RELATIONSHIP BETWEEN DEGREE OF BILINGUALISM AND MATHEMATICAL WORD
PROBLEM SOLVING

John I. Jean, Ed.D., Ph.D.

Southern Connecticut State University

# ABSTRACT

The purpose of this study was to determine the extent to which degree of
bilingualism influences bilingual students\' reversal error generation
on compare word problems that have two different levels of consistency
in two languages of testing. It was hypothesized that bilingualism has
positive, cognitive consequences depending on the levels of linguistic
proficiency reached in both languages. Also, the consistency hypothesis
assumed that inconsistent language problems in which the unknown
variable is the object of the relational sentence are more error-prone
than consistent language problems in which the unknown variable is the
subject of the relational sentence. Seventy seventh graders and 55
eighth graders from two urban school districts were subjects in the
study. Two mathematics tests were used as outcome measures to assess
subjects\' mathematical problem solving performance in both Haitian and
English. The results of this study showed that both bilingualism and
language consistency significantly influenced student performance in
word mathematical problem solving.

## Introduction

Bilingualism, defined as possessing two languages, has always been a
controversial issue in society. During the early 1900s, bilingualism was
considered an unwelcome topic among American professionals and
politicians. Educators rendered bilingualism responsible for immigrant
children's failure in school subject matter. Employers believed that
immigrants, due to their low competence in English, did not fit the
requirements needed to become part of the United States workforce.
Psychologists regarded bilingualism as a handicap to cognitive
development; it was assumed that bilingualism was a barrier affecting
verbal intelligence (Darcy, 1963). Policy makers who accorded great
importance to psychologists' works declared that bilingual education
should be banned in schools and at the workplace since it did not
benefit learning growth and productivity in industry and business. Foes
of bilingualism and bilingual education elicited support for their
assumption; they wanted to demonstrate that bilingualism was a handicap
to the functioning of society and turned to social science research
(e.g., Saer, 1923). The result of such a research was accepted with
little attention given to how the results were attained.

Early bilingual researchers did not consider the psychometric standards
that are presently guiding the conduct of research (Diaz, 1985; Nanez,
Padilla, & Lopez-Maez, 1992). For example, researchers used tests that
were developed for the White American middle-class. Therefore, scores
that were obtained from these tests had to be interpreted according to
norms relevant to the White American population. The use of these tests
with the bilingual population would immediately minimize their validity
and reliability. The majority of bilinguals, living in the United States
at that time, were recently arrived from Mexico, Asia, and
non-English-speaking European countries; these immigrants were from
different cultures and social classes. The assessment of cognitive
abilities of immigrants with translated tests was viewed as violating
basic principles of measurement and research. Peal and Lambert (1962),
who conducted a study in which they controlled for age and
socio-economic factors, found evidence that bilingualism had positive
effects on cognition, contrary to the results of Saer's (1923) study.
Peal and Lambert (1962) found that the "balanced bilingual" enjoyed a
"mental flexibility, a superiority in concept formation, and a more
diversified set of mental abilities" (p. 3). Peal and Lambert, however,
failed to demonstrate in their study the cognitive consequences of
different levels of bilingualism.

Other scholars were concerned with the methodological flaw in Peal and
Lambert (1962) study. Diaz (1985) suggests that studies on bilinguals
should be conducted within class membership in order to freely observe
the effects of levels of bilingualism on cognition. At present, the
majority of bilingual studies follow Diaz's (1985) suggestions for
studying the effects of bilingualism in specific cognitive areas
including mathematics. Several studies (Clarkson & Galbraith, 1992;
Hernandez, 1983; Tuck, 1983) have found support for the positive
relationship between bilingualism and mathematics; however, the
literature on bilingualism includes few studies examining the effect of
degree of bilingualism in mathematical word problem solving and the ways
in which balanced bilinguals demonstrate their "mental flexibilities"
when faced with challenging mathematical word problems expressed in two
languages of testing (as referred to languages in which a test is
administered). Research by Lewis and Mayer (1987) and Stern (1993)
showed that compare word problems, one of the four classes of arithmetic
word problems, are very difficult for children (Riley & Greeno, 1985).
Compare word problems are characterized by a static relation between two
variables. They come in two special classes: consistent language
problems in which the unknown variable is the subject of the second
sentence, and inconsistent language problems in which the unknown
variable is the object of the second sentence. The solution of
inconsistent language problems requires the rearrangement of the
relational sentence to fit the format of the relational sentence of
consistent language problems. The straightforward prediction is that
reversal error will be likely to occur during mental transformation of
inconsistent language problems. There are no studies that look at levels
of bilinguals and compare word problem solving. This study is the first
to investigate the influence of degree of bilingualism in compare word
problem solving.

