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Affordances of Graphical and Symbolic Representations in Algebraic Problem Solving

BY

MARTA K. MIELICKI

B.S., Boston University, 2007

M.A., City University of New York – Brooklyn College, 2013

THESIS

Submitted as partial fulfillment of the requirements

for the degree of Masters of Arts in Psychology

in the Graduate College of the

University of Illinois at Chicago, 2015

Chicago, Illinois

Defense Committee:

Jennifer Wiley, Chair and Advisor

James Pellegrino

Mara Martinez, Mathematics, Statistics, and Computer Science

AFFORDANCES OF REPRESENTATION IN ALGEBRA

ii

TABLE OF CONTENTS

CHAPTER PAGE

1. INTRODUCTION……………………………………………………………………….… 1

2. METHOD…………………………………..…………………………………………….... 6

2.1. Participants………………………………………………………………………….… 6

2.2. Materials………………………………………………………………………............. 7

2.2.1. Algebra Problems…………………………………………………………….... 7

2.2.1.1. Counterbalancing Manipulations……………………………………..... 9

2.2.2. Final Survey…………………………………………………………………... 10

2.3. Procedure………………………………………………………….………………..... 11

3. RESULTS………………………………………………………………………………... 13

3.1. Preliminary Analyses………………………..……………………………………..... 13

3.2. Main Analyses for Representation and Problem Type…………………………........ 13

3.2.1. Accuracy……………………………………………………………………... 14

3.2.2. Time to Correct Solution…………………………………………………….. 14

3.2.3. Summary……………………………………………………………………... 15

3.3. Item Subtype Analyses…………………………………………………...…………. 15

3.3.1. Computation Subtype Analyses….…………………………………………... 15

3.3.2. Interpretation Subtype Analyses….………………………………………….. 16

3.3.3. Summary……………………………………………………………………... 17

3.4. Additional Measures………...……………………………….……………...……..... 17

3.4.1. Difficulty Ratings…………………………………………………………….. 18

3.4.2. Configurations…………………………………………………………..….… 18

3.4.3. Interview Responses………………………………………………………...... 19

3.4.3.1. Relations with Performance………………………………………….. 21

3.4.4. Rephrasing Responses………………………………………………………... 23

3.4.5. Final Survey Responses……………………………………………………..... 25

4. DISCUSSION…………………………………………………………………………….. 27

5. REFERENCES………………………………………………………………………….... 33

6. FOOTNOTES…………………………………………………………………………...... 36

7. APPENDICES...………..……………………………………………………………....… 37

7.1. Appendix A……………………………………………………………….…....... 37

7.2. Appendix B……………………………………………………………….…....... 49

7.3. Appendix C………………………………………………….……………........... 53

7.4. Appendix D……………………………………………………….…………....... 56

8. HUMAN SUBJECTS COMMITTEE PROTOCAL APPROVAL………………….…... 57


iii

TABLE OF CONTENTS (continued)

CHAPTER PAGE

9. CURRICULUM VITAE………………………………………………………………..... 58


iv

LIST OF TABLES

TABLE PAGE

I. DESCRIPTIVE RESULTS FROM FINAL SURVEY

6

II. EXAMPLES OF COMPUTATION AND INTERPRETATION

PROBLEMS

8

III. LINEAR FUNCTION CONFIGURATIONS WITH CORRESPONDING

GRAPHICAL AND SYMBOLIC REPRESENTATIONS

9

IV. ACCURACY AND TIME TO CORRECT SOLUTION BY

CONFIGURATION

19

V. PERCENTAGES OF RESONSES TO INTERVIEW QUESTIONS

20

VI. REASONS FOR REPRESENTATION PREFERENCES

21

VII. ACCURACY ON INTERPRETATION PROBLEMS PRESENTED IN

SYMBOLIC FORMAT AS A FUNCTION OF REPRESENTATION

PREFERENCE

23

VIII. ACCURACY ON INTERPRETATION PROBLEMS PRESENTED IN

GRAPHICAL FORMAT AS A FUNCTION OF REPRESENTATION

PREFERENCE

23

IX. CODING SCHEME FOR INTERPRETATION PROBLEM REPHRASE

RESPONSES

24

X. SUMMARY OF CODING FOR REPHRASING RESPONSES FOR

INTERPRETATION PROBLEM SUBTYPES BY REPRESENTATION

24

XI. CORRELATIONS BETWEEN OVERALL ACCURACY AND

MEASURES FROM THE FINAL SURVEY

25

XII. FINAL SURVEY MEASURES AND ACCURACY BY

REPRESENTATION PREFERENCE

26


v

LIST OF FIGURES

FIGURE PAGE

1. Mean accuracy (a) and completion time for correct items (b) for

interpretation and computation problems presented in graphical and

symbolic format.

14

2. Mean accuracy (a) and completion time for correct items (b) for

computation and interpretation problem subtypes presented in graphical

and symbolic format.

17

3. Mean difficulty ratings for interpretation and computation problems

presented in either graphical or symbolic format.

18


vi

LIST OF ABBREVIATIONS

ACT American College Testing

ANOVA Analysis of Variance

F Test Statistic Based on the Fisher-Snedecor Distribution

K-12 Kindergarten through 12th Grade

M Mean

ω2 Omega-squared Measure of Effect Size

N Number in group

Ns Non-significant

P Point Comparison

S Slope Comparison

SD Standard Deviation

SES Socioeconomic Status

T Test Statistic Based on Gasser’s Student Distribution


vii

Summary

Successful algebraic problem solving entails adaptability of solution methods using different

representations. Research suggests that students are more likely to choose symbolic solution

methods over graphical ones even when graphical methods are more efficient. However,

existing research has not addressed whether the efficiency of solution method varies depending

on the nature of the problem solving task. This masters project addressed the question of

whether symbolic and graphical solution methods provide different affordances with respect to

computation and interpretation problems by presenting students with either symbolic or a

graphical representation. Graphical representation was found to facilitate problem solving,

particularly for interpretation problems. Participants experienced the most difficulty with

interpretation problems that required the comparison of slopes when these problems were

presented in symbolic format. Results suggest that this difficulty may be caused by symbolic

representation biasing participants towards incorrect strategies. Participants rated graphical

problems to be less difficult than symbolic problems, and the majority of participants reported a

preference for graphical representation. These results set the stage for follow-up studies that will

explore whether the nature of the problem solving task affects students’ choice of solution

method.

AFFORDANCES OF REPRESENTATION IN ALGEBRA 1

1. Introduction

Most mathematical concepts can be represented in a variety of ways. A number can be

represented verbally, as a numeric character, as a set of objects, or as a quantity in the context of an

algebraic equation. Representation plays an important role in mathematical problem solving (Kaput,

1991). Representation can be used to guide understanding of the constraints of a problem (Hall et al.,

1989), to record steps and subgoals in the problem-solving process (Pape & Tchoshanov, 2001), to

support abstraction and complex computation (Kaput, 1998), to monitor and evaluate a solution method,

and to communicate about the problem with others (Stylianou, 2011). Representations take on an

increasingly important role as students transition from arithmetic to algebra. During this transition,

students must grapple with concepts that are increasingly abstract and removed from experience and

thus only accessible via conventional representations (Duval, 2006; Kaput, 1991). Because of the

complexity of algebraic problem solving, it is important for students to understand and appreciate the

affordances of different representations, and to know when to utilize solution strategies that rely on a

specific representation (Kaput, 1991).

Several researchers have proposed that students progress from procedural to structural

understanding of algebraic concepts (see Kieran, 1992 for a review). Moschkovich, Schoenfeld, and

Arcavi (1993) consider student interactions with multiple representations in algebraic problem solving

within a framework that distinguishes between a process and an object view of algebraic functions. In

the process view the function is treated as a link between x and y, with each value of x yielding a unique

y value. In this view, functions are perceived as rules for computation (e.g., solve for y when x is

given). In the object view, a function is considered to be an entity which can be manipulated.

According to Sfard (1991), students typically progress from a procedural to a structural view of a

concept through the process of reification, and exposure to multiple representations of a concept may

support this process. There is some evidence that curricula that emphasize multiple representations of


functions through the use of graphing software may facilitate the reification process relative to

traditional curricula that primarily emphasize symbolic manipulation (Hollar & Norwood, 1999).

An algebraic function can be expressed symbolically, graphically, or as a table of ordered pairs,

and different representations of functions are thought to engender problem-solving approaches more

consistent with either the process or object view (Kaput 1998; Sfard, 1991). Symbolic (equation-based)

representations are thought to be associated with approaches grounded in the process view because these

representations facilitate computation of exact values. Graphical representations are thought to be

associated with object view approaches because graphs facilitate perception of the function as a whole.

An important characteristic of competence in algebra is understanding the connection between multiple

representations of algebraic concepts (NCTM, 2000). Multiple representations can serve as tools to aid

students in the development of algebraic understanding. Schliemann, Goodrow, and Lara-Roth (2001)

found that instruction emphasizing multiple representations (tables and graphs) facilitated the shift from

computation of unknown values (process view) to generalization of patterns between variables (object

view) for a group of second graders, which is noteworthy because in the U.S. the concept of algebraic

functions is typically not introduced until middle school. Brenner et al. (1997) found that students

exposed to pre-algebra instruction that emphasized multiple representations were more successful at

problem representation when presented with algebra word problems. Although researchers and

educators tend to agree that understanding connections between multiple representations of algebraic

concepts is vital to algebraic understanding, several researchers have observed that students struggle

with making these connections (Leinhardt, Zaslavsky, & Stein, 1991; Romberg, Fenema, & Carpenter,

1993).

A particular set of findings that has been used to suggest that students do not understand the

connections between multiple representations of algebraic functions is that students do not seem to base

solution strategies on the affordances of a specific representation. Students generally choose symbolic


representations over graphical ones even when the latter are thought to be more efficient. Knuth (2000)

gave high school students several function problems to solve while thinking aloud, and provided both a

symbolic and a graphical representation for each problem. Knuth found that students generally chose

symbolic strategies over graphical strategies when solving function problems, even when the former

might be expected to be less efficient than the latter. Knuth attributed this strategy preference to

instruction that emphasizes symbolic representations over graphical ones (Yerushalmy & Chazan,

2002).

Taking a similar approach, Herman (2007) examined the strategies used on algebraic function

problems by college students. The students had completed a course which emphasized multiple

representations, and had been trained to use a graphing calculator. Herman found that symbolic

strategies were still overwhelmingly preferred by students when solving algebraic function problems

even after students had received this instruction. Herman also conducted follow-up interviews with the

students, and these interviews suggest that students considered symbolic manipulation to be a more

important mathematical skill than graphing, and perceived an instructor bias towards using symbolic

representations over graphical ones.

Although the results of both Knuth (2000) and Herman (2007) demonstrate a student preference

for symbolic methods over graphical methods when solving problems pertaining to functions, many

questions remain about why students may have these preferences, such as whether a graphical problem-

solving approach is actually a more efficient approach for students to take when solving problems

pertaining to functions, whether it might only be a more effective approach when problems require an

object-view approach, or whether students are just generally unable to use graphical representations. To

answer this question, in the present study students were presented with either graphical or symbolic

representations, and were given two types of problems: those that required computation of specific

points for a single linear function versus those that required observation of trends or patterns among


multiple linear functions. This dichotomy is based on a distinction made between local and global graph

interpretation tasks by Guthrie, Shelley, and Kimmerly (1993). Local search tasks entail location of

specific details within a graph, while global search tasks entail identification of trends or patterns within

a graph. These types of search tasks have analogs in algebraic problem solving with linear functions

(Leinhardt, Zaslavsky, & Stein, 1991). Students typically progress from tasks requiring identification of

specific points on a graph (local) to tasks requiring general inferences and pattern recognition (global)

(Bieda & Nathan, 2009), and algebra problems involving linear functions that students typically

encounter can be classified as requiring either computation (local) or interpretation (global). In the

present study, the computation tasks entailed solving a single linear equation for specific values whereas

the interpretation tasks involved observing general patterns and relationships among three linear

functions.

Knuth (2000) suggested that students’ reluctance to use graphical strategies stems from the

difficulty of isolating specific coordinate points in graphical representations, and from a lack of

understanding that if a point falls on the line in a graph then that point is a solution to the algebraic

equation of the line. However, Knuth only administered problems that required a process view of

functions, and the process view is associated with a computational approach. The post-test used in

Herman’s (2007) study also featured only computation problems. For interpretation problems that

require comparison of multiple functions, graphical representation may facilitate problem solving by

encouraging an object view of the functions – that is, encouraging the solver to attend to properties of

the function itself and not just computation of values. Because it is possible that problem type might

affect the efficiency of a graphical approach, it was important to use both computation and interpretation

problem solving tasks in the present study.

The main goal of this experiment was to test the question whether graphical representations

could actually be seen to facilitate performance on problems that entail comparing multiple linear


functions (interpretation problems) relative to problems that require computation of a value from a

single linear function (computation problems). Although students have been shown to generally prefer

symbolic representations, it can be assumed that symbolic and graphical representations should provide

different affordances with respect to computation and interpretation problems. Graphical representation

should allow for easier visualization of overall patterns than symbolic representation, while symbolic

representation should allow for ease in computation of exact points (Hall et al., 1989). Thus, the main

prediction was a representation by problem type interaction, with higher accuracy and faster solution

times for interpretation problems presented in graphical format than interpretation problems presented in

symbolic format. A main effect of problem type was also expected, with higher accuracy and faster

solution times for computation problems relative to interpretation problems across both representations.

This is because computation problems only require participants to consider a single linear function

whereas interpretation problems require comparison of multiple functions, which is more

computationally demanding and introduces more opportunities for error.


2. Method

2.1. Participants

A sample of 32 students (11 males) recruited from the University of Illinois at Chicago subject

pool participated in this experiment in exchange for course credit. This allowed for two replications of

sixteen different running program versions (described below) with two participants completing each

version. Three participants were replaced because they did not complete one or more items in the task.

Data from one participant was replaced due to the participant’s failure to engage meaningfully with the

task (resulting in overall accuracy of 50%).

Descriptive data describing the sample from the final survey are presented in Table 1.

Table I.

Descriptive Results from Final Survey

Task M (SD) Range

SES Composite Score (out of 6) 3.27 (1.22) 1-5.75

Parental Education (out of 8) 4.05 (1.81) 1-7.5

Income (out of 4) 2.50 (1.22) 1-4

Math ACT Score (out of 36) 24.39 (5.32) 15-36

Number of Math Courses Taken in College 1.50 (1.74) 0-9

Grade in Math Courses (out of 4.0) 2.75 (1.11) 0.0-4.0

Math Confidence (out of 6) 4.07 (1.23) 1-6

The majority of participants (66%) completed all of their K-12 education in the U.S. Of the participants

who completed some portion of their education outside the United States, 4 participants completed it in

India, 2 participants in Pakistan, and 1 participant each in Canada, Iran, Jordan, Poland, and Vietnam.


The number of grades completed in other countries ranged from 2 to 11 (M = 6.18, SD = 2.82). Overall

accuracy on the task did not differ depending on whether participants completed all of their K-12

education in the United States (M = .79, SD = .12) or abroad (M = .74, SD =.12), t(30) = .99, ns. Four

students reported pursuing math-related majors (accounting, computer science, finance, and economics),

three students reported minoring in mathematics, and two students reported math-related minors

(finance, computer science). Out of these students, two students reported majors and minors that were

both math-related (economics/mathematics, accounting/finance). Other majors included science

(chemistry, biology, neuroscience), health (pre-med, pre-dental, kinesiology, occupational therapy,

nursing-related), psychology, communications, and gender studies.

2.2. Materials

2.2.1. Algebra Problems

Each participant was presented with 24 problems, which were divided equally by representation

format (graphical, symbolic) and problem type (interpretation, computation): 6 interpretation

problems in graphical format, 6 interpretation problems in symbolic format, 6 computation problems in

graphical format, and 6 computation problems in symbolic format. Computation problems are defined

as requiring computation of a value for a single linear function. Computation problems were divided

into two subtypes: problems that require solving for the dependent variable, (solve for y), and problems

that require solving for the independent variable, (solve for x). Interpretation problems are defined as

requiring comparison of more than one linear function, and this study focused on two subtypes: slope

comparison (S) and point comparison for three functions (P). Slope comparison problems require

comparing the slopes of the three linear functions to determine which has the largest or smallest value.

Point comparison problems require participants to compare the y values for all three functions across

some range of x values. The problems were modeled on problems involving linear functions found in


Pearson Hall Connected Mathematics materials. Examples of each problem type are presented in Table

2.

Table II.

Examples of Computation and Interpretation Problems

Computation Problems Interpretation Problems

Bob is participating in a walkathon, and

he has gotten three sponsors to donate

money to charity for every kilometer that

he walks. Each sponsor has a different

pledge plan for how much money they

will donate.

Solve for x

If sponsor A

donates $35, how

many kilometers

did Bob walk?

Solve for y

How much will

sponsor C donate

if Bob walks 6

kilometers?

Malik is comparing three cab companies. Each

company has a different fare structure for charging

customers.

Point Comparison (P)

Which company offers

the best deal if Malik

wants to travel over 5

miles?

Slope Comparison (S)

Which company has the

lowest rate per mile?

Three configurations of linear functions were used in this experiment: zero slope, middle

intersection, and converging. Two cover stories were created for each configuration. Graphs and

equations for the three configurations are presented in Table 3. These three configurations were selected

to provide a variety of linear functions in order to be able to generalize the results of this study to

different types of problems. However, certain features of linear functions were avoided in order to

minimize difficulty for participants (i.e. all linear functions have positive or zero slopes and y-

intercepts). Eight problems were associated with each configuration: 4 computation problems and 4

interpretation problems, and the problems were presented in both graphical and symbolic format.


Table III.

Linear Function Configurations with Corresponding Graphical and Symbolic Representations

Configuration Graphical Symbolic

Zero Slope

Provider A: y = 0.5x + 15 Provider B: y = 0.25x + 25

Provider C: y = 35

Middle

Intersection

Sponsor A: y = 3x + 5

Sponsor B: y = 2x + 10

Sponsor C: y = x + 15

Converging

Jonah: y = 3x + 20 Tim: y = 4x + 10

Anchee: y = 5x

2.2.1.1. Counterbalancing Manipulations. In an effort to keep the number of versions to a

reasonable amount, some elements of the design were not varied across participants. The three

configurations (middle intersection, converging, zero slope) were presented in the same order to all

participants across pairs of scenarios (including one graphical and one symbolic representation

scenario), resulting in two full repetitions. In addition, computation problems that required solving for x

were always paired with slope comparison interpretation problems; and computation problems that

required solving for y were always paired with point comparison interpretation problems, with all four

occurring once within each pair of scenarios. Although there was no reason to expect that answering


one type of computation problem will influence performance on one type of interpretation problem (or

vice versa), extra care was taken in designing the pairs of computation and interpretation problems to

ensure that the answer to one was not connected to the answer of the other.

Two problems (one computation and one interpretation) were presented for each problem

scenario. Scenarios were presented in pairs using one of two different cover stories so that each

participant saw each configuration in both the graphical and symbolic representation formats in each

pair (See Appendix A). The pairing of cover story to representation type was counterbalanced across

participants, with half of the participants receiving graphical representation for cover story A and half

receiving graphical representation for cover story B for each instance of each configuration. This

counterbalancing was necessary to ensure that cover stories did not differentially influence performance

on problems with the same configuration. The order of representation presentation was also

counterbalanced, with half of the participants receiving graphical representations first within pairs of

problem scenarios and the other half receiving symbolic representations. The order of presentation of

problems pairs was counterbalanced across participants so that half received solve for x

computation/slope comparison interpretation pairs first and half received solve for y computation/point

comparison interpretation pairs first. The order of computation-interpretation subtype pairs was

reversed for each participant on the second half of the scenarios. Finally, the order of presentation for

interpretation and computation problem types was counterbalanced, with half of the participants

receiving computation problems first within each cover story scenario and the other half receiving

interpretation problems first. This 2 x 2 x 2 x 2 counterbalancing scheme resulted in 16 different

versions of the task. The entire counterbalancing design can be found in Appendix B.

2.2.2. Final Survey

The final survey (see Appendix C) included items designed to assess participants’ math ability,

experience, and confidence, as well as socioeconomic status (SES). The SES items included parental


education and household income. Each parent received an education score from 1 to 8 (1=less than high

school to 8=Ph.D./M.D./J.D.) and the education scores for both parents were averaged to obtain a

composite parent education score. The composite parent education score was then averaged with the

score for household income (on a scale from 1=under $45,000 to 4=over $60,000), to yield a composite

SES score that can range from 1 to 6.

The math items provided data regarding participants’ aptitude in math (ACT scores and grades in

math classes taken since graduating high school), and experience with math (major/minor, number of

math courses, and other courses involving math). The survey also featured items designed to assess

participants’ confidence in math on a scale from 1 to 6 with higher scores indicating lower confidence.

2.3. Procedure

After signing an agreement to participate, students began the problem solving task. All

problems were presented on a computer, but participants received an answer booklet which contained

the equations and graphs required to solve the problems. This allowed participants to make full use of

both representations by being able to manipulate them on paper rather than mentally from a computer

screen. Participants were first presented with an instruction screen with the following instructions: “In

this experiment you will be asked to solve some math problems. Please write the answer to each

problem in the answer booklet provided, and work as quickly and accurately as possible.” Problems

were presented one at a time on the computer screen. Participants wrote their answers down in the

answer booklet and pressed a key to move on to the next problem. Solution time for each problem was

recorded.

After the participants completed the problems, the experimenter collected the answer booklet.

Participants were then presented with an interview booklet containing all of the problems that they had

just completed (in the same order). Participants were prompted to rate the difficulty of each problem on

a scale of 1-6 (1 being extremely easy and 6 being extremely difficult). These difficulty ratings were


used to gauge participants’ subjective experience of problem difficulty, and to investigate the extent to

which this subjective experience is reflected in performance. Once the difficulty rating was completed,

the experimenter asked each participant to rephrase 8 of the problems using the prompt “please tell me

what you think the problem is asking you to do.” Participants’ ability to reword the problems correctly

was used as an indicator of how well participants understood the problem. The 8 problems were all

taken from the middle-intersection configuration items (so the first 4 problems and then problems 13

through 16). This allowed for a sample of problems that included all problem types (solve for x, solve

for y, slope comparison, and point comparison) presented in both graphical and symbolic format.

After participants completed the rephrasing portion the experimenter asked a series of interview

questions which provided data about participants’ experience reading graphs and their confidence in

their ability to use graphs when solving problems (see Appendix D). Participants’ responses to the

rephrasing portion and the interview questions were recorded and subsequently transcribed. After

answering the interview questions participants were asked to complete the final survey on paper and

were subsequently debriefed. The whole session took under an hour.


3. Results

3.1. Preliminary Analyses

Several preliminary analyses were conducted in order to ensure that any observed differences in

performance on the task were not due to the effects of the counterbalancing manipulations. A 2 x 2 x 2

x 2 between subjects analysis of variance (ANOVA) was conducted for overall accuracy with

representation order (graphs first, equations first), item type order (interpretation first, computation

first), item pair order (solve for x/compare slopes pair first, solve for y/compare points pair first), and

cover story order (story A first, story B first) as independent variables. No significant effects were seen

for representation order on accuracy, F < 1. Significant effects were seen for item type order, F(1,31) =

12.17, p < .05, cover story order, F(1,31) = 5.00, p < .05, and item pair order, F(1,31) = 5.00, p < .05.

Further inspection revealed that these effects may have been driven by the performance of a few

individuals with extreme ACT math scores. One participant did not report an ACT math score, two had

extremely high ACT math scores (32 and 36), and one had an extremely low score of 15i. When ACT

math score was entered as a covariate in the analysis, the effects for item type order, F(1,31) = 2.71, p =

.12, and cover story order, F(1,31) = 1.12, p = .31, were no longer significant, and item pair order was

marginal, F(1,31) = 3.71, p = .08. The effect of item pair order may be the result of pairing the more

difficult computation item, solve for x, with the more difficult interpretation item, slope comparison (see

analysis below).

3.2. Main Analyses for Representation and Problem Type

To test whether graphical representations leads to more accurate or efficient problem solving, a 2

x 2 within-subjects ANOVA was conducted with representation (graphical, symbolic) and problem type

(interpretation, computation) as independent variables for both accuracy and time to correct solution.

Results are provided in Figure 1.


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Interpretation Computation

Pro

po

rtio

n C

orr

ect

Problem Type

Graphical Symbolic

a. b.

Figure 1. Mean accuracy (a) and completion time for correct items (b) for interpretation and

computation problems presented in graphical and symbolic format. Error bars represent standard error.

3.2.1. Accuracy. The ANOVA on the accuracy data revealed a main effect of representation,

F(1,31) = 20.19, p < .001, ω2 = .08. Participants solved both types of problems more accurately when

problems were presented with graphs than with equations. The analysis also revealed a main effect of

problem type, F(1,31) = 60.31, p < .001, ω2 = .36. Participants solved computation problems more

accurately than interpretation problems, regardless of presentation format. These significant main

effects were qualified by a representation by problem type interaction, F(1,31) = 4.90, p < .05, ω2 = .02.

Planned comparisons revealed that graphical representation facilitated performance for interpretation

problems, t(31) = 4.01, p < .001, but not for computation problems, t(31) = 1.50, p = .14

3.2.2. Time to Correct Solution. The analysis of completion time data for correctly solved

items also revealed a main effect for both representation, F(1,30) = 44.06, p < .001, ω2 = .26, and

problem type, F(1,30) = 15.30, p < .001, ω2 = .14. Participants correctly solved both types of problems

faster using graphs than equations, and participants took longer to correctly solve interpretation

problems than computation problems regardless of representation. There was also a representation by

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00


Co

mp

leti

on

Tim

e (

sec)

Problem Type

Graphical Symbolic


problem type interaction, F(1,30) = 11.28, p < .01, ω2 = .03. Planned comparisons revealed that

graphical representation resulted in faster time to correct solution for both interpretation, t(30) = 6.47, p

< .001, and computation problems, t(31) = 4.67, p < .001, but this facilitation effect was greater for

interpretation problems than computation problems.

3.2.3. Summary. The analyses of accuracy and time to correct solution suggest that

interpretation problems were more difficult for participants overall, and these problems were particularly

difficult when presented with equations relative to graphs. However, it is possible that the two subtypes

of each problem type varied in their levels of difficulty. Additional analyses were conducted to explore

this possibility. Since the nested problem subtypes were not parallel, separate analyses were conducted

for each problem type.

3.3. Item Subtype Analyses

3.3.1. Computation Subtype Analyses. Within the computation items, there were two problem

subtypes. Half of the problems required solving for x and the other half required solving for y. For

computation problems, solving for y requires less algebraic manipulation than solving for x when

problems are presented with symbolic representation (because all problems were presented in slope-

intercept form), so there was a possibility that accuracy or time to correct solution might vary by

subtype. Figure 2 shows mean accuracy and mean time to correct solution for each problem subtype. A

2 x 2 within-subjects ANOVAs with representation (graphical, symbolic) and computation subtype

(solve for x, solve for y) as independent variables, revealed that accuracy was similar across subtypes,

F(1,31) = 1.06, p = .31, and across representations, F(1,31) = 2.26, p =.14 , and there was no interaction,

F<1.

Analysis of time to correct solution revealed that there was a main effect of subtype for

computation problems with solve for y problems being solved faster than solve for x problems, F(1,29)

= 14.13, p < .05, ω2 = .09. There was also a main effect of representation, both problem subtypes were


solved faster with graphs than equations, F(1,29) = 18.55, p < .05, ω2 = .17. In addition there was a

subtype by representation interaction, F(1,29) = 6.14, p < .05, ω2 = 03. Follow-up paired-samples t-tests

revealed that graphical representation resulted in faster time to correct solution for solve for x problems,

t(30) = 4.23, p < .05, and for solve for y problems, t(30) = 2.83, p < .05, but this effect was stronger for

solve for x problems.

3.3.2. Interpretation Subtype Analyses. Within the interpretation items, there were also two

problem subtypes. Half of the problems required a point-to-point comparison and the other half required

a comparison of slopes. Comparing points may be more computationally demanding than comparing

slopes when problems are presented with equations relative to graphs, which could lead to differences in

accuracy or time to correct solution. For accuracy on interpretation problems, a main effect of subtype,

F(1,31) = 27.52, p < .05, ω2 = .22, was observed with point comparison problems being solved more

accurately than slope comparison problems overall. In addition, there was a main effect of

representation, F(1,31) = 16.13, p < .05, ω2 = .10, with both subtypes being solved more accurately with

graphical representation. There was no subtype by representation interaction, F(1,31) = 2.05, p = .16.

Analysis of time to correct solution revealed that there was no main effect of subtype, F < 1.

There was a main effect of representation, F(1,19) = 39.79, p < .05, ω2 = .39, with graphical problems

solved faster than symbolic problems overall. There was a representation by subtype interaction,

F(1,19) = 7.97, p < .05, ω2 = .05. Follow up paired-samples t-tests indicate that both slope comparison,

t(19) = 2.41, p < .05, and point comparison, t(30) = 9.13, p < .05, problems were solved faster with

graphs than with equations, however the effect was stronger for point comparison problems. Additional

paired-sample t-tests revealed no difference in completion time between subtypes presented in symbolic

format, t(20) = -1.32, p = .20, and that point comparison problems were solved faster in graphical format

than slope comparison problems, t(28) = 2.77, p = .01.


a. b.

Figure 2. Mean accuracy (a) and completion time for correct items (b) for computation and

interpretation problem subtypes presented in graphical and symbolic format. Error bars represent

standard error.

3.3.3. Summary. The analyses of problem subtypes suggest that the affordances of

representation type varied by subtype. Although accuracy was high for both computation problem

subtypes in both representations, solve for x problems took more time to solve than solve for y

problems, specifically when the problems were presented with equations. For interpretation problems,

slope comparison problems emerged as more difficult than point comparison problems, but graphical

representation facilitated performance for both subtypes. Performance on both problem types and three

out of four subtypes benefitted from graphical representation.

3.4. Additional Measures.

Results from the analyses of accuracy and time to correct solution suggest that participants

struggled most with interpretation problems, particularly when these problems were presented in

symbolic format and required slope comparisons. Several other measures were analyzed to provide

insight into possible factors contributing to difficulty of these items.

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

Solve for x Solve for y SlopeComp.

PointComp.

Co

mp

leti

on

Tim

e (

sec)

Problem Subtype

Graphical Symbolic

0.000.100.200.300.400.500.600.700.800.901.00

Solve for x Solve for y SlopeComp.

PointComp.

Pro

po

rtio

n C

orr

ect

Problem Subtype

Graphical Symbolic


3.4.1. Difficulty Ratings. A 2 x 2 within-subjects ANOVA was conducted with representation

(graphical, symbolic) and problem type (interpretation, computation) as independent variables for

participant difficulty ratings. Figure 3 shows the average difficulty ratings that participants provided for

each problem on a scale of 1 (extremely easy) to 6 (extremely difficult). Average rating was around

1.75, which suggests that most participants found the task very easy. Overall, participants gave higher

difficulty ratings to problems presented with equations than problems presented with graphs, F(1, 31) =

24.41, p < .001, ω2 = .18. Participants also rated interpretation problems to be more difficult than

computation problems overall, F(1,31) = 62.41, p < .001, ω2 = .29. There was no representation by

problem type interaction, F(1,31) = 2.60, p = .12. This pattern of results for difficulty ratings parallels

that of both the accuracy and time to correct solution analyses.

Figure 3. Mean difficulty ratings for interpretation and computation problems presented in either

graphical or symbolic format. Error bars represent standard error.

3.4.2. Configurations. A one-way within-subjects ANOVA was conducted for both accuracy

and time to correct solution to determine whether configuration had an impact on either variable.

Accuracy did not differ based on configuration, F<1. As shown in Table 4, time to correct solution did

0.00

1.00

2.00

3.00

4.00

5.00

6.00


Dif

ficu

lty

Rat

ing

(ou

t o

f 6

)

Problem Type

Graphical Symbolic


differ by configuration, F(2,30) = 4.19, p < .05, ω2 = .08. Post-hoc pairwise comparisons revealed that

time to correct solution for the converging configuration was less than for middle intersection, t(31) =

2.75, p < .05, or zero slope, t(31) = 2.17, p < .05, and time to correct solution for the latter two

configurations did not differ, t < 1. No significant differences in difficulty ratings were seen across

configurations, F(2,30) = 2.62, p = .09.

Table IV.

Accuracy and Time to Correct Solution by Configuration

Middle Intersection Converging Zero Slope

M (SD) M (SD) M (SD)

Accuracy .78 (.15) .77 (.15) .77 (.19)

Time to Correct

Solution (sec) 38.43 (17.66) 30.76 (15.88) 37.99 (18.54)

Difficulty Ratings 1.82 (.58) 1.71 (.58) 1.87 (.66)

3.4.3. Interview Responses. As shown in Table 5, when asked which representation they

preferred, most participants reported a preference for graphical representation. When asked whether

they felt that problems were easier when presented with a particular representation, most participants

responded that the problems presented with graphs were easier than those presented with equations. The

number of participants who reported more experience with graphs was similar to the number of

participants who reported more experience with equations. For the students who reported a preference

for graphical representation, 80% also reported that the problems presented with graphs were easier than

problems presented with equations. For the participants who reported a preference for symbolic

representation, only 42% reported that symbolic problems were easier than graphical problems.


Table V.

Percentages of Responses to Interview Questions

Graphical Symbolic Same

Which do you prefer? 62.5 37.5 -

Which was easier? 59.4 18.7 21.9

Which do you have

more experience with? 44.0 47.0 9.0

Participants reported high confidence in the ability to use graphs to solve problems (M = 4.98 out

of 6, SD = 1.02), and 75% of participants responded “yes” when asked whether they had mastered

reading graphs. Of the participants who reported having mastered reading graphs, half reported that this

occurred in grades 6-8 and the other half reported in grades 9-12.

When asked when their first exposure to graphs occurred, 66% of participants reported grades 2-

5, 28 reported grades 6-8, and 6% reported grade 9. Most participants, 75%, reported that this first

exposure had occurred in the context of a math class (16% reported science class and 9% reported both).

All participants reported some exposure to graphs in non-math classes. The most commonly reported

classes were science (chemistry, biology, physics, environmental science, geology), followed by

psychology, history, and English. In response to the question of whether the majority of their

experience with graphs was in math courses or in science courses, 53% responded math, 6% responded

science, and 41% responded that experience with graphs had been equal in math and science courses.

As shown in Table 6, the main reasons participants cited in favor of graphical representation

were that the visual nature of graphical representation makes the solution to a problem more apparent,

that graphs involve “less math”, and that participants liked not having to choose or execute an algebraic

procedure. The main reasons participants cited in favor of symbolic representation included: that graphs


require more interpretation, that symbolic representation is more efficient and leads to more precise

solutions, and that participants liked being able to execute procedures (e.g. “plug in a number”).

Table VI.

Reasons for Representation Preferences

Reason for Preference Percentage Example

Graphical N=20

The visual nature makes the

solution to a problem more

apparent

95 “you can easily observe the shape of the curve or

it’s just more visual… just look at the points”

Involves “less math” 25

“ it's easier to like read it and interpret it as opposed

to like the equation that you have to like solve it

and really think about it more”

Does not require choice or

execution of an algebraic

procedure

35 “you don't have to solve anything you can just look

for the certain point and then find the value”

Symbolic N=12

Requires less interpretation

(than graphical) 50

“it's mostly you just plug it in and get an answer

where then in a graph you kind of have to interpret

it and like there's different ways to look at it”

More efficient and leads to

more precise solutions 33

“when you get the answer it's for sure and you can

plug the numbers again to make sure that they're

100% correct”

Choice and execution of

algebraic procedure is clear 75

“I can actually like plug in a number for each of the

I can actually plug in the number and then you get a

definite answer I guess”

3.4.3.1. Relations with Performance. Because interpretation problems, particularly those

presented in symbolic format, had lower accuracy as well as higher difficulty ratings than computation

problems, performance on these problems was explored further in order to determine whether any of the

factors addressed in the interview questions might be related to performance. The means for accuracy

on interpretation problems presented in symbolic format broken down by response to several interview

questions are presented in Table 7 and the means for interpretation problems presented in graphical

format are in Table 8. Participants who indicated that they preferred graphs to equations scored higher


on symbolic interpretation problems than participants who indicated a preference for equations.

Participants who reported that graphical problems were easier scored slightly higher on symbolic

interpretation problems than participants who reported that symbolic problems were easier or did not

report a difference in difficulty based on representation. Participants who reported a preference for

equations scored higher on graphical interpretation problems than symbolic interpretation problems, and

participants who reported finding symbolic problems easier overall also performed better on

interpretation problems presented with graphs than those presented with equations. These findings are

surprising because they suggest that participants’ preference for symbolic representation was not

associated with better performance on symbolic interpretation problems. The findings also suggest that

appreciating the affordances of graphical representation (by indicating a preference for graphs over

equations) may be associated with higher performance on interpretation problems in both

representations.

A second interesting finding was that participants who reported equal exposure to both

representations outperformed participants who reported more exposure to graphs or equations on

interpretation problems presented in symbolic format. This is consistent with the idea that instruction

that emphasizes multiple representations facilitates algebraic understanding (Brenner et al., 1997;

Schliemann, Goodrow, & Lara-Roth, 2001).


Table VII.

Accuracy on Interpretation Problems Presented in Symbolic Format as a Function of Representation

Preference

Graphical

Representation

Symbolic

Representation Same


Which would you

choose? .60 (.24) .47 (.21) -

Which problems were

easier? .58 (.26) .50 (.11) .52 (.26)

Which do you have

more exposure to? .54 (.22) .53 (.25) .72 (.25)

Table VIII.

Accuracy on Interpretation Problems Presented in Graphical Format as a Function of Representation

Preference

Graphical

Representation

Symbolic

Representation Same


Which would you

choose? .73 (.22) .76 (.24) -

Which problems were

easier? .76 (.24) .78 (.14) .67 (.25)

Which do you have

more exposure to? .73 (.23) .76 (.23) .78 (.19)

3.4.4. Rephrasing Responses. Participants were asked to respond to the prompt “please tell me

what you think the problem is asking you to do” for a subset of problems, and participants’ responses

were coded and analyzed in order to better understand why symbolic representation led to lower

accuracy on interpretation problems. The coding scheme is detailed in Table 9 and the total number of

codes for each interpretation problem subtype can be found in Table 10.


Table IX.

Coding Scheme for Interpretation Problem Rephrase Responses

Code Description Example

Incorrect Procedure Indicated a solution strategy

that was incorrect

Figure out who pays the least

amount each month so plug in I

guess maybe any number for x and

then solve for y

Referenced Representation Mentioned the representation

but no solution strategy

Basically from the equations who

would save the most per month

Paraphrased

Simply restated the problem

with no indication of

solution strategy and no

mention of representation

Who saved the most money with the

car that they purchased per month

Incomplete Procedure

Indicated a solution strategy

that may have been correct

but was incomplete

You want to figure out which one

has the lowest rate by… just figuring

out the number associated with the

variable

Correct Procedure/ Mention

of Concept

Indicated a correct solution

strategy or referenced the

appropriate mathematical

concept

It’s asking you to find which one of

these has the lowest slope

Note: The examples are all taken from slope comparison problems presented with equations.

Table X.

Summary of Coding for Rephrasing Responses for Interpretation Problem Subtypes by Representation

Problem Type Incorrect

Procedure

Referenced

Representation Paraphrased

Incomplete

Procedure

Correct

Procedure/

Concept

Slope Comparison

Graphical 7 2 13 1 9

Slope Comparison

Symbolic 13 3 10 1 5

Point Comparison

Graphical 0 6 12 3 11

Point Comparison

Symbolic 0 2 14 4 12


Descriptive analysis of the coded data suggested little difference in strategy based on

representation for point comparison problems; however, there were differences in strategy based on

representation for slope comparison problems. When slope comparison problems were presented with

symbolic representation, participants used incorrect procedures more often than when the problems were

presented with graphs. Conversely, when slope comparison problems were presented with graphs

participants executed correct procedures (or referenced the appropriate mathematical concept) more

often than when these problems were presented with equations. Further inspection of the responses

coded as incorrect procedures for symbolic slope comparison problems suggests that participants were

more inclined to “plug something in” in order to solve these problems. Thus, symbolic representation

appears to have biased participants towards an incorrect strategy of substituting values for unknowns.

3.4.5. Final Survey Responses. Measures from the final survey were analyzed in order to

observe which, if any, were associated with performance on the task. Correlations between overall

accuracy on the task and measures from the final survey are reported in Table 11. The only measures

that significantly correlated with overall accuracy were ACT Math score and math confidence.

Table XI.

Correlations between Overall Accuracy and Measures from the Final Survey

Measure 1 2 3 4 5 Overall Accuracy

1. ACT Math Score - -.17 .45* .62** .42* .63**

2. Number of Math Courses - - .13 -.03 -.28 .15

3. Grade in Math Courses - - - .45* .33 .12

4. Math Confidence - - - - .25 .35*

5. SES Score - - - - - .16

Note: *p < .05 **p < .01


Since performance on the most difficult problems (symbolic slope comparison) varied by

reported representation preference, final survey measures were explored to see whether these measures

varied by reported preference as well (see Table 12). Independent-samples t-tests were conducted to

determine whether there were differences in any of the measures between participants who reported a

preference for either graphs or equations. None of the measures differed significantly, but participants

who reported a preference for graphs over equations did report marginally lower math confidence than

participants who reported a preference for equations, t(30) = -1.58, p = .13, and had marginally lower

SES composite scores, t(30) = -1.44, p = .16.

Table XII.

Final Survey Measures and Accuracy by Representation Preference

Graphical Symbolic

M (SD) M (SD)

Symbolic Slope Comparison Problem Accuracy .47 (.38) .22 (.26)

Math ACT (out of 36) 24.45 (6.17) 24.27 (3.55)

SES Composite Score (out of 6) 3.04 (1.36) 3.67 (.86)

Number of Math Courses Taken in College 1.45 (1.99) 1.58 (1.31)

Grade in Math Courses (out of 4.0) 2.58 (1.17) 2.97 (1.04)

Math Confidence (out of 6) 3.81 (1.23) 4.50 (1.16)


4. Discussion

The primary question tested in this study was whether graphical representation would be more

efficient than symbolic representation for problems pertaining to linear functions, and whether the

affordances of graphical representation would be limited to particular problem types. The prediction that

graphical representation would specifically facilitate problem solving for interpretation problems was

borne out, with higher accuracy and less time to correct solution for interpretation problems presented in

graphical format relative to symbolic format. Although there were no differences in accuracy for

computation problems presented with graphs versus equations, graphical representation did lead to less

time to completion than symbolic representation for computation problems. The prediction that students

would be more accurate and efficient overall on computation problems than interpretation problems was

also supported.

Further analysis of problem subtypes indicated that the affordances of graphical representation

varied by subtype. Accuracy for computation problems was high for both representations, but the

difference in time to completion between problems that entailed solving for x and problems that entailed

solving for y did vary by representation. Solve for x problems took longer to complete when presented

with equations, and this difference can be attributed to the additional algebraic manipulation that is

required to solve for x in symbolic format. For interpretation problems graphical representation led to

higher accuracy and lower completion time than symbolic representation for both subtypes. Students

struggled most with slope comparison problems presented in symbolic format. Analysis of participants’

rephrasing responses suggests that symbolic representation may engender a problem solving approach

that is more consistent with a process view of functions than an object view – participants were more

likely to approach slope comparison problems with an incorrect procedure (e.g. “plug something in”)

than to consider the properties of the function (specifically, the slope) when it was presented in symbolic

format.


These findings are consistent with the assumptions made by Knuth (2000) and Herman (2007)

that graphical representation should facilitate algebraic problem solving. However, both Knuth (2000)

and Herman (2007) used computation problems in their studies, and the results of the current study

suggest that the affordances of graphical representation are most apparent for interpretation problems.

At the same time, it is possible that the difficulty level of the computation problems in the current study

resulted in a ceiling effect for accuracy, and that an advantage of graphical representation might be

observed with more difficult computation problems. The results of this study also support Knuth’s

(2000) and Herman’s (2007) contention that students do not appreciate the connection between multiple

representations. This is most evident in performance on the slope comparison problems, which had the

greatest difference in performance between representations despite being the same problems. Unlike

point comparison problems, where the amount of computation required varied by representation,

symbolic slope comparison problems did not require additional computation beyond that required by

graphical slope comparison problems because the equations were in slope-intercept form. Therefore the

difficulty of the symbolic slope comparison problems cannot be attributed to differences in cognitive

demand based on representation, but rather must be due to the representations themselves engendering

different approaches to problem solving.

Analysis of the interview data revealed that, contrary to Knuth (2000) and Herman (2007), many

participants did recognize the affordances of graphical representation. Most participants reported that

they would choose graphs over equations for problem solving in general, and that problems presented

with graphs in the study were easier to solve than those presented with equations. Recognition of the

affordances of graphical representation was also reflected in participants’ ratings of difficulty, with

problems presented with graphs being rated as less difficult than problems presented with equations.

The results of this study did not support Knuth’s (2000) contention that student choice of symbolic

representation over graphical representation may have been partially driven by more exposure to


equations than graphs during instruction, because the number of participants who reported more

exposure to equations was similar to the number of participants who reported more exposure to graphs.

The main reasons participants gave for preferring graphs to equations were that graphs involve

“less math” because they are visual and obviate the need to choose and execute any kind of algebraic

procedure. Several of these participants admitted to the experimenter that they did not consider

themselves “good at math” and therefore preferred graphs. This echoes Herman’s (2007) findings that

students considered symbolic solution methods to be valued higher in their math courses than graphical

solution methods. In contrast, participants who reported a preference for equations felt that equations

lead to more precise solutions than graphs, and equations are easier than graphs because the correct

procedure is apparent. Interestingly, participants who reported a preference for equations also felt that

graphs require more interpretation than equations, which was not reflected in the accuracy or time to

correct solution data obtained in this experiment.

Participants who reported a preference for equations over graphs also reported marginally higher

math confidence. This is also consistent with the interview data from Herman (2007) – if symbolic

representation is considered “true math” then participants who are comfortable with symbolic

representation may also experience high confidence in their mathematical abilities. At the same time,

these results suggest that these students may be over-confident because participants who reported a

preference for symbolic representation also had lower accuracy on symbolic interpretation problems

than participants who preferred graphs. This suggests a disconnect between representation preference

and success in problem solving – a reported preference for equations was not associated with better

performance on symbolic interpretation problems than a reported preference for graphs.

Participants who reported equal exposure to both graphs and equations had higher accuracy on

symbolic interpretation problems than participants who reported more exposure to either representation,

which supports the notion that exposure to multiple representations in instruction leads to gains in


algebraic understanding (Brenner et al., 1997; Schliemann, Goodrow, & Lara-Roth, 2001). Exposure to

mathematical concepts in multiple representations may facilitate the reification process proposed by

Sfard (1991) where the individual proceeds from a process view of a concept to an object view. If

participants in this experiment who reported equal exposure to graphs and equations had also acquired

an object view of algebraic functions, this could explain their high performance on the symbolic

interpretation items which entailed considering the properties of the function (object view) as opposed to

treating it as a means to generate output from a given input (process view).

In the upcoming Fall semester, data from three outlier participants (with extreme Math ACT

scores) will be replaced and the new data set will be reanalyzed to ensure that the effects reported above

are still present, and to address the observed effects from the counterbalancing manipulations.

A possible reason for the divergence in findings pertaining to student preference of

representation between this study and Knuth (2000) and Herman (2007) is the context of the study itself.

Both Knuth (2000) and Herman (2007) conducted their research within math classrooms, whereas this

study was conducted independent of any math course. It is possible that students might feel compelled

to choose symbolic solution methods over graphical ones in the context of a math class, but that this

preference was not expressed in the current study because participants did not feel the need to

demonstrate their math ability to the experimenter. Also, unlike Knuth (2000) and Herman (2007),

participants in the current study were not actually given the option to choose a representation for solving

the problems. It is possible that when students are presented with both representation options they may

choose to solve problems symbolically. Although the self-report data from this study suggested that

many students preferred graphical representations, it is important to follow-up this finding with studies

that actually give students a choice between representations to test whether this reported preference is

reflected in participants’ choice of representation during problem solving.


There is some prior research that suggests students may change strategies based on task

demands, which supports the prediction that students may choose different representations for

computation and interpretation problems. Hall et al. (1989) administered algebraic story problems to

college students and found that many students switched from formal algebraic (symbolic) solution

methods to model-based reasoning (visual/graphical) if they reached an impasse using the first method.

Huntley et al. (2007) observed that although students generally use solution strategies involving

symbolic manipulation when solving sets of linear equations (of the form ax ± b = cx ± d), students were

more likely to spontaneously shift from symbolic solution strategies to graphical ones when given

problems that yield unexpected results (i.e. identities and parallel lines). The results of these studies,

along with the results of the present study, suggest that students may be more sensitive to the

affordances of different solution methods than Knuth (2000) and Herman (2007) found, and that this

sensitivity may also translate into the selection of a more efficient representation for a given problem

type.

The Common Core State Standards Initiative (2010) includes the expectation that high school

students should be able to solve problems with algebraic functions using different representations. In

order for students to be able to select the best representation for solving a given problem, they must

understand the affordances of different representations. The results of this study demonstrate that

graphical representation is particularly beneficial for problems that require the comparison of multiple

algebraic functions along some dimension, and these findings could prove useful for educators and

practitioners. It is also important to understand student perceptions of multiple representations, and the

relationship that these perceptions have with math confidence, because these may play a role in students’

mathematical practices. Herman (2007) found that students perceive symbolic solution methods to be

more “true math” than solution methods involving other representations, and the results of this study

suggest that students who express a preference for symbolic representations may be over-confident in


their ability to solve symbolic problems. It is important for educators to communicate the affordances of

different representations through instruction, without also communicating a bias towards one

representation or another.


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6. FOOTNOTES

i. Data from these participants is included in the analysis, but will be replaced when the subject pool

opens in the Fall 2015 semester.


7. APPENDICES

Appendix A

Middle Intersection Configuration 1.

Story A: Bob is participating in a walkathon, and he has gotten three sponsors to donate money to

charity for every kilometer he walks. Each sponsor has a different pledge plan for how much money

they will donate.

Graph: Here is a graph of each sponsor’s pledge.

Equations: Here are the equations for each sponsor’s pledge where y is the amount of money donated in

dollars and x is the distance walked in kilometers.

Sponsor A: y = 3x + 5

Sponsor B: y = 2x + 10

Sponsor C: y = x + 15

Interpretation (P): Which sponsor will donate the least if Bob walks over 5 kilometers?

Interpretation (S): Which sponsor will give Bob the most money per kilometer?

Computation (y): How much will sponsor C donate if Bob walks 10 kilometers?

Computation (x): If sponsor A donates $35, how many kilometers did Bob walk?


Middle Intersection Configuration 1.

Story B: Malik is comparing three cab companies. Each company has a different fare structure for

charging customers.

Graph: Here is a graph of each cab company’s fare structure.

Equations: Here are the equations for each cab company’s fare structure where y is the cost of the ride

in dollars and x is the distance traveled in miles.

Company A: y = 3x + 5

Company B: y = 2x + 10

Company C: y = x + 15

Interpretation (P): Which cab company offers the best deal if Malik wants to travel over 5 miles?

Interpretation (S): Which cab company has the lowest rate per mile?

Computation (y): How much does company B charge for traveling 10 miles?

Computation (x): If company C charges Malik $25, how many miles did he travel?


Middle Intersection Configuration 2

Story A: Alan, Don, and Judy are each saving up to buy a car. They each have a different monthly

savings plan.

Graph: Here is a graph of each person’s car fund savings.

Equations: Here are equations for each person’s car fund savings where y is the amount of money saved

in dollars and x is the number of months.

Alan: y = 30x + 700

Don: y = 60x + 400

Judy: y = 50x + 500

Interpretation (P): Who will have the most money after the first 3 months?

Interpretation (S): Who saves the most money per month?

Computation (y): How much money has Don saved at 5 months?

Computation (x): If Judy has $1200 saved, how many months has it been?


Middle Intersection Configuration 2

Story B: Alex, David, and Jessica each recently purchased new computers. They have different payment

plans.

Graph: Here is a graph of each person’s payment plans for their computers.

Equations: Here are equations for each person’s computer payment plans where y is the amount of

money paid in dollars and x is the number of months.

Alex: y = 30x + 700

David: y = 60x + 400

Jessica: y = 50x + 500

Interpretation (P): Who will have paid the least amount of money after the first 7 months?

Interpretation (S): Who pays the least amount of money per month?

Computation (y): How much money has Jessica paid at 8 months?

Computation (x): If Alex has paid $1000, how many months has it been?


Converging Configuration 1.

Story A: Anchee, Jonah, and Tim earn weekly allowances for doing chores over the summer. Each

person has a different arrangement for how they get their earnings.

Graph: Here is a graph of each person’s earnings from doing chores.

Equations: Here are equations for each person’s earnings from doing chores where y is the amount of

money earned in dollars and x is the number of chores completed.

Anchee: y = 5x

Jonah: y = 3x + 20

Tim: y = 4x + 10

Interpretation (S): Who earns the least money per chore completed?

Interpretation (P): Who earns the least money for doing 3 or fewer chores?

Computation (y): How much money will Anchee earn for doing 4 chores?

Computation (x): How many chores will Tim have to do to earn $30?


Converging Configuration 1.

Story B: Alice, Jamal, and Tony are working summer jobs handing out flyers for different companies.

Each company has a different way of paying the employees.

Graph: Here is a graph of each person’s earnings from handing out flyers.

Equations: Here are equations for each person’s earnings from handing out flyers where y is the amount

of money earned per day and x is the number of flyers handed out.

Alice: y = 5x

Jamal: y = 3x + 20

Tony: y = 4x + 10

Interpretation (P): Who earns the most money handing out 5 to 10 flyers?

Interpretation (S): Who makes the most money per flyer?

