Math anxiety and learn helpless

97
UNIVERSITY OF MIAMI MATHEMATICS ANXIETY AND LEARNED HELPLESSNESS By Joseph Franke Kolacinski A DOCTORAL TREATISE Submitted to the Faculty of the University of Miami in partial fulfillment of the requirements for the degree of Doctor of Arts Coral Gables, Florida August 2003 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Transcript of Math anxiety and learn helpless

Page 1: Math anxiety and learn helpless

UNIVERSITY OF MIAMI

MATHEMATICS ANXIETY AND LEARNED HELPLESSNESS

By

Joseph Franke Kolacinski

A DOCTORAL TREATISE

Submitted to the Faculty

of the University of Miami

in partial fulfillment of the requirements for

the degree of Doctor of Arts

Coral Gables, Florida

August 2003

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 2: Math anxiety and learn helpless

UMI Number: 3096372

Copyright 2003 by

Kolacinski, Joseph Franke

All rights reserved.

®

UMIUMI Microform 3096372

Copyright 2003 by ProQuest Information and Learning Company.

All rights reserved. This microform edition is protected against

unauthorized copying under Title 17, United States Code.

ProQuest Information and Learning Company 300 North Zeeb Road

P.O. Box 1346 Ann Arbor, Ml 48106-1346

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 3: Math anxiety and learn helpless

©2003 Joseph Franke Kolacinski

All Rights Reserved

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 4: Math anxiety and learn helpless

UNIVERSITY OF MIAMI

A treatise submitted in partial fulfillment of the requirements for the degree of

Doctor of Arts

MATHEMATICS ANXIETY AND LEARNED HELPLESSNESS

Joseph Franke Kolacinski

Approved:

Gilbert Cuevas, Ph.D.Professor of Teaching and Learning Co-Chair of the Treatise Committee

Marvin V. Mielke, Ph.D.Professor of Mathematics Co-Chair of the Treatise Committee

Robert L Kelley, Ph.D.Associate Professor of Mathematics Associate Chair of Mathematics Co-Chair of the Treatise Committee

Steven G. Ullmann, Ph.D Dean of the Graduate School

Scott Ingold, Ed Associate Dea and Registrar

Enrollment

Shulim Kaliman, Ph.D Professor of Mathematics

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 5: Math anxiety and learn helpless

KOLACINSKI, JOSEPH FRANKS (D.A., Mathematics)Mathematics Anxiety and Learned Helplessness (August 2003)

Abstract of a doctoral treatise at the University of Miami.

Treatise supervised by Professor Gilbert Cuevas, Professor Marvin Mielke and Professor Robert L. Kelley.No. of pages in text. 85

Two factors that have been shown to interfere with the learning of

mathematics are mathematics anxiety and learned helplessness. Mathematics

Anxiety is a negative emotional state associated with low mathematical

achievement. Learned helplessness is a response to uncontrollable adverse

stimulus that leads to motivational and cognitive deficits. This study explores the

relationship between these two phenomena.

In the first phase of this study, respondents were separated into four

attributional styles. Categories A1, A2, A3 and A4 consisted of the respondents

who tend to attribute failure in mathematics to lack of effort, environmental

factors, task difficulty and lack of ability respectively. Attribution Theory tells us

that the likelihood that a respondent would experience a helplessness response

increases as the index increases. It was therefore predicted that the mean

mathematics anxiety score of each of these categories would also increase as

the index increases. This study demonstrated this up to the limitations of the

data. Nothing could be inferred about category A2, because too few respondents

fell into that category, but otherwise the mean anxiety score of each category

showed a statistically significant increase coinciding with the known increased

likelihood of a helplessness response.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 6: Math anxiety and learn helpless

In the second phase of this study, students were exposed to an

intervention consisting of several methods known to alleviate or prevent

helplessness responses. Here it was predicted that the mathematics anxiety

score, a would decrease significantly between a pre-intervention survey and a

post-intervention survey. This did not happen. However, the mean value of a of

the comparison group increased significantly between the pre-intervention survey

and the post-intervention survey. This may indicate that the intervention

prevented a normal increase of mathematics anxiety for the experimental group.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 7: Math anxiety and learn helpless

In memory of my mother,

Mary Franke Kolacinski

i i i

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 8: Math anxiety and learn helpless

Acknowledgments

There were many people who played a roll in the completion of this

project. All of them have my thanks whether I mention them explicitly or not.

I’d first like to recognize my family and friends for their support and

encouragement over the years. Aside from the folks who will be mentioned

below, there are many people I’d like to thank here, but I will restrain myself and

mention only three; my brother, Richard Gadd, who always steers me in the right

direction when I ask him for advice, and two of my closest friends, Jason Foster

and Vicki Pearlman.

Probably the most difficult part of this process for me personally was

completing the paperwork for the Human Studies Committee. I’d like to thank

Ken Goodman, Julia Beutler and Linda Belgrave for their assistance with that

process.

Virtually everyone in the mathematics department was supportive over the

years. Professor Subramanian Ramakrishnan helped me to figure out which

statistical tests were appropriate to use and gave me some useful insight into

how to handle the data. Marta Alpar, Roneet Merkin, Patty Rua and Jay Stine

were gracious enough to allow me to collect data in their classes. Marta, Patty,

Jim Kell and Leticia Oropesa also helped collect data. All of them have my

gratitude. I would particularly like to thank Patty Rua for her assistance and

company.

Mike Rubino, a former student of mine, assisted by making a presentation

to the two classes in the experimental group. Mike is now himself in graduate

school and I wish him the best.

I count myself fortunate to have worked with each of the members of my

committee. Professors Marvin Mielke and Robert Kelley were co-chairs of the

iv

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 9: Math anxiety and learn helpless

committee, while Professors Shulim Kaliman and Scott Ingold were committee

members. Their advice and suggestions uniformly made this a better project.

The two people who probably heard the most about this project over the

years are also two of my best friends, Elvira Loredo and John Beam. Both of

them were great sounding boards, made solid suggestions and were extremely

helpful.

My most profound thanks go to the third co-chair of my treatise committee,

Professor Gilbert Cuevas. Dr. Cuevas always made himself available and taught

me a great deal as this project progressed. He was supportive of the decisions I

made, wise in his council and very patient. It due to his assistance that this

treatise has turned out as well as it has.

v

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 10: Math anxiety and learn helpless

Contents

1. Introduction 1

2. Review of Relevant Literature

Part 1 - Learned Helplessness 3

Treatment of Learned Helplessness 8

Part 2 - Mathematics Anxiety 12

Part 3 - Known Links and Some Speculations 19

3. The Study 32

4. Results, Conclusions and Recommendations 39

Instruments 39

Phase 1: The Correlational part of the Study 41

Developing an Attributional Model for Mathematics Anxiety 48

Phase 2: The Interventional part of the Study 52

Conclusions 56

References 63

Appendix A: The Phobos Inventory 66

Appendix B: The Mathematics Attribution Scale 68

Appendix C: Human Research Protocol Form 71

Appendix D: Informed Consent Memos 79

Tables and FiguresFigure 1 - Mean anxiety scores by attributional style 42

Figure 2 - Mean anxiety scores with confidence intervals 43

Figure 3 - The main effects plot 48

v i

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 11: Math anxiety and learn helpless

Table 1 - Participants in correlational study by course

Table 2 - Typical attributions, locus of control and

Susceptibility to Learned Helplessness

Table 3 - Attributional styles of the respondents in phase 1

Table 4 - ANOVA results

Table 5 - Tukey-Kramer paired comparison test results

Table 6 - Kruskal-Wallis test results

Table 7 - Mann-Whitney Test Results

Table 8 - Results of two-sample t-test assuming unequal

variance, A1 vs. A3

Table 9 - Results of two-sample t-test assuming unequal

variance, A3 vs. A4

Table 10 - Correlation Coefficients

Table 11 - Linear regression, a as a function of F-A, F-T,

F-EN and F-EF

Table 12 - Linear regression, or as a function of F-A and F-T

Table 13 - Results of the paired, two-sample t-test for means

comparing pre and post intervention values for the

experimental group

Table 14 - Results of the paired, two-sample t-test for means

comparing pre and post intervention values for the

comparison group

33

34

41

42

43

45

45

47

47

49

50

51

54

55

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 12: Math anxiety and learn helpless

■ Chapter I - Introduction

This study deals with some of the challenges and obstacles that students

face as they attempt to learn mathematics. More specifically, it addresses a

number of issues from the realm of “affective variables.” Laurie Hart Reyes says

that,affective refers to students’ feelings about mathematics, aspects of the

classroom, or about themselves as learners of mathematics. The definition is not intended to limit the affective domain to general feelings such as liking/disliking of mathematics, nor is it meant to exclude perceptions of the difficulty, usefulness and appropriateness of the mathematics as a school subject. [Reyes, 1984, p. 558]

Over the last few decades, affective variables, particularly mathematics anxiety,

have been studied in the context of such things as gender and cultural

differences in mathematics achievement and shortages of qualified people in

technical fields. [Evans, 2000] The most compelling reason to study affective

variables is to find ways to help students learn mathematics more effectively.

[Reyes, 1984]

Affective Variables include things like mathematics anxiety, mathematical

self-concept, the perceived usefulness of mathematics, the perceived difficulty of

mathematics, learned helplessness and attributional styles. [Evans, 2000; Reyes,

1984] Of these, this study focuses on mathematics anxiety and learned

helplessness. Studies have shown that there is a negative correlation between

mathematics anxiety and achievement in mathematics. [Evans, 2000] Other

1

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 13: Math anxiety and learn helpless

studies have established that learned helplessness produces a deficit in

mathematical performance. [Gentile & Monaco, 1986] A reasonable relationship

to explore is that between learned helplessness and mathematics anxiety.

The working hypothesis of this study is that there is indeed a relationship

between these two variables.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 14: Math anxiety and learn helpless

■ Chapter II - Review of Relevant Literature

Part 1 - Learned Helplessness

The term "learned helplessness" was first coined by M. E. P. Seligman

and J. B. Overmier (1967) to describe the effects of uncontrollable stimulus on

animals. In their earliest experiments a box was partitioned into two

compartments so that an electric shock could be administered through the floor

in either side of the box. This is referred to as a "shuttle box" because the shock

can be avoided by shuttling to the other side of the box. When a shock was

applied to one side of the box, a dog with no preconditioning that was placed in

the box would scramble across the barrier and escape the shock. Other dogs

were preconditioned with a series of shocks that they could not avoid. These

dogs thrashed about wildly when the shock was first applied, but would ultimately

lie down quietly and whine, failing to escape the shock. [Seligman & Maier,

1976]

In later studies it was determined that it was the uncontrollable nature of

the shocks, rather than the trauma of the shocks themselves that caused this

helplessness effect. To demonstrate this, M. E. P. Seligman and S. F. Maier

developed a triadic test design. Three groups of eight dogs were used. The

3

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 15: Math anxiety and learn helpless

4

dogs in the "escape group" were placed in a harness and were able to terminate

any shocks that were administered by pressing a panel with their nose. A

second group, the "yoked group" was administered shocks that they were unable

to control. Each dog in the yoked group received shocks identical in number,

duration and pattern to the shocks given to a dog in the escape group. The

control group received no shock in the harness. Twenty-four hours later the dogs

were tested in the shuttle box. The escape group and the control group were

able to learn to avoid the shocks easily, while six of the eight dogs in the yoked

group failed completely to avoid shock. Thus it was the uncontrollable nature of

the shocks rather than the trauma of the shocks themselves that caused the

helplessness effect. [Seligman & Maier, 1976]

In fact, a helplessness response can be evoked even when the adverse

events are not totally unavoidable. As long as the reinforcement is non­

contingent on the response, a helplessness effect is produced. An outcome is

non-contingent on a response if the probability that the response triggers the

desired outcome is equal to the probability that the outcome follows the absence

of the response. The subjects will therefore receive reinforcement whether they

respond or not. Thus a helplessness response is produced when the

reinforcement is or appears to be totally random and uncontrollable. [Mikulincer,

1994]

A study done by D. S. Hiroto in 1974 replicated the above results in

college students. Using a triadic design, the students were separated into three

groups. The students in the "escape group" were subjected to a painfully loud

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 16: Math anxiety and learn helpless

noise, which they were able to turn off by pressing a button. The "inescapable

group" was subjected to the same noise, but had no means of shutting off the

noise. The control group, also called the “no-noise group”, was not subjected to

the noise at all. All three groups were then subjected to the same type of noise

in a setting where they had the ability to turn the noise off. Subjects in the control

group and the escape group were able to quickly learn how to shut off the noise,

while the members of the inescapable group did not learn to escape the noise.

Most merely sat passively and endured the noise. [Hiroto & Seligman, 1975]

In addition to these motivational deficits caused by exposure to

uncontrollable adverse events, cognitive deficits have been demonstrated as

well. For example, college students with the same triadic test conditions as

above were given a series of 25 five-letter anagrams, each with the same

pattern. Two types of cognitive deficits were observed. A student from the

escape group or the no-noise group was able to solve each anagram much more

readily than a student exposed to inescapable noise, who would try and fail to

solve the first problem in the allotted 100 seconds. [Seligman & Maier, 1976]

Furthermore, a student from the escape group or the no-noise group

discovered the pattern after about three consecutive successes, while a subject

from the inescapable noise group needed approximately seven consecutive

successes before noticing a pattern. Thus exposure to uncontrollable events

made each individual anagram more difficult and inhibited the subject's ability to

notice or learn a pattern. Subsequent experiments showed that exposure to

unsolvable problems produced exactly the same effect as did inescapable noise.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 17: Math anxiety and learn helpless

6

[Seligman & Maier, 1976]

Studies on learned helplessness in humans indicate that the helplessness

response is related to a number of consequences of a person's personality, in

particular, the subject’s involvement in the unsolvable problem and their

expectations of control over their environment. [Dor-Shav & Mikulincer, 1992]

The causal attributions made by a subject in interpreting his or her failure

determine both the subject’s involvement in the problem and his or her

expectation of control. There are three dimensions to causal attribution: locus,

stability and globality. The locus dimension determines if the person will attribute

failure to internal or external causes. Stability is concerned with whether the

causes will remain steady over time and globality with whether the causes will

remain the same over many situations. In particular, one would expect that

attributing failure to internal, stable and global causes would lead to

helplessness, while attributing failure to external, unstable and specific causes

would lead to reactance, an improvement in performance on subsequent tasks.

[Abramson, Seligman & Teasdale, 1978]

A subject’s attributional style determines that subject’s characteristic

response to frustration, which can be categorized in four ways. Intrapersistent

subjects will respond to frustration by trying to reach their goals through their own

devices. Extrapersistent subjects will try to attain their goals through outside

assistance. Intrapersistance and Extrapersistance are collectively referred to as

"need persistence." Intrapunative subjects will respond to frustration by blaming

themselves, while extrapunative subjects will defend their egos by blaming an

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 18: Math anxiety and learn helpless

outside force. Together, these responses are known as "ego defensiveness."

[Dor-Shav & Mikulincer, 1992]

One of the salient differences between need persistence and ego

defensiveness subjects is their expectation of success after failure. Need

persistence subjects will expect to succeed in future tasks, while the ego

defensiveness subjects will expect failures to recur. In addition, an internal locus

is assumed to increase ego involvement, while an external locus is assumed to

decrease ego involvement since either the responsibility or the solution is shifted

to external forces. [Dor-Shav & Mikulincer, 1992]

Thus intrapersistant subjects will attribute failure to lack of effort, an

internal and unstable cause. This should lead to an optimistic attitude and ego

involvement and in turn reactance. Extrapersistant subjects will typically attribute

failure to bad luck, which is external and unstable. These subjects should then

show an optimistic attitude and a decrease in ego involvement, leading to

reactance, but to a lesser degree than the intrapersistant group.

The intrapunative group will blame failure on a lack of ability, leading to a

pessimistic attitude and ego-involvement and hence to helplessness. Finally the

extrapunative subjects should attribute failure to task difficulty, causing a

pessimistic attitude and low ego involvement. These subjects, then, should

display helplessness at a lower level than the intrapunative group. [Dor-Shav &

Mikulincer, 1992]

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 19: Math anxiety and learn helpless

Treatment of Learned Helplessness

It has been established that Learned Helplessness is a predictor of low

academic achievement. Consider a college student who fails his or her first

examination in a college course. If the student attributes the failure to an

unstable cause such as a headache, helplessness deficits will be short lived

because the headache will go away. If the student attributes the failure to a

stable cause such as lack of ability the deficits will tend to persist because the

perceived cause will persist. [Peterson, Maier & Seligman, 1993]

Several studies have also shown that interventions based upon

helplessness theory can be used to improve the grades of college students.

[Peterson, Maier & Seligman, 1993] Wilson and Linville used the reasoning

above to develop just such an intervention, which improved the grades of college

students. A group of college students were told that their grades would tend to

improve over the course of their college career. This encouraged them to

attribute initial failures and disappointments to unstable causes. In comparison

to the control group, which was not given this intervention, the grades of the

experimental group did, in fact, improve. [Wilson & Linville, 1985]

There are many techniques that are recommended to either treat an

existing helplessness response or to lessen the likelihood of a helplessness

response in a normal subject. The relevant question here, then, is whether these

techniques will also alleviate mathematics anxiety.

Gentile and Monaco created a list of strategies to prevent helplessness

responses in mathematics students. Many of their preventative methods deal

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 20: Math anxiety and learn helpless

9

with attributional style and frustration tolerance. Even early in a student’s

“mathematics career” it is useful to provide him or her with attributions involving

effort, persistence and strategy. Students should be made to understand that

certain mathematical concepts are difficult but can be mastered with adequate

effort and the correct strategy. Attributions that emphasize inherent ability and

aptitude should be avoided. [Gentile & Monaco, 1986]

Previous work should be reviewed and students should be shown how it

leads to the concepts that are currently being studied. Students should then be

given some early success experiences to help then fit the new concepts into their

cognitive schemata. [Gentile & Monaco, 1986]

It is also important to de-emphasize the dichotomy between success and

failure. First of all, the students’ work should be evaluated, with good thinking

and partially correct work noted. Problems should not merely be graded as right

or wrong. In addition, students should be required to correct their errors allowing

them to learn to do the problems correctly and giving them an eventual

successful outcome for their effort. Explanation of errors should give the

students a sense that they have some control over the outcomes of their exams,

instead of success and failure seeming to be random. [Gentile & Monaco, 1986]

Another way to circumvent a helplessness response is to provide the

subject with experiences in which he or she can control the outcome. This

process is sometimes known as “immunization.” This is equivalent to exposing

the dogs in the shuttle box experiments to escapable shocks before treating them

with inescapable shocks. In cases where animals were initially exposed to

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 21: Math anxiety and learn helpless

10

escapable shocks, the helplessness effect was virtually eliminated. [Peterson,

Maier & Seligman, 1993] Methods for treating a helplessness response are

similar to the methods for preventing such a response.

Depression and helplessness are most likely to occur and will be most

severe whenever:

(a) The perceived probability of a positive outcome is very low or the

perceived probability of an adverse outcome is very high.

(b) The outcome in question is either highly adverse or highly desirable.

(c) The subject suspects that the outcome is uncontrollable, or...

(d) The subject attributes the uncontrollability to global, stable and internal

factors. [Abramson, Seligman & Teasdale, 1978, pp. 68-70]

These four areas suggest methods for treating a helplessness response.

If the perceived probability of an adverse outcome is very high or the

perceived probability of a positive outcome is very low, one can attempt to

change the subject’s perception of the probabilities of these events. This can be

accomplished by manipulating the environment to remove adverse outcomes or

provide for desirable outcomes. If it is possible, the actual probabilities of the

outcomes can be altered. If the outcome in question is either highly adverse or

highly desirable, the subject’s perceptions can again be modified. In this case

the adverse outcome can be made to seem less adverse, while the very

desirable outcome can be made to seem less desirable.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 22: Math anxiety and learn helpless

11

These things can be accomplished through helping the subject reevaluate

unattainable goals and encouraging him or her to replace them with realistic

and/or attainable alternatives. [Abramson, Seligman & Teasdale, 1978]

In the event that the subject perceives an outcome to be beyond his or her

control, this perception can be changed. If a successful response is not yet

within the person’s ability, the necessary skills can be learned. Otherwise, the

subject’s erroneous belief that his or her responses will fail must be modified.

One means to this end is to provide for the performance of relevant and

successful responses from the subject. This is equivalent to repeatedly dragging

the dog in the shuttle box across the barrier until it learns that the shock is in fact

avoidable. After this therapy is repeated 30 or 40 times, the dog begins to

respond on its own. [Peterson, Maier & Seligman, 1993]

Changing the subject’s attribution of past failures from a lack of ability to a

lack of effort can also increase a subject’s expectation of controllability. This has

been shown to lead to more successful responses. This is generally referred to

as “attributional retraining.” Attributional retraining also forms the basis for

treatment in the situation in which a subject has attributed uncontrollability to a

global, stable and internal factor. If this attribution can be replaced with an

attribution to a specific, unstable and external factor, the subject should become

less likely to exhibit a helplessness response and more likely to show reactance.

[Abramson, Seligman & Teasdale, 1978] Most of the available studies involving

academic applications have used some form of attributional retraining. As

mentioned above, Wilson and Linville showed that changing attributions from

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 23: Math anxiety and learn helpless

12

stable to unstable led to increased academic performance among college

students. Other studies have worked with changing the “lack of ability” attribution

to an attribution of “lack of effort”, also a shift from stable to unstable. Evidence

also exists that a change in attribution from an internal factor to an external one

has led to an alleviation of a helplessness response. [Wilson & Linville, 1982]

Part 2 - Mathematics Anxiety

Unlike "learned helplessness" which is clearly defined and carefully

researched, Mathematics Anxiety lacks even a universally accepted definition

[Reyes, 1984, p. 563]. In order to better understand what is involved with

mathematics anxiety, we will begin our discussion with a look at the general

concept of anxiety as viewed by psychologists.

The notion of “anxiety” as a psychological construct dates back at least as

far as Freud, who characterized it by motor innervations or discharges, a

consciousness of these and feelings of “unpleasure.” An essential part of

Freud’s understanding of anxiety was that, because it is unpleasant, anxiety is

likely to be repressed or internalized and reside mainly in the unconscious. Once

there, the effects of the anxiety become masked or distorted, making the source

difficult to discern. Because, in Freud’s view, anxiety can be unconscious,

subjects cannot report their anxiety in a meaningful way, nor can it be readily

observed. [Evans, 2000]

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 24: Math anxiety and learn helpless

13

After the Second World War, the philosophy of the psychology community

shifted in part to make psychological work more scientific. So that data could be

more readily collected, anxiety was assumed to be an observable phenomenon

and as something that subjects could reliably report. Thus, “anxiety” came to be

understood merely as manifest or expressed anxiety. If a subject did not report

anxiety, it was assumed there was none. Freud’s ideas fell into disuse. [Evans,

2000]

As this reinterpretation of anxiety continued, psychologists began to

question whether to view anxiety as a constant personality trait or as a response

to stimulus. Spielberger (1970, 1972) attempted to answer this question when

he defined anxiety as “a palpable, but transitory, emotional state or condition

characterized by feelings of tension and apprehension and heightened

autonomic nervous system activity.” Although this has some similarity to Freud’s

definition it is important to note that Spielberger’s ideas, like other late 20th

century work, view anxiety purely as a manifest phenomenon. [Evans, 2000] In

Spielberger’s work, two types of anxiety are discussed. The first, the “A-state” or

state anxiety is defined as the “unpleasant emotional state or condition which is

characterized by activation or arousal of the autonomic nervous system.” State

anxiety occurs specifically when an individual is confronted with an experience

that he or she finds threatening or stressful. The second type, trait anxiety, or the

A-trait, can be described as a propensity toward experiencing state anxiety.

Thus, an individual with a high level of trait anxiety will tend to experience the A-

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 25: Math anxiety and learn helpless

14

state more severely and more frequently that an individual who is less trait

anxious. [Reyes, 1984].

The term “anxiety” has numerous definitions in the psychological literature.

In many expositions, “anxiety” is distinguished from “fear” even though these

terms are used interchangeably in other contexts. In some expositions, fear is

the intellectual process of assessing something and recognizing or believing that

it is a threat. Anxiety is the emotional and physiological response to

encountering that, which is feared. Aaron Beck writes: “A person with a fear of

small animals perceives these animals to be dangerous. However, he does not

experience anxiety until he finds himself exposed to a small animal or imagines

himself in such a situation.” [Beck, Emery & Greenberg, 1985, p. 9] Notice that

in both these definitions the primary definition of anxiety refers to an immediate

response to stimulus. Other expositions invert these notions of fear and anxiety.

In Learned Helplessness. A Theory for the Age of Personal Control, by Peterson,

Maier and Seligman, fear is described as a set of reactions, emotional,

physiological and behavioral, triggered by explicit signals that danger is present.

Anxiety, on the other hand, involves similar emotional, physiological and

behavioral reactions, but without an overt or clearly definable stimulus. Under

these definitions, the intensity of the reactions evoked by anxiety are not justified

by the perceived cause of the anxiety. [Peterson, Maier & Seligman, 1993, p. 70]

Anxiety or state anxiety provokes a number of reactions, both behavioral

and physiological. The immediate physiological reactions can include sweating,

trembling, muscle tension, increased heart rate and other symptoms and will

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 26: Math anxiety and learn helpless

15

cause an individual to cope with the perceived threat through various possible

methods. These methods include, but are not limited to, inaction,

combativeness, repression and rationalization. Some of these responses will

improve an individual’s performance but, more often, a response to the A-state

will cause a person’s performance to decrease and may even have a negative

impact on the individual. Anxiety that leads to a positive outcome such as

increased performance is called “facilitative” while, if the consequences are

negative the anxiety is said to be “debilitative.” [Reyes, 1984].

The most commonly accepted definition of mathematics anxiety,

introduced by Richardson and Suinn in 1972 draws on these notions.

“Mathematics Anxiety,” they say, “involves 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." Another

widely accepted definition, by Byrd (1982), includes both facilitative and

debilitative anxiety occurring in any situation in which mathematics is

encountered. Both of these definitions view mathematics anxiety as a form of

state anxiety experienced in mathematical situations. Other researchers view

mathematics anxiety merely as a lack of confidence in one’s ability to learn

mathematics. [Reyes, 1984, p. 565]. Some even take a broader view. In her

book Defeating Math Anxiety. Anita Kitchens writes, “Any feeling that prevents

you from learning mathematics in a natural way as you did as a young child, or

from performing in a way that demonstrates what you have learned, is math

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 27: Math anxiety and learn helpless

16

anxiety.” [Kitchens, 1995, P. 7] This interpretation, one could argue, could

include the entire range of affective variables.

People suffering from mathematics anxiety typically recall their initial

realization that they were having difficulty with mathematics as feeling like

"sudden death." This reaction has been documented at all levels of mathematics

education from elementary to graduate school. It is generally coupled with a

feeling that some new concept or operation is completely incomprehensible.

Many have described it as a curtain being drawn or running into an impassible

wall. The feeling is described as "instant and frightening."

This sudden onset is thought to stem from students' tendency to resort to

learning by rote when faced with difficult concepts. They will try to memorize all

the information that they need to get by without trying to understand the

underlying concepts. This causes a latency period in mathematics anxiety during

which, because grades are not at first affected, neither the student nor the

teacher realizes that there is a problem with the student's understanding of the

material. When the pattern of reiterated failure inevitably occurs, it is sudden and

dramatic. No one is aware that the difficulty may have been building for years.

[Tobias, 1992]

It is possible that, prior to the "sudden and dramatic” realization above, a

student’s initial response to these repeated failures will be to become over­

anxious and try harder to comprehend or memorize the formulas the student

feels will help him or her to regain control. The student’s anxiety becomes

debilitative, actually exacerbating the problem and diminishing the student’s

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 28: Math anxiety and learn helpless

17

ability to get back on track. The more anxious the student becomes, the harder

he or she tries. The harder he or she tries, the worse the student is able to do.

The worse the student is able to do, the more anxious he or she becomes. Thus

a "vicious cycle" is set into motion. This is an example of the Yerkes-Dodson

law, which states that increased motivation is actually detrimental to performance

in more complex tasks. The greater the complexity of the task, the lower the

degree of motivation that leads to optimal performance. [Skemp, 1987]

These repeated failures then cause further exposures to tasks related to

these concepts to become highly charged with emotion. We then have a

situation in which it is previous experience, rather than the task at hand, that is

the strongest determining factor in the student's success or failure. People

exposed to repeated failure "find that there is a great deal of inertia in even

attempting a problem." [Buxton, 1991, p. 115]

There have been several attempts to build a cohesive theory of

mathematics anxiety. Laurie Buxton built a model around what he calls “math

panic.” Buxton considered the interaction of reason and emotion. In a situation

in which a subject feels a threat of impending failure, such a strong negative

emotional response is evoked that the person is incapable of reasoning

effectively, shutting off all reasonable thought about the problem. [Buxton, 1991]

Another model, developed by Byrd (1982), took more of a case study approach.

It was built from interviews with six college students who showed signs of

mathematics anxiety. These students experienced anxiety in a variety of settings

and in virtually every setting it was debilitating. This anxiety caused these

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 29: Math anxiety and learn helpless

18

students to avoid courses, majors, jobs, careers and colleges with stringent

mathematics requirements. In both of these models we again see mathematics

anxiety viewed as a form of state anxiety. [Reyes, 1984] Joseph and Nancy

Martinez, in their book Math Without Fear cast a wider net. Mathematics anxiety

“is not a discrete condition.” they write, it is a complex construct. It seldom has a

single cause or a single effect. They continue, “It has multiple causes and

multiple effects, interacting in a tangle that defies simple diagnosis and simplistic

remedies. [Martinez & Martinez, 1996, p. 2]

One factor that is consistent is the belief that there is a relationship

between mathematics anxiety and mathematical achievement. Studies have

found a consistent negative correlation between these two factors with low

anxiety linked to high achievement, although a cause and effect relationship has

not been demonstrated. [Reyes, 1984]

Further confounding the issue, a number of weaknesses have been

pointed out regarding these studies. Jeff Evans points out one of the most

significant in his book Adults' Mathematical Thinking and Emotions. Evans

looked at the two most commonly used measures of mathematics anxiety, the

Mathematics Anxiety Scale (MAS) (Fennema and Sherman, 1976) and the

Mathematics Anxiety Rating Scale (MARS) (Richardson and Suinn, 1972). In a

review of the items on each inventory, he found that in both cases mathematics

anxiety was viewed as debilitating and that the mathematics anxiety measured

was assumed to be a form of chronic or trait-anxiety. [Evans, 2000]

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 30: Math anxiety and learn helpless

19

Thus, even putting aside competing definitions, there are a number of

reasonable concerns with the notion of anxiety in general and mathematics

anxiety in particular. One is that modern conception of anxiety disregards the

effects of anxiety in the subconscious, although these may be relevant. Another

is that, when a model of mathematics anxiety is specific enough to be potentially

useful, that model views mathematics anxiety as a form of state anxiety, while

the instruments we use to study mathematics anxiety view it as a form of trait

anxiety limiting the scope and the usefulness of the studies. [Evans, 2000]

Clearly, better models are called for.

Part 3 - Known Links and Some Speculations

Even in the early animal experiments that formed the basis of learned

helplessness theory there was evidence of a relationship between the notions of

fear and anxiety on the one hand and the notion of learned helplessness on the

other. In studying animals that were exposed to uncontrollable stressors, such

as the inescapable shocks that were part of the shuttle box experiments, it was

observed that these stressors cause fear in the animals. These generated more

fear, in general, than controllable stressors and following the exposure to the

uncontrollable stressors, the animals showed signs of anxiety for an additional 48

to 72 hours. [Peterson, Maier & Seligman, 1993]

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 31: Math anxiety and learn helpless

20

There are experiments that indicate that increased levels of fear and

anxiety at the time of the exposure to the inescapable shock are necessary to

cause a helplessness effect. Tranquilizers such as diazepam, which reduce

feelings of fear and anxiety, have been administered to rats prior to their

exposure to inescapable shock. In these rats, the subsequent deficits in shuttle

box learning were eliminated and no helplessness effect was observed.

[Peterson, Maier & Seligman, 1993]

Observations done at the time of the shuttle box testing readily

demonstrated that both fear and helplessness responses could be measured in

the same animals and that the two seemed to coincide. In order to determine if

the learning deficits attributed to helplessness theory were actually the effects of

fear and anxiety, a number of rats were exposed to either inescapable shock or

no shock at all. Twenty-four hours later these same rats were tested in a

standard shuttle box environment. Before the shuttle box tests, the rats were

injected with diazepam, naltrexone or a control substance. The researchers

observed the rats for the presence and intensity of behavior associated with fear

and anxiety. Diazepam eliminated the fear experienced by the rats when the

shock occurred, but the subsequent helplessness effect was unaffected.

Naltrexone was already known to alleviate the helplessness response. In this

study the rats injected with naltrexone experienced a greater level of fear and

anxiety yet the helplessness effect was still eliminated and these rats learned at

the same rate as those that did not receive the inescapable shock. Thus,

although uncontrollable stressors produce high levels of fear and anxiety, it

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 32: Math anxiety and learn helpless

21

appears that the level of fear and anxiety at the time of the shuttle box testing

does not cause the helplessness effect. [Peterson, Maier & Seligman, 1993]

Let us now turn our attention to mathematics anxiety. Consider the

prevalent views of mathematics in the population. Many parents and teachers

often treat mathematics as something difficult and mysterious. These people

present mathematics as something painful and worthy of fear while showing that

they fear it themselves. Mathematics is also presented as being very important

for success in later life, but something only within the grasp of the very bright.

[Tobias, 1992, pp. 52-53] In addition, an inability to do mathematics is treated as

a normal state of affairs and nothing to be ashamed of. People will readily

discuss their inability to do mathematics problems, while no one will freely admit

to being unable to read. [Zaslavsky 1994, p. 5] Although these attitudes are cited

as causes of mathematics anxiety, they also coincide with the conditions that

make a helplessness response more likely or more severe. We can speculate

that the emphasis on the importance of mathematics to success in later life

would, to the student struggling with mathematics, make an adverse outcome

seem to be very likely. Abramson lists this among the things that increase the

likelihood and severity of a helplessness response. [Abramson, Seligman &

Teasdale, 1978, pp. 68-70] The beliefs that only the very bright can do

mathematics and that it is normal for people to have difficulties in mathematics

perpetuate the notion that some people can learn mathematics and others

cannot. This encourages those who have trouble in mathematics to attribute

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 33: Math anxiety and learn helpless

22

their failure to a lack of ability, the internal and stable cause that makes a

helplessness response most severe.

Within this context, then, consider the impact of constant failures on a

student trying to learn mathematics. Suppose that student worked a problem

incorrectly, and the reason that the work was incorrect was not explained, or

explained in a way he or she did not understand. To the student, the adverse

event, getting a bad grade or losing points because of an incorrect answer would

be non-contingent on whatever effort the student put into the problem or into

studying. Consider the impact of seemingly getting problems wrong or right at

random. It is reasonable to expect that this apparent lack of control on the part of

the student could manifest itself as a helplessness response. Like the dog or the

rat that is exposed to inescapable shocks, the student may stop trying because

whatever he tries seems to have no effect on the eventual outcome.

Experiments in learned helplessness have shown that exposure to

unsolvable problems leads to a decreased performance on subsequent tasks.

Extrapolating this situation to math students, exposing a student to a question

beyond his or her ability could well trigger a helplessness response and thus the

ordering of questions on an exam should have a meaningful impact on student

test performance. Towle and Merrill (1972) found this was the case on a test

consisting of mathematical items. In their study, students who were given a test

on which the items were ordered from difficult to easy scored significantly worse

than those who took exams on which the ordering of the questions was either

random or from easy to difficult. [Spies-Wood, 1980] Richard Skemp, in his

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 34: Math anxiety and learn helpless

23

book, The Psychology of Learning Mathematics, attributes the same effect to

anxiety:

...a good teacher can, by initially asking questions that the learner can answer, reduce anxiety and build up confidence, and thereby improve performance: a bad teacher can reduce an averagely intelligent pupil to tongue- tied incompetence. [Skemp, 1987, p. 94]

The genesis of much of the ongoing research into mathematics anxiety

was actually a study of “math avoidance” especially by females. In 1974, Lucy

Sells circulated a report on the mathematical preparation of entering freshmen at

the University of California, Berkeley. The study showed that 57 percent of the

male students in the freshman class had completed four years of high school

mathematics. Only eight percent of the female students had a comparable math

background. Two years later, Sheila Tobias began to popularize the concept of

mathematics anxiety to partially explain this math avoidance and other gender

differences in mathematics. Her work on math anxiety formed the basis for a

number of “math clinics” and other interventions aimed at attracting more women

into mathematical fields. [Tobias & Weissbrod, 1980, p. 64] Tobias’ work had an

immediate impact. For the first time, educators began looking at mathematics

anxiety as a psychological state that could explain these differences rather than

focusing on skills deficits. It was quickly shown that native ability was not the

underlying cause of the gender differences in math avoidance and performance.

Unfortunately, mathematics anxiety also fails to explain these gender

differences. Although women have been shown to display, or at least report,

significantly higher levels of mathematics anxiety, these do not explain gender

differences in mathematical performance or math avoidance. In fact, at least

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 35: Math anxiety and learn helpless

24

among high school students, these “anxiety effects” have been shown to

manifest themselves more strongly in male students. Possible explanations for

this apparent contradiction include a greater willingness on the part of girls and

women to admit to feelings of anxiety and the possibility that women are more

capable of coping with feelings of anxiety than are men. [Hembree, 1990]

Early in the study of these gender differences it was established that it

was persistence rather than ability that accounted for the gender differences in

mathematics performance. Numerous studies in the late seventies showed that

there is no biological explanation for these differences in mathematical

performance. At the same time, other factors were shown to be linked to gender.

Prominent among these were expectations of success and attributional style. In

general, women and girls were shown to be more likely to attribute failure in

mathematics to a lack of intelligence, while men and boys tend to attribute such

failures to a lack of effort. [Tobias & Weissbrod, 1980]

Notice that this would be consistent if it were a helplessness effect rather

than mathematics anxiety that was at the root of these gender differences. Thus,

even if levels of mathematics anxiety were equal, men and women would have

very different ways of responding to them. Since women tend to attribute failure

to ability, they are, as we have seen, much more likely to exhibit a helplessness

response. [Dweck & Licht, 1980] They would tend to give up and avoid working

at and studying mathematics because they believe that whatever effort they

invest in the process will have little effect on the eventual outcome. Men on the

other hand, tend to attribute failures in mathematics to a lack of effort. This tends

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 36: Math anxiety and learn helpless

25

to make them less likely to exhibit a helplessness response and more likely to

experience reactance, leading to the increased effort and the greater persistence

identified by researchers as being responsible for the greater mathematical

performance of men.

The mathematics anxiety explanation for math avoidance follows a similar

vein. Although it has never been put forth as a single and unique explanation for

math avoidance, many researchers believe that, for some people, previous

experiences with mathematics could be severe enough and unpleasant enough

to effect their decisions regarding academic programs and career goals. The

general relationship is that, to avoid the unpleasant feelings of mathematics

anxiety, some students will stop studying mathematics. [Tobias & Weissbrod,

1980] Again this could be consistent with a helplessness response as those

earlier unpleasant experiences could be such that they convince a person that

their successes and failures in mathematics are not contingent on their actions.

In this case the person in question would avoid mathematics due to a

helplessness response.

Much has been made in recent years about the differences in the

mathematical achievements of Asian children and American children. When we

look at the attributional styles of these two groups, we see the same thing that we

saw in the comparison of American men and women. When asked why some

students don’t do as well in mathematics as others, American children point to a

lack of ability, while Asian children appeal to a lack of effort. [Tobias, 1992] As is

the case when comparing the mathematical achievement of men and women, it

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 37: Math anxiety and learn helpless

26

is reasonable to suspect that helplessness theory can help to explain these

differences. Although the information is not readily available, it would be

interesting to discover if there is a corresponding difference between the

mathematics anxiety scores of these two groups.

Ann B. Oaks, at the 1989 National Conference on Women in Mathematics

and the Sciences proposed that there is a cognitive dimension to mathematics

anxiety and, in a number of ways, makes statements that support the view that

much of what is attributed to mathematics anxiety is explainable as a

helplessness response.

In her view, the root cause of mathematics anxiety lies in the resistance

that some students have toward gaining a conceptual understanding of

mathematics. These students do not look at mathematics as a cohesive

discipline unto itself, they view it as a meaningless set of procedures for

transforming one set of symbols into another. Their main goal in the

mathematics classroom is to learn how to perform these manipulations easily.

They do not see the usefulness of reasoning and creativity in mathematics and

so, they are unable to generalize what they have learned to even slightly different

circumstances and their verification skills are lacking. In this view, working hard

to learn mathematics consists merely of trying to memorize algorithms. Since

very few people are capable of retaining such a large amount of information

without putting it into some meaningful framework, these students fail repeatedly.

Since their failure follows hard work, they conclude that the failures are a result of

a lack of ability or mathematics anxiety itself. Indeed, many of the students

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 38: Math anxiety and learn helpless

27

interviewed by Oaks claimed that math anxiety was the primary factor preventing

them from being successful at mathematics. This caused them to avoid

mathematics, rather than to work at it. [Oaks, 1989] As she puts it:

Lefcourt (1982) explained that anxiety is produced by a combination of two factors: negative results coupled with a lack of control over the situation. The students in our study have learned that in the mathematics classroom they are likely to fail, and they know from experience that there is nothing they can do to keep this from happening. Therefore, they experience the anxiety normal for an individual in this situation.

In addition, because they have worked hard and have not experienced success, these students are forced to shift their understanding of what causes success for them away from controllable factors (such as effort and getting outside help) to uncontrollable factors such as ability and (as they view it) math anxiety. [Oaks, 1989, p. 198]

As before, consider Oaks’ views on mathematics anxiety through the lens

of helplessness theory. At the core of her view of mathematics anxiety is a

fundamental lack of understanding on the part of the students. Since the

students are focused on naive symbol manipulations, they will have little

understanding why, for example, 2(x + y) = 2x + 2y, but (x + y f * x2 + y2 since

these appear very similar when viewed purely as operations on symbols without

the context that understanding the meanings of the symbols brings. Without this

understanding and even an aversion on the part of the students to gain a

conceptual understanding of the underlying mathematics, the students will be

incapable of understanding any explanation that their teacher might give them as

to why certain answers are correct and others are incorrect. The students’

successes and failures would then seem to be arbitrary and uncontrollable to the

students, which is the sort of experience that leads to a helplessness response.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 39: Math anxiety and learn helpless

28

Furthermore, as Oaks points out, the continued failures of these students,

especially subsequent to what they consider to be hard work, causes them to

change their attributions of failure from lack of effort and other controllable factors

to uncontrollable factors such as a lack of intelligence or ability. Those who

attribute failure to such stable and internal factors are the most likely to

experience a helplessness effect. That these students subsequently stop

working at mathematics could be evidence of a helplessness effect taking place.

Finally, many of Oaks’ students blamed their lack of success in

mathematics on mathematics anxiety itself. This rational may be more

comfortable to the students’ self-image than attributing their failures to a lack of

intelligence, but it is still an internal and stable attribution which would have the

same effect as these other attributions toward bringing about a helplessness

response. [Oaks, 1989]

We can also see evidence of the possible usefulness of helplessness

theory when we examine some of the interventions that have been used to

alleviate mathematics anxiety.

Oaks states that, in her view, interventions that try only to alleviate anxiety

are unlikely to have an effect on a student’s long-term problems with

mathematics. In her course, she attempts to increase her students

understanding by having them work both alone and in groups in a non­

threatening atmosphere. The students share their discoveries with each other

and write papers describing their reasoning and thoughts as they tried to solve

certain problems. It is possible for her students to earn an “A” on these papers

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 40: Math anxiety and learn helpless

29

without ever generating the correct answers. The results are said to be

encouraging, with the students gaining a new sense of control over the

mathematics they do. [Oaks, 1989] Oaks appears to be attempting to shift

attributions from a lack of ability to a lack of effort and lessening the dichotomy

between success and failure for the students. Both of these techniques have

been shown to alleviate helplessness responses.

At American University, Weissbrod and Adams attempted an intervention

that provided students with positive mathematical experiences. Groups made up

their own syllabi and engaged in group work in which each member of the group

selected two problems, one that the student could do and one that he or she

could not. These problems then formed the basis of the group work. [Tobias

and Weissbrod, 1980] In allowing the students to work on problems they were

capable of solving, Weissbrod and Adams provided each group with the sort of

success experience that is known to “immunize” against a helplessness effect.

Aurelia Skiba, a high school mathematics teacher wrote about her

approach to working with math anxious students in the March 1990 issue of the

Mathematics Teacher. Again, large portions of her technique could actually be

aimed at alleviating a helplessness effect. When she states, “If I start with easy

problems and praise each step that is correct, I find that the students can

accomplish significantly more than [their] memories would suggest”, she is giving

the students early success experiences and “immunizing” them against a

helplessness response.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 41: Math anxiety and learn helpless

30

Furthermore, when these students practice techniques, they do so with a

teacher’s guidance throughout. Corrections are made throughout the process

and so end with a high rate of success. Not only would this continue the

“immunization” process but also, it seems that this would lessen the dichotomy

between success and failure, further alleviating a helplessness response.

Finally, appealing to the example regarding the Asian and American

children, Skiba places heavy emphasis on the importance of effort in learning

mathematics. “I try to instill this concept of the ‘virtue of effort’ in my students."

She writes, “Once they believe they are working hard and trying their best, they

succeed.” [Skiba, 1990] Again we see a technique, attributional retraining,

recognized in alleviating helplessness responses applied to treat mathematics

anxiety.

Gentile and Monaco (1986) may have been the first to actually propose

that there is a link between mathematics anxiety and learned helplessness. They

write:

Given the large number of people who show signs of math anxiety, it seems to us to be worth exploring their causal explanations of their own difficulties. However, if learned helplessness is to be a reasonable account of how they might have acquired their math anxiety, there needed to be, in our view, at least one objective demonstration that the learned helplessness manipulation produces a mathematics performance deficit as well as producing certain varieties of attributions. [Gentile & Monaco, 1986, p. 166]

They were able to show exactly this. In their experiment, sixty-four high school

students were given a set of multiplication problems to solve and were given

failure feedback. These students subsequently showed a helplessness deficit in

a parallel set of multiplication problems as well as on a set of pattern recognition

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 42: Math anxiety and learn helpless

31

problems of the form “25 is to 5 as is to 8.” Thus, helplessness effects

were shown to occur in a mathematical context. Furthermore, the attributions

made by these students were similar to those in other research. In this case,

girls were more likely to attribute their failures to internal factors such as lack of

ability or effort while boys were likely to blame their failures on external factors

like problem difficulty or clarity. Interestingly, there was not a corresponding

difference in the magnitude of the helplessness effect, the performance of the

boys and the girls were depressed an equal amount. [Gentile and Monaco, 1986]

Gentile and Monaco did not try to measure or quantify the mathematics anxiety

of these students.

There is ample evidence, anecdotal and otherwise, to suggest interaction

between the concepts of mathematics anxiety and learned helplessness.

Notions from helplessness theory appear relevant and continue to further

illuminate the study of mathematics anxiety both in theory and in practice. The

study that is the basis of this paper further investigates this relationship.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 43: Math anxiety and learn helpless

■ Chapter III - The Study

The purpose of this study was to investigate the relationship between

Mathematics Anxiety and Learned Helplessness. This was done in two parts.

The first phase of this project was a correlational study in which a large sample of

students was partitioned by attributional style to indicate the students’ propensity

toward learned helplessness. The mean mathematics anxiety scores of these

attributional styles were then examined to determine if the known differences in

helplessness responses corresponded to differences in mathematics anxiety

scores. The second phase was an interventional study. A smaller group of

students was exposed to an intervention consisting of several methods known to

lessen helplessness responses. After the students were exposed to the

intervention, they were again given questionnaires to discover if there was a

corresponding decrease in their mathematics anxiety scores.

Nine mathematics classes taught at the University of Miami were chosen

to participate in the correlational part of the study. These classes were selected

to give a cross section of the freshmen and developmental level mathematics

courses offered at the university. The students in these classes who were

eighteen years of age or older were asked to participate in the study by

32

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 44: Math anxiety and learn helpless

33

completing both the Phobos Inventory, which measures levels of Mathematics

Anxiety, and the Mathematics Attribution Scale, which measures students’

attributions regarding success and failure in mathematics. In all, 193 students

agreed to complete the questionnaires and returned usable information. Table 1

shows the courses the respondents were recruited from and indicates the

number of respondents participating from each course.

Table 1 - Participants in correlational study by course

Course Number of RespondentsMTH 099 - Intermediate Algebra 30MTH 101 - Algebra for College Students 19MTH 103 - Finite Mathematics 43MTH 107 - Precalculus I 45MTH 108 - Precalculus II 26MTH 1 1 1 - Calculus I 30

TOTAL 193

Several responses were deemed unusable because the information returned was

incomplete or illegible or because the respondent was under the age of eighteen.

Recall that it has been shown that there is a relationship between the

attributions a person makes regarding failure and his or her susceptibility to

learned helplessness. [Dor-Shav & Mikulincer, 1992] The researcher used this

knowledge to investigate the link between learned helplessness and

mathematics anxiety. There are three dimensions to causal attribution: locus,

stability and globality. The locus dimension determines if the person will attribute

failure to internal or external causes. Stability is concerned with whether the

causes will remain steady over time, and globality with whether the causes will

remain the same over many situations. Specifically, if someone attributes failure

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 45: Math anxiety and learn helpless

to an internal and stable cause, such as a lack of ability, that person expects

failure to recur and helplessness results. If the perceived cause is internal and

unstable, such as a lack of effort, the subject tends to have an optimistic attitude

leading to reactance, an increase in performance on subsequent tasks. The

notion of globality is less useful here, since the study only relates to a single

topic, mathematics. The different attributional styles and their propensity toward

learned helplessness are summarized in table 2. [Dor-Shav & Mikulincer, 1992]

Table 2 - Typical attributions, Locus of Control and susceptibility to Learned

Helplessness:

Category A1 (F-EF) Category A4 (F-A)

Internal Interpersistent Subjects Intrapunative Subjects

“Lack of Effort” “Lack of Ability”

Locus High Reactance High Helplessness

Category A2 (F-EN) Category A3, (F-T)

External Extrapersistent Subjects Extrapunative Subjects

“Bad Luck” “Task Difficulty”

Low Reactance Low Helplessness

Unstable Stable

Stability

The attributional style of each of the 193 respondents was established

using the Mathematics Attribution Scale. The MAS generates eight scores

indicating how likely a subject is to attribute success and failure to four

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 46: Math anxiety and learn helpless

35

attributions based on locus and stability. Thus there are scores indicating how

likely it is that a subject will attribute failure to effort (F-EF), environmental factors

(F-EN), task difficulty (F-T) and innate ability (F-A). Similar scores are generated

for success attributions. The largest of the four failure scores was used to

determine in which of 4 categories a respondent was placed. In the event of a

tie, that respondent was not categorized.

Thus, the outcome of this phase of the study was to stratify the sample

into the four categories of attributional styles that indicate a person's propensity

for learned helplessness from low (A1) to high (A4). pAi is defined to be the

mean mathematics anxiety score for attributional style Ai. A hypothesis test with

null hypothesis, “H0: pai = |M(i+i)” and alternative hypothesis

“Ha: pAi = m-a (i+ i) is not so” was conducted using an Analysis of Variance

(ANOVA). The Tukey-Kramer paired comparison test was then used to

determine if there was a significant pairwise difference in these means.

It was expected that the mean anxiety score would increase significantly

as the index increases, that is, the working hypothesis of this part of the study

was that pAi < PA2 < PA3 < PA4-

The second or interventional phase of this study attempted to show that

methods known to alleviate learned helplessness will also alleviate mathematics

anxiety.

To this end, four classes taught by the researcher were asked to

participate. These classes consisted of two sections of MTH 103 (Finite

Mathematics) and two sections of MTH 107 (Precalculus I). One section of each

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 47: Math anxiety and learn helpless

36

course was included in an experimental group while the other was kept as a

comparison group. The students in these sections were tested at the beginning

of the semester using the Phobos Inventory of Mathematics Anxiety and the

Mathematics Attribution Scale. These scores were also used as part of the

correlational part of this study. The sections making up the experimental group

were exposed to information and activities aimed at preventing or lessening

helplessness responses throughout the semester.

The intervention was based upon two facts: exposure to adverse events

that are perceived as uncontrollable often leads to a helplessness response; and

a helplessness response is most severe when a person attributes failure to

internal and stable causes. The researcher attempted to both prevent and

alleviate helplessness responses.

Attributional retraining is the primary method for alleviating a helplessness

response. Thus, the students in the experimental group were encouraged

throughout the semester to adopt attributions involving effort and strategy, and to

attribute failure to unstable causes rather than stable ones. Several methods

were used to encourage students in the experimental group to adopt attributions

involving effort and strategy. A former student of the researcher made a

presentation to each class in the experimental group emphasizing the importance

of effort and strategy to success in mathematics. He spoke of his own

experiences in taking MTH 107, emphasizing that, although his early grades in

the class were not as high as he would have liked them to be, he was eventually

able to succeed in the class through a combination of hard work and good study

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 48: Math anxiety and learn helpless

habits. He then described these habits and reminded the students of the on-

campus resources available to assist them with their mathematics classes. The

classes in the comparison group heard no such presentation. The experimental

group watched the Nova episode “The Proof which chronicles Dr. Andrew Wiles

quest to prove Fermat’s Last Theorem. The discussion following the video

emphasized attributions involving effort, particularly noting how Dr. Wiles solved

what is possibly the most difficult problem in all of mathematics through 7 years

of intense work. The experimental group was also given an extra credit

assignment in which the students were asked to watch and review the film Stand

and Deliver. In the film a high school math teacher inspires underachieving

students to work hard and tap potential they never knew they had. The

comparison group participated in similar activities that did not encourage

attributions toward effort but instead involved attributions of inherent ability.

These activities were built around the short film “N is a Number”, a biography of

Paul Erdos and the feature film A Beautiful Mind. To further emphasize the

importance of effort to learning mathematics, the students in the section of MTH

107 that was a part of the experimental group were provided with several “extra

practice” worksheets to supplement their regular homework assignments. These

worksheets were made available to the students in the comparison section of

MTH 107 but they were not handed out.

Methods for preventing a helplessness response were also employed. To

prevent a helplessness response, one can attempt to lessen the impact of

adverse events and give the student some expectancy of control. Recall that

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 49: Math anxiety and learn helpless

38

one factor that makes a helplessness response more likely or more severe is the

dichotomy between success and failure. [Gentile & Monaco, 1986] Thus, this

part of the intervention included giving students in the experimental group extra

credit assignments in which they corrected and explained any errors they had

made on their exams. This activity should make a helplessness response less

likely by lessening the dichotomy of success versus failure with respect to the

exams. The corresponding activity for the comparison group was sets of

additional challenging problems relating to the material on each test. The

experimental group was also told, following their first test that grades usually

improve on the second test. This encouraged them to attribute any failures to

unstable causes. The comparison group was not given this information.

At the end of the semester the students were again tested with the

Phobos Inventory and the Mathematics Attribution Scale. Methods for verifying

changes in paired data were then applied to determine if there was a significant

difference between the pre-experiment and post-experiment Mathematics

Anxiety scores. The working hypothesis of this portion of the study is that the

mathematics anxiety scores of the experimental group should decrease

significantly from the pre-intervention survey to the post intervention survey. The

results of the study are presented in chapter 4.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 50: Math anxiety and learn helpless

■ Chapter IV - Results, Conclusions and

Recommendations

Instruments

Two instruments were used in this study. The Phobos Inventory

(Appendix A) is designed to measure levels of mathematics anxiety. It is a thirty

question Likert style questionnaire that Ronald Ferguson (1986) adapted from

the Mathematics Anxiety Rating Scale (MARS) developed by Richardson and

Suinn (1972). The first ten items on the inventory are designed to measure

numerical or computational anxiety. Items 11 through 20, measure mathematics

test anxiety, while questions 21 through 30 measure abstraction anxiety. Each

item on the inventory describes a situation involving mathematics and the

respondent is asked to indicate how much that item frightens him or her on a

scale ranging from 1 (not at all) to 5 (very much). [Ferguson, 1986] Traditionally,

the responses are totaled to generate a math anxiety score ranging from 30 to

150. Under this scheme, unanswered questions would skew the results toward a

low anxiety result. To allow for an occasional blank or indeterminate response,

the researcher generated three sub-scores by finding the mean of the actual

responses from questions 1 through 10,11 through 20 and 21 through 30

respectively. These sub-scores were then summed, generating math anxiety

39

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 51: Math anxiety and learn helpless

40

scores that range from 3 to 15 with each sub-score contributing between 1 and 5

points to the total. This anxiety score will be denoted by a.

The attributional styles of the respondents were determined using the

Mathematics Attribution Scale (Appendix B), developed by Elizabeth Fennema,

Patricia Wolleat and Joan Daniels Pedro (1979). Recall that there are four

attributional styles that are relevant here. Characterized by their standard

attributions, these can be classified as “effort” (A1), “environmental factors” (A2),

“task difficulty” (A3), and “ability” (A4). The MAS is a thirty-six-item Likert style

questionnaire. Nine events involving mathematics are described. After each

event is a list of four possible causes or attribution statements for that event, one

for each of the four relevant attributional styles. The respondents are asked to

determine whether each cause could be an explanation for the associated event.

The possible responses range from “1” meaning “strongly disagree” to “5”

meaning “strongly agree.” Thus, the higher the value, the more strongly the

respondent believes that the attribution statement could describe the cause of

the event in question. Eight of the events are used in the evaluation of the

instrument; the ninth is not scored. Four of these describe successful outcomes

and the other four describe unsuccessful outcomes. The MAS therefore

generates eight scores, four indicate how likely the respondent is to attribute

failure to effort, environmental factors, task difficulty and innate ability, (F-EF, F-

EN, F-T and F-A respectively) while the remaining scores measure the same

attributions for success. [Fennema, Wolleat & Pedro, 1979] As was done in the

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 52: Math anxiety and learn helpless

41

case of the Phobos Inventory, the mean of the responses actually received for

the relevant items was used in place of the sum of those responses.

Phase 1: The Correlational part of the study

In the corelational part of the study, the 193 respondents were classified

by their attributional styles with regard to failure. Recall from chapter 2, part 1,

that it was a person’s attributions of and responses to failure that determined his

of her susceptibility to learned helplessness. Each student was categorized

according to his or her maximum failure score. Forty students could not be

categorized because of ties in their largest scores. The results are summarized

in table 3. It is striking that so few students fell into category A2. Further

research should be conducted to determine if this is a statistical abnormality or if

this is a normal characteristic of college students.

Table 3 - Attributional Styles of the respondents in phase 1

Category Largest Failure Scorefrom the MAS

StandardAttribution

# o fRespondents

Mean value of or

A1 F-EF Effort 58 6.47A2 F-EN Environment 5 6.86A3 F-T Task 48 7.04A4 F-A Ability 42 8.61

The corresponding mean total anxiety score from the Phobos Inventory for each

of these categories is shown in table 3. As predicted in chapter 3, the mean

anxiety score increased with the index. And so we have:

M-A1 < PA2 < PA3 < PA4.

This is shown graphically in figure 1.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 53: Math anxiety and learn helpless

42

Figure 1 - Mean anxiety scores by attributional style

Category

A single factor AN OVA was performed to determine if these increases are

significant. The results of the ANOVA, shown in table 4, show a p value of

.0000075, indicating that there is a statistically significant difference between the

means of at least two of the four categories at a confidence level greater than

99%. As we can see in figure 2, if the means of each category are compared

using a 95% confidence interval, it appears that pAi and pA3 are significantly

smaller than pA4.

Table 4 - ANOVA results

Source of Variation SS df MS F P-vaiue F critBetween Groups 115.69 3 38.56 9.65 0.0000075 2.67Within Groups 595.51 149 3.00

Total 711.19 152

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 54: Math anxiety and learn helpless

43

Figure 2 - Anxiety score means of A1 through A4 with confidence intervals

Ind iv idual 95% CIs For Mean Based on Pooled StDev

CategoryA1

N58

Mean6.474

StDev -----1.866

A2 5 6 . 860 1.784 { - - -- _ - _*____ _.---------- )A3 48 7 . 042 1.842 ( ------ * - - - )A4 42 8.612 2 .342 ( ------ * -------)

Pooled StDev = 1.999 6.0 7.2 8.4

This was further investigated using a Tukey-Kramer paired comparison test. As

the results in table 5 indicate, it is indeed the case that pAi and jj,A3 are

significantly smaller than pA4. The confidence intervals containing the differences

of the means between A1 and A4 and between A3 and A4 do not contain zero

and therefore the means of the anxiety scores of these categories are

significantly different at a confidence level of at least 95%.

Table 5 - Tukey-Kramer paired comparison test results.

Tukey's pairwise comparisons

Family error rate = 0.0500 Individual error rate = 0.0104

Critical value = 3.67

Intervals for (column level mean) - (row level mean)

A2

A3

A4

A1 A2 A3

-2.805 2 . 032

-1.580 -2.6200.444 2.256

-3.189 -4.206 -2.666-1.087 0 .703 -0.474

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 55: Math anxiety and learn helpless

44

And so, even though this does not support the entire of the hypothesis that

Pai < pa2 < PA3 < pa4 at a 5% level of significance, it does support the overall

notion that there is a relationship between the phenomena of mathematics

anxiety and learned helplessness. In particular, the attributional style most prone

to a helplessness response displayed levels of anxiety significantly higher than

the group least prone to such a response. The ability attribution may be the most

salient here, as the mean anxiety score for that attribution was significantly

higher than the mean anxiety scores of the effort and task attributions, which

were not significantly different from each other according to these tests. It is

possible that the anxiety score for the ability attribution would be significantly

higher than the environment attribution as well, but it is difficult to infer anything

meaningful about the environmental attribution, because so few respondents fell

into category A2.

Both the ANOVA and the Tukey-Kramer Paired Comparison Test assume

that the data being analyzed is normally distributed with equal variance. [Rice,

1995] Since this is not necessarily the case here, it is prudent to bolster these

findings using some non-parametric methods. A Kruskal-Wallis Test was

performed on the data to determine if there existed a significant difference in the

median anxiety scores of categories A1, A2, A3 and A4. Let j]Ak be the median

anxiety score of attributional style Ak. The difference is indeed significant as we

can see in table 6. With p < 0.05, the median anxiety scores are different at a

confidence level greater than 95%.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 56: Math anxiety and learn helpless

45

Table 6 - Kruskal-Wallis Test Results

Category N Median Ave Rank ZA1 58 6.500 61 . 0 -3 . 50A2 5 7 . 000 70 .2 -0 .35A3 48 7 . 050 74 .4 -0.49A4 42 8.400 102 . 9 4 .45Overall 153 77 . 0

H = 22.26 DF = 3 P = 0 . 000H = 22.27 DF = 3 P = 0.000 (adjusted for ties)

As is the case with the ANOVA, this tells us that there is a significant

difference between at least two of the categories, but it does not give us

information as to which of the categories are different. Thus Mann-Whitney

Tests were conducted on pairs of attributional styles. Again, no meaningful

information could be inferred about category A2, since so few respondents fell

into that category. The results regarding the other attributional styles however,

were more definitive than those from the ANOVA and Tukey-Kramer tests. The

relevant results are given in table 7.

Table 7 - Mann-Whitney Test results

Mann-Whitney Test and Cl: A3, A1

A3 N = 48 Median = 7.050A1 N = 58 Median = 6.500Point estimate for rjA3 - rjA1 is 0.60090.0 Percent Cl for rjA3 - r/A1 is (-0.000,1.299)W = 2837.5Test of r]A3 = rjA1 vs rjA3 > rjA1 is significant at 0.043 9 The test is significant at 0.0438 (adjusted for ties)

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 57: Math anxiety and learn helpless

46

Table 7 (continued) - Mann-Whitney Test results

Mann-Whitney Test and Cl: A4, A3

A4 N = 42 Median = 8.400A3 N = 48 Median = 7.050Point estimate for r]Ai - rjA3 is 1.52890.0 Percent Cl for T]A4: - i]A3 is (0.722,2.300)W = 2313.0Test of /7a4 = ?7a3 v s rjAi > rjA3 is significant at 0.0006 The test is significant at 0.0006 (adjusted for ties)

Thus ?7ai < 77A3 is shown with a 5% level of significance. The results for the other

relationship are stronger, and show that tja3 < 77A4 is true at a better than 1% level

of significance. Recall that the hypothesis of this phase of the study was that

Pai < pa2 < pa3 < P-A4- This relationship is true for the medians of the anxiety

scores for every attributional style that was large enough to yield useful results.

It is reasonable to conclude that the Mann-Whitney and Kruskal-Wallis tests give

better information in this situation because they do not assume any underlying

distributions on the data and, the Mann-Whitney allows one-sided tests as well

as two-sided tests. In addition, testing for results with respect to the median

rather than the mean makes the results less sensitive to outliers.

Two-sample t-tests were conducted to determine if this relationship could

be demonstrated with regard to the means of the anxiety scores for each of the

attributional styles using one-tailed tests. The results were similar to the results

from the Mann-Whitney and Kruskal-Wallis tests. Once more, nothing

meaningful could be determined regarding the environmental attribution,

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 58: Math anxiety and learn helpless

47

category A2, but the hypothesis was otherwise supported by these tests with

respect to the other attributional styles as is shown in tables 8 and 9.

Table 8 - Results of two-sample t-test assuming unequal variance, A1 vs. A3

A1 A3Mean 6.47 7.04Variance 3.48 3.39Observations 58 48Hypothesized Mean Difference 0df 101tS tat -1.57P(T<=t) one-tail 0.06t Critical one-tail 1.66P(T<=t) two-tail 0.12t Critical two-tail 1.98

Here, using a one-tailed test, since p « 0.06 < 0.10 we can reject the null

hypothesis that pAi = pa3 and accept the alternate hypothesis that ^Ai < pA3 at a

level of significance less than 10%. The result that p,A3 < |aA4 is much stronger

and can be accepted at a better than 1% level of significance.

Table 9 - Results of two-sample t-test assuming unequal variance, A3 vs. A4

A3 A4Mean 7.04 8.61Variance 3.39 5.48Observations 48 42Hypothesized Mean Difference 0df 78tStat -3.5P(T <=t) one-tail 0.0004t Critical one-tail 1.66P(T<=t) two-tail 0.0008t Critical two-tail 1.99

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 59: Math anxiety and learn helpless

48

The hypothesis of phase one of the study is therefore verified up to the limitations

of the data. That is, jmi < pa3 < pa4 at a 10% or better level of significance.

Developing an attributional model for mathematics anxiety

To further explore this relationship; the failure scores from the MAS were

compared to a, the anxiety score from the Phobos Inventory, for all 193

respondents. The main effects plot is shown in figure 3, below. It shows the

mean value of a for each value of the four scores, F-EF, F-EN, F-T and F-A from

the MAS, plotted against that value. Examining this figure, there appears to be

little relationship between the scores F-EF and F-EN and the anxiety scores.

The scores F-T and F-A however, appear to have linear relationship with a, with

the correlation appearing stronger with respect to F-A.

Figure 3 - The Main Effects Plot

F-EF F-EN F-T F-A a v e

2? 10.0

8'T ) 8.5too■5 7.0o

< 4.0

<5.0 1.01.0 4.5 2.0 5.0 1.0

MAS SCORES

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 60: Math anxiety and learn helpless

49

The correlation coefficients further support this observation. As is evident

in table 10, the correlation coefficient increases as we progress from the scores

related to the attributional styles associated with reactance to those related to

helplessness. Thus, the more likely an attributional style is to have a

helplessness response, the more closely the related MAS score correlates with

the mathematics anxiety score. The correlation coefficient p .53 between a and

F-A indicates a fairly strong relationship considering the self-reporting nature of

the data at hand.

Table 10 - Correlation Coefficients

MAS Score Correlation coefficient with anxiety score, a

F-EF 0.028678F-EN 0.181144F-T 0.397659F-A 0.530908

An attempt was made, using linear regressions to find a function that

would model a in terms of F-A, F-T, F-EN and F-EF. The initial attempt, involving

all four of these factors is shown in table 11.

This model is significant at a confidence level of at least 99%, given that

the value of “Significance P « 3.90 x 10 ~9 < 0.01. The value of “Adjusted R2” ^

0.29 is again acceptable given the nature of the data. A better model is possible

however. The large P-values for F-EN and F-EF as well as the small values of

the coefficients of these variables indicate that these factors contribute little to the

model.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 61: Math anxiety and learn helpless

50

Table 11 - Linear regression, or as a function of F-A, F-T, F-EN and F-EF

Regression StatisticsMultiple R 0.550R Square 0.302Adj. R Square 0.287Standard Error 1.815Observations 193

ANOVAdf SS MS F Significance F

Regression 4 268.13 67.03 20.34 6.15 x 10 ~14Residual 188 619.62 3.30Total 192 887.75

CoefficientsStandard

Error tS tat P-valueIntercept 2.110 0.895 2.357 0.019F-A 1.045 0.169 6.179 3.90 x 10 ~9F-T 0.539 0.271 1.990 0.048F-EN 0.096 0.218 0.442 0.659F-EF -0.117 0.151 -0.772 0.441

Other models were tried. The significance of interactions, (factors such as

F-A x F-T) was tested as was the significance of quadratic factors such as (F-A)2.

None of these made a significant contribution to the model. The model based on

just F-A and F-T was the best fit and is shown in table 12.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 62: Math anxiety and learn helpless

51

Table 12 - Linear regression, a as a function of F-A and F-T

Regression StatisticsMultiple R 0.547R Square 0.299Adj. R Square 0.292Standard Error 1.809Observations 193

ANOVAdf SS MS F Significance F

Regression 2 265.796 132.898 40.599 2.09 x 1 0 ~ 1bResidual 190 621.951 3.273Total 192 887.747

CoefficientsStandard

Error tS tat P-valueIntercept 1.917 0.776 2.471 0.014F-A 1.043 0.168 6.190 3.63 x 10 ~9F-T 0.556 0.255 2.181 0.030

Thus, in addition to the relationships noted above, there is a model

describing a that is based upon the scores for the two attributions, ability and

task difficulty, that lead to a helplessness response. Recall that the other two

attributional styles, effort and environment, lead to reactance rather than

helplessness.

or = 1.917 + 1.043(F-A) + 0.556(F-T) + e

It may be possible to use this model to predict mathematics anxiety.