## Language and Bilingualism

Language plays a major role in thinking. It serves as a mediator for the
connection of thoughts and ideas. Vytgotsky (1962) asserts that inner
speech provides an individual with the opportunity to examine the
liaison that takes shape between identical elements that several
separate thoughts share together. The higher-order thought that results
from the merge of these several individual thoughts allows a thinker to
explore new areas of ideas (Anderson, 1995). The role of language is
more than a passive host for the enhancement of thinking. It plays an
active role in the production of metacognitive thinking. It creates a
state of alertness for the thinker to check his or her productive
thoughts. Thus, a lack of language proficiency can limit a thinker's
awareness of contemplating new ideas that can emanate from his previous
thoughts or monitor his thinking process (Baker, 1995; Bialystok, 1988).
Cummins (1987) also entertains the cognitive advantages of bi-lingual
proficiency. He asserts that bilingual individuals who reach a minimum
level of language proficiency in both first and second languages are
capable to demonstrate great thinking skills. Greater range of language
proficiency would depend on the extent to which a bilingual person can
transfer his thoughts from one language to the other. Coffeen (1982)
asserts a strong command in two languages would lead an individual to
more linguistic information, greater storage and retrieval of
information abilities, and the abilities to contrast linguistic systems
in developing conceptual thought processes. However, Cummins (1987)
recognizes the cognitive limitations of bilingualism when a bilingual
person does not develop yet the minimal level of language proficiency in
either his first language or the second language. In the same vein,
Cummins hypothesizes that individuals who develop high linguistic
proficiency in both first and second language (called balanced
bilinguals) are capable to enjoy the positive cognitive benefits of
bilingualism. Needless to say, bilingualism can have a positive
influence on problem solving since it is considered as the highest form
of cognition (Mayer, 1991). Several studies explored the relationship
between bilingualism and arithmetic problem solving (Mestre, 1986)

## Mathematical word problem solving

Mathematical word problems consist of story problems that incorporate
numbers. These story problems always end with question(s). The success
of a problem solver is related to his ability to accurately represent
the situation that is being told in the story and to select the most
appropriate mathematical operation to find the solution (Kintsch &
Greeno, 1985). The representation of a word problem is broken into two
parts (Mayer, 1991). First, the problem solver decodes the textual
components of the problem text into propositional representations. Next,
he integrates these representations with his existing knowledge domain
to create a broader representation of the problem. During this process,
the problem solver actively searches his previous knowledge to select a
model that can best integrate the linguistic composition of the problem
text and the contextual situation that emanates from the text. Thus, an
accurate translation of a problem text is as important as the problem
solver's previous knowledge in understanding the problem.

The degree of difficulty met during the comprehension process varies
with the type of word problems. The literature counts four problem
prototypes (Carpenter & Moser, 1982; Riley & Greeno, 1988; Riley et al.,
1983). They respond to change problems in which a set of objects is
transformed to another set, combine problems in which two sets of
objects merge together to become a larger set, and compare problems in
which two sets of objects are compared. Interestingly, compare word
problems are to determine one of the two sets of objects, given that one
set of objects and the difference set of objects are known.

In particular, compare word problems are built with a relational
statement that brings together the two sets to be compared (Lewis &
Mayer, 1987). It is this relational statement that is the locus of
attention in representing compare word problems. The relational
statement can be presented in two symmetric forms that would require the
same mathematical operation that leads to the solution of the problem.
In one form, the relational statement is consistent with the problem
solver's information processing, in which the unknown variable is the
subject of the statement (e.g., *John has five cookies. Janet has two
cookies less than John. How many cookies does Janet have*?). Thus,
problems of this form are called consistent language problems. In the
other form, the relational statement is inconsistent with the problem
solver's information processing, in which the unknown variable is the
object of the relational statement (e.g., *John has 5 cookies. He has
two cookies more than Janet. How many cookies does Janet have*?). Word
problems that pertain to this form are called inconsistent language
problems.

Lewis and Mayer (1987) hypothesize that problem solvers are more likely
to use a mathematical operation that is irrelevant to the situation
described in the problem when solving inconsistent language problems
than when solving consistent language problems. They coin a problem
solver's *mis*-representation of an inconsistent language problem as
reversal error. A problem solver will commit reversal error when she
failed to mentally rotate the relational statement of the inconsistent
language problem to fit the relational statement of the equivalent
consistent language problem. Thus, the problem solver tends to use a
mathematical operation that is relevant to the situation described in
the problem text.