Computation (y): How much money will Alice earn if she hands out 7 flyers?

Computation (x): How many flyers would Jamal have to hand out in order to earn $35?


Converging Configuration 2

Story A: Jamie, Sarah, and Angel are growing plants for a biology project. Each plant grows a little

differently.

Graph: Here is a graph of each person’s plant growth.

Equations: Here are equations for each person’s plant growth where y is the plant height in centimeters

and x is the number of weeks.

Jamie: y = 0.5x + 1.25

Sarah: y = 0.75x

Angel: y = 0.25x + 2.5

Interpretation (P): Whose plant is shortest during the first 3 weeks?

Interpretation (S): Whose plant is growing at the fastest rate?

Computation (y): How tall is Angel’s plant at 4 weeks?

Computation (x): If Jamie’s plant is 2.25 cm tall, how many weeks has it been?


Converging Configuration 2

Story B: John, Samantha, and Adnan are each filling their pools up with water. Each person’s pool fills

differently.

Graph: Here is a graph of each person’s pool filling progress.

Equations: Here are equations for each person’s pool where y is the height of the water in feet and x is

the number of hours.

John: y = 0.5x + 1.25

Samantha: y = 0.75x

Adnan: y = 0.25x + 2.5

Interpretation (P): Whose pool will have the most water in the first 2 hours?

Interpretation (S): Whose pool is filling at the slowest rate?

Computation (y): How high is the water in Samantha’s pool at 3 hours?

Computation (x): If the water in Adnan’s pool is 3.75 feet high, how many hours has it been?


Zero Slope Configuration 1.

Story A: Jose, Mario, and Melanie work in different departments in the same store. Since some

departments work on commission, each person has different daily earnings depending on how much he

or she sells.

Graph: Here is a graph that shows the daily earnings for each person.

Equations: Here are equations for each person’s daily earnings where y is the amount of money earned

in dollars and x is the number of sales.

Jose: y = 5x + 20

Mario: y = 60

Melanie: y = 6x + 12

Interpretation (S): Who earns the most per sale?

Interpretation (P): Will Mario earn more than Melanie and Jose if they each make fewer than 8 sales?

Computation (y): How much will Melanie earn if she makes 13 sales?

Computation (x): How many sales would Jose have to make in order to earn $40?



Story B: John, Mark, and Monique are personal trainers. Each person has a different monthly payment

plan for charging customers.

Graph: Here is a graph that shows each person’s payment plan.

Equations: Here are equations for each person’s payment plan where y is the monthly cost in dollars and

x is the number of sessions.

John: y = 5x + 20

Mark: y = 60

Monique: y = 6x + 12

Interpretation (S): Who charges the least per session?

Interpretation (P): Will Mark charge more than Monique and John if they each have over 8 sessions?

Computation (y): How much will Monique charge for a month with 3 sessions?

Computation (x): If John charges $80 for a month, how many sessions did he have?



Story A: Stella is comparing three different cell phone providers. Each provider offers a different

monthly plan.

Graph: Here is a graph of each cell phone plan.

Equations: Here are equations for each provider’s cell phone plan where y is the monthly cost and x is

the number of minutes used for calls.

Provider A: y = 0.5x + 15

Provider B: y = 0.25x + 25

Provider C: y = 35

Interpretation (S): Which provider charges the most per minute?

Interpretation (P): Which provider is best if Stella usually uses more than 40 minutes a month?

Computation (x): If Stella chose provider A and her monthly bill was $40, how many minutes did she

use?

Computation (y): If Stella chose provider B, how much would her monthly bill be if she used 80

minutes?



Story B: Max is comparing 3 different car rental companies. Each company has a different pricing plan.

Graph: Here is a graph of each company’s pricing plan.

Equations: Here are equations for each company’s pricing plan where y is the cost of the rental and x is

the number of miles driven.

Company A: y = 0.5x + 15

Company B: y = 0.25x + 25

Company C: y = 35

Interpretation (P): Which company offers the best deal if Max travels fewer than 40 miles?

Interpretation (S): Which company’s plan has the lowest rate per mile?

Computation (y): If Max chooses Company C and travels 30 miles, how much will the rental cost?

Computation (x): If Max chose Company B and they charged him $45, how many miles did he travel?


Appendix B

C=configuration

S=cover story (A or B)

G= graphical representation E=symbolic representation

LX=solve for x, LY=solve for y, GS=slope comparison, GP= point comparison

Version 1 Version 2 Version 3 Version 4

Item 1 C1_SA_G_LX C1_SB_G_LX C1_SA_E_LX C1_SB_E_LX

Item 2 C1_SA_G_GS C1_SB_G_GS C1_SA_E_GS C1_SB_E_GS

Item 3 C1_SB_E_LY C1_SA_E_LY C1_SB_G_LY C1_SA_G_LY

Item 4 C1_SB_E_GP C1_SA_E_GP C1_SB_G_GP C1_SA_G_GP









Item 13 C1a_SA_G_LY C1a_SB_G_LY C1a_SA_E_LY C1a_SB_E_LY

Item 14 C1a_SA_G_GP C1a_SB_G_GP C1a_SA_E_GP C1a_SB_E_GP

Item 15 C1a_SB_E_LX C1a_SA_E_LX C1a_SB_G_LX C1a_SA_G_LX

Item 16 C1a_SB_E_GS C1a_SA_E_GS C1a_SB_G_GS C1a_SA_G_GS











Item 1 C1_SA_G_LY C1_SB_G_LY C1_SA_E_LY C1_SB_E_LY

Item 2 C1_SA_G_GP C1_SB_G_GP C1_SA_E_GP C1_SB_E_GP

Item 3 C1_SB_E_LX C1_SA_E_LX C1_SB_G_LX C1_SA_G_LX

Item 4 C1_SB_E_GS C1_SA_E_GS C1_SB_G_GS C1_SA_G_GS









Item 13 C1a_SA_G_LX C1a_SB_G_LX C1a_SA_E_LX C1a_SB_E_LX

Item 14 C1a_SA_G_GS C1a_SB_G_GS C1a_SA_E_GS C1a_SB_E_GS

Item 15 C1a_SB_E_LY C1a_SA_E_LY C1a_SB_G_LY C1a_SA_G_LY

Item 16 C1a_SB_E_GP C1a_SA_E_GP C1a_SB_G_GP C1a_SA_G_GP










Appendix C

FINAL SURVEY

1. Please circle the highest level of education reached by your mother.

Less than High School

High School

Professional Training

Some College

College

Some Graduate School

Masters

Ph.D./M.D./J.D.

Other: ____________________

2. Please circle the highest level of education reached by your father.

Less than High School

High School

Professional Training

Some College

College

Some Graduate School

Masters

Ph.D./M.D./J.D.

Other: ____________________

3. Please circle the option that best describes your parents’ household income.

Under $45,000 $45,000-$50,000 $50,000-$60,000 Over $60,000


4. Have you taken the SAT or ACT in the past 5 years? (Circle one) Yes No

5. What was your highest score on the math portion of the test?

ACT (out of 36): _________________ b. SAT (out of 800): _________________

6. Did you learn how to read graphs in school? (Circle one) Yes No

7. Please rate your confidence in your ability to read graphs and use graphs to answer questions. (Circle

one)

1 2 3 4 5 6

Not Confident at All Extremely confident

8. If given the choice, would you prefer to use graphs or equations to solve problems? ________

9. Did you complete any of your K-12 schooling outside the US? (Circle one) Yes No

If you answered "yes" to the previous question, please indicate which country and how many years.

Country: ______________________________________ Number of years: ________

10. What is your major concentration in school? If “undecided” what course of study do you think you

might want to pursue?

11. If you are considering a minor concentration, what is it?

12. Please list the math courses you have taken since graduating high school below, when you took

them, and what grade you received for each.

Course Title Semester (e.g. F14) Grade (e.g. B+)

13. Please list the non-math courses that involve math (ex. physics, accounting) that you have taken

since graduating high school below, when you took them, and what grade you received for each.

Course Title Semester (eg. F14) Grade (e.g. B+)


Please rate the accuracy of each statement below (1 = extremely accurate, 6 = doesn’t describe me at all)

14. I am good at solving math problems.

1 2 3 4 5 6

Describes Me Doesn’t Describe me at all

15. I have generally had good grades in my math classes.

1 2 3 4 5 6


16. I am comfortable with math.

1 2 3 4 5 6


17. I get anxious/uncomfortable when asked to do math.

1 2 3 4 5 6


18. I avoid using math whenever possible.

1 2 3 4 5 6


19. Please use the section below to provide any other information you would like to disclose (e.g. If

you have been diagnosed with a learning disability).


Appendix D

1. If you were given a choice to solve problems like the ones in this experiment using graphs or

equations, which would you choose? Why?

2. Overall, did you find the graph problems in this experiment to be easier, more difficult, or the

same level of difficulty as the equation problems? Why?

3. If you had to rate your confidence in using graphs to solve problems on a scale of 1 – 6 (1 being

not confident at all and 6 being extremely confident) how would you rate yourself?

4. Do you feel like you’ve mastered reading graphs? If so, what age/grade were you? If not, why

not?

5. When did you first learn how to read graphs (what age/grade)? Was it in math class or another

class? Which class?