It is interesting that F-EF, the score for the effort attribution, does not

correlate with the anxiety score and that it contributes little to the model of a in

table 11. One might expect to find a strong negative correlation between F-EF

and a since the effort attribution leads to high reactance rather than

helplessness. That this is not the case is an interesting question warranting

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 63: Math anxiety and learn helpless

52

further research. Recall that intrapersistent subjects, who attribute failure to lack

of effort, expect success rather than failure on future tasks. [Dor-Shav &

Mikulincer, 1992] It is therefore possible that their attributions regarding failure

have little effect on their overall attitudes about mathematics and on their level of

mathematics anxiety in particular. Oaks mentions a phenomenon that is another

possible explanation. When students become convinced that they cannot

succeed at mathematics, they avoid working at it. This allows a student to blame

failure on lack of effort, allowing him or her save face. [Oaks, 1989] Oaks

proceeds to quote a student who explains her feelings on the matter.

When I feel myself starting to slip, I put up the books because I don’t want to slip and I’m scared. It’s easier to put up the books because then you say, “Well, I didn’t study, and I got this bad grade. But it’s O. K. because l didn’t study and I didn’t deserve it anyway.”

But that little fear of “I studied; I tried and look at my grade” is scary. I’m scared that if I do open the book and I do try to study, that I’m not going to get it and that I’m going to be a failure anyway. You don’t want to find out that that you’re stupid and you couldn’t handle it.[Oaks, 1989, pg. 199]

Students in this situation may have inflated F-EF scores, masking a relationship

between a and F-EF that would otherwise be discernable.

Phase 2: The Interventional Part of the Study

In phase 2 of the study students in an experimental group were exposed

to an intervention designed to lessen helplessness responses. There were thirty-

six students in the experimental group and twenty-seven in the comparison group

who responded to both the pre-intervention questionnaire and the post­

intervention questionnaire. Recall that the hypothesis of this phase of the study

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 64: Math anxiety and learn helpless

53

was that although the intervention was designed to lessen a helplessness

response, there would be a corresponding decrease in the mathematics anxiety

score, a of the students in the experimental group. There was also a tacit

assumption that the scores of the students in the comparison group would not

change. Neither of these things happened. For the experimental group, there

was no discernable change in a at all, much less a statistically significant one.

There was also no significant change in any of the failure scores from the MAS.

These facts were established using a paired two-sample t-test for means. The

results for the experimental group are summarized in table 13.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 65: Math anxiety and learn helpless

54

Table 13 - Results of the paired, two-sample t-test for means comparing pre and

post intervention values for the experimental group.

Pre anx Postanx Pre F-EF Post F-EF Pre F-EN Post F-ENMean 7.53 7.55 3.57 3.60 2.68 2.78Variance 3.84 4.31 0.74 0.80 0.35 0.38Observations 36 36 36 36 36 36Pearson Correlation 0.74 0.55 0.59Hypothesized Mean Diff. 0 0 0df 35 35 35tStat -0.08 -0.17 -1.07P(T<=t) one-tail 0.47 0.43 0.15t Critical one-tail 1.69 1.69 1.69P(T<=t) two-tail 0.94 0.87 0.29t Critical two-tail 2.03 2.03 2.03

Pre F-T Post F-T Pre F-A Post F-A

Mean 3.63 3.55 3.41 3.42Variance 0.34 0.40 0.75 0.80Observations 36 36 36 36Pearson Correlation 0.62 0.83Hypothesized Mean Diff. 0 0df 35 35tStat 0.81 -0.08P(T<=t) one-tail 0.21 0.47t Critical one-tail 1.69 1.69P(T<=t) two-tail 0.42 0.94t Critical two-tail 2.03 2.03

Hembree (1990) did an overview of research done on mathematics anxiety and

noted that whole-class interventions were not effective in the treatment of

mathematics anxiety. Phase two of this study may be another example in

support of Hembree’s findings. However, examining the data for the comparison

group, as shown in table 14, gives us another possible explanation.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 66: Math anxiety and learn helpless

55

Table 14 - Results of the paired, two-sample t-test for means comparing pre and

post intervention values for the comparison group

Pre anx Postanx Pre F-EF Post F-EF Pre F-EN Post F-ENMean 6.96 7.61 3.17 3.56 2.64 2.68Variance 3.02 5.93 0.69 0.65 0.42 0.39Observations 27 27 27 27 27 27Pearson Correlation 0.73 0.77 0.35Hypothesized Mean Difference 0 0 0df 26 26 26tStat -2.00 -3.63 -0.26P(T<=t) one-tail 0.03 0.00006 0.40t Critical one-tail 1.71 1.71 1.71P(T<=t) two-tail 0.06 0.0012 0.79t Critical two-tail 2.06 2.06 2.06

Pre F-T Post F-T Pre F-A Post F-AMean 3.46 3.69 2.88 3.20Variance 0.23 0.34 0.57 0.78Observations 27 27 27 27Pearson Correlation 0.50 0.68Hypothesized Mean Difference 0 0df 26 26t Stat -2.14 -2.54P(T<=t) one-tail 0.02 0.01t Critical one-tail 1.71 1.71P(T <=t) two-tail 0.04 0.02t Critical two-tail 2,06 2.06

Notice that there is a statistically significant increase between the pre and post

intervention values of a, F-A, F-T and F-EF. For the one-tailed tests, all these

results are significant at a level of 5% or better. For the two-tailed tests, the

increases in F-A, F-T and F-EF are significant at the 5% level, while the increase

in a is significant at the 10% level. One possible explanation for this is that an

increase in anxiety is normal at the end of a semester. The corresponding

increases in F-A and F-T are consistent with this according to the linear

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 67: Math anxiety and learn helpless

regression model given in table 12, and although an increase in F-EF might be

suspected of relating to a decrease in a, the regression models did not support

that fact. Verifying that an increase in mathematics anxiety is normal over the

course of a semester is an obvious subject for further research. If this is indeed

the case, these results would seem to indicate that the intervention was

successful in preventing a normally occurring increase in mathematics anxiety.

Further research should be done to verify this assumption. If such an increase in

mathematics anxiety is normal, it might be fruitful to revisit some of the studies

reviewed by Hembree. Treatments that appeared to be ineffective in treating

mathematics anxiety may have, in actuality had a significant preventative effect.

Conclusions

This study has shown a relationship between mathematics anxiety and

learned helplessness. Separating students into attributional styles showed that,

with the exception of the one style for which there was insufficient data, as the

likelihood of a helplessness response increased, the mean and median

mathematics anxiety score of that group would increase as well. A regression

model was developed that predicted the mathematics anxiety score as a function

of the two attribution scores most closely related to helplessness. It also appears

that an intervention aimed at lessening helplessness responses prevented a

normal increase in mathematics anxiety.

To summarize the findings, in the correlational phase of the study, the

respondents were separated into four attributional styles. Category A1 consisted

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 68: Math anxiety and learn helpless

57

of those who tend to attribute failure to lack of effort. Categories A2, A3 and A4

contained the respondents who tend to attribute failure in mathematics to

environmental factors, task difficulty and lack of ability respectively. Attribution

Theory tells us that the likelihood that a respondent would experience a

helplessness response should increase as the index increases. It was therefore

predicted that the mean mathematics anxiety score of each of these categories

would also increase as the index increases. That is,

|M 1 < HA2 < PA3 < M-A4-

This was established up to the limitations of the data. Nothing could be inferred

about category A2, because too few respondents fell into that category. With

that exception there was significant evidence that this relationship exists, and so,

PA1 < PA3 < HA4

was shown with a better than 10% level of significance. This was further

supported by non-parametric methods. This relationship holds true for the

medians of these categories as well as the means. Thus the relationship,

T]A1 < T1A3 < T1A4

was established with a better than 5% level of significance.

It was also shown that the scores generated by the Mathematics

Attribution Scale could be used to model a, the mathematics anxiety score

generated by the Phobos Inventory. The scores most closely related to

helplessness, F-A and F-T, measuring the likelihood that a respondent would

attribute failure in mathematics to lack of ability and task difficulty were the only

scores that had a significant effect on the model and so,

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 69: Math anxiety and learn helpless

58

a = 1.917 + 1.043(F-A) + 0.556(F-T) + e

is a model which might be used to predict mathematics anxiety. The adjusted R-

square value of 0.292 is large enough to be considered meaningful given the

nature of the data at hand.

Finally, in the interventional phase of the study, students in an

experimental group were exposed to an intervention consisting of several

methods known to alleviate or prevent helplessness responses. Here it was

predicted that the mathematics anxiety score, a would decrease significantly

between a pre-intervention survey and a post-intervention survey. This did not

happen. However, the mean value of a of the comparison group increased

significantly between the pre-intervention survey and the post-intervention

survey. This may indicate that the intervention prevented a normal increase of

mathematics anxiety for the experimental group.

The results of this study support a re-examination of the concept of

mathematics anxiety. Recall that a number of weaknesses in the theory of

mathematics anxiety were explored in part 2 of chapter 2. These included the

fact that there are competing definitions of mathematics anxiety and that the

instruments available do not measure precisely what the various models define

mathematics anxiety to be. This study has shown that it is possible to reinterpret

mathematics anxiety using the theories of other affective variables that do not

share these weaknesses. A more cohesive, well-formed and useful model can

be built.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 70: Math anxiety and learn helpless

59

How can educators use this information to help students displaying what is

now interpreted as mathematics anxiety? It has been shown in various studies

that learned helplessness effects can be treated and immunized against.

[Abramson, Seligman & Teasdale, 1978; Gentile & Monaco, 1986] This study

has provided evidence that these methods, particularly those proposed by

Gentile and Monaco may provide a mitigating effect on mathematics anxiety.

Further research should be conducted to determine the value of incorporating

these methods on a large scale and to study other methods of treating learned

helplessness and adapting their use in the mathematics classroom.

The most important thing to remember is that helplessness occurs when

negative outcomes such as bad grades become non-contingent on a student’s

efforts. If a student does not understand why he or she is receiving poor marks

helplessness can result.

This has direct consequences in the area of testing. The quality and

nature of the feedback a student receives can affect the development of a

helplessness response. When a student misses a problem, mathematics

instructors must take care to express as clearly as possible the reason the

problem is incorrect and what the student must do in the future to work similar

problems correctly. Errors must be explained so that students maintain some

expectancy of control. It therefore seems reasonable that multiple-choice exams,

on which problems are simply marked right or wrong, should be avoided

altogether. At least the manner in which they are graded should be altered to

provide meaningful feedback to the students.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 71: Math anxiety and learn helpless

60

No feedback will be meaningful unless the students have a sufficient

conceptual understanding of the material. Thus mathematics instructors need to

be certain that they teach, not only processes and algorithms, but the reasoning

and rationale behind those procedures as well. To quote Laurie Buxton:

I have often watched a teacher, with sound control, a lively manner, excellent relationships with the class, and a clear unhesitating presentation - and wondered what was wrong. I then saw that there was no discussion of concepts, and yet, something was being described. It was the behavior of the symbols. The sort of thing that I have heard is as follows:

Here's our equation:3x - 6 = 14 - 2x

Take the -6 to the other side and it becomes +6 3x = 20 - 2x

Now bring the -2x over... [Buxton, 1995, p. 205]

Buxton finishes this description and summarizes, "The teacher is talking about

what the symbols are doing, not what the operation is about and the concepts on

which it is based." [Buxton, 1995, p. 206] Without this explanation students will

be unable to discern when it is appropriate to use certain techniques and when it

is not and any resulting feedback will seem arbitrary and random.

Part of this is a curriculum issue that needs to be addressed in elementary

and mathematics education programs. Educators in general need to make

certain that the teachers placed in the classroom are knowledgeable enough and

skilled enough to properly explain the material at hand. They should also be able

to judge when it is appropriate to quote rules and theorems and when it is not

and to discern when a student has given a reasonable response and when he or

she has not.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 72: Math anxiety and learn helpless

61

Unfortunately, many teachers in the United States lack these

qualifications. Liping Ma conducted a study that compared the mathematical

understanding of elementary school teachers in the United States and China. In

her study she interviewed twenty-three U. S. teachers who were considered

“better than average” and found some disturbing results. Many had only a

procedural knowledge of the topics they were teaching and made statements that

were mathematically incorrect. For example, twenty-one of these teachers

attempted the computation 1% - >2 but only nine of them completed their

computations and reached the correct answer of 314 Worse, only one of the

twenty-three teachers was capable of creating a conceptually correct

representation of the problem and this representation was pedagogically

unsound because it involved a fractional number of children. A number of the

teachers displayed a strong commitment to “teaching for understanding” but this

was undermined by the their limited knowledge of mathematics. [Ma, 1999]

This is an untenable situation. If teachers lack a sufficient understanding

of mathematics to explain the rationale behind different algorithms to their

students, their feedback will seem random and unconnected to the matter at

hand. If students try hard to do well, yet they are unable to understand their

mistakes and learn how to correct them, their effort will seem non-contingent on

their successes and failures. In this event, helplessness and anxiety can result

and interfere with the students’ efforts to learn mathematics. If, on the other

hand, there were knowledgeable teachers in the classroom leading their students

to a genuine, conceptual understanding of mathematics, these difficulties could

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 73: Math anxiety and learn helpless

62

be eliminated. Students with a deep understanding of mathematics would be

able to learn from their errors rather than repeating them and would avoid the

frustrations that lead to helplessness in mathematics. Thus it is important that

the teachers placed in the classroom are as knowledgeable as possible.

Mathematics teachers must also understand how students learn if they are

to be effective. Not simply how students process information and conceptualize

ideas, teachers must understand the challenges and obstacles that students face

as they attempt to learn mathematics. This study has begun to do that. With

sufficient effort and study, interpreting mathematics anxiety through learned

helplessness and other models might lead to more useful insights. It could well

allow educators to better understand the point of view of the so-called “math

anxious” student and open other avenues leading to means of alleviating the

problem.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 74: Math anxiety and learn helpless

ReferencesAbramson, L, Seligman, M. E. P. & Teasdale, J. (1978) Learned Helplessness in

Humans: Critique and Reformulation. Journal of Abnormal Psychology, 87 (1)49-74.

