Claire Lenehan - Final Honours Thesis.pdf

IMPLICIT MATH-GENDER STEREOTYPE IN PROCEDURAL AND CONCEPTUAL

FRACTION LEARNERS

BY

CLAIRE A. LENEHAN

A Thesis submitted to the Psychology Department

In partial fulfillment of the

Requirements for the Honours

Degree, Faculty of Arts

Department of Psychology

Memorial University of Newfoundland

April 2014

MATH-GENDER STEREOTYPE i

Abstract

Recent research suggests that students can be classified as either conceptual or procedural

fractions learners, and that these two groups differ on several motivational variables (Bakhtiar,

2013). This follow-up study examined potential relationships between conceptual vs. procedural

learners, motivation, and gender, by introducing an IAT of math-gender stereotype. Contrary to

Bakhtiar (2013), a number of three-way interactions between motivation, cluster membership

and gender were discovered. These findings suggest that there are some differences between

conceptual and procedural learners with regard to motivational variables, but more research must

be conducted to determine the nature and extent of these differences.

Keywords: conceptual and procedural knowledge; academic motivation; math-gender stereotype;

IAT

MATH-GENDER STEREOTYPE ii

Acknowledgements

This project would not have been possible without the unwavering support of my

supervisor, Dr. Darcy Hallett, who provided excellent guidance and assistance. Deep gratitude is

also due to Kyle Morrissey for his patience and mentorship. I thank you both for your time and

commitment. Thank you also to the members of the Research Centre for the Development of

Mathematical Cognition. Finally, sincere thanks to the teachers, and students who participated in

this research and made the study possible.

MATH-GENDER STEREOTYPE iii

Table of Contents

Page

Abstract…………………………………………………………………………………………... i

Acknowledgements……………………………………………………………………………… ii

Table of Contents…………………………………………………………………………………iii

List of Tables….………………………………………………………………………………..... iv

List of Figures……………………………………………………………………………………. v

Introduction……………………………………………………………………………………….. 1

Individual Differences between Conceptual and Procedural Knowledge ……………… 2

Motivational Differences …………………………………………………..……………3

Implicit Gender Associations With Math……………………………………………….. 5

The Current Study………………………………………………………………………..6

Method……………………………………………………………………………………………. 7

Participants…………………………………………………………………………….... 7

Measure…………………………………………………………………………………..7

Procedure……………………………………………………………………………... 10

Results…………………………………………………………………………………………… 12

Discussion..………………………………………………………………………………………32

References………………………………………………………………………………………. 37

Appendices…………………………………….…………………………………………………40

MATH-GENDER STEREOTYPE iv

List of Tables

Page

Table 1. Means and Standard Deviations for Each Scale…………………….……..…………..15

Table 2. Inter-correlation Matrix……………………………...………………….………..……16

Table 3. Language and Math IAT Correlations for Males and Females…………….……….….17

Table 4. Regression Analyses Predicting Conceptual Scores………………….....….………….19

Table 5. Regression Analyses Predicting Conceptual Scores (With IAT)…..………………...…24

Table 6. Regression Analyses Predicting Procedural Scores………………..…………………..26

Table 7. Regression Analyses Predicting Procedural Scores (With IAT)…………….…………30

MATH-GENDER STEREOTYPE v

List of Figures

Page

Figure 1. Cluster solution for residualized conceptual and procedural scores…………………..13

Figure 2. Three-way interaction between verbal self-concept, cluster membership and gender,

with conceptual scores as the dependent variable……………………………………………..…20

Figure 3. Three-way interaction between external attribution for math success, cluster

membership and gender, with conceptual scores as the dependent variable…………………….21

Figure 4. Three-way interaction between performance avoidance orientation, cluster membership

and gender with conceptual scores as the independent variable…………………………………22

Figure 5. Two-way interaction between performance orientation and gender, with conceptual

scores as the dependent variable…………………………………………………………………25

Figure 6. Three-way interaction between verbal self-concept, cluster membership and gender,

with procedural scores as the dependent variable.……………………………………………….27

Figure 7. Three-way interaction between performance avoidance orientation, cluster membership

and gender, with procedural scores as the dependent variable. …………………………………28

Figure 8. Three-way interaction between external attribution of math success, cluster

membership and gender, with procedural scores as the dependent variable..………...…………31

MATH-GENDER STEREOTYPE 1

Implicit Math-Gender Stereotype in Procedural and Conceptual Fraction Learners

Mathematical cognition researchers have described two types of mathematical knowledge:

conceptual and procedural (Hiebert and LeFevre, 1986). Conceptual knowledge refers to

understandings of how pieces of knowledge or entities relate to one another. True conceptual

knowledge requires the ability to see interconnections between separate units of information. This

knowledge is developed through constructing an understanding of the relationships between

different types information (Hiebert & LeFevre, 1986). In contrast, procedural knowledge consists

of knowing how to follow a sequence of actions in order to achieve a goal. In mathematics,

procedural knowledge presents as the ability to follow mathematical algorithms and rules to arrive

at the correct answer to a problem (Hallett, Nunes, & Bryant, 2010). The ability to perform

procedures does not depend on a complete conceptual understanding of a given mathematical

problem, but instead only the steps necessary in order to complete or solve a problem. For

example, a student could memorize the procedural steps required for multiplying fractions, without

necessarily understanding what the fractions represent. As described by Hiebert and LeFevre

(1986), these procedural steps may be executed linearly and independent of meaning.

Early research on conceptual and procedural knowledge of mathematics investigated the

developmental order in which these two types of knowledge emerge, with contradictory findings.

Some researchers have suggested that conceptual knowledge emerges before procedural

knowledge (Byrnes & Wasik, 1991), while others have suggested that children either learn

procedures first (Peck & Jencks, 1981), or that both types of knowledge develop in tandem and

serve to reinforce each other in the process of learning (Rittle-Johnson, Siegler, & Alibali, 1998).

More recent research (Hallett et al., 2010, Hallett, Nunes, Bryant & Thorpe, 2012, Hecht &

Vagi, 2012) has suggested instead that there may be individual differences in how students


combine these two types of mathematical knowledge. In other words, there are some children who

rely more on conceptual knowledge and some children who rely more on procedural knowledge.

Although the existence of these different groups of children may explain some of the earlier

contradictions of previous research, it has also opened up new questions. If there are children who

differ on their relative reliance on conceptual and procedural knowledge, how else do they differ?

Is the conceptual-procedural profile the only thing that differentiates these children from each

other?

The purpose of this paper is to investigate whether more conceptual children differ from

more procedural children on a set of motivational variables and gender associations. Bakhtiar

(2013) has already begun to ask some of these questions in regards to motivational questions, but

her results included some interesting interactions with gender. For this reason, the research

reported here will extend her work by considering how implicit gender associations about math

(i.e., boys are good at math, girls are good at language) may also differ between these groups.

First, it is necessary to more thoroughly review the previous research in this field.

Individual Differences in Conceptual and Procedural Knowledge

Hallett et al. (2010) were the first researchers to investigate individual differences in how

children combine conceptual and procedural knowledge of fractions. Their study investigated the

hypothesis that there are individual differences in the way that children combine conceptual and

procedural knowledge of fractions. The participants in their study, Grade 4 and 5 students,

completed a test of fractions understanding, which included both procedural and conceptual

fractions questions. A cluster analysis was performed using the students’ residualized scores and it

was determined that the children fell into one of 5 clusters – lower procedural, lower conceptual,

higher procedural-lower conceptual, higher conceptual-lower procedural and higher.


Hecht and Vagi’s (2012) follow-up study also tested students in Grade 4 and 5 and had

similar results. They found that a 5-cluster solution was most appropriate for Grade 4 students,

while a 7-cluster solution was best for Grade 5 students. They also discovered that the highest-

performing group was the cluster with higher conceptual and higher procedural scores, followed

by the cluster with higher conceptual and lower procedural scores, the cluster with higher

procedural and lower conceptual scores, and lastly, the cluster with both lower conceptual and

procedural math scores.

Hallett and colleagues (2012) tested students in Grade 6 and 8 using measures of

conceptual and procedural knowledge of fractions, general fraction knowledge, general conceptual

ability and general procedural math ability. Among Grade 6 students, four distinct clusters were

identified: students who were strong in both procedural and conceptual knowledge, students who

were weak in both procedural and conceptual knowledge, students who were stronger in

procedural knowledge, and students who were stronger in conceptual knowledge. This Grade 6

cluster pattern was similar to what was found with Grade 4 and 5 students by both Hallett and his

colleagues (2010) and Hecht and Vagi (2012). The cluster pattern with Grade 8 students, however,

was much different, as only two clusters of Grade 8 students were identified: students who were

stronger in conceptual knowledge and those who were stronger in procedural knowledge. Hallett

and his colleagues were unable to say why Grade 8s exhibited this simpler cluster pattern.

Nevertheless, it provided an opportunity to explore how children who differ in their relative

strength in conceptual versus procedural knowledge may also differ in other ways.

Motivational Differences

In a follow-up to Hallett and colleagues (2012), Bakhtiar (2013) conducted the first study

to examine whether conceptual and procedural learners differ in terms of their academic


motivation. Students in Grade 8 were tested using Hallett and his colleagues’ (2012) measure of

procedural and conceptual fraction knowledge. Students were separated into clusters that were

either stronger in conceptual knowledge or stronger in procedural knowledge, as performed in

Hallett and colleagues (2012), and results were examined in relation to motivation. Motivation was

examined through three constructs that help to form academic motivation – self-concept, self-

attribution and goal orientation.

Self-concept was examined in broad terms as the perception of oneself, as defined by

Shavelson, Hubner, and Stanton (1976). Self-concept is formed through one’s experiences and can

change over time. It can explain and/or predict how a person may act in a given situation. Bakhtiar

(2013) tested both math and verbal self-concept. Bakhtiar (2013) was interested in whether

students perceived the causes for their success and failure in mathematics to be related to ability,

effort, or external factors. Bakhtiar (2013) defined goal orientation as an individual’s purposes or

aims with respect to developing competence at an activity or set of activities. An individual’s

achievement goals can be categorized as either mastery or performance oriented (Ames, 1992).

Bakhtiar (2013) performed regression analyses to determine whether there is a relationship

between the grouping of individuals as either conceptual or procedural learners and motivation.

Several motivational characteristic differences were discovered between the two groups.

Procedural and conceptual learners differed in how they perceive their math ability. The two

groups were also different in the way they attribute their math failure.

Of particular interest to the current study, Bakhtiar (2013) found that the relationships

between motivation and cluster assignment interacted with gender; whether motivation was related

to conceptual and procedural knowledge depended on participant gender. Due to the scope of her

study, the issue of gender was not explored or tested specifically. Bakhtiar called for further


research into how gender, procedural vs. conceptual math orientation, and math-gender stereotype

may influence attitudes towards math fractions performance.

Implicit Gender Associations with Math

In our society, math is associated more strongly with boys than with girls (Cvencek,

Meltzoff, & Greenwald, 2011). This stereotype can sometimes be detected through explicit

measures, such as questionnaires, but individuals do not always accurately report their own

feelings on socially sensitive topics, and are sometimes unaware of their own biases (Greenwald &

Banaji, 1995). To measure these unconscious aspects of stereotyping, implicit math-gender

stereotype is typically measured through the use of a computer-based Implicit Association Test

(IAT; Greenwald, McGhee, & Schwartz, 1998). IATs were designed to measure the strength of

associations between two pairs of concepts. For example, in the current study the associations

between male/female and math/language are examined. Specifically, IATs are designed to measure

how quickly these associations are made in comparison with one another.

The endorsement of math-gender stereotype appears to vary depending on factors such as

age, gender and environment. Nosek, Banaji, and Greenwald (2002) found that both male and

female university students endorsed implicit math-gender stereotypes equally. Cvencek et al.

(2011) tested children between 6 and 10 years of age using measures of both implicit and explicit

math-gender stereotype. They found that even the youngest children demonstrated an endorsement

of math-gender stereotype on both measures. However, Muzzati and Agnoli (2007) found that

explicit gender stereotyping of mathematics was present in Grade 4 students, but disappeared by

Grade 8. The researchers speculated that older children had internalized stereotypes and that the

stereotypes were still present implicitly.


Steffens, Jelenec and Noack (2010) tested students in Grade 4, 7 and 9 and identified

implicit math-gender stereotype in female students, and explicit math-gender stereotype in both

male and female students. They found that the endorsement of math-gender stereotype is related to

poorer math achievement in girls and better math achievement in boys. Additionally, girls’ implicit

stereotypes uniquely predicted math self-concepts (Steffens et al., 2010).

These studies are important to the current study as we expect to find evidence of math-

gender stereotype which may enable further understandings of the relationships between individual

differences in fractions learning, motivation and gender.

The Current Study

Building on Bakhtiar (2013), this study examines potential relationships between

individual differences in fractions knowledge (conceptual vs. procedural learners), motivation and

gender. Specifically, this study will explore the potential effect of math-gender stereotype on these

relationships.

All four measures used in Bakhtiar (2013) are replicated in this current study. These

measures are the Procedural and Conceptual Fraction Measure (Hallett et al., 2010), Math and

Verbal Self-Concept Subscales (Marsh & Shavelson, 1985), Sydney Attribution Scale (Marsh &

Parker, 1984), and Personal Achievement Goal Orientation Subscale (Midgley et al., 2000). The

results of these four measures will contribute to furthering her findings regarding the relationships

between individual differences in fractions learners and motivation.

In addition to these four measures, an additional measure pertaining to implicit math-

gender stereotype is being evaluated as a possible adjunct to this study. This measure is the

Implicit Association Test (IAT) of Math-Gender Stereotype. This test was constructed specifically

for this current study. The IAT is used to as a measure of the degree to which ideas are


unconsciously associated with one another. In the context of math-gender stereotypes, this will be

a measure of how much more difficult it is to match math-related words with girls names versus

boys names, and reading-related words with boys names versus girls names. The purpose of this

test is to gain a better understanding of any potential effects of gender as it relates to motivation

and individual differences in fractions learning.

Method

Participants

The participant sample for this study consisted of 73 Grade 8 students from a single school

in the Newfoundland and Labrador English School District. Seven students were excluded from

the original sample due to incomplete data. The final sample of 66 students (39 males and 27

females) had a mean age of 13.74 years. This grade level was chosen because the procedural and

conceptual measure used in this study is appropriate for clustering Grade 8 students into two

groups distinct groups – more procedural and more conceptual learners (Hallett et al., 2012). This

is also an appropriate age for identifying implicit math-gender stereotype, as this stereotype

appears to peak in early adolescence (Steffens et al., 2010).

Measures

Procedural and Conceptual Fraction Measure (Hallett et al., 2010). This measure was

presented as a written task containing 24 items. The measure was designed to assess students’

procedural and conceptual knowledge of fractions. The conceptual items in this measure assess the

participants’ understanding of equivalency, ability to judge whether one fraction is larger than

another, and knowledge of other characteristics associated with understanding the concepts of

fractions and their relationships. The procedural items require students to solve a problem by


following a series of computational steps they have previously been taught in school (Bakhtiar,

2013).

Implicit Association Test (IAT) of Math-Gender Stereotype. This measure of implicit

math-gender stereotype is a computer-based test that consists of a set of four blocks of trials. In

each trial, a participant is asked to classify a given word as either male or female, or to classify

them as relating to either math or language. Blocks one and two are for practice and include an

error-feedback prompt to help participants learn the rules of the test. The first block includes a set

of eight male names and eight female names presented on the screen one at a time in random

order. Participants are asked to classify them as either male or female using assigned keys on the

keyboard. This was followed by a second block with eight math words and eight language words,

which participants were asked to classify as either Math or Language. In block three, the congruent

trial, participants are randomly shown the 32 stimuli from blocks one and two, and one-by-one

they classify them as either male or female if one of the first block words appears on the screen

and as either Math or Language if one of the second block words appears on the screen. On the left

side of the screen, categories are labeled as “math” and “male”, with “language” and “female” on

the right hand side. This means that if a participant wants to say that a word is “male” or “math”,

he or she would use the assigned key on the left side of the keyboard. If a participant wants to say

that a word is either “female” or “language”, then he or she would use the assigned key on the

right side of the keyboard. For block four, the incongruent trial, the same method as block three

applies; only now the categories “language” and “math” have switched sides so that the left key

indicates is now used to indicate either “male” or “language” while the right key indicates either

“female” or “math”.


In total, there are 64 observations recorded in the experimental trials (i.e., the trials from

Blocks 3 and 4). Incorrect trials are replaced with the block mean plus a 600ms error penalty as

described in Greenwald, Nosek and Banaji (2003). Of these, a difference score is calculated with

reaction times of math and language words in the congruent trial being subtracted from the

incongruent trial. This yields a language word difference score and a math word difference score

for each participant. These difference scores are each divided by their respective participant’s

overall standard deviation of their reaction time. The resulting outcome of the test is a language

and a math IAT Cohen’s d effect size measure for each participant. Outlier treatment is adapted

from Greenwald et al. (2003). Participants are excluded if they respond in less than 300ms for

more than 10% of total responses. In addition, individual reaction times longer than 9.5 seconds

are also excluded. This task includes no further outlier treatment and only one participant was

excluded on this basis.

Math and Verbal Self-Concept Subscales (Marsh & Shavelson, 1985). The math and

verbal self-concept subscales are a paper and pencil measure designed to be appropriate for

students from Grade 7 to Grade 12. The full measure has 102 items measuring 11 different factors

of self-description, but for the purpose of this research, only students’ verbal and math self-

concept were tested. For each of 20 items (10 math and 10 verbal), participants were required to

respond on a six-point Likert scale, with 1 being “False” and 6 being “True.” The items were both

positively and negatively worded, and negatively worded items were reversed scored during data

analysis. A high score on this scale reflected a more positive self-concept, while a low score

reflected a more negative self-concept (Bakhtiar, 2013).

Sydney Attribution Scale (Marsh & Parker, 1984). This written measure evaluated

students’ perception of the causes of their academic success and failure. It included 24 items with


three possible responses for each item. The three responses were designed to reflect causal

attributions that are loaded on ability, effort or some external factor, which meant that there were

three separate subscales for each one of these outcomes: math-failure, math-success, reading-

failure, and reading-success. The scale on this measure ranged from 1 = “False”, to 5 = “True”.

Although the measure includes both reading and math attribution scales, the scores on the reading

scores were not included in the data analyses (Bakhtiar, 2013).

Personal Achievement Goal Orientation Subscale (Midgley et al., 2000). This subscale

was adopted from the original Patterns of Adaptive Learning Scales (PALS). The complete version

of PALS could be broken down into two scales: one for students and the other for teachers. In this

study, the items were rephrased to reflect goals that were specific to the math domain. These tests

were administered in accordance with the most recent test manual provided by the scale

developers (Midgly et al., 2000). A five-point Likert scale was used to collect responses for each

item on the measure, with responses ranging from 1 = “Not at all true” to 5 = “Very true”. The

order of the items was randomized without identification of a specific goal orientation, which

resulted in three separate subscales indicating the level of agreement for each individual with each

goal orientation (Bakhtiar, 2013).

Procedure

A total of 12 junior high schools in the St. John’s Metropolitan Area were contacted by

mail and telephone and invited to participate in this study. Upon approval from the school

principal, teachers were asked to distribute information and consent forms to parents/guardians for

their review and signature. Parents/guardians were assured that their child’s identifying data would

remain anonymous and confidential during reporting and dissemination of study results.


Data was collected in two testing sessions. In the first session, the Procedural and

Conceptual Fraction Measure (Hallett et al., 2010) was administered. This written measure was

administered as a group during a regular class period. Students were told that the researcher was

interested in knowing how comfortable the students were with fractions, and that they therefore

were not permitted to use calculators. Students were reminded also that they would not be graded

on this test, and that they should try to do the best that they could. This first data collection session

took approximately 25 minutes.

For the second session, students were tested in smaller groups of approximately 6 students.

This grouping allowed for ease of movement between the written and computer-based portions of

this testing session. This session consisted of two parts – one written and one computer-based. In

the first written part, students were positioned physically so that they were not sitting in close

proximity to each other. This physical configuration was organized as such because some of the

questions included in this session were sensitive to social comparisons (e.g. seeing and comparing

responses and peer influences). Students were asked to complete all three written motivational

measures: Math and Verbal Self-Concept Subscales (Marsh & Shavelson, 1985), Sydney

Attribution Scale (Marsh & Parker, 1984), and Personal Achievement Goal Orientation Subscale

(Midgley et al., 2000). The order of the three written motivational measures was counterbalanced

across participants, using a Latin-Square ordering, to account for order effects. On these written

measures, students were encouraged to be honest with their answers, and they were reassured that

their responses would remain anonymous and that no questions had right or wrong answers.

In the second part of this session, the IAT of Math-Gender Stereotype (computer-based)

was administered. Students were given verbal instructions and encouraged to ask questions if they

did not understand. Including both the written and computer-based parts, this second data


collection session took approximately 30 minutes. At the end of the second session, each student

participant was given $10 in cash for their participation in this study.

Results

Cluster Analysis

Using a similar procedure as Hallett and his colleagues (2012), students’ conceptual and

procedural scores were regressed against one another, and the standardized residual scores were

saved in each case. These residual scores were generated in order to remove variability due to

overall math performance. Residualized procedural scores reflect the extent to which procedural

performance deviated from what their conceptual performance would predict, rather than simply

overall procedural ability. The same idea applies to residualized conceptual scores. This allowed

the clustering to use scores that reflect participants’ relative strength in conceptual or procedural

knowledge.

A k-means cluster analysis was performed using these residualized scores in order to sort

participants into two clusters. A two-cluster solution was used, as Hallett et al. (2012)

demonstrated that this solution is most appropriate for Grade 8 students, and the solution is

presented in Figure 1. As this figure indicates, the first cluster can be considered the more

conceptual cluster and the second cluster can be considered the more procedural cluster. Of the 66

students included in this study, 45 students (30 males and 15 females) were classified as more

conceptual learners, and 21 students (9 males and 12 females) were classified as more procedural

learners. A goodness of fit chi-square analysis determined that the assignment of clusters was not

even, with twice as many in the conceptual group, χ2(1) = 8.73, p = .0031.


Figure 1. Cluster solution for residualized conceptual and procedural scores. Error bars represent standard error of the mean.

Math-Gender Stereotype Implicit Association Test (IAT)

The purpose of using the IAT in the current study was to measure the existence and the strength of

math-gender stereotype. Students who take significantly longer to respond to Math-Female and

Language- Male associations than Math-Male and Language-Female associations are said to

exhibit math-gender stereotype. In the current study, students took, on average, approximately

277.13ms longer in the incongruent condition when sorting math related words to the same side as

girls’ names, which indicates that these Grade 8 students endorse math-gender stereotype (Cohen’s

d = 0.49). For language and reading words, participants took an average of 157.54ms longer in the

incongruent condition, where participants were required to sort these words in the same column as

boys’ names (Cohen’s d = .28). Given that the congruent condition came first, any practice effects

would be expected to result in faster performance in the incongruent condition, which makes it


notable that participants actually took longer in the incongruent condition.. The longer response

times in the incongruent condition were therefore more likely due to stereotype interference. Male

participants exhibited a somewhat larger stereotype effect on average, however this did not

approach significance for either the math words, nor for the language words. Participants varied in

how much math-gender stereotype caused interference. Students ranged from being about .78 SD

faster for the incongruent condition, to being as much as 1.1 SD slower for the incongruent

condition. This range of scores provides a good basis for later analyses included in this study.

Regression Analyses

Description information about the variables are first presented below. Table 1 presents the

means and standard deviations for all variables. Table 2 presents the zero-order correlations

between the motivational variables and the procedural and conceptual scores. Table 3 presents the

correlations between gender, IAT scales, and motivational variables.

Similar to Bakhtiar (2013), group differences were investigated using relational analyses

rather than mean-difference analyses. It is expected that the various motivational variables would

be related to both conceptual and procedural knowledge. The purpose of the regressions, however,

was to explore whether the relation between the motivational variables interacted with cluster

when predicting either conceptual or procedural knowledge. The regressions described in more

detail below are meant to ask this question.


Table 1. Means and Standard Deviations for Each Scale

Scale N Mean Standard Deviation

1. Total Conceptual (scored out of 24) 66 16.42 4.58 2. Total Procedural (out of 14) 69 5.04 3.53 3. Total Fraction (out of 38) 69 20.75 8.36

4. Math Self-Concept (out of 60) 69 40.07 11.06

5. Verbal Self-Concept (out of 60) 69 45.22 8.32

6. Math Success Ability (out of 30) 69 19.80 5.14 7. Math Success Effort (out of 30) 69 22.01 4.67

8. Math Success External (out of 30) 69 17.91 5.35

9. Math Failure Ability (out of 30) 69 14.46 5.37 10. Math Failure Effort (out of 30) 69 17.30 4.23

11. Math Failure External (out of 30) 69 17.00 3.48

12. Mastery Goal (out of 25) 69 18.71 3.96

13. Performance Approach (out of 25) 69 12.64 4.78 14. Performance Avoidance (out of 20) 69 11.01 4.05

15. Math IAT 69 .49 .56

16. Language IAT 69 .28 .57


Table 2. Inter-correlation Matrix

Notes: * p < .05. ** p <.01. Concept is Conceptual Fractions Score, Proced is Procedural Fraction Score, OFraction is Overall Fraction Score, Math SC is Math Self-Concept, Verbal SC is Verbal Self-Concept, MSuccA is Ability Attribution for Math Success, MSuccEff is Effort Attribution for Math Success, MSuccExt is External Attribution for Math Success, MFailA is Ability Attribution for Math Failure, MFailEff is Effort Attribution for Math Failure, MFailExt is External Attribution for Math Failure, PerApp is Performance Approach, PerAvoid is Performance Avoidance, MathIAT is Math IAT Score and LangIAT is Language IAT Score

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 1. Concept - .62** .93** .43** .09 .31** .11 -.04 -.34** -.08 .19 .06 .11 .23 -.12 .11 2. Proced - - .86** .48** .16* .33** .13 -.23 -.44** -.15 -.37** .10 -.13 -.15 .01 .13

3. OFraction - - - .43** .12* .31** .09 -.30* -.35** -.13 -.36** .17 .02 .04 -.03 .14

4. Math SC - - - - .43** .56** .35** -.27* -.71** -.16 -.33** .28* .18 -.03 .14 .15

5. Verbal SC - - - - - .31** .27** -.03 -.22 .05 -.21 .17 .04 -.03 .13 .09

6. MSuccA - - - - - - .49** -.34** -.38** .12 -.15 .34** .06 .02 .05 .15

7. MSuccEff - - - - - - - -.16 -.09 .08 -.37** .45** -.02 .03 .16 .12

8. MSuccExt - - - - - - - - .22 .07 .43** -.14 .10 .07 -.03 -.10

9. MFailA - - - - - - - - - .46** .24* -.12 -.08 .05 -.24 -.05

10. MFailEff - - - - - - - - - - .17 .06 -.05 .02 -.21 -.22

11. MFailExt - - - - - - - - - - - -.25* .25* .16 -.15 -.22

12. Mastery - - - - - - - - - - - - .20 .16 .23 .26*

13. PerApp - - - - - - - - - - - - - .65** -.05 -.12

14. PerAvoid - - - - - - - - - - - - - - .03 -.21

15. MathIAT - - - - - - - - - - - - - - - .22

16. LangIAT - - - - - - - - - - - - - - - -


Table 3. Language and Math IAT Correlations for Males and Females

Male Female

Language IAT Math IAT Language IAT Math IAT

Concept .13 -.07 .09 -.22

Proced .28 .08 -.09 -.10

Math SC .20 .22 .11 .08

Verbal SC -.02 .14 .24 .12

RSuccA .11 .25 .16 .20

RSuccEff .16 .13 .32 .20

RSuccExt -.34* -.26 -.08 -.06

RFailA -.23 -.29 -.14 -.13

RFailEff -.20 -.416** -.39* -.17

RFailExt .08 .14 -.20 -.27

MSuccA .18 .08 .13 .03

MSuccEff .08 .18 .19 .15

MSuccExt -.31* .08 .08 -.15

MFailA -.09 -.33* -.01 -.13

MFailEff -.25 -.26 -.17 -.10

MFailExt -.19 -.12 -.27 -.20

Mastery .13 .28 .45* .22

PerApp -.10 -.08 -.14 -.01

PerAvoid -.19 .09 -.26 -.05

LangIAT - .19 - .28

Notes: * p < .05. ** p <.01. Concept is Conceptual Fractions Score, Proced is Procedural Fraction Score, Math SC is Math Self-Concept, Verbal SC is Verbal Self-Concept, MSuccA is Ability Attribution for Math Success, MSuccEff is Effort Attribution for Math Success, MSuccExt is External Attribution for Math Success, MFailA is Ability Attribution for Math Failure, MFailEff is Effort Attribution for Math Failure, MFailExt is External Attribution for Math Failure, PerApp is Performance Approach, PerAvoid is Performance Avoidance, MathIAT is Math IAT Score and LangIAT is Language IAT Score


Predicting conceptual scores. Regression analyses were used to examine possible

interactions between motivation, cluster and gender. Each variable was tested for a three-way

interaction. If the three-way interaction was significant, the full model was retained. If the three-

way interaction was not significant, separate regressions were carried out for each possible two-

way interaction. Each regression model was subject to separate outlier treatment, and participants

who had a Mahalanobis distance greater that the chi-square value at p = .001 were eliminated

from that particular analysis. Table 4 reports the final models involving each of the motivational

variables. Three significant three-way interactions were identified between motivational scores,

cluster membership and gender when predicting conceptual scores. The first of these three-way

interactions occurs between verbal self-concept, cluster membership and gender. Verbal self-

concept is a highly significant predictor of conceptual fractions scores for male students in the

procedural cluster. This interaction is shown in Figure 2.

Another significant three-way interaction occurs between external attribution of math

success, cluster membership and gender. For those in the procedural cluster, external attribution

of math success is negatively associated with conceptual fractions scores for males and

somewhat positively for females. For those in the conceptual cluster, there is no relation between

external attribution of math success and conceptual fraction scores. This interaction is shown in

Figure 3.

In addition, a significant three-way interaction was identified between performance

avoidance orientation, cluster membership and gender. Performance avoidance orientation is

positively associated with conceptual math scores in conceptual females and, to a lesser extent,

in procedural males. Performance avoidance orientation is negatively associated with conceptual

math scores in procedural females. This three-way interaction is depicted in Figure 4.


Table 4: Regression Analyses Predicting Conceptual Scores

*p<.05, **p<.01 a. Participants 10 and 67 excluded as outliers b. Participants 13 and 28 excluded as outliers c. Participants 10 and 13 excluded as outliers

Mot β

Cluster β

Gender β

Mot x Cluster β

Cluster x Gender

β

Mot x Gender

β

Mot x Cluster x Gender

β Motivation (Mot) Scale Math Self-Concept .533** .481** -.085 - - - -

Verbal Self-Concept a 1.082* .638** .413 -.890 -.350 -1.781* 1.711*

Ability Attribution for Math Success .349** .398** -.022 - - - -

Effort Attribution for Math Success .180 .392** .016 - - - -

External Attribution for Math Success b .196 .251 .093 -.150 -.025 -1.007** .889*

Ability Attribution for Math Failure -.439** .464** -.030 - - - -

Effort Attribution for Math Failure -.129 .369** .053 - - - -

External Attribution for Math Failure -.242* .399** .023 - - - -

Mastery Orientation .085 .366** .169 - - - -

Performance Approach Orientation .033 .348** .044 - - - -

Performance Avoidance Orientation c -.316 .374 .123 .715* .028 .632 -.894*


Figure 2. Three-way interaction between verbal self-concept, cluster membership and gender, with conceptual scores as the dependent variable.

0"

5"

10"

15"

20"

25"

10" 20" 30" 40" 50" 60"

Conceptual+Scores+

Verbal+Self+Concept+

Male+

Conceptual"Group"Conceptual"Group"Procedural"Group"Procedural"Group"

0"

5"

10"

15"

20"

25"

10" 20" 30" 40" 50" 60"

Conceptual+Scores+

Verbal+Self+Concept+

Female+Conceptual"Group"Conceptual"Group"Procedural"Group"Procedural"Group"


Figure 4. Three-way interaction between performance avoidance orientation, cluster membership and gender with conceptual scores as the independent variable.

0"

5"

10"

15"

20"

25"

0" 5" 10" 15" 20" 25"

Conceptual+Scores+

Performance+Avoidance+Orientation+

Male+

Conceptual"Group"Conceptual"Group"Procedural"Group"Procedural"Group"

0"

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10"

15"

20"

25"

0" 5" 10" 15" 20" 25"

Conceptual+Scores+

Performance+Avoidance+Orientation+

Female+Conceptual"Group"Conceptual"Group"Procedural"Group"Procedural"Group"


IAT influence in predicting conceptual scores. The previous analyses mirror what was

done by Bakhtiar (2013) to analyze the effects of gender and cluster in the relation between the

motivation variables and conceptual scores. For the next set of analyses, Math and Language

IAT were added to the regressions to see if they changed the results of the regressions. Table 5

illustrates the interactions that occur when the math and language IAT entered into the regression

with the main effects, before the interactions. All three significant three-way interactions found

in the previous regression retain their significance with the introduction of the IAT scores.

Higher Math IAT scores appear to predict lower conceptual math scores when included with the

main effect of ability attribution for math failure. As depicted in Figure 5, a significant two-way

interaction between performance approach orientation and gender also occurs.


Table 5: Regression Analyses Predicting Conceptual Scores (With IAT)

*p<.05, **p<.01 a. Participants 10 and 67 excluded as outliers b. Participant 13 excluded as an outlier c. Participant 46 excluded as an outlier d. Participants 10 and 13 excluded as outliers

Mot β

Cluster β

Gender β

Math IAT β

Language IAT β

Mot x Cluster β

Cluster x Gender

β

Mot x Gender

β


β Motivation (Mot) Scale Math Self-Concept .547** .479** -.096 -.192 .872 - - - -

Verbal Self-Concept a 1.001* -.615** .354 -.092 .154 -.815 -.285 -1.668* 1.616*

Ability Attribution for Math Success .336** .396** -.027 -.138 .108 - - - -

Effort Attribution for Math Success .186 .395** .005 -.158 .149 - - - -

External Attribution for Math Success b .211 .241 .033 -.104 .130 -.160 .042 -939** .869*

Ability Attribution for Math Failure -.491** .474** -.050 -.250* .155 - - - -

Effort Attribution for Math Failure -.131 .371** .043 -.153 .149 - - - -

External Attribution for Math Failure -.242* .398** .015 -.153 .124 - - - -

Mastery Orientation .079 .369** .013 -.146 .152 - - - -

Performance Approach Orientation c .365 -.242 .041 -.142 .168 - - -.391* -

Performance Avoidance Orientation d .570* -.374 .151 -.104 .213 .715* -.028 -.632 -.894*


Figure 5. Two-way interaction between performance orientation and gender, with conceptual scores as the dependent variable.

Predicting procedural scores. Table 6 illustrates the interactions between motivational

scores, cluster membership and gender when predicting procedural fractions scores. As shown in

Figure 6, a significant three-way interaction was found between verbal self-concept, cluster

membership and gender. For females in the procedural cluster, verbal self-concept is a highly

significant predictor of procedural fractions scores in a positive direction while procedural males

have a negative relation between these two variables. In Figure 7, the significant three-way

interaction between performance avoidance orientation, cluster membership and gender is

apparent. Performance avoidance was negatively associated with math scores in procedural

females, but positively associated in conceptual females. Among males, however, this pattern is

reversed..

0

5

10

15

20

25

0 5 10 15 20 25

Con

cept

ual S

core

se

Performance Approach Orientation

Boys Girls


Table 6: Regression Analyses Predicting Procedural Scores

*p<.05, **p<.01 a. Participants 10 and 67 excluded as outliers b. Participants 10 and 13 excluded as outliers

Mot β

Cluster β

Gender β

Mot x Cluster β

Cluster x Gender

β

Mot x Gender

β


β Motivation (Mot) Scale Math Self-Concept .498** -.235* -.129 - - - -

Verbal Self-Concept a 1.491** -.072 .295 -1.374** -.186 -1.671* 1.667*

Ability Attribution for Math Success .339** -.311** -.073 - - - -

Effort Attribution for Math Success .100 -.333** -.024 - - - -

External Attribution for Math Success -.086 -.342** -.005 - - - -

Ability Attribution for Math Failure -.445** -.242* -.083 - - - -

Effort Attribution for Math Failure -.095 -.343** -.001 - - - -

External Attribution for Math Failure -.268* -.305* -.031 - - - -

Mastery Orientation .006 -.353** -.009 - - - -

Performance Approach Orientation

-.084 -.338** -.004 - - - -

Performance Avoidance Orientation b -.482 .348 -.009 .676 .181 .489 -.834*


Figure 6. Three-way interaction between verbal self-concept, cluster membership and gender, with procedural scores as the dependent variable.