## Test Language

In measurement and research, test language refers to the language in
which subjects are tested. A test taker is expected to respond to test
items in the same manner whether they are written in the first language
or second language. Psychometric qualities of a test can be negatively
affected if the test language does not convey clearly the objectives of
the test items. The negative consequences of test language are more felt
when a test is designed to assess students' achievement or intelligence
in a language in which they do not have a strong command (Gunnarsson,
1978). Such an inconvenience would increase measurement errors that
will, in turn, minimize the reliability and validity of the test.
Gunnarson (1978) states " Psychometric tests are clearly biased against
speakers of non-majority varieties of English" (p. 34). To control the
effect of test language on test takers' achievement, tests must be
administered and assessed in takers' first and second languages
(Pletcher et al., 1978; Simon et al., 1977).

## Purpose of the Study

The purpose of this study was to determine the extent to which degree of
bilingualism influences bilingual students' reversal error generation on
compare word problems having two different levels of consistency in two
languages of testing. More specifically, the study is designed to answer
these research questions:

> 1\. Does bilingualism influence reversal error generation on
> mathematical compare word problems?
>
> 2\. Does test language affect production of reversal errors on
> mathematical compare word problems?
>
> 3\. Does language consistency influence generation of reversal errors
> on mathematical compare word problems?
>
> 4\. Is the effect of bilingualism on reversal error commitment
> moderated by test language?
>
> 5\. Is the effect of bilingualism on reversal error commitment
> moderated by language consistency?
>
> 6\. Is the effect of language consistency on reversal error generation
> moderated by test language?
>
> 7\. Is the effect of bilingualism on reversal error commission
> moderated by the interaction of test language and language
> consistency?

# METHOD OF THE STUDY

## Participants

The study was conducted using 125 subjects chosen from two urban school
districts. These subjects were Haitian/English bilingual students living
in the urban school district. They were 70 seventh graders and 55 eighth
graders attending their last month of the school year. Four-fifths of
the students who participated in the study were born in Haiti. The
length of residence of the subjects in the United States of America
varied from 1 to 12 years. An examination of the data from the personal
data questionnaire revealed that English-dominant students have been
living in the United States for 3 to 9 years, balanced bilingual
students for 1 to 5 years, and Haitian-dominant students for 1 year to 4
years. The average age of the subjects was 13 years.

Since the sample was made up of two age groups, it is assumed that age
or mental development would be an extraneous factor that could affect
the relationship between bilingualism and compare word problem solving.
Therefore, the investigator examined the association between grade
(seventh and eighth grade) and error category. The results of the
investigation showed that developmental growth was not a major threat to
the results of the study since seventh and eighth graders tended to
achieve differentially in only two out of eight compare problem types.

## Preparation of Writing Topics and Questionnaire

The investigator prepared two topics from which the subjects were to
choose one on which to write in Haitian in one period and in English in
another period. Since the content of a topic was irrelevant, the
investigator selected the optional topics that could offer the subjects
the greatest latitude in expressing their ideas through writing skills
(abundant vocabulary, grammatical rules, organization and sequences of
ideas, illustration of ideas in appropriate examples). In the topic, the
students were asked to make use of the relational terms (more than, less
than, as many) that were relevant to the content of the word problems
they were going to solve. The contents of the topics "shopping" and
"household work" were based on the fact that most of the students had
experienced shopping and witnessed various activities their relatives
did at home. Students did not have to search for novel ideas that would
delay their writing ability.

The other screening measure was a questionnaire that the investigator
prepared for students to rate their own proficiency in both Haitian and
English. The four linguistic skills (listening, speaking, writing, and
reading) in both languages were the areas of self-evaluation. A
four-option scale was used, ranging from "Very Well," "Well," "Quite
Well," to "Not At All.

## Administration of Screening Measures

The writing activity embraced two important dimensions: timing and
language distribution. Students wrote on a topic in Haitian in one
period and in English in another period. The regular 60-minute period
was given for each time. Also, the investigator divided the subjects
into two groups of equal numbers. One group was requested to write their
topic in Haitian, the other group in English. At the end of the first
period, the investigator collected all students' papers. During the
second period, the groups switched writing language. The group who had
been writing in Haitian now wrote in English; the group who had been
writing in English now wrote in Haitian. If someone needed additional
time to complete a composition, the investigator accorded the student a
5- to 10-minute extension of time.

In the prelude to the essay writing, the investigator distributed to the
students the sheets containing the topic on which to write (the topic
was written in both Haitian and English) and two blank sheets of paper.
Instructions about the topic were read to the subjects. The classroom
teacher assisted the investigator in encouraging the whole class to
collaborate and participate in the writing activity which was an
important part of the research study.

The second section of the screening was the language questionnaire which
took place during another time. The questionnaire completion was done in
25 minutes. First, the investigator distributed the questionnaires to
the students to fill out. He then instructed them how they would proceed
in completing the questionnaire. Next, the investigator read the
questions in both Haitian and English, and the subjects checked the
answer box that was appropriate to their feelings. The subjects were
required to listen carefully to the instructor before they checked any
answer box. When the students answered all items in the questionnaire,
the investigator collected the filled-out questionnaires while the
students remained in their seats.

