children’s mathematics anxiety and its effect on …

CHILDREN’S MATHEMATICS ANXIETY AND ITS EFFECT ON THEIR

CONCEPTUAL UNDERSTANDING OF ARITHMETIC AND THEIR ARITHMETIC

FLUENCY

A Thesis

Submitted to the Faculty of Graduate Studies and Research

In Partial Fulfillment of the Requirements

For the Degree of

Master of Arts

in

Experimental and Applied Psychology

University of Regina

By

Jill Alexandra Beatrice Price

Regina, Saskatchewan

August, 2015

© Copyright 2015: J. A. B. Price

UNIVERSITY OF REGINA

FACULTY OF GRADUATE STUDIES AND RESEARCH

SUPERVISORY AND EXAMINING COMMITTEE

Jill Alexandra Beatrice Price, candidate for the degree of Master of Arts in Experimental & Applied Psychology, has presented a thesis titled, Children’s Mathematics Anxiety and Its Effect on Their Conceptual Understanding of Arithmetic and Their Arithmetic Fluency, in an oral examination held on August 26, 2015. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material. External Examiner: Dr. Kathleen T. Nolan, Faculty of Education

Supervisor: Dr. Katherine Arbuthnott, Deparment of Psychology

Committee Member: Dr. Jeff Loucks, Department of Psychology

Committee Member: Dr. Tom Phenix, Department of Psychology

Chair of Defense: Dr. Martin Beech, Department of Physics

i

Abstract

Research shows that children’s mathematics anxiety negatively impacts their

performance on mathematics tests (e.g., Chernoff & Stone, 2012). However, no research

to my knowledge has investigated how children’s mathematics anxiety impacts their

conceptual understanding of arithmetic. The current study investigated the characteristics

and development of Grades 4, 5, and 6 children’s mathematics anxiety and how it

impacts their conceptual understanding and application of arithmetic using arithmetic

concepts. For comparison, the current study also investigated how children’s

mathematics anxiety impacts their arithmetic fluency using timed mathematics tests. As

an exploratory and secondary component, the current study examined teachers’

mathematics anxiety and how it impacts their students’ mathematics anxiety, conceptual

understanding of arithmetic, and arithmetic fluency. Results showed that children with

higher mathematics anxiety had weaker conceptual understanding of arithmetic and

weaker arithmetic fluency compared to children with lower mathematics anxiety. In

particular, children’s mathematics anxiety had a greater impact on their arithmetic

fluency compared to their conceptual understanding of arithmetic. It also showed that

females had higher mathematics anxiety compared to males. However, there was no

significant effect of grade on children’s mathematics anxiety. The exploratory

component showed that teachers’ mathematics anxiety did not impact their students’

mathematics anxiety, conceptual understanding of arithmetic, or arithmetic fluency

compared to teachers with lower mathematics anxiety. Grade 4 teachers had higher

mathematics anxiety compared to Grade 5 teachers. However, there was no significant

difference between Grades 5 and 6 teachers’ mathematics anxiety. Female teachers also

ii

had higher mathematics anxiety compared to male teachers. Lastly, students were

unaware of their teachers’ level of mathematics anxiety. To help improve children’s

performance on mathematics tests, future research should focus on early identification

and corresponding interventions for children and teachers with mathematics anxiety.

iii

Acknowledgements

I would like to thank all organizations that made the completion of this thesis possible. I

would like to thank the Government of Saskatchewan and the University of Regina for

supporting me in the form of the Saskatchewan Innovation Scholarship in the Fall 2014

and Winter 2015 semesters, the Faculty of Graduate Studies and Research for supporting

me in the form of the Graduate Students’ Association Scholarship and the Graduate

Teaching Assistantship for the Fall 2014 semester as well as the Graduate Student Travel

Awards in the Spring/Summer 2014 semester, and the Psychology Association of

Saskatchewan for supporting me in the form of the Psychology Association of

Saskatchewan Student Scholarship in the Fall 2014 semester. Thank you for your

generous contributions!

iv

Dedication

I would like to express my sincere gratitude to everyone who contributed to the

completion of my master’s thesis. In particular, I would like to thank the mentorship of

my academic supervisor, Dr. Katherine Robinson. Thank you for all your guidance. I

would also like to acknowledge my grandmother, Mildred Price, who passed away one

year ago. She taught me patience, gratitude, and determination; all of which helped me

tremendously throughout the completion my thesis. Most importantly, she taught me the

value and privilege of education. I dedicate this body of work in honour of her.

v

Table of Contents

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

Acknowledgements………………………………………………………………………iii

Dedication………………………………………………………………………………...iv

Table of Contents………………………………………………………………………….v

List of Tables……………………………………………………………………………viii

List of Figures………………………………………………………………………..……x

List of Appendices………………………………………………………………………xiv

1. CHAPTER ONE: Introduction…………………………………………………………1

1.1 Overview…………………………………………………………………………...1

1.2 Mathematics Anxiety………………………………………………………………1

1.2.1 The onset of mathematics anxiety…………………………………………….1

1.2.2 Consequences of mathematics anxiety………………………………………..2

1.2.3 Causes of mathematics anxiety……………………………………………….3

1.3 Children’s Mathematics Anxiety and Performance on Mathematics Tests………..4

1.3.1 Children’s mathematics anxiety and conceptual understanding of arithmetic..6

1.3.2 Children’s mathematics anxiety and arithmetic fluency…………………….13

1.3.3 Grade differences in children’s mathematics anxiety ...…………………….15

1.3.4 Sex differences in children’s mathematics anxiety.…………………………16

1.4 Exploratory Analysis of Teachers’ Mathematics Anxiety………………………..17

2. CHAPTER TWO: Method…………………………………………………………….19

2.1 Participants………………………………………………………………………..19

2.2 Materials and Procedure…………………………………………………………..19

vi

3. CHAPTER THREE: Results…………………………………………………………..23

3.1 Children’s Performance on Mathematics Tests.………………………………….23

3.1.1 Children’s conceptual understanding of arithmetic………………………….23

3.1.1.1 Accuracy………………………………………………………………..23

3.1.1.2 Reaction time…………………………………………………………...29

3.1.1.3 Strategy use……………………………………………………………..34

3.1.2 Children’s arithmetic fluency………………………………………………..38

3.1.2.1 Number of problems attempted……………………………………...…38

3.1.2.2 Accuracy on the number of problems attempted……………………….41

3.2 Children’s Mathematics Anxiety…………………………………………………43

3.3 Children’s Mathematics Anxiety and its Effect on their Performance on

Mathematics Tests...…………………………………………………………………..45

3.3.1 Children’s mathematics anxiety and conceptual understanding of

arithmetic…………………………………………………………………………..45

3.3.2 Children’s mathematics anxiety and arithmetic fluency…………………….47


4. CHAPTER FOUR: Discussion………………………………………………………..55

4.1 Children’s Mathematics Anxiety and Performance on Mathematics Tests……....55

4.1.1 Characteristics and development of children’s mathematics anxiety………..55

4.1.1.1 Grade differences in children’s mathematics anxiety ………………….56

4.1.1.2 Sex differences in children’s mathematics anxiety …………………….58

4.1.2 Children’s performance on mathematics tests...……………………………..60

4.1.2.1 Children’s conceptual understanding of arithmetic…………………….60

vii

4.1.2.2 Children’s arithmetic fluency…………………………………………..63

4.1.3 Summary…………………………………………………………………….65


4.2.1 Teachers’ mathematics anxiety and its effect on their students’ mathematics

anxiety……………………………………………………………………………..65

4.2.2 Teachers’ mathematics anxiety and its effect on their students’ performance

on mathematics tests...……………………………………………………………..66

4.2.3 Grade differences in teachers’ mathematics anxiety..……………………….67

4.2.4 Sex differences in teachers’ mathematics anxiety.…..………………………67

4.3 Future Research…………………………………………………………….……..68

4.4 Limitations……………………………………...………………………………...69

4.5 Conclusion……………………………………………….………………………..69

5. References……………………………………………………………………………..71

6. Appendices………..…………………………………………………………………...85

viii

List of Tables

Table 1. Children’s Conceptual Understanding of Arithmetic on the Problem-Solving

Task by Problem Type (Identity, Negation, Commutativity, Inversion, Associativity, and

Equivalence) and Sex (Male and Female)……………………………..………………...24

Table 2. Children’s Conceptual Understanding of Arithmetic on the Problem-Solving

Task by Problem Type (Identity, Negation, Commutativity, Inversion, Associativity, and

Equivalence) and Operation Type (Additive and Multiplicative)………………………24

Table 3. Children’s Conceptually-based Strategy Use (%) on the Problem-Solving Task

by Problem Type (Identity, Negation, Commutativity, Inversion, Associativity, and

Equivalence), Operation Type (Additive and Multiplicative), and Grade (4, 5, and

6)………………………………………………………………………………..……......37

Table 4. Descriptive Statistics of Children’s Mathematics Anxiety on the MASC……...44

Table 5. Correlation Matrix Showing the Relationships Between Children’s Mathematics

Anxiety and their Mathematics Performance on the Problem-Solving Task and Timed

Mathematics Task………………………………………………………………………..46

Table 6. Descriptive Statistics of Teachers’ Mathematics Anxiety on the A-MARS……49

Table 7. Correlation Matrix Showing the Relationships Between Teachers’ Mathematics

Anxiety and their Students’ Mathematics Anxiety and their Students’ Mathematics

Performance on the Problem-Solving Task and Timed Mathematics Task……………..50

ix

List of Figures

Figure 1. Children’s Conceptual Understanding of Arithmetic on the Problem-Solving

Task by Operation Type (Additive and Multiplicative)…………………………………25

Figure 2. Children’s accuracy and strategy use on the problem-solving task for sex (male

and female)..……………………………………………………………………….……..26

Figure 3. Children’s accuracy on the problem-solving task for problem-type (identity,

negation, commutativity, inversion, associativity, and equivalence) and sex (male and

female).…..………………………………………………………………………………27

Figure 4. Children’s accuracy on the problem-solving task for operation type (additive

and multiplicative) and sex (male and female)..…………………………………………28

Figure 5. Children’s accuracy on the problem-solving task for problem type (identity,

negation, commutativity, inversion, associativity, and equivalence) and operation type

(additive and multiplicative).…………………………………………………………….29

Figure 6. Children’s reaction time on the problem-solving task for problem type (identity,


(additive and multiplicative).……………………………………………………….……31

Figure 7. Grade 4 children’s reaction time on the problem-solving task for problem type

(identity, negation, commutativity, inversion, associativity, and equivalence) and sex

(male and female).…...…………………………………………………………………..32



(male and female).……………………………………………………………………….33

x



(male and female)………………………………………………………………………..34

Figure 10. Children’s strategy use on the problem-solving task for problem type (identity,


(additive and multiplicative).…………………………………………………………….36

Figure 11. The number of problems children attempted on the timed mathematics task for

operation type (addition, subtraction, multiplication, and division) and sex (male and

female).…………………………………………………………………………………..39

Figure 12. The number of problems children attempted on the timed mathematics task for

operation type (addition, subtraction, multiplication, and division) and grade (4, 5, and

6)…………………………………………………………………………………………40

Figure 13. Children’s arithmetic fluency on the timed mathematics task for sex (male and

female)…………………………………………………………………………………...41

Figure 14. Children’s accuracy on the number of problems attempted on the timed

mathematics task for operation type (addition, subtraction, multiplication, and division)

and sex (male and female)……………………………………………………………….42

Figure 15. Children’s mathematics anxiety score on the MASC for sex (male and

female)…………………………………………………………………………………...45

Figure 16. Teachers’ mathematics anxiety score on the A-MARS for grade (4, 5, and

6)…………………………………………………………………………………………53

Figure 17. Teachers’ mathematics anxiety score on the A-MARS for sex (male and

female)…………………………………………………………………………………...54

xi

List of Appendices

Appendix A. University of Regina Research Ethics Board Certificate of Approval…….85

Appendix A. Addition Arithmetic Concept Problems…………………………………...86

Appendix B. Multiplicative Arithmetic Concept Problems…….…..………………..…..88

Appendix C. Addition Timed Mathematics Test……………………………………..….90

Appendix D. Subtraction Timed Mathematics Test...………………………………..….91

Appendix E. Multiplication Timed Mathematics Test………………………………..…92

Appendix F. Division Timed Mathematics Test…………………………….………..….93

Appendix G. Mathematics Anxiety Scale for Children (MASC) Questionnaire……..….94

Appendix H. Abbreviated Mathematics Anxiety Rating Scale (A-MARS)

Questionnaire………………………………………………………………………….....95

1

CHAPTER ONE

Introduction

1.1 Overview

Mathematics is fundamental in both academic and professional success (Chernoff

& Stone, 2012; Maloney, Waechter, Riskoc, & Fugelsang, 2012; Wu, Barth, Amin,

Mclarne, & Menon, 2012). Mathematics success depends not only on cognitive abilities

but also on positive attitudes (Dowker, Bennett, & Smith, 2012). Mathematics anxiety

inhibits cognitive abilities and creates negative attitudes about mathematics often

resulting in negative long-term academic and professional consequences (Beilock, 2008;

Geist, 2010; Hembree, 1990; Preston, 2008; Wu et al., 2012). Mathematics anxiety is

defined as “feelings of tension and anxiety that interfere with the manipulation of

numbers and the solving of mathematical problems in a wide variety of ordinary life and

academic situations” (Richardson & Suinn, 1972, p. 551). It can also create fear,

helplessness, shame, and mental disorganization in individuals when faced with

mathematical challenges (Gresham, 2007; Wood, 1988). It is thus important to

understand the characteristics and development of children’s mathematics anxiety for

early identification and corresponding interventions (Ramirez, Gunderson, Levine, &

Beilock, 2013; Scarpello, 2007; Wu et al., 2012).

1.2 Mathematics Anxiety

1.2.1 The onset of mathematics anxiety. Anxiety disorders are the most

common mental disorders in children (Beesdo, Knappe, & Pine, 2009). For instance,

generalized anxiety disorder, specific phobias, and social phobia can all be observed in

children (e.g., Harari, Vukovic, & Bailey, 2013). Mathematics anxiety can also be

2

observed in children (e.g., Gierl & Bisanz, 1995; Jackson & Leffingwell, 1999; Suinn,

Taylor, & Edwards, 1988; Zakaria & Nordin, 2008). However, the age of onset of

mathematics anxiety is unclear. Some research claims that mathematics anxiety emerges

in Grade 6, when children have encountered a complex mathematics curriculum, have

had sufficient opportunity to internalize negative feedback, and have the ability to

successfully report mathematics anxiety (Ashcraft, Krause, & Hopko, 2007). Yet,

Jackson and Leffingwell (1999) found that 16% of children report experiencing their first

traumatic mathematics experience between Grades 3 and 4. This may be because it is in

Grade 3 that children begin to learn complex mathematical operations and concepts (e.g.,

Wu et al., 2012). Most recently, research suggests that mathematics anxiety is detectable

as early as Grade 1 (Harari et al., 2013). Nevertheless, the majority of research claims

that the generally accepted age of onset of mathematics anxiety is Grade 3 (Ashcraft &

Moore, 2009; Jackson & Leffingwell, 1999; Wu et al., 2012).

The current study investigated mathematics anxiety in Grades 4, 5, and 6 children.

This age range was selected to ensure that children were within the generally accepted

age of onset of mathematics anxiety and that they were able to report their mathematics

anxiety. Research also shows that this is a critical stage for the development of children’s

attitudes about mathematics (Newstead, 1998). The aim of the current study, however,

was not to confirm the age of onset of mathematics anxiety but to examine the

characteristics and development of children’s mathematics anxiety and to explore its

effect on their performance on mathematics tests.

1.2.2 Consequences of mathematics anxiety. Individuals with mathematics

anxiety enjoy mathematics less, have lower perceptions of their mathematics abilities,

3

and participate less during mathematics class (Beilock, 2008; Harari et al., 2013). They

are also less likely to see the value of learning mathematics and avoid taking mathematics

courses (Harari et al., 2013). Most concerning, children with high mathematics anxiety

are significantly slower and less accurate on mathematics problems compared to children

with low mathematics anxiety (Ashcraft & Moore, 2009; Wu et al., 2012). Given the

consequences associated with mathematics anxiety, it is important to understand what

causes children’s mathematics anxiety and how it impacts their performance on

mathematics tests.

1.2.3 Causes of mathematics anxiety. Mathematics anxiety can be caused by

many factors. For instance, parents with mathematics anxiety can pass on a genetic

predisposition for mathematics anxiety to their children (e.g., Wang et al., 2014).

Children with parents or siblings with negative attitudes about mathematics are also often

at risk for developing mathematics anxiety (e.g., Dowker et al., 2012). The majority of

research, however, suggests that mathematics anxiety is created in the classroom (e.g.,

Ashcraft et al., 2007; Finlayson, 2014; Lerner & Friesema, 2013; Newstead, 1998; Wu et

al., 2012). In particular, research shows that mathematics tests have been shown to

contribute to the development and maintenance of mathematics anxiety in children (e.g.,

Lerner & Friesema, 2013). The current study, therefore, explored the relationship

between children’s mathematics anxiety and their performance on mathematics tests.

Research shows that mathematics anxiety could also be caused by teachers (e.g.,

Dowker et al., 2012; Finlayson, 2014; Gresham, 2007; Jackson & Leffingwell, 1999;

Newstead, 1990). For instance, Newstead (1998) explains that teachers’ use of

memorization techniques rather than understanding and reasoning can promote

4

mathematics anxiety in their students. Furthermore, Finlayson (2014) found that

children’s mathematics anxiety is highly influenced by teachers’ own mathematics

anxiety. Teachers with high mathematics anxiety can unintentionally encourage negative

attitudes about mathematics to their students (e.g., Jackson & Leffingwell, 1999).

However, little research has examined the impact of teachers’ mathematics anxiety on

their students. As part of an exploratory and secondary component, the current study,

therefore, investigated teachers’ mathematics anxiety and its impact on their students.

1.3 Children’s Mathematics Anxiety and Performance on Mathematics Tests

Mathematics problem solving relies on three types of knowledge: conceptual,

procedural, and factual knowledge (Voutsina, 2012). Conceptual knowledge allows

children to understand why a procedure works by forming a relationship between existing

and new information, procedural knowledge allows children to understand how to

perform a task by applying rules, algorithms, and procedures, and factual knowledge

allows children to recall answers directly from memory without performing any

calculations (Voutsina, 2012). Previous research, however, has only examined the impact

of mathematics anxiety on procedural (e.g., Ramirez et al., 2013) and factual knowledge

(e.g., Royer, Tronsky, Chan, Jackson, & Marchant III, 1999). For instance, Ramirez and

colleagues (2013) studied children’s procedural knowledge and found that Grades 1 and 2

children with higher mathematics anxiety were less accurate on mathematics tests

compared to children with lower mathematics anxiety. Similarly, Royer and colleagues

(1999) studied children’s factual knowledge and found that Grades 5 to 8 children with

higher mathematics anxiety were slower and less accurate compared to individuals with

lower mathematics anxiety. Overall, research typically shows a negative relationship

5

between mathematics anxiety and performance on mathematics tests (e.g., Beilock, 2008;

Chernoff & Stone, 2012; Devine, Fawcett, Szûcs, & Dowker, 2012).

The current study expands on previous research by investigating how children’s

mathematics anxiety impacts their conceptual understanding of arithmetic (i.e.,

conceptual knowledge) because no research to my knowledge has ever explored this

relationship. Studying the impact of children’s mathematics anxiety on their conceptual

understanding of arithmetic is important because conceptual understanding of arithmetic

allows children to learn complex mathematical skills and concepts (e.g., Allard, 2011).

Allard (2011) elaborates that the foundation of conceptual understanding of arithmetic is

arithmetic fluency. As a result, the current study also investigated children’s arithmetic

fluency (i.e., factual knowledge) and expands on previous research by investigating how

children’s mathematics anxiety impacts their arithmetic fluency as little research has

studied this relationship in elementary school. Studying the impact of children’s

mathematics anxiety on their arithmetic fluency is important because it facilitates the

development and understanding of advanced mathematical strategies and skill (Carr &

Alexeev, 2011). Overall, the current study contributes novel information about the

characteristics of children’s mathematics anxiety and how it impacts both their

conceptual understanding of arithmetic and arithmetic fluency.

Some research, however, shows no relationship between mathematics anxiety and

performance on mathematics tests (e.g., Dowker et al., 2012). For instance, Dowker and

colleagues (2012) found that Grades 3 and 5 children’s mathematics anxiety was not

related to their procedural mathematics performance. Instead, children’s mathematics

anxiety was impacted by their perception of their mathematics ability (Dowker et al.,

6

2012). Similarly, Tsui and Mazzocco (2006) suggest that weaker performance on

mathematics tests is due to weaker mathematics competence rather than higher

mathematics anxiety. Mathematics perception and competence, however, cannot explain

all aspects of poor performance on mathematics tests that have been associated with

mathematics anxiety. For instance, individuals with higher mathematics anxiety have

more effortful processing on procedural mathematics problems compared to individuals

with lower mathematics anxiety (Ashcraft & Kirk, 2001). Overall, there is stronger

evidence that poor performance on mathematics tests is linked with mathematics anxiety.


arithmetic. The current study explored the effects of children’s mathematics anxiety on

their conceptual understanding of arithmetic using six arithmetic concept problems (i.e.,

identity, negation, commutativity, inversion, associativity, and equivalence; see

Appendices A, B, C, and D). Arithmetic concepts refer to mental mathematical

representations that organize experience (Gelman, 2009) and are usually measured by

participants’ accuracy, reaction time, and/or strategy use (e.g., Dubé & Robinson, 2010;

Robinson & Dubé, 2009a).

Two of the most basic arithmetic concepts are the identity and negation concepts

(e.g., Baroody, Lai, Li, & Baroody, 2009; Baroody, Torbeyns, & Verschaffel, 2009;

Price, 2013). The identity concept is an arithmetic function in which the identity of a

number remains unchanged (Baroody, Torbeyns, et al., 2009). If children understand

additive (e.g., 9 – 0 = ?) or multiplicative (e.g., 9 x 1 = ?) identity concepts, they will state

the first number (i.e., 9) without performing any calculations. Research shows that

7

elementary school children typically have good conceptual understanding of the identity

concept (e.g., Price, 2013).

The negation concept is an arithmetic function in which the total identity of a

number is removed (Baroody, Torbeyns, et al., 2009). If children understand additive

negation concepts (e.g., 9 – 9 = ?), they will state ‘0’ without performing any

calculations. If children understand multiplicative negation concepts (e.g., 9 ÷ 9 = ?),

they will state ‘1’ without performing any calculations. Research shows that elementary

school children also typically have good conceptual understanding of the negation

concept (e.g., Price, 2013).

The commutativity concept is more difficult for children to understand compared

to the identity and negation concepts (Price, 2013; Squire, Davies, & Bryant, 2004). It is

an arithmetic function in which the group of quantities, connected by the same operation,

remains the same regardless of the order (Baroody & Dowker, 2009). If children

understand additive (e.g., 7 + 4 = 4 + ?) or multiplicative (e.g., 7 x 4 = 4 x ?)

commutativity concepts, they will simply state the first number (i.e., 7) without

performing any calculations. Research shows that Grades 5 and 6 children have greater

difficulties with multiplicative commutativity problems compared to the additive

commutativity problems (Squire et al., 2004). Nevertheless, elementary school children

typically have some conceptual understanding of the commutativity concept (e.g.,

Canobi, Reeve, & Pattison, 2002; Price, 2013; Squire et al., 2004; Wilkins, Baroody, &

Tiilikainen, 2001).

The inversion concept is also more difficult for children to understand compared

to the identity and negation concepts (Baroody & Lai, 2007; Baroody, Lai, et al., 2009;

8

Canobi & Bethune, 2008; Price, 2013) and even the commutativity concept (Canobi &

Bethune, 2008). The inversion concept is an arithmetic function in which the identity of

a number remains the same due to operations of inversion relationships (e.g., addition and

subtraction; Baroody, Torbeyns, et al., 2009; Dubé & Robinson, 2010).

There are three strategies to solving inversion problems (e.g., 2 + 8 – 8 = ?): the

left-to-right strategy, the negation strategy, and the inversion strategy (Dubé & Robinson,

2010; Robinson & Dubé, 2009a; 2009b; 2009c; Robinson, Ninowski, & Gray, 2006).

The left-to-right strategy is the preferred strategy by children on inversion problems, and

it involves calculating the problem without using any shortcuts. The use of this strategy

demonstrates no conceptual understanding of the inversion concept. The negation

strategy involves calculating the first part of the problem (i.e., 2 + 8) and then realizing

that no calculations are necessary to solve the problem. The use of this strategy

demonstrates partial conceptual understanding of the inversion concept (Robinson &

Dubé, 2009b). The inversion strategy involves stating the first number of the problem

(i.e., 2) without performing any calculations. The use of this strategy demonstrates good

conceptual understanding of the inversion concept (e.g., Robinson & Dubé, 2009a). The

inversion strategy is also the fastest and most accurate strategy for inversion problems

(Robinson & Dubé, 2009b; 2009c). Research shows that Grades 3 to 5 children have

greater difficulties understanding the multiplicative inversion problems compared to the

additive inversion problems (Robinson & Dubé, 2009a; 2012). Nevertheless, elementary

school children typically have some conceptual understanding of the inversion concept

(e.g., Price, 2013; Robinson & LeFevre, 2012).

9

The associativity concept is more difficult for children to understand compared to

the commutativity (e.g., Patel & Canobi, 2010) and inversion concepts (e.g., Robinson &

LeFevre, 2012). The associativity concept is an arithmetic function in which the order

that operations are performed is independent of the answer (Robinson, Ninowski, et al.,

2006). If children understand additive (e.g., 7 + 9 – 3 = ?) or multiplicative (e.g., 7 x 9 ÷

3 = ?) associativity concepts, they will begin with the second part of the associativity

problem (i.e., 9 – 3 or 9 ÷ 3) to simplify their work (e.g., Baroody, Lai, et al., 2009).

Research shows that Grade 6 children have greater difficulties understanding

multiplicative associativity problems compared to additive associativity problems

(Robinson & Dubé, 2009c). Overall, elementary school children typically have weak

conceptual understanding of the associativity concept (e.g., Robinson & Dubé, 2009a;

Robinson, Ninowski, et al., 2006).

The equivalence concept is generally difficult for children to understand (Alibali,

1999; McNeil, 2007; Price, 2013; Rittle-Johnson, Matthews, Taylor, & McEldon, 2011).

More specifically, Price (2013) shows that the equivalence concept is the most difficult

arithmetic concept for children to understand. The equivalence concept is an arithmetic

function in which both sides of the problem have the same total value (e.g., McNeil,

2007; Rittle-Johnson et al., 2011). If children understand additive (e.g., 2 + 6 + 4 = 2 + ?)

or multiplicative (e.g., 2 x 6 x 4 = 2 x ?) equivalence concepts, they will only perform the

second part of the problem (i.e., 6 + 4 or 6 x 4). Knuth, Alibali, McNeil, Weinberg, and

Stephens (2005) also found that children who have greater understanding of the equal

sign have greater understanding of additive equivalence problems. Still, children

10

typically have weak conceptual understanding of the equivalence concept (e.g., Alibali,

1999; McNeil, 2007; Robinson & Dubé, 2012).

Children’s conceptual understanding of arithmetic is partly based on their

understanding of additive (i.e., addition and subtraction) and multiplicative (i.e.,

multiplication and division) operations. Additive and multiplicative reasoning allow

children to solve problems more quickly and more accurately (Caddle & Brizuela, 2011).

Research shows that children have greater difficulty solving subtraction problems

compared to addition problems and division problems compared to multiplication

problems (e.g., Dixon, Deets, & Bangert, 2001), and children also have greater difficulty

understanding multiplicative problems compared to additive problems (e.g., Robinson &

Ninowski, 2003; Robinson & Dubé, 2009a; 2009c). This latter point may be because

multiplicative problems are learned later than additive problems (e.g., Robinson & Dubé,

2009c; Robinson & Ninowski, 2003). Improving children’s understanding of additive

and multiplicative operations improves their speed and accuracy on arithmetic concept

problems (Robinson & Dubé, 2009c; Robinson & Ninowski, 2003; Smith & Smith,

2006).

Greater conceptual understanding of arithmetic allows children to solve

mathematics problems more quickly, more accurately, and with greater flexibility (Squire

et al., 2004). For instance, children’s use of conceptually-based strategies demonstrates

their level of conceptual understanding of arithmetic (Gilmore & Bryant, 2006). Children

who fail to use conceptually-based strategies do not necessarily lack conceptual

understanding of arithmetic; they may be aware of the conceptually-based strategies but

simply choose not to use them (Robinson & Dubé, 2009a; 2012). Children sometimes

11

even consider the use of conceptually-based strategies a form of cheating (Robinson &

Dubé, 2009b). Nevertheless, research primarily shows that children with greater

conceptual understanding of arithmetic have greater arithmetic concept problem-solving

skills compared to children with weaker conceptual understanding of arithmetic (Gilmore

& Bryant, 2006; Robinson & Dubé, 2012).

Older children typically have greater conceptual understanding of arithmetic

compared to younger children (e.g., Baroody, Lai, et al., 2009; Canobi et al., 2002;

Robinson, Ninowski, et al., 2006). For instance, Dubé (2014) showed that older children

had greater conceptual understanding of inversion and associativity concepts compared to

younger children. Gelman (1999) explains that arithmetic concepts are derived from

experience. In other words, older children likely have greater conceptual understanding

of arithmetic because they have had more practice solving arithmetic concepts compared

to younger children. Still, some research shows that conceptual understanding of

arithmetic is independent of age (e.g., Canobi, 2004; Robinson & Dubé, 2009a).

Robinson and Dubé (2009a) showed that grade did not improve Grades 2, 3, or 4

children’s conceptual understanding of additive inversion and associativity problems

suggesting that the development of their conceptual understanding of additive inversion

and associativity problems is slow across these grades. They elaborate that formal

schooling is not responsible for improvements in children’s conceptual understanding of

additive inversion and associativity problems and may actually interfere with its

development (Robinson & Dubé, 2009a). In other words, factors other than grade may

actually be responsible for improvements in children’s conceptual understanding of

arithmetic and arithmetic fluency. Nevertheless, further research is needed to clarify

12

these grade differences. The current study, therefore, investigated children’s conceptual

understanding of arithmetic to see how it changes over Grades 4, 5, and 6.

Research primarily shows no sex differences in children’s conceptual

understanding of arithmetic (e.g., Beck, Ghesquière, Lagae, & Smedt, 2014; Nunes,

Bryant, Hallett, Bell, & Evans, 2009; Robinson & Dubé, 2012). This suggests that males

and females are equally capable of understanding why a mathematical procedure works.

Other research, however, shows that males have greater conceptual understanding of

arithmetic compared to females (e.g., Carr, Steiner, Kyser, & Biddlecomb, 2008; Geary,

Saults, Lui, & Hoord, 2000; Royer et al., 1999). For instance, Carr and Davis (2001)

found that Grade 1 males had greater conceptual understanding of addition and

subtraction problems compared to Grade 1 females and attribute this male advantage to

their preference for retrieval of mathematical facts compared to females’ preference for

computation. These sex differences, though, declined on more difficult problems (Carr &

Davis, 2001). Similarly, Hyde, Fennema, and Lamon (1990) found that sex differences

were evident in early elementary school and disappeared around Grade 3. Overall, the

majority of research suggests that sex differences in children’s conceptual understanding

of arithmetic are minimal. Nevertheless, the current study investigated children’s

conceptual understanding of arithmetic to see if it differs between males and females in

Grades 4, 5, and 6.

The hypotheses for children’s conceptual understanding of arithmetic and how it is

impacted by their mathematics anxiety were based on previous research. Based on

previous research, it was expected that children would have good conceptual

understanding of identity and negation, some conceptual understanding of commutativity

13

and inversion, and weak conceptual understanding of associativity and equivalence (e.g.,

Price, 2013). Older children would have greater conceptual understanding of arithmetic

on the problem-solving task compared to younger children (e.g., Patel & Canobi, 2010).

No sex differences in children’s conceptual understanding of arithmetic on the problem-

solving task were expected (e.g., Alibali, 1999; Robinson, Arbuthnott, et al., 2006;

Robinson & Dubé, 2009a). Lastly, given that previous research shows a negative

relationship between children’s mathematics anxiety and their performance on

mathematics tests (e.g., Maloney et al., 2012), it was expected that children with higher

mathematics anxiety would also have weaker conceptual understanding of arithmetic

compared to children with lower mathematics anxiety.

1.3.2 Children’s mathematics anxiety and arithmetic fluency. The current

study explored the effects of children’s mathematics anxiety on their arithmetic fluency

using timed mathematics tests for the four arithmetic operations of addition, subtraction,

multiplication, and division (see Appendices E, F, G, and H). Arithmetic fluency allows

children to retrieve mathematical facts more quickly and more accurately and is measured

by participants’ speed and accuracy on timed mathematics tests (Lerner & Friesema,

2013). Timed mathematics tests are also important tools for improving mathematical

skill and arithmetic fluency (Geist, 2010). For instance, they promote retrieval of

mathematical facts (Lerner & Friesema, 2013). However, timed mathematics tests can

undermine children’s natural thinking process (Geist, 2010) and prevent an accurate

assessment of children’s mathematical skill (Walen & Williams, 2002). For instance,

Walen and Williams (2002) found that individuals focused on the passing of time rather

than on the mathematical tasks. Timed mathematics tests can also increase anxiety,

14

decrease accuracy, and create negative attitudes about mathematics (e.g., Ashcraft, 2002).

As a result, the relationship between mathematics anxiety and performance on

mathematics tests is much stronger in timed conditions compared to untimed conditions

(Royer et al., 1999; Tsui & Mazzocco, 2006; Walen & Williams, 2002).

Older children generally have greater arithmetic fluency compared to younger

children (e.g., Calhoon, Robert, Flores, & Houchins, 2007; Kamii, Lewis, & Kirkland,

2001; Robinson, Arbuthnott, et al., 2006). For instance, Robinson, Arbuthnott, and

colleagues (2006) found that Grades 4 through 7 children became faster and more

accurate on simple division problems as they aged. Alternatively, McNeil (2007)

explains that mathematics development can be U-shaped, with performance on

mathematics tests sometimes declining before it improves. Other research shows no

grade differences in children’s arithmetic fluency (e.g., Robinson & Dubé, 2009b).

Robinson and Dubé (2009b) found no grades differences in Grades 6, 7, and 8 children

on multiplicative inversion problems suggesting that factors other than grade are also

responsible for improvements in children’s arithmetic fluency on multiplicative

problems. Although research is unclear about how grade influences children’s arithmetic

fluency, the majority of research shows that arithmetic fluency typically increases with

age. The current study, therefore, investigated children’s arithmetic fluency using timed

mathematics tests to see how children’s arithmetic fluency changes over Grades 4, 5, and

6.

Males typically have greater arithmetic fluency compared to females (Carr et al.,

2008; Geary et al., 2000; Nunes et al., 2009; Royer et al., 1999). Research shows that

retrieval of mathematical facts allow children to solve problems faster and more

15

accurately compared to computation (Carr et al., 2008). Research also shows that males

prefer retrieval strategies whereas females prefer computation strategies (Carr & Jessup,

1997). This could explain why males are faster and more accurate compared to females.

However, males’ advantage on arithmetic fluency over females has been disputed. For

instance, other research shows no sex differences in children’s arithmetic fluency

suggesting that males and females are equally capable of retrieving mathematical facts

(Beck et al., 2014; Carr & Davis, 2001). Further research is needed to help clarify the

impact of sex on children’s arithmetic fluency. The current study, therefore, investigated

children’s arithmetic fluency using timed mathematics tests to see if and how children’s

arithmetic fluency differs between males and females in Grades 4, 5, and 6.

The hypotheses for children’s arithmetic fluency and how it is impacted by their

mathematics anxiety were based on previous research. Based on previous research, it

was expected that children would have greater arithmetic fluency on the addition and

subtraction tests compared to the multiplication and division tests (e.g., Calhoon et al.,

2007; Cates & Rhymer, 2003), older children would have greater arithmetic fluency

compared to younger children (e.g., Calhoon et al., 2007; Walen & Williams, 2002), and

males would have higher arithmetic fluency compared to females (e.g., Beilock, 2008).

Lastly, it was expected that children with higher mathematics anxiety would have weaker

arithmetic fluency compared to children with lower mathematics anxiety (e.g., Maloney

et al., 2012; Preston, 2008).

1.3.3 Grade differences in children’s mathematics anxiety. Research shows

that mathematics anxiety, if not resolved, becomes more pronounced over time (e.g.,

Furner & Berman, 2003; Gierl & Bisanz, 1995; Ma, 1999). This suggests that older

16

children have higher mathematics anxiety compared to younger children. This

relationship appears to increase across grade (Beesdo et al., 2009; Ma, 1999), peaking in

Grades 9 and 10 (Ashcraft & Moore, 2009; Beilock, 2008; Hembree, 1990). This

increase in mathematics anxiety is partly attributed to an increasingly difficult

mathematics curriculum (Ashcraft et al., 2007), and also to older children becoming

increasingly concerned about the consequences of their performance on mathematics tests

(Gierl & Bisanz, 1995). Other research, however, shows no grade differences in

mathematics anxiety (Dowker et al., 2012; Wigfield & Meece, 1988). For instance,

Wigfield and Meece (1998) found no impact of grade on Grade 5 through 12 children’s

mathematics anxiety. Dowker and colleagues (2012) and Wigfield and Meece (1998)

attribute changes in children’s mathematics anxiety over time to perceptions in their

mathematics ability rather than grade. Nevertheless, the majority of research suggests

that older children have higher mathematics anxiety compared to younger children. The

current study, therefore, examined the effect of grade on children’s mathematics anxiety.

1.3.4 Sex differences in children’s mathematics anxiety. Sex differences can

also influence the development of children’s mathematics anxiety. Anxiety disorders, as

a whole, occur more frequently in females compared to males (e.g., Beesdo et al., 2009).

In particular, research shows that females report higher mathematics anxiety compared to

males (e.g., Geist, 2010; Wang et al., 2014). Females tend to feel less confident about

their mathematics answers, enjoy mathematics less, and are more susceptible to the

negative effects of timed mathematics tests (Geist, 2010). They are also more willing to

report their anxiety compared to males (Hembree, 1990). Some research, however,

reports no sex differences in children’s mathematics anxiety (e.g., Cates & Rhymer,

17

2003; Jameson, 2014; Wigfield & Meece, 1988; Wood, 1988). For instance, Jameson

(2014) found no sex differences in Kindergarten through to Grade 6 children’s

mathematics anxiety. Hembree (1990) alternatively found that males reported higher

mathematics anxiety compared to females. Inconsistencies in sex differences in

children’s mathematics anxiety can be attributed to children’s willingness to admit that

they have mathematics anxiety. For instance, research shows that females are more likely

to admit that they experience mathematics anxiety compared to males (Hembree 1990).

Nevertheless, research primarily shows that sex differences are an important contributing

factor to children’s mathematics anxiety and that females typically have higher

mathematics anxiety compared to males. The current study, therefore, investigated the

impact of sex on children’s mathematics anxiety.

1.4 Exploratory Analysis of Teachers’ Mathematics Anxiety

Teachers with mathematics anxiety can reportedly influence the development of

children’s mathematics anxiety (e.g., Beilock & Willingham, 2014; Chernoff & Stone,

2012; Finlayson, 2014; Geist, 2015; Hadley & Dorward, 2011). Little research, however,

has explored the characteristics and development of teachers’ mathematics anxiety. In

addition, little research has explored the effects of teachers’ mathematics anxiety on their

students. Research shows that children internalize teachers’ interest and enthusiasm for

subjects (Finlayson, 2014; Furner & Berman, 2003; Gresham, 2007; Jackson &

Leffingwell, 1999). Teachers with mathematics anxiety, though, are often incapable of

generating interest and enthusiasm for mathematics (Wood, 1988) causing children to

internalize teachers’ negative attitudes about mathematics (Finlayson, 2014; Geist, 2015;

Jackson & Leffingwell, 1999). As an exploratory and secondary component, the current

18

study investigated teachers’ mathematics anxiety and its effect on their students’

mathematics anxiety, conceptual understanding of arithmetic, and arithmetic fluency.

This was considered exploratory due to the small sample size.

The hypotheses for teachers’ mathematics anxiety and how it impacts their

students were based on previous research. First, the current study explored the effects of

teacher’ mathematics anxiety on their students’ mathematics anxiety. Teachers with

higher mathematics anxiety were expected to have students with higher mathematics

anxiety compared to teachers with lower mathematics anxiety (e.g., Beilock, Gunderson,

Ramirez, & Levine, 2009; Beilock & Willingham, 2014; Chernoff & Stone, 2012; Furner

& Berman, 2003; Geist, 2015; Hadley & Dorward, 2011). Second, the current study

explored the effects of teachers’ mathematics anxiety on their students’ conceptual

understanding of arithmetic and on their arithmetic fluency. Teachers with higher

mathematics anxiety were expected to have students with lower conceptual understanding

of arithmetic and lower arithmetic fluency compared to teachers with lower mathematics

anxiety (e.g., Dowker et al., 2012; Furner & Berman, 2003; Newstead, 1998; Wu et al.,

2012). Third, the current study explored children’s awareness of their teacher’s level of

mathematics anxiety. Children were expected to be aware of their teacher’s level of

mathematics anxiety (e.g., Finlayson, 2014).

19

CHAPTER TWO

Method

2.1 Participants

Data was collected from 43 Grade 4 children (18 Male, 25 Female), 41 Grade 5

children (21 Male, 20 Female), and 36 Grade 6 children (15 Male, 21 Female) from five

schools. Grade 4 participants had a median age of 9 years, 10 months (range: 9 years, 3

months to 10 years, 3 months). Grade 5 participants had a median age of 10 years, 11

months (range: 10 years, 3 months to 11 years, 4 months). Grade 6 participants had a

median age of 11 years, 10 months (range: 11 years, 6 months to 13 years, 4 months). In

addition, data from 17 teachers (5 Male, 12 Female) from the participating classes were

collected. Consent was obtained from principals of the participating schools, teacher

participants, and parents. Assent was also obtained from child participants. Child

participants received a gel pen for their participation and teacher participants received a

box of chocolates. Ethical approval was obtained from the University of Regina Ethics

Board (see Appendix A) and the Regina Public School Board and data was obtained from

February to June 2014.

2.2 Materials and Procedure

Children participated in two sessions that lasted approximately 45 minutes in total.

In the first session, children completed a problem-solving task. In the second session,

children completed a timed mathematics task followed by a mathematics anxiety

questionnaire. On average, the timed mathematics task and mathematics anxiety

questionnaire were administered 5 days after problem-solving task. It is important that

the mathematics anxiety questionnaire was administered last to avoid any influence on

20

children’s scores on the mathematics tests. Alternatively, teachers only participated in

one session that lasted approximately 10 minutes. During this session, teacher

participants self-administered a mathematics anxiety questionnaire tailored for adults.

Children’s conceptual understanding of arithmetic was assessed on the problem-

solving task that lasted approximately 30 minutes. This assessment was administered

individually to each child participant. Half of the participants in each grade began with

the first condition (i.e., additive problems first; see Appendix B) and the other half of the

participants began with the second condition (i.e., multiplicative problems first; see

Appendix C). All participants completed both additive and multiplicative problems.

Each child viewed four problems of each arithmetic concept (i.e., identity, negative,

commutativity, inversion, associativity, and equivalence), for a total of 24 problems per

condition. Problems were presented on a computer using E-Prime software. Participants

were instructed to hit the space bar when they knew the answer to the problem (i.e.,

reaction time), state their answer (i.e., accuracy), and then describe how they got that

answer (i.e., strategy use). If they were unable to provide an explanation, they were

given cues (e.g., “where did you start this problem?”). Each problem disappeared after

30 seconds. This ensured that, on this task, children did not feel pressured for time.

Accuracy, reaction time, and strategy use were collected for each problem.

Children’s arithmetic fluency was assessed on a timed mathematics task that lasted

approximately 5 minutes. Four timed mathematics tests (i.e., addition, subtraction,

multiplication, and division; see Appendices D, E, F, and G) were administered to groups

of participants (ranging from 2 to 8 children) using pen and paper. After the instructions

were read to the participants, children had one minute each to answer as many problems

21

as possible on each test. Number of problems attempted and accuracy on the number of

problems attempted were calculated for each timed mathematics test.

Children’s mathematics anxiety was assessed using the Mathematics Anxiety Scale

for Children (MASC; Chiu & Henry, 1990; see Appendix H) that lasted approximately 10

minutes. The MASC is used to identify mathematics anxiety in children between Grades

4 to 8. It is an internally consistent and suitable measure for children (Chiu & Henry,

1990). This assessment was also administered to groups of participants (ranging from 2

to 8 children) using pen and paper. Instructions were read to the participants followed by

the 22-items (e.g., taking a quiz in math). Children rated their level of anxiety

considering each statement on a 4-point Likert-type scale ranging from 1 (not nervous) to

4 (very very nervous). When requested, participants were provided clarification of any

item. Each statement response was awarded 1 to 4 points, respectively, and then summed

for all 22-items. Based on Chiu and Henry (1990), possible scores ranged from 22 to 88

points. Scores 27 and lower represented low mathematics anxiety, scores 28 to 51

represented some mathematics anxiety, and scores 52 and above represented high

mathematics anxiety. In this study, an additional question was added to the end of the

original MASC instructing children to rate their teacher’s level of mathematics anxiety on

a 4-point Likert-type scale ranging from 1 (not nervous) to 4 (very very nervous). This

statement was also awarded 1 to 4 points. A score of 1 represented low teacher

mathematics anxiety, scores 2 and 3 represented some teacher mathematics anxiety, and a

score of 4 represented high teacher mathematics anxiety. This question assessed

children’s awareness of their teacher’s level of mathematics anxiety.

22

Teachers’ mathematics anxiety was assessed using the Abbreviated Mathematics

Anxiety Rating Scale (A-MARS; Alexander & Martray, 1989; see Appendix I) that lasted

approximately 10 minutes. The A-MARS is an internally consistent measure that is less

time consuming, easier to administer, and faster to score than the 98-item Mathematics

Anxiety Rating Scale (MARS; Alexander & Martray, 1989). Teachers self-administered

this assessment. They were given the questionnaire at the beginning of the data

collection at their school and were instructed to begin by reading the instructions

followed by the 25-items (e.g., taking a final in a math course). Teachers then rated their

level of anxiety considering each statement on a 5-point Likert-type scale ranging from 1

(not at all) to 5 (very much). Each statement response was awarded 0 to 4 points,

respectively, and then summed for all 25-items. Possible scores ranged from 0 to 100

points. Based on Ashcraft and Moore (2009), scores 20 and lower represented low

mathematics anxiety, scores 21 to 51 represented some mathematics anxiety, and scores

52 and above represented high mathematics anxiety. After completion of the

questionnaire, teachers were instructed to seal it in a provided envelope. Envelopes were

collected from the teachers at the end of the data collection at each school.

23

CHAPTER THREE

Results

Participant performance was assessed through mixed model analyses of variance

(ANOVA), one-way ANOVAs, multiple regression analyses, cluster analyses,

discriminant analyses, and correlation analyses. Significant results had an alpha level of

.05 or lower unless reported otherwise.

3.1 Children’s Performance on Mathematics Tests

3.1.1 Children’s conceptual understanding of arithmetic.

3.1.1.1 Accuracy. A 3 (Grade: 4, 5, 6) x 2 (Sex: Male, Female) x 2

(Operation type: additive, multiplicative) x 6 (Problem type: identity, negation,

commutativity, inversion, associativity, equivalence) mixed model ANOVA was

conducted to assess children’s accuracy on the problem-solving task. A main effect was

found for problem type, F(5, 570) = 105.49, p < .001, MSE = 99004.58, 𝜂p2 = .48. Mean

differences showed that children were more accurate on identity, negation, and inversion

problems compared to commutativity, associativity, and equivalence problems, HSD =

13.31 (see Table 1). A main effect was found for operation type, F(1, 114) = 154.41, p <

.001, MSE = 140330.98, 𝜂p2 = .58. Children were more accurate on additive problems

compared to multiplicative problems (see Table 2). A main effect was also found for

grade, F(2, 114), p = .024, MSE = 19929.15, 𝜂p2 = .063. Mean differences showed that

Grade 5 children were more accurate on the problem-solving task compared to Grade 4

children but no differences were found between Grades 5 and 6 children or Grades 4 and

6 children, HSD = 7.97. Lastly, a main effect was found for sex, F(1, 114) = 10.13, p =

24

.002, MSE = 52498.45, 𝜂p2 = .083 (Figure 1). Males were more accurate on the problem-

solving task compared to females.

Table 1: Children’s Conceptual Understanding of Arithmetic on the Problem-Solving Task by Problem Type (Identity, Negation, Commutativity, Inversion, Associativity, and Equivalence) and Sex (Male and Female)

Table 2. Children’s Conceptual Understanding of Arithmetic on the Problem-Solving Task by Operation Type (Additive and Multiplicative).

25

Figure 1. Children’s accuracy and strategy use on the problem-solving task for sex (male and female). There were four significant interactions. First, an interaction was found between

problem type and sex, F(5, 570) = 2.86, p = .015, MSE = 2684.11, 𝜂p2 = .024 (see Figure

2). Mean differences showed that the male advantage was largest on inversion and

associativity problems and smallest on identity problems, HSD = 22.51. Second, an

interaction was found between operation type and sex, F(1, 114) = 6.86, p = .010, MSE =

6230.38, 𝜂p2 = .057 (see Figure 3). Mean differences showed that the male advantage

was larger on multiplicative problems compared to additive problems, HSD = 26.66.

Third, an interaction was found between problem type and operation type, F(5, 570) =

14.58, p < .001, MSE = 7362.48, 𝜂p2 = .12 (see Figure 4). Mean differences showed that

children’s advantage on addition problems compared to multiplicative problems was

26

largest on associativity and negation problems and smallest on identity and

commutativity problems, HSD = 29.34.

Figure 2. Children’s accuracy on the problem-solving task for problem-type (identity, negation, commutativity, inversion, associativity, and equivalence) and sex (male and female).

27

Figure 3. Children’s accuracy on the problem-solving task for operation type (additive and multiplicative) and sex (male and female).

28

Figure 4. Children’s accuracy on the problem-solving task for problem type (identity, negation, commutativity, inversion, associativity, and equivalence) and operation type (additive and multiplicative). Fourth, a three-way interaction was found between problem type, operation type,

and grade, F(10, 570) = 3.81, p < .001, MSE = 1888.50, 𝜂p2 = .063 (see Figure 5). Mean

differences showed similar results as found in the interaction between problem type and

operation type, HSD = 18.88. However, it revealed that Grade 4 children were relatively

as competent on additive commutativity problems as multiplicative commutativity

problems. Results also showed that as children age, their advantage on additive problems

remains relatively consistent for basic arithmetic concepts (i.e., identity, negation,

commutativity, and inversion) but gradually faded for complex arithmetic concepts (i.e.,

associativity and equivalence). Therefore, children’s conceptual understanding of

associativity and equivalence improved over Grades 4, 5, and 6 suggesting that these are

29

critical ages for the development of associativity and equivalence problems. No

interactions were found between problem type and grade, F(10, 570) = 1.64, p = .092,

MSE = 1537.96, 𝜂p2 = .028, between problem type, grade, and sex, F(10, 570) = .30, p =

.980, MSE = 285.20, 𝜂p2 = .005, between operation type and grade, F(2, 114) = .42, p =

.657, MSE = 382.78, 𝜂p2 = .007, between operation type, grade, and sex, F(2, 114) = .043,

p = .958, MSE = 38.75, 𝜂p2 = .001, or between problem type, operation type, and sex,

F(5, 570) = 2.23, p = .050, MSE = 1107.12, 𝜂p2 = .019.

Figure 5. Grade 4 children’s accuracy on the problem-solving task for problem type (identity, negation, commutativity, inversion, associativity, and equivalence) and operation type (additive and multiplicative). 3.1.1.2 Reaction time. A 3 (Grade: 4, 5, 6) x 2 (Sex: Male, Female) x 2


30


conducted to assess children’s reaction time on the problem-solving task. Results should

be interpreted with caution as children’s reaction time on the problem-solving task was

only included in the analyses if participants had at least one correct response for each

problem type (i.e., identity, negation, commutativity, inversion, associativity, and

equivalence). Consequently, less than half of the participants (n = 46) were included.

A main effect was found for problem type, F(5, 200) = 143.16, p < .001, MSE =

1036177933, 𝜂p2 = .78. Mean differences showed that children had fast reaction times on

identity and negation, moderate reaction times on inversion and commutativity, and slow

reaction times on associativity and equivalence problems, HSD = 1169.01 (see Table 1).

No main effects were found for operation type, F(1, 40) = .35, p = .558, MSE =

1651431.03, 𝜂p2 = .010, grade, F(2, 40) = 1.95, p = .155, MSE = 66094116.27, 𝜂p

2 = .089,

or sex, F(1, 40) = 2.17, p = .149, MSE = 73484402.72, 𝜂p2 = .051.

There were two significant interactions. First, an interaction was found between

problem type and operation type, F(5, 200) = 5.83, p < .001, MSE = 39215276.29, 𝜂p2 =

.13 (see Figure 6). Mean differences showed that children’s advantage on additive

problems compared to multiplicative problems was largest for associativity problems and

smallest for identity problems, HSD = 1127.28. Surprisingly, children had a large

advantage on multiplicative equivalence problems compared to additive equivalence

problems. Children also had a small advantage on multiplicative commutativity

problems compared to additive commutativity problems.

31

Figure 6. Children’s reaction time on the problem-solving task for problem type (identity, negation, commutativity, inversion, associativity, and equivalence) and operation type (additive and multiplicative). Second, a three-way interaction was found between problem type, grade, and sex,

F(10, 200) = 3.17, p = .001, MSE = 22931324.40, 𝜂p2 = .14 (see Figure 7). Mean

differences showed that Grade 4 males’ advantage was largest on commutativity

problems and smallest on inversion and associativity problems, HSD = 2080.80. Grade 4

females, surprisingly, had a large advantage over males on equivalence problems. Grades

5 and 6 males demonstrated a relatively consistent advantage over females on all

problems types (see Figures 8 and 9). In particular, Grade 5 males’ advantage was largest

on equivalence problems and smallest on commutativity problems. Grade 6 males’

advantage was largest on equivalence problems and smallest on identity and negation

problems. Overall, males generally had faster reaction times compared to females. No

32

interactions were found between problem type and grade, F(10, 200) = .80, p = .630,

MSE = 5821548.72, 𝜂p2 = .039, between problem type and sex, F(5, 200) = .98, p = .430,

MSE = 7123901.63, 𝜂p2 = .024, between operation type and grade, F(2, 40) = .44, p =

.650, MSE = 2075329.74, 𝜂p2 = .021, between operation type and sex, F(1, 40) = 1.75, p

= .190, MSE = 8288588.56, 𝜂p2 = .042, between operation type, grade, and sex, F(2, 40)

= 1.37, p = .270, MSE = 537.65, 𝜂p2 = .064, between problem type, operation type, and

grade, F(10, 200) = 1.34, p = .210, MSE = 9014467.86, 𝜂p2 = .063, or between problem

type, operation type, and sex, F(5, 200) = .21, p = .960, MSE = 1421345.70, 𝜂p2 = .005.

Figure 7. Grade 4 children’s reaction time on the problem-solving task for problem type (identity, negation, commutativity, inversion, associativity, and equivalence) and sex (male and female).

33

Figure 8. Grade 5 children’s reaction time on the problem-solving task for problem type (identity, negation, commutativity, inversion, associativity, and equivalence) and sex (male and female).

34

Figure 9. Grade 6 children’s reaction time on the problem-solving task for problem type (identity, negation, commutativity, inversion, associativity, and equivalence) and sex (male and female). 3.1.1.3 Strategy use. A 3 (Grade: 4, 5, 6) x 2 (Sex: Male, Female) x 2



conducted to assess children’s strategy use on the problem-solving task. There were

three main effects. A main effect was found for problem type, F(5, 570) = 201.31, p <

.001, MSE = 227192.44, 𝜂p2 = .64. Mean differences showed that children used

conceptually-based strategies more often on identity, negation, and commutativity

problems compared to inversion, associativity, and equivalence problems, HSD = 14.60

(see Table 1). A main effect was found for operation type, F(1, 114) = 80.65, p < .001,

35

MSE = 60527.12, 𝜂p2 = .41 (see Figure 1). Children used conceptually-based strategies

more often on additive problems compared to multiplicative problems. Lastly, a main

effect was found for sex, F(1, 114) = 5.59, p = .020, MSE = 19580.17, 𝜂p2 = .047 (see

Figure 2). Males used conceptually-based strategies more often compared to females.

No main effect was found for grade, F(2, 114) = 2.40, p = .095, MSE = 8425.39, 𝜂p2 =

.040.

There were two significant interactions. First, an interaction was found between

problem type and operation type, F(5, 570) = 43.70, p < .001, MSE = 26734.76, 𝜂p2 = .28

(see Figure 10). Mean differences showed that children’s advantage on additive

problems compared to multiplicative problems was largest on negation problems and

smallest on identity and associativity problems, HSD = 10.75. Again, children’s

advantage was larger on multiplicative commutativity problems compared to additive

commutativity problems. Second, a three-way interaction was found between problem

type, operation type, and grade, F(10, 570) = 2.71, p = .003, MSE = 1658.18, 𝜂p2 = .045.

Mean differences showed that the additive advantage was largest on negation problems

and smallest on identity, associativity, and equivalence problems for Grade 4 children,

largest on equivalence problems and smallest on identity and associativity problems for

Grade 5 children, and largest on negation problems and smallest on equivalence problems

for Grade 6 children, HSD = 17.69 (see Table 3). No interactions were found between

problem type and grade, F(10, 570) = 1.74, p = .069, MSE = 1963.50, 𝜂p2 = .030, between

problem type and sex, F(5, 570) = .43, p = .830, MSE = 481.98, 𝜂p2 = .004, between

problem type, grade, and sex, F(10, 570) = .36, p = .960, MSE = 407.66, 𝜂p2 = .006,

between operation type and grade, F(2, 114) = .60, p = .550, MSE = 452.75, 𝜂p2 = .010,

36

between operation and sex, F(1, 114) = 1.12, p = .290, MSE = 837.25, 𝜂p2 = .010,

between operation type, grade, and sex, F(2, 114) = .07, p = .940, MSE = 49.29, 𝜂p2 =

.001, between problem type, operation type, and sex, F(5, 570) = 1.74, p = .120, MSE =

1066.55, 𝜂p2 = .015, or between problem type, operation type, grade, and sex, F(10, 570)

= 1.63, p = .094, MSE = 998.22, 𝜂p2 = .028.

Figure 10. Children’s strategy use on the problem-solving task for problem type (identity, negation, commutativity, inversion, associativity, and equivalence) and operation type (additive and multiplicative).

37

Table 3: Children’s Conceptually-Based Strategy Use (%) on the Problem-Solving Task by Problem Type (Identity, Negation, Commutativity, Inversion, Associativity, and Equivalence), Operation Type (Additive and Multiplicative), and Grade (4, 5, and 6)

To summarize, children had good conceptual understanding of identity and

negation, some conceptual understanding of commutativity and inversion, and weak

conceptual understanding of associativity and equivalence problems. In particular, these

findings showed that Grades 4, 5, and 6 are critical ages for the development of

associativity and equivalence. Children also had greater conceptual understanding of

additive problems compared to multiplicative problems. Males had greater conceptual

understanding of arithmetic compared to females on the problem-solving task. Lastly,

inconsistent with expectation, grade did not have a significant effect on children’s overall

conceptual understanding of arithmetic on the problem-solving task.

38

3.1.2 Children’s arithmetic fluency.

3.1.2.1 Number of problems attempted. A 3 (Grade: 4, 5, 6) x 2 (Sex:

Male, Female) x 4 (Operation type: addition, subtraction, multiplication, division) mixed

model ANOVA was conducted to assess the number of problems children attempted on

the timed mathematics task. There were three main effects. A main effect was found for

operation type, F(3, 342) = 73.93, p < .001, MSE = 940.83, 𝜂p2 = .39 (see Figure 11).

Mean differences showed that children attempted more multiplication and division

problems compared to addition and subtraction problems, HSD = 1.43. A main effect

was also found for grade, F(2, 114) = 6.34, p = .002, MSE = 705.52, 𝜂p2 = .10 (see Figure

12). Mean differences showed that Grades 5 and 6 children attempted more problems

than Grade 4 children on the timed mathematics task but there was no significant

difference between Grades 5 and 6, HSD = 2.02. Lastly, a main effect was found for sex,

F(1, 114) = 15.72, p < .001, MSE = 1749.94, 𝜂p2 = .12 (see Figure 13). Males attempted

more problems compared to females.

39

Figure 11. The number of problems children attempted on the timed mathematics task for operation type (addition, subtraction, multiplication, and division) and sex (male and female).

40

Figure 12. The number of problems children attempted on the timed mathematics task for operation type (addition, subtraction, multiplication, and division) and grade (4, 5, and 6).

41

Figure 13. Children’s arithmetic fluency on the timed mathematics task for sex (male and female). An interaction was found between operation type and sex, F(3, 342) = 4.06, p =

.007, MSE = 51.45, 𝜂p2 = .034 (see Figure 11). Mean differences showed that males’

advantage was largest on multiplication and division problems and smallest on addition

and subtraction problems, HSD = 2.89. No interaction was found between operation type

and grade, F(6, 342) = .81, p = .561, MSE = 10.34, 𝜂p2 = .014, or between operation type,

grade, and sex, F(6, 342) = .41, p = .875, MSE = 5.16, 𝜂p2 = .007.

3.1.2.2 Accuracy on the number of problems attempted. A 3 (Grade: 4, 5,

6) x 2 (Sex: Male, Female) x 4 (Operation type: addition, subtraction, multiplication,

division) mixed model ANOVA was conducted to assess children’s accuracy on the

number of problems attempted on the timed mathematics task. There were two main

42

effects. A main effect was found for operation type, F(3, 342) = 23.45, p < .001, MSE =

8010.11, 𝜂p2 = .17 (see Figure 14). Mean differences showed that children were more

accurate on addition and multiplication problems compared to subtraction and division

problems, HSD = 7.42. A main effect was also found for sex, F(1, 114) = 6.08, p = .015,

MSE = 6517.06, 𝜂p2 = .051 (see Figure 13). This means that males were more accurate

on the number of problems they attempted compared to females. No main effect was

found for grade, F(2, 114) = 4261.31, p = .142, MSE = 2130.66, 𝜂p2 = .034.

Figure 14. Children’s accuracy on the number of problems attempted on the timed mathematics task for operation type (addition, subtraction, multiplication, and division) and sex (male and female). An interaction was found between operation type and sex, F(3, 342) = 5.05, p =

.002, MSE = 1724.96, 𝜂p2 = .042 (see Figure 14). Mean differences showed that males’

43

advantage was largest on division problems and smallest on multiplication and

subtraction problems, HSD = 16.68. Conversely, there was no sex difference in

children’s accuracy on the number of problems attempted on the addition timed

mathematics test. No interaction was found between operation type and grade, F(6, 342)

= 1.03, p = .407, MSE = 351.08, 𝜂p2 = .018, or between operation type, grade, and sex,

F(6, 342) = .54, p = .777, MSE = 341.60, 𝜂p2 = .009.

Overall, males had greater arithmetic fluency on the timed mathematics task

compared to females. Results also showed that Grades 5 and 6 children attempted more

problems on the timed mathematics task compared to Grade 4 children, however, there

was no significant effect of grade on children’s accuracy on the number of problems they

attempted on the timed mathematics task. This suggests that Grades 5 and 6 children are

attempting more problems than Grade 4 children but they are not more accurate on the

number problems they attempted. This may demonstrate that Grades 5 and 6 children are

guessing more often than Grade 4 children or this could display that Grades 5 and 6

children have a misunderstanding of mathematics facts.

3.2 Children’s Mathematics Anxiety

Children’s mathematics anxiety was assessed on the MASC (see Table 4). It was

expected that older children would have higher mathematics anxiety compared to

younger children (e.g., Ashcraft & Moore, 2009). It was also expected that females

would have higher mathematics anxiety compared to males (e.g., Jameson, 2014).

44

Table 4: Descriptive Statistics of Children’s Mathematics Anxiety on the MASC

Note. MA = mathematics anxiety A 3 (Grade: 4, 5, 6) x 2 (Sex: Male, Female) mixed model ANOVA was conducted

to assess the effects of grade and sex on children’s mathematics anxiety. There was a

main effect of sex, F(1, 119) = 10.05, p = .002, MSE = 1409.29, 𝜂p2 = .081 (see Figure

15). Females had higher mathematics anxiety compared to males, HSD = 4.54. No main

effect was found for grade, F(2, 119) = .79, p = .458, MSE = 110.40, 𝜂p2 = .014, and no

interaction was found between grade and sex, F(2, 119) = .94, p = .394, MSE = 131.74,

𝜂p2 = .016. Therefore, consistent with expectations, sex had a significant effect on

children mathematics anxiety but, inconsistent with expectations, grade did not.

45

Figure 15. Children’s mathematics anxiety score on the MASC for sex (male and female). 3.3 Children’s Mathematics Anxiety and its Effect on their Performance on

Mathematics Tests


arithmetic. Correlation analyses were conducted to assess the relationships between

children’s mathematics anxiety and their conceptual understanding of arithmetic on the

problem-solving task. Negative correlations were found between children’s mathematics

anxiety and their accuracy and strategy use on the problem-solving task (see Table 5).

Children with higher mathematics anxiety had lower accuracy and used conceptually-

based strategies less often on the problem-solving task compared to children with lower

mathematics anxiety. A positive correlation was also found between children’s

mathematics anxiety and their reaction time on the problem-solving task suggesting that

46

children with higher mathematics anxiety had slower reaction times on the problem-

solving task compared to children with lower mathematics anxiety. Overall, children

with higher mathematics anxiety had weaker conceptual understanding of arithmetic on

the problem-solving task compared to children with lower mathematics anxiety.

Table 5: Correlation Matrix Showing the Relationships Between Children’s Mathematics Anxiety and their Mathematics Performance on the Problem-Solving Task and Timed Mathematics Task

MA

Problem-Solving Task Timed Mathematics Task

Accuracy Reaction Time

Strategy Use

Number of Problems Attempted

Accuracy

MA 1 -.275** .192* -.179* -.392** -.446

Problem-Solving Task

Accuracy -.275** 1 -.172 .911** .443** 476**

Reaction Time .192* -.176 1 -.234* -.194* -.202*

Strategy Use -.179* .911** -.234* 1 .376** .343**

Timed Mathematics

Task

Number of Problems Attempted

-.392** .443** -.194* .376** 1 .380**

Accuracy -.446** .476** -.202* .343** .380** 1 **. Correlation is significant at the 0.01 level (2-tailed). *. Correlation is significant at the 0.05 level (2-tailed). Note. MA = mathematics anxiety. A direct-entry multiple regression analysis was also conducted to assess the effects

of children’s mathematics anxiety on their conceptual understanding of arithmetic on the

problem-solving task. Children’s conceptual understanding of arithmetic had a

significant effect on their mathematics anxiety, F(3, 119) = 6.25, p = .001, MSE = 824.27,

r = .373. More specifically, children’s accuracy, b = .70 [1.29, .33], p = .001, reaction

time, b = .19 [.000, .003], p = .036, and strategy use, b = .51 [.12, 1.28], p = .019, on the

problem-solving task had a significant effect on their mathematics anxiety. To check for

47

multicollinearity, the variance inflation factor (VIF) values were assessed. Results

showed that the VIF values for accuracy (i.e., 5.94), reaction time (i.e., 1.07), and

strategy use (i.e., 6.10) were less than 10. These findings suggest that multicollinearity

was not violated. However, the average VIF value of 4.37 was substantially greater than

1, which means that this regression model may be biased. These findings suggest that

children with higher mathematics anxiety had weaker conceptual understanding of

arithmetic on the problem-solving task compared to children with lower mathematics

anxiety. Overall, children’s mathematics anxiety had the largest impact on their accuracy

compared to their reaction time and strategy use on the problem-solving task.

3.3.2 Children’s mathematics anxiety and arithmetic fluency. Correlation

analyses were conducted to assess the relationships between children’s mathematics

anxiety and their arithmetic fluency on the timed mathematics task. Negative correlations

were found between children’s mathematics anxiety and the number of problems

attempted and their accuracy on the number of problems attempted on the timed

mathematics task suggesting that children with higher mathematics anxiety attempted

fewer problems and had lower accuracy on the number of problems they attempted on the

timed mathematics task compared to children with lower mathematics anxiety (see Table

5). Overall, children with higher mathematics anxiety had weaker arithmetic fluency on

the timed mathematics task compared to children with lower mathematics anxiety.

A direct-entry multiple regression analysis was also conducted to assess the effects

of children’s mathematics anxiety on their arithmetic fluency on the timed mathematics

task. Children’s arithmetic fluency had a significant effect on their mathematics anxiety,

F(2, 119) = 20.20, p < .001, MSE = 2282.12, r = .507. More specifically, the number of

48

problems children attempted, b = .26 [.628, .130], p = .003, and children’s accuracy on

the number of problems attempted, b = .35, [.377, .129], p < .001, on the timed

mathematics task had significant effects on children’s mathematics anxiety. To check for

multicollinearity, the VIF values were assessed. Results showed that the VIF value for

number of problems attempted (i.e., 1.17) and accuracy on the number of problems

attempted (i.e., 1.17) on the timed mathematics task were less than 10. These findings

suggest that multicollinearity was not violated. In addition, the average VIF value of 1.17

was not substantially greater than 1, which means that this regression model was likely

not biased. These findings suggest that children with higher mathematics anxiety had

weaker arithmetic fluency compared to children with lower mathematics anxiety.

Overall, children’s mathematics anxiety had the most significant effect on their accuracy

on the number of problems attempted compared to the number of problems they

attempted on the timed mathematics task. The current study showed that children’s

mathematics anxiety had a significant effect on their conceptual understanding of

arithmetic on the problem-solving task. Children with higher mathematics anxiety had

lower accuracy, slower reaction time, and used conceptually-based strategies less often

compared to children with lower mathematics anxiety. However, results should be

interpreted with caution as this regression model may be biased. Children’ mathematics

anxiety also had a significant effect on their arithmetic fluency on the timed mathematics.

Children with higher mathematics anxiety attempted fewer problems and had lower

accuracy on the number of problems attempted compared to children with lower

mathematics anxiety. Overall, the relationship was stronger between children’s

mathematics anxiety and their arithmetic fluency on the timed mathematics task

49

compared to the relationship between children’s mathematics anxiety and their



As part of an exploratory and secondary component, the current study explored

teachers’ mathematics anxiety and its effect on their students (see Table 6). A correlation

analysis was conducted to assess the relationship between teachers’ mathematics anxiety

and their students’ mathematics anxiety. However, no significant correlation was found

(see Table 7). This means that teachers’ mathematics anxiety did not impact their

students’ level of mathematics anxiety.

Table 6: Descriptive Statistics of Teachers’ Mathematics Anxiety on the A-MARS

Note. MA = mathematics anxiety.

50

Table 7: Correlation Matrix Showing the Relationships Between Teachers’ Mathematics Anxiety and their Students’ Mathematics Anxiety and their Students’ Mathematics Performance on the Problem-Solving Task and Timed Mathematics Task

Teachers’ MA

Students’ MA

Problem-Solving Task Timed Mathematics Task Students’

Awareness of their

Teachers’ MA

Accuracy

Reaction Time

Strategy Use

Number of

Problems Attempte

d

Accuracy

Teachers’ MA 1 .068 -.298** -.098 .244** -.015 -.149 -.001

Students’ MA .068 1 -.260** .236** -.246** -.370** -.399** .283**

Problem-Solving

Task

Accuracy -.298** -.260** 1 -.175 .916** .107 .535** .008

Reaction Time -.098 .236** -.175 1 -.225* -.268** -.254** .094

Strategy Use -.244** -.246** .916** -.225* 1 .081 .421** .002

Timed Mathematic

s Task

Number of

Problems Attempte

d

-.015 -.370** .107 -.268** .081 1 .353** -.107

Accuracy -.149 -.399** .535** -.254** .421** .353** 1 -.072

Students’ Predicted Level of their Teachers’

MA -.001 .283** .008 .094 .002 -.107 -.072 1

**. Correlation is significant at the 0.01 level (2-tailed). *. Correlation is significant at the 0.05 level (2-tailed). Note. MA = mathematics anxiety.

Correlation analyses were conducted to assess the relationships between teachers’

mathematics anxiety and their students’ conceptual understanding of arithmetic on the

problem-solving task. Negative correlations were found between teachers’ mathematics

anxiety and their students’ accuracy and strategy use on the problem-solving task

suggesting that teachers with higher mathematics anxiety had students with lower

accuracy and that used conceptually-based strategies less often on the problem-solving

task compared to teachers with lower mathematics anxiety (see Table 7). However, no

significant correlation was found between teachers’ mathematics anxiety and their

students’ reaction time on the problem-solving task. Overall, teachers with higher

51

mathematics anxiety had students with weaker conceptual understanding of arithmetic on

the problem-solving task compared to teachers with lower mathematics anxiety.

A direct-entry multiple regression analysis was also conducted to assess the effects

of teachers’ mathematics anxiety on their student’s conceptual understanding of

arithmetic on the problem-solving task. Results showed that teachers’ mathematics

anxiety had a significant effect on their students’ conceptual understanding of arithmetic

on the problem-solving task, F(3, 116) = 4.86, p = .003, MSE = 1563.08, r = .338.

However, when looked at more in depth, teachers’ mathematics anxiety impacts their

students’ accuracy, b = .44 [.647, .003], p = .052, but it does not impact their students’

reaction time, b = .15 [.004, .000], p = .110, or strategy use, b = .12, [.278, .489], p =

.587. Therefore, teachers’ mathematics anxiety did not impact their students’ overall


Correlation analyses were conducted to assess the relationships between teachers’

mathematics anxiety and their students’ arithmetic fluency on the timed mathematics

task. No significant correlations were found between teachers’ mathematics anxiety and

the number of problem their students’ attempted or between teachers’ mathematics

anxiety and their students’ accuracy on the number of problems attempted on the timed

mathematics task (see Table 7). Therefore, teachers’ mathematics anxiety also did not

impact their students’ arithmetic fluency on the timed mathematics task.

Furthermore, a correlation analysis was conducted to assess students’ ability to

predict their teachers’ level of mathematics anxiety. No significant correlation was found

between teachers’ mathematics anxiety and their students’ predicted level of their

teachers’ mathematics anxiety suggesting that students were unaware of their teachers’

52

mathematics anxiety (see Table 7). Interestingly, a positive correlation was found

between children’s mathematics anxiety and their predicted level of their teachers’

mathematics anxiety. Mean differences showed that children with higher mathematics

anxiety were more likely to rate their teachers’ level of mathematics anxiety as higher

compared to children with lower mathematics anxiety. Overall, these findings prompt

many questions for future research.

Lastly, a 3 (Grade: 4, 5, 6) x 2 (Sex: Male, Female) mixed model ANOVA was

conducted to assess the effects of grade and sex on teachers’ mathematics anxiety. There

were two main effects. First, there was a main effect for grade, F(2, 116) = 3.85, p =

.024, MSE = 1133.14, 𝜂p2 = .065. Mean differences showed that Grade 4 teachers had

higher mathematics anxiety compared to Grade 5 teachers but there was no significant

difference between Grades 5 and 6 or between Grades 4 and 6, HSD = 7.43 (see Figure

16). A main effect was also found for sex, F(1,116) = 9.98, p = .002, MSE = 2941.29, 𝜂p2

= .082. Female teachers had higher mathematics anxiety compared to male teachers (see

Figure 17). No interaction was found between grade and sex, F(2, 116) = 2.28, p = .107,

MSE = 671.78, 𝜂p2 = .039. Overall, grade and sex each had a significant effect on

teachers’ mathematics anxiety.

53

Figure 16. Teachers’ mathematics anxiety score on the A-MARS for grade (4, 5, and 6).

54

Figure 17. Teachers’ mathematics anxiety score on the A-MARS for sex (male and female).

55

CHAPTER FOUR

Discussion

The current study primarily focused on children’s mathematics anxiety and its

effect on their conceptual understanding of arithmetic and arithmetic fluency. Results

showed that children’s mathematics anxiety not only impacts their arithmetic fluency

(e.g., Tsui & Mazzocco, 2006), it also impacts their conceptual understanding of

arithmetic. This finding is critical because conceptual understanding of arithmetic allows

children to learn complex mathematics skills and concepts (e.g., Allard, 2011).

Children’s mathematics anxiety, therefore, inhibits their performance on mathematics

tests more significantly than previously thought. Results also confirm that females have

higher mathematics anxiety compared to males (e.g., Geist, 2010). Sex differences were

also evident in children’s conceptual understanding of arithmetic and their arithmetic

fluency. Furthermore, no grade differences were found in children’s mathematics

anxiety, arithmetic fluency, or conceptual understanding of arithmetic. Results displayed

that teachers’ mathematics anxiety did not interfere with teaching their students retrieval

of mathematical facts or why a mathematical procedural works. Teachers’ level of

mathematics anxiety also did not influence their students’ level of mathematics anxiety

and students were unaware of their teachers’ level of mathematics anxiety. Lastly, Grade

4 and female teachers had higher mathematics anxiety compared to Grade 5 and male

teachers. These findings and implications are discussed further.

4.1 Children’s Mathematics Anxiety and Performance on Mathematics Tests

4.1.1 Characteristics and development of children’s mathematics anxiety.

Research readily emphasizes the importance of cognitive abilities in performance onn

56

mathematics tests (e.g., Dowker et al., 2012). However, cognitive abilities are not the

only characteristic that influences performance on mathematics tests. Mathematics

attitudes, such as mathematics anxiety, also critically impact performance on

mathematics tests (e.g., Beilock, 2008; Harari et al., 2013). The current study examined

children’s mathematics anxiety and its effect on their conceptual understanding of

arithmetic and arithmetic fluency. The majority of previous research suggests that

mathematics anxiety emerges in Grade 3 (e.g., Wu et al., 2012). Wu and colleagues

(2012) attribute this to the Grade 3 mathematics curriculum that begins to focus on

complex mathematics concepts and operations. Even though the aim of the current study

was not to confirm the age of onset of mathematics anxiety, it does verify that Grades 4,

5, and 6 children do experience mathematics anxiety.

4.1.1.1 Grade differences in children’s mathematics anxiety. The general

consensus has been that mathematics anxiety, if not resolved, becomes more pronounced

over time (e.g., Ashcraft & Moore, 2009, Dowker et al., 2012; Walen & Williams, 2002).

The current study, however, does not support these findings. Instead, it found that there

was no effect of grade on children’s mathematics anxiety suggesting that mathematics

anxiety is relatively stable throughout Grades 4, 5, and 6. Beilock (2008), however,

found that mathematics anxiety begins to increase in Grade 5, peaking in Grades 9 and

10. Research attributes children’s increasing mathematics anxiety to the progressively

difficult mathematics curriculum (Ashcraft et al., 2007) as well as children becoming

more concerned about the consequences of their performance on mathematics tests (Gierl

& Bisanz, 1995). These findings, however, suggest that these factors may not have such

57

a prominent influence on children’s mathematics anxiety as previously thought; at least,

not in Grades 4, 5, and 6.

Conceptual understanding of arithmetic (e.g., Geary, 1994; Robinson & Ninowski,

2003; Sears, 2008) and arithmetic fluency (e.g., Geary, 1994; Ramos-Christian et al.,

2008) also typically improve over time. The current study found that Grade 5 children

were more accurate on the problem-solving task compared to Grade 4 children and that

Grades 5 and 6 children attempted more problems on the timed mathematics task

compared to Grade 4 children. However, there was no effect of grade on children’s

overall conceptual understanding of arithmetic or overall arithmetic fluency. Some

research supports these findings. For instance, Vilette (2002) found that children as

young as preschool had some conceptual understanding of arithmetic suggesting that

formal schooling does not influence children’s conceptual understanding of arithmetic.

Robinson and Dubé (2009b) similarly found that Grades 6, 7, and 8 did not influence

children’s arithmetic fluency. These findings could suggest that factors other than grade

may be responsible for improvements in children’s conceptual understanding of

arithmetic and arithmetic fluency. For instance, LeFevre, Skwachuk, Smith-Chant, Fast,

and Kamawar (2009) found that preschool children’s early experience playing with board

games was associated with their arithmetic fluency in Kindergarten, Grade 1, and Grade

2. Wolfgang, Stannard, and Jones (2001) similarly found that preschool children’s early

experience playing with blocks was associated with their performance on standardized

mathematics tests in high school. In other words, early mathematics experiences could

have a more meaningful impact on children’s performance on mathematics tests than

grade. However, other research has found grade differences in children’s conceptual

58

understanding of arithmetic (e.g., Dubé & Robinson, 2010; Robinson & Dubé, 2009a)

and arithmetic fluency (e.g., Calhoon et al., 2007). For instance, Dubé and Robinson

(2010) found that Grade 8 children used conceptually-based strategies more often than

Grade 6 children. Calhoon and colleagues (2007) found that older children consistently

had greater arithmetic fluency compared to younger children between Grades 2 through

6. This discrepancy in grade differences in children’s performance on mathematics tests

suggests that factors such as teaching strategies may also have a greater impact than

grade. For instance, Robinson and Dubé (2009a) explain that teachers often promote

methodological approaches to solving mathematical problems (i.e., left-to-right strategy),

which can interfere with children’s appeal to use conceptually-based strategies. As a

result, older children may actually have greater conceptual understanding of arithmetic

compared to younger children but fail to demonstrate such on mathematics assessments

because they are adhering to the rigid left-to-right strategies that they have been taught

(Robinson & Dubé, 2009a). Overall, these findings suggest that, at least in this sample,

formal schooling did not improve Grades 4, 5, and 6 children’s conceptual understanding

of arithmetic or arithmetic fluency and that children’s conceptual understanding of

arithmetic and arithmetic fluency develops slowly across Grades 4, 5, and 6.

4.1.1.2 Sex differences in children’s mathematics anxiety. Independent

from grade, females had higher mathematics anxiety compared to males. This is

consistent with previous research (e.g., Beilock et al., 2009; Geist, 2010; Wang et al.,

2014). Females’ higher mathematics anxiety has been attributed to lower confidence in

their mathematics ability (Geist, 2010) or possible mediating variables (e.g., spatial

ability; Maloney et al., 2012). However, it is unclear when sex differences become

59

evident in children’s mathematics anxiety. Hembree (1990) claims that sex differences in

children’s mathematics anxiety are evident across all grades. Other research shows that

sex differences do not emerge until adulthood (e.g., Harari et al., 2013). Gierl and Bisanz

(1995), conversely, found no sex differences in Grades 3 to 6 children’s mathematics

anxiety. Nevertheless, the findings from this study suggest that females had higher

mathematics anxiety compared to males.

The current study found that males had greater conceptual understanding of

arithmetic compared to females. This is consistent with some research (e.g., Baroody,

Lai, et al., 2009; Geary et al., 2000). However, the majority of research shows no sex

differences in children’s conceptual understanding of arithmetic (e.g., Dubé, 2014;

Robinson, Arbuthnott, et al., 2006; Robinson & Dubé, 2009b; Robinson & Ninowski,

2003). This could be explained by the negative relationship between mathematics

anxiety and performance on mathematics tests, which shows that children with higher

mathematics anxiety have weaker performance on mathematics tests compared to

children with lower mathematics anxiety (e.g., Beilock & Willingham, 2014; Maloney et

al., 2012). This relationship, though, has never been investigated with children’s

conceptual understanding of arithmetic. Overall, the current study shows that females

had greater difficulty understanding why a mathematical procedure works compared to

males.

The current study also found that males had greater arithmetic fluency compared to

females. This suggests that males more easily retrieved mathematical facts compared to

females. These findings are consistent with some research (e.g., Carr & Alexeev, 2011;

Royer et al., 1999) but not all (e.g., Carr & Davis, 2001; Geary et al., 2000). Carr and

60

colleagues (2008) found that Grade 2 males had greater arithmetic fluency compared to

Grade 2 females. This sex difference is often associated with strategy preferences with

males relying on retrieval strategies and females relying on computational strategies (e.g.,

Carr & Jessup, 1997; Carr et al., 2008). Conversely, Carr and Davis (2001) found no sex

differences in Grade 1 children’s arithmetic fluency suggesting that sex differences do

not emerge until Grade 2. Overall, the findings from this study support previous research

showing that sex differences in children’s arithmetic fluency are evident in Grades, 4, 5,

and 6.

4.1.2 Children’s performance on mathematics tests.

4.1.2.1 Children’s conceptual understanding of arithmetic. Conceptual

understanding of arithmetic allows children to solve mathematics problems more quickly,

more accurately, and with greater flexibility (Squire et al., 2004). Studying children’s

conceptual understanding of arithmetic is important because it is the foundation for which

children learn complex mathematical skills and concepts (e.g., Allard, 2011). The current

study examined children’s conceptual understanding of arithmetic using six arithmetic

concept problems (i.e., identity, negation, commutativity, inversion, associativity, and

equivalence) for both additive (i.e., addition and subtraction) and multiplicative problems

(i.e., multiplication and division).

Research shows that children with good conceptual understanding of arithmetic

recognize the inverse relationship between additive operations and between multiplicative

operations (Robinson, Ninowski, et al., 2006). Consistent with previous research (e.g.,

LeFevre & Robinson, 2010; Robinson, Ninowski, et al., 2006), the current study found

that children had greater conceptual understanding of additive problems compared to

61

multiplicative problems. According to Gelman (2009), conceptual understanding of

arithmetic is derived from experience. Multiplicative problems are, therefore, more

difficult for children compared to additive problems because multiplication and division

are learn after addition and subtraction (e.g., Dixon et al., 2001). Children with good

conceptual understanding of additive problems are more likely to also have good

conceptual understanding of multiplicative problems (e.g., Robinson, Ninowski, et al.,

2006). However, good conceptual understanding of additive problems does not guarantee

good conceptual understanding of multiplicative problems (e.g., LeFevre & Robinson,

2010). For instance, the current study found that children had greater conceptual

understanding of additive problems compared to multiplicative problems. These

findings, therefore, demonstrate that Grades 4, 5, and 6 children still struggle to achieve

good conceptual understanding of arithmetic. In particular, they struggle the most with

conceptual understanding of multiplicative problems. Spending more time on

multiplication and division problems in the classroom could help improve children’s

understanding of these operations and their inverse relations, consequently, improving

their overall conceptual understanding of arithmetic.

Children’s conceptual understanding of identity, negation, commutativity,

inversion, associativity, and equivalence were also examined. Children had good

conceptual understanding of identity and negation, some conceptual understanding of

commutativity and inversion, and weak conceptual understanding of associativity and

equivalence problems. This is consistent with previous research (e.g., Price, 2013).

Robinson and LeFevre (2012) explain that children have greater conceptual

understanding of arithmetic concepts that develop earlier compared to ones that develop

62

later. Consequently, children have good conceptual understanding of identity and

negation because they are the easiest arithmetic concepts to understand (e.g., Price,

2013). Research also shows that identity and negation can develop before formal

schooling and often simultaneously explaining why children’s conceptual understanding

of these arithmetic concepts is so similar (e.g., Baroody, Lai, et al., 2009). Nevertheless,

these findings support previous research that children have greater conceptual

understanding of identity compared to negation (Baroody, Lai, et al., 2009; Price, 2013).

Children also demonstrated greater conceptual understanding of commutativity

compared to inversion. This finding is consistent with previous research (e.g., Canobi

and Bethune, 2008). Squire and colleagues (2004) explain that children’s conceptual

understanding of commutativity can develop without formal schooling. For instance,

preschool children demonstrate some conceptual understanding of commutativity (Patel

& Canobi, 2010). Siegler and Stern (1998), conversely, show that children’s conceptual

understanding of inversion is slow and often requires help of formal schooling. More

specifically, children begin to develop conceptual understanding of inversion around

kindergarten (e.g., Bryant, Christie, & Rendu, 1999). Overall, these findings support

previous research that children have greater conceptual understanding of commutativity

compared to inversion (e.g., Price, 2013).

The most difficult arithmetic concepts for children to understand are associativity

and equivalence. Associativity and equivalence are likely the hardest arithmetic concepts

children learn because, unlike the rest of the arithmetic concepts, they require some

calculations. The current study found that children had greater conceptual understanding

of equivalence compared to associativity. Alternatively, Price (2013) found that Grades

63

5, 6, and 7 children had greater conceptual understanding of associativity compared to

equivalence. This discrepancy is likely due to children’ inconsistent development of

conceptual understanding of associativity (Gilmore & Bryant, 2006). The current study

expands on this by suggesting that the development of children’s conceptual

understanding of equivalence is also inconsistent. These findings support McNeil (2007)

who states that, although mathematics development typically increases with age or

experience, occasionally it declines before further improving. Overall, early cognitive

skills can have long-term impacts on children’s mathematics competency (Carr &

Alexeev, 2011). Therefore, it is important for children to begin learning these six

arithmetic concepts as early as possible for optimal conceptual understanding of

arithmetic.

4.1.2.2 Children’s arithmetic fluency. Arithmetic fluency allows children

to retrieve mathematical facts more quickly and more accurately (Lerner & Friesema,

2013). Studying children’s arithmetic fluency is important for developing and

understanding more advanced mathematical strategies and skill (Carr & Alexeev, 2011).

In particular, arithmetic fluency provides the foundation for development of children’s

conceptual understanding of arithmetic (Allard, 2011). The current study examined

children’s arithmetic fluency for simple addition, subtraction, multiplication, and division

problems.

Recognizing the inverse relationship between additive and multiplicative operations

is not only important for children’s conceptual understanding of arithmetic, it is also

important for children’s arithmetic fluency as it allows children to simplify their

mathematical tasks (Gilmore & Bryant, 2006). The current study found that children had

64

greater arithmetic fluency of addition compared to subtraction and greater arithmetic

fluency of multiplication compared to division. This is consistent with previous research

(e.g., Dixon et al., 2001, Robinson & Dubé, 2009b). Robinson and Dubé (2009b) explain

that multiplication and division are more difficult for children to understand because they

are discovered later and more slowly than addition and subtraction. Subsequently, it was

expected that children would have greater arithmetic fluency of addition and subtraction

problems compared to multiplication and division problems (e.g., Robinson & Dubé,

2009a; 2009b). The current study, however, found that children actually had good

arithmetic fluency on multiplication problems, some arithmetic fluency on addition

problems, and weaker arithmetic fluency on subtraction and division problems. This

suggests that Grades 4, 5, and 6 children struggle the most with subtraction and division.

This may be because children are less experienced with subtraction (e.g., Robinson &

Dubé, 2009a) and division (e.g., Robinson & Dubé, 2009b). The current study elaborates

that children’s low number of subtraction problems attempted and low accuracy on the

number of subtraction problems attempted suggests that children lack understanding of

this operation. Furthermore, children’s high number of division problems attempted and

low accuracy on the number of division problems attempted suggests that children either

also lack understanding of this operation or they are guessing. Nevertheless, these

findings further suggest that Grades 4, 5, and 6 children do not fully understand the

inverse relationship between addition and subtraction and between multiplication and

division. Improving children’s understanding these inverse relationships will not only

improve their arithmetic fluency, it will also improve their conceptual understanding of

arithmetic (Ramos-Christian et al., 2008).