Beck, A. T., Emery, G. & Greenberg, R. L. (1985). Anxiety Disorders and Phobias, A Cognitive Perspective. Basic Books

Buxton, L. (1991). Math Panic. Portsmouth, NH: Heinemann

Dor-Shav, N. & Mikulincer, M. (1992). Learned Helplessness, Causal Attribution and Response to Frustration. Journal of General Psychology, 117 (1), 47-58.

Dweck, C. & Licht, B. (1980). Learned Helplessness and Intellectual Achievement. In Garber, J, & Seligman, M. E. P. (Ed.), Human Helplessness Theory and Application (pp. 197-222). New York, NY: Academic Press

Evans, J. (2000). Adults’ Mathematical Thinking and Emotions, A Study of Numerate Practices. New York, NY: Routledge-Farmer

Fennema, E., Wolleat, P. & Pedro, P.O. (1979). Mathematics Attribution Scale: An Instrument Designed to Measure Students ’ Attributions of the Causes of Their Successes and Failures in Mathematics. Corte Madera, CA:Select Press

Ferguson, R. D. (1986). Abstraction Anxiety: A Factor of Mathematics Anxiety, Journal for Research in Mathematics Education, 17(2), 145-150

Gentile, J. R. & Monaco, N. M. (1986). Learned Helplessness in Mathematics, What Educators Should Know, Journal of Mathematical Behavior, 5, 159-178

Hembree, R. (1990). The Nature, Effects, and Relief of Mathematics Anxiety. Journal for Research in Mathematics Education, 21 (1), 33-46

Hiroto, D. S. & Seligman, M. E. P. (1975). Generality of Learned Helplessness in Man, Journal of Personality and Social Psychology, 31 (2), 311-327

Kitchens, A. N. (1995). Defeating Math Anxiety. Chicago, IL: Irwin Career Education Division

63

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 75: Math anxiety and learn helpless

64

Ma, L. (1999) Knowing and Teaching Elementary Mathematics, Teachers’Understanding of Fundamental Mathematics in China and the UnitedStates. Mahwah, NJ: Lawrence Erlbaum Associates

Martinez, J. G. R, & Martinez, N. C. (1996). Math Without Fear: A Guide for Preventing Math Anxiety in Children. Boston, MA: Allyn and Bacon

Mikulincer, M. (1994). Human Learned Helplessness.New York, NY: Plenum Press

Oaks, A. B. (1989). A Cognitive Root to Math Anxiety. In Keith, S. Z. & Keith, P. (Ed.), Proceedings of the National Conference on Women in Mathematics and the Sciences (pp. 197-200). St. Cloud, MN.

Peterson, C., Maier, S. F., & Seligman, M. E. P., (1993). Learned Helplessness - A Theory for the Age of Personal Control. New York, NY: Oxford University Press

Reyes, L. H. (1984). Affective Variables and Mathematics Education. The Elementary School Journal, 84 (5), 558-581.

Rice, J. A. (1995). Mathematical Statistics and Data Analysis. Belmont, CA: Duxbury Press

Seligman, M.E.P. & Maier, S.F. (1976). Learned Helplessness: Theory and Evidence. Journal of Experimental Psychology: General, 105 (1), 3-46.

Skemp, R. (1987). The Psychology of Learning Mathematics.Hillsdale, NJ: Lawrence Erlbaum Associates

Skiba, A. E. (1990). Reviewing an Old Subject: Math Anxiety. Mathematics Teacher, 87, 188-189.

Spies-Wood, E. (1980). Learned Helplessness and Item Difficulty Ordering. Psychologia Africana, 19, 29-40.

Tobias, S. (1993). Overcoming Math Anxiety. New York, NY:WW Norton and Co.

Tobias, S. & Weissbrod, C. (1980). Anxiety and Mathematics, an Update. Harvard Educational Review, 50 (1), 63-70.

Wilson, T. D. & Linville, P. W. (1982). Improving the Academic Performance of College Freshmen: Attribution Theory Revisited. Journal of Personality and Social Psychology, 42 (2), 367-376.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 76: Math anxiety and learn helpless

65

Wilson, T. D. & Linviile, P. W. (1985). Improving the Performance of CollegeFreshmen With Attributional Techniques. Journal of Personality and Social Psychology, 49 (1), 287-293.

Zaslavsky, C. (1994). Fear of Math, Howto Get Over it and Get On With Your Life. New Brunswick, NJ: Rutgers University Press.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 77: Math anxiety and learn helpless

Appendix APHOBOS INVENTORY

[Adapted from Ferguson (1986)]

DIRECTIONS: For each of the following items, indicate how much that item frightens you or causes you stress nowadays. Use a five point scale ranging from:

1______ 2______ 3 4______ 5not at all very much

Circle the selected responses on the answer sheet provided.

1. Determining the amount of change you should get back from a purchase involving several items.

2. Listening to a salesman show you how you would save money buying his higher priced item because it reduces long term expenses.

3. Listening to a person explain how he or she figured out your share of the expenses on a trip, including meals, transportation, etc.

4. Reading your W-2 form showing your annual earnings and taxes.

5. Figuring the sales tax on an item that costs more than $1.

6. Hearing some friends make bets on a game as they quote the odds.

7. Juggling class times around to determine the best course schedule.

8. Deciding which courses to take in order to come out with the proper number of credit hours for graduation.

9. Working on a concrete, everyday application of mathematics that has meaning to you, such as figuring how much money you can spend on recreation after paying the bills.

10. Figuring the monthly budget.

11. Signing up for a math course.

12. Walking into a math class.

66

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 78: Math anxiety and learn helpless

67

13. Raising your hand in a math class to ask a question.

14. Thinking about a final examination in a math class.

15. Thinking about an upcoming math examination one day before.

16. Thinking about an upcoming math examination one hour before.

17. Waiting to have a math test returned.

18. Realizing that you have a certain number of math classes to take in order to fulfill therequirements for graduation.

19. Receiving your final math grade in the mail.

20. Being given a "pop" test in a math class.

21. Having to work a math problem that has x's and y's instead of 2's and 3's.

22. Being told that everyone is familiar with the Pythagorean Theorem.

23. Realizing that an instructor has just written some algebraic formulas on the chalkboard.

24. Being asked to solve the equation x - 5x + 6 = 0.

25. Being asked to discuss a proof of a theorem about triangles.

26. Trying to read a sentence full of symbols such as:A = {x|x2 - 2x = 3, x in R}.

27. Listening to a friend explain something they have just learned in calculus.

28. Opening up a math book and not seeing any numbers, only letters, on the entire page.

29. Reading a description in the Undergraduate Bulletin of the topics to be covered in a mathcourse.

30. Having someone lend me a calculator to work a problem and not being able to tell whichbuttons to push to get the answer.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 79: Math anxiety and learn helpless

Appendix BMathematics Attribution Scale

Elizabeth Fennema, Patricia Wolleat and Joan Daniels Pedro University of Wisconsin - Madison

You are going to read about an event which could have happened to you. In addition, you are going to see four possible causes o f that event. You are going to respond to how you feel about whether the causes listed could really explain the event if it had happened to you. Each event and its possible causes are listed in a group. In each group an event is followed by four possible causes. You are to read the event carefully and then respond to how you feel about each o f the causes o f the event.

EVENT A: A part o f your math homework was wrong.

CAUSES:1. You just can't seem to remember to do the steps.2. You were careless about completing it.3. The part marked wrong included a step which was more difficult.4. You were unlucky.

Event A says "A part o f your math homework was wrong." Numbers 1, 2, 3 and 4 are possible causes for that event. Look at Number 1. Think about whether this could be a cause for Event A, "A part o f your math homework was wrong." It says, "You just can't seem to remember to do the steps." Do you STRONGLY AGREE, AGREE, DISAGREE or STRONGLY DISAGREE that this could be a cause for Event A, or are you UNDECIDED? Indicate how you feel about Number 1 as a possible cause for the event. Circle the correct letter; A = STRONGLY AGREE, B = AGREE, C = UNDECIDED,D - DISAGREE and E = STRONGLY DISAGREE.

Now look at Number 2, "You were careless about completing it." Do you STRONGLY AGREE, AGREE, DISAGREE or STRONGLY DISAGREE that this could be a cause for Event A, or are you UNDECIDED? Indicate how you feel about Number 2 as a possible cause for the event. Circle the correct letter on your answer sheet. Now mark how you feel about Numbers 3 and 4 as possible causes o f Event A. Then go to each o f the remaining events and mark on your answer sheet how you feel about each possible cause for that event.

68

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 80: Math anxiety and learn helpless

69

EVENT B: You got the grade you wanted for the semester in Algebra.

5. The content of the class was easy.6. You spent a lot of time each day studying Algebra.7. The teacher is good at explaining Algebra.8. You have a special talent for math.

EVENT C: You had trouble with some of the problems in the daily assignment.

9. There was no time to get math help because o f a schedule change for the day.10. You don't think in the logical way that math requires.11. You didn't take the time to look at the book.12. They were difficult word problems.

EVENT D: You have not been able to keep up with most of the class in Algebra.

13. Students sitting around you did not pay attention.14. You haven't spent much time working on it.15. The material is difficult.16. You have always had a difficult time in math classes.

EVENT E: You have been able to complete your last few assignments easily.

17. The Problems were more interesting.18. The effort you put into homework at the beginning helped.19. You're a very able math student.20. You lucked into working with a helpful group.

EVENT F: You were able to understand a difficult unit o f Algebra.

21. The way your teacher presented the unit helped.22. Your ability is more obvious when you are challenged.23. You put hours o f extra study time into it.24. The problems were easy because they had been covered before.

EVENT G: You received a low grade on a chapter test.

25. You're not the best student in math.26. You studied, but not hard enough.27. There were questions you'd never seen before.28. The teacher had spent too little class time on the chapter.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 81: Math anxiety and learn helpless

70

EVENT H: You have passed most tests with no trouble.

29. The teacher made learning math interesting.30. Like everyone says, you’re good at math.31. But, you spent hours of extra time on this class.32. The units were the beginning group, the easy ones.

EVENT I: There were times when you were not able to solve equations.

33. It was a task that did not interest you.34. Despite studying, you didn’t understand it well enough.35. Your friends’ lack of attention in class was part of the problem.36. But then you didn’t spend time doing homework.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 82: Math anxiety and learn helpless

Appendix C.UNIVERSITY QF MIAMI

HUMAN RESEARCH PROTOCOL FORMBEHAVlQRALSCiEMCES SUBCOMMITTEE

1. Title of Project

Mathematics Anxiety and Learned Helplessness

2. Principal Investigator and Collaborators:

Principal Investigators:

Dr. G. Cuevas, Professor Dept, of Teaching and Learning 305.284.5192

Dr. M. Mielke, Professor Dept, of Mathematics305.284.2575

Dr. R. Kelley, Professor Dept, of Mathematics305.284.2575

Collaborators:

Joseph Franke Kolacinski, Lecturer Department of Mathematics 305.284.2308

3. Performance Site:

University of Miami, Main Campus

4. Proposed Start Date:

August 2002

5. Funding Agency:

not applicable

71

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 83: Math anxiety and learn helpless

72

6. Project Objectives

The purpose of this study is to establish if a significant relationship exists between Mathematics Anxiety and Learned Helplessness. This will be done in two parts. One group of students will be recruited for a simple correlational study. A smaller group will be exposed to interventions known to lessen helplessness responses. These students will then be surveyed to discover if there is a corresponding drop in mathematics anxiety.

7. Recruitment Procedure:

(x) Other (specify):A variety of classes will be selected at the University of Miami, ranging in difficulty from MTH 099 (Intermediate Algebra) to MTH 111 (Calculus I). Students in these classes will be surveyed. In addition, two sections of two courses, MTH 103 and MTH 107 taught by J. Kolacinski will participate in the longitudinal portion of the study. One section of each course will serve as an experimental group while the other will be a control group.

8. Methods and Procedures:

The purpose of this study is to investigate the relationship between mathematics anxiety and learned helplessness.

Learned helplessness is a response to uncontrollable adverse events. Simply put, if a subject believes that he or she is unable to affect the environment, he or she will passively endure adverse events, making no attempt to change or improve the outcome. Helplessness responses lead to a number of cognitive and motivational deficits.

It has been shown that there is a relationship between a person's attributional style and his or her susceptibility to learned helplessness. There are three dimensions to causal attribution: locus, stability and globality. The locus dimension determines if the person will attribute failure to internal or external causes. Stability is concerned with whether the causes will remain steady over time, and globality with whether the causes will remain the same over many situations. Specifically, if someone attributes failure to an internal and stable cause, such as a lack of ability, that person expects failure to recur and helplessness results. If the perceived cause is internal and unstable, such as a lack of effort, the subject tends to have an optimistic attitude leading to reactance, an increase in performance on subsequent tasks.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 84: Math anxiety and learn helpless

73

Typical attributions, Locus of Control and susceptibility to Learned Helplessness:

Internal

Locus

External

There will be two phases to this study, a correlational phase and a longitudinal phase.

In the first phase of this study, we attempt to determine if there is a correlation between mathematics anxiety and learned helplessness. As stated previously, it has been shown that there exists a relationship between a person's attributional style and their susceptibility to learned helplessness. We will therefore test approximately 200 students enrolled in freshman and developmental level mathematics classes with both the Phobos Inventory, which measures levels of mathematics anxiety, and the Mathematics Attribution Scale, which measures students’ attributions regarding success and failure in mathematics. The outcome of this phase of the study will be a stratification of the sample into the four attributional styles that indicate a person's propensity for learned helplessness from low (A1) to high (A4). It is expected that the mean anxiety score will increase significantly as the index increases.

Students will be given these surveys in class, but the instructor of any given class will not survey the students and the instructor will not be present in the classroom when the students are surveyed.

The second phase of this study will attempt to show that the methods known to alleviate learned helplessness will also alleviate mathematics anxiety.