## Scoring of Screening Measures

To score students' writing tests, the investigator used the following
criterion writing skills: abundant vocabulary, grammatical rules,
organization and sequence of ideas, and illustration of ideas in
appropriate examples. Students who showed high performance on those
skills in both languages (Haitian and English) were tentatively
classified as balanced bilinguals. Students who showed high performance
on the criterion skills in only one language were tentatively
categorized as Haitian-dominant or English-dominant bilinguals,
depending on the language of dominance. Students who showed poor
performance in both languages were considered limited bilinguals.

To score the questionnaires, the investigator categorized students'
responses ("Very Well" and "Quite Well") as "favored" and ("With Many
Errors" and "Not At All") as "not-favored." Students who favored both
Haitian and English for speaking, listening, writing, and reading were
tentatively classified as balanced bilinguals. Students who favored
Haitian more than English for speaking, listening, writing, and reading
were tentatively classified as Haitian-dominant. Finally, students who
favored English more than Haitian for speaking, listening, writing, and
reading were tentatively classified as English-dominant.

The three categories of bilingual students resulting from the direct
assessment (writing proficiency test) were compared with the three
categories of bilingual students obtained from the indirect observation
(questionnaire) to evaluate the consistency of the two instruments. It
resulted that 92 students were classified as either balanced bilingual,
English-dominant, or Haitian-dominant students by both assessing
instruments. The investigator used the results of the writing test alone
to categorize the other nine students whose placements did not match by
both the writing test and the self-report questionnaire. Three of these
students were categorized as Haitian-dominant, and the other six
students were classified as English-dominant. A total of 36 students
were classified as balanced bilingual, 32 students as Haitian-dominant,
and 33 students as English-dominant. Finally, the remaining 24 students
who demonstrated poor performance in both languages on the writing tests
were not considered for further categorization. This group was rejected
from the study.

##  Problem Solving Tests

The investigator prepared the test items based on Lewis and Mayer's
(1987) Consistency Hypothesis. Each problem comprised two steps: a
compare problem as the first step and a direct variation problem as the
second step. The compare portions of the problems correspond,
respectively, to eight problem types: consistent addition (i.e.,
requires adding two numbers and the problem says "more"), inconsistent
addition (i.e., requires adding two numbers and the problem says
"less"), consistent subtraction (i.e., requires subtracting the second
number from the first and the problem says "less"), inconsistent
subtraction (i.e., requires subtracting the second number from the first
and the problem says "more"), consistent multiplication (i.e., requires
multiplication of two numbers and the problem says "n times as many"),
inconsistent multiplication (i.e., requires multiplication of two
numbers and the problem says "1/n times as many"), consistent division
(i.e., requires division of two numbers and the problem says "1/n times
as many"), and inconsistent division (i.e., requires division of two
numbers and the problem says "n times as many").

Each problem consisted of three sentences. The first sentence was an
assignment statement expressing the value of some variable (e.g., "At
Apple Tree butter sells 65 cents per stick"). The second sentence was a
relational statement, expressing the value of a second variable in
relation to the first variable (e.g., "Butter at C-Town sells 8 cents
more per stick than butter at Apple Tree"). The third sentence asked a
question about the value of some quantity in terms of the second
sentence (e.g., "If you want to buy 15 sticks of butter, how much will
you pay at C-Town?"). Answering the question always involved
multiplication of the value of the second variable by a quantity given
in the third sentence (Verschaffel et al., 1992).

The materials (either words or numbers) that comprised the tests were
chosen by the investigator to be relevant to the subjects' previous
knowledge, since the purpose of the study was not to evaluate students'
vocabulary knowledge and arithmetic operations. In consultation with an
expert in applied linguistics, the investigator examined the cultural
and grammatical relevance of the words that entered in the construction
of each problem so that they would not affect the subjects'
understanding process. The arithmetic operations that were used in the
tests (addition, subtraction, multiplication, and division) were
familiar to the population of seventh and eighth graders so that they
were not a major obstacle to the subjects' comprehension of the test
items.

## Test Administration 

The investigator distributed the mathematics tests in two groups. Each
group had eight mathematics problems (four English and four Haitian) in
which the language of testing was alternated (English, Haitian, English,
Haitian, etc.). In the first group, the mathematics test items were of
the following types: consistent addition, inconsistent addition,
consistent subtraction, and inconsistent subtraction. In the second
group, the mathematics test items were of the following types:
consistent multiplication, inconsistent multiplication, consistent
division, and inconsistent division. In each group, the items were
arranged in four different blocks so that each item would have the same
chance of being presented to the examinees. The four blocks (made of
ordered items) were:

\(a\) 1, 2, 3, 4, 5, 6, 7, 8;

\(b\) 8, 7, 6, 5, 4, 3, 2, 1;

\(c\) 4, 3, 2, 1, 8, 7, 6, 5;

\(d\) 5, 6, 7, 8, 1, 2, 3, 4.

Each block made a test booklet that contained nine regular sheets of
paper, with one arithmetic word problem typed on each sheet and a front
sheet that enlisted the test instructions to the examinee. Also, each
block of items within each separate group was equally distributed to the
examinees.

The administration of the mathematics tests was done at two different
times. During the first time, the examinees were presented with
consistent addition, inconsistent addition, consistent subtraction, and
inconsistent subtraction problems in both Haitian and English. During
the second time, examinees were given consistent multiplication,
inconsistent multiplication, consistent division, and inconsistent
division problems in both Haitian and English. The order of the testing
times was different for each participating school. The investigator read
the testing instruction to the examinees and informed them that they had
a 60-minute period to finish the test; nevertheless, students who needed
more time to complete the test were given extra time. The classroom
teacher assisted the researcher in encouraging the students to try each
problem, and especially to show all their work.

## Scoring of Problem Solving Tests

Each mathematics exam was scored "correct" if the proper numerical
answer was given; otherwise, the solution was scored as belonging to one
or more of the following error categories: "reversal error" is a
representation error in which the numerical operation for the first step
in the problem is reversed (i.e., addition for subtraction, subtraction
for addition, multiplication for division, and division for
multiplication); "arithmetic error" occurs when the computation in the
first or second step of the problem is carried out incorrectly, and
"goal monitoring error" occurs when the second step (i.e., direct
variation) of the problem is omitted. The investigator computed
frequencies and percentages of each category of error types and examined
the occurrence of reversal errors in order to respond to the research
questions.

## Further Scoring of Problem Solving Tests

Four scores were computed for each subject based on the two languages of
the test (English and English) of the consistent and inconsistent
problem types. In other words, subjects were given an English consistent
score (Y1), Haitian consistent score (Y2), English inconsistent score
(Y3), and Haitian inconsistent score (Y4). Since there were four
consistent and inconsistent problems presented in each of the two
languages, a subject could be scored 0, 1, 2, 3, or 4 if he/she solved
none of the four problem types incorrect, one of them incorrect, two of
them incorrect, three of them incorrect, or all four incorrect,
respectively, within each consistency level and each language level.

## Design of the Study

The design of the study was a repeated measures design consisting of one
between-subjects factor and two within-subjects factors. The
between-subjects factor was bilingualism with three levels (Haitian
dominance, balanced bilingualism, and English dominance). The
within-subjects factors were language consistency with two levels
(consistent language and inconsistent language) and test language with
two levels (English and Haitian). The dependent variable of the study
was reversal errors. Hence, the design was a 3×2×2 repeated measures of
analysis of variance.

# RESULTS AND DISCUSSION

The present study sought to examine the relationship between reversal
error generation in mathematical compare word problem solving and degree
of bilingualism, language consistency, and test language. The results of
the study were consistent with Cummins' (1976) Threshold Theory and
Lewis and Mayer's (1987) Consistency Hypothesis. The data showed that
dominant bilingual students (mean = 2.20), more than balanced bilingual
students (mean = 1.13), generated a greater proportion of reversal
errors in solving compare word problems, \[F(1, 98) = 52.03, p \< .05\].
Similarly, the data showed a significant main effect between
English-dominant and Haitian-dominant students on compare word problems,
\[F(1, 98) = 8.28, p \< .05\]. The results indicated that
English-dominant students (mean = 1.96) were less likely than
Haitian-dominant students (mean = 2.46) to commit reversal errors on
compare word problems.

The error generation on compare word problems demonstrated that the
sample of balanced bilingual students and that of the dominant bilingual
students used in the study represented two different subpopulations
(having different cognitive and linguistic abilities) of the bilingual
population. This finding was in line with previous studies (Cathcart,
1980, 1982; Clarkson & Galbraith, 1992; Dawe, 1983; Hernandez, 1983;
Tuck, 1983) seeking to show evidence to the threshold hypothesis (to
demonstrate the cognitive superiority of balanced bilinguals, as
compared to non-balanced bilinguals, in mathematics). This finding gave
supported to Arnberg's (1981) argument that cognitive development is
additive, and that the further the child moves towards balanced
bilingualism, the more he accumulates cognitive advantages. Although one
would intuitively believe that balanced bilinguals are cognitively more
flexible than dominant bilinguals, the finding may be surprising given
the fact that the United States Policy has in general supported programs
whose goal is transitional rather than full bilingualism (Hakuta, 1987).