65

4.1.3 Summary. To conclude, children with higher mathematics anxiety had

weaker conceptual understanding of arithmetic and weaker arithmetic fluency compared

to children with lower mathematics anxiety. This relationship was stronger in timed

mathematics task compared to the problem-solving task suggesting that mathematics

anxiety has a greater negative impact when there is a greater time pressure. Mathematics

anxiety, therefore, does not impact all types of mathematics performance equally.

Overall, minimizing time pressure on mathematics tests could help alleviate some of

children’s mathematics anxiety and, consequently, improve their retrieval of

mathematical facts and their understanding for why a mathematical procedure works.


As part of an exploratory and secondary component, the current study investigated

teachers’ mathematics anxiety and its impact on their students’ mathematics anxiety and

performance on mathematics tests. Although research on teachers’ mathematics anxiety

is limited, it does offer a valuable starting point for the current study. Interpretation of

these results should be done with caution due to the small sample size.

4.2.1 Teachers’ mathematics anxiety and its effect on their students’

mathematics anxiety. Teachers’ level of mathematics anxiety did not influence their

students’ level of mathematics anxiety. This is inconsistent with previous research (e.g.,

Hadley & Dorward, 2011). For instance, Hadley and Dorward (2011) demonstrate this

relationship between Grades 1 and 6 children. Finlayson (2014) explains that teachers

with higher mathematics anxiety report being less confident and less knowledgeable

about mathematics skills compared to teachers with higher mathematics anxiety, which

can impact students’ mathematics anxiety by not instilling confidence or knowledge in

66

their own performance on mathematics tests. This discrepancy could be due to the small

sample size. Alternatively, teachers’ mathematics anxiety may only impact their

students’ mathematics anxiety if they are aware of their teachers’ level of mathematics

anxiety. Finlayson (2014) found that students were aware of their teachers’ level of

mathematics anxiety and, consequently, teachers’ mathematics anxiety had an effect on

their students’ mathematics anxiety. Given that students in this sample were unaware of

their teachers’ level of mathematics anxiety, this could explain why teachers’

mathematics anxiety did not impact their students’ mathematics anxiety. Lastly, children

demonstrated biases in their predictions of their teachers’ level of mathematics anxiety,

associating it to a level of mathematics anxiety similar to their own. For instance,

children with higher mathematics anxiety were more likely to predict that their teacher

also had higher mathematics anxiety whereas children with lower mathematics anxiety

were more likely to predict that their teacher also had lower mathematics anxiety. This

relationship has never been addressed in previous research to my knowledge.

4.2.2 Teachers’ mathematics anxiety and its effect on their students’

performance on mathematics tests. Not only can teachers’ level of mathematics

anxiety impact their students’ level of mathematics anxiety, it can also impact their

students’ performance on mathematics tests (e.g., Dowker et al., 2012). The current

study uniquely examined the effects of teachers’ mathematics anxiety on their students’

conceptual understanding of arithmetic using arithmetic concept problems. For

comparison, it also examined the impact of teachers’ mathematics anxiety on their

students’ arithmetic fluency using timed mathematics tests. Results showed that

teachers’ mathematics anxiety did not impact their students’ conceptual understanding of

67

arithmetic or their arithmetic fluency. This suggests that teachers, despite their level of

mathematics anxiety, are capable of effectively teaching their students retrieval of

mathematical facts as well as why a mathematical procedure works. Interpretation of

these results, however, should be done with caution due to the small sample size.

4.2.3 Grade differences in teachers’ mathematics anxiety. The current study

found that Grade 4 teachers had higher mathematics anxiety compared to Grade 5

teachers. This is consistent with previous research, which shows that teachers in lower

grades have higher mathematics anxiety compared to teachers in higher grades (e.g.,

Finlayson, 2014). Ashcraft and Moore (2009) found that university students with higher

mathematics anxiety are most frequently enrolled in the faculty of education compared to

university students with lower mathematics anxiety because it is the undergraduate

degree with the least amount of mathematics requirements. However, Grade 6 teachers

did not have greater mathematics anxiety compared to Grades 4 and 5 teachers. This

discrepancy could be due to the small sample size. Nevertheless, these findings may

suggest that university students with higher mathematics anxiety are not only enrolling in

the faculty of education, they may also be choosing to teach grades with lower

mathematics expectations compared to university students with lower mathematics

anxiety.

4.2.4 Sex differences in teachers’ mathematics anxiety. Teachers’ mathematics

anxiety is also strongly impacted by sex. Consistent with previous research (e.g., Hadley

& Dorward, 2011), female teachers have higher mathematics anxiety compared to male

teachers. More specifically, the current study found that 25% of female teachers had high

mathematics anxiety whereas no male teachers had high mathematics anxiety. The lack

68

of male teachers with high mathematics anxiety is probably due to the small sample size

and potentially due to males being less willing to admit their mathematics anxiety

compared to females. Research elaborates that sex differences in mathematics anxiety

could also be due to family socialization (e.g., Finlayson, 2014). For instance, Finlayson

(2014) found that a mother’s attitude and encouragement towards mathematics was an

important predictor of children’s mathematics anxiety. Nevertheless, the current study

shows that females continue to report higher mathematics anxiety compared to males

well into adulthood.

4.3 Future Research

The current study encourages several directions for future research. Further

investigation into children’s conceptual understanding of associativity and equivalence

will help clarify their developmental trajectories and why these arithmetic concepts are

the most difficult for children to understand. Children’s mathematics anxiety should be

further examined. LeFevre and colleagues (2009) found that numeracy activities, such as

mathematics games, in preschool promote children’s arithmetic fluency in elementary

school. Given that children with higher performance on mathematics tests have lower

mathematics anxiety compared to children with lower performance on mathematics tests

(e.g., Maloney et al., 2012), early numeracy activities may also help alleviate children’s

mathematics anxiety. Therefore, research could explore the role of early mathematics

experience, rather than grade, on children’s mathematics anxiety. Replicating this study

could also prove to be beneficial. For instance, it could help clarify sex differences in

children’s conceptual understanding of arithmetic. Exploring the relationship between

children’s mathematics anxiety and their conceptual understanding of arithmetic prior to

69

Grade 4 and beyond Grade 6 could also explain how this relationship changes in

elementary school. Lastly, teachers’ mathematics anxiety and its effects on their students

should be further explored with a larger sample size for more meaningful results.

4.4 Limitations

Despite best efforts, there were a few possible limitations to the current study. No

grade differences were found in children’s conceptual understanding of arithmetic,

arithmetic fluency, or mathematics anxiety. This could be due to the portion of the Grade

6 sample (n = 9) that was recruited from an inner city community school whereas the rest

of the participants were recruited from residential public schools. These Grade 6 children

showed a considerable deficit in their mathematics skills compared to the rest of the

participants. Another possible limitation of the current study was assessing children’s

mathematics anxiety in groups of participants. This could be a limitation because a

couple children verbalized their answers during the assessment, which could have altered

other children’s answers on the MASC. The lack of participants (n = 46) included in the

analyses of children’s reaction time on the problem-solving task is also a possible

limitation as participants must have had at least one correct answer for each problem type

(i.e., identity, negation, commutativity, inversion, associativity, and equivalence). Lastly,

an obvious limitation of the current study was the small number of teacher participants in

the exploratory component of the current study.

4.5 Conclusion

In summary, there are two main findings. The current study extends previous

research by showing that children’s mathematics anxiety impacts not only their arithmetic

fluency (e.g., Tsui & Mazzocco, 2006) but also their conceptual understanding of

70

arithmetic. This is important because children’s conceptual understanding of arithmetic

is based on existing mathematical facts and relationships (e.g., Allard, 2011). As a result,

children who struggle to achieve conceptual understanding of arithmetic will likely not

develop understanding of higher, more complex mathematical concepts (e.g., Allard,

2011). Reducing children’s mathematics anxiety could, then, not only improve children’s

retrieval of mathematical facts, it could also improve their conceptual understanding of

why a mathematical procedure works. The current study also uniquely shows that the

relationship between mathematics anxiety and performance on mathematics tests is

stronger in conditions with a greater time pressure compared to conditions with less of a

time pressure. Results also showed that teachers, despite their level of mathematics

anxiety, are capable of effectively teaching their students retrieval of mathematical facts

and why a mathematical procedure works. Overall, the current study emphasizes the

impact of mathematics anxiety on children’s conceptual understanding of arithmetic. It

also highlights the significance of early identification of mathematics anxiety and

encourages further research on children and teachers’ mathematics anxiety as well as

corresponding interventions.

71

5. References

Alexander, L., & Martray, C. (1989). The development of an abbreviated version of the

Mathematics Anxiety Rating Scale. Measurement and Evaluation in Counseling

and Development, 22, 143-150. Retrieved from

https://www.researchgate.net/publication/232590588_The_development_of_an_abb

reviated_version_of_the_Mathematics_Anxiety_Rating_Scale

Alibali, M. W. (1999). How children change their minds: Strategy change can be gradual

or abrupt. Developmental Psychology, 35, 127-145. doi:10.1037/0012-

1649.35.1.127

Allard, E. (2011). Handbook of Latinos and Education: Theory, Research, and Practice.

doi:10.1111/j.1548-1492.2011.01133.x

Ashcraft, M. H. (2002). Math anxiety: Personal, educational, and cognitive consequences.

Current Directions in Psychology Science, 11, 181-185. doi:10.1111/1467-

8721.00196

Ashcraft, M. H., & Kirk, E. P. (2001). The relationship among working memory, math

anxiety, and performance. Journal of Experimental Psychology, 130, 224-237.

doi:10.1037/0096-3445.130.2.224

Ashcraft, M. H., Krause, J. A., & Hopko, D. R. (2007). Is math anxiety a mathematical

learning disability? In D. B. Berch & M. M. M. Mazzocco (Eds.), Why is math so

hard for some children? The nature and origins of mathematical learning

difficulties and disabilities (pp. 329-348). Baltimore, MD: Paul H. Brookes.

72

Ashcraft, M. H., & Moore, A. M. (2009). Mathematics anxiety and the affective drop in

performance. Journal of Psychoeducational Assessment, 27, 197-205.

doi:10.1177/0734282908330580

Baroody, A. J., & Dowker, A. (2009). The development of arithmetic concepts and skills:

Constructing adaptive expertise. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

Baroody, A. J., & Lai, M.-L. (2007). Preschoolers’ understanding of the addition-

subtraction inverse principle: A Taiwanese sample. Mathematics Thinking and

Learning, 9, 131-171. doi:10.1080/10986060709336813

Baroody, A. J., Lai, M.-L., Li, X., & Baroody, A. E. (2009). Preschoolers’ understanding

of subtraction-related principles. Mathematical Thinking and Learning, 11, 41-60.

doi:10.1080/10986060802583956

Baroody, A. J., Torbeyns, J., & Verschaffel, L. (2009). Young children’s understanding

and application of subtraction-related principles. Mathematical Thinking and

Learning, 11, 2-9. doi:10.1080/10986060802583873

Beck, L. V., Ghesquière, P., Lagae, L., & Smedt, B. D. (2014). Left fronto-parietal white

matter correlates with individual differences in children’s ability to solve additions

and multiplications: A tractography study. NeuroImage, 90, 117-127.

doi:10.1016/j.neuroimage.2013.12.030

Beesdo, K., Knappe, S., & Pine, D. S. (2009). Anxiety and anxiety disorders in children

and adolescents: Developmental issues and implications for DSM-V. Psychiatric

Clinics of North America, 32, 483-524. doi:10.1016/j.psc.2009.06.002

Beilock, S. L. (2008). Math performance in stressful situations. Current Directions in

Psychological Science, 17, 339-343. doi:10.1111/j.1467-8721.2008.00602.x

73

Beilock, S. L., Gunderson, E. A., Ramirez, G., & Levine, S. C. (2009). Female teachers’

math anxiety affects girls’ math achievement. Proceedings of the National

Academy of Sciences, 107, 1860-1863. doi:10.1073/pnas.0910967107

Beilock, S. L., & Willingham, D. T. (2014). Math anxiety: Can teachers help students

reduce it? American Educator, 38, 28-32. Retrieved from

http://hpl.uchicago.edu/sites/hpl.uchicago.edu/files/uploads/American%20Educator

,%202014.pdf

Bryant, P., Christie, C., & Rendu, A. (1999). Children’s understanding of the relation

between addition and subtraction: Inversion, identity, and decomposition. Journal

of Experimental Child Psychology, 74, 194-212.

doi:10.1016/j.cognition.2007.12.007

Caddle, M. C., & Brizuela, B. M. (2011). Fifth graders’ addition and multiplication

reasoning: Establishing connections across conceptual fields using a graph. The

Journal of Mathematical Behavior, 30, 224-234. doi:10.1016/j.jmathb.2011.04.002

Calhoon, M. B., Robert, W. E., Flores, M., & Houchins, D. E. (2007). Computational

fluency performance profile of high school students with mathematics disabilities.

Remedial and Special Education, 28, 292-303. doi:10.1177/0741932507028005041

Canobi, K. H. (2004). Individual differences in children’s addition and subtraction

knowledge. Cognitive Knowledge, 19, 81-93. doi:10.1016/j.cogdev.2003.10.001

Canobi, K. H., & Bethune, N. E. (2008). Number words in young children’s conceptual

and procedural knowledge of addition, subtraction and inversion. Cognition, 108,

675-682. doi:10.1016/j.cognition.2008.05.011

74

Canobi, K. H., Reeve, R. A., & Pattison, P. E. (2002). Young children’s understanding of

addition concepts. Educational Psychology, 22, 513-532.

doi:10.1080/0144341022000023608

Carr, M., & Alexeev, N. (2011). Fluency, accuracy, and gender predict developmental

trajectories of arithmetic strategies. Journal of Educational Psychology, 103, 617–

631. doi:10.1037/a0023864

Carr, M., & Davis, H. (2001). Gender differences in arithmetic strategy use: A function

of skill and preference. Contemporary Educational Psychology, 6, 330-347.

doi:10.1006/ceps.2000.1059

Carr, M., & Jessup, D. L. (1997). Gender differences in first-grade mathematics strategy

use: Social and metacognitive influences. Journal of Educational Psychology, 89,

318-328. doi:10.1037/0022-0663.89.2.318

Carr, M., Steiner, H. H., Kyser, B., & Biddlecomb, B. (2008). A comparison of predictors

of early emerging gender differences in mathematics competency. Learning and

Individual Differences, 18, 61-75. doi:10.1016/j.lindif.2007.04.005

Cates, G. L., & Rhymer, K. N. (2003). Examining the relationship between mathematics

anxiety and mathematics performance: An instructional hierarchy perspective.

