Comparison of males and females on math item performance

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Comparison of males and females on math item performance analysis of response patterns
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Feliciano, Sonia, 1938-
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Sex differences   ( lcsh )
Foundations of Education thesis Ph. D
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Thesis (Ph. D.)--University of Florida, 1986.
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Bibliography: leaves 107-115.
Statement of Responsibility:
by Sonia Feliciano.
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Typescript.
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Vita.

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COMPARISON OF MALES AND FEMALES
ON MATH ITEM PERFORMANCE:
ANALYSIS OF RESPONSE PATTERNS





By

SONIA FELICIANO


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1986











ACKNOWLEDGMENTS


I would like to acknowledge my gratitude to several

people who have influenced my formal education and/or made

this study possible. My special thanks to Dr. James

Algina, Chairman of the doctoral committee who contributed

to the development of my love for research and statistics.

He has been insuperable as professor and valued friend; his

help and guidance in the preparation and completion of this

study were invaluable. I extend my thanks to Dr. Linda

Crocker for her advice and help during my doctoral studies

at University of Florida. Thanks also go to Dr. Michael

Nunnery, member of the doctoral committee. To Dr. Wilson

Guertin, who was a friend for me and my family, I extend my

special thanks. Thanks are also extended to Dr. Amalia

Charneco, past Undersecretary of Education of the Puerto

Rico Department of Education, for her continuous support.

To my sister Nilda Santaellar who typed the thesis, I give

my sincere thanks. Special thanks go to my family and to

those friends who provided encouragement throughout this

critical period of my life.










TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS.... .......................... ii

LIST OF TABLES .. ................ ................... v

LIST OF FIGURES...................... ............... vii

ABSTRACT....... ...................................... viii

CHAPTER

I INTRODUCTION.................................. 1

Purpose of the Study ............... ........ 6
Significance of the Study................... 6
Organization of the Study................... 8


II REVIEW OF THE LITERATURE ...................... 9

Sex-related Differences in Incorrect
Response Patterns........................ 10
Sex-related Differences in Problem
Solving............... ..... ........... .... 14
Cognitive and Affective Variables that
Influence Sex Differences in Mathematics
Learning and Achievement................ 23
Differences in Formal Mathematics
Education....................... 24
Differences in Spatial Ability.......... 26
Differentiated Effect of Affective
Variables................... ....... ... 30

Problem Solving Performance and Related
Variables.... ..................... ......... 37
Computational Skills and Problem Solving
Performance. .......................... 38
Reading and Problem Solving Perfor-
mance........................... .... .. 44
Attitudes Toward Problem Solving and
Problem Solving Performance........... 50


iii










III METHOD..... ... ...... ...... ..... ............... 54

The Sample ................ ................... 54
The Instrument............. ................ 55
Analysis of the Data....................... 57
Analysis of Sex by Option by Year
Cross Classifications................ 57
Comparison of Males and Females in
Problem Solving Performance........... 66


IV RESULTS........... ..... ....... ...... .. ........ 68

Introduction ............................. 68
Sex-related Differences in the
Selection of Incorrect Responses.......... 68
Sex-related Differences in Problem
Solving Performance.................... 81
Summary.................................... 93


V DISCUSSION..................................... 101

Summary and Interpretation of the
Results ................................... 101
Implications of the Findings and
Suggestions for Further Research.......... 103
Sex-related Differences in
Incorrect Responses................ 104
Sex-related Differences in Problem
Solving Performance................... 105

REFERENCES.......................................... 107

BIOGRAPHICAL SKETCH................................ 116









LIST OF TABLES


Table Page

3.1 Hypothetical Probabilities of Option
Choice Conditional on Year and Sex.......... 58

3.2 Hypothetical Probabilities of Option
Choice Conditional on Year and Sex and
Arranged by Sex......... ...... ........... 58

3.3 Hypothetical Joint Probabilities of Yearr
Option, and Sex............................. 64

3.4 Hypothetical Probabilities of Option
Choice Conditional on Sex and Year.......... 66

4.1 Summary of Significant Tests of Model 3.2..... 71

4.2 Chi-square Values for the Comparison of 75
Models (3.2) and (3.3) with Actual Sample
Sizes and with the Corresponding Number
of Subjects Needed for Significant Results 75

4.3 Means, Standard Deviations, and t-Test for
the Eight Mathematical Variables............ 82

4.4 ANCOVA First Year: Multiplication Covariate 85

4.5 ANCOVA First Year: Division Covariate....... 85

4.6 ANCOVA Second Year: Subtraction Covariate... 86

4.7 Reliability of the Covariates for Each
of the Three Years of Test Administra-
tion........................................ 91

4.8 ANCOVA First Year: Other Covariates......... 92

4.9 Adjusted Means on Problem Solving, by
Covariate and Sex.......................... 94

4.10 ANCOVA Second Year: Other Covariates........ 95









Table


Page


4.11 Adjusted Means on Problem Solving,
by Covariate and Sex........................ 96

4.12 ANCOVA Third Year........................ 97

4.13 Adjusted Means on Problem Solving,
by Covariate and Sex....................... 98










LIST OF FIGURES


Figure Page


3.1 Number of Questions by Skill Area in a
Basic Skills Test in Mathematics-6.......... 56

4.1 Sex by Multiplication Interaction............. 87

4.2 Sex by Division Interaction ................ 88

4.3 Sex by Subtraction Interaction................ 89


vii














Abstract of Dissertation Presented to the Graduate
School of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy

COMPARISON OF MALES AND FEMALES
ON MATH ITEM PERFORMANCE:
ANALYSIS OF RESPONSE PATTERNS

By

Sonia Feliciano

August, 1986


Chairman: James Algina
Major Department: Foundations of Education

The first objective of this study was to investigate

sex differences in the selection of incorrect responses on

a mathematics multiple-choice test, and to determine

whether these differences were consistent over three

consecutive administrations of the test. A second

objective was to compare male and female performance in

problem solving after controlling for computational skills.

The responses of all 6th grade students from the

public schools in Puerto Rico who took the Basic Skills

Test in Mathematics-6 ("Prueba de Destrezas B&sicas en

Matem&ticas-6") during three academic years were used in

the analyses relevant to the first objective.


viii









Log-linear models were used in the analysis of

incorrect responses. The results of the analyses showed

that for 100 of the 111 items of the test, males and

females selected different incorrect options, and this

pattern of responses was consistently found during the

three years of test administration. However, for the vast

majority of the 100 items the male-female differences were

relatively small, considering the fact that the number of

subjects needed to obtain statistical significance was very

large.

The responses of approximately 1,000 randomly selected

students per academic year were analyzed in the comparison

of male and female performance in problem solving. Females

outperformed males in problem solving and in six of the

seven computational variables. Males showed superiority in

equivalence in all the three years, but statistical

significance was obtained in only one of the years.

Analysis of covariance (ANCOVA) was used in the

comparison of male and female performance in problem

solving after controlling for computational skills. Seven

analyses of covariance tests were conducted, one for

each of the covariates. Estimated true scores for observed

scores were used in the analyses. The results tend to show

that for examinees with similar levels of computational









skills, sex-related differences in problem solving

performance do not exist. Females retained their

superiority in problem solving when equivalence (in all

three years) and subtraction (in one year) were the

controlling variables.


The question of whether male-female differences in

problem solving depend on computational skills was

answered, partially, in the affirmative.













CHAPTER I
INTRODUCTION



Sex differences in mathematics learning and

achievement have been the subject of intensive research.

Research done before 1974 has shown that male performance

on mathematical achievement tests is superior to female

performance by the time they reach upper elementary or

junior high school (Fennema, 1976, p. 2). The literature

strongly suggests that at the elementary level females

outperform males in computation and males excel in

mathematical reasoning (Glennon & Callahan, 1968; Jarvis,

1964; Maccoby, 1966).

Since 1974, research findings have been less

consistent. Fennema (1974), after reviewing 36 studies,

found that during secondary school or earlier, sex-related

differences in mathematics achievement are not so evident,

but that when differences are founds they favor males in

high level cognitive tasks (problem solving) and females in

low level cognitive tasks (computation). As a result of a

further review of the literature, Fennema (1977) concluded

that at the elementary level, sex-related differences do

not exist at all cognitive levels, from computation to

problem solving.











Many variables, cognitive affective, and educational,

have been investigated since 1974 in relation to sex

differences in mathematics learning and achievement.

Fennema and Sherman (1977) investigated the effect of

differential formal mathematics education. After

controlling for the number of years of exposure to the

study of mathematics, they found sex differences in only

two of the four schools under study. However, in those

schools where boys scored higher than girls, differences

were also found in their attitudes toward mathematics.

Hilton and Berglund (1974) found significant sex

differences after controlling for the number of mathematics

courses taken, and attributed them to sex differences in

interests. "As the boys' interests in science increase

related to the girls', their achievement in mathematics

increases relative to that of the girls" (p. 234).

Wise, Steel, and McDonald (1979) reanalyzed test data

collected in a longitudinal study of 400,000 high school

students (Project Talent). They found that when the effect

of the number of high school mathematics courses was not

controlled, no sex differences emerged for 9th graders, but

that gains made by boys during the next three years were

more than twice that of the girls. These differences

between the sexes disappeared when the number of

mathematics courses taken was controlled. Results of the

1978 Women and Mathematics National Survey, Survey I,










indicated no significant sex differences for 8th grade

students on measures of problem solving or algebra.

However, females outperformed males in computation and

spatial visualization. For the 12th grade students,

statistically significant sex differences favoring males

were found in problem solving, but not in algebra,

computation, or spatial visualization. For males and

females who had enrolled in courses beyond general

mathematics and who had taken or were enrolled in courses

such as pre-calculus, calculus, or geometry, differences in

problem solving or spatial visualization did not exist.

Sex differences favoring males were found on a total score

obtained summing across the computation, problem solving,

and algebra subtests (Armstrong, 1979). The mathematics

data collected in the second survey by the 1978 National

Assessment of Educational Progress showed significant sex

differences for both 13- and 17-year-old students. The

13-year-old females outperformed males in the computational

subtest and males outscored females by 1 1/2 percentage

points in problem solving (statistically significant). No

statistically significant differences were found in

algebra. No sex differences were found for the 17-year-old

group either in the computation subtest or in the algebra

subtest. Males surpassed females in problem solving. A

reanalysis of the data from the 17-year-old group confirmed

male superiority in problem solving after controlling for











mathematics preparation. Males who were enrolled or had

completed algebra II outperformed the females in

computation and problem solving but not in algebra. Males

who studied beyond algebra II outscored females on all

three subtests: computation, algebra, and problem solving

(Armstrong, 1979).

Carpenter, Lindquist, Mathews, and Silver (1984)

analyzed the results of the Third National Assessment of

Educational Progress (NAEP), and compared them with the

First and Second Surveys. Between 1978 and 1982, the

differences between the average performance of males and

females remained stable at each age level. At ages 9 and

13, the overall performance of males and females was not

significantly different. At age 17r males scored higher

than females by about 3 percentage points. When course

background was held constant, achievement differences still

existed at age 17. For each category of course background,

male achievement exceeded female achievement. Consistent

with previous assessments, sex differences in problem

solving in favor of males were found for the 17-year-old

sample. At ages 9 and 13, no large differences were found

between the sexes within any level of course background.

Marshall (1981, 1984) investigated sex differences in

mathematics performance. She found that males and females

excel each other in solving different types of problems.











Females were better on items of computation and males were

more successful on word-story problem items (problem

solving). She also found that females successful

performance in the problem solving items was more dependent

on their successful performance in the computation items.

Males did not need, as much as females, to succeed in the

computation items in order to answer correctly the problem

solving items.

Although the general findings seem to support sex

differences in mathematics learning and achievement, the

research done does not consistently support superiority for

either sex. Most of the research has been concerned with

how the sexes differ on subtests or total test scores in

mathematics. Moreover, the great majority of the studies

deal with correct responses. Sex differences in incorrect

responses at the item level have not been fully researched.

Only two studies dealing with sex differences in incorrect

responses at the item level were found in the research

literature (Marshall, 1981, 1983). Marshall investigated

whether boys and girls made similar errors in computation

and story problems. She analyzed boys' and girls' answers

to six mathematics items and found that the sexes made

different errors, possibly reflecting different problem

solving strategies. Her original findings were supported

when she studied the same problem using a large number of

items three years later.











Purpose of the Study

The purpose of this study was twofold. The first was

to investigate sex-related differences in the selection of

incorrect responses in a multiple-choice achievement test

in mathematics. For each test item, the research questions

were as follows:

1. Is there a difference in the proportion of males

and females choosing each incorrect option?

2. Is the same pattern of differences found in data

obtained in three different administrations of the test?

The second objective was to investigate sex-related

differences in test scores in mathematics problem solving.

The following questions were studied:

1. Do males and females differ in problem solving

performance?

2. Do sex differences in problem solving persist

after controlling for computational skills, and does the

differential success of males and females on problem

solving items depend on their success on the computation

items?

Significance of the Study

Item response patterns are very useful techniques in

the assessment of mathematics learning and achievement.

Total test scores can be very misleading in the assessment

of student performance and provide no diagnostic

information about the nature and seriousness of student











errors (Harnisch, 1983). Item response patterns are

valuable for the identification of large group differences,

including district-to-district, school-to-school, and

classroom-to-classroom variations on different subsets of

items. The response patterns can provide diagnostic

information about the type of understanding the student has

on various mathematics topics (e.g.r problem solving).

Marshall (1981, 1984) has used the item response

pattern technique and her findings indicate sex differences

in mathematics performance at the item level. Females

outperformed males in computation and males outscored

females in problem solving. Also the success of girls in

the problem solving items was dependent upon their success

in the computation items; for boys, success in the problem

solving items did not depend as much on their computational

performance. Marshall (1981, 1983) has also reported that

males and females differ in the selection of incorrect

responses, reflecting differences in reasoning abilities.

In Puerto Rico, a high percentage of children promoted

to the 7th grade in the public schools does not master the

basic skills in mathematics. If 6th grade male and female

children can be diagnosed as having different problem

solving abilities, as Marshall found with California

children, teachers may need to provide tailor-made

mathematics instruction for each sex, in order to ensure

equal access to formal education and enhance mathematics











achievement. Since there are no investigations reported in

sex differences in item response patterns in Puerto Rico,

research is needed.


Organization of the Study

A review of the literature on sex differences in

mathematics performance is reported in Chapter II. The

research methodology is presented in Chapter III. Research

questions, sample, instrument, and data analysis are

discussed in that chapter. Chapter IV is an exposition of

the results of the study. Chapter V contains a summary and

interpretation of the results of the study and the

implications of the findings together with suggestions for

further research.













CHAPTER II
REVIEW OF THE LITERATURE



Sex differences in mathematics learning and

achievement have been a subject of concern for educators

and psychologists. Many studies found in the literature

support the existence of these differences. Boys show

superiority in higher level cognitive tasks (problem

solving or mathematical reasoning) in the upper elementary

years and in the early high school years (Fennemar 1974;

Maccoby & Jacklin, 1974).

Almost all the research carried out has dealt with

analysis of total correct scores in mathematics aptitude

and achievement tests or scores in subtests. The

literature related to sex differences in incorrect

responses, the main subject of the present study, is

surprisingly sparse. For the most part, the studies have

investigated the differences between the sexes in

mathematics learning and achievement and the underlying

variables causing the differences. Cognitive and affective

variables have been the matter at issue in the

establishment of sex differences.

Although research in mathematics problem solving, the

secondary subject of this investigation, is extensive, most











of the studies consider sex differences incidental to the

major study findings. The available literature offers very

little research directly related to the problem of sex

differences in this area.

The review of the literature has been divided in four

sections. The first section consists of a detailed summary

of the available research on sex differences in incorrect

responses. The second section deals with sex-related

differences in problem solving performance. These sections

are directly related to the objectives of the study. The

third section is more peripheral, and contains a discussion

of the more prevalent issues about the influence of

cognitive and affective variables on sex differences in

mathematics learning and achievement. The fourth is a

summary of the research dealing with variables considered

as influential to mathematics problem solving performance.



Sex-related Differences in Incorrect Response Patterns



Research findings tend to suggest that boys and girls

may be approaching problem solving differently (e.g.,

Fennema and Sherman, 1978; Marshall, 1981, 1983; Meyer,

1978, among others). Marshall (1981) investigated whether

6th grade boys and girls approach mathematical problem

solving with different strategies. Her specific interest

was whether the sexes made the same errors.











She analyzed the responses of 9,000 boys and 9,000

girls to 6 selected items, 2 computation items, and 1 story

problem item from each of 2 of the 16 forms of the Survey

of Basic Skills test administered during the academic year

1978-79. The Survey is a 30-item achievement test

administered every year to all 6th grade children in

California through the California Assessment Program.

There are 16 forms of the test, to which approximately

9,000 boys and 9,000 girls respond each year. Of the 160

mathematics items contained in the 16 forms of the test, 32

are on measurement and graphing, 28 on number concepts, 28

on whole number arithmetics, 20 on fraction arithmetics, 20

on decimal arithmetics, 20 on geometry, and 12 on

probability and statistics.

The item analysis performed on the 1978-79 data showed

that boys and girls tended to select different incorrect

responses. In the first computation item (Form 1 of the

test) both sexes reflected similar mistakes in carrying,

but in different columns. In the second computation item,

both sexes ignored the decimal points and selected the same

incorrect response. However, more girls than boys chose

this response.

In the first computation item (Form 2 of the test) the

incorrect choice of both sexes was option cr but the

second most frequently selected option was a for boys

and b for girls. In the second computation item of this











test form, no sex differences were found in response

patterns. Approximately 45% of each sex selected option

c. The next popular choice for both sexes was option

d, selected by approximately 35% of both boys and girls.

On the story problem of Form 1, males and females

responded alike. Their most popular incorrect response

choice was option a for both males and females. The

second most popular incorrect choice was option c for

both sexes.

Response to the story problem in Form 2 showed sex

differences in response choice. Including the correct

option, 33% of the girls selected option a, 20% chose

option c, and 20% option d. For males approximately

25% selected option a and the same percent selected

option d.

Marshall concluded that although the analysis of

incorrect responses does not explain why boys and girls

differ in their responses, the analysis shows that boys and

girls approach problems in different ways and these varying

strategies can be useful in identifying how the sexes

differ in reasoning abilities.

Two years later, Marshall (1983) analyzed the

responses of approximately 300,000 boys and girls to

mathematics items contained in the 16 test forms of the

Survey of Basic Skills during the years 1977, 1978, and

1979. She used log-linear models (explained in Chapter












III) to investigate sex related differences in the

selection of incorrect responses, and the consistency of

such differences over three years of administration of the

test.

Based on her findings that sex differences were found

in 80% of the items, Marshall classified the students'

errors according to Radatz' (1979) five-category error

classification. The categories are language (errors in

semantics), spatial visualization, mastery, association,

and use of irrelevant rules.

It was found that girls' errors are more likely to be

due to the misuse of spatial information, the use of

irrelevant rules, or the choice of an incorrect operation.

Girls also make relatively more errors of negative transfer

and keyword association. Boys seem more likely than girls

to make errors of perseverance and formula interference.

Both sexes make language-related errors, but the errors are

not the same.

Available research is not extensive enough to make

definite judgments about the sex-related differences

observed in incorrect responses. Clearly more research is

needed.











Sex-related Differences in Problem Solving



It has already been acknowledged that the subject of

problem solving has been extensively researched. However,

as early as 1969, Kilpatrick criticized the fact that the

study of problem solving has not been systematic; some

researchers have studied the characteristics of the problem

while others have given their attention to the

characteristics of the problem solvers. Moreover,

differences in the tests used to measure problem solving

performance also constitute an obstacle when trying to

compare the results of the studies carried out.

In order to avoid this pitfall and provide a basis for

comparison, the studies reviewed in this section, dealing

with sex-related differences in problem solving have been

divided in two groups. The first comprises those studies

that used the Romberg-Wearne Problem Solving Test. The

second contains other relevant studies in which problem

solving performance has been measured by means of other

instruments.

The Romberg-Wearne Problem Solving Test merits special

mention because it was the first attempt "to develop a test

to overcome the inadequacies of total test scores in

explaining the reasons why some students are successful

problem solvers and others are not" (Whitaker, 1976, pp. 9,

10).











The test is composed of 23 items designed to yield 3

scores: a comprehension score, an application score, and a

problem solving score. The comprehension question

ascertains whether a child understands the information

given in the item stem. The application question assesses

the child's mastery of a prerequisite concept or skill of

the problem solving question. The problem solving question

poses a question whose solution is not immediately

available, that is, a situation which does not lend itself

to the immediate application of a rule or algorithm. The

application and problem solving parts of the test may refer

to a common unit of information (the item stem) but the

questions are independent in that the response to the

application question is not used to respond to the problem

solving question.

Meyer (1978) Whitaker (1976), Pennema and Sherman

(1978), and Sherman (1979) have used the Romberg-Wearne

test in their studies. Meyer (1978) investigated whether

males and females differ in problem solving performance and

examined their prerequisite computational skills and

mathematical concepts for the problem solving questions. A

sample of 179 students from the 4th grade were administered

19 "reference tests" for intellectual abilities and the

Romberg-Wearne test. The analysis showed that males and

females were not significantly different in the

comprehension, application, and problem solving questions











of the test. The sexes differed in only 2 of the 19

reference tests, Spatial Relations and Picture Group

Name-Selection. A factor analysis, however, showed

differences in the number and composition of the factors.

For females, a general mathematics factor was determined by

mathematics computation, comprehension, application, and

problem solving. For males, the comprehension and

application parts determined one factor; problem solving

with two other reference tests (Gestalt and Omelet)

determined another factor.

Meyer concluded that comprehension of the data and

mastery of the prerequisite mathematical concepts did not

guarantee successful problem solving either for males or

for females. Problem solving scores for both sexes were

about one third their scores in comprehension and one

fourth their scores in application.

She also concluded that the sexes may have approached

the problem solving questions differently. The methods

used by females for solving problem situations may have

paralleled their approach to the application parts.

Males may have used established rules and algorithms for

the application parts, but may have used more of a Gestalt

approach to the problem solving situation.

Whitaker (1976) investigated the relationship between

the mathematical problem performance of 4th grade children

and their attitudes toward problem solving, their teachers'











attitude toward mathematical problem solving, and related

sex and program-type differences. Although his main

objective was to construct an attitude scale to measure

attitudes toward problem solving, his study is important

because his findings support Meyer's regarding the lack of

significant sex-related differences in problem solving

performance. Performance in the problem solving questions,

for both males and females, was much lower than performance

in the application questions, and much lower than

performance in the comprehension questions. In fact, the

mean score for each part of the item, for both males and

females, was almost identical to the mean scores obtained

by males and females in Meyer's study. Whitaker noted that

each application item is more difficult than its preceding

comprehension item, and that each problem solving item is

more difficult than its preceding application item. No

significant sex-related differences were found for any of

the three parts of the item (comprehension, application, or

problem solving).

Fennema and Sherman (1978) investigated sex-related

differences in mathematics achievement and cognitive and

affective variables related to such differences. They

administered the Romberg-Wearne Problem Solving Test to a

representative sample of 1320 students (grades 6-8) from

Madison, Wisconsin, predominantly middle-class, but

including great diversity in SES. The sample consisted of











students who had taken a similar number of mathematics

courses and were in the top 85% of the class in mathematics

achievement. They were tested in 1976. Four high school

districts were included. In only one of the high school

districts were sex-related differences in application and

problem solving founds in favor of males. They concluded

that when relevant factors are controlled, sex-related

differences in favor of males do not appear often, and when

they dor they are not large.

Sherman (1980) investigated the causes of the emerging

sex-related differences in mathematics performance, in

favor of males, during adolescence (grades 8-11). She

wanted to know if these differences emerge as a function of

sex-related differences in spatial visualization and

sociocultural influences that consider math as a male

domain. In grade 8, she used the Romberg-Wearne Test and,

in grade 11, a mathematical problem solving test derived

from the French Kit of Tests.

The analysis showed that for girls, problem solving

performance remained stable across the years. Mean problem

solving performance for boys, however, was higher in grade

11 than in grade 8. No sex-related differences were found

in grade 8, but boys outperformed girls in grade 11, where

the Stafford test was used.

Sherman found that for both sexes problem solving

performance in grade 8 was the best predictor of problem











solving performance in grade 11. Spatial visualization was

a stronger predictor for girls than for boys. Mathematics

as a male domain was a good predictor for girls only; the

less a girl stereotyped mathematics as a male domain in

grade 8, the higher her problem solving score in grade 11.

Attitudes toward success in mathematics in grade 8 was a

more positive predictor of problem solving performance for

boys than for girls; the more positive the attitudes toward

success in mathematics in grade 8, the higher their

performance in problem solving in grade 11.

None of these four studies, all of which used the

Romberg-Wearne Mathematics Problem Solving Test, show

statistically significant sex-related differences in

problem solving performance. In later studies other tests

were used to measure this variable (Kaufman, 1984;

Marshall, 1981, 1984).

Kaufman (1983/1984) investigated if sex differences in

problem solving, favoring males, exist in the 5th and 6th

grades and if these differences were more pronounced in

mathematically gifted students than in students of average

mathematical ability. The Iowa Test of Basic Skills and a

mathematics problem-solving test were administered to 504

subjects. Males in the average group as well as males in

the gifted group outperformed females, but only the gifted

group showed statistically significant differences.











As a result of her investigations, Marshall (1981)

concluded that sex-related differences in mathematics

performance may be the result of comparing the sexes on

total test scores. If the test contains more computation

items than problem solving items, girls will perform better

than boys, but if the test contains more problem solving

items than computation ones, boys will outperform girls.

With this in mind, Marshall investigated sex-related

differences in computation and problem solving by analyzing

the responses of approximately 18,000 students from grade 6

who had been administered the Survey of Basic Skills Test:

Grade 6, during the academic year 1978-79.

Two of the 16 test forms of the Survey were used to

assess skills such as concepts of whole numbers, fractions,

and decimals. These skills were tested both as simple

computations and as story problems (problem solving).

Two computation items and one story problem item were

selected because they were particularly related; both

computation items required skills needed in solving the

corresponding story problem. It was assumed that correct

solution of the computation item correlates with solving

the story problem because the story problem requires a

similar computation.

Marshall found that girls were better in computation

and boys were better in problem solving. She also found

that boys were much more likely than girls to answer the











story problem item correctly after giving incorrect

responses to both computation items. Apparently, mastery

of the skills required by the computation items is more

important for girls than for boys. If girls cannot solve

the computation items, they have little chance of solving

the related story problem item. For girls, the

probability of success in the story problem item after

giving successful answers to both computation items is

almost 2 1/2 times the probability of success after giving

incorrect responses to both computation items. For boys,

the probability of success in the story problem item after

successful responses to the computation items is about

1 3/4 times the probability of success on the story problem

item after incorrect responses to the computation items.

Three years later, Marshall (1984) analyzed more in

depth these phenomena of sex-related differences. Her

interest was twofold. First, she wanted to know if there

were differences in the rate of success for boys and girls

in solving computation and story problem items. Second,

she examined additional factors that interact with sex to

influence mathematics performance, such as reading

achievement, socio-economic status (SES), primary language,

and chronological age. Two questions were raised: Do the

probabilibities of successful solving of computation and

story problem items increase with reading score? Are these

probabilities different for the two sexes?











Approximately 270,000 students from the 6th grade were

administered the Survey of Basic Skills of the California

Assessment Program, during the years 1977, 1978, and 1979.

Responses were analyzed using log-linear models.

Successful solving of computation items was positively

associated with successful solving of story problems.

Girls were more successful in computation than boys, and

boys were more successful than girls in solving story

problems. This finding supports reports from the National

Assessment of Educational Progress (NAEP) (Armstrong,

1979).

To investigate the effects of reading, SES, language

and chronological ager only those test forms containing 2

computation items and 2 story problems were considered for

analysis; 32 items from 8 test forms were included in the

analysis.

The results of these analyses showed that at every

level of reading score, 6th grade children were more

successful in computation than in story problems.

Although the differences were not larger at every reading

score boys consistently had higher probabilities of

success in story problems than did girls, and girls

consistently showed higher probabilities of success in

computation than boys. Also, as the reading score

increased, the difference between the probability of

success in story problems and the probability of success in











computation grew larger. This difference grew larger for

girls than for boys.

Although SES was a major factor in solving computation

and story problem items successfully, the effect was

similar for each sex. Sex-related differences by primary

language or chronological age were not large.

This research carried out by Marshall with elementary

grade children supports previous research findings that

males are better than females in mathematics problem

solving (a higher order skill) and females are better than

their counterpart males in computation (a lower level

skill). Marshall's research also brought out a different

aspect of this question: the notion that girls find it more

necessary than boys to succeed in the computation items in

order to successfully solve the story problem items.


Cognitive and Affective Factors That
Influence Sex Differences in Mathematics Learning
and Achievement


The research reviewed in the literature does not

provide evidence of any unique variable that could serve

as an explanation for the observed sex-related differences

in mathematics learning and achievement. However, some

issues have been discussed, among which the most prevalent

are that sex-related differences in mathematics learning

and achievement are a result of differences in formal

education; that sex-related differences in mathematics











learning and achievement arise from sex differences in

spatial visualization; and that sex-related differences

result from a differentiated effect of affective variables

on the mathematics performance of males and females.


Differences in Formal Mathematics Education (Differential
Coursework Hypothesis)


The basis for the differential coursework hypothesis

is the fact that sex-related differences in mathematical

learning and achievement show up when comparing groups

which are not equal in previous mathematics learning.

Atter the 8th grader boys tend to select mathematics

courses more otten than girls. Therefore, girls show lower

achievement scores in mathematics tests because their

mathematics experience is not as strong as the boys'

(Fennemar 1975; Fennema & Sherman, 1977; Sherman, 1979).

Fennema and Sherman's study (1977) lends additional

support to the feasibility of viewing sex differences in

mathematics learning and achievement as reflecting

something other than a difference in mathematics aptitude.

After controlling for previous study of mathematics, they

found significant sex differences in mathematics

achievement in only two of the four schools under study,

making the attribution to sex per se less likely.

Controlling for the number of space visualization-related

courses, the sex-related differences which originally












emerged in spatial visualization scores became

non-significant. In the two schools where sex differences

in mathematics achievement were found, differences between

the sexes were also found in their attitudes toward

mathematics.

Researchers like Backman (1972), who analyzed data

from Project Talent, and Allen and Chambers (1977) have

also hypothesized that sex-related differences in

mathematics achievement may be related to different

curricula followed by males and females. Allen and

Chambers attributed male superiority in mathematics problem

solving to differences in the number of mathematics courses

taken in high school.

This issue has been seriously questioned by Astin

(1974) Fox (1975ar 1975b), and Benbow and Stanley (1980),

among others. Astin and Fox have reported large

differences in favor of males among gifted students taking

the Scholastic Achievement Test. These differences occur

as early as grade 7, when there are no sex differences in

the number of courses taken. Benbow and Stanley (1980)

compared mathematically precocious boys and girls in the

7th grader with similar mathematics background, and found

sizeable sex-related differences favoring boys in

mathematical reasoning ability. Five years later, they

conducted a follow-up study which showed that boys











maintained their superiority in mathematics ability during

high school. While Fox attributed sex-related differences

in mathematical achievement to differential exposure to

mathematical games and activities outside school, Benbow

and Stanley suggested that sex-related differences in

mathematics performance stem from superior mathematical

ability in males, not from differences in mathematics

formal education.

The differential coursework hypothesis is not totally

convincing and, as reported before, it has been challenged

by researchers such as Benbow and Stanley (1980).

However, Pallas and Alexander (1983) have questioned the

generalizaoility of Benbow and Stanley's findings based on

the fact that they used highly precocious learners. The

differential coursework hypothesis can be accepted only as

a partial explanation of differences in mathematics

performance found between the sexes.

Differences in Spatial Ability

The basic premise in this issue is that males and

females differ in spatial visualization and this explains

differential mathematics learning and achievement. Until

recently, sex differences in spatial ability in favor ot

males were believed to be a fact and were thought by some

to be related to sex differences in mathematical

achievement.











Research findings in this area have been inconsistent.

In 1966, Maccoby stated that "by early school years, boys

consistently do better (than girls) on spatial tasks and

this difference continues through the high school and

college years" (p.26). In 1972, Maccoby and Jacklin said

that the differences in spatial ability between the sexes

"remain minimal and inconsistent until approximately the

ages of 10 or 11, when the superiority of boys becomes

consistent in a wide range of populations and tests"

(p.41). In 1974, after a comprehensive literature search,

Maccoby and Jacklin concluded that sex differences in

spatial visualization become more pronounced between upper

elementary years and the last year of high school, the

years when sex-related differences in mathematics

achievement favoring boys emerge.

Guay and McDaniel (1977) supported in part Maccoby and

Jacklin's 1974 findings. They found that among elementary

school children, males had greater high level spatial

ability than females, but that males and females were equal

in low level spatial ability. This finding is inconsistent

with that portion of Maccoby and Jacklin's review that

suggests that sex differences become evident only during

early adolescence. Cohen and Wilkie (1979) however, stated

that in tests measuring distinct spatial tasks, males

perform better than females in early adolescence and

throughout their life span. Most studies carried out after











1974 have failed to support these sex differences in

spatial abilities (Armstrong, 1979; Connor, Serbin, &

Schackman, 1977; Fennema & Sherman, 1978; Sherman, 1979).

Fennema and Sherman (1978) and Sherman (1979) have

explored sex-related differences in mathematical

achievement and cognitive and affective variables related

to these differences. In a study involving students from

grades 6, 7, and 8, from four school districts, Fennema and

Sherman found that spatial visualization and problem

solving were highly correlated for both sexes (.59 and

.60). Even in the school district where sex differences

were found in problem solving, no significant sex-related

differences were found in spatial visualization.

When Sherman (1980) compared groups of males and

females in two different grades, 8 and 11, she found no

sex-related differences in problem solving or in spatial

visualization in grade 8. In grade 11, however, although

the sexes differed in their problem solving performance, no

sex-related differences were found in spatial

vizualization. Even though spatial visualization in grade

8 was the second best predictor of problem solving

performance in grade 11, sex differences in grade 11 were

not a result of spatial visualization since no differences

were found in that skill.

In spite of the fact that no sex differences were

found in spatial abilities, it is evident that males and











females may use them in a different way. Meyer (1978),

with an elementary grade sample, and Fennema and Tartree

(1983) with an intermediate level sampler found that the

influence of spatial visualization on solving mathematics

problems is subtler and that males and females use their

spatial skills differently in solving word story problems

(problems that measure problem solving ability or

reasoning). Fennema and Tartree (1983) carried out a

three-year longitudinal study which showed that girls and

boys with equivalent spatial visualization skills did not

solve the same number of items, nor did they use the same

processes in solving problems. The results also suggested

that a low level of spatial visualization skills was a more

debilitating factor for girls than for boys in problem

solving performance.

Landau (1984) also investigated the relationship

between spatial visualization and mathematics achievement.

She studied the performance of middle school children in

mathematical problems of varying difficulty, and the extent

to which a diagramatic representation is likely to

facilitate solution. She found that spatial ability was

strongly correlated to mathematical problem solving and

that the effect of spatial ability was more influential for

females. Females made more use of diagrams in the solution

of problems, reducing the advantage of males over females

in problem solving performance.











The issue of sex-related differences in spatial

visualization ability as an explanation for sex differences

observed in mathematics achievement is less convincing and

the findings more contradictory than in the issue of sex

differences in formal education. Besides these cognitive

issues, other issues, mostly affective in nature have also

been studied in trying to explain the origin of these sex

differences in mathematics achievement and learning. The

studies dealing with these affective variables are reviewed

in the next section.

Differentiated Effect of Affective Variables

Researchers have attempted to explain the effect of

sex differences in internal beliefs, interests, and

attitudes (affective variables) on mathematics learning and

achievement. A brief statement of each explanation

precedes the summary of studies conducted that support the

explanation.

Confidence as lerners of mathematics. Females, more

than males, lack confidence in their ability to learn

mathematics and this affects their achievement in

mathematics and their election of more advanced mathematics

courses.

Maccoby and Jacklin (1974) reported that self-

confidence in terms of grade expectancy and success in

particular tasks was found to be consistently lower in

women than in men. In 1978, Fennema and Sherman reported











that in their study involving students from grades 6

through 12, boys showed a higher level of confidence in

mathematics at each grade level. These differences between

the sexes occurred in most instances even when no

sex-related differences in mathematics achievement were

found. The correlation between confidence in mathematics

performance and mathematics achievement in this study was

higher than for any other affective variable investigated.

Sherman reported a similar finding in 1980; in males,

the most important factor related to continuation in

theoretical mathematics courses was confidence in learning

mathematics. This variable weighed more than any of the

cognitive variables: mathematics achievement, spatial

visualization, general ability, and verbal skill. In the

case of females, among the affective variables, confidence

in learning mathematics was found to be second in

importance to perceived usefulness of mathematics. Probert

(1983) supported these findings with college students.

A variable that needs discussion within the context of

sex differences in confidence as learners of mathematics is

causal attribution. Causal attribution models attempt to

classify those factors to which one attributes success or

failure. The model proposed by Weiner (1974) categorizes

four dimensions of attribution ot success and failure:

stable and internal, unstable and internal, stable and

external, and unstable and external. For example, if one











attributes success to an internal, stable attribute, such

as ability, then one is confident of being successful in

the future and will continue to strive in that area. If

one attributes success to an external factor such as a

teacher, or to an unstable one, such as effort, then one

will not be as confident or success in the future and will

cease to strive. Failure attribution patterns work this

way: if failure is attributed to unstable causes, such as

effort, failure can be avoided in the future and the

tendency will be to persist in the task. However, if

failure is attributed to a stable cause, such as ability,

the belief that one cannot avoid failure will remain.

Studies reported by Bar-Tal and Frieze (1977) suggest

that males and females tend to exhibit different

attributional patterns of success and failure. Males tend

to attribute their success to internal causes and their

failures to external or unstable ones. Females show a

different pattern; they tend to attribute success to

external or unstable causes and failures to internal ones.

The pattern of attributions, success attributed externally

and failure attributed internally, has become hypothesized

to show a strong effect on mathematics achievement in

females. Kloosterman's (1985) study supported these

findings. According to Kloosterman, attributional

variables appear to be more important achievement mediators

for females than for males, as measured by mathematics word

problems. More research is needed in this area.












Mathematics as a male domain. Mathematics is an

activity more closely related to the male sex domain than

to the female sex domain (Eccles et al., 1983). Thus, the

mathematical achievement or boys is higher than that of

girls.

According to John Ernest (1976) in his study

Mathematics and Sex, mathematics is a sexist discipline.

He attributed sex-related differences in mathematical

achievement to the creation by society of sexual

stereotypes and attitudes, restrictions, and constraints

that promote the idea of the superiority of boys in

mathematics. Ernest reported that boys, girls, and

teachers, all believe that boys are superior in

mathematics, at least by the time students reach

adolescence. Bem and Bem (1970) agree and argue that an

American woman is trained to "know her place" in society

because or the pervasive sex-role concept which results in

differential expectations and socialization practices.

Plank and Plank (1954) were more specific. They

discussed two hypotheses related to this view: the

differential cultural reinforcement hypothesis and the

masculine identification hypothesis. The differential

cultural reinforcement hypothesis states that society in

general perceives mathematics as a male domain, giving

females less encouragement for excelling in it. The

masculine identification hypothesis establishes that











achievement and interest in mathematics result from

identification with the masculine role.

A study related to the differential cultural

reinforcement hypothesis is that of Dwyer (1974).

Dwyer examined the relationship between sex role standards

(the extent to which an individual considers certain

activities appropriate to males or females) and

achievement in reading and arithmetic. Students from

grades 2, 4, 6, 8, 10, and 12 participated in this study.

She found that sex role standards contributed significant

variance to reading and arithmetic achievement test scores

and that the effect was stronger for males than for

females. This led to her conclusion that sex-related

differences in reading and arithmetic are more a function

of the child's perception of these areas as sex-appropriate

or sex-inappropriate than of the child's biological sex,

individual preference for masculine and feminine sex roles,

or liking or disliking reading or mathematics.

In a study which agrees with the masculine

identification hypothesis, Milton (1957) found that

individuals who had received strong masculine orientation

performed better in problem solving than individuals who

received less masculine orientation. Elton and Rose (1967)

found that women with high mathematical aptitude and

average verbal aptitude scored higher on the masculinity

scale of the Omnibus Personality Inventory (OPI) than those

with average scores on both tasks.











It is not until adolescence that sex differences in

the perception of mathematics as a male domain are found

(Fennemar 1976; Stein, 1971; Stein & Smithless, 1969;

Verbeker 1983). In a study with 2nd, 6th and 12th

graders, Stein and Smithless (1969) found that students'

perceptions of spatial, mechanical, and arithmetic skills

as masculine became more defined as these students got

older. Fennema (1976) considers that the influence each

sex exerts upon the other on all aspects of behavior is

stronger during adolescence. Since during these years

males stereotype mathematics as a male domain, they send

this message to females who, in turn, tend to be influenced

in their willingness to study or not to study mathematics.

Before that stage, girls consider arithmetic feminine,

while boys consider it appropriate for both sexes (Bobber

1971).

Usefulness of mathematics. Females perceive mathema-

tics as less useful to them than males do, and this

perception occurs at a very young age. As a results

females exert less effort than males to learn or elect to

take advanced mathematics courses.

Many studies reported before 1976 found that the

perception of the usefulness of mathematics for one's

future differs for males and females, and is related to

course taking plans and behavior (Fox, 1977). If females

do not perceive mathematics as useful for their future,











they show less interest in the subject than counterpart

males. These differences in interest are what Hilton and

Berglund (1974) suggest to account for sex-related

differences in mathematics achievement.

Although the perception of the usefulness of

mathematics is still an important predictor of course

taking for girls, there is a growing similarity between

males and females regarding the usefulness of mathematics

(Armstrong & Pricer 1982; Fennema & Sherman, 1977; Moller,

1982/1983). Armstrong and Price investigated the relative

influence of selected factors in sex-related differences in

mathematics participation. Both males and females selected

usefulness of mathematics as the most important factor in

deciding whether or not to take more mathematics in high

school. Moller's study revealed that both males and

females based mathematics course-taking decisions on career

usefulness. A Fennema and Sherman (1977) study showed only

slight differences between males and females in their

feelings about the usefulness of mathematics. In her study

of this variable among college students, Probert

(1983/1984) did not find any sex-related differences

either.

These have been the main affective variables

researched in attempting to explain the underlying causes

of sex-related differences observed in mathematics learning

and achievement. In spite ot the great diversity of studies











dealing with both cognitive and affective variables, there

are no clear-cut findings to render unequivocal support to

a particular variable as accounting for these sex-related

differences. However, everything seems to point to the

fact that affective, rather than cognitive variables play a

more significant role in the sex-related differences

observed in mathematics performance and learning. In most

of the studies dealing with affective variables, findings

consistently show that these factors influence mathematics

performance in females more than in males. In at least one

area confidence as learners of mathematics, Sherman (1980)

found that this variable influenced course election more

than all the cognitive variables previously discussed.

The case for the societal influences on sex roles and

expectations to account for the differences in mathematics

learning is also supported in one way or another in the

studies reported in the literature.



Problem Solving Performance and Related Variables

Problem solving has been perhaps the most extensively

researched area in mathematics education. Published

reviews by Kilpatrick (1969) Riedesel (1969), and Suydam

and Weaver (1970-1975) attest to this. Much of the

research done has focused on identifying the

determinants of problem difficulty and the problem features

that influence the solution process.












At presents no set of variables has been clearly

established as a determinant of problem difficulty.

Several researchers have investigated the effect or reading

and computation on problem solving performance. Others

have studied the effect of student attitudes toward problem

solving in problem solving learning and achievement.

Typically, correlational methods have been used to

investigate these questions.

Computational Skills and Problem Solving Performance

One of the first researchers to study the effect of

computation and reading on problem solving performance was

Hansen (1944). He investigated the relationship of

arithmetical factors, mental factors, and reading factors

to achievement in problem solving. Sixth grade students

were administered tests in problem solving and categorized

as superior achievers (best problem solvers) and inferior

archievers (poorest problem solvers). The two groups were

compared in selected factors believed to be related to

success in arithmetic problem solving: arithmetical, mental

and reading factors. After controlling for mental and

chronological age, the superior achievers in problem

solving surpassed the inferior achievers in mental and

arithmetical factors. The superior group did better in

only two of the six items under the reading factors:

general language ability and the reading of graphs, charts,

and tables.











The findings suggest that reading factors are not as

important as arithmetic and mental factors in problem

solving performance. However, these findings should be

taken cautiously, as the content of the Gates tests (used

to measure reading) is literary and does not include

mathematical material.

Chase (1960) studied 15 variables in an effort to find

out which ones have significant influence on the ability to

solve verbal mathematics problems. Only computation,

reading to note details, and fundamental knowledge were

primarily related to problem solving. Computation

accounted for 20.4% of the 32% variance directly associated

with problem solving.

Chase concluded that a pupil's ability in the

mechanics of computation, comprehension of the principles

that underline the number systems, and the extent to which

important items of information are noticed when reading,

are good predictors of the student's ability in solving

verbal problems.

Balow (1964) investigated the importance of reading

ability and computation ability in problem solving

performance. He objected to the approaches used by other

researchers who in their analyses dichotomized research

subjects as "poor" or "good" students, and who ignored the

recognize effect of intelligence on reading and on

mathematics achievement. Balow administered the Stanford











Achievement Test (subtests of reading, arithmetic, and

reasoning) and the California Short-Form test of mental

ability to a group of 1,400 children from the 6th grade.

All levels of achievement were included in the analysis.

Analysis of variance and covariance were used and compared.

He confirmed the findings of other researchers to the

effect that there is a direct relationship between I.Q. and

reading ability and between I.Q. and computational skills.

The results of the analysis of variance revealed that

increases in computation ability were associated with

higher achievement in problem solving. A relationship

between reading ability and problem solving was also found,

but it was not as strong. Significant differences in

problem solving performance associated with computational

ability were found when intelligence was controlled.

Balow concluded that computation is a much more

important factor in problem solving than reading ability,

and that when I.Q. is taken into consideration, the degree

of the relationship between reading and problem solving

ability becomes less pronounced. Intelligence tends to

confound the relationship between these two variables.

Knifong and Holtan (1976, 1977) attempted to

investigate the types of difficulties children have in

solving word problems. They administered the word problem

section of the Metropolitan Achievement Test to 35 children

from the 6th grade. Errors were classified in two











categories. Category I included clerical and computational

errors. Category II included other types of errors, such

as average and area errors, use of wrong operation, no

response, and erred responses offering no clues. It the

student's work indicated the correct procedure and yet the

problem was missed because of a computational or clerical

errors it was assumed that the problem was read and

understood.

An analysis of frequencies showed that clerical errors

were responsible for 3% of the problems incorrectly solved,

computational errors accounted for 49%, and other errors

for 48% of the erred problems.

Knifong and Holtan concluded that "improved

computational skills could have eliminated nearly half or

the word problem errors" (p. 111). These computational

errors were made in a context where other skills such as

reading, interpretation of the problem, and integration of

these skills necessary for the solution of word problems,

might interact. However, Knifong and Holtan state that

their findings neither confirm nor deny that improvement or

reading skills will lead to improvement in problem solving.

They conclude that "it is difficult to attribute major

importance to reading as a source of failure" (p. ll).

In a later analysis, looking for evidence of poor

reading abilities affecting children's success in word

problems, Knifong and Holtan (1977) interviewed the











children whose errors fell under the category or "other

errors." Students were asked to read each problem aloud

and answer these questions: What kind of situation does

the problem describe? What does the problem ask you to

find? How would you work the problem?

Ninety five percent of the students read the problem

correctly; 98% explained the kind of situation the problem

described in a correct manner; 92% correctly answered what

the problem was asking them to find, and 36% correctly

answered the question of how to work the problem.

The fact that a large percent of the students whose

errors were classified as "other errors" (in which reading

skills might have been a factor) correctly stated how to

work the problem, is strong evidence of their ability to

read and interpret the problems correctly. The errors made

by this group of students had a distinct origin, unrelated

to reading ability.

Zalewski (1974) investigated the relative contribution

of verbal intelligence, reading comprehension, vocabulary,

interpretation of graphs and tables, mathematical concepts,

number sentence selection, and computation to successful

mathematical word problem solution, and the relationship of

the dependent variable to the eight independent variables.

She worked with a group of 4th grade children who

were administered the subtests of the Iowa Test of Basic

Skills (ITBS) and the Weschler Intelligence Scale for











Children (WISC). Multiple regression analysis was

performed. A correlation of .769 was found between word

problem solving and the eight independent variables.

Correlations between word problem solving and the

independent variables ranged from .363 (verbal

intelligence) to .674 (mathematical concepts).

Correlations between the independent variables ranged from

.369 (verbal intelligence and computation) to .749 (reading

comprehension and vocabulary). Mathematical concepts,

computation, and number sentence selection were almost as

effective as all eight independent variables in predicting

achievement in mathematical word problem solving.

Mathematical concepts, computations number sentence

selection, and reading comprehension accounted for 58% of

the variance, whereas all eight predictors accounted for

59% of the variance. The two best predictors were

mathematical concepts and computations which accounted for

54% variance. Other variables accounted for about 40% of

the variance.

The author recommends that the findings of this study

be interpreted cautiously because the correlation between

the eight independent variables was high, and, according to

Zalewski,


in a study of this nature where the interest is
primarily in the influence of several variables
on one dependent variable, a low correlation
between the independent variables is required.
(p. 2804)











In a more recent investigations Exedisis (1983)

studied the contribution of reading ability, vocabulary,

mathematical concepts, computation, sex, and race on

problem-solving performance. The Iowa Test of Basic

Skills was administered to a group of 6th, 7th, and 8th

grade anglo and black Chicago male and female adolescents.

Problem solving was highly correlated to an understanding

of basic mathematical concepts, somewhat correlated to

race, and weakly correlated to computational and

vocabulary skills, sex, and reading ability.

Although the findings of these studies show a

relationship between computational skills and problem

solving achievement, this relationship is not strong enough

to be considered the most determinant factor in problem

solving achievement, as some of the researchers have been

careful to point out. In spite ot the dismissal of reading

as a determinant factor in problem solving achievement by

some of these same researchers, more recent studies in this

area have led others to hold different views.

Reading and Problem Solving Performance

Martin (1964) studied the contribution of reading

comprehension, computation, abstract verbal reasoning, and

arithmetic concepts to arithmetic problem solving

performance. Fourth and 8th grade students were

administered the Iowa tests of Basic Skills and the

Lorge-Thorndlike intelligence test (verbal).











He found that in the 4th grade the correlations

between problem solving and abstract verbal reasoning,

reading comprehension, arithmetic concepts, and computation

were .61, .64, .66, and .60 respectively, and .56, .68,

.69, and .63 in the 8th grade. When computation was held

constant, the correlation between problem solving and

reading was .52 in grade 4 and .54 in grade 8. When

reading was held constant the correlation between problem

solving and computation was .43 in grade 4 and .42 in

grade 8.

Creswell (1982) worked with a sample of anglo and

black adolescents from Chicago. Each subject was

administered the California Achievement test. Multiple

regression was used to analyze the data. The analysis

showed that reading is more important than computation in

predicting student performance in problem solving. Reading

accounted for 49.5% of the variance; computation accounted

for 14.6% of the variance.

Ballew and Cunningham (1982) worked with 6th grade

students in an attempt to find what proportion of students

have as their main source of difficulty with word problems

each of the following factors: a) computation skills,

b) interpretation of the problem, c) reading and,

d) integrating these skills in the solution of problems.

They also wanted to know if a student can be efficiently

diagnosed as having one of the four categories as his/her

main difficulty with mathematics word problems.












Their study is important because it represents an

attempt to demonstrate that multiple factors can interact

in the correct solution of a mathematics word problem.

They constructed three graded tests from a basal

mathematics series for grades 3 through 8. For test 1i the

problems were set in pure computational form (the effects

or reading, interpretation, as well as the necessity for

integration were removed in an effort to measure the

computational skills required by the word problems).

For test 2, the effects of reading and computation

were removed by reading the problems to the students and by

giving scores based on whether or not the students set them

up properly, in an attempt to measure problem

interpretation alone. For test 3, the effect of

computation was removed. The test yielded two scores--one

by grading the students on whether or not they set up the

problems properly and another by grading on the basis of

the correct answer.

The tests were administered to all 244 students from

the 6th grade in two different schools. A diagnostic

profile was obtained for each of the 217 students for which

complete data were available: a computational score, a

problem-interpretation score, a reading score, and a

reading-problem solving score.

They assumed that if the reading-problem

interpretation score was lower (one or more levels lower)











than the problem-interpretation score, the difficulty was

due to reading ability. If the score of the lowest of the

three areas (computation, problem interpretation, and

reading-problem interpretation score) was the same as the

reading-problem solving score, the student's area of

greatest immediate need was either computation, problem

interpretation, or reading. If the reading-problem solving

score was lower than the lowest of the other three scores,

the student's area of greatest immediate need was

integration.

Analysis of the data revealed that for 19% of the

students, problem interpretation was their major

difficulty; for 26% of the students, integration (total

problem solving) was their greatest immediate need; for

another 26%, computation was the major weakness; and for

29%, reading was their greatest immediate need.

Seventy five percent of the students demonstrated

clear strength in computation, 21% in problem

interpretation, and 4% in reading-problem interpretation.

An analysis across all students (including those without

complete data) showed that 26% of the subjects could not

work word problems at a level as high as that at which they

could computer interpret problems, and read and interpret

problems, when those areas were measured separately. This

led them to conclude that knowing the skills or the

components of solving word problems is not sufficient for











success, since the components must be integrated into a

whole process (mastery learning of the components cannot

assure mastery of the process).

Their analysis also led them to conclude that, in the

case of 6th graders, inability to read problems is a major

obstacle in solving word problems. Only 12% ot the

subjects could read and set up problems correctly at a

higher level than they could computer while 60% could

compute correctly at a higher level than they could read

and set up problems; 44% could set up problems better when

they heard them read than when they read the problem

themselves. Only 13% could set up problems better when

they read them than when they heard them read.

Muth (1984) investigated the role of reading and

computational skills in the solution of word problems. A

group of 200 students from the 6th grade were administered

a test of basic skills and a mathematics word problem test.

The word problem test consisted of 15 sample items supplied

by the National Assessment of Educational Progress. The

items were adapted to include some extraneous information

and complex syntactic structure. Four versions of the test

were constructed by combining two versions of problem

information (absence vs. presence of extraneous

information) with two versions of syntactic structure

(simple vs. complex syntax). Task performance was measured

by means of the number of problems answered correctly,











number of problems set up correctly, and amount of time

spent taking the test.

Reading ability and computational ability were both

positively correlated with number of correct answers and

with number of problems correctly set up, and negatively

correlated with test-taking time. Presence of extraneous

information was negatively correlated with correct answers

and correct set ups and positively correlated with

test-taking time. Syntactic complexity was not

significantly correlated with any of the performance

measures.

Results of a multiple regression analysis showed that

reading accounted for 46% of the variance in total correct

answers and computation accounted for 8%. Reading ability

and computational ability uniquely accounted for 14% and 8%

of the variance in the number of correct answers,

respectively. Extraneous information added significantly

to the variance explained in the number of correct answers,

but syntactic structure did not. Reading ability accounted

for 5% of the variance in test-taking time, but computation

did not add significantly to the variance explained by

reading.

Muth concluded that reading and computation both

contribute significantly to success in solving arithmetic

word problems, but that reading plays a more significant

role than does computation.











The studies reviewed in this section show a positive

relationship between reading and problem solving

performance, but in the case of Ballew and Cunningham

(1982) this relationship is not viewed singly but rather

as one among the interacting factors that produce

successful problem solving.

The third variable reviewed is the effect of student

attitudes toward problem solving on problem solving

performance. Many researchers have tried to demonstrate

that this variable is a determinant factor in problem

solving achievement.


Attitudes Toward Problem Solving and Problem Solving
Performance

Research studies support the existence of positive and
rather stable relationships between student attitudes and

achievement in mathematics. Aiken (1970) has suggested

that an individual's attitude toward one aspect of the

discipline (mathematics), such as problem solving, may be

entirely different from his/her attitude toward another

phase of the discipline, such as computation.

Research, however, has been directed to the use of

single, global measures of attitudes toward mathematics

rather than to the investigation of attitudes toward a

particular phase of the discipline.

The studies described below are only part of the few

investigations which have examined the relationship between











student attitudes and performance in the area of problem

solving.

Carey (1958) constructed a scale to measure attitudes

toward problem solving. Her interest was in general

problem solving rather than in mathematical problem

solving. Her work constitutes the first attempt to

construct a measure of attitudes toward problem solving.

The scale was used with a group of college students, and

she found, among other things, that problem solving

performance is positively related to problem solving

attitudes and that, in the case of females, positive

modification of attitudes toward problem solving brings a

significant gain in problem solving performance.

Lindgren, Silvar Faraco, and DaRocha (1964) adapted

Carey's scale of attitudes toward problem solving and

applied it to a group of 4th grade Brazilian children.

Students also answered an arithmetic achievement test, a

general intelligence test, and a socioeconomic (SE) scale.

A low but significant positive correlation was found

between arithmetic achievement and attitudes toward problem

solving. A near zero correlation was found between

attitudes toward problem solving and intelligence. Since

problem solving is one aspect of the discipline of

mathematics, this correlation between attitudes and

arithmetic achievement can lead to a conclusion or a strong












correlation between attitudes toward problem solving and

problem solving performance.

Whitaker (1976) constructed a student attitude scale

to measure some aspects of 4th grade student attitudes

toward mathematic problem solving. He included statements

reflecting children's beliefs about the nature of various

types of mathematical problems, the nature of the problem

solving process, the desirability or persevering when

solving a problem, and the value of generating several

ideas for solving a problem.

He correlated student attitudes toward problem solving

with their scores in a mathematical test which yielded a

comprehension score, an application score, and a problem

solving score. He found a significant positive

relationship between problem solving performance and

student attitude scores on the subscale which measured

reactions to such things as problem solving techniques or

problem situations, or to the frustration or anxiety

experienced when confronted with problem solving

situations.

In another part of this study, Whitaker investigated

the relationship between the attitudes of 4th grade

teachers toward problem solving and their students'

performance in problem solving. A very weak and

nonsignificant negative correlation was found between the

teacher's attitudes toward problem solving and student

performance.











The studies reviewed have confirmed the relationship

between problem solving performance and attitudes toward

problem solving (Carey, 1958; Lindgren et al., 1964;

Whitaker, 1976). However, the results reported in the

studies that investigated the relationship between problem

solving performance and computation and between reading and

problem solving fail to be consistent in their conclusions.

Hansen (1944), Chase (1960) Balow (1964), Knifong and

Holtan (1976, 1977), and Zalewski (1974) concluded that

computation is more strongly related to problem solving

than is reading. Martin (1964), Creswell (1982), Ballew

and Cunningham (1982), and Muth (1984), concluded that

reading ability and mathematical problem solving show a

stronger relationship than computation and problem solving.

Exedisis's (1983) findings led to the conclusion that the

effect or reading and computation in problem solving

performance is unimportant.













CHAPTER III
METHOD



The first objective of this study was to investigate

sex differences in the selection of incorrect responses in

a mathematics multiple-choice achievement test, and to

determine whether these differences were consistent over

three consecutive administrations of the test. The second

objective was to investigate whether males and females

differ in problem solving performance, if these differences

persist after accounting for computational skills, and if

the male-female differences depend on the level of

computational skills. This chapter contains descriptions

of the sample, the test instrument, and the statistical

analysis used in achieving the above mentioned objectives.



The Sample
To achieve the first objective of the study, all the

students who took the Basic Skills Test in Mathematics-6

("Prueba de Destrezas Basicas en Matem&ticas-6") in

each or the three years were included in the study. To

achieve the second objective of the study, approximately











1,000 students were selected randomly for each year. For

the first year, 1,002 were selected (492 boys and

510 girls); for the second year, 1,013 students were

selected (504 boys and 509 girls); and, for the third years

1,013 students were selected (509 boys and 504 girls).

The student population in Puerto Rico includes

children from the urban and rural zones and comprises

children from low and middle socioeconomic levels.

Findings can be generalized only to this population.



The Instrument

The Basic Skills Test in Mathematics-6 is a

criterion-referenced test used in the Department ot

Education of Puerto Rico as part of the annual assessment

program. The test measures academic achievement in

operations, mathematical concepts, and story problems. It

has a reported split half reliability or .95. The test was

designed specifically for Puerto Rico. Its contents and

the procedures followed for its development were formulated

and reviewed by educators from the mathematics department

of the Department of Education of Puerto Rico, in

coordination with the Evaluation Center ot the Department

of Education and mathematics teachers from the school

districts. The emphasis placed on each skill area is

depicted in Figure 3.1.


























































Fig. 3.1 Number of Questions by Skill Area in the Basic
Skills Test in Mathematics 6













Analysis of the Data

Analysis of Sex by Option by Year Cross Classifications



Log-linear models were used to analyze the sex by

option by year cross classifications for each item. Two

topics are addressed in this section, the hypotheses tested

using log-linear models and a comparison of the hypotheses

tested in this study with those tested by Marshall (1981,

1983).

The object of the analysis was to test two hypotheses:

1. The proportion of males and the proportion of

females choosing each incorrect option does not vary from

year to year. Note that this hypothesis is stated in the

null form.

2. Assuming that the first hypothesis is correct, the

proportion of males who choose each incorrect option is

different from the proportion of females who choose each

incorrect option. This hypothesis is stated in the

alternate form.

In Table 3.1 a hypothetical cross classification of

sex, option, and year is presented. Hypothesis 1 is true

for this three dimensional contingency table. In Table 3.2

the three dimensional contingency table is rearranged to

show the year by option contingency table for each gender.












Table 3.1

Hypothetical Probabilities of Option Choice
Conditional on Year and Sex


Option
Year Sex 1 2 3


First M .7 .2 .1
F .5 .3 .2

Second M .7 .2 .1
F .5 .3 .2

Third M .7 .2 .1
F .5 .3 .2



Table 3.2

Hypothetical Probabilities of Option Choice
Conditional on Year and Sex and Arranged by Sex


Option
Sex Year 1 2 3


M First .7 .2 .1
Second .7 .2 .1
Third .7 .2 .1

F First .5 .3 .2
Second .5 .3 .2
Third .5 .3 .2











Inspection of the year by option contingency tables shows

that year and option are independent for each gender.

Thus, hypothesis 1 is equivalent to the hypothesis that

year and option are independent conditional upon gender.

Hypothesis 2 is also true for Table 3.1 and Table 3.2.

Therefore, when hypothesis 1 is correct, hypothesis 2

is equivalent to the hypothesis that sex and option choice

are dependent.

In his discussion of the analysis of three dimensional

contingency tables, Fienberg (1980) presents the following

saturated model for the data:



log mijk = Ui+Uj+Uk+Uij+Uik+Ujk+Uijk. (3.1)


In this model, mijk is the expected value of the

frequency in cell ijk of the three dimensional table. The

model states that all three classification factors for a

three dimensional contingency table are mutually dependent.

In the present research i is the year index, j is the

option index, and k is the sex index. Fienberg shows that

deleting the terms Uij and Uijk yields a model in which

year and option are independent conditional upon sex. This

model is


log mijk = Ui+Uj+Uk+Uik+Ujk.


(3.2)











Fienberg also shows that deleting the Ujk term from (3.2)

to obtain



log mijk= Ui+Uj+Uk+Uik. (3.3)



yields a model that specifies that option is independent of

sex.

Based on Fienberg's presentation, an appropriate

analysis for testing the hypotheses is

1. Conduct a likelihood ratio test of the adequacy of

model (3.2). If this test is nonsignificant, then the

data support the adequacy of the model, and the hypothesis

that conditional on gender, options and year are

independent. Because model (3.1) is a saturated model,

testing the adequacy of model (3.2) is the same as

comparing the adequacies of models (3.1) and (3.2).

2. Conduct a likelihood ratio test comparing .the

adequacies of models (3.2) and (3.3). If this test is

significant, then model (3.2) fits the data better than

model (3.3) and the data support the hypothesis that the

choice of incorrect option is dependent on sex.

To summarized if the first test is nonsignificant and

the second test is significant, then the choice of option

is dependent on sex, and the pattern of dependency is the

same for all three years.











A problem arises in interpreting this analysis with

the data used in this research. Over the three years,

there were responses available from 135,340 students. Even

if only 5% of the students answered an item incorrectly,

the responses of 7,767 students would be used in analyzing

this item. On the other hand, if 90% answered an item

incorrectly, the responses of 121,806 students would be

used in analyzing the data. As a result of the large

sample size the tests described above are likely to be

very powerful. In step 1 of the analysis, then, even a

very small change from year to year in the proportion of

males or females who choose an option is apt to be

detected, and the results will indicate that

hypothesis 1 is not supported. For step 2, even a very

small dependence of option choice on sex is likely to be

detected and hypothesis 2 is likely to be supported. In

brief, the problem caused by the large sample size is that

practically insignificant differences may yield statistical

significance.

Fortunately, the form of the test statistic used in

the likelihood ratio test suggests a reasonable solution to

this problem. The test statistic is


G2 = 2 ZFo loge (Fo/Fe).


(3.4)











Here the summation is over all the cells in the contingency

table Fo refers to the observed frequency in a cell, and

Fe refers to the estimated expected frequency in a cell.

Denoting the observed proportion in a cell by Po and the

estimated expected proportion in a cell by Per the test

statistic can be written


G2 = 2 N TPo loge (Po/Pe)- (3.5)



where N is the total number of subjects. This form of the

test statistic suggests the following strategy. For any

significant G2, using Po and Pe calculated from the

total data set available for an item, calculate the minimum

N required for G2 to be significant. If the minimum N is

very larger this suggests that the statistically

significant result is not practically significant since it

can only be detected in very large samples. Of course, the

question remains as to what can be considered a minimum

large N. Although there is room for argument, it seems

reasonable to claim that if an average of 1000 subjects per

year is required to show significance, then the result is

not likely to be practically significant. On the basis of

this reasoning, it was proposed to ignore all significant

results that would be nonsignificant if there were less

than 3000 subjects available. In addition, all log-linear











model tests were conducted using a .01 level of

significance.

Since this research is based on Marshall (1981, 1983),

it is important to compare the method of analysis used in

this study to the one used by Marshall. Marshall also used

a two-step analysis. In the first step of her analysis she

deleted the Uijk term from (3.1) and tested the adequacy

of the model,


log mij k = Ui+Uj+Uk+Uij+Uik+Ujk. (3.6)


Following this, she deleted the Ujk term to obtain



log mijk = Ui+Uj+Uk+Uij+Uik. (3.7)


and compared these two models using a likelihood ratio

test. If the first test was nonsignificant and the second

significant, Marshall claimed that option choice was

dependent on sex and the pattern of dependency was the same

from year to year.

This is the same claim that this study sought to

establish. However, the approach used here was to

present evidence that model (3.2) fits the data while

Marshall tried to show that model (3.6) fits the data.

The major difference between the two approaches

concerns the operationalization of the concept that the








64



gender-option dependency is the same from year to year. In

this study, a three dimensional table was considered to

exhibit the same year to year pattern of gender-option

dependency if, conditioned on gender, the same proportion

of students chose each incorrect response over the three

year period. This seems to be a straightforward and

natural way to operationalize the concept. To illustrate

how Marshall operationalized the concept in question, a

hypothetical set of probabilities was constructed

conforming to the pattern specified by Marshall. This is

displayed in Table 3.3.




Table 3.3

Hypothetical Joint Probabilities of Year, Option,
and Sex


Option

Year Sex 1 2 3



------------------------
First M .120 .045 .015
F .060 .060 .030
I------------ I
Second M .144 .022 .019
F .072 .030 .039
I----------------------~
Third M .132 .040 .016
F .066 .054 .039



Note: Probabilities reported are truncated to three
places.











One important characteristic of this table involves

two-by-two subtables of sex cross classified with two of

the options for each year. For example, consider the three

two-by-two tables obtained by cross classifying sex and

option choices one and two for each year. These tables are

indicated by dotted lines on Table 3.3. For this table,

the ratio of the odds of a male choosing option one to the

odds of a female choosing option one is the same (within

truncating error) for each year. For example, this odds

ratio for the first year is (.120/.045) / (.060/.060) =

2.67. Within the error caused by reporting truncated

probabilities the odds ratio for years two and three is

the same as that for year one. The equality of these odds

ratios is Marshall's operationalization stage of the year-

to-year gender-option dependency.

To show that the odds ratio can be constant over

years, but that the probabilities of option choice

conditional on sex and year can change from year to year,

for both males and females, the probabilities in Table 3.3

were converted to the probabilities of option choice

conditional upon sex and year. These conditional

probabilities are reported in Table 3.4. Unlike the

probabilities in Table 3.2, those in Table 3.4 change from

year to year for both males and females.









Table 3.4

Hypothetical Probabilities of Option Choice
Conditional on Sex and Year


Option
Year Sex 1 2 3

First M .666 .250 .083
F .400 .400 .200

Second M .774 .120 .104
F .510 .212 .276

Third M .698 .214 .087
F .431 .352 .215


Note: Probabilities reported are truncated to three
places.



Which procedure is more appropriate to test the claim

that the pattern of male and female option choice remains

the same from year to year? It seems more reasonable to

test for a pattern like that in Table 3.2 than to test for

a pattern like that in Table 3.4, and, consequently, this

was the strategy adopted in this study.

Comparison of Males and Females in Problem Solving
Performance

One object of the study was to compare the performance

of males and females on problem solving. Two questions

were addressed. First, do males and females differ in

problem solving performance? Second, do these differences

persist when computational skill is controlled for, and do

these differences depend on the level of computational

skill? Seven analyses of covariance (ANCOVA) tests were











conducted, one with each of addition, subtraction,

multiplication, division, addition of fractions, decimals

in subtraction, and equivalence as covariates. A well

known problem that arises in the use of ANCOVA is that an

unreliable covariate can cause spurious differences between

the sex groups. To solve this problem, Porter (1967/1968)

proposed the use of estimated true scores for observed

scores. Porter (1967/1968) conducted a simulation that

gave empirical support to the adequacy of this strategy.

Hunter and Cohen (1974) have provided theoretical support

for this strategy.












CHAPTER IV
RESULTS


Introduction

The data gathered from 6th grade students from the

public schools in Puerto Rico who took the Basic Skills

Test in Mathematics-6 during three consecutive years were

analyzed in this study. The first objective was to

investigate whether boys and girls differ in the

selection of incorrect responses, and if the pattern of

differences was consistent throughout the three years in

which the test was administered. The second objective was

to investigate whether males and females differ in problem

solving performance, if these differences persist after

accounting for computational skills, and if the male-female

differences depend on the level of computational skills.

The results are discussed in two sections. Study

findings in the area of sex differences in the selection of

incorrect responses are discussed in the first section.

The second section is devoted to an exposition of the

findings in the area of sex-related differences in problem

solving.

Sex-Related Differences in the Selection of
Incorrect Responses

As indicated in Chapter III, there are two models of

interest. The first model indicates that year and option











are independent, conditional upon sex,



log mijk = Ui+Uj+Uk+Uik+Ujk. (3.2)



Substantively, this model implies that the pattern of male

and female option choices is consistent over the three

years of test administration.

The second model indicates that option is independent

of sex,



log mijk = Ui+Uj+Uk+Uik- (3.3)


This model implies that the pattern of option choice is the

same for males and females.

In order to determine if males and females differed in

the selection of incorrect responses and if these

differences were stable across the three years of test

administration, a two-step test was performed. First, a

likelihood ratio test of the adequacy of model (3.2) was

conducted for each of the 111 items of the test. Under

this test a model fits the data when the X2 value

obtained is nonsignificant at a specified alpha level. For

50 (45%) of the items, model (3.2) fitted the data

adequately. For these items the pattern of male and female

option choices was consistent for the three years. The X2

values for the 61 items for which the likelihood ratio











tests were significant at the .01 level are reported in

Table 4.1. Also reported in this table are the actual

sample sizes and the minimum sample sizes necessary for the

likelihood ratio tests to be significant. Of the 61 items,

59 had minimum sample sizes greater than 3,000. Thus,

although for the three years both males and females samples

had inconsistent option choices on these 61 items, on 59 of

the items the inconsistency was relatively minor.

Consequently, these 59 items were included in step 2 along

with the initial 50 items.

In step 2, a likelihood ratio test was performed

comparing the adequacy of the models in (3.2) and (3.3).

The X2 values associated with models (3.2) and (3.3) for

each of the 109 items subjected to step 2 are reported in

columns b and c of Table 4.2. Also reported in Table 4.2

is the difference between the two X2 values (see

column d). This latter figure is the test statistic for

comparing the adequacies of models (3.2) and (3.3).

Significant X2 statistics are indicated by asterisks.

The X2 statistics were significant for 100 of the items

indicating a male-female difference in option choice for

these items. In column e of Table 4.2 the actual sample

sizes are reported. In column f, the minimum sample sizes

necessary for significance are reported. For those 100

items which had significant X2 statistics reported in

column d, 94 (94%) had minimum sample sizes greater than

3,000.











TABLE 4.1

Summary of Significant Tests of Model 3.2


Item Xe for Actual Sample Min. Sample
Number Model 3.2* Size Size for
Significance

(a) (b) (c) (d)


8

13

23

25

28

29

30

31

34

36

37

42

45

48

49

52

54

56

57


46.15

25.74

29.25

35.90

46.00

61.48

25.19

23.26

58.25

84.36

23.70

23.26

22.41

179.25

27.82

26.01

41.41

21.62

28.51


16,796

15,718

60,377

26,707

84,847

88,396

87,473

32,870

82,257

86,137

37,068

59,140

63,098

97,593

16,766

39,858

74,502

44,107

53,075


3,352

5,624

19,011

6,852

16,988

13,242

31,982

13,015

13,006

9,404

14,405

23,417

25,932

5,014

5,550

14,114

16,570

18,789

17,146












TABLE 4.1 Continued


Item Xe for Actual Sample
Number Model 3.2* Size


(a) (hi (


50.96

66.43

21.39

55.03

158.36

88.01

56.28

50.51

34.28

48.01

75.29

44.39

14,425.97

24.61

22.75

99.68

79.38

64.56

23.98


---


Min. Sample
Size for
Significance

(d)

7,560

8,387

15,342

8,485

3,028

7,660

6,943

14,352

13,935

12,128

7,647

14,011

54**

19,969

34,077

6,485

7,693

10,957

18,700


41,829

60,494

35,632

50,699

52,068

73,196

42,429

78,708

51,862

63,223

62,511

67,528

84,368

53,358

84,175

70,183

66,304

76,806

48,690











TABLE 4.1 Continued


Item XZ for Actual Sample Min. Sample
Number Model 3.2* Size Size for
Significance
(a) (b) (c) (d)

80 75.01 55,277 6,787

81 95.51 57,710 5,565

82 51.99 71,315 12,633

83 20.12 81,021 37,088

84 38.62 78,959 18,830

85 1240.78 41,180 306**

86 102.85 39,903 3,573

87 66.75 48,194 6,650

88 63.01 38,742 5,663

89 43.26 39,648 8,441

90 85.53 53,557 5,767

91 82.53 62,954 7,025

93 78.39 80,723 9,484

94 70.15 87,241 11,454

95 40.59 83,969 19,053

96 29.31 74,287 23,343

98 29.87 69,052 21,291

101 23.77 83,782 32,462

102 61.06 75,708 11,419

104 40.47 72,517 16,503












TABLE 4.1 Continued


Item XZ for Actual Sample Min. Sample
Number Model 3.2* Size Size for
Significance
(a) (b) (c) (d)

105 28.66 60,712 19,510

108 26.56 86,261 29,912

111 30.71 86,190 25,849


* p<.Ol

** Minimum sample size less than 3,000











TABLE 4.2


Chi-square Values for the Comparison of Models (3.2) and
(3.3) with Actual Sample Sizes and with the Corresponding
Number of Subjects Needed for Significant Results


Item Xk for X2 for X1 for Actual
Num- Model Model Difference Sample
ber 3.2 3.3 Size


lal

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17


(b)

16.11

17.54

20.06

7.54

14.75

11.86

4.60

46.15

10.13

9.54

5.89

12.58

25.74

11.19

16.63

14.51

17.90


(c)

30.94

103.22

47.51

21.87

70.40

23.13

35.73

329.98

109.99

65.29

19.06

198.46

62.37

80.50

17.33

84.29

44.20


(_d)

14.83*

85.68*

27.45*

14.33*

55.65*

11.27*

31.13*

283.83*

99.86*

55.75*

13.17*

185.88*

36.63*

69.31*

0.70

69.78*

26.30*


(e)

7,427

11,018

11,497

6,016

7,706

13,046

8,832

16,796

23,075

35,060

24,908

40,423

15,718

30,238

42,889

35,114

43,962


Minimum
Sample
Size for
Signif.

(f)

10,061

2,583**

8,733

8,414

2,782**

23,256

5,700

1,189**

4,642

12,634

37,996

4,369

8,621

8,765

1,230,914

10,109

33,582











TABLE 4.2. Continued


Item Xe for X for X2 for Actual Minimum
Num- Model Model Difference Sample Sample
ber 3.2 3.3 Size Size for
Signif.
(a) (b) (c) (d) (e) (f)

18 11.71 15.68 3.97 52,340 264,864

19 9.37 18.77 9.40* 41,363 88,402

20 12.37 13.46 1.09 55,707 1,026,746

21 11.51 19.03 7.52 49,240 131,547

22 9.63 83.74 74.11* 51,724 14,022

23 29.25 153.74 124.49* 60,377 9,744

24 10.35 42.93 32.58* 68,426 42,194

25 35.90 147.40 111.50* 26,707 4,812

26 18.84 47.30 28.46* 56,717 40,037

27 8.78 222.87 214.09* 78,353 7,353

28 46.00 242.78 196.78* 84,847 8,662

29 61.48 170.22 108.74* 88,396 16,331

30 25.19 53.84 28.65* 87,473 61,338

31 23.26 47.42 24.16* 32,870 27,333

32 15.51 41.69 26.18* 48,777 37,430

33 7.57 48.84 41.27* 39,470 19,214

34 58.25 324.67 266.42* 82,257 6,203

35 6.91 225.66 218.75* 88,069 8,088












TABLE 4.2 Continued


Item X2 for a for Xk for Actual Minimum
Num- Model Model Difference Sample Sample
ber 3.2 3.3 Size Size for
Signif.
(a) ( (c) (d) (e) (f)


36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54


84.36

23.70

13.64

13.35

6.67

12.65

23.26

16.29

3.55

22.41

6.56

7.82

179.25

27.82

14.23

16.37

26.01

6.69

41.41


134.56

76.85

84.24

757.60

701.39

603.42

189.11

21.65

115.84

55.90

16.30

76.15

452.64

59.75

240.27

224.54

96.14

27.49

95.41


50.20*

53.15*

70.60*

744.25*

694.72*

590.77*

165.85*

5.36

112.29*

33.49*

9.74*

68.33*

273.39*

31.93*

226.04*

208.17*

70.13*

20.80*

54.00*


mE


86,137

37,068

38,870

61,057

56,438

57,145

59,140

33,712

67,192

13,098

56,727

56,555

97,593

16,766

25,410

24,715

39,858

44,694

74,502


34,472

14,011

11,0618

1,648**

1,630**

1,943**

7,154

126,357

12,021

37,851

117,007

16,628

7,172

10,549

2,258**

2,385**

11,418

43,168

27,718











TABLE 4.2 Continued


Item Xz for Xa for Xl for Actual Minimum
Num- Model Model Difference Sample Sample
ber 3.2 3.3 Size Size for
Signif.
(a) (b) (c) (d) (e) (f)


55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

72

73


90.87

70.14

35.78

51.69

73.68

148.39

394.54

300.95

278.34

246.48

94.11

113.88

180.60

225.46

251.66

271.35

208.90

37.80


163.01 140.26*


71.34*

48.52*

7.27

34.20*

22.72*

81.96*

373.15*

245.92*

119.98*

158.47*

37.83*

63.37*

146.32*

177.45*

176.37*

226.96*

191.49*

13.19*


19.53

21.62

28.51

17.49

50.96

66.43

21.39

55.03

158.36

88.01

56.28

50.51

34.28

48.01

75.29

44.39

17.41

24.61


11,770

18,263

146,668

25,873

36,987

14,828

1,918**

4,142

8,719

9,279

22,532

24,953

7,121

7,158

7,121

5,977

8,831

81,271


41,796

44,107

53,075

44,044

41,829

60,494

35,632

50,699

52,068

73,196

42,429

78,708

51,862

63,223

62,511

67,528

84,175

53,358


74 22.75


84,175 12,057











TABLE 4.2 Continued


Item X1 for X1 for Xa for Actual
Num- Model Model Difference Sample
ber 3.2 3.3 Size


(aL

75

76

77

78

79

80

81

82

83

84

86

87

88

89

90

91

92

93

94


(b)

8.69

99.68

79.38

64.56

23.98

75.01

95.51

51.99

20.12

38.62

102.85

66.75

63.01

43.26

85.53

82.53

16.36

78.39

70-15


(c)

237.07

167.72

432.36

127.04

72.59

128.13

106.82

108.12

95.18

69.05

433.47

430.40

126.27

289.14

405.04

85.48

93.12

229.02

137.27


(d)

228.38*

68.04*

352.98*

62.48*

48.61*

53.12*

11.31*

56.13*

75.06*

30.43*

330.62*

363.65*

63.26*

245.88*

319.51*

2.95

76.76*

150.63*

67.12*


(e)

67,783

70,183

66,304

76,806

48,690

55,277

57,710

71,315

81,021

78,959

39,903

48,194

38,742

39,648

53,557

62,954

88,231

80,723

87,241


Minimum
Sample
Size for
Signif.

(f)

5,963

20,723

3,774

24,696

20,123

20,906

102,511

25,525

21,685

52,129

2,425**

2,662**

12,304

3,240

3,368

428,727

23,092

10,766

26,113











TABLE 4.2 Continued


Item X1 for Xa for X for Actual Minimum
Num- Model Model Difference Sample Sample
ber 3.2 3.3 Size Size for
Signif.
(a) (b) (c) (d) (e) (f)


68.41

212.34

126.96

41.77

9.94

41.48

315.62

437.09

7.97

283.94

48.59

1,682.71

99.20

79.33

18.33

49.11

477.44


27.82*

183.03*

118.36*

11.90*

0.50

30.13*

291.85*

376.03*

1.85

243.47*

19.93*

1,669.34*

86.07*

52.77*

12.60*

31.31*

446.73*


83,969

74,287

50,028

69,052

72,647

60,931

83,782

75,708

50,663

72,517

60,712

71,946

72,482

86,261

76,800

70,980

86,190


60,638

8,154

8,492

116,576

2,918,956

40,627

5,767

4,045

550,173

5,984

61,199

866**

16,918

32,840

112,453

45,544

3,876


* P<.01


** Minimum sample size less than 3,000


95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111


40.59

29.31

8.60

29.87

9.44

11.35

23.77

61.06

6.12

40.47

28.66

13.47

13.13

26.56

5.73

17.80

30.71










Sex-related Differences in Problem
Solving Performance


In this part, results for the two questions related to

the second objective of the study are presented. These

questions serve as the framework for the presentation.

Each question is stated, followed by the results pertaining

to that question.



Question 1: Do males and females differ in problem solving

performance?

The responses of 492 males and 510 females who took

the Puerto Rico Basic Skills Test-6 during the spring of

the first year were analyzed in this study. Also, data

from 504 males and 509 females tested in the second year

and from 509 males and 504 females tested in the third year

were included in the analysis.

The mean performance scores and the standard

deviations for each of the eight variables are presented in

Table 4.3. Results of t-tests are also presented in this

table. Females outperformed males in problem solving, a

finding consistently present in all the three years of test

administration. Over the three-year period the mean

differences favored females in all variables except

equivalence. The sex-related differences in problem

solving were significant (p<.01) for all three years.













TABLE 4.3

Means, Standard Deviations, and t-Tests for the Eight
Mathematical Subtests



Year/ Males Females
Subtest X SD X SD t


First

Problem Solving

Addition

Subtraction

Multi pl i ca ti on

Division

Fracadd

Decsub

Equivalence




Second

Problem Solving

Addition

Subtraction

Multiplication

Division

Fracadd


3.436 2.32

5.412 1.00

4.550 1.66

4.014 1.80

3.136 1.87

2.475 1.75

2.648 1.83

.77 .77

N = 492


3.632

5.470

4.843

4.148

3.242

2.565


2.28

.98

1.52

1.73

1.77

1.74


3.862

5.592

4.852

4.433

3.580

2.743

3.321

.73


2.45

.86

1.56

1.68

1.91

1.81

1.86

.76


2.82*

3.05*

1.22

3.81*

3.72*

2.32**

5.77*

- .83


N = 510


4.015

5.603

4.923

4.550

3.632

2.903


2.36

.86

1.44

1.67

1.86

1.91


2.65*

2.29**

.86

3.76*

3.42*

2.94*













TABLE 4.3 Continued


Year/ Males Females
Subtest X SD X SD t



Decsub 2.863 1.87 3.440 1.80 5.00*

Equivalence .76 .76 .65 .75 -2.32**

N = 504 N = 509
Third

Problem Solving 3.927 2.49 4.341 2.44 2.67*

Addition 5.536 .84 5.704 .64 3.59*

Subtraction 4.836 1.52 4.958 1.46 1.30

Multiplication 4.168 1.74 4.541 1.68 3.47*

Division 3.343 1.88 3.795 1.83 3.87*

Fracadd 2.819 1.89 3.117 1.85 2.53**

Decsub 3.021 1.93 3.448 1.85 3.60*

Equivalance .830 .82 .800 .78 .60

N = 509 N = 504


Note: The number of items in the problem solving subtest was 9.
In each computation subtest, the number of items
was 6. An item was included in the computation subtest
only if it measured a computation skill required to solve
a problem solving item.


* p <.01
** p <.05











Consistent significant differences were also found for

addition, multiplication, division, addition of fractions,

and subtraction of decimals. For subtraction the

difference was not statistically significant.



Question 2: Do sex-related differences in problem solving

persist when computational skills are

controlled for, and is the male-female

differences in problem solving dependent on

level of computational skills?

To address the question of dependence of male-female

problem solving differences in computational skills for

each year and computation subtest, the possibility of an

interaction was investigated. For the first year,

statistically significant interactions were found between

sex and multiplication, F (1,998) = 8.59, p<.01; and sex

and division, F (1,998) = 4.25, p<.01. A significant

interaction was found between sex and subtraction in the

second year, F (1,1009) = 6.39, p<.05. No significant

interactions were found in the third year. Analysis of

covariance summary tables are shown as Tables 4.4, 4.5, and

4.6. Also, the three interactions are depicted in

Figures 4.1, 4.2, and 4.3. Each figure indicates that at

lower levels of computational skills, males outperformed

females in problem solving, with the reverse happening at

higher levels of computational skills.











TABLE 4.4

ANCOVA Summary Table: Multiplication Covariate

First Year


Source df SS MS F




Multiplication (M) 1 1195.90 1195.90 265.66*

Sex (S) 1 31.95 31.95 7.07*

M x S 1 38.80 38.80 8.59*

Error 998 4509.00 4.51



p <.01


TABLE 4.5

ANCOVA Summary Table: Divison Covariate

First Year



Source df SS MS F



Division (D) 1 1195.90 1195.90 264.66*

Sex (S) 1 13.94 13.94 3.22

D x S 1 18.39 18.39 4.25*

Error 998 4317.00 4.32


* p<.01












TABLE 4.6

ANCOVA Summary Table: Subtraction Covariate

Second Year


Source df SS MS F



Subtraction (S) 1 93.14 593.14 122.80*

Sex (S) 1 18.58 18.58 3.85*

SxS 1 30.85 30.85 6.39*

Error 1009 4873.40 4.83


*p <. 5










87



9




8




7




6 .




0
I~ 5


o

,. 4





3




2





1






1 2 3 4 5 6

Multiplication

Fig. 4.1 Sex by Multiplication Interaction.










88




9




8.




7



6




5




4




3




2




1






1 2 3 4 5 6


Division
Fig. 4.2 Sex by Division Interaction










89





9




8.





7.




6





: 5
*r*
O
U)




















1
4
























1 2 3 4 5 6


Subtraction

Fig. 4.3 Sex by Subtraction Interaction











The significant interactions found between sex and

multiplications and sex and division (first year), and sex

and subtraction (second year), answered, in part, the

question of whether male-female differences in problem

solving performance depend on computational ability.

However, the evidence is quite weak. Of 21 possible

interactions, only three were significant. No variable

exhibited a significant interaction for each of the three

years.

The analysis of covariance was also used to determine

if sex-related differences exist after controlling for

computational skills. Analyses were conducted for

those variables that did not exhibit significant

interactions with sex. As discussed in Chapter III,

estimated true scores were used for observed scores to

adjust for unreliability of the covariates (the

computational subtests). Reliability coefficients

calculated for each covariate are shown in Table 4.7.

Summaries of the analyses of covariance for the first

year are reported in Table 4.8. The results show that

females retained their superiority in problem solving

performance when equivalence was the controlling variable

in the analysis of covariancer the only variable in which

males outperformed females (nonsignificant). When the

controlling variables were addition, subtraction, addition

of fractions, and subtraction with decimals, female




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