The data revealed that bilingual students tended to commit more reversal
errors when solving inconsistent language problems (mean = 2.71) than
when solving consistent language problems (mean = .98), \[F(1, 98) =
244.27, p \< .05\]. In other words, bilingual students were more prone
to commit reversal errors (e.g., using an arithmetic operation that was
primed by the relational term in the problem) when solving inconsistent
language problems than when solving consistent language problems that
already fit their preferred format (Lewis & Mayer, 1987). This finding
corroborated with previous studies in compare word problem solving,
showing that students committed more reversal errors on inconsistent
language problems than on consistent language problems (Cummins et al.,
1988; Hegarty et al., 1992; Lewis & Mayer, 1987; Morales et al., 1985;
Riley & Greeno, 1988; Riley et al., 1983; Stern, 1993).

The examination of the significant two-way interaction between language
consistency and bilingualism illuminates the impact of the independent
variables (bilingualism and language consistency) on reversal error
generation. In the study, the trend of reversal error generation across
the three levels of bilingualism was not linear across the two levels of
language consistency. The study showed that the direction of the error
difference between balanced and dominant bilingual students remained the
same across both consistent and inconsistent language problems. In other
words, balanced bilingual students committed less reversal errors than
dominant bilingual students on both consistent and inconsistent language
problems, F(1, 98) = 25.60, p \< .05; however, the data showed that the
direction of error difference between Haitian-dominant and
English-dominant students changed across consistent and inconsistent
language problems. In other words, Haitian-dominant students tended to
generate a higher rate of reversal errors on consistent language
problems (mean = 1.89) than English-dominant students (mean = .68),
while the latter tended to commit a higher rate of reversal errors on
inconsistent problems (mean = 3.03) than did Haitian-dominant students
(mean = 3.21). This finding further provided evidence that balanced
bilinguals' superior problem solving capability over dominant bilinguals
was not influenced by the degree of difficulty of the mathematical
problems at hand. However, the superior performance of Haitian-dominant
students over English-dominant students was influenced by the degree of
difficulty of compare word problems. While English-dominant students
showed greater problem-solving ability on consistent language problems,
as compared to Haitian-dominant students, the latter were more
proficient than English-dominant students on inconsistent problems that
were more challenging than consistent problems. The unequal performance
of the two dominant bilingual groups on compare word problems came
encounter to the assumption that students whose linguistic proficiency
is below the second threshold level in both languages should have
demonstrated the same cognitive strengths when faced with challenging
academic endeavors (Cummins, 1987). There are three various plausible
explanations for the uneven performance between Haitian- and
English-dominant bilingual students in inconsistent language problem
solving. First, the interactive effect of bilingual dominance with
language consistency in reversal errors could be an artifact of the
sample in the study. The subjects came into the study as preselected
Haitian-dominant and English-dominant, with their own previous
backgrounds that were not controlled. Prior schooling, unequal tutoring
service given to the students at home, and students' socio-economic
status and individual mathematical ability could account for the unequal
performance. The second explanation for the uneven compare word problem
performance between Haitian- and English-dominant students could be due
to the nature of arithmetical thinking that characterized the problems
that comprised the tests. The problems were additive and multiplicative
types. It was suggested that multiplicative thinking that is more
cognitively demanding than additive thinking could have had an
interactive effect with the relationship between bilingual dominance and
language consistency. The third explanation for the superiority of
Haitian-dominant students over English-dominant students in inconsistent
language problems could be provided through the analysis of linguistic
components (personal pronoun, key-words, language symmetry) of compare
word problems that is addressed later in this paper.

This study found that test language did not influence bilingual
students' reversal error generation. Students committed reversal errors
on compare problem solving across both English and Haitian versions of
the test with the same magnitude. This finding was not consistent with
the assumption that the majority language more than the minority
language would favor a greater acceptance of learning and experience
(Coffeen, 1982; Cummins, 1976). One would expect the subjects in this
study to be more prone to generate more reversal errors in Haitian (the
minority language) than in English (the majority language). This finding
sheds light on the possible syntactic similarities between English and
Haitian that capture the comprehension of compare word problems. For
example, subjects might represent at an equal rate, in both English and
Haitian, the phrases "more than," "less than," and so forth in their
model of problem comprehension. Further research studies are suggested
to take on the relationship between the syntactic and semantic
structures of English and Haitian in the context of mathematical word
problems in order to be more accurate about the influence of test
language on word problem solving.