Journal of Behavioral Education, 12, 23-34. doi:10.1023/A:1022318321416

Chernoff, E. J., & Stone, M. (2012). An examination of math anxiety research. Journal of

Educational Psychology, 52, 29-31. Retrieved from

http://search.proquest.com/docview/1563633518?accountid=13480

Chiu, L.-H., & Henry, L. L. (1990). Development and validation of the mathematics

anxiety scale for children. Measurement and Evaluation in Counseling and

75

Development, 23, 121. Retrieved from

https://www.academia.edu/11783024/The_development_and_validation_of_the_Ch

ildrens_Anxiety_in_Math_Scale

Devine, A., Fawcett, K., Szûcs, D., & Dowker, A. (2012). Gender differences in

mathematics anxiety and the relation to mathematics performance while controlling

for test anxiety. Behavioral and Brain Functions, 8, 33-41. doi:10.1186/1744-9081-

8-33

Dixon, J. A., Deets, J. K., & Bangert, A. (2001). The representations of the arithmetic

operations include functional relationships. Memory & Cognition, 29, 462-477.

doi:10.3758/BF03196397

Dowker, A., Bennett, K., & Smith, L. (2012). Attitudes to mathematics in primary school

children. Child Development Research, 2012, 1-8. doi:10.1155/2012/124939

Dubé, A. K. (2014). Adolescents’ understanding of inversion and associativity. Learning

and Individual Differences, 36, 49-59. doi:10.1016/j.lindif.2014.09.002

Dubé, A. K., & Robinson, K. M. (2010). The relationship between adults’ conceptual

understanding of inversion and associativity. Canadian Journal of Experimental

Psychology, 64, 60-66. doi:10.1037/a0017756

Finlayson, M. (2014). Addressing math anxiety in the classroom. Improving Schools, 17,

99-115. doi:10.1177/1365480214521457

Furner, J. M., & Berman, B. T. (2003). Math anxiety: Overcoming a major obstacle to the

improvement of student math performance. Childhood Education, 79, 170-174.

doi:10.1080/00094056.2003.10522220

76

Geary, D. C., Saults, S. J., Lui, F., & Hoord, M. K. (2000). Sex differences in spatial

cognition, computational fluency, and arithmetic reasoning. Journal of

Experimental Child Psychology, 77, 337-353. doi:10.1006/jecp.2000.2594

Geist, E. (2010). The anti-anxiety curriculum: Combating math anxiety in the classroom.

Journal of Instructional Psychology, 37, 24-31. Retrieved from

http://wendyharp73.wiki.westga.edu/file/view/MATH+anxiety+article.pdf

Geist, E. (2015). Math anxiety and the “mind gap”: How attitudes towards mathematics

disadvantages students as early as preschool. Education, 135, 328-336. Retrieved

from

http://go.galegroup.com.libproxy.uregina.ca:2048/ps/i.do?id=Gale%7CA40616256

5&v=2.1&v=uregina.lib&it=r&p=ITOF&sw=w&asid=39cb36c96d82ef4b66a0c38

63f1fb0fd

Gelman, S. A. (2009). Learning from others: Children’s construction of concepts. Annual

Review of Psychology, 60, 115-140. doi:10.1146/annurev.psych.59.103006.093659

Gierl, M. J., & Bisanz, J. (1995). Anxieties and attitudes related to mathematics in grade

3 and 6. Journal of Experimental Education, 63, 139-158.

doi:10.1080/00220973.1995.9943818

Gilmore, C. K., & Bryant, P. (2006). Individual differences in children’s understanding of

inversion and arithmetic skill. The British Psychological Society, 76, 309-331.

doi:10.1348/000709905X39125

Gresham, G. (2007). A study of mathematics anxiety in pre-service teachers. Early

Childhood Education Journal, 35, 181-188. doi:10.1007/s10643-007-0174-7

77

Hadley, K. M., & Dowker, J. (2011). Investigating the relationship between elementary

teacher mathematics anxiety, mathematics instructional practices, and student

mathematics achievement. Journal of Curriculum and Instruction, 5, 27-44.

doi:10.3776/joci.2011.v5n2p27-44

Harari, R. R., Vukovic, R. K., & Bailey, S. P. (2013). Mathematics anxiety in young

children: An exploratory study. The Journal of Experimental Education, 81, 538-

555. doi:10.1080/00220973.2012.727888

Hembree, R. (1990). The nature, effects, and relief of mathematics anxiety. Journal for

Research in Mathematics Education, 21, 33-46. Retrieved from

http://www.jstor.org.libproxy.uregina.ca:2048/stable/749455?pq-

origsite=summon&seq=1#page_scan_tab_contents

Hyde, J. S., Fennema, E., & Lamon, S. J. (1990). Gender differences in mathematics

performance: A meta-analysis. Psychological Bulletin, 107(2), 139-155.

doi:10.1037/0033-2909.107.2.139

Jackson, C. D., & Leffingwell, R. J. (1999). The role of instructors in creating math

anxiety in students from kindergarten through college. The Mathematics Teacher,

92, 583-586. Retrieved from

http://www.jstor.org.libproxy.uregina.ca:2048/stable/27971118?pq-

origsite=summon&seq=1#page_scan_tab_contents

Jameson, M. M. (2014). Contextual factors related to math anxiety in second-grade

children. The Journal of Experimental Education, 82, 518-536.

doi:10.1080.00220973.2013.813367

78

Kamii, C., Lewis, B. A., & Kirkland, L. D. (2001). Fluency in subtraction compared to

addition. The Journal of Mathematics Behavior, 20, 33-42. doi:10.1016/S0732-

3123(01)00060-8

Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2005).

Middle school students’ understanding of core algebraic concepts: Equivalence and

variable. Zentralblatt für Didaktik der Mathematik, 37, 68-76.

doi:10.1007/BF02655899

LeFevre, J., & Robinson, K. M. (2010). Use of conceptual knowledge among adults:

Experimental evidence. Unpublished manuscript.

LeFevre, J., Skwarchuk, S., Smith-Chant, B. L., Fast, L., & Kamawar, D. (2009). Home

numeracy experiences and children’s math performance in early school years.

Journal of Behavioural Sciences, 41, 55-66. doi:10.1037/a0014532

Lerner, J., & Friesema, A. (2013). Do timed tests hurt deep math learning? American

Teacher, 97, 3. Retrieved from http://sks.sirs.pnw.orc.scoolaid.net

Ma, X. (1999). A meta-analysis of the relationship between anxiety toward mathematics

and achievement in mathematics. Journal for Research in Mathematics Education,

30, 520-540. doi:10.2307/749772

Maloney, E. A., Waechter, S., Risko, E. F., & Fugelsang, J. A. (2012). Reducing the sex

differences in math anxiety: The role of spatial processing ability. Learning and

Individual Differences, 22, 380-384. doi:10.1016/j.lindif.2012.01.001

McNeil, N. M. (2007). U-shaped development in math: 7-year-olds outperform 9-year-

olds on equivalence problems. Developmental Psychology, 43, 687-695.

doi:10.1037/0012-1649.43.3.687

79

Newstead, K. (1998). Aspects of children’s mathematics anxiety. Educational Studies in

Mathematics, 36, 53-71. doi:10.1023/A:1003177809664

Nunes, T., Bryant, P., Hallett, D., Bell, D., & Evans, D. (2009). Teaching about the

inverse relation between addition and subtraction. Mathematical Thinking and

Learning, 11, 61-78. doi:10.1080/10986060802583980

Patel, P., & Canobi, K. H. (2010). The role of number words in preschoolers’ addition

concepts and problem-solving procedures. Educational Psychology, 30, 107-124.

doi:10.1080/01443410903473597

Prather, R. W., & Alibali, M. W. (2009). The development of arithmetic principle

knowledge: How do we know what learners know? Developmental Review, 29,

221-248. doi:10.1016/j.dr.2009.09.001

Preston, R. (2008). Mathematics anxiety: Research and implications for middle school

students and teachers. Conference Proceedings of the Masters in Teaching Program

2006-2008, Olympia, Washington.

Price, J. A. B. (2013). Children’s understanding of arithmetic concepts (Honour’s thesis).

University of Regina, Regina.

Ramirez, G., Gunderson, E. A., Levine, S. C., & Beilock, S. L. (2013). Math anxiety,

working memory, and math achievement in early elementary school. Journal of

Cognition and Development, 14, 187-202. doi:10.1080/1532-7647

Ramos-Christian, V., Schleser, R., & Varn, M. E. (2008). Math fluency: Accuracy versus

speed in preoperational and concrete operational first and second grade children.

Early Childhood Education Journal, 35, 543-549. doi:10.1007/s10643-008-0234-7

80

Richardson, F. C., & Suinn, R. M. (1972). The Mathematics Anxiety Rating Scale:

Psychometric data. Journal of Counseling Psychology, 19, 551-554.

doi:10.1037/h0033456

Rittle-Johnson, B., Matthews, P. G., Taylor, R. S., & McEldon, K. L. (2011). Assessing

knowledge of mathematical equivalence: A construct-model approach. Journal of

Educational Psychology, 103, 85-104. doi:10.1037/a0021334

Robinson, K. M., Arbuthnott, K. D., Rose, D., McCarron, M. C. Globa, C. A., &

Phonexay, S. D. (2006). Stability and change in children’s division strategies.

Journal of Experimental Child Psychology, 93, 224-238.

doi:10.1016/j.jecp.2005.09.002

Robinson, K. M., & Dubé, A. K. (2009a). Children’s understanding of addition and

subtraction concepts. Journal of Experimental Child Psychology, 103, 532-545.

doi:10.1016/j.jecp.2008.12.002

Robinson, K. M., & Dubé, A. K. (2009b). Children’s understanding of the inverse

relation between multiplication and division. Cognitive Development, 24, 310-321.

doi:10.1016/j.cogdev.2008.11.001

Robinson, K. M., & Dubé, A. K. (2009c). A microgenetic study of the multiplication and

division inversion concept. Canadian Journal of Experimental Psychology, 63,

193-200. doi:10.1037/a0013908

Robinson, K. M., & Dubé, A. K. (2012). Children’s addition concepts: Promoting

understanding and the role of inhibition. Learning and Individual Differences, 23,

101-107. doi:10.1016/j.lindif.2012.07.016

81

Robinson, K. M., & LeFevre, J. (2012). The inverse relation between multiplication and

division: Concepts, procedures, and a cognitive framework. Educational Studies in

Mathematics, 79, 409-428. doi:10.1007/s10649-011-9330-5

Robinson, K. M., & Ninowski, J. E. (2003). Adults’ understanding of inversion concepts:

How does performance on addition and subtraction inversion problems compare to

performance on multiplication and division inversion problems? Canadian Journal

of Experimental Psychology, 57, 321-330. doi:10.1037/h0087435

Robinson, K. M., Ninowski, J. E., & Gray, M. L. (2006). Children’s understanding of the

arithmetic concepts of inversion and associativity. Journal of Experimental Child

Psychology, 94, 349-362. doi:10.1016/j.jecp.2006.03.004

Royer, J. M., Tronsky, L. N., Chan, Y., Jackson, S. J., and Marchant III H. (1999). Math-

fact retrieval as the cognitive mechanism underlying gender differences in math test

performance. Contemporary Educational Psychology, 24, 181-266.

doi:10.1006/ceps.1999.1004

Scarpello, G. (2007). Helping students get past math anxiety. Techniques, 82, 34-35.

Retrieved from http://files.eric.ed.gov/fulltext/EJ775465.pdf

Sears, A. Y. (2008). An examination of the additive concepts of inversion and

associativity in children (Honour’s thesis). University of Regina, Regina.

Siegler, R. S., & Stern, E. (1998). Conscious and unconscious strategy discoveries: A

microgenetic analysis. Journal of Experimental Psychology, 127, 377-397.

doi:10.1037/0096-3445.127.4.377

82

Smith, S. Z., & Smith, M. E. (2006). Assessing elementary understanding of

multiplication concepts. School Science and Mathematics, 106, 140-149.

doi:10.1111/j.1949-8594.2006.tb18171.x

Squire, S., Davies, C., & Bryant, P. (2004). Does the cue help? Children’s understanding

of multiplication concepts in different problem contexts. British Journal of

Educational Psychology, 74, 515-532. doi:10.1348/0007099042376364

Suinn, R. M., Taylor, S., & Edwards, R. W. (1988). Suinn mathematics anxiety rating

scale for elementary school students (MARS-E): Psychometric and normative data.

Educational and Psychological Measurement, 48, 979-986.

doi:10.1177/0013164488484013

Tsui, J. M., & Mazzocco, M. M. M. (2006). Effects of math anxiety and perfectionism on

timed versus untimed math testing in mathematically gifted sixth graders. Roeper

Review, 29, 132-139. doi:10.1080/02783190709554397

Vilette, B. (2002). Do children grasp the inverse relationship between addition and

subtraction? Evidence against early arithmetic. Cognitive Development, 17, 1365-

1383. doi:10.1016/S0885-2014(02)00125-9

Voutsina, C. (2012). Procedural and conceptual changes in young children’s problem

solving. Educational Studies in Mathematics, 79, 193-214. doi:10.1007/s/10649-

011-9334-1

Walen, S. B., & Williams, S. R. (2002). A matter of time: Emotional responses to timed

mathematics tests. Educational Studies in Mathematics, 49, 361-378.

doi:10.1023/A:1020258815748

83

Wang, Z., Hart, S. A., Kovas, Y., Lukowski, S., Soden, B., Thompson, L. A., Plomin, R.,

McLoughlin, G., Bartlette, C. W., Lyons, I. M., & Petrill, S. A. (2014). Who is

afraid of math? Two sources of genetic variance for mathematical anxiety. Journal

of child Psychology and Psychiatry, 55, 1056-1064. doi:10.1111/jcpp.12224

Wigfield, A., & Meece, J. L. (1988). Math anxiety in elementary and secondary school

students. Journal of Educational Psychology, 80, 210-216. Retrieved from

http://www.rcgd.isr.umich.edu/garp/articles/wigfield88.pdf

Wilkins, J. L., Baroody, A. J., & Tiilikainen, S. (2001). Kindergarteners’ understanding

of additive commutativity within the context of word problems. Journal of

Experimental Child Psychology, 79, 23-36. doi:10.1006/jecp.2000.2580

Wolfgang, C. H., Stannard, L. L., & Jones, I. (2001). Block play performance among

preschoolers as a predictor of later school achievement in mathematics. Journal of

Research in Childhood Education, 15, 173-180. doi:10.1080/02568540109594958

Wood, E. F. (1988). Math anxiety and elementary teachers: What does research tell us?

For Learning of Mathematics, 8, 8-13. Retrieved from

http://www.jstor.org.libproxy.uregina.ca:2048/stable/pdf/40248135.pdf?acceptTC=

true

Wu, S. S., Barth, M., Amin, H., Mclarne, V., & Menon, V. (2012). Math anxiety in

second and third graders and its relation to mathematics achievement. Frontiers in

Psychology, 3, 1-11. doi:10.3389/fpsyg.2012.00162

Zakaria, E., & Nordin, N. M. (2008). The effects of mathematics anxiety on matriculation

students as related to motivation and achievement. Eurasia Journal of

84

Mathematics, 4, 27-30. Retrieved from

http://www.ejmste.com/v4n1/eurasia_v4n1_zakaria_nordin.pdf

85

6. Appendices

Appendix A: University of Regina Research Ethics Board Certificate of Approval

86

Appendix B: Additive Arithmetic Concept Problems

87

Appendix B: Additive Arithmetic Concept Problems (Continued)

88

Appendix C: Multiplicative Arithmetic Concept Problems

89

Appendix C: Multiplicative Arithmetic Concept Problems (Continued)

90

Appendix D: Addition Timed Mathematics Test

91

Appendix E: Subtraction Timed Mathematics Test

92

Appendix F: Multiplication Timed Mathematics Test

93

Appendix G: Division Timed Mathematics Test

94

Appendix H: Mathematics Anxiety Scale for Children (MASC) Questionnaire

95

Appendix I: Abbreviated Mathematics Anxiety Rating Scale (A-MARS) Questionnaire

children’s mathematics anxiety and its effect on …

Documents