To this end, two or more sections of a college mathematics course such as MTH 101, MTH 103 or MTH 107 will be chosen. The students in these sections will be tested at the beginning of the semester using the Phobos Inventory of

“Lack of Effort” “Lack of Ability”(A1) (A4)

High Reactance High Helplessness

“Bad Luck” “Task Difficulty”(A2) (A3)

Low Reactance Low Helplessness

Unstable Stable

Stability

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 85: Math anxiety and learn helpless

Mathematics Anxiety and the Mathematics Attribution Scale. As was the case with the students in the correlational phase of the study, students will be given these surveys in class, but the instructor of any given class will not survey the students and the instructor will not be present in the classroom when the students are surveyed. The section or sections that make up the experimental group will be exposed to information and activities aimed at preventing or lessening helplessness responses throughout the semester.

The intervention is based upon two facts: exposure to adverse events that are perceived as uncontrollable often leads to a helplessness response; and a helplessness response is most severe when a person attributes failure to internal and stable causes. We will attempt to both prevent and alleviate helplessness responses.

Attributional retraining is the primary method for alleviating a helplessness response; thus, the experimental group will be encouraged throughout the semester to adopt attributions involving effort and strategy, and to attribute failure to unstable causes rather than stable ones. Methods of accomplishing this will include:

1) a presentation from a former student emphasizing the importance of effort to success in mathematics.

2) a showing of the Nova episode “The Proof which chronicles Dr.Andrew Wiles 7-year effort to prove Fermat’s Last Theorem.3) an extra credit assignment in which the students review the film Stand and Deliver. In the film a high school math teacher inspires underachieving students to work hard and tap potential they never knew they had.

4) After the first test, the students will be informed that grades usually improve on the second test, encouraging them to attribute any failures to unstable causes.

Methods for preventing a helplessness response will also be employed. To prevent a helplessness response, we attempt to lessen the impact of adverse events and give the student some expectancy of control. This part of the intervention will include:

1) Extra credit assignments in which the students in the experimental group correct and explain any errors they made on their exams. This activity should make a helplessness response less likely by lessening the dichotomy of success verses failure with respect to the exams.

2) An attempt will be made to provide the students in the experimental

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 86: Math anxiety and learn helpless

75group with success experiences, giving them a greater expectancy of control. When a new topic is covered in class the initial examples will be simple and straightforward, and will only later move to more challenging problems.

3) On exams the test items will be ordered from easiest to most difficult. This should either prevent a helplessness response during the course of the exam, or at least postpone it until late in the test.

Much of the intervention with the experimental group will take the form of extra credit assignments, but these assignments are all variations of standard assignments that have been used in University of Miami Mathematics classes. Students have traditionally been eager to participate in these assignments. It will be made clear to the students that the data is being collected anonymously and that any choice they make about filling out the survey will have no effect on their ability to participate in the extra credit assignments. Students in the control group will have the opportunity to earn the same number of extra credit points as those in the experimental group through similar assignments that do not encourage particular attributions.

At the end of the semester the students will again be tested with the Phobos Inventory and the Mathematics Attribution Scale. Methods for verifying changes in paired data will be applied to determine if there is a significant difference between the pre-experiment and post-experiment Mathematics Anxiety scores.

To accomplish this, each participant’s pre and post intervention scores must be linked so that they can be compared. To maintain anonymity each student will be identified by a randomly generated five-digit number. Two copies of this number will be sealed in separate envelopes and, prior to the initial survey, both envelopes in each pair will be marked with a particular student’s name. One envelope will be given to the student during each survey and the student will be asked to place the number on the survey’s cover sheet. No records will be kept identifying which 5-digit number is sealed in which pair of envelopes and no attempt will be made to match a particular student to a particular 5-digit number or to a particular set of responses.

9. Participants - check all that apply

(x) University of Miami Students

10. Federal Regulations have established guidelines for the inclusion of women,

minorities, and children in research involving human subjects, whether or not it is supported by NIH.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 87: Math anxiety and learn helpless

76Please check those you are including: (x) Women (x) Minorities

(x) Participants under the age of21

11. Number of Participants to be recruited: 150 - 250Age of participants: 18 and over (depends on classes selected)Sex of participants: both (proportions depend on classes selected)

12. Records:

Participants’ records regarding this protocol will be maintained

(x) other (specify):

In the files of Joseph Kolacinski

13. Confidentiality:

Describe the provisions which have been made for preservation of anonymity or confidentiality in the transmittal of data:

(x) Other (specify):

For the correlational part of this study, data will be anonymous; no identifying information will be collected. In the longitudinal part of the study, there is a need to link pre-experiment scores and post-experiment scores, but data will still be collected anonymously. Each participant will be given a randomly generated five-digit number in a sealed envelope. Each participant will copy this number onto his or her survey. No record of these numbers will exist and there will be no attempt to link a particular set of answers to a particular participant. In addition, no instructor will survey his or her own students and a class’s instructor will not be present when surveys are conducted.

14. Deceptive Techniques:

(x) Not applicable.

In order to avoid any possible deception, slightly different versions of the informed consent memo will be given to the control group and the experimental group. This will avoid potential disruptions to the learning process, such as students from the different groups comparing extra credit assignments and attempting to determine how the different assignments relate to the study.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 88: Math anxiety and learn helpless

77

After the second questionnaire is given, the students will be debriefed and told explicitly how the activities of the two groups differed, and how the activities of the experimental group might have an effect on mathematics anxiety.

15. Investigator’s Evaluation of Potential Physical, Psychological, or Social Risk to

Subjects:

None.

16. Informed Consent: (attach to this form)

(x) to be sent out as a non-returned cover memo.

Please also refer to item 14.

17. Study Results:

Describe the procedure that will be used to inform the subjects of the results of the study.

Subjects will be able to get information about the results of the study by contacting Joseph Kolacinski after the study’s completion.

18. Medical facet (check one):

(x) This research does not have any medical facet.

19. Assurances

I affirm that no change will be made in the methods of procedure or the informed consent statement of this study without prior approval of the reviewing committee.

I affirm that the Principal Investigator will prepare a summary of the project annually, including all information specified by the Guidelines for Behavioral Research Involving Subjects at the University of Miami.

I affirm that I have received a copy of the University of Miami’s guidelines for Behavioral Research involving Human Subjects, and agree to follow and abide by them.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 89: Math anxiety and learn helpless

78I received a copy of the Multiple Project Assurance of Compliance with DHHS Regulations for Protection of Human Research Subjects.

The Undersigned are fulltime faculty members who assume responsibility for this study.

Principal Investigator ______________________ _____Dr. Gilbert Cuevas (Signature) (Date)Dept, of Teaching and Learning

Principal Investigator ______________________ _____Dr. Marvin Mielke (Signature) (Date)Dept, of Mathematics

Principal Investigator ______________________ _____Dr. Robert Kelley (Signature) (Date)Dept, of Mathematics

Department Chair ______________________ _____Dr. Alan Zame (Signature) (Date)Dept, of Mathematics

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 90: Math anxiety and learn helpless

Appendix D

Informed Consent Memos

The following materials were handed out to the students participating in

this study.

The first version of the informed consent memo was given to the students

who were involved only in the correlational part of the study. No identifying

information was needed on these questionnaires.

The remaining two versions were provided to the students who are

participating in the interventional part of the study. Each of these students was

identified using a random number as described in the second and third versions

of the informed consent memo. The second version of the memo was given to

the comparison group; the third was given to the experimental group.

79

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 91: Math anxiety and learn helpless

80

Mathematics Anxiety and Learned HelplessnessInformed Consent

[Version 1 - used only in the correlational part of the study]

PurposeYou are being asked to participate in a study concerning college students’ attitudes about mathematics. This study is part of a doctoral thesis and is being conducted to investigate one possible cause of Mathematics Anxiety.

ProceduresAs a participant in this study, you will be requested to answer the questions on the enclosed questionnaire honestly and carefully. It should take approximately 20 minutes to complete the questionnaire.

RisksThere are no anticipated risks to you if you participate in this study.

BenefitsNo benefit can be promised to you from your participation in this study.

CompensationYou will not be compensated for your participation in this study.

AlternativesYou may refuse to participate in this study. Nothing bad will happen to you if you refuse to participate in this study.

EligibilityStudents under the age of 18 may not participate in this study.

ConfidentialityTo protect your right to privacy as a volunteer participant in this study, all information is being collected anonymously. There will be no attempt to link a particular set of answers to a particular student.

Right to WithdrawYour participation is voluntary; you have the right to withdraw or to skip any questions if you want to.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 92: Math anxiety and learn helpless

81

Further InformationIf you wish to find out the results of this study, you may contact Joseph Kolacinski at (305) 284-2308 or <[email protected]> in the Department of Mathematics. The study should be completed by January, 2003.

Questions regarding this study or your participation should be directed to Joseph Kolacinski or Dr. G. Cuevas at (305) 284-3141. If you have questions about your rights as a research participant, you may call Maria Arnold, Institutional Review Board Director at (305) 243-2079. You may keep this copy of this cover letter for your records.

Thank you for your assistance.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 93: Math anxiety and learn helpless

82

Mathematics Anxiety and Learned HelplessnessInformed Consent

[Version 2 - given to the comparison group]

PurposeYou are being asked to participate in a study concerning college students’ attitudes about mathematics. This study is part of a doctoral thesis and is being conducted to investigate one possible cause of Mathematics Anxiety.

ProceduresAs a participant in this study, you will be asked to fill out a questionnaire, both now and toward the end of the semester. Please fill out the questionnaire honestly and carefully. It should take approximately 20 minutes to complete the questionnaire.

RisksThere are no anticipated risks to you if you participate in this study.

BenefitsNo benefit can be promised to you from your participation in this study.

CompensationYou will not be compensated for your participation in this study.

AlternativesYou may refuse to participate in this study and you may end your participation at any time. Nothing bad will happen to you if you refuse to participate or if you stop participating in this study.

EligibilityStudents under the age of 18 may not participate in this study.

ConfidentialityThe information gathered in this study is being collected anonymously. To protect your right to privacy as a volunteer participant in this study, the following precautions have been taken:

• Since the results of your survey need to be compared to the results from the end of the semester, you will be identified by a randomly generated five-digit number.

• Two slips of paper containing this number will be sealed in envelopes. You will be given one of these envelopes each time you complete the questionnaire. You will write this number on the cover sheet of your questionnaire and then discard the envelope and the slip of paper containing the number when you are done.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 94: Math anxiety and learn helpless

83

• No record of these numbers will exist.

• There will be no attempt to link a particular set of answers to a particular student.

• Your instructor will not be present when the surveys are conducted and will not know which students participated in the survey nor which students completed which questionnaire.

Right to WithdrawYour participation is voluntary; you have the right to withdraw or to skip any questions if you want to.

Further InformationIf you wish to find out the results of this study, you may contact Joseph Kolacinski at (305) 284-2308 or <[email protected]> in the Department of Mathematics. The study should be completed by January, 2003.

Questions regarding this study or your participation should be directed to Joseph Kolacinski or Dr. G. Cuevas at (305) 284-3141. If you have questions about your rights as a research participant, you may call Maria Arnold, Institutional Review Board Director at (305) 243-2079. You may keep this copy of this cover letter for your records.

Thank you for your assistance.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 95: Math anxiety and learn helpless

84

Mathematics Anxiety and Learned HelplessnessInformed Consent

[Version 3 - given to the experimental group]

PurposeYou are being asked to participate in a study concerning college students’ attitudes about mathematics. This study is part of a doctoral thesis and is being conducted to investigate one possible cause of Mathematics Anxiety.

ProceduresAs a participant in this study you will be asked to:

• Fill out a questionnaire, both now and toward the end of the semester both honestly and carefully. It should take approximately 20 minutes to complete the questionnaire.

• Participate in certain extra credit assignments.

• Listen to information presented throughout the semester.

RisksThere are no anticipated risks to you if you participate in this study.

BenefitsNo benefit can be promised to you from your participation in this study.

CompensationYou will not be compensated for your participation in this study.

AlternativesYou may refuse to participate in this study and you may end your participation at any time. Nothing bad will happen to you if you refuse to participate or if you stop participating in this study. In particular, you will be able to participate in any extra credit assignment whether or not you decide to participate in the survey.

EligibilityStudents under the age of 18 may not participate in this study. Students under the age of 18 will have the same opportunities for extra credit as every other student.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 96: Math anxiety and learn helpless

85

ConfidentialityThe information gathered in this study is being collected anonymously. To protect your right to privacy as a volunteer participant in this study, the following precautions have been taken:

• Since the results of your survey need to be compared to the results from the end of the semester, you will be identified by a randomly generated five-digit number.

• Two slips of paper containing this number will be sealed in envelopes.You will be given one of these envelopes each time you complete the questionnaire. You will write this number on the cover sheet of your questionnaire and then discard the envelope and the slip of paper containing the number when you are done.

• No record of these numbers will exist.

• There will be no attempt to link a particular set of answers to a particular student.

• Your instructor will not be present when the surveys are conducted and will not know which students participated in the survey nor which students completed which questionnaire.

Right to WithdrawYour participation is voluntary: you have the right to withdraw or to skip any questions if you want to.

Further InformationIf you wish to find out the results of this study, you may contact Joseph Kolacinski at (305) 284-2308 or <[email protected]> in the Department of Mathematics. The study should be completed by January, 2003.

Questions regarding this study or your participation should be directed to Joseph Kolacinski or Dr. G. Cuevas at (305) 284-3141. If you have questions about your rights as a research participant, you may call Maria Arnold, Institutional Review Board Director at (305) 243-2079. You may keep this copy of this cover letter for your records.

Thank you for your assistance.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page 97: Math anxiety and learn helpless

Vita

Joseph Franke Kolacinski, the son of William Victor Kolacinski, a police

officer and Mary Franke Kolacinski, an artist, was bom in Brooklyn, New York on

24 January 1964. He is named for his maternal grandfather who was one of the

great pen and ink illustrators of the 1920’s and 30’s.

Joseph grew up, mostly, in Lake Worth, Florida where he had many

strange and fascinating adventures. He graduated from Lake Worth High School

and did his undergraduate work at Palm Beach Junior College and Florida

Atlantic University. He began studying at F. A. U. with the goal of teaching high

school mathematics, but his first education class at that institution inspired him to

continue on to do graduate work. In mathematics. He eventually earned a

Doctor of Arts degree in Mathematics from the University of Miami in August,

2003.

Joseph currently resides, along with his two cats, in Miami, Florida where

he teaches Mathematics to college students and hopes to have many more

strange and fascinating adventures.

Permanent Address: Post Office Box 24-8805Coral Gables, FL 33124 [email protected]

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.