## Gender and Mathematics: What is Known and What do I Wish was Known?

**Gender and Mathematics**

**Complexity!** After studying issues related to gender and mathematics for about three 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 behaviour 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 flavour 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 behaviours. When I use the word *gender* I am inferring social or environmentally causation of behaviours 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 [

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 enrol 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 [

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 [*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 ([*differential course-taking hypothesis*, became a point of attack by Julian Stanley and Camilla Benbow [

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 [

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 [

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 [

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 behaviours, and characteristics of classrooms and teaching behaviours 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 [

However, when more subtle examples of teacher-pupil behaviour were studied, Peterson and I ([

In connection with this series of studies, Peterson and I proposed the Autonomous Learning Behaviours 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 [

Identifying behaviours 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 ([

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 [

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 [

**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 [

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 [

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 [

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**

- Gender differences in mathematics may be decreasing.
- Gender differences in mathematics still exist in:
- learning of complex mathematics;

- personal beliefs in mathematics;

- career choice that involves mathematics.

- Gender differences in mathematics vary:
- by socioeconomic status and ethnicity;

- by school;

- by teacher.

- Teachers tend to structure their classrooms to favour male learning.
- Interventions
*can*move towards achieving equity in mathematics.

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.

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" [Cognitive Science and its Impact

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 [

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 behaviours 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 [

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 [

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.

One extensive study, Cognitively Guided Instruction (CGI), was done by Tom Carpenter, me, and several others (Fennema, Carpenter, Jacobs, Franke, and Levi, [What Cognitive Science has Taught Us about Girls' and Boys' Mathematics

*strategies*used to solve problems, with girls tending to use more concrete strategies like modelling 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* **27** (1998), 4-22.) While one scholar was somewhat sceptical 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 [*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* [

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 [

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., modelling 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. Modelling 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 modelling 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* [

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 [*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 [

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 [

Another theme that informs many of the feminist perspectives is the necessity for females' voices to be heard [

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 [

**What Do I Know?**

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 [The Complexity

**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 [

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 [

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?**

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.The Future Contributions of Research

*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.

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 21st 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 [Values and Research

If it were true that girls are less interested than boys in mathematics, *so what*? What would follow? Clearly, we still could not judge the next female or male who walks into our 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 mathematics 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 mathematics 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 valorised. 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 three decades. The entire educational community - composed of practitioners, researchers, and policymakers - need 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:

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

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

**References**for this article is at THIS LINK.

JOC/EFR November 2017

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