers indicated that they were responding to the task as though it were an exercise familiar to
them from past instructional experiences. Students seemed confident with naming a reIaàonship
found in a table or graph and then represented that reiationship in the form of an equation or
verbal description. Most students who used this strategy did not attempt to find a funetion that
Can you think of r hnction thac would cross the function seen in the
i
gnph below? What is
n you thought of!
Explain why the cquiitiwi yw chose is a good one
Correct Strategies
Partially Correct Strategies
Incorrect Sirategies
y=x+7
Figure 1 1. Sample Correct, Partialiy Correct, and incorrect Strategies for Item 3.
would cross the function given. Rather, they focused on finding and explaining the relationship
in the graph itself. This strategy is illustrated in the middle column of Figure 11.
The Grade 10 control group's pretest correct solutions and the incorrect algorithmic
approach used by both Grade 10 groups on the pretest were related in terrns of strategy. An
algorithmic approach was generally characterizai by students attempting to find the negative
reciprocal of the given function, and then applying that new slope to an equation. This strategy
depended on students knowing fiom previous years' instruction that the slopes of perpendicular
lines are negative reciprocais of one another. For those 40% of students in the Grade 10 control
group who got item 3 correct, this was the strategy of choice. However, many ofthese students
seemed to have oniy a partial recollection of the procedure they were trying to use and created
solutions that were not only incorrect but were also inconsistent with the graphs drawn to
illustrate their equations (see the fim example in the third column of Figure 11). This
inconsistency illustrated that these students' numeric and spatial understandings of hctionai
relationships had not been integrated in their previous years of instruction, and their
nurneric/algebraic understandings twk precedence in their approach to this problem. Responses
uhere students were unsuccessful with this procedure accounted for onequarter of al1 Grade 10
pretest mistakes.
The second common strategy used by both Grade 10 groups on the pretest was a strategy
coded as "Generai Equation." Solutions in this category included some uninstantiated variations
on the general equations y = mx + b and y =a? + b. An example of this sort of solution is h
d
in the second sarnple incorrect response in the third column of Figure 11.
Item 3 uosttest results. Posttest result showed that al1 three experimentalgroups readiIy
passed this item (mean correct score of 69% and mean partiaiiy correct score of 7%), whiIe the
Grade 10 control group barely passed it (53% c o m t and 18% partiaiiy correct) and the Grade 8
control group did not approach a passing score (13% correct and 22% partially correct). The
Grade 8 control group's correct responses will not be considered when compaxing strategies for
correct responses on the posttest since there were so few of them.
Item 3 was the only item that showed quahtatively different correct strategies among the
experimental groups. The Grade 6 students gave correct increasing Linear fiinctions more ofien
than they did decreasing tiinctions (38% vetsus 29%, respectively). The other two experimental
goups did the reverse, that is used decreasing h e a r fiinctions more often than increasing ones
(43% and 17% respectively for the Grade 8 experimerital group and 50% and 6% for the Grade
10 experimentai group, respectively). An increasing linear function required the basic notion h t
the slope of any new fiinction had to be sufficiently steep so as to p a s through the original
tùnction. A decreasing linear function involved an added conceptual complexity of working with
functions that integrated negative dopes and positive y-intercepts. Both of the older experixnental
groups primarily used decreasing lines and ihus showed greater familiarity and comfort witb tbat
complexity. The Grade 10 students' experiences with perpendicular lines and with operating
tvith negative numbers and including negative nurnbers in their algebraic constructions may have
contributed to their choices. But, this does not explain the Grade 8 experïmental group's
preference for a decrerising function because they had not had these particuiar experiences. Thus,
a developrnental explanation that points to a more greatly elaborated conceptual structure for
linear functions arnong the Grade 8 and Grade 10 experimental students is viable.
A main ciifference between the correct responses given by the experbentai groups and
those given by the Grade 10 conml groul, was in choosing a ünear versus a aoa-hear fuaction
for the solution. Most of the experimental d e n t s chose either an increasing or decreasing linear
function to cross the one seen in the graph (see the fkst and second examples in the b t column
of Figure 1 1). About half of the correct response given by the Grade 10 control group were
quadratic hnctions (24% of the totai responses) (see the third example in the first column of
Figure 1 1). This choice was likely inauenced by the focus the control curriculum had on
quadratic hnctions toward the end of the unit.
A second qualitative difference between the experimental groups and the Grade 10
control group had to do with the strategy described above for solving the problem by fùiding the
negative reciprocal of the dope of the given Cunction and substituting that slope into the equation
for it. This strategy w u used by most of the Grade 10 contra1 students who gave a cor~ect
equation for a decreasing linear firaction. This strategy did not actually require that students
understood hnctional relationships, or even bat they were able to graph the resulting function.
CVhat they needed to know was how to coastnict an equation for the given function (which was a
ski11 found at the Grade 8 level on the pretest), how to find the slope in that function, how to find
the negative reciprocal of that vaIue, and how to substitute the new slope into the equation for the
original function. These requirements are not maIl feats. However, each one is associateci with a
particular procedure found arnong the topics of instruction for the control curricula
The poor performance of the Grade 8 control group on the posttest is worth commenthg
on since their results do not suggest that even an initial integration was established, Almost onethird of this p u p iefi this item blank and another one-third gave solutions that were not only
incorrect but also sufficiently vague such that their strategies were indiscemiile or obscure. StiU
23% of this group gave the equationy =x + 7 (the given hction) as the solution. No other
goup had a high concentration of respomeson the psttest in this category. It is still possible
that these students misunderstood the task.However, it is more likely tbat they responded with
the same strategy used on the pretest that reflectedthe sorts of exmises done in the traditional
instruction units: to find the relationsbip shown in the graph and represent that relationship as an
equation.
Summarv of item 3. This item ~ a i i requireû
y
ody the most basic of integratiom such
that students could answer cotrectly witb a simpley = nrr construction, where m is a positive
value that allows for a function steep enough to pass through the one given. This basic
construction was dominant in the Grade 6 posttest solutions, and demonstrated the basic 6rst-
order integration. However, the Grade 8 and Grade 10 students who received the experimentd
training showed that they had moved beyond that initial integration to include both spatial and
numeric elaborations. The numeric elaborati011~
aliowed for the inclusion of a positive yintercept and a negative value for the dope. The spatial elabmation allowed for a decreasing
rather than
an increasing Iine. With these elaborations, a full undentanding of the y = mx + b
family of functions was shown and suggests that a furthet second-order integration had taken
place. Furthenore, these snidents demonstrated a more advanceci understanding of, and
flexibirity with, the spatial requirements of this task by giving solutions that were more diverse in
terms of their choices for slope and y-intercept. The patterns ofcorrect, partiaüy correct, and
incorrect strategies between the experimentai and control instnrctiond conditions pouited to the
impact that the di fferent curricula and prior experiences had on students' reasoning.
Transitionai Items 4 and 5
Table 9 shows the percentage of students in each group who got items 4 and 5 correct and
partially correct. These items represent students' understandings as they transition h m Level2
ro Level3.
Item 4
item 4 and sarnple correct, partiaily correct, and incorrect solutions are shown in Figure
Item 4 ~reteststrateeies. Pretest scores for item 4 were ail low. The mean pretest correct
score for the five groups was 18% and the mean pretest partiaily correct score was 8%. One kind
of incorrect solution stood out on the pretest for the Grade 6 group and both the Grade 10 groups.
This solution was one where the table and the graph were inconsistent with each other. Students
who gave this sort of response generally labeled the axes of the graph and gave a table of values
that had no apparent connection to the graph (see the f i t example in the hird column of Figure
12). These students knew how to label a gaph and knew how to record coatdinate pairs in a
table, but did not dernonstrate an understanding of how the two are connected. The Grade 6
group had worked with "t-tables" (tables of values) earlier in their schaol year but did not have
extensive expenence with coordinate graphiug. Thus, their inability to connect the two
representations may have resulted h m a lack of experience in doing so. The Grade 10 students,
on the other hand, had had several years of graphmg coordinate pairs h
m a table of values in
earlier instructional units on functions. However, as suggested by the Grade 8 control
curriculum, students were likely not given contexts or activities which eucouraged their
Pcrcetitegc of Siiidenls Who Ciiivc Correct atid f'arlially Coriccl Soliiiioiis 10 ilein 4 aiid 5 (Traiisiiioii froni Lcvel 2 lo Lcvc! 3)
Grade 6
Experiniental
Prc
Pos t
Experiniental
Pre
Post
Coiitrol
Prc I'ost
(46)
(48)
(22)
(23
(22)
(23)
Correcl
9
54
27
87
14
30
19
88
20
24
Partially Correct
O
19
9
4
5
4
13
12
13
24
Total
9
73
36
91
19
34
32
100
33
48
Correct
O
56
O
70
18
13
6
63
7
18
Partial1y Correct
O
15
9
9
5
O
O
6
O
O
Total
O
71
9
79
23
13
6
69
7
18
1teiii
Ciratle 8
Grade 10
Expcrimental
Cotit rol
Pre
Post
Pre
Posl
(16)
(1'3
(15)
(17)
4. Make a table for increasiiig
liiiear funclion with a negaiive
, - inlercept
5. Oive a Iùnction Ihat generales
2, 5,8, 11, 14
Make a table of values that would produce the fwiction seen below.
4 7 , .
X
Correct Strate~ies
,
Partially Correct Strategies
Incorrect Strategies
Figure 12. Sarnple Correct,Partiaiiy Cornet and Incorrect Solutions for Item 4.
analyzing iinear relationships within a table of values. Ratber, "linearity" of a function was a
teature found in and characterized by its graph and its equation (y = nur + b construction), but not
its table. Thus, opportunities to connect linear patterns of nwnbers with linearpatterns in
equations and graphs were not made explicit. The Grade 8 groups did not use this strategy much
on the pretest. The Grade 8 experimental did not have one salient incorrect strategy, and 50% of
the Grade 8 control students lefl this item blank on the pretest.
Item 4 ~osttestresults. There were large posttest differences between the experimentai
and control groups on this item. The experimental groups had a posttest mean correct score of
76% and a posttest mean partially correct score of 6%. The control groups had a posttest mean
correct score of 27% and a posttest mean partially correct score of 14%.
Among the experimental groups, few qualitative differences were found for strategies
that led to a correct response on the posttest. Despite the relatively large diffemces between the
percentage of Grade 6 students who got this item correct and the percentage of Grade 8 and
Grade 10 expenmental students who got this item correct (differences of 33% and 34%,
respectively), there were no predominating enor categones for the Grade 6 group. However,
within the separate incomect response categories, these younger students showed that they were
struggring with aspects of negative numbers such as adding with negative numbers in order to
produce a constant increasing linear pattern within the table of values. This problem was
especially prevalent as results of computation appmachedy = O. Students were unsure of how to
add up to O f?om a negative amount and tended to want to skip the zero entirely. Consequendy,
students' responses resulted in tables that were either inconsistent with the graph or did not
present a number pattern representing a constant siope.
The differences in posttest strategies undertaken by the experimental groups compared to
the control groups were striking. Correct posttest solutions for al1 the experimental groups were
generally of the sort found in the h t example in the k t column of Figure 12. Students
estimated the approximate value of the y-inteccept of the given function and then constnicted a
table of values with the first coordinate pair king (O, estimated value), foliowed by a pattern of
CO-variationbetween the variables that dearly shows aconstant slope (Confrey & Smith, 1994,
1995). This linear nurneric representation is C
O M ~ Cto~the
~ linear pattem
found in the graph.
The concentration of posttest incorrect sûategies for the Grade 8 control students was on
soiutions in which the table and the graph were inconsistent (22%). This strategy was described
above as the most a common pretest error (see the first example in the third column of Figure
1 2). 1t is interesting that the same general m
r the experienced Grade 10 students had before
instruction, was the same general emr the Grade 8 control students had a#er traditional
instruction. This pattern suggests that the traditional instnictional approach did not encourage the
integration of the graphic (spatial) and tabular (numeric) representations of a fùnction. Because
the textbooks used were popular and modem, and most of the students in the sample were hi&achieving, and the teachers were motivating, progressive, and enthusiastic, it is unlikely that
these
results are confined to just this study.
The Grade 10 control snideats supedicially connected the numbers in theu tables to
niimbers they perceived to be on graph. However they did not represent the implications of
linearity within their tables. That is, these stuâents estimateci the approximate position of
coordinate pairs dong the line of the graph and then rmrded them in a table. They did not,
however, ensure that the resulting table represented a Iinear refationship in which there was a
constant slope. Twenty-four percent of the Grade IO control students used this strategy, which is
shown in the second incorrect example in the third column of Figure 12. This "paintwise"focus
(Leinhardt et al, 1990; Mansfield, 1985; Orton, 1971) showed that these students had not
developed a key notion in the understanding the graphs of functions. That is, the exact location
of points is not necessary if the goal of a graph is to represent a hctions' general relationship
(Alson, 1992).
The Grade 10 control students' problems were aiso the result of an alarming nurnber of
them not being able to identiQ coordinate pairs for points on a line correctly and then to record
[hem in a table. One example of this is given in the third example in the third colurnn of Figure
17. This student has taken the x-coordi~teh m the x-intercept and they-coordinate h m they-
intercept and recorded them in the table as a single coordinate (6, -9). He subsequently estimated
a second point and did the same. One implication of this erroneous strategy is that students then
made futile attempts to use the common rise/run algorithm to detmine the dope of the hction.
ObviousIy this strategy also cannot result in a table of values that accurately reflects the graph,
and emphasizes that these students had not integrated theu numeric and spatial understandings io
form an intepted conceptuai h e w o r k for linear functions. Schoenfeld et ai ( l m ) showed
how a student's ill-constructed scheme for identifjhg coordinate points on a Cartesian plane
"caused major difficultieswith ail aspects of slope" (p. 89) and that m in y = mx + b did not have
immediate graphical entailrnents for this student.
Summarv of item 4. Item 4 seemed to be a pivotal item with respect to cevealing the
effects of the two curricula between the experimentai and control groups and the effects of grade
level among the experimental groups. This is the first item that required d e n t s to show an
elabonted understanding of an integrated conceptuai structure fof fiuictiom by including
negative y-values in both the table and the gtaph. Furthennore, although item 3 (see Figure II)
was designed to test students' integration of the spatial and numeric aspects of a fiuiction, it
could be solved using an aigebraic procedure. Item 4 absolutely required that students
demonstrated an understanding of how the spatial and nurnenc aspects of linear functions are
connected.
Drarnatic qualitative and quantitative différences were found between the experimental
and control groups on this item that are linked ta the different instructionai experiences students
had with respect to understanding the spatial and numeric implications of linearity. A notable
quantitative difference was also found between the Grade 6 experimental group and the other
two cxpenmental groups, This 6nding supports the hypothesis that there are limitations to
students' ability to solve a problem requiruig more thana basic integration of the primary
numeric and spatial aspects of a functioa. Forty-six percent of ihe sixth grade students were
unable to solve this problem, Iikely because they had insufficient expenences that support the
speci fic elaborations required for success with item 4 (in this case, negative numbers).
Item 5
Item 5 and sample correct, partially correct, and incorrect solutions are shown in Figure
Item 5 uretest stratecies. The overd mean c o m t score on the pretest for the five groups
was 6% and the overall mean partiaily correct score was 1%. Cleariy, on the pretest students
could not answer this question conectIy.
The most common mistake made by di groups on the pretest was the obvious error of
stating that the repeated +3 increase in the sequence was the relationship between x and y, that is,
-. = .r + 3.
Half of ail students gave this tesponse on the pretest. Because this was aIso the
m
Look at the foiiowiag sequence of numbers:
2,5,8,11, 14, 17 ...
Write an equation for a function that would generate this pattern of values.
Correct Response
Partiallv Correct Reswnse
Incorrect Response
Figure 13. Sample Correct, Partiaiiy Correct and Incorrect Solutions for Item 5.
most cornmon posttest incorrect solution given by both experimental and control students and it
was a strategy that separated the two instructional p u p s on the posttest, this strategy will be
discussed with the posttest patterns.
One-third of the sixth graders used a particular incorrect strategy on the pretest that was
categorized as "+3, +3, +3.. ."In these responses students tried to represent the idea that +3 was
a recursive feature in the sequence of numbers. The 6rst example in the third column of Figure
13 shows one such attempt. This approach suggests two things. First, the sixth-grade students
were not yet farniliar with algebraic constmctions as a way of generalizing arithmetic (Usiskin,
1 988). Second, students were not yet thinking in terms of two variables. This is an additive
strategy that may trip up students when it cornes COtheir needing to think multipiicatively with
respect to rate of change and dope (Seymour & Shedd, 1997). However, students who leamed
about functional relationships with the experimental cirrriculum were given the opportunity to
apply that additive strategy within a mode1 of CO-variationof reiating variables (Confrey &
Smith, 1995). Such a mode1 allowed for "adding three" to successive numbers to mean the "up
by" amount, and it was also related to a multiplicative expression in the equation y = 3x, which
represents an increase of three for each unit change in x, or a dope of t h e . An average of 10%
of the students in the cemainhg four groups used this strategy on the pretest. No students in any
of the experimental groups used this strategy on the posttest and only 5% of the control groups
used it on the posttest.
Item 5 posttest results. An average of 63% of îhe experimental students as a whole got
this item correct on the posttest (10% wece partially correct). This is compared to an average of
16% of the control students giving a correct solution on the posttest, and there were no partially
correct solutions given by control students on the posttest,
Students in the Grade 6 group stniggledmore with this item than the other two
experimental groups (54% correct for the sixth graders versus 70% and 63% for the eighth and
tenth graders, respectively). However, correct strategies among the experimental groups were
consistent across grsdes. The favored response was y = 3x + 2. Students found the slope of 3 for
the function by determining a difference of +3 between successivey-values (for which they
assumed a unitized domain). They then f o d the y-intercept by making the first .r value equal to
O. If the first y value is 2 and the slope is 3 and the first x value is O, then students rnust add 2 to
3.r. or y = 3.r + 2 (see the tüst example in the fintcolumn of Figure 13). An alternate solution
was for students to make the first x value qua1to 1,
then 3x - 1 = 2 and the solution is y = 3.r - 1
(see the second example in the first column of Figure 13). Fifly-three percent of the experimental
students as a whole used the first strategy fory = 3x + 2 and five of the six control d e n t s who
got this item correct also used it. For the expetimental students, choosing this strategy was likely
linked to their being accustomed to beginning any sequence of x values with x = O. Ten percent
of the experimental group began with x = 1 and gave y = 3x - 1.
There were no outstanding posttest ermrs for the Grade 6 and Grade 8 experimental
goups. However, 25% of the Grade 10 experirnentai group at posttest still gave y =x + 3 as their
solution. These students did not provide any explanations or insight into theu strategies and thus
in terpreting why these students fided was not possible. The lower passing rate of the Grade 6
goup may reflect that the mental complexity of integratingy = mx with y =x + b to get a
function in the form y = mr + 6 was tao m c d t for the Grade 6 group in general, as predicted
b y the mode1 of leaming. As expected, both the Grade IO and Grade 8 experimental pups were
mostly able to make those connections following instruction.
Most of the control students gavey = x + 3 as &eu pomest response. Students who gave
this solution were not thinking of a M o n as a rdationship between two différent variables.
There were two general strategies for arriving at the solutiony =x + 3. The k t was uscd mostly
b y the Grade 8 control group (sec the second example in the third column of Figure 13). These
students did recognize the need for two sets of numbers, but created pairs of numbers from
within the given sequence. Doing this did Qive a table inwhich 3 was added to eachx value, but
the resulting relationship between the twa sets of numbers could not be.r + 3 because there was
neither a dope of 1 nor a y-intercept of3. These stuàents created a table of vaiues they felt fit the
circumstances but did not mentaily coordinate the two columns in a way that demonstrateci an
understanding of functions as a relationship between two co-varyhg quantities.
Furthemore, students did not recognize the inconsistency between the equation and
table. Not recognizing this inconsistency suggested that students did not have a well-consmcted
"sense" for how the representations of a function cwrdinate (Case, 1998; Eisenùerg, 1992).
Developing such a "sense" has been attributai ta students developing a central conceptual
structure for functions (Kalchman et al., 200 1). Indeed, since this group was generaliy
unsuccessful with item 3, which was intended to test for an integrated conceptuai stmcture, they
likely had not developed such a concepnial Eramework.
The second approach to y =x + 3 was used mainiy by the Grade I O conmi group. In this
strategy, students defined x and y as successive Aues fiom within the given sequence and
claimed that .r + 3 =y (see the thini example in the third co[umnof Figure 13). These studeats
may have taken the task to be one in which they were to generalize the arithmetic rather than to
express the relationship between two variabIes. Regardless, the essence of a fùnction as a
relationship bebveen CO-varyingquantities was overIooked or not mcorporated into these
solutions. 1t appeared as though these students had not fiilly elaborated theü primitive digital
schema, which involves going h m recursive computation of a single operation on a string of
whole numbers to iterative computation on a string of whole numbers to generate a second set of
numbers.
Summarv of item 5. The large qualitative and quantitative differences between the
expenmental and control groups for this item were accompanied by large differences in
strategies. Like item 4, this item seemed pivotal and revealed a significantdeficit in the control
students' post-instructional integrated conceptual fiamwvork for functions. The spatial
entailments of this seerningly numeric problem were welI articulated in the student's solution
found in the first colurnn of Figure 13 when she wrote: "if on a graph y-int [sic] would be (0,2)."
The usefulness of a conceptual structure in which spatial and numeric understandings of
functions are integrated, even when the problem does not specifically cal1 for a graphical
representation, is identified in this probkm.
Level3 Items
Table 10 shows the percentage of studenrs in each p u p who got items 6 and 7 correct
and partially correct.
Item 6
Item 6 and sample correct, parîïally correct, and incorrect solutions are shown in Figure
14.
Item 6 Dretest strategies. Pretest scores were again low for this item. Nine percent of al1
students got this item correct on the pretest and 20% got it partially correct on the pretest. The
only rernarkable pretest strategy was au incorrect one used by both of the Grade 10 groups.
Approximately one quarter of the entire Grade IO sample gave the general linear equation,~=
nix + h, as its rcsponse. This response gave no hints as to whether or not students understood
how to instantiate the general linear equation using appropriate values for slope and y-intercept.
That is, it is not known if students understood that the problem required a solution of two
equations, each of ". ..which [has] a positive slope with a positive intercept."
Item 6 uosttest results. There were large quantitative and qualitative differences for this
item as well. Fifty-two percent of the experimentai students gave a correct solution on the
posttest, and 35% of them gave a partidy correct solution. This is compared to 21% of the
control studmts giving a correct posttest response and 30% giving a partially correct posttest
response. The partially correct solutions are infomtive for this item because the combination of
correct and partially correct made a considerable difference with respect to understanding
students' strategies.
The combined posttest correct and partiaüy correct resuits for the experimental students
showed that most of the students were abte to generate at least one equation that wouid correctly
represent the given graph. This suggests that students had at least a basic understanding of how
the spatial and elaborated numerïc aspects of a h c t i o n are connected, The youngest studenîs,
however, did not demonstrate the same level of conceptual facility with generaüzing that
-
--
Make up 2 equations that would produce a function with the following shape:
Correct Response
Partiw Correct Response
Incorrect Response
Chanoed v-interceot oniv
Chanoed doue oniy
'
01
Changed both
a
Figvre 14. Sample correct, partiaiiy correct, and incorrect solutions for item 6.
integration ta a family of linear îünctioas that the Grade 10 snrdents did. The Grade 8
experimental students showed slightly more facility than the Grade 6 group with the
e laborations, inteptions and generalizatiom necessary for wnstn~ctinga conceptual Eramework
for who le farnilies of functions rather than individual fiinctiom.
The most common correct strategy was for students to give two equatiom for wbich only
the y -intercept was di fferent (see the fhst example in the first column of Figure 14). The example
shows how the same "space" between the origin and the y-intetcept could be partitioned in two
ways: first by six and then by two. This student changed the value of the y-intercept to get two
distinct relationships. Some students also gave a pair of equations with the same y-intercept and
di fferent slapes (see the second example in the Esst column of Figure 14). The third correct
strategy was one where students changed both the y-intercept and the dope to derive a pair of
equations (see the third example in the first coIumn of Figure 14). This student made a tahie of
values to represent y = lx + 2, and then drew a graph in which he adjusted the scale on both axes
from his first example to get the equation for another fimction. Students who were successtùl
wi th this item demonstrated a gwd understanding of the general nature of linear iünctions and of
the roies that dope and y-intercept play in that family of fiuictions. Furthemiore, these students
displayed a good "sense" of linear hctions, which the control students generally did not.
There were two main kinds of partidly correct solutions. The 6rst was where students
gave only one equation as a solution, but it was a correct equation. The second was where
students gave the same relationship in two forms (see examples in the second column of Figure
14). These two partiaily correct strategies were used in WtuaIly the same proportions for b t h
the Grade 6 and Grade 8 experimentai groups. The two Grade 10 groups, however, concentrateci
on one or the other for a partiaiiy correct srrategy.
One-quarter of the Grade 10 experimental group usai the first partially correct strategy
and gave only one correct equation. Students who did this generally wrote in a scale for the y-
axis and constructed an equation based on that scde. These students showed they were able to
instantiate the graph in an elaborated numeric form (usiag an equation). They were not able,
however, to generalize the situation to include a whole family of hctions. This ùiability to
generalize may be related to difficulties students are reported to have with scale (Goldenberg,
1 988; Hillel et al., 1992; Kerslake, 1977). Goldenberg (1988) concluded that if students are to
derive meaning tom the visual presentation of a function, it is essential that they understand the
e Kec t of a chosen scale, and the interaction between the scale and the shape of the function.
In a microgenetic anaiysis of one student's understanding of linear fhctionss,Schoenfeld
et
al. ( 1990) found that the traditional instruction this bright student had been experiencing did
not foster an understanding of the meaning of slope or y-intercept across representations. The
control students in the present study demonstrated sirnilar difficuities. Schoenfeld et al. (1990)
identified the "Cartesian Comection" as understanding that "specific algebraic expressions have
gaphical identities." When students have made this comection, they have devebped "the
underlying structure of the Cmesian plane and the mathematical conventions for interpreting the
properties of linear graphs" (p. 108). This connection was achieved in this study by many more
s'cperimental students than control students.
Alrnost one-third of the Grade 10 control students (29%) gave partially correct responses
by giving the same reiationship in two f o m . These students gave pair'; of equations that
represented the sarne relationsbip between x and y. The examples shown in the middle column of
Figure 14 illustrate four different strategies for aniving at this sort of solution. The b t and the
most popular strategy among the Grade 10 control class was to multiply both sides of a pmper
linear equation by a fixed amount (see the first example in the middle colurnn of Figure 14).
Grade 10 control students did this in their instructional unit to solve linear systems in two and
three variables. The second strategy was to re-present the slope in some way (see the second
example in the middle coiumn of Figure 14). in this strategy, students showed their flexibitity
with moving between fractions and decimals, but did not change the relatioaship between
variables ço it was stiil the same function. In the tbird strategy, students sirnply rearranged the
order the terrns appeared in the equation without changing the relationslip (see the thüd example
in the rniddle colurnn of Figure 14). Fourth, students gave one correct equation and then an
cquation that at first seemed to express a linear function with a negativey-intercept. For
cxample, y = x* 1 + 4 and y = .r* 1 - 6 are the two found in the fourth example in the second
CO lumn
of Figure t 4. Students then extended the line to a point they estimateci to be the x-
intcrcept. In this case it was x = -6. The slope h m the h t equation was maintaineci and the -6
added on. In the fourth example shown in the middle column of Figure 14, the student explained
"The first equation starts at the y-intercept; the second equation starts at the x-intercept-"
AIthough the second representation is flawed, the strategy results in giving two equations that
conceptualIy represent the sarne function.
it is difficult to Say if students who gave the same relatioaship in different foms fiilly
understood the question, although each task was discussed with students before both the pretest
and posttest were taken. For students who did understand the task, it is possiile that they did not
understand that changing the order in which the tenus of an equation appear or the way in which
any number is represented do not alter the relatiomhip inherent to the expression. In Wagner's
( 1981 )
study on high school and middie school students' ability to conserve in an equation, she
found that some of these students beiieved that merely changing the letter in an equation
indicated a whole new reiationship between variables. Although Wagner's study was
inconclusive with respect to its generaiizabiiity, it may be the case that students in the preseut
study who gave "the sarne equation" twice betieved they were expressing two d i f f i t
relationships. The control students' weak pst-instructionai conceptions of dope and y-intercept
found in earlier items support this idea that the conml students were not entuely clear about the
spatial and nurneric implications of each parameter ofy = mr + b.
Twenty-six percent of the responses to item 6 h m the Grade 8 control group on the
posttest were anomalous, and a m
e
r 35% were left blank. Students in this p u p showed a
v e l lirnited undentandhg of how to approach this pmblem in either a totally correct or partiaily
correct way.
Summarv of item 6. This item was difficdt for al1 groups. Even though the Grade 6
group did not p a s this item and the Grade 8 expetiaieatal p u p only just passed, the
experimentai groups were more successfirl in general than the control groups. The Grade 6 and 8
s s p erimen ta1 groups'
partially correct posttest responses showed that even for those students
who could not answer the question completely conectly, they still used strategies that showed an
integrated conceptuai structure for hear fullctiom. From item 4, we know that the Grade IO
controI students were unable to link a table to the graph of a hear h c t ï o n . Their partialiy
correct responses for item 6, however, showed tfia they were better able to connect a graph and
an equation. This competency may be misteading in temis of concluding that students had
connected their spatial and numeric understandings of a fiinction. It seems tbat ihey were able to
connect one part of the numeric schema (the equation) to the graph, but were missingthe
foundation of the numeric aspect of a h c t i o n - recognition of the spatid pattem found in a
table oCnumbers. Furthmore, it was their partidly correct mswers that show this ability and
their low scores for correct responses suggests that they had not abstracted that connection to
linear functions in geneial.
Item 7
Item 7 and sample correct, partially correct, and incorrect solutions are shown in Figure
1tem 7 uretest stratepies. The overall pretest means were 26% correct and 5% partially
correct. Both of the Grade 10 groups app-hed
a passing g o & on the pretest (44% and 40%
correct for the experirnental and control groups, respectively). Both of these p u p s of students
approached the problem by attending to one parameter of both functions, usually the y-intercept.
That is, students claimed that the fiuictions codd not be the same because y = x - 10 has a yin tercept of "- 10" and the graph shown has a positive y-intercept. Students then correctly
sketched a graph ofy = x - 10 (see the tint example in the 6rst column of Figure 15).
Item 7 uosttest strategies. This item was unique in that it is the oniy item that both Grade
10 groups passed and the only item Piat the Grade 10 control group passed and the Grade 6 and
Grade 8 experimental groups did not. In fact, the integration of concepts required for this item
proved resistant to either instructional condition for the two younger gracies, with al1 thtee
younger goups scoring between 30% and 40% correct (see Table 10). However, the Grade IO
experimental students did much better than the Grade 10 control students (88% correct versus
58% correct). Students' partially correct solutions were not informative for this item, The
qualitative differences on posttest tespouses between the experimental and control conditions m
general are distinct and reflect a qualitativety differeat set of post-instructional understandings
Itcm7
Look at the hinetion bdow. Couid it rcpmm y =x - 107 Why a why not?
if you ihink it couM w skmh what you ibink it look likt.
Correct Response
No hecause of the Y-
Partiaüy Correct Rcspoosc
incorrect Rcsponsc
intefCebt
Table & maph do not q&l
j. r-o
Fi-
15. Correct, Partially Correct anci Incorrect Solutions for Item 7.
Half of the Grade 10 experimmtal students clairnecl on the pusttest that the two functions
could not be the same because neither the slopes nor the y-intmepts match (see the second
example in the first column of Figure 15). Thtrty-seven percent of the Grade 10 experimental
clriss
and 36% O l the Grade 10 control class claimed on the posttcst t b t the functions could not
be the same because they have diffant y-intmepts.
In both cases, students correctly sketched a
g a p h of y = .r - 10. Seventy-nine percent of the correct solutions for the Grade 6 students and the
two Grade 8 groups also usd this strategy of comparing y-intetcepts. Both of these correct
strategies demo nsnated that students had elaborated their already integrated conceptuai
s tmct ures to the extent that they were facile with negative values dong the y-axis.
In most of the responses given by the Grade 10 control ciass, students made a table fory
= s - 1O
and compared the coordinate points found in that table to points they assumeci to be on
the line of the given graph (see the third example in the first column of Figure 15). This
cornparison yielded a conflict in the two rppresentationsand students subsequently graphed the
coordinate points they had generated on the set of axes provided for the sketch. These students
did lot demonstrate the same de-
of discriminatory power with respect to confUsable
functions as did their experimentd counterparts. That "poww" cornes h m students' ability to
relate variants of their elaborated and integrated conceptual structure and discern incompatibility
between and among representations of a fimctioa. By contrast, these controi students relied on a
poinnvise and primarily numeric approach to the problem (Leinhardt et ai, 1990, Mansfield,
1983; Orton, 1971; Thomas, 1975).
The Grade 6 and Grade 8 experimental groups bad a common and notable psttest
incorrect solution strategy that was coded as "-1 O as slope" (see the third column of Figure 15).
In this strategy, students treated the -10 as the slope. This solution was the result of two possible
sorts of reasoning. The first was that students may have mixed up the position of dope andyintercept in an equation. This is a common mistake and has been reporteci in other studies (e.g.,
Schoenfeld et al, 1990). The second sort of incorrect strategy that resulted in "-IO as slope" was
one where students drew detailed graphs and made tables that indicated a clear understanding of
what it means nurnerically and graphically for a fimction to have a dope of -10. What these
students were not clear on was the algebraic representation of those patterns, at least in the
context of a function with a negative y-intercept. Several eigbth and sixth graders in general had
di fficulty understanding why "-1W
and "x-10" had different meanings. These students were
j ust learning how to combine letters and numbers in equations without using signs for
multiplication and were experiencing difficultieswith comatenations (Lee& Mesmer, 2000).
Students generalty needed to be reminded in class whether 'k-IO" meant multiply the x by -10,
or take away 10 from x. Thus, in fact, some of these students were dernonstrating a sufficiently
intcgated understanding of the question. However, they had not yet elaborated their conceptual
frameworks to include some of the conventional notation structures for hctions that include
negative integers. Consequently, these students could not interpret the question correctly.
This central e m r on the posttest for the Grade 6 and Grade 8 experimental students -
"-
1 O as slope" - may have been the result of a cognitive obstacle (Herscovics, 1989; Sierpinska,
1993). Thompson (1994) descnies a cognitive obstacle as "a way of knowing sometbg that
sets in the way of something else" (p. 37). Because neither Grade 10 group demonstrated this
same process, this particuiar instructional obstacle is likely transitory and is overcome through
students' experiences with algebra
One other incorrect strategy is worthy of mention because it also deals with cognitive
obstacles. When students begin working with aigebra, it is counter-intuitive to imagine that y =x
-
10 has a graph that increases. An average of 21% of the eighth-grade sample responded to item
7 by saying that the two finctions could be the same because '&-IO makes the b c t i o n go down."
Only 4% of the sixth graders and none of the tenth graders gave this solution on the pretest.
These latter results are likely because the sixth-graders were not even at a place on the pretest
where they could make sense of an equation in two variables and comect it to a graph, and the
tenth graders had probably had enough experience with equations and integers to have a better
understanding of how the two relate in the context of algebra Thus, the "-10" presented a
cognitive obstacle primarily for the eighth-graderswho were beginning to elaborate theu
understandings and experiences with negative integers in f'unctional equations. On the posttest,
26% of the Grade 8 control students, and 12% of the experimentai students gave a response in
the category "Yes, because it's going down," thus showing that the control cunicula did not aid
in students' overcoming this obstacle. However, none of the experimental Grade 8 or Grade 10
students and only 10% of the Grade 6 students gave this kind of response, suggesting that the
experimental curriculum did help students overcome the obstacle, even if they could not fully
elaborate their conceptual structures to Uiclude integen.
Summarv of item 7. item 7 was cleariy too hard for the Grade 6 and Grade 8 students in
~eneralregardless of instruction, but the Grade 10 experimmtal p u p was highly successfUl
with it. The posttest errors of the two younger experimentai groups mdicated that they were still
approaching the problem h m an integrated perspective even though they h a .not yet achieved
the elaborations of that integrated perspective to successflllly anmer the question. This was the
item on which the Grade 8 experimental group's d t s dropped off and where their concepnial
frameworks were limiting. This seeming inabiIity to incorporate negative integers into an
integrated conceptual framework for undeistanding functions may be explaineci with respect to
students' di fficukies with the "-10"
ia the equationy = x - IO. The "-10" may have illuminated
some cognitive obstades that impeded the eighth grade students' further progress into Level3
and their ability to fully coordinate the spatial and numeric implications of integers into their
integrated conceptuai structures for functions, One such cognitive obstacle was descnbed with
respect to concatenations. A second cognitive obstacle was relatai to students' scbemas for
subtraction.
Lwel4 Items
Table 1 1 shows the percentage of d e n t s who got the Level4 items (8-12) correct and
partially correct. As expected, students at al1 grade levels and in both instructional conditions had
considerable di ff~cultywith items at this level. These items were predicted to be difficult men
for the Grade IO experirnental students. ïhese items required that students had integrated al1 of
the vanants of the conceptual structures developed in Level3, and had built a compretiensive
framework for understanding funcîions as a mathematical system. Because correct strategies
cannot be compared among and between groups, and incorrect strategies were concentrated in
the "Incorrect homaious" category, these items will not be analyzed and discussed in the saine
detai l as the first seven items. However. some pst-instnictionaisûategy choices among the
goups are interesting because they further demonstrate the differences in the ways each of the
instructional conditions influenceci students' reasoning.
Item 8
Figure 16 shows item 8 dong with a correct, partially correct, and incorrect response.
This item was originally designated as a Level3 item, but pst-hoc analyses showed that the
mental connections necessary for success with this item were beyond the g e n d expectations of
students at Level3 in their development. However, when students' partially correct responses
were considered, notable differences were found between the experïmentai and control students'
responses. Students in the experimental groups showed consistently their understanding of the
graphic implications of changing the degree of an exponent in a simple power hction. They
were able to represent spatially the idea that a fiction "is less steeper [when] it is multiplied by
i tsel f lesser tirnes." That is, the lower the exponent, the less rapid the rise of the c w e because
the dependent variable does not increase as rapidly for each unit change in x. The first and third
examples in the middle colurnn of Figure 16 show this reasoning. These students, however, did
not show that they had elaborated that understanding to include computation and graphing with
negative .r values. It is not possible to say ifthese students did not know how to mdtiply
negative x-values by odd exponents to get negative y-values, or if the complexity of integratioas
was
too great for these students to process.
Some of the experimental students, however, did make the connection between cubing
negative x-values and the spatial embodiment of that operation. These studentswere able ta give
a correct cubic shape for the fimction, but did not incorporate the diffetetice in the rates of
change between the two functions into their solutions. This oversight meant that the upper-right
quadrant of the graph was not correct, and aiso mggestecithat students had made sipifkant
progress in one necessary elaboration but could not mentaiiy orchestrate aii of the connections
necessary for an entirely correct solution.
Correct Response
Partidly Correct Response
Incorrect Response
Full-P
y - x4
Figure 16. Sarnple Correct, Partidy Correct, and incorrect Scrategies for Item 8.
It is interesting to note that a large proportion of buth Grade 10 groups at ptetest drew
parabolas that were entirely inside the given one and r a i d to have ay-intercept above zero.
Students were asked der the study why they did tfiat. The consensus was that "3 is çmdler than
3, so the whole graph had to be smaller." On the posttest, almost half of the Grade 10 control
group,
and over a fifth of the Grade 8 contra1 group drew parabolas entirely inside the given one,
with the vertex at the origin. This strategy suggests tbat these students had mpde some progress
in ternis of knowing that the graph wodd not be situated above the origin. However, it also
suggests that students still believed that the iower expanent made the graph closer to the y-ais
and "the bigger the exponent, the M e r away it gets."
The conceptual gains between instructionai conditions are noteworthy. Many of the
cxperimentai students had generally developed an understanding of the numeric implications of
changing the exponent of a function, and had C O M ~ & ~that understanding to the corresponding
spatial representation. in fact, the partially correct resuIts v a r d little h m expe-rhentalgroup to
experimental group (65%, 7O0h,and 63% across grades). However, students in dl grades showed
t hat they
were relatively weak in computation involving expanents, negative numbers, and
especially a combination of the two. With respect to the conîrol groups, students not onIy
showed dificukies with the computation, but a h showed IittIe understanding of the
spatiaVgraphical effects of changing the degree of an exponent of a fiuiction. This difference
likeIy occurred because neither control curricula placed any empbasis on cornparhg the graphs
and number sequences of tùnctions with different lead exponents. Consequently, the graphs and
tables were no t directly related and the impiications of one for the other were not explored or
surnmarized.
The experimental curriculum, an the 0 t h haad, included mariy oppartimitiesboth ui the
c lassroorn and on the computer for students to observe the spatial and numeric effets of
c hanging the exponent of a function One set of computer activities was dedicated to students'
witnessing and recording their observations of how the curve and table for y =? changed when
the exponent was increased and decreased. These activities foliowed pencil and paper
experiences of inventing rules of spansorship that would generate increasingly steep Cumes. This
type of activity allowed students to be constantly connecting the changes they were making in
the equations with the reçulting changes in the graphs and tables.
Items 9 and 10
Items 9 and 10 are being considered together because they both represent items that
reflect instructional concentrations in the Grade 10 control cumculum. The items and sample
correct solutions are found in Figure 17,
Item 9 is an interesting one to review because of the ciifference in the percent cocrect
scores between the Grade 10 groups (13% Tor the experimental group and 41% for the control
goup). The Grade 10 control group spent two full classes (160 minutes) on transfomiations ofy
= .Y'. They spent one
hl1 period using a computer program to shift vertically and lateraily the
graph ofy = x' and to cornpress and expand it. Students recordeci which part of the equationy =
Ü(J-~)'+ q
they altered and how they altered it in order to eEect each chauge. However, students
did not record or even see ttie tables for each new function they created. The second of the two
periods was spent "taking up" or rwiewing students' findings with respect to the computer
activity. The upshot of this activity was t
hstudents were to memorize what sort of
Item 10
Item 9
1
1
Wrire an equation for a hinctiaa haî
would have the foilowing shape:
4t what points would the function y =
lair -x2 cross the x axis? Please show
fl of yow work.
I
l
Correct Respoase
Fi-me 17. Sample Correct Responses for Items 9 and 10.
transformation resulted h m altering a certain parameter in a particuhr way. Given the relevance
of this experience, a 41% success rate may even be considemi low for this grog of students.
However, this approach to leaming about the behavior of quadratic hctions helps to explain
this group's results for item 8 where they were very unsuccessful at differentiating the graphic
implications ofy =.? from y =A?.
The Grade 10 control students also had instruction and exercises dkctiy related to item
1 0. The Grade 10 control students had been taught an aigorithm for finding the mots (or .r-
intercepts) of a quadratic equation. The students in this group that got this item correct used that
algorithm successfully to find at what points the function y = 10x -.? crosses thex-axis.
However, 53% of the students in the group used the algorithm incorrectly and gave atgorithmic
r look k e . For exampk
solutions that were inconsistent with what the graph ofy = l ~ -9
based on their incorrect algorithmic solutions, some students clairneci that the fimction never
crosses the x-auis because no real mots couId be found. In contrast, those experimental students
across grades who did get this answer correct used the approach of making a table of vaiues and
graph for the function and then describhg the points at which the graph and/or table reachedy =
O. The experimental students did not have an algorithm to use for this problem and had to use
their competence with computation and graphing and their understandingthat the particular
niimber pattern found in the table made an upside d o w parabola. Thus, they combined their
spatial and numeric structures in a very sophisticated way.
Items 11 and 121
1tems 1 1 and 12 and sample correct responseare found in Figure 18. These problems
were beyond the capabilities of all students with respect to their understanding both the graphic
and numeric implications of the retationship between x andy. StiU, there were çome
notable patterns found between the gmups at posttest. Forty-one percent of the exptximental
students across grades who got this item wrong explained that the sbape of the fiuictiony = lOOx
- 25x' + x3 would be curved because of the exponents in the equation. Although the complexity
of the computation and the graphing was tou difficult, these students still made the connection
between a nurneric representation (the equation) and the graph. No such conceptual fhuewotk
for the problem was shown by the control students.
Item 12 was an interesting window uito students' conceptions of rationai numbers. A
major difficulty with describing "what happeas to y in the b c t i o n y =
-1 as x increases" was
X
that students did not necessarily know that as the denominator of a fraction increases the value of
the number decreases. Thus, even if studenîs understood the substitution process, they stiU
determined that the y increases as a result of tbnrdifaicuities with understanding the rational
number system (e.g., Kieren, 1992; Moss, 2ûûû; Moss & Case, 1999). Furthemore, even if
students did understand how to order tiactions, the notion of a b t i o n being undehed when the
denominator is O was either unknown or not remembered by students. Consequently, students
assumed that a fraction with a denominatorof O is equal to O and thus, the function would have a
' 1tem 12 is vimally the sme as one s u g g d by Usiskm (IN%),
who uscd it as an example for showing the
ditlkulty students have with algebraic pattcfns wilh ~spectto mtuoetion hlwcen a tiinction's variables. Whm the
functions test was devdopd and subscguauly impiemcnped the eximnrc of this item m U s W s wock was
unknown.
Item 11
Item 12
What shape would the graph of the function
y = 1OOx - ?5.? + .$ likely have? Explain why
you think so.
.
D-bc
1
what happens in the hioction y = ;
.
Correct Remonse
as x increases.
Correct Remonse
Fieure 18. Sample Correct Responses for Items 11 and 12.
y-intercept of O and potentiaUy increase or decrase h m there. Finally, asymptotic behavior
was unfamiliar to most students. Thus, although 34% of the entire experirnental sample and 40%
of the control students correctly gave a numeric intqtetation of the problem, the graphic
representations offered were inconsistent with and disconnected h m that interpretation.
Discussion and Conclusions
Two questions were driving the strategy analyses reported here: (1) 1s there evidence
from responses on the bctions test that students in the experimental groups had integrated their
spatial and numeric understandings of functions? And, is it the case that the control
students did not demonstrate such integration? (2) Were there différences among the correct
strategies of the experimental groups? And, where differences in passing rates were found among
the experimental groups, was there evidence that the differences in the correct and incorrect
strategies were attributable to the students' level of development as predicted by the mode1 for
Icming functionsdescribed in Chapter 2?
The distribution of various correct, partially correct, and incorrect strategies used by the
experimental and control groups on the fimctiom test showed that the former were more flexible
and fluid with problems requiring an integrated conceptual fiamework for understanding how the
multiple representations of a hction are both constnicted and related. Students in the control
groups mainly dernonstrateci cornpetence with the numeric aspects of fiinctions (tables,
equations, coordinate pairs). Moreover, any elaboratiom achieved by the controt snidents of the
numeric aspects of a hction were manifest in ways that suggested that they had not cornectecl
those elaborations with their spatid implications. LieWise, any spatial elabrations fostered by
the control cumcula were achieved separate h m the relevant numeric implications.
Furthermore, there were no conspicuous differences with respect ta the type of correct
solution strategies used among the three experimental groups for those items where al1 gmups
ivere generally successfii. Quantitative and qualitative diffefences among these gmups did
appear on the items chat were at the junctures behveen the proposeci levels of understandmgs.
Unsuccessfu1 students' strategies at these junctures indicated difficulties with an added
elaboration or integration inherent to the task in particular and the level of understanding in
general. These unsuccessiÙi students were dispmportionately in the younger grades, thus roughly
supporting the deveiopmenta1analyses.
Effects of the Cumcula
The qualitative differences found between the solution strategies used by audents in the
experimental groups and the snidents in the controi groups may be considered in light of the
groups ' respective curricular emphases. Each curriculum (1) made different ciroices for
emphases of and relations among instructional topics, (2) used contexts in different ways, and (3)
assigned different sorts of activities. These Merences iafiuenced the ways in which
participatin; students were encourageci to thmk about and internaiiy (mentdly) and extenially
(physically) construct concepts and representations.
Instructional Touics
Experimental curriculum. The sequencing of the experirnental curriculum involved
introducing, consolidating, and applying the "big ideas" of functions such as slope, intercept,
linearity, and dependent relationships betweea variable quantities. That is, each new topic was
trcated as a generd concept inherent to Functions that interdates with other important domain-
specific concepts, al1 of which eventuaiiy integrate to form the underpinnings for an overdl
understanding of functions. As concepts were introduced, students were encouraged to
consolidate and apply them two ways. First, they were given opportunities to inscribe their
Jevcloping mental representations and understandings using conventional notation (Roth&
bIcGim, 1998). Second, as students applied and connected each new concept to other domain-
related concepts, they were continually supported in working toward integrating their emerging
knowledge structures to form a central conceptuai structure for understanding hctions as a
comprehensive and complex mathematical system.
Control Curricula. The control curricula, on the other hand, focused on helping children
leam the techniques, methods, definitions, and conventional notation stnictures and
representations for functions largely independeatly of an ovenll conceptual h e w o r k for
applying those topics. Ftuthermore, there were many topics in the control curricula that were not
conceptually supported.
An example of the lack of relationships even within topics was in the Grade 8 coatrol
cumculum. Y-intercept was part of instruction at three dBerent times in the unit as three distinct
topics: Intercepts (x and y), Partial Variation, and y = mx + 6.(These topic names are found in
Tables 1,2, and 3, which were developed for summariPIlg instructional foci for each of the
cumcula used in the study.) These topics were the 2"', 6", and 7" topics, respectively.
Consequently, early in the unit, students were told that to find the x-intercept of a fiinction they
should make y = O in the equation for that hction, Likewise, to h d the y-intercept they should
make x = O. Much later on, partial variation was defined as "a linear relation that has a fixed
amount, which causes the graph to meet the y-axis at a place other than (O, O)." The instruction
and activities for this Iesson were not related to the earlier lesson on intercepts. Then, y = mir + b
was introduced as another way of writing "y = number*~+ another number." That other number,
h, is the fixed amount that was describecl in partial variation and is also the value one gets when
making x = O in the equation for a function. However, students were not encouraged to make that
connection. Thus, a general knowledge structure for y-intercept was not promoted in this
cumculum. The control students subsequentiy demonstrated a lack of an integrated conceptual
framework not only for the range of individuai concepts found in functions but aiso for the ways
in which these concept inter-relate within the broder conceptuai field (Vergnaud, 1994).
A more concise example of not relating topics is h m the Grade IO control cumculum.
Quadratic functions were introduced to students as functions whose graphs are parabolas and
wliose equations are in the form ofy = LU?
+ bx + c. Students subsequently worked with
quadratic functions in a variety of ways includiig doing translation tasks with technology, and
leaming the procedure for "cornpleting the square" in order to put a function's equation in the
form y
At
= a(x - p)2 + q in order to
facilitate the identi6cation of the vertex @, q) of the parabola.
no time, however, did instruction turn to why quadratïc functions rn parabolas or why the
equations of quadratic functions have the particular features and structure they do. Answers to
ihese "why" questions stem fiom an examination of how the numeric and spatial aspects of
quadntic functions connect, and how that comection is related to the shapes and ninaber
patterns of linear functions and other polynomiais. Thus, without this gromding of the topic
within a larger conceptual framework for understanding its meaning, students were at a Ioss for
so lving many if not
most of the tasks on the fimctions test that required an integrated knowledge
structure for understanding functions - linear and non-linear.
Use of Context
ExperimentaI curriculum. The two iastructiond conditions used contexts in very different
ways. The experimental cumculum was derived h
m the theoretical assumptions descnied in
the mode1 for Ieaming functions, which States thaî for students to consüuct a sound conceptual
framework for understanding functions as a complex system of inter-comected concepts and
representations, they must progress through a series of deveIopment stages b u g h o u t which
they elaborate and integrate the numeric and spatial understandings inherent to the domain.
Instruction and curriculum are thus intended to completnent tfiat position by helping d e n t s
engage their pnor knowledge within a powerful instructional context farniliar to students in order
to
make those necessary elaborations and inteptions. The roIe of the context thus is central and
pivotal to students' Ieaming. The situation of a walkathon was used as the context for the
experimental cumculum.
The context chosen for any curriculum using Case's theory is considered a bridging
context for two reasons. First, it bridges studenis primary spatial and numeric understandings as
they coordinate to form the relevant central conceptuai structure (Kalchman & Case, 1998;
Kalchrnan et ai., 2001). Second, the context also semes as a bridge between sntdenb' everyday
out-Of-school experiences and th& schwl-based learning (Fuson& Kakhan, 2001).
The wakathon serveci both bndging purposes. F i building a bridge between students'
numeric and spatial understandingswas facilitateci easily and naairatly using the &thon.
The
money earned for the distance walked was a relationship that students could record equaily well
in a verbal story situation, a graph, a table, and an equation. Furthemore, students could
consequently see how a single relationship between two variable quiintities can be represented in
these various ways. The extemal and internai mappings between the representations connect the
spatial and numeric aspects of each representation in particular and of the concept of a fiuiction
in general as students build their understandings and work through the curriculum.
The experimental approach to teaching fiuictions paid particuiar attention to general
number patterns found in the tables of individual and of families of fiuictions. It was by focusing
on these number patterns that students were encouraged to make the spatial and numeric
connections that integrate the numeric bais of a fiuiction with its graph (Confrey & Smith,
1 995). Both the walkathon situation and the spreadsheet activities were instmctional contexts
that gave students the opportunity to privilege the use of the tabular representation as a
mathematical object (Confrey & Smith, 1994).
The walkathon was also a bridge between students' everyday experiences and the
classroom. Most students were familiar with the notion of an "a-thon," whether it be a
walkathon, readathon, bikeathon, etc. They also understood the relationship between the distance
gone (or pages read) and the money earned as a bction of some specified rule of sponsorship.
These familiar circurnstances grounded students' leaming in their informal expenences with
mathematics and supported their learning as they progressively absmcted, or fonnaked, their
knowledge to more conventional sorts of mathematical notations and tenus generally found in
the classroom (Koedinger, Alibali & Nathan, 2001; National Research Council, 1999).
The sequencing of the experimental curriculum was such that students transition4 h m
the walkathon situation, in which they did mostly pencil and paper activities, to working with
technolo~.However, both coaventional and contextual notations were used throughout the
actitities that students did with the spreadsheets so that students could ground their leamhg in
context if necessary. The younger students in particular showed evidence in their responses on
the fumions test that they were grounding their solutions in the walkathon context. This
evidence was in the form of using the symôols "kmnand 'T'for constructing equations and
labeling columns in tables. One sixth-grade student explained for item 3 (giving an equation for a
function that would cross y =x + 7 shown graphicaily [see Figure 111) that y =.r*9 would work
because "after one kilometer you already have niue dollars." Students mostly grounded theu
solutions in this context for item 3, which was the fint item that required a complete integration
of their elaborated numenc and spatial understandings of a fiinction. This use of the walkathon
situation for solving a problem showed the power and utility of the bridging context.
Control Curricuia The control curricula on the other hand used a variety of contexts
throughout the units as examples of mathematical consûucts in the "mi world." These contexts,
however, were generally unrelateci to students' lives and t h e were no particular contexts on
which students could refer for grounding solutions and understandings across the curriculum. In
sorne cases, the concept a context was meant to introduce was lost in the exarnple or else did not
Iink the situation to students' ieaming in the classroom.
For exarnple, students were Qiven the exarnple of direct costs of production for explaining
putial variation. In this context, students not ody needed to understand the meaning of partiai
~wiation(which was not relatai to earlier leaming ofy-intercept), but aiso they k e l y aeeded to
understand the meaning and implications of a h e d cost in mass production of a comodity. The
complexity of the intended concept was thus mecessady complicated by the difficulties of
identifjmg with the context.
Likewise, slope was introduced to the Grade 8 control students in the context of tfie
gradient of a hi11 and how the ratio of the nse to the nin in the circumstance defines the steepness
of the hill. The context was in fact rather spatial. A hi11 connotes steepness, which is a qualitative
measure of slope. However, this spatial context was not conducive to seeing the numeric
implications of steepness and slope. The nm in the case of a hi11 (the horizontal distance) is
di fficult to discern when the hiIl is part of a road. Thus the spatial and numenc aspects were lefi
isolated. Furthemore, this context was not revisited when students moved into the leaming of
direct variation, which was situated more in a ratesf-change context. Finally, when the equation
J* = n u
+ b was introduced neither the context of the hiIl nor fixed cost was reviewed to ground
students' understanding.
The use of even those select and isolated contexts was scarce in the Grade 10 control
curriculum compared to the Grade 8 unit. The former was much more abstract thughout and
focused mainly on how equations can represent physical phenomena that were generally
unrelated to students' lives. Little attention was paid to the mathematicai meaning of those
phenomena and why a particular sort of equation was a p r e f d representation over another.
In general, neither control curriculum used contexts to help students bridge their
understandings of concepts introduced or to help them bridge their everyday experiences with
classroorn mathematics learning. Consequently, students did not make the mental connections
arnong concepts that would facilitate the sort of flexi'bility and fluency necessary for bridging
their numeric and spatial understandings of a fimction. W y on one control test did a student
attempt to ground his answer. He tned to explain his tespoase to item 3 (again the item requinng
a first integration) with respect to 6 x 4 cost. However, he did not know how to constmct an
cquation that rvould represent his thinking about how steep he wanted his new function to be and
where he wanted the y-intercept. Intemhgly, the Grade IO students in general on the pretest
and the Grade control group on the posttest tried to "ground" many of theu responses in their
experiences with algorithms and general equations. However, as we saw in the description of
how the control students work with equations that are disengaged h m theu graphs and nwnber
patterns, the "ground" is soft and unable to support an infktnicture for reasoning about
functiom.
Assignment of Activities
Exuerimental Curriculum. The experimeutd curriculum had three main sorts of activities
t hat
contributed to students developing an integrated conceptual framework for understanding
and applying fùnction concepts: (1) inçlass activities that relatecl to the walkathon context; (2)
spreadsheet activities; and (3) presentatioas.
( 1 ) The in-class activities that relatecito the walkathon were those for which students
were challenged to work within the context and to extend and apply their learning in creative
ways. For example, after Ieaming about slope, y-intercept and non-linear functions in the context
of the walkathon, students were given the situation of needing to raise % 153.00 for a shelter over
a 10 kilometer wakathon. They were asked to corne up with three ditrérent mles of sponsorship
that would provide for their eaming exactly that amount. Two of those mles had to produce
linear functions and one had to produce a non-hear bction. Students aIso had to record their
solutions in tables, graphs, and equations, and then identiQ they-intercept of each bction. For
the hvo linear relationships, they also were asked to ident* the dope of the fimctions. These
sorts of activities helped students mentaily connect the concepts they had leamed as well as
helped them relate the representations of tables, graphs, equations, and verbal story situations.
(2) nie technology component of the curriculum was drivm by hvo forces. First, many of
the recent curricular innovations for the teaching of fiinctions to students from middle scbool
tlirough to post-secondary level have been techaology-onented. One central aim of most of the
computer applications has been to facilitate students' movement among the representation of a
function. Computers have an unprecedented ability to do tedious calculations almost instantly,
leaving students to attend to larger concepts and to relationships (Dugdale, 1982; Goldenberg,
1988). Computers generate graphs quickly, and they give students immediate feedback on their
ideas and actions (Goldenberg, 1988; Hillel et ai, 1992). Furthermore, students c m manipulate
any one fom of a function and witness the effects of that manipulation in another representation.
The second force is less utilitarian and more psychological. Pea (1987) promotes the use
of instructional technologies because they are likely to be of value for learning translation skills.
Software for leaming functions can be very motivating for students (Dugdaie, 1982; 1984). It
makes otherwise static representations dynamic. Thus, students are said to be able to compare the
tables, graphs, and equations of multiple functions and consolidate their understandings of how
the spatial and numeric aspects of slope, intercept, and even linearity interact to generate
particular functions and families of fùnctions. Furthemore, much has been written and discussed
about the value of students' developing an object-understandimgcf tùnctious (Dubinsky & Harel,
1992; Kalchman & Case, 2000; Sfard, 1991,1992). That is, students are able to see any fundon
as a mathematical and conceptual object (Kalchman & Case, 2000) that has distinct properties
and behaviors. AIthough convincing data remains scarce, using technology may heIp students
reify functions and families of functions and help students conceive of functions as conceptual
entities (Greeno, 1983).
Students also develop their meta-cognitive skills using computers by making hypotheses
about the effects of certain manipulations, getting instant feedback on those hypothesis,
reflecting on their predictions, and adjusting their understandings accordingly (Kalchman &
Koedinger, ZOO 1). in doing these sorts of activities, students are said to develop more versatile
and less mechanical approaches to the study of local and global behaviors of functions
(Goldenberg, 1998; Hillel et al, 1992).
The role of the cornputer activities in the experimental curriculum is not instruction per
se. Meic henbaum and Biemiller (1998) outlime a leaming sequence of (1) instruction, (2)
consolidation, and (2) application. In the instruction phase students are introduced to a new
concept. In the consolidation phase students engage in activities and pmblems that strengthen
their understandings of the new concept and connect those understand'igs to existing
understandings and experiences. In the application phase, students apply their new
understandings to novel circurnstances. The spreadsheet activities chat studenîs do in the
experimental cumculum are of the consolidation and application sort- That is, students are
extending and building on the concepts introduced in their waikathon experiences. The
spreadsheet activities afford students al1of the opportunities discussd above with twpect to the
tec hnical capabilities of the technology and the opportunities for mdents to make connections
among representations. Furthemore, students are able to reflect on their understandings, and to
apply those understandings in novel and creative ways. Goldenberg (1988) has cautioned
educators about the premature use of technology. That is, students need to 6rst have experience
with the concepts and ideas of any topic to be explored using technology andbuild on those
experiences using effective software.
Presentations. The presentations that students made at the end of theù unit are also key
when considering which aspects of the experimentai curricuium most promoted learning. There
are of course meta-cognitive skills students need for preparing and presenting their
understandings. They m u t be able to reflect on the understandings; to idemi& the important and
central ideas; and to cornmunicate those to their classrnates adequately. Communicating
mathematical ideas, representations, and principles clearly and accurately has been identified by
NCTM as an essential component of any rnathernatics program h m primary through to
secondary schooling (NCTM, 2000).
The presentations also gave students an oppornmity to become experts on a particular
aspect of their learning and to teach their peers (Brown & Carnpione, 1994). Meichenbaum and
Biemiller ( 1998) have shown how the acquisition and communication of speciaiized expertise
has both motivationai and cognitive advantages. Motivationally, al1 students had the o p p o d t y
to be experts and to teach those who are less knowledgeable. For those students who are usually
in the bottom half of the ciass, this was an opportunity to shine. Cognitively, groups of students
presented their work and the clws as a whole continued to consolidate their understandings of a
wide range of functions and theü pmperties (Kalchman & Case, 1999).
Control Curricula
The activities the control groups did were primarily pend and paper exercises h m the
textbook or teacher-prepared worksheets. Most of these exercises were variations of examples
given either in the textbook or by the teacher at the start of the lesson.
The only extraordinary feature of either curriculum was the Grade 10 group's use of
technology. They spent two p e n d (160 minutes)dohg activities in which they were to
discover how to transform quadratic fmtions vertically, Iaterally, and how to compress and
expand them. There were two main différences between the experiences of the Grade 10 control
group with technology and the experiences of the experimentai group. First, the recording that
the control group did with respect to their observations and discoveries were limited to how the
rquation ofy = 3 needed to be manipulated in order for the panbola to move in particular
directions. Students did not examine the way in which the table of values changed as a result of
those manipulations. Thus, not only were the spatial and numenc implications of translating a
quadratic fiinction not integrated, but the numeric aspect of quadratic functions was not
developed. This helps explain why students in this gmup had marginal success with item 9 on
the functions test (giving an equation for an upsidedown parabla with a negative y-intercept
and vertex in the upper-right quadrant [seFigure 171) and virnially no success, partia! or hll,
with item 8 (sketching the graph oiy =x' c o m p d to y =.2 [see Figure 161). Students who
were successful with item 9 but not item 8 Wrely did not understand what made parabolas have
di fferent shapes, sizes and positions, only how to change the shape, size and position.
The second difference between compter activities was that the Grade 10 control students
were using the technology in the insmictional phase of their leamhg about transfomations of
quadratic functions. As has been described throughout this work, the Grade IO control students
clearly did not enter into instruction with a sound conceptual fiamework for understanding
functions. Thus, leaming about transfocmations andquadratic functions was a ta11 order when
precursor understandings h m which to build were not in place. Consequently, it is not possible
to conclude with any cenainty what the e£fécîs of using techology as an instmctional tao1
(versus a consolidation or application twi) were in this study. It is also not clear what would
have happened has these students done these same actinties as consolidation and application
exercises. However, because the results of this study indicate that students did not have an
inregrated conceptual framework for fiuictions following their experimces with the control
curriculum in general, and this particular set of activities did not promote such an integration, it
is unlikely that recasting these activities for consolidation and application would have made a
di fierence.
Summarv of the Effects of the Curricula
The Grade 10 control group only marginally passed items 3 (53%) and 7 (59%) (see
Figures 1 1 and 15, respectively), and nearly passed item 9 (41%) (see Figure 17). Item 3 was the
first item that was intended to test for an integrated cognitive structure for fiinctions. item 7 was
passed "out of sequence" with respect to the progression of results found among the
experimenta1 groups. Item 9 was directly related to the computer activities these students didThere was li ttle evidence that the students had constnicted an integrated understanding of these
three problems. Rather, their success and near-success with these items was achieved with a
primariiy numeric approach to the pmblem such as finding the slopes of perpendicular lines,
graphing equations in the form y = mr + 6.These numeric approaches reflected the sorîs of
techniques and procedures the Grade IO control students practiced in k i r assignments and
activities.
Some of the Grade 10 contml students' success may be the resuit of 'tmgrounded
competence" (Ken Koedinger, personal communication Febniary 12,2001; Kaedinger, Aübaii &
Nathan, 200 1j. That is, the successful students rnay have mastered an "equation solving" sttategy
for finding slopes of perpendicuiar lines, coordùlate graphing, and transforming quadtatic
functions, without having a larger conceptuai h e w o r k for understanding the mathematicd
-
relationships. The present hypothesis that such a h e w o r k is built t?om opportunities for
students to merge their spatial and numenc understandings within a context that also p u n d s
their lexming in earlier experiences - is complemented by this idea Koedinger et al's notion of
"grounding" is analogous to the dimension of the instructional bridging context that relates to
hridg ing students' everyday experiences with their school-based Ieaniing. Thus, students success
that resulted from sophisticated skills for solving equations was likely not representative of their
understanding of the interactive nature of the numeric and spatial aspects of a fùnction.
Effects of Emerience
The qualitative and quantitative differences found among the experimental groups may
bc explained by a combination of the experimental curriculum and the developmental Icvels of
the students. The correct, many of the partiaily correct, and even some of the incorrect strategies
used by the experimental students on the posttest were rooted in an integrated understandingof
the pro blems. ïhis showed that the type of Ieaming fostered by the experimental cwiculum in
general was consistent.
The developmental factor showed up in the levels at which students' learning bmke
down. The main pmblems for the Grade 6 students first appeared on items that int-ed
bettveen Levels 2 and 3, and for the Grade 8 students, within Level3. The particuiar difficulties
students exhibited at these critical points, correspondecito their expected and respective places in
the developmental trajectory describeci in Chapter 2 for leaming functions. This correspondence
behveen each expenmental group's item-by-item performance and the hypothesized
development sequence supports the idea that students at each grade Ievel were somehow iimited
with respect to the quantity and quality of their learning of functions. Because the design of tbis
study was cross-sectional and not longitudinal, it is not possible to confinn that these limitations
were the result merely of the age or grade of the shidents. However, the quantitative and
qualitative patterns for the expenmental pups' posttest results suggest that studmts'
experiences with rnathematics, at least in school contribuml to their overail acbievement.
Specificaily, students' experiences with topics Like rational numbers, integers, dgebraic
constructions in two variables, and cwrdinate graphing in four quadrants seemed to have a major
impact on students' progress through the hypothesizedteamhg sequence.
[n sum, the differences between the ways in which the experhental and contra1 students
approached items on the functions test may be atûibuted to their respective instructional
experiences. Specifically, diffaences accuned in (1) the topics and sequence of instruction, (2)
the use of context for punding students' learning, and (3) the assignment of particular sorts of
activities throughout the curricular sequenees.
CELUTER 4
GeneraI Summary and Conclrisions
Throughout this work,the grnerat fiamewark and principles of Case's theriiy of
conceptua1 development and Iearning were used for descniing and interpreting the leaming of
mathematical functions for students between the ages of 10 and 18. A four-stage lemhg
sequcnce was proposed:
Level 1: Children begin with relevant primary aurneric aad spatial understandings
(iterative computation schema and bar graph schema, respectively),
Level2: They etaborate the two schemas in isolation and then integrate them ta form a
base central conceptual structure for huictions. Fmm this base structure, students understand the
inter-relation of the numenc and spatial features of individual parameten of a function such as
slopc, y-intercept, or Iinearity.
Level3 : Students integrate their conceptuai structures for these individuai parameters of
functions and elaborate their intepted structure as a whole. They also begin to apply the
integated stnicture to novel circumstances and thus constnrct variants of the origuial structure to
include integers and rational numbers for graphing and computation.
Level4: Students integrate the variants of the integrated structure and consuuct a
framework for understanding the numeric and spatial entailments of h c t i o n s as a full
conceptual system.
This mode1of leaming was then used as a guide for the design of an experimental
curriculum for teaching functions to students in Grades 6 through 10. The intent of the
curriculum was to facilitate students elaboratmg and then integraihg their primary numeric and
spatial structures and to facilitate students' progression tbrough the proposed levels of
development. The curriculum was based in the situation of a walkathon, which served as a
context for bridging students'
relevant primary numeric and spatial structures and for bridging
their everyday expenences with the mathematics classroom. A set of cornputer exercises was
also designed both to motivate students and tc give them opportunities to extend, consolidate,
and apply their walkathon-based l d g to new situations. Finally students becarne "experts"
on one aspect of their learning and shared their expertise with classrnates in reports on "theif
function.
The leaming sequence and the curriculum were then tested empirically in ihe context o f
three instructional studies at each ofGrades 6.8, and 10. The Grade 6 study was a treatment-only
study because no comparable standard curriculurn was available in the region w k e the study
was carried out. The Grade 8 and Grade 10 studies were treatment-control studies in which the
control groups were taught hnctions using the school's standard textbook-based curriculum for
the teaching of functions at each respective grade level. Ail experimental groups were taught
using the experimentd functioas curriculum. Sîudents' pre- to gmt-instructionai leamhg was
measured using a specially designed set of items h w n in Appendix A. The bctions test was
originally designed with four items at each of Levei 2, Level3, and Level4 understanding (items
I -4, 5-8, and 9-12, respectively).
Quantitative and qualitative aaaiyses were conducteci (1) to compare pretest to posttest
resulrs for students in the experimental gmups to those of the students in the controi p u p s , and
(2) to see if any differences in the experimental groups' results were indicative of the Ievels of
understandings proposed for each grade in the hypothesized leamin5 sequence. The theoretical
tiamework was used to interpret the pretest to posttest results between the experimental and
control students and to describe the expected sequence of leaming for the experimental groups.
Results reported in Chapter 2 showed that al1 p u p s irnproved significantly h m pretest
to posttest on the functions test regardless of instnictional condition. For the two treatmentcon trol studies, ANOVA analyses showed a main effect of instructional p u p in favor of the
experimental group and a significant $gg x instructional m n u ~interaction also in favor of the
tsperimental group at posttest. When the Grade 6 leiiniing was compared to that of the Grade 8
and Grade 10 control groups, it was show that the Grade 6 students went from a pretest score
be low that for each of the conml groups and finished with a posttest score significmtly a b v e
i he Grade 8 controt group's
score and statistically equivalent to the Grade IO control gmup (the
rziw scores actually favored the sixth graders). Large efféct sizes in favor of the experimental
groups were calculated for the Grade 8 and Grade 10 studies (138 and 1.00, respectively) and
for the cornpuison between the Gnde 6 experimentd group and the Grade 8 control group's
results ( 1.31). ResuIts for the experimeiital groups showed that the pretest to posttest gains for
each group were nearly equivaient and thus no interaction effects were found There were main
effects for time and grade oniy.
The percentage of students in each p u p at each time who got each item correct was
calculated. These results showed hion a per item basis, there were large differences ktween
the experimental and control groups on the posttest (but not the pretest) boih in terms of the
percentage of students who got each item conect and in terms of the nurnber of items passed by
the experimentd versus the control groups, The Grade 8 control group passed (at least 50% of
students were correct) only one item. The Grade 10 controt group passed the first three items and
item 7. The Grade 6, Grade 8, and Grade 10 experimental p u p s passed the k t 56,and 7
items, respectively.
The pattern of posttest results found among the individuai items for the experimental
groups led to an examination of those items on which groups of students fell short of, or
rxceeded, the predicted levels of success. This examination led to three items being reclassified
based on their levei of cornplexit-: Items 4 and 5 were narned as transitional items (between
Levels 2 and 3), and item 8 was redassified h m Level3 to Level4.
In Chapter 3, a quaiitative strategy analysis was done to examine further the differences
arnons the students in terms of the strategies they used for answenng the questions. Cornparisons
w r e made between the expenmental and control groups' reasoning on each item. The anaiysis
looked for any pre- and post-instructionai evidence in students' item-by-item strategies of the
sorts of elaborations and integrations of the numenc and spatial understandings of functions
described in the proposed leamhg sequerice. Specifically, students' responses to each item on
thc pretest and the posttest were coded, and categories of particuiar correct, partiaily correct, and
incorrect strategies for each item were determinai. Patterns of outstanding strategies were
examined from pretest to posttest between the experimental and control groups and among the
cxperimental groups.
Ail three experimental p u p s and the Grade tO control group passed the 6rst three items
( L e i d 2 items). These required an elaboration of the numerk schema (item l), and elaboration
of the spatial schema (item 2), and an integrationof the two (item 3). The Grade 8 control group
was only successful with the hrst item. The Grade 10 contml p u p ' s posttest passing rate on the
item testing for an integration of the primary numeric and spatial structures was oniy 53%
(compared to 75% for the Grade 6 group and 70% and 63% for the Grade 8 and Grade 10
experimental groups, respectively). Among that 53% correct were some strategies in which
studenrs used a mainly numenc or aigebraic approach to the problem that did not demonstrate an
integrated understanding of the problem.
Al1 three experimentai groups passed the transition items (item 4 and 5), but the bottom
h d f of the Grade 6 classes struggled with these pst-Level2 items (the three experimental
groups had a mean correct score of 70% for items 4 and 5). Neither of the control groups were
even close to passing these items (an average of 21% of the control students passed both items).
Tliese items proved pivotal and informative with respect to seeing the differences in the
cxpenmental versus control students' post-instructional thinking. The control students were
clearly weak in their understanding of how linearity is representable in a number pattern.
A full Levei 3 understanding (items 6 and 7) was achieved only by the Grade 10
expenmental group. The Grade 8 experimental group was successful with item 6, and the Grade
6
group was successful with neither. Both of these youager experimentd groups, however,
sliowed in their partially correct and incorrect strategies for items 6 and 7 that they were still
approaching the problem from an integrated standpoint but stmggling with making al1 of the
necessary mental connections to arrive at completely correct solutions. The Grade 8 control
goup's strategies on the Level3 items showed no evidence of an integrated conceptual
ti-amework for iûnctions. The Grade 10 conml group did pass item 7 (59%), but like item 3, did
so rnainly with strategies suggesting a strongly numeric approach and did not show evidence of
understanding the ways in which the numeric and spatiai aspects of a function inter-reIate.
None of the groups passed the Level4 items (8-12), which required an understanding of
îùnctions as a conceptual system that is composeci of inter-related eIaborated and integcated
spatial and numeric conceptual structures.
The quantitative and qualitative differences found between the sîmtegies used by the
experirnental groups and the control groups were discussed and explained in the context of three
ksy differences between the experimental and control curricula: (1) the choice of topics; (2) the
use of context; and (3) the sorts ofactivities and assignments.
The experimentai curriculum focused on topics as concepts that come together to create a
general conceptual framework for understanding functions in generai. The controi cunicula
focused on techniques, methods, definitions, and conventional notation structures and
representations for functions largely independently of an overaIt conceptual framework for
applying those topics.
The experimental curriculum used the walkathon situation as an instructional context for
liclping students bridge the primary spatial and numeric structures necasary for constructing the
central conceptual structure for functions, and also for helping students bridge their everyday
experiences with the mathematics classroorn. The control curricula on the other hand used a
variety of contexts, many of which were generally unrelated to students' lives and did not
prornote understanding of how different mathematicai inscriptions (tables, m h s , equations) are
used to represent the concept of function in general.
The experimental curriculum used a variety of dif'férentkinds of activities ta keep
students motivated, grounded in their experiences, and continuaily consolidating, extending, and
ripp l ying their understandings. The major types of activities were ( 1) those using the walkathon
context, (2) the compter spreadsheet activities that linked functiom' tables, algebraic equaiions
and graphs, and (3) the presentations that elicited students' understandings of the links among
the representations for one particular function or aspect of a fiinction. In the control cumcula, on
the other band, students were mainly assigned exercises fiom the textbook or teacher-prepared
worksheets. The Grade 10 control goup did do a set of computer activities that introduced
siudents to transformations of quadratic functions. This "learning", however, did not seem
hitful because only 41% of this group passed item 9 on the posttest (a problem that düectly
rclated to the set of computer activities).
Discussion and Conciusions
Comparing qualitative and quantitative results h m the cross-sectionai investigation
showed that the type of leaming fostered by the experimental cturiculum in generd was
consistent acmss ag,e gmups. The differences iound among the experimental groups may be
sxplained, at least in part, by the deveiopmental hypotheses. The correspondence between each
experimental group's item-by-item performance and the hypothesized developmentd sequence
supports the idea that students at each grade level were somehow limited in their leaming of
functions during the available tirne. Because the more advanceci items required mathematical
knowledge not yet mastered by the younger students, they were only able to sdve these items
partially. The number of the later items the younger students could pass if they had an
opportunity to learn the necessary mathematics is an empirical question for the future.
The differences between the experimental and control groups' pretest to posttest gains
and their post-instructional reasoning and solution strategies on the functions test indicated that
the former were better able to solve problems that required eiaborated and integrated numeric
and spatial schemas for functions. The conceptual b e w o r k , or the centrai conceptuai structure,
that is presurned to have resulted h m these elaborations and integrations, allowed students to
apply a set of interna1 cognitive processes to a variety of externai situations and representations
ssociated with the mathematics of funetions.
There is an important distinction between tables, graphs, and equations as multiple
representations of a function, and the concept of function as representable in tables, graphs, and
equations (Kaput, 1993; Thompson, 1994). Thompson (1994) dcmibed the need h r students to
sec something "the same"
as they move among tables, graphs, and equations. This "sameness"
then increases the likelihood that students will see the representations as embodiments of a
common concept rather than as "'topics' leamed in isolation9*(p. 39). He suggested that
curriculum designers should "find situations that are sufficiently propitious for engendering
multitudes of representational activity [and k t ] orient students toward drawing comectionç
among their representational activities in regard to the situation that engendered t h d (p. 39).
The walkathon situation is an exampIe of how situation and reprrsectationwere
coordinated to allowed students to see how the concept of hction in general is representable in
a variety of ways. Furthemore, the activities and applications that correspondeci to the
walkathon situation helped students constnrct the rich and varied coanections (Moschkovich et
al., 1993) that fostered their understanding of how the externai representations of grapbs, tables,
and equations are connected. R d t s h m the treatment-conml groups illustrateci that when the
extemal representations of a iunction were taught as isolated topics, students did not construct an
interior network of mental connections. This networkof menta1 connections may be thought of
as the central concepnid structure for bctions. Without this structure, &dents did not
understand how and why each of the different represeutations were equivalent expressions of the
same mathematical relationship in particular, and the same mathematical concept in general.
Case's theory as it applies to the development of cbildren's understanding of hctions is
complemented by other accounts of how the concept of h c t i o n develops as a mathematical
construct. The most central of these accounts are also not mutually exclusive and include the
process-to-object paradigm (Dubinsky & Harel, 1992; Sfard, 1991,1992). the CO-variation
cersus correspondence approach (Confiey & Smith, 1994, I995), and the construction of the
"Cartesian Connection" (Schoenfeld et al., 1990). These models for understanding have dso
been used to organize and build corresprinding instructional progmns, most of which have been
technology-based.
Process-Obiect Paradim
The process-to-object paradigm is ernbedded within the learning sequence proposed for
the present study. Students use a process or procedural approach to functions in Level 2 as they
&borate their primary numeric schema by generating tables of numbers and ptotting the
h 2 and 3, they progress toward a
resulting coordinate pairs. As students move t t ~ ~ u gLevels
Level4 understanding, which involves thinking of a fùnction as an object which can then be
used as a primary schema for learning more advanced mathematicai ideas. In the literature on the
proccss-object paradigm, however, any descriptions of the cognitive processes that underlie ais
conceptual transition have been rather vague (kichman & Case, 2000). One aspect of the
prcsent work was to specify the cognitive mechanisms that spur students h m a process to an
object understanding, (that is, to go h m a Level 2 to a Levei 4 understanding of hcîioons) and
to optimize that progression through i n s t ~ ~ t i o n .
The cunicular supports for the process-to-object paradigm have corne aimost exclusively
in the fom of computer activities (Breidenbach et al, 1992; Goldenberg, 1988; Goldenberg, et
al., 1992). However, the success of the walkathon context suggests that some introductory work
with a bridging context might set up, or at least facilitate further, the process-tosbject passage.
Co-variation Ammach
The recent work by Confiey and her colleagues using a CO-variationapproach to the
teaching of functions also complements the present wotk (Borba & Confrey, 1996; Confrey &
Smith, t 995). These researchers have concluded that the conventional conespondenceapproach
to the teaching of bctions neglects the role that tables play in students' constnicting an
understanding of functions. As a result, they promote the tabular representation of a function as a
valid mathematical object from which students are encouraged to make the spatial and numeric
connections that integrate the graph with the more conventionally numeric features of a hction
(the table and equation).
In the Grade 10 control curriculum, students were specifically given a correspondence
de finition of a function. In contrast to this, the experimental curriculum presented bnctions in
the frarne of variation rather than correspondence. Support h m the present work for the CO-
variation over the correspondence approach to fMetions was found particularly in the results for
items 4 and 5 on the functions test. Compad to the experimental groups, students in both
control groups were particularly unsuccessîùiwith these items, which required that they
understand how linearity is represented in a hction's table, or in other words, the patterns of
variation between -r and y. A clear advantage the experimental students had over the control
students was in the former's understanding of the spatial characteristics representable in a table
of numbers, which is a feature of a CO-viiriationappmach to the topic.
The sohvare program Function Probe (Confrey, 1992) is the result of years of teaching
and research conducted by the Mathematics Education Research Group at CorneII University. It
has also been shown to be effective in facilitating many of the same sorts of integrations between
tables and graphs and equations descnied in the present mode1 for learning hction (Boha &
Con frey, 1996). Function Probe is a commercial product that, unlike spreadsheets, has been
designed specifically for working within a context of the multiple representations of functions.
Consequentiy, it uses more conventional mathematical notation for equations, and has more
features specific to working with fiinctions than the spreadsheet program used in the
esperimental cumculum. For exarnple, with Function Probe, students can operate on the graph
more readily by using the rnouse to sketch graphs and by plotting graphs point-by-point (Bocba
Sr Confrey, 1996). .
Using the spreadsheet software, however, has its advantages. First, most schoots have a
spreadsheet program as part of their school's software packages, and thus it is readily available.
Second, many of cunicular mandates for teaching mathematics in Ontario include the use of
spreadsheets for data analysis and charting, and thus the experimentd curriculum helps teachers
meet t heir professional requirements (Ministry of Education and Training, 1997, 1999). Many
kachers and students are thus either already farniliar with spreadsheet langage or rnust become
fmiliar with it for other topics in the required curriculum.
Although the leamhg goais are similar for the two technologies, there is a difference in
the way Function Probe and the spreadsheet activities are used. Function Robe is considered
instructional technoiogy in that is for the ''instruction" phase of instruction, consolidation,
application sequence described by Meichenbaurn and Biemiller (1998). in this case, the
technology is the main medium for leamingland the designers have incorporated issues and
ideas from students' classroom learning into the activities to be used with the software. This is in
contrast to the technology portion of the experimental cmiculum being for the consolidation and
application phase of learning. Furthemore, Function Probe has been used successfully with
students at different grade levels, with clear leaming goals defineci for each with respect to
concept development. However, no comprehensive leaming trajectory for students with respect
to their building concepts across those grades has been described or comected to the activities
associated with the software.
Cartesian Connection
Finally, some of the ideas inherent to the Cartesian Connection (Schoenfeld et ai, 1990)
are also applicable to the present work. The Cartesian Comection has two parts, Connection A
and Connection B. Connection A states that "The point (a,
yo) is on the graph of the functiony =
Ax) if and only if the point satisfies the equation, that isyo
=fTxo).
Comection B states that "in
the Carresian plane, specific algebraic expressîons have graphical identities. For example, (y2 ) is a directed line segment with both direction and magnitude specified by mathematical
convention" (p. 109). It is Connection B that is most relevant to this work.
Students in the experimental groups showed they had achieved the Cartesian Connection
with their strong showing on item 6 (anaverage of 83% correct or partiaiiy correct), for which
they were asked to give two equations that would produce a gmph of an increasing linear
function with a positive y-intercept. Likewise, the Grade 10 control group was also fairly
successful with this task and had a total of 71% correct or partiaiiy correct responses. Although
the Grade 10 control group appeared to have made the Cartesian Connection, item 4 showed that
unlike the experimental groups, they were unable to make tables that would represent a linear
function. Thus, a crucial yet missing piece to the Cartesian Connection analysis is students'
understanding of the connection between the table and the graph and the equation. More
specifically, it is arguable that to conclude that students really understand how the numeric and
spatial aspects of a function are comected, they must recognize not onîy that algebraic
expressions have graphic identities, but also that tabularlnumeric expressions of fiinctions have
spatiaVgraphic identities. For example, a table of aumbers in which they-values decrease by 2
for every unit change in x is also "a directed line segment with both direction and magnitude."
Furthemore, non-linear graphs are not addressed in the Cartesian Connection, and the present
results show that the experimentai students and the Grade 10 conml students were able to
identi fy the graphical identities of non-linear functions given as algebraic expressions.
There are sirnilarities in the types of activities students do with the Grapher software and
the types of activities students do in the experixnental curriculum with the spreadsheets, In both
prograrns, students predict needed values for the slope and y-intercept of linear t'unctions in order
to "move" lines through pre-arranged targets (Dugdale, 1982). They then make algebraic
alterations that they believe will correspond to theirpredications and the computer gives them
instant and automatic feedback on theù ideas. Students do not explore non-linear fiuictions,
however. with Grapher, and thus are not connecting their understandings of, for example, yintercept in a linear function, with the role of y-intercqt in a fiinction in general.
The role of the software Grapher in the work of Schoenfeld and his coiieagues is also
di fferent fiom the role of the spreadsheet activities in the eXpenrnentai cu~culum.Grapher
seerns to be primarily a diagnostic tml. That is, students' reasoning is analyzed while they use
the software to explore linear fimctions. The spreadsheet activities are designed ta be extensions
of students' classroom learnuig begm with the context of the walkathon.
Results of this study indicate that rnappiag out a developmental najeciory and
cornp lementary curriculum that attend to issues of both concept development and human
development are practical and effective for helping stuàents construct a comprehensive
conceptual framework for understanding functions. Moreuver, al1 of the experimental groups'
teachers took readily to the cunicdum. They saw the immediate and the longtemi value in
having students l e m functions in this i n t e m g manner, and found the language and use of
rcpresentations in the curricuIum to be easily adopted. indeed, the Grade 6 ieachers reported that
the cumculum helped them understand the mathematics of functious. in fact, the exphentai
groups' teachers reported that the curricuium helped them to see not only the importance of
helping students relate previously isolated topics in the curriculum, but also ways these topics
c m be related pedagogicalIy.
Case's theory offers four crucial inter-related contributions to the ieaching and learning
of functions. First, it offers a framework fordescriiinghow students relate the numeric and
spatial aspects of a function both i n t d l y (in tenns of how they are conceptuaiiy connected)
and extemally (in terms of how the comections are i n S c n i ) . Second, it c m guide the
cumcular design of activities for facilitatingthose connections and for helping students develop
interna1 and extemal fluency and flexiiility with moving among multiple repfe~e~ltations,
Third,
it offers a comprehensive and systernatic fiamework for supportmg and testing cbildren's
learning over the course of many years. This h e w o r k helps cesearchers and ducators describe
and interpret students' reasoning about fiinctions at different points both in their domain-specific
conceptual development and in their more generai phase of cognitive development. Existing
frameworks and cumcular innovations are not designed to address the learning needs of students
across a range of ages and who bring a range of everyday and school-based leaniing experiences
to the mathematics classroom. And fourth, although the theory offers d l of the above for
knowledge specific to functions, it is also applicable across a wide range of mathematical topics
as well as other non-mathematical topics students encounter in their Ieamhg experiences.
Cumcula need to be developed that capture these mapping relations across this range of
knowledge domains.
Future Work
The most striking result of this work is the success the Grade 6 students had with the
concepts. Many of these students showed deep understandings of slope, y-intwcept, linearity in
general and even non-linearity, which is a topic not typically addresseci until secondary Ievel
mathematics instruction. Furthemore, these youngcr amdents dernonstrated that they had
integrated the primary numeric and spatial schemas in a way that even the Grade 10 conmi
students had not. Also, even for items that the Grade 6 students could not solve completely, they
had solution strategies consistent in sophistication with those of the Grade 8 and Grade 10
experimental groups.
However, these results mut be considered in light of the fact that the Grade 6 group was
comprised of hi@-achieving students as measured by the CATl2, The appticability ofthe
leaming sequence and the insüuctional approach and curriculum to less-advanced sixth graders
needs IO be expIored. Furthemore, no work has been done with respect to developing the
precursor schemas with the students. Thus, research that explores the theoreticai and practical
implications of beginning the leamhg sequence earlier, in fourth or fifth grade, is also important.
By first focusing on developing the precusor schemas and then revising the cmicular sequence
to accommodate these younger students, existing Limitations of the proposed leaming sequence
and cumculum cari be illuminated.
Finally, more assessment items need to be developed that capture students'
understandings at key points in their leaming. That is, more items are necessary for idecfying
smaller steps students take within al1 levels, and for identifying more precisely the nature of the
elriborations and integrations at each of those steps. The reclassification of items on the current
test resulted From inconsistencies that were identified between what the experimental srudents
understood on the posttest at different grade levels and what was proposeci in the modei of
development. Additional assessment items would inform revisions of the iearning sequence with
respect to content expectations within each age level and with respect to the assignment of age
intervals to respective levels. New items would also inform revisions of the curriculum, which
uill feedforward (Cobb, in press) into m e r theoretical and curricuiar revisions.
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Appendices
FUNCTIONS
1.
What shape would the graph of the functiony
=.?+ 1 have? Draw it below.
Can you think of a function that would cross the function seen in the graph
below? What is the equation of the hction you thought of?
2.
O
1
2
3
4
5
6
7
0
9
1
0
Explain why the equation you chose is a g d one.
Suppose 1agree to pay you $30.00 for every how you work. Give a function
t hat we could use to calculate the total amount of money you have eamed afler
you have finished x hours of work.
3.
4.
Make a table o f values that would produce the b c t i o n seen below.
5.
Make up 2 equaaons that would produce a function with the following
shape:
.
Look at the hnction below. Could it represent y = x - 1O? Why or why not?
y axis
!
X axis
1i'you think it could not, sketch what you t h d it looks like.
y axis
7.
Look at the following sequence of nurnbers:
2,5,8, l l , l 4 , l 7 ...
Write an equation for a fùnction that would generate this pattern of values.
8.
Sketch in approximately the graph of y = x3.
171
9.
Write an equation for a function that would have the foIlowing shape:
At what points would the function y = 10.r - .$ cross the x a i s ? Please
show a11 of your work.
10.
i 1.
-
What shape would the graph of the function y = lOOx 25x2 + x3 likely
have?
Explain why you think so.
i
z.
I
Desçnbe what happens in the hction y = as x increases.
Appendix Tuble A
Solution Siratcgies for Responses to Item I
Grade 6
Strategy
Single mpmmtation only (Iiblo, grapb,
apiurlori, varbol dorcription)
Experimental
Experimental
Pm
Re
Post
Gradc 10
Grade 8
Post
Conwol
Pm
Pst
Experimental
PFe
Post
Control
Pre
Posl
Appendix ï'ablc U
Solution Siratcgics for Respoiiscs ta Item 2
Grade 6
Gradc 8
Exprimenial
Straiegy
Experimental
Gradc 10
Control
Experimcntal
Conirol
Post
Pm
Posi
Pre
Pmt
8
4
6
X
13
X
2
13
13
6
6
7
17
W
18
39
63
X
66
X
Pre
Post
Cunicd line h m(O, O)
X
Partiilly C
Prc
Post
Pre
Cwve fmm (O,!)
Full P ~ b o l r
Curve stutUig irom (1.2)
m AnonUlm
21
42
Appridix 'l'iible C
Soliiiion Siratcgics for Kesporiscs ro Iicin 3
Grade 6
Gnde 8
Grade 10
Pm
Post
Pre
Posi
Prc
Pust
X
X
X
X
X
X
Increasing Linaar Function
D a m s i n # Lincai Funciion
Curving Funclion
Incomct algodthmic approach
Pm
Pas1
Prt
Post
Appendix Table D
Solution Sirntegics for Kcsponses IO hein 4
Grade 6
Strategy
Experimental
Grade 10
Grade 8
Experimental
Control
Experimental
Control
Pre
Post
Pre
Post
Re
Post
Pre
Post
Prc
X
19
9
4
5
4
13
12
13
91
27
64
9
82
65
69
X
67
X
10
Post
Slopc of 1
Comct Anomalous
Puiially Corrcct Anomaiws
No constant slope
Table and graph are inconsistent
Incorrect Anomalous
10
!l3
Appcndix Tablc E
Solution Straiegics for Responses io Item 5
Grade 6
Siraiegy
Experimental
Pre
Post
Grade 8
Experimental
Pre
Pas1
Grade 10
Control
Pm
Post
Experi menta1
Pre
Post
Conirol
Prc
Post
Appcndix Tabla F
Solution Strategies for Kesponses to Iteni 6
Grade 8
Grde 6
Grade 10
Exptrimental
Experimcntnl
Control
Pre
Posi
Pre
Post
Pm
Post
Qnly ans cquation, corrcct
2
15
14
13
14
9
Sune mlationship in two forms
2
17
14
13
5
4
19
6
PPrtially Correct Anomalous
X
6
X
4
X
X
X
6
Expcnmental
Pre
Post
Contml
Pm3
Post
13
18
I3
X
m
C h g c d slope OR y-iniercept only
Chuigcâ both dope rnd y-inlcrccpt
Onc comct equation, one incorrect
Gcnenl Equation
Incorrect Anumalous
blank
X
Appcntlix I'üblc G
Soluiioii Strategies for Kesponses to Iteiii 7
Grade 6
Stralegy
Griide 8
Grade 10
Experimenial
Experimcntal
Control
Pre
Post
Pre
Post
Pre
No k a u m of s l o p and 3-iniempî
2
4
X
9
X
No, table uid gnph don't match
X
2
X
X
5
Putially COITCC~
Anornalaus
4
4
X
X
89
63
77
13
X
9
Experi mental
Control
Post
Pre
Post
Pre
Post
X
5
6
X
13
12
65
82
51
50
12
47
29
X
14
9
X
X
X
X
No because of single p a r a m e r
-10 as the slope
"Yes, h a u s e it's going down,"
Cwrdinatc graphing probtcms
lnconrxt Anomdous
Blank
Appcndix Table H
Solution Stratcgics for Hesponses
to Itciii
8
Grade 8
Grade 6
Strategy
Experimental
Pre
Conect Rcpmsentation in UppcrRight Quadnn( Only
2
Coniect Cubic Shape lnsidc y = x4
X
Full Purbols Ouuidc y = x4
4
97
Full Parabola lnside y = x4
Pmbola Inside y = x4 and raised
lncorrcct Anomalous
Post
Expcrimentnl
Pre
Post
Grade 10
Control
Pre
Post
Experimental
Pre
Post
Conirol
Pre
Post
Appendix Table 1
Solution Straiegies for Kcsporises io Item 9
Grade 6
Strategy
Only missing the lateral shift
Focus on Negative y-inbrcept
A to a Negative Exponent
General Equation
Grade 8
Experimental
Experimental
Pm
(46)
Grade 10
Control
Experimental
Control
Pre
Post
Pte
Post
Re
Post
(22)
O
(23)
O
(22)
O
(23)
O
(16)
O
(16)
Pre
(15)
O
Post
(48)
6
13
O
Post
(17)
41
X
X
X
4
X
X
X
100
94
100
%
100
100
100
50
100
35
X
X
X
7
X
X
2
X
Appendix Table J
Solution Strategics for Rcsponscs to Item 10
Grade 6
Grade 8
Experimental
Experimenial
Pre
Post
Pre
Conceptual Approach (made! a table
and/or graph)
X
10
X
Algorlthmic Approach (complcting
Lhe square or finding mots)
X
X
X
2
17
2
Siralegy
Partially Comcî Anomalws
Incomt Algorithmic Approach
Incomt Anomalous
Blank
Grade 10
Control
-
Contml
Pre
Post
Pre
Post
X
X
6
X
X
X
X
X
6
X
27
O
6
19
7
12
X
6
19
7
12
60
100
94
69
93
65
X
9
Pre
Post
5
X
5
13
17
5
13
97
73
95
60
X
X
X
4
Posi
Experirnental
1
Appendix Tüblc K
Solution Strategies for Responses io ltcm 1 1
Grade 6
Strategy
Ex peri mental
Pre
Up an Down Because of Signs in
the Equation
Curved Because of the Exponent
Inctcasing Straight-line
Incorrect Anomalous
Post
Grade 10
Grade 8
Experimental
ne
Post
Control
Pre
Post
Experimental
Pre
Post
Control
Pre
Post
Appendix Table L
Solution Strategies for Responses io Item 12
Grade 6
Strategy
Spatial and Numeric Rcpresentatian
Experimental
Grade 10
Grade 8
Experimenid
Contsol
Experimental
Contml
Re
Pas1
Pre
Post
Pre
Past
Rc
Post
Prt
Post
X
X
X
X
X
X
6
X
X
12
86
M
72
70
44
31
67
47
Numeric R~asoningOnly
As x incrosses y increues
Incornt Anamalous