0"

2"

4"

6"

8"

10"

12"

10" 20" 30" 40" 50" 60"

Procedural*Scores*

Verbal*Self*Concept*

Male*Conceptual"Group"Conceptual"Group"Procedural"Group"Procedural"Group"

0"2"4"6"8"10"12"

0" 10" 20" 30" 40" 50" 60"

Procedural*Scores*

Verbal*Self*Concept*

Female*Conceptual"Group"Conceptual"Group"Procedural"Group"Procedural"Group"


Figure 7. Three-way interaction between performance avoidance orientation, cluster membership and gender, with procedural scores as the dependent variable.

0"2"4"6"8"10"12"

0" 5" 10" 15" 20" 25"

Procedural*Scores*

Performance*Avoidance*Orientation*


0"

2"

4"

6"

8"

10"

12"

0" 5" 10" 15" 20" 25"

Procedural*Scores*

Performance*Avoidance*Orientation*

Female* Conceptual"Group"Conceptual"Group"Procedural"Group"Procedural"Group"


IAT influence in predicting procedural scores. Table 7 depicts the interactions when

math and language IAT scores are included in the regression analyses. The three-way interaction

between performance avoidance orientation, cluster membership and gender loses its

significance when IAT scores are introduced. However, a significant three-way interaction

between external attribution for math success, cluster membership and gender becomes

significant when IAT scores are included in the analyses. As shown in Figure 8, external

attribution of math success is negatively associated with procedural fractions scores for males

and somewhat positively correlated for females.


Table 7: Regression Analyses Predicting Procedural Scores (With IAT)

*p<.05, **p<.01 a. Participants 10 and 67 excluded as outliers b. Participant 13 excluded as an outlier

Mot β

Cluster β

Gender β

Math IAT β

Language IAT β

Mot x Cluster β

Cluster x Gender

β

Mot x Gender

β


β Motivation (Mot) Scale Math Self-Concept .510** -.238* -.136 -.117 .034 - - - -

Verbal Self-Concept a 1.452 -.082 .254 .000 .113 -1.348** -.458 -2.326* 2.301*

Ability Attribution for Math Success .334** -.313* -.075 -.066 .048 - - - -

Effort Attribution for Math Success .098 -.330* -.029 -.074 .098 - - - -

External Attribution for Math Success b .365 -.633** -.273 -.012 .104 -.300 .394 -.890* .817*

Ability Attribution for Math Failure -.483** -.237* -.097 -.176 .095 - - - -

Effort Attribution for Math Failure -.092 -.340** -.007 -.075 .095 - - - -

External Attribution for Math Failure -.266* -.306* -.035 -.084 .060 - - - -

Mastery Orientation -.010 -.350** -.011 -.058 .111 - - - -

Performance Approach Orientation -.075 -.335* -.010 -.061 .101 - - - -

Performance Avoidance Orientation -.050 -.339** -.011 -.055 .098 - - - -


Figure 8. Three-way interaction between external attribution of math success, cluster membership and gender, with procedural scores as the dependent variable.

0"2"4"6"8"10"12"

0" 10" 20" 30" 40" 50" 60"

Procedural*Scores*

External*Attribution*of*Math*Success*


0"

2"

4"

6"

8"

10"

12"

0" 10" 20" 30" 40" 50" 60"

Procedural*Scores*

External*Attribution*of*Math*Success*

Female*Conceptual"Group"Conceptual"Group"Procedural"Group"Procedural"Group"


Discussion

As the data suggest, the introduction of the math-gender stereotype IAT provides some

insight into the relationships between individual differences in fractions learning, motivation and

gender. When participants are separated by gender, IAT scores are significant predictors of

several motivational variables, as shown in Table 3. In male students, Language IAT

significantly predicts external attribution of reading and math success in boys, but it is the Math

IAT that predicts ability and effort for reading and math success. For female students, Language

IAT predicts effort attribution for reading failure. Language IAT is also a significant predictor of

mastery for female students, while Math IAT is a better predictor of mastery than Language IAT

for male students.

The math and language IAT scores seemed to predict attribution of both reading and

math success, as well as attribution of math ability, but this prediction interacted somewhat with

gender. These data exhibited a trend that suggests the IAT measure’s utility is not only related to

the strength of implicit stereotypical beliefs, but whether or not those beliefs apply to the gender

of the person who holds them. These correlations between the results of the IAT and

motivational variables suggest that the IAT is a valuable tool for gaining an understanding of

math-gender stereotype and how it relates to different types of math learning and academic

motivation. In general, the results suggest that we must take the gender of the participant into

account when we are investigating the influence of math-gender implicit associations.

When examining how cluster and gender may interact with motivation variables in

predicting conceptual and procedural knowledge, the results are more mixed. Bakhtiar (2013)

found that procedural and conceptual learners differ based on several, but not all, motivational

factors investigated. However in this study the interactions of motivation, procedural/conceptual


cluster and gender, as predictors of procedural and conceptual scores, were not the same factors

identified by Bakhtiar. Contrary to Bakhtiar’s findings that conceptual and procedural learners

differ in terms of their math self-concept and how they attribute math failure, the current study

determined that the two groups of learners differ in their verbal self-concept and how they

attribute math success. While Bakhtiar identified a number of two-way interactions between

cluster membership and gender, those findings were not replicated in the current study. Instead,

in this study, three-way interactions occurred between motivation and gender, and motivation

and cluster membership. Additionally, this study revealed a number of three-way interactions

between motivation, cluster membership and gender, whereas only one three-way interaction

between these variables was noted in Bakhtiar’s (2013) study. Bakhtiar’s study revealed a three-

way interaction between math self-concept, cluster membership and gender, but this interaction

was not present in the current study. Instead, when predicting conceptual fractions scores three

three-way interactions occurred: verbal self-concept, cluster membership and gender; external

attribution for math success, cluster membership and gender; and performance avoidance, cluster

membership and gender. When predicting procedural fractions scores, three-way interactions

occurred between verbal self-concept, cluster membership and gender, and performance

avoidance, cluster membership and gender. Additionally, when math and language IAT scores

are entered into the regression, the three-way interaction between external attribution for math

success, cluster membership and gender achieves significance.

There are several potential explanations for the discrepancies between the findings of the

current study and those of Bakhtiar (2013). The sample used in the current study was slightly

larger than that of Bakhtiar. The current study included 66 participants (39 males and 27

females), while Bakhtiar included 58 participants (33 males and 25 females). While both studies


involved Grade 8 students from the Newfoundland and Labrador English School District, the

current sample is more homogenous, as all students were from the same school. In particular, the

students in the current sample demonstrated much higher conceptual scores than the participants

in Bakhtiar’s study. Most boys in the current study (30 out of 39) were classified as conceptual

learners, while the students in Bakhtiar’s were split nearly evenly between the conceptual and

procedural groups. Because all the participants in the current study attend the same junior high

school, this phenomenon could be due to factors such as specific teaching methods used at that

school or at its feeder elementary school.

Although the current study had a larger sample size than Bakhtiar (2013), the difference

in group composition could be a factor in the discrepant findings. Whereas each of Bakhtiar’s

four groups (male conceptual, male procedural, female conceptual, female procedural) included

between 12 and 17 participants, the same groups in the current study had between 9 and 30

participants respectively. To account for the possibility of disparities in group sizes, it will be

important in further research to utilize an overall larger participant sample. A larger sample

would provide for more consistency and generalizability of results.

The results of the current study in comparison with those of Bakhtiar (2013) demonstrate

the possible influence of socioeconomic factors on the findings. Bakhtiar’s sample mostly came

from one school that is known to have low SES students (Bakhtiar, personal communication),

while the sample reported here came from a higher SES school. Further research could include

the study of relationships between socioeconomic factors and individual differences in fractions

learning, academic motivation and/or math-gender stereotype.

Despite these contradictory and sometimes confusing results, the data below offer some

qualified support for the main hypothesis of this paper – conceptual and procedural learners do


differ on motivational variables. The analysis here suggests that conceptual learners and

procedural learners do vary in the strength and direction of the relation between motivational

variables and conceptual and procedural knowledge, even if this is only true for one gender and

not the other.

Finally, it is worth noting that one of the outcomes of the current study is the

development of an IAT that seems to capture some aspects of implicit math-gender stereotypes.

Math-gender stereotype was found in both male and female students. This result is not

unexpected, as similar results were found by Nosek et al. (2002), who tested adults, and Cvencek

et al. (2011), whose sample consisted of 6- to 10-year-old children. However, in contrast to the

findings of this study, Steffens et al. (2010) found that German girls in grades 4, 7 and 9 endorse

math-gender stereotype, but that boys do not. This may suggest that there may be cultural

differences in the development of math-gender stereotype. As an aside, Cvencek et al. (2011)

suggested that math-gender stereotype develops very early and influences the emergence of math

self-concept, but the current study did not find any direct correlation between math-gender

stereotype and our measure of math self-concept.

In conclusion, this study offers tentative and qualified support for the contention that

conceptual and procedural learners differ in terms of various motivational variables. Although

differences were found on some of these variables, these relations always interacted with gender,

and were not consistent with the findings of Bakhtiar (2013). Adding the IAT to the regressions,

which had the effect of controlling for implicit math-gender stereotype, resulted in some

additional effects, but only a few, and gender continued to be involved in the interactions.

Overall, the data suggest that there are some differences between conceptual and procedural


learners with regard to motivational variables, but more research must be conducted to determine

the nature and extent of these differences.


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Appendices

Appendix A

Procedural and Conceptual Fraction Measure Sample

Example of a conceptual item:

1. Shade in two-thirds of each of these shapes:

(a) (b) (c)

Example of a procedural item:

2. Please solve this and show your workings:

€

47

+914

=


Appendix B

Self-Concept Measure Sample

Likert scale used for all items on the Math and Verbal Self-Concept scales:

Example of Math Self-Concept question:

1. I have always done well in Mathematics Example of Verbal Self-Concept question: 2. I do badly on test that needs a lot or reading ability

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Appendix C

Sydney Attribution Scale Sample

Example question for math failure outcome:

Example question for math success outcome:

1. Suppose you get a maths question wrong in class. It is probably because:

False

Mostly False

Some-times False, Some-times True

Mostly True

True

a) you often have trouble in maths ............................... ! ! ! ! !

b) the question was hard ............................................... ! ! ! ! !

c) you never pay any attention in maths lessons……………………………………

! ! ! ! !

2. Suppose you are chosen from your school to take part in a state maths competition. This is probably because:

a) you will try your best .............................................. ! ! ! ! !

b) you were lucky ........................................................ ! ! ! ! !

c) you are good at maths ............................................. ! ! ! ! !


Appendix D

Personal Achievement Goal Orientation Sample

Num. Items Not at all True

Somewhat True

Very True

Mastery-Oriented 1. Learning a lot of new things is what is

important to me in math. 1 2 3 4 5

2. One of my main goals in math is to improve my skills.

1 2 3 4 5

3. My main goal in math is to learn as much as I can.

1 2 3 4 5

4. Really understanding my math work is important to me.

1 2 3 4 5

5. Learning new skills in math is one of my goals. 1 2 3 4 5 Performance-Approach 1. In math, doing better than other students is

important to me. 1 2 3 4 5

2. My goal in math is to look smarter than other students.

1 2 3 4 5

3. One of my goals is to show others that math is easy for me.

1 2 3 4 5

4. It is important to me that others think I am good at math.

1 2 3 4 5

5. My goal in math is to do better than other students.

1 2 3 4 5

Performance-Avoidant 1. It is important to me that I don’t look stupid in

math class. 1 2 3 4 5

2. One of my goals is to keep others from thinking I am not smart in math.

1 2 3 4 5

3. I do my math work so that my teacher doesn’t think I know less than others.

1 2 3 4 5

4. My goal in math is to avoid looking like I can’t do my work.

1 2 3 4 5


Appendix E

Implicit Association Test of Math-Gender Stereotype Word List

Male Female Math Language

Jessica Joshua Literature Equation

Madison Matthew Reading Addition

Ashley Jacob Story Subtraction

Megan Ryan Poem Numbers

Olivia Jessica Books Division

Emily Ethan Writing Multiplication

Sarah Michael Grammar Fractions

Hannah Benjamin Spelling Decimals

Claire Lenehan - Final Honours Thesis.pdf

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