The interpretation of the two-way interaction between test language and
bilingualism better explains the main effects of the independent
variables on the dependent variable (reversal error generation). The
data showed no significant interaction between bilingualism and test
language, (p \> .05). In other terms, the direction of error difference
between balanced and dominant bilingual students and between
Haitian-dominant and English-dominant students remained the same across
both levels (English and Haitian) of test language. Even though the
interaction was not significant, it was worth studying since it would
provide a better way to interpret the main effects already discussed.
Indeed, balanced bilingual students tended to generate fewer reversal
errors than dominant bilingual students, whether the language of the
test was English or Haitian. This finding showed that balanced bilingual
students' superior performance in compare problem solving, as compared
to dominant students, was not influenced by either English and Haitian
across both consistent and inconsistent language problems.

The data revealed no interaction between language consistency and test
language, (p \> .05). The direction of the main effect vector
(consistency vs. inconsistency) of the language consistency factor was
the same across both levels (English and Haitian) of test language. In
other words, subjects tended to commit more reversal errors when solving
inconsistent language problems than when solving consistent language
problems across both language versions (English and Haitian) of the
mathematics tests. Again, Lewis and Mayer's (1987) consistency
hypothesis was supported by the inherent characteristics of inconsistent
language problems to cause more reversal errors than consistent language
problems in compare word problem solving, regardless of the language
version of the test. In other words, the possibility to err in
rearranging relational terms in inconsistent problems has more to do
with logico-mathematical knowledge that requires problem solvers to
mentally manipulate the symmetry of these operations (n more x than y
and n less y than x) than with linguistic knowledge that simply rests
upon the meanings of more, less, and than separately. Indeed,
logico-mathematical knowledge invariably persists through natural
language while linguistic knowledge may vary from one language (e.g.,
English) to another (e.g., French).

# CONCLUSIONS

The results of the study showed that degree of bilingualism had a
significant, positive relation with error generation on compare word
problems. The data revealed that both Haitian- and English-dominant
students were more likely than balanced bilingual students to generate
reversal errors in compare word problems. This finding gave support to
the assumption that balanced bilingualism is causal to cognitive
development (Arnberg, 1981). However, while Haitian-dominant students
were more likely than English-dominant to commit reversal errors on
consistent language problems, the latter outperformed English-dominant
students in inconsistent language problem solving that are assumed to be
more difficult than consistent language problems. Three explanations
were given to the fact that Haitian-dominant students generated less
reversal errors than English-dominant students on inconsistent language
problems. First, the two dominant bilingual groups were preselected in
the study; they came with their own prior knowledge and mathematical
ability that were not controlled. It is suggested that these
uncontrolled factors might contribute to the unequal performance of the
Haitian- and English-dominant groups. Second, the nature of the
arithmetical thinking on the problems tests could also influence the
uneven performance of the two groups. Since the mathematics tests were
comprised of additive and multiplicative problems, it is suggested that
multiplicative thinking could have an interactive effect with the
relationship between bilingual dominance and language consistency.
Third, the complex processing of linguistic components (personal
pronoun, key-words, and language symmetry) of compare word problems may
also explain the unequal performance between Haitian- and
English-dominant students. It seems that Haitian-dominant students, as
compared to English-dominant students, processed at a greater rate the
search for a referent to the personal pronoun that is embedded in
inconsistent language problems, the representation of inconsistent
language problems that is model-centered instead of key-word strategies,
and the rearrangement of the relational sentence of inconsistent
language problems to fit the format of the second sentence of consistent
language problems that is parallel to human cognitive processing.

The analysis of variance showed that test language did not have a
significant relation with reversal error generation on compare word
problems. In other words, students tended to commit reversal errors in
compare word problems at an equal rate in both Haitian and English
versions of the test. This result was not in line with the assumption
that the majority language would be more likely than the minority
language to favor a greater acceptance of learning and experience
(Coffeen, 1982; Cummins, 1976). One would expect the subjects in the
study to be more prone to commit reversal errors in Haitian (the
minority language) than in English (the majority language). This finding
suggested linguistic similarities between English and Haitian that
embrace the comprehension of compare word problems. The interaction
between bilingualism and test language was not significant. In other
words, the reversal error differences found among the three bilingual
groups did not depend upon the level of test language, whether English
or Haitian. Again, this finding showed that the cognitive superiority of
balanced bilinguals was not influenced by test language; the cognitive
advantages of balanced bilinguals allowed them to transcend the
superficial, linguistic embodiment of word problems to attain the
situation that emanates from the text input (Kintsch & Greeno, 1983).
The data revealed no significant interaction between language
consistency and test language factors. Subjects tended to generate more
reversal errors when solving inconsistent language problems than when
solving consistent language problems, whether the test language was
Haitian or English. This finding lent support to the inherent nature of
inconsistent language problems to be more error prone than consistent
language problems, regardless of language of testing (Lewis & Mayer,
1987). Finally, the data showed no significant three-way interaction
among bilingualism, language consistency, and test language. The two
main effects of bilingualism that captured the error differences between
balanced and dominant bilingual students and between Haitian-dominant
and English-dominant students did not statistically vary with the
interaction between language consistency and test language.

