Editor's Note: At the 1993 Woodrow Wilson Gender Equity in Math and Science Congress, several speakers addressed the participants on gender equity issues in math and science from the standpoint of their experience and research. Below is an imagined dialogue among four of these speakers on some of the key issues raised at the Congress: Joan Countryman, Elizabeth Fennema, Judith Jacobs, and Sheila Tobias. An activity based on this dialogue follows.
Q. What is the problem of gender equity in mathematics and science?
Fennema (F). I first identified it in a 1974 paper in the Journal of Research in Mathematics Education as a result of reviewing the literature of the preceding 20 years. The literature showed that boys were doing better in mathematics than girls, with the greatest gender difference in problem solving, and I wanted to know why.
Jacobs (J). The longer we teach math and science, the less students like it, more so girls than boys. On top of that, only 50% of the boys who like math feel they do well at it, and this percentage is even lower for girls.
Countryman (C). I have also found that boys have both more wrong answers and right answers on math tests than girls - they're bigger risk takers.
Tobias (T). Many "math anxious" students - with high verbal scores and low math scores on SATs - are girls. "Math anxious" students feel out of control. Similarly, another group I have been researching - students who could succeed in science but have not chosen science majors - have felt alienated by science courses.
F. Very often these students are inconsistent. I remember one student who subtracted 23 from 70 correctly with base ten blocks, yet insisted that the "53" he got "from the rules" was correct.
J. That's like the student who agreed that a bill for $2 and another bill for $7 would make her $9 poorer, yet claimed that -2 added to -7 gave +9, because "two negatives make a positive." This is an example of what I call received knowing.
Q. What is "received knowing"?
J. It's knowing based on returning the words of authority; the speaker is not the source of knowledge. This is one of a sequence of types of knowing. Before "received knowing" comes silence, which is mere acceptance of authority. After it comes "subjective knowing" (knowing because it "feels right"), followed by "procedural learning" and "constructed learning." "Constructed learning" is receiