**GENDER
AND MATHEMATICS:
WHAT IS KNOWN AND WHAT DO I WISH WAS KNOWN?**

Elizabeth
Fennema

Wisconsin Center for Education Research

May
2000

Prepared
for the Fifth Annual Forum of the National Institute for Science Education, May
22-23, 2000, Detroit, Michigan.

COMPLEXITY! After studying issues related to gender
and mathematics for about 3 decades, I have decided complexity most adequately
describes what I know. It also is a major component of the questions that I wish
I knew the answers to. Gender differences in learning mathematics are complex;
the multiplicity of forces and environments that operate within our Society to
influence that learning are complex; it is complex to design effective
intervention programs; the role that biological factors might or might not play
are complex; it is certainly complex to conduct research about gender and
mathematics; it is even more complex to interpret research for practitioners.
While research over the past three decades has made significant contributions to
both defining and understanding this complexity, there is much left to know.
What I am trying to do in this paper is share with you what I have come to know
and understand through my study of gender and mathematics. To do this, I rely on
my knowledge and understanding of extant scholarship, societal forces, and
personal beliefs as well as my perception of how all of these interact to
influence the development and maintenance of gender differences in mathematics.

**Research
and Personal Beliefs
**

All
scholarship is strongly influenced by personal beliefs. Sometimes beliefs are
overt and explicitly defined, but in many cases the beliefs are covert and not
easily recognized. In the instance of scholarship related to gender, beliefs
exert a strong influence. Gender is a vital part of each human being that cannot
be ignored. Although I once wrote about the necessity for teachers to become
sex-blind, part of my journey has led me to know that is impossible. We cannot
eliminate our sex-role identify from our learning, our beliefs, or our
scholarship.

Some
definitions would be helpful because words communicate beliefs. Two different
words have been used that imply beliefs about *causation* of differences between females and males, i.e. *sex
*and *gender*. And my own writing, as well as the writing within the field,
reflects scholars’ maturing view of the complexity of causation of differences
between females and males. Work published before about 1970 used the phrase *sex
*differences when research results were reported. This phrase contributed to
the implication that any found differences were biologically, and thus
genetically, determined. According to this belief, these differences were
immutable and could not be changed. Therefore, schools could accept the
difference as non-changeable and not work to change them. Work published during
the 70s and 80s often used the term *sex-related*
differences that many hoped would be interpreted as an indication that while the
behavior of concern was clearly related to the sex of the subjects, it was not
necessarily genetically determined. More recently most scholars have discussed *gender
*differences believing that such a term has a stronger flavor of social or
environment causation of differences that are observed between the sexes. While
the meaning and use of words is a murky area, when I use the word *sex*,
I am referring to biologically determined behaviors. When I use the word *gender*
I am inferring social or environmentally causation of behaviors that differ for
females and males. Of course, it is impossible to totally separate social and
biological influences and perhaps it isn’t always necessary. However, I shall
try to be consistent even though I am sure that my mixing of the two words will
reflect the complexity of sorting out causation of learning differences between
males and females.

My
own value position, or set of personal beliefs, has strongly influenced what I
have done and what I say here. It is important (to me at least) that I make some
of my beliefs explicit. I have always believed that through scholarly activity,
I can learn how to better facilitate the learning of mathematics by females and
males. Coupled with this is my belief that classrooms play a major role in
determining what an individual learns. It follows that if one identifiable group
of people is not learning as well as another group, then the educational
environment can and should be modified to ensure that these group differences
are eliminated. Another of my beliefs that seems to be particularly relevant to
this paper has to do with the place of mathematics in education. I believe that
all pre-university students should learn mathematics, not just for the sake of
learning one of the most important bodies of knowledge that humans have
developed, but because mathematical knowledge provides power in understanding
the world as well as the possibility of choice. Without learning mathematics,
one cannot chose to pursue graduate study in many fields, change careers, or do
many other things. Not all people will chose to do careers where the knowledge
of mathematics is essential, but I believe they should have the option to make
that choice. My entire professional career has been predicated on the
over-arching belief that women deserve equity with men in all walks of life, and
that belief has been reflected in a significant part of my scholarly activities
in the area of gender and mathematics since all of my intersecting beliefs can
be easily seen in my scholarship. Does this mean that I am not a very good
scholar or only report what I want to believe? Not at all!
It just means that the questions that I chose to investigate and the
methods I chose to use were strongly influenced by those beliefs. (For a more
complete discussion of this point of view, see Lagemann & Shulman, 1999.)

One
of my original, naïve ideas had to do with what *equity* is. While it appeared easy to define, this has not proved to
be the case. Does equity mean that females and males should have an equal
opportunity to learn whether or not they avail themselves of that opportunity?
This is the definition adopted in much federal legislation that has dealt with
equity such as Title IX. No one can restrict access to mathematics courses on
the basis of sex. Both girls and boys should be able to enroll in the same
mathematics courses, textbooks should portray males and females in identical
roles, girls and boys should have equal access to computers, etc. While these
overt things have been fairly well accomplished, this definition of equity has
not been achieved as can be seen in the various studies that looked at
teacher-pupil interactions. Researchers have gone into classrooms and counted
the number of times teachers interact with boys and girls and attempted to
document well defined educational experiences. For example, how many times do
teachers call on students of each sex to solve mathematical problems, praise
them, etc. I know of no study that does not indicate that teachers interact with
boys more than with girls, so there is not equality of educational experiences.

Instead
of equal educational experiences, equity could mean *equality of outcomes*, i.e. that females should learn exactly the
same mathematics as do males, be able to perform the same on various measures of
mathematical learning, and have the same personal feelings toward oneself and
mathematics. Under this definition when equity is achieved, girls will be as
confident about learning mathematics as are boys, girls will believe that they
have as much control of their mathematics learning as do boys, etc., and there
would be no differences found on such tests as the SAT or local, state,
national, or international measures of achievement. While other papers will
address gender differences in attitudes toward and learning mathematics and so I
won’t be expansive about it, I read the literature to indicate that whenever
higher level cognitive skills are measured, girls are still not performing as
well as boys, nor do they hold as positive an attitude toward mathematics.

Some
have suggested that equality of outcomes can be achieved by different
instructional methods for females and males. It is based on the belief that
males and females learn in different, but equitable ways. E.g., some research
suggests that girls learn better in cooperative groups and boys learn better in
competitive groups, or that single-sex classrooms adapted for a female learning
style should be established. This particular definition has received a great
deal of scholarly and media attention in the last few years. (Most of the
Gilligan work in based on it.) While the implied definition of equity in the
mathematics *Standards* directly rejects
this definition when it says that “All students should learn mathematics,”
it then goes ahead to recommend one type of mathematics and instructional
procedures for all groups.

Others
have suggested that equity has to do with social justice for all in learning
mathematics. (And that seems to me just to bring in another poorly understood
term.) As can readily be seen,
merely understanding the word *equity*
is complex. (See Secada, 1989; and Leder, Forgasz, & Solar, 1996 for an
expanded discussion of these ideas.) There is no correct definition or
understanding of what it is. Certainly research won’t help much in resolving
it. While various kinds of philosophical discussion can enrich our
understanding, defining equity comes down to what one personally believes.

What
I believe is that equity in mathematics education will be achieved when there
are no perceivable differences between the mathematics known, or how females and
males feel about themselves and mathematics. If in order to achieve this goal, it is necessary to have
separate instructional methods, they are acceptable. If teachers have to treat
boys and girls identically, they can be trained to do so. I believe that equity
using such a definition means that equity is achievable. It also follows
logically that research will help in understanding how it can be achieved as
well as providing educational guidelines for achieving it.

**Research
from 1970-1990
**

During
the years between 1970 and 1990, there were probably more research studies
published concerned with gender and mathematics than in any other area (Leder,
1996). This work has been well reviewed elsewhere and I won’t duplicate these
reports. To give a flavor of the work, I will briefly review my own work that
was done in association with a variety of colleagues. In 1974, my first article
about gender, a review of extant work on sex difference in mathematics, was
published in the *Journal of Research in
Mathematics Education.* In this article, I concluded that while many studies
had been poorly analyzed and/or included sexist interpretations, there was
evidence to support the idea that there were differences between girls’ and
boys’ learning of mathematics, particularly in activities that required
complex reasoning; that the differences increased at about the onset of
adolescence; and that these differences were recognized by many leading
mathematics educators. As an aside, it was really the writing of that 1974
article that turned me into an active feminist. It compelled me to recognize the
bias that existed towards females, which was exemplified by the recognition and
acceptance by the mathematics education community at large of gender differences
in mathematics as legitimate.

The
Fennema-Sherman studies (Fennema & Sherman, 1977, 1978; Sherman &
Fennema, 1977), sponsored by the National Science Foundation and published in
the mid 1970s, documented sex-related differences in achievement and
participation in Grades 6 to 12. Although there were many subtle results, these
findings basically agreed with those of my original review. In addition, Sherman
and I found gender differences in the election of advanced level mathematics
courses. When we coupled the achievement differences with the differential
course enrollment, we hypothesized that if females participated in advanced
mathematics classes at the same rate that males did, gender differences would
disappear. Many things are learned as one does research, and from the stating of
this hypothesis, I learned that what you write and say stays with you a long
time. This hypothesis, labeled by others as the *differential
course-taking hypothesis*, became a point of attack by Julian Stanley and
Camilla Benbow (Stanley & Benbow, 1980), who used interpretations of some of
their studies as a refutation of our hypothesis. They went ahead to use their
work as evidence that gender differences in mathematics are genetic. Although
widely attacked and disproved, the publication of their claims in the public
media did have unfortunate repercussions (Jacobs & Eccles, 1985).

Affective
or attitudinal variables were also examined in the Fennema-Sherman studies.
Identified as critical were beliefs about the usefulness of mathematics and
confidence in learning mathematics, with males providing evidence that they were
more confident about learning mathematics than were females, and males believing
that mathematics was, and would be, more useful to them than did females. It
also became clear that while young men did not strongly stereotype mathematics
as a male domain, they did believe much more strongly than did young women that
mathematics was more appropriate for males than for females. The importance of
these variables, their long-term influence, and their differential impact on
females and males was reconfirmed in many of our later studies, as well as by
the work of many others (Leder, 1992).

One
cognitive variable also studied in the Fennema-Sherman studies was spatial
skills or spatial visualization, which I continued to investigate in a
three-year longitudinal study in collaboration with Lindsay Tartre (Fennema
& Tartre, 1985). Differences between females and males in spatial skills,
particularly spatial visualization or the ability to visualize movements of
geometric figures in one’s mind, had long been reported (Maccoby & Jacklin,
1974). Since items that measure spatial visualization are so logically related
to mathematics, it has always appeared reasonable to believe that spatial skills
contributed to gender differences in mathematics. We found that while spatial
visualization was positively correlated with mathematics achievement (that does
not indicate causation), not all girls are handicapped by inadequate spatial
skills, but perhaps only those girls who score very low on spatial tasks.

Although
they were not particularly innovative nor did they offer insights that others
were not suggesting, the Fennema-Sherman studies had a major impact due to a
variety of reasons. They were published in highly accessible journals just when
the concern with gender and mathematics was growing internationally. Partly
because the studies were accessible, not generally controversial, and because
they employed fairly traditional methodology, their findings have been accepted
by the community at large. The studies were identified by two independent groups
(Walberg & Haertel, 1992; Anonymous, in preparation) as among the most often
quoted social science and educational research studies during the 80s and 90s.
Each week, I still receive at least one request for information about the
Fennema-Sherman Mathematics Attitude Scales that were developed for those
studies. The problems of gender and mathematics were defined and documented in
terms of the study of advanced mathematics courses, the learning of mathematics,
and certain related variables that appeared relevant both to students’
selection of courses and learning of mathematics. Many have used them as
guidelines for planning interventions and other research.

After
completing the Fennema-Sherman studies, with the indispensable aid of many
others (Laurie Reyes Hart, Peter Kloosterman, Mary Koehler, Margaret Meyer,
Penelope Peterson, and Lindsay Tartre), I broadened my area of investigation to
include other educational variables, particularly teachers, classrooms, and
classroom organizations. We studied teacher-student interactions, teacher and
student behaviors, and characteristics of classrooms and teaching behaviors that
have been believed to facilitate females’ learning of mathematics.

The
series of studies dealing with educational variables, reported and summarized in
the book edited by Gilah Leder and me (1990), suggested that many studies have
documented that it is relatively easy to identify differential teacher
interactions with girls and boys. In particular, teachers interact more with
boys than with girls, praise and scold boys more than girls, and call on boys
more than girls. However, the impact of this differential treatment is unclear
and difficult to ascertain. The data that resulted from the studies reported in
this book do not support the premise that all differential teacher treatment of
boys and girls is very closely related to gender differences in mathematics.
(See also Eccles & Blumenfeld, 1985; Koehler, 1990; Leder, 1982).

However,
when more subtle examples of teacher-pupil behavior were studied, Peterson and I
(Fennema & Peterson, 1986; Peterson & Fennema, 1985) found that small
differences in teacher behavior combined with the organization of
instruction, made up a pattern of classroom organization that appeared to favor
males. For example, competitive activities encouraged boys’ learning and had a
negative influence on girls’ learning, while the opposite was true of
cooperative learning. Since competitive activities were much more prevalent than
cooperative activities, it appeared that classrooms we studied were more often
favorable to boys’ learning than to girls’ learning.

In
connection with this series of studies, Peterson and I proposed the Autonomous
Learning Behaviors model, which suggested that because of societal influences
(of which teachers and classrooms were main components) and personal belief
systems (lowered confidence, attributional style, belief in usefulness), females
do not participate in learning activities that enable them to become independent
learners of mathematics (Fennema & Peterson, 1985). This model still appears
valid, although my understanding of what independence is has grown, and I
believe that independence in mathematical thinking may be learned through
working in cooperation with others to solve mathematical problems.

Identifying
behaviors in classrooms that influence gender differences in learning and
patterns has been difficult. Factors that many believed to be self-evident have
not been shown to be particularly important, and I do not believe that we have
sufficient evidence that would allow us to conclude that teachers interacting
more or differently with girls than with boys is a major contributor to the
development of gender differences in mathematics. Many intervention programs
have been designed to help teachers recognize how they treat boys and girls
differently. Unfortunately, such programs do not appear to have been successful
in eliminating gender differences in mathematics. I believe that differential
teacher treatment of boys and girls is merely one piece of the complexity of the
causes of gender differences in mathematics.

My
next set of studies was conducted with Janet Hyde. For these studies, we did a
series of meta-analyses of extant work on gender differences reported in the US,
Australia, and Canada (Hyde, Fennema, & Lamon, 1990; Hyde, Fennema, Ryan,
& Frost, 1990). These results indicated that while gender differences in
mathematics achievement might have decreased, they still existed in tasks that
required functioning at high cognitive levels. It also seemed that when tests
measured problem solving at the most complex cognitive level, the more apt there
were to be gender differences in mathematics in favor of males. The
international assessment reported by Gila Hanna (Hanna, 1989) showed results
that basically confirmed this.

My
work has not been the sole chain of inquiry that has occurred during the last
two decades. The work of Jacquelynne Eccles, Gilah Leder (Leder, 1992), and
others has been conducted independently and closely parallels the major themes I
have addressed. An overly simplistic and not inclusive summary suggests that
during this time, scholars documented that differential mathematics achievement
and participation of females and males existed; some related educational and
psychological variables were identified; explanatory models were then proposed;
and finally (or concurrently in some cases) interventions, based on the
identified variables, were designed to alleviate the documented differences.

One
line of inquiry that I have not pursued, but that adds a significant dimension
and more complexity to the study of gender and mathematics, is the work that has
divided the universe of females into smaller groups. In particular, the work of
the High School and Beyond Project (a large multi-year project that documented
gender differences in mathematics as well as many other areas) as interpreted by
Secada (1992), and the work of Reyes (Hart) and Stanic (1988) have investigated
how socio-economic status and ethnicity interact with gender to influence
mathematics learning. The US, as many other countries, is a highly heterogeneous
society, made up of many layers, divisions, and cultures. The pattern of female
differences in mathematics varies across these layers and must be considered.

**Intervention
Studies
**

By
about 1980, there were some rather consistent findings that described gender
differences in mathematics. Based on these findings and with the help of three
others (Joan Daniels Pedro, Patricia Wolleat, and Ann Becker DeVaney), I
developed an intervention program called Multiplying Options and Subtracting
Bias (Fennema, Wolleat, Becker, & Pedro, 1980) This program lasted
approximately one-hour and was focused on parents, teachers, counselors, boys,
and girls. It was composed of sharply focused videotapes to be viewed by the
target group, and a workshop guide that included suggestions for follow-up
activities. It was extensively evaluated, particularly with regard to its
effectiveness in increasing girls’ participation in advanced secondary school
mathematics classes and its impact on confidence and perceived usefulness. We
found that this short intervention that helped the target audiences recognize
the importance of mathematics and the stereotyping of mathematics that was
prevalent, resulted in more girls, and more boys, electing to take mathematics
courses. A more negative finding, however, was that pointing out the sexism that
existed in classrooms and environment increased the girls’ anxiety about
mathematics.

Since
*Multiplying Options and Subtracting Bias* was completed a plethora of
other intervention programs have been designed and implemented. The *Women
in* *Science* program of the National Science Foundation funded 136
projects between 1976 and 1981, and *The
Women in Engineering* Programs reported 395 interventions in 1975 and 859 in
1991 (Leder, Forgasz, & Solar, 1996). Many other programs have been
developed by other groups that range from short-term interventions of one day or
less to long-term programs that focus on curriculum and instructional change,
total school re-organization (changing from coed to single sex classrooms), or
teacher education. The population targeted has included many diverse groups:
teachers, university faculty that work with pre-service teachers, parents,
females in general or specific groups of females such as those who are
traditionally disadvantaged. Sometimes boys are the focus. Pre-school to
graduate programs have been developed. The content of the interventions has
ranged from curriculum content to hands on activities illustrative of what users
of mathematics do.

While
some of these programs have been evaluated, the effectiveness of many of the
programs has not been well documented. Several excellent summaries are available
and I urge reading of them for more information (see for example Campbell;
Clewell and her colleagues; or Gilah Leder and her colleagues, 1996). Even
without specific evidence of total effectiveness with the various targeting
populations, it is accurate to say that the programs have been extremely
effective in educating the general public about the importance of learning
mathematics to both boys and girls. Unfortunately, the message that has been
received by many is that “girls can not do mathematics” (Eccles, 1985).

In
summary, because I was asked so often to speak about gender and mathematics, I
compiled yearly lists of what I had concluded that research had shown. Following
is a portion of the list I made in 1990.

**Gender
Differences in Mathematics: 1990**

1.
Gender
differences in mathematics may be decreasing.

2.
Gender
differences in mathematics still exist in:

·
learning of
complex mathematics

·
personal
beliefs in mathematics

·
career
choice that involves mathematics

3.
Gender differences in mathematics vary:

·
by
socioeconomic status and ethnicity

·
by school

·
by teacher

4.
Teachers tend to structure their classrooms to favor male learning.

5.
Interventions *can* move towards
achieving equity in mathematics.

On the basis of an examination of the five items on
this list, it is clear that in the years following my original review in 1974,
my understanding of gender and mathematics had grown as far as related variables
were concerned, but the same gender differences, albeit perhaps smaller, still
existed. I could describe the problem more precisely. I knew that large
variations between groups of females existed; I knew that there were differences
among schools and teachers with respect to gender and mathematics issues; I knew
that females and males differed with respect to personal beliefs about
mathematics; and I knew that some interventions could make a difference. I was
increasing my understanding that the issue of gender and mathematics was
extremely complex.

Now before I sound too pessimistic, it should be
noted that there were many females who were achieving in mathematics and are
currently pursuing mathematics-related careers. However, let me reiterate that
in spite of some indications that achievement differences were becoming
smaller—and they were never very large anyway—they still existed in those
areas involving the most complex mathematical tasks. These differences became
more evident as students progressed to middle and secondary schools. There were
also major differences in participation in mathematics-related careers. Many
women, capable of learning the mathematics required, chose to limit their
options by not studying mathematics. And while I have no direct data, I strongly
suspected that the learning and participation of many women, who might be in the
lower two-thirds of the achievement distribution, have not progressed at all. I
had to conclude that many of the differences that were reported in the 1970s,
while smaller overall than they were then, still existed in 1990.

**The
1990s and Beyond
**

My
personal odyssey with gender and mathematics has continued until the present
day. Although I took a hiatus that resulted in two changes of direction. While I
continued to accept without question the basic premise of the International
Commission for Mathematics Instruction Study Conference (1992) that “there is
no physical or intellectual barrier to the participation of women in
mathematics,” about 1990 I stopped doing any original work on gender and
mathematics. I had worked hard for about 25 years, but in spite of all that work
and the additional work done by many dedicated educators, mathematicians, and
others, the *problem* still existed in
much the same form that it did in 1974. Not only was I discouraged, but I was
convinced a new perspective on the research about women, girls, and mathematics
was needed. Fortunately, I was not alone in recognizing that research on gender
and education needed to change. And direction for the change came from two
directions: cognitive science and feminist scholarship.

*Cognitive Science and its Impact*

One of the emerging and productive educational
research paradigms has been cognitive science, or the “scientific study of
mental events, primarily concerned with the contents of the human mind, . . .
and the mental processes in which people engage” (Brown & Borko, 1992).
One major assumption underlying much of cognitive science research is that the
principles of cognitive development, particularly in mathematics, can be applied
universally to all people, both females and males. And indeed, findings that
indicate the universality of the emerging theories of cognitive development have
been reported from a number of countries and cultures. (See for example, Adetula,
1989; Olivier, Murray, & Human, 1990; Secada, 1991.)

Within the mathematics education research
community, cognitive science methodologies have been used to investigate both
students’ and teachers’ thinking. Often when students are studied, they are
asked to report mental strategies they have used to solve various kinds of
mathematical problems derived from a precise definition of a mathematical
domain. Particularly in the elementary school, the robust body of knowledge
about children’s thinking in arithmetic has had major impact on the
instructional program. Effective professional development programs have also
been built on knowledge derived from cognitive science studies. These programs
usually help teachers to understand their students’ thinking and to build
their instructional programs on what the students know. (For example see
Carpenter, Fennema, Franke et al., 1999).

The question of gender differences doesn’t appear to
have been interesting to cognitive science scholars, perhaps because they
believed that the patterns of mental activity they were finding were universal
and thus there were no gender differences in cognitive behaviors to be found.
But, there have been a few studies that have indicated that the assumption of no
gender differences in mathematical thinking may not be true. Carr and Jessup
(1997) reported that girls in Grades 1-3 tended to use observable or overt
strategies such as counting, while the same age boys tended to use mental
strategies. Gallagher and DeLisi (1994) reported that while there were no
overall differences in the number of selected SAT items answered by females or
males, females tended to use more conventional or commonly taught strategies,
while males tended to use more untaught strategies. Some working within this
paradigm have chosen not to report the gender differences that they found (Reys,
Rybolt, Bestgen, & Wyatt, 1982), “perhaps because they were unable to
provide any reasons for their finding, and they decided to put it aside” (Sowder,
1998, p. 12). As a result, there are not many studies related to gender that
have been done using a cognitive science perspective.

Once again, I turn to my own and my colleagues’
work, most of which has come from the Cognitively Guided Instruction Project
(CGI). CGI is a professional development program designed to help teachers
understand their student’s mathematical thinking and to use this understanding
to design instruction. The program development and related research was
supported by the National Science Foundation for about 10 years and resulted in
many studies that focused on teachers, instruction, and students’ learning
when they had been in CGI classrooms. Overall, the results indicated that
teachers could learn to accurately assess their children’s thinking using some
cognitive science methodologies. When the teachers gathered knowledge about
their students’ strategies for solving mathematics problems, they modified
their instruction rather dramatically so that their students’ knowledge and
mental processes became a significant part of the instructional programs.
Students who had learned in CGI classrooms learned significantly more than their
non-CGI counterparts.

One study investigated teachers’ knowledge of and
beliefs about boys’ and girls’ success in mathematics (Fennema, Peterson,
Carpenter, & Lubinski, 1990). Although teachers thought the attributes of
girls and boys who succeeded in mathematics were basically similar, teachers’
knowledge about which boys were successful was more accurate than teachers’
knowledge about which girls were successful, and teachers attributed the boys’
successes more to ability and girls’ successes more to effort. Linda
Weisbeck’s (1992) results add some interesting dimensions to our knowledge of
teachers’ cognitions. During stimulated-recall interviews, teachers reported
that they thought more about boys than about girls during instruction. However,
the characteristics they used to describe girls and boys were very similar.

It appears that teachers were very aware of whether
the child they were interacting with was a boy or a girl. However, they didn’t
think that there were important differences between girls and boys that should
be attended to as they made instructional decisions. Boys just appeared to be
more salient in the teachers’ minds. Teachers appeared to react to pressure
from students, and they got more pressure from boys. Interventions designed with
this finding in mind would be very different from interventions that assume that
teachers are sexist.

*What Cognitive Science has Taught Us about
Girls’ and Boys’ Mathematics*

One extensive study, Cognitively Guided Instruction
(CGI), was done by Tom Carpenter, me, and several others (Fennema, Carpenter,
Jacobs, Franke, & Levi, 1998). In a 3-year longitudinal study we studied
teachers and their students as they progressed from Grade 1 through Grade 3 (Fennema,
Carpenter, et al., 1998). Once or twice each year, children who had learned
their mathematics in CGI classrooms, were asked to solve a variety of problems
(number facts, addition/subtraction word problems, non-routine, and extension
problems) and to report how they solved the problems. We found no gender
differences in correctly solving number fact, addition/subtraction, or
non-routine problems throughout the three years of the study. This finding was
in agreement with literature where it has been widely reported, as well as
believed, that gender differences do not emerge until early adolescence. In our
study, however, each year from Grade 1 to Grade 3 we found strong and consistent
gender differences in the *strategies*
used to solve problems, with girls tending to use more concrete strategies like
modeling and counting and boys tending to use more abstract strategies that
reflect conceptual understanding. In other words, the mental processing of boys
and girls were different, and we also found some significant achievement
differences in solving extension problems.

By the end of the third grade, the girls used more standard algorithms than did the boys. On the problems that required flexibility in extending one’s problem solving procedures, boys were more successful than were girls. The ability to solve the extension problems in the third grade appeared to be related to the use of invented rather than procedural algorithms in earlier grades, as both girls and boys who had used invented algorithms early were better able to solve the extension problems than those who had not.

Because
these results were so unexpected to us, we asked 3 prominent scholars who had
worked in different areas to interpret the results and to speculate about the
results’ importance, causation, and potential impact on future mathematical
learning. (See *Educational Researcher,* 1998, *27*, pp. 4-22). While
one scholar was somewhat skeptical that the results were large enough to be
important, the others felt that they were critically important, and might
presage the gender differences that are found to increase as students move into
advanced mathematics. The importance of the findings was reflected in Judith
Sowder’s words:

Children who can invent strategies for computational tasks show a more advanced grasp of basic mathematical concepts that those children who are dependent on (other strategies). The children who can invent strategies are more likely to find sense in the mathematics they are learning and come to believe that mathematics makes sense and to seek out sense in the mathematics they continue to learn. Their understanding will lead to deeper confidence in their ability to do mathematics.

They have a better chance of succeeding mathematically. (p. 13) (Italics added)

The
mathematical understanding that was indicated by the strategies used more by the
boys’ than the girls’ is important for development of fundamental concepts
and students’ ability to be flexible in new situations. Thus, the more
abstract strategies that children invent to solve various problems is probably
related to their future understanding of mathematics, and could indeed help to
explain the gender differences in older learners that had been evident for many
decades. Major gender differences in performance usually don’t appear until
sometime in adolescence when they are more often exhibited in complex
mathematics tasks, particularly on tests of problem solving. The gender
differences that were reported in this study strongly suggest that more girls
than boys were following a pattern of mathematical development and learning that
was not based on understanding. And the lack of understanding becomes more
critical as students progress through school. While it is possible to learn to
do arithmetic procedures in the early grades without understanding, it becomes
more and more difficult to learn advanced ideas unless a foundation of
understanding is present from the very beginning.

Overall
this study suggests that gender differences appear earlier and are more complex
than had previously been recognized. The results certainly call into question an
assumption that is prevalent in the various recommendations for reform in
mathematics teaching and learning. It is widely believed that one reformed
curriculum with its accompanying instructional design and methodology will
suffice for all children. However, it seems to me that the results of this study
suggest that without explicit attention to traditionally underachieving groups,
all groups of children will **not **learn
mathematics equitably. Many have identified classrooms such as the ones in which
these children were learning as epitomizing needed reforms in mathematics
teaching. These CGI classrooms emphasized complex mathematical tasks (problem
solving), communication about mathematics, and learning with understanding—all
of which are major tenets of mathematics education reform. And it is clear that
the students who learned their mathematics in these classrooms did learn and
understand significantly more than did children in more traditional classrooms,
but there were still dramatics gender differences.

Many
advocates of basing curriculum on understanding as well as most scholars who
study teaching and learning believe that equity issues can be addressed by
improving mathematics instruction for all. (See for example Carey, Fennema,
Carpenter, & Franke, 1995). Pervasive in the mathematics education community
there is an underlying assumption that one program based on understanding will
enable all students to learn in an equitable fashion. Based on this study, *this
assumption may not be valid. *The teachers in this study were participating
in a professional development program that emphasized reform ideas and included
discussion about equity issues. Much of the instruction approached the ideal of
furthering children’s understanding in a way that is consistent with current
recommendations. We did intense observations in each classroom and saw no
evidence of teachers treating girls and boys differently. Yet, strong gender
differences were manifested by children who had learned in these classrooms for
three years. While the differences found were probably no greater and could have
been less than what exists in traditional classes, the fact remains that they
existed. And most would agree that these differences could lead to stronger
differences in the future. The results call into question the suggestion of the *Standards
*(National Council of Teachers of Mathematics, 1989, 1991) that major
curriculum reform, by itself, will provide equity in mathematics education.

Explanations
of why boys, more than girls, developed mathematical understanding as they moved
through Grades 1-3 can only be speculative. All the scholars believed that
something was taking place in the classrooms that encouraged these gender
differences to emerge. It was addressed most directly by Hyde and Jaffee who are
social and feminist psychologists. They suggested that the differences were a
result of differential treatment of girls and boys by the teachers (Hyde &
Jaffee, 1998). They suggested that because the type of mathematical program used
in the CGI classrooms was based on teachers having the freedom to make
instructional decisions, the teachers’ stereotypical beliefs about gender and
mathematics could have led them to interact differently with boys and girls, and
that in turn led to the differences found.

Another
hypothesis has to do with the children’s choice of strategies to report.
Children in these classrooms had a great deal of freedom in deciding how to
solve problems and also in deciding what strategies to report about their
problem solutions. It is clear that children often have a variety of strategies
to use to solve problems, and strategy use is a matter of preference. For some
reason, did the girls prefer to use and report strategies that would have an
influence on the development of their understanding?
This choice may have inhibited the development of fluency with more
abstract strategies. Hyde and Jaffee inferred that this may have been so and
suggest that the freedom to choose may have permitted the children’s
stereotypical beliefs to influence strategy use and thus the development of
understanding in these classrooms.

Perhaps
girls chose to use strategies that could make their ideas clear (e.g., modeling
with concrete materials) partly because the teachers and peers wanted to
understand each child’s thinking. Hyde and Jaffee suggested that girls, more
than boys, are more socially aware of others’ responses and are considerate of
others’ needs and/or are more compliant. It was a basic tenet of these
classrooms that the teacher’s understanding of each child’s thinking was
essential. It was obvious in the classrooms that teachers wanted to know what
the child had done, and children were equally eager to make sure that the
teachers understood. Modeling strategies are easier to report and to understand
than are invented algorithms. In order to comply with teachers’ wishes or to
help them and other students to understand their responses, did girls tend to
report modeling strategies rather than other kinds? Or did the girls simply
prefer reporting the less abstract strategies? Sowder supported this idea when
she suggested that although the girls may have understood invented strategies as
well as the boys did, they may have just preferred less abstract strategies.
Many believe that student preferences are important, but in this case, it may
have been that using such strategies inhibited the development of more abstract
strategies.

In
summary, it is clear that cognitive science methodologies are providing tools
for us to gain deeper understanding of the complexity of gender differences. We
are just beginning to understand differences in mental activities between girls
and boys and to assess their impact on learning. We also know that teachers’
thoughts about girls and boys influence their instructional decisions.
Understanding teachers’ beliefs and knowledge about girls and boys will
provide important information as we plan interventions to achieve equity.

**Feminist
Perspectives
**

While
it is impossible to expound here very deeply on various feminists theories that
are being used to shape research, it is safe to say that these theories are
influencing many people’s world-view. I am no expert on feminist theories and
their accompanying research paradigms, but it seems to me that people working in
feminist perspectives share one common component. Without exception, they focus
on interpreting the world and its components from a female’s point of view,
and the resulting interpretations are dramatically different from world-views
that used to be accepted.

Feminist
scholars argue very convincingly that most of our beliefs, perceptions, and
scholarship, including most of our scientific methodologies and findings, have
been and are dominated by male perspectives or interpreted through masculine
eyes. According to feminist scholars, this perspective has resulted in a view of
the world that is incomplete at best and often wrong. If females’ actions and
points of view had been considered over the last few centuries, according to
many of these feminist scholars, our perceptions of life would be much different
today. A basic assumption of feminist work is that there are basic differences
between females and males that are more prevalent than the obvious biological
ones. These differences result in males and females interpreting the world
differently. Many of these scholars present convincing arguments about how the
world influences males and females differently. It appears irrelevant whether
these differences are inherent or environmentally caused, and most feminist
writers that I have read are basically uninterested in whether or not such
differences are genetic or related to socialization. It is enough that the
differences exist. These differences become stronger over time and influence
one’s entire world-view and life. For those who are just thinking about this
idea for the first time, I recommend that you find a little book called *The
Yellow Wallpaper* (Gilman, 1973). Written over 100 years ago, it gives a
picture of one woman’s view of her world and, at the same time, the picture of
that same world from her husband’s viewpoint. Both pictures impress the reader
with their accuracy—and they are dramatically different.

Feminist
scholars work in many areas and almost all of them are outside mathematics
education. Some are trying to interpret a basic discipline of science (such as
biology or history) from a female, rather than a male, point of view. They argue
that almost all scholarship, including the development of what is called science
and mathematics, has been done by men from a masculine viewpoint, utilizing
values that are shared by men, but not by women. Those major bodies of knowledge
that appear to be value-free and to report universal truths are in reality based
on masculine values and perceptions. Since males’ roles and spheres in the
world have been so different from females’ roles and spheres (Greene, 1984),
these bodies of knowledge do not reflect 50 percent of human beings and thus are
incomplete and inaccurate. Jim Schuerich (1992) has suggested that a feminist
science is better than a value-free science. To support this, he draws from
Charol Shakeshaft’s (1987) work on educational administration and Carol
Gilligan’s (1982) work on the development of moral judgment. Each of them has
demonstrated quite conclusively that research on male-only populations has
produced results that were not only incomplete but *wrong*. Similar approaches to history and literature have resulted in
deeper, richer understanding.

The
idea of masculine-based interpretations in areas such as history or literature,
and even in medical science, is not too difficult to illustrate nor even to
accept. After all, until recently history didn’t bother to include many
females except for those few who happened to be queens or were burned at the
stake. For example, how many even knew who Sacajawea was or her contributions to
the opening of our American West until very recently. (Did you know that was her
image on the new dollar coin?) Many
conclusions in medical research have been based solely on male subjects; their
inaccuracy is easy to illustrate. History has been presented as if most of our
ancestors were male and as if important things in the public arena happened
predominantly because of and to males. The use of male names by female writers
in order that their writing be accepted, or even published, is commonly known.

Does
the prevalence of the idea of a masculine or feminine world-view apply to what
mathematics is and if so, how? Can mathematics be seen as masculine or feminine?
Is not mathematics a logical, value-free field? The idea of a masculine or
feminine mathematics is difficult to accept and to understand, even for many who
have been concerned about gender and mathematics. But a few people are working
to explicate what a gendered mathematics might be—in particular, Suzanne
Damarin (1995), and Zelda Isaacson (1986), are struggling to define what a
feminist approach to the study of mathematics education might be.

One
way to approach the problem of a gendered mathematics is not to look at the
subject, but to examine the way that people think and learn within the subject.
The work of Belenky and her colleagues (Belenky, Clinchy, Goldberger, &
Tarule, 1986) in identifying women’s ways of thinking and knowing is
provocative as we consider these questions. Do females learn math differently
than do males? Should we develop special instructional programs for females? It
is beyond the scope of this paper to explicate these ideas, but we should
consider this. Earlier I alluded to the idea that most mathematics classrooms
appear to be organized to be more appropriate for boys than for girls and even
quoted some research that supports that idea. Others have interpreted
discussions arising from the belief about a female world-view and applied the
ideas to describing female-friendly instruction. Such instruction usually
includes such things as the greater inclusion of cooperation rather than of
competition in classrooms, small group rather than individual work, more
communication, and/or more socially relevant mathematics. Others have argued for
single-sex schools oriented to the mathematics instruction of females. Running
through these suggestions, it seems to me, is a basic belief that females learn
differently and perform differently in mathematics than do males. This belief is
dramatically different than the belief of a universal way of human thinking
espoused by the cognitive scientists.

Another
theme that informs many of the feminist perspectives is the necessity for
females’ voices to be heard (Campbell & Greenberg, 1993). To scholars with
this conviction, it is not enough that scholars identify important research
questions that are studied objectively using a positivist approach. Females must
have a hand in the identification of the questions and the research itself so
that females’ life experiences can take center stage. Thus, the world can, and
will be, interpreted from a female perspective. Scholars who work within this
belief system often use women subjects as co-investigators, have females
reporting their own experiences, and use females as the main subjects under
investigation who also help to interpret results. There are not many of these
studies available currently in mathematics education, but I predict that we
shall see increasing numbers of them as the importance of female voices is
recognized. Those that are available indicate that females often have a very
negative view of mathematics, how it is taught, and mathematicians.

It is
too early to be able to assess the impact that studies using feminist
methodologies will have on our understanding of the relationship between gender
and mathematics, both the identification of the problem and its solutions. But,
it appears logical to me that as I try to interpret the problem from a feminist
standpoint, the focus used in my earlier work changes. I do not interpret the
challenges related to gender and mathematics as involving *problems*
of females (e.g females are deficient because they are less confident, don’t
believe mathematics is useful, lack spatial skills, etc.) or design
interventions based on the masculine world-view of changing the females so they
are equal to males. Instead, I begin to look at how a male view of mathematics
has been destructive to females. I begin to articulate the problem that lies in
our current views of mathematics and its teaching. I am coming to believe that
females have recognized that mathematics, as currently taught and learned,
restricts their lives rather than enriches them.

I
must say at this point that the current reform movement has strong feminine
overtones (and that is an anathema to many people). But the emphasis on
students’ views (their thinking), communication, social relevant mathematics
could have come straight from many feminist scholars.

Whatever
our own value position about feminism and mathematics, I believe that we need to
examine carefully how feminist perspectives can add enriched understanding to
our knowledge of mathematics education. And, indeed, we should be open to the
possibility that we have been so enculturated by the masculine-dominated society
we live in that our belief about the gender neutrality of mathematics as a
discipline may be wrong or, at the very least, incomplete. Perhaps we have been
asking the wrong questions as we have studied gender and mathematics. Could
there be a better set of questions, studied from feminist perspectives, that
would help us understand gender issues in mathematics? What would a feminist
mathematics be? Is there a female way of thinking about mathematics? Would
mathematics education, organized from a feminist perspective, be different from
the mathematics education we currently have? Suzanne Damarin (1995) stated that
we need to “create a radical reorganization of the ways that we think about
and interpret issues and studies of gender and mathematics.” Many scholars
believe that only as we do this will we be able to understand gender issues in
mathematics. Perhaps my beginning to believe that the decision by females not to
learn mathematics or enter mathematics-related careers because mathematics has
not offered them a life they wish to lead is an indication that my old view
about learning and teaching mathematics, as well as about gender and
mathematics, was immature and incomplete. An examination of what the female
voices in the new research are saying will help me—and perhaps others—to
understand teaching, learning, gender, and mathematics better.

**What Do I Know?**

*The Complexity*

Throughout
this paper, I have been expounding on the complexity of dealing with gender and
mathematics. Nothing appears to be simple and listing what I really know is
difficult. That females participate
in mathematics-related careers less than do males is one of the few accepted
facts. That differences exist in the learning of mathematics seems clear also,
although many scholars believe either that the differences are diminishing or
that any differences that exist are unimportant. Females appear to hold more
negative values about mathematics and their own relationship to mathematics than
do males but there is some evidence that these differences are decreasing (Forgasz,
Leder, & Vale, 1999). But I caution everyone about such simplistic
statements. What mathematics is being measured in tests where gender differences
are either shown or not shown? How was the information about values obtained?
Were females’ voices a part of the data gathering procedures? Too often
research dealing with these issues provides an incomplete picture at best and
only helps to perpetuate the belief that females are somehow inadequate in
relation to learning and doing mathematics.

**Dilemmas
for Practice**

Two
of my colleagues and I have identified some dilemmas that we face as we
interpret a variety of kinds of research and reform recommendations that are
appropriate for organizing classroom instruction (Ambrose, Levi, & Fennema,
1997). Many of these appear logically to apply equally to girls and boys. But a
closer examination reveals that nothing to do with gender is simple.

Consider
the reform recommendation that has to do with encouraging students to
communicate their mathematics thinking by presenting their ideas and convincing
peers of their correctness by arguing, questioning, and disagreeing. It is
widely believed that those who enter into this kind of debate will learn better.
But will girls enter into this kind of communication as willing as do boys? Many
teachers have reported informally that girls will not for a variety of reasons.
Perhaps this even helps to explain some of the gender differences we reported in
the CGI study. Will boys tend to dominate such discussions and not listen as
well as girls?

Another
major reform recommendation has to do with the use of technology in the
classroom. Others at this conference have discussed this as it relates to girls
and boys. It is clear that boys have more experience with technological toys
than do girls. Does this reflect interest? Does this mean boys have more
knowledge? How do teachers take these ideas into consideration?

The
*Standards* recommend that mathematics
be situated in problem-solving contexts that are socially relevant.
Unfortunately many textbooks and teachers are more aware of contexts that are
from male dominated fields such as projectiles for parabolic equations, or
sports for statistics. One interesting study that I did not review earlier
suggests that gender differences in problem solving skills were eliminated when
girls were familiar with the context in which the problem was situated
(Marshall, 1984). But will boys willingly participate in problems from
female-dominated fields?

Should
classrooms be competitively organized or organized around cooperative
activities? Certainly the most visible reward in most mathematics classrooms is
grades that are highly competitive. The Fennema-Peterson studies quoted earlier
suggest that young boys learned better in a competitive situation while young
girls learned better in a cooperative situation. Is that finding true for older
students? Is the solution to have single-sex classrooms? And would the
experience we have had with black/white schools be repeated and females
classrooms become inevitably less adequate?

Thoughtful,
reasonable practitioners can probably create solutions to each of the dilemmas
presented. But it is the role of researchers to help them identify the potential
problems that may exist and aid them in evaluating any solutions that are
created.

**What Do I Wish Was Known?**

*The Future Contributions of Research*

Research
into gender and mathematics must continue. We should continue to monitor the
best we can learning, attitudes, and participation in mathematics.
In addition, we need to develop new paradigms of research that will
provide insight into why gender differences occur. In other words, gender as a
critical variable must enter the mainstream of mathematics education research.
It is insufficient to say and to believe that the study of gender differences
can be left to those who are specifically interested in gender. *That
is not just nor fair*! Aren’t
we all interested in how ALL learn mathematics? And ALL includes that 50% of the
student body who happen to be female. Fairness and justice demand that ALL
researchers be concerned with ALL the students even when results are obtained
that can not be easily interpreted and understood.

Specifically, we need to continue the study of
gender in relation to mental processing of both students and teachers. As
research on teachers continues to mature and improve, we must include gender as
a variable. We probably cannot study how the sex of the teacher influences
instruction because of the limitations imposed by the number of male teachers
available. However, we can study teachers’ beliefs and knowledge about girls
and boys and the impact that teachers’ cognition has on instructional
decisions for both girls and boys.

Classrooms that reflect the various demands for reform are beginning to become more and more prevalent. But are they equally effective for boys and girls? The CGI study discussed earlier provides some evidence that just reforming classrooms without specific attention to traditionally under achieving groups is insufficient to achieve equity. The learning that results from these reformed classrooms needs to be carefully monitored. Perhaps as we do this, we will begin to develop an image of what equitable mathematics education is.

*Values and Research*

Personal
values dominate the doing and interpreting of research in gender and
mathematics. I think I became an educational researcher because I believed that
I would discover TRUTH. That has not happened and I believe that if truth can be
found from educational research, it is not in the area of gender and
mathematics. But, research has deepened our knowledge about gender and
mathematics and the many, many studies about gender have provided some insight
into the inequities that have existed and that has led to heightened awareness
of things that need to be changed. But there are some questions about gender and
mathematics for which research cannot provide the answers. Is mathematics really
necessary for a life of value in the 21^{st} century? This is a
heretical question coming from a mathematics educator but one that needs to be
addressed. It appears to me that I may have been attaching the worthwhileness of
an individual to whether or not she or he learns mathematics. Now, in fairness
to me, I have spent my professional career in trying to assist traditionally
underachieving groups to learn mathematics, but Nel Noddings (1998) has led me
to examine my own beliefs in this regard. She had the courage to raise a very
difficult question about major differences between boys and girls and
mathematics as she addressed the differences found in the CGI study she relates
the results to the possibility “that girls are just less interested in math
than (are) boys.”

If
it were true that girls are less interested than boys in math, *so
what*? What would follow? Clearly, we still could not judge the next female
or male who walks into out classroom on the basis of this generalization. The
next female may be Hypatia reincarnated and the next male Forrest Gump. Further,
the generalization in itself doesn’t tell us what to do. . . . A positive
answer to the question about gender differences in interest in math might lead
to further exploration of an idea that repels many of us, i.e., the question of
genetic differences. . . . But the genetic argument does not seem particularly
helpful to us as educators and launching the argument about interest in math
would enable us to examine the question of gender differences in a way that
might be helpful. . . . Why do we see it as a problem if young women are less
interested than young men in mathematics? Why *don’t*
we see it as a problem if young men are less interested than young women in
early childhood education, nursing, elementary teaching, and full-time
parenting? The easy answer to the issue posed in this fashion is that
proficiency in mathematics opens the doors to professional success and financial
well-being. There’s no money in the other activities. But consider what is
being valorized. Why is there so little financial compensation and prestige in
fields traditionally associated with women? . . . Do we approve of a social
structure that values competence in mathematics over competence in child-care? .
. . No student’s self-worth should depend on her or his interest or capability
in mathematics, and we should not endorse the propaganda that mathematics is
essential in almost all worthwhile occupations. . . . We must explore the
unpleasant possibility that many girls do not *want*
to be part of the math crowd because many of its members seem socially inept or
aloof. (pp. 17-18)

I
shall end with some personal soul searching that I have been engaged in. There
are no *right* answers but perhaps we should consider the following. Is it
possible that I, and others who have been doing work related to gender and
mathematics, have been doing a major injustice to females by pursuing issues
related to gender and mathematics? Are we just making the chosen roles of
females in society (that often don’t involve mathematics) less important, less
adequate, or of less value than the chosen roles of males (that often include
mathematics)? Is it critical for everyone to learn mathematics? Are those who
learn mathematics at lower levels of less value than those who learn at higher
levels?

Research
on gender and mathematics has provided a powerful scientific discourse during
the past 3 decades. The entire educational community¾composed
of practitioners, researchers, and policymakers¾need to continue to engage in this discourse about
and to explore ways to deepen our understanding of what equity is and how it can
be achieved. It is in discourse about philosophical questions as well as
research questions that our understanding of gender and mathematics will grow.

**Endnote
**

Some
of this paper was excerpted from:

Fennema,
E. (1996). Mathematics, gender and research. In G. Hanna (Ed.), *Towards
gender equity in mathematics education* (pp. 9-26). Amsterdam: Kluwer.

Fennema,
E., & Carpenter, T. P. (1998). New perspectives on gender differences in
mathematics: An introduction and a reprise. *Educational Researcher, 4*(11),
19-22.

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