## Changes to the Study

Methodological changes that should be effected in the study are relevant
to test item composition, data collection, test scoring, and sample
analysis. An increase of test items is important for test reliability
improvement. The regulations of schools in which the study was conducted
allowed the investigator to use a short period of time in order to
intervene as little as possible with students' learning time. The
investigator was forced to administer a short test that could fit a
60-minute period. Thus, each of the eight problem types could appear
only once on the test. Test takers could succeed or fail each problem
type only once. The scoring grading showed that the test reliability
could be hampered as students had reduced opportunity to show their
genuine understanding of each problem type. Chance might play a great
role in students' decision making in answering the questions. Hence, it
is highly suggested to re-conduct the same study in an environment where
a lengthened test, that will be more reliable, can be used in the
research study (Anastasi, 1986). Improved reliability would enhance the
validity of the test, which would make more precise the relations that
are sought between bilingualism, test language, language consistency,
and reversal error generation.

The second way of improving the relation between bilingualism and
compare problem solving would be to minimize foreign factors that would
affect data collection and measurement. There are a number of
determinants of error variance in this study such as noise, students'
end-of-year preoccupations (final exams, graduation rehearsal, and field
trips), teachers' indifference or excessive zeal in motivating their
students to participate in the study, age, grade level, and a single
examiner to correct students' exams. The effects of these factors are
twofold: a reduction of experimental variance and a decrease in the
reliability of the writing and mathematics tests. "Thus minimizing error
variance has two principal aspects: (1) the reduction of errors of
measurement through controlled conditions of an experiment, and (2) an
increase of the reliability of measures" (Kerlinger, 1986, p. 290).
Conducting the same study at the beginning of the school year or during
the mid-school year is encouraged since these are times when students'
attention is preoccupied with fewer school-related duties (e.g., final
exams) in a noise-free atmosphere. The subjects would be drawn from the
same age or grade population. Also, it is suggested that teachers be
informed to keep a neutral and unobtrusive behavior during test
administration. While teachers' interest in the study (in terms of
encouraging students to participate in the study) is appreciated, they
may influence students' scores in providing them cues (verbal or body
gestures).

Likewise, one examiner can influence students' scores for different
reasons: fatigue, misreading of a sentence, and so forth. In replicating
the same study, it is suggested that two examiners grade students'
papers; they would then compare the two sets of scores to see if they
measure the same thing (e.g., reversal error) with the same scale. The
given recommendations would reduce error variance and give systematic
variance (e.g., experimental variance due to bilingualism) a chance to
show itself.

In the present study, it is assumed that the preselection of bilingual
groups could account for their unequal performance in compare word
problem solving. In replicating the study, it is suggested that one
control the sample more effectively; one would collect data on subjects'
prior schooling (whether bilingual or not), date of arrival in the
United States, tutoring service given after school or at home,
socio-economic status, and mathematical ability.

## Educational Implication of the Study

The results of the study can be used in two ways: fostering bilingual
education in the school system and training dominant bilingual students
to represent mathematical compare word problems. The outstanding
performance of balanced bilingual students over dominant bilingual
students in compare problem solving indicated the positive cognitive
benefits of bilingualism in academic endeavors. Therefore, it is
suggested that United States educational policy makers should encourage
bilingual education in the educational system. Funds should be made
available to bilingual programs in order to provide bilingual students
with appropriate learning environments and competent bilingual educators
who will allow them to develop their cognitive abilities in academic
endeavors.

The second way of implementing the results of the study (whereby
balanced bilingual students outperformed dominant bilingual students in
compare problem solving) is to train dominant bilingual students to
represent mathematical word problems. Previous studies have indicated
that the majority of students' errors in word problems are
misrepresentations of problem structure rather than computational errors
(Anand & Ross, 1987; DeCorte, Verschaffel, & DeWinn, 1985). It is
suggested that teaching dominant bilingual students in their two
languages how to represent word problems may improve their performance
in compare word problems.

## Further Research

With respect to future research in mathematics and bilingual education,
the results of the present study suggest the following questions:

1\. To what extent do linguistic components (the personal pronoun, key
words, and language symmetry) of compare word problem influence the
relation between degree of bilingualism and reversal errors?

2\. To what extent do problem-types (additive and multiplicative)
influence the relation between degree of bilingualism and compare word
problem solving?

3\. To what extent do first-language dominant and second-language
dominant bilingual students demonstrate the same cognitive strengths in
solving compare word problems?

4\. To what extent do linguistic similarities between two languages
(e.g., Haitian and English) influence mathematics problem solving
learning among bilinguals?

5\. Does the relationship between degree of bilingualism and
mathematical word problem solving persist across all age and grade
levels?

These questions await other investigators.

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