6. Have you encountered graphs in non-math classes? If so, what kinds of classes?

7. Would you say your main experience with graphs has been in math classes or other classes? If

other, which classes?

8. Would you say you have more experience with graphs or equations?


8. HUMAN SUBJECTS COMMITTEE PROTOCOL APPROVAL

This research was approved by the University of Illinois Human Subjects Institutional Review Board

under protocol 2001–0489.


9. CURRICULUM VITAE

Marta K. Mielicki

University of Illinois at Chicago

Department of Psychology (M/C 285)

1007 W. Harrison St.

Chicago, IL 60607

Phone: (610) 609-1737

[email protected]

____________________________________________________________________________________

EDUCATION

2017 PhD. in Cognitive Psychology (expected)

2015 M.A. in Cognitive Psychology (expected)

University of Illinois, Chicago; Chicago, IL

Thesis: Affordances of Graphical and Symbolic Representations in Algebraic Problem Solving

2013 M.A. in Experimental Psychology

City University of New York – Brooklyn College; Brooklyn, NY

Thesis: Executive Control in Mathematical Problem Solving: Is There a Bilingual Advantage?

2007 B.S. in Mathematics Education

Boston University School of Education; Boston, MA

____________________________________________________________________________________

RESEARCH INTERESTS

I am interested in mathematical problem solving and the cognitive factors that can influence

performance. I am particularly interested in the development of algebraic understanding,

students’ use of multiple representations in algebraic problem solving, as well as the intersection

of mathematics and language and how linguistic factors influence mathematical problem solving

performance.

____________________________________________________________________________________

GRANTS, HONORS, AND AWARDS

2014 CADRE Fellowship recipient, NSF funded

2013 Travel Award from the International Workshop on Bilingualism and Cognitive Control

____________________________________________________________________________________

PUBLICATIONS & MANUSCRIPTS IN PREPARATION

Mielicki, M. K., & Wiley, J. (in progress). Algebraic problem solving with multiple representations

Mielicki, M., Koppel, R., Valencia, G. & Wiley. J., (in preparation). Negative consequences of oral

administration of letter-number sequencing tasks for bilingual young adults.

Mielicki, M. K., & Wiley, J. (in preparation). Bilingualism and symbolic thought


PRESENTATIONS

Mielicki, M. K., & Wiley, J. (2015, August). Plotting out success: Students can use graphs! Poster to be

presented at the 19th Annual Junior Researchers of EARLI Conference. Limassol, Cyprus.

Cushen, P. J., Jarosz, A. F., Mielicki, M. K. & Wiley, J. (2015, May). Bilingualism and the creative side

of cognition. Paper to be presented at the 10th Annual International Symposium on Bilingualism. New

Brunswick, New Jersey.

Mielicki, M. K., & Wiley, J. (2015, April). Affordances of graphical and symbolic representation in

algebraic problem solving. Poster to be presented at the 87th Annual Meeting of the Midwestern

Psychological Society. Chicago, Illinois.

Mielicki, M. K., & Wiley, J. (2014, November). Obstacles to algebraic understanding. Poster presented

at the 55th Annual Meeting of the Psychonomic Society. Long Beach, California.

Mielicki, M. K., & Wiley, J. (2014, April). Executive control as a predictor of performance on a novel

algebraic problem solving task. Paper presented at the Chicago Psychology Graduate Student Research

Symposium. Chicago, Illinois.

Mielicki, M. K., Ouchikh, Y., & Kacinik, N. (2013, May). Executive Control in Mathematical Problem

Solving: Is There a Bilingual Advantage? Poster presented at the International Workshop for

Bilingualism and Cognitive Control. Krakow, Poland.

____________________________________________________________________________________

RESEARCH PROJECTS AND EXPERIENCE

NSF Funded Project: Improving Formative Assessment Practices, May 2014 – present

Graduate Research Assistant, LSRI, University of Illinois at Chicago

This project is developing a learning trajectory for linear functions in order to inform

instructional practice and professional development for teachers. It is funded by a DRK-12 grant

from NSF (DRK-12 Project: Improving Formative Assessment Practices: Using Learning

Trajectories to Develop Resources that Support Teacher Instructional Practice and Student

Learning in CMP2) to Alison Castro Superfine PI (Susan Goldman, James Pellegrino, and Mara

Martinez Co-PIs). My responsibilities include assessment development and administration, data

collection, development of student interview protocols, and analysis.

Multiple Representation in Algebraic Problem Solving, August 2014 – present

Graduate Research Assistant, Wiley Cognition and Eyetracking Lab, University of Illinois at Chicago

This project is exploring students’ difficulties in using graphical representations for linear

functions and serves as my Masters project. Data collection is complete and defense is expected

this summer, although a follow-up study may be needed for publication.


Individual Differences in Performance on the Symbol Math Task, August 2013 – present

Graduate Research Assistant, Wiley Cognition and Eyetracking Lab, University of Illinois at Chicago

This project explored benefits of bilingualism and executive control on performance a novel

algebraic problem solving task. Data collection and analyses are complete and the manuscript is

the in last stages of preparation for publication.

Cognitive Neuroscience of Language Lab, June 2011 - June 2012

Graduate Research Assistant, Brooklyn College

I assisted Dr. Natalie Kacinik and her doctoral student Rita El-Haddad in a study exploring

dyslexia in first degree relatives. I ran participants, administered subtests from the Woodcock

Johnson III, WRAT 4 and WAIS-III. I also assisted with data entry and contributed to

decisions about scoring ambiguous items.

Social Cognition Lab, February 2011 - May 2011

Research Assistant, Brooklyn College

I assisted Dr. Curtis Hardin and his doctoral student Karla Felix in her study exploring automatic

religious prejudice. I ran participants and assisted with data entry.

Cognitive Development Project, September 2009 - February 2010

Undergraduate Research Assistant, Villanova University

I worked with Dr. Rebecca Brand on several studies exploring effect of “motionese” on infant

behavior. I recruited and ran participants.

Memory and Cognition Lab, October 2009 - February 2010

Undergraduate Research Assistant, Villanova University

Under the supervision of Dr. Thomas Toppino I coordinated and administering studies exploring

the testing effect on memory. I also administered a preliminary study exploring the effect of

environment on recall.

____________________________________________________________________________________

SYNERGISTIC ACTIVITIES

Ad Hoc Reviewer:

The Journal of Problem Solving (2014)

____________________________________________________________________________________

TEACHING EXPERIENCE

Teaching Assistant

Fall 2013 to Spring 2014


Course: Introductory Psychology


Mathematics/Test Preparation Tutor

September 2009 to February 2010

MJ Test Prep (Bryn Mawr, PA) and Mathnasium (Rosemont, PA)

I Tutored students (K-12) in basic math, algebra, geometry, pre-calculus and calculus. I also

prepared students for mathematics and verbal sections of the SAT, PSAT, SSAT, ISEE, and

HSPT.

Teacher of English as a Foreign Language

August 2008 to July 2009

Jazykova Skola Spevacek (Language School), Prague, Czech Republic

I prepared and taught lessons in conversational and business English to students of various ages

and abilities. I also taught Cambridge Exam and IELTS preparation courses.

High School Teacher of Mathematics

August 2007 to June 2008

Brighton High School, Boston, MA

I taught two 10th grade Geometry sections and one 9th Grade algebra section. I prepared original

lesson plans and activities within an established mathematics curriculum. I participated in after-

school tutoring with students 5 hours a week and prepared students for the Massachusetts State

Examinations (MCAS) math section.

____________________________________________________________________________________

UNDERGRADUATE MENTORING

Farrukh Karimov (F12) Executive Control in Mathematical Problem Solving: Is There a

Bilingual Advantage?

Yasmine Ouchikh (S13) Executive Control in Mathematical Problem Solving: Is There a

Bilingual Advantage?

Sonia Soto (S14) Cognitive Factors that Predict Performance on a Novel Algebraic Problem

Solving Task

____________________________________________________________________________________

OTHER ACTIVITIES/SKILLS

Full MS Office proficiency

SPSS data analysis

Working knowledge of ePrime 2.0

Working knowledge of R

Fluent in written and conversational Polish

LiveCode programming experience

TEFL Certification in 2008.


____________________________________________________________________________________

PROFESSIONAL REFERENCES

Jennifer Wiley, Ph. D.

Professor

Department of Psychology


1007 W. Harrison St. (M/C 285)

Chicago, IL 60607

Phone: (312)355-2501

Email: [email protected]

James W. Pellegrino, Ph. D.

Distinguished Professor

Department of Psychology and Education

Co-Director Learning Sciences Research

Institute


1007 W. Harrison St. (M/C 285)

Chicago, IL 60607

Phone: (312)412-2320


Mara Martinez, Ph. D.

Assistant Professor

Mathematics, Statistics, and Computer

Science


322 Science and Engineering Offices

(M/C 249)

851 S Morgan St.

Chicago, IL 60607-7045

Phone: (312) 996-6168


Alison Castro Superfine, Ph. D. Associate Professor, Director Mathematics

Education

Mathematics, Statistics, and Computer

Science


322 Science and Engineering Offices

(M/C 249)

851 S Morgan St.

Chicago, IL 60607-7045

Phone: (312) 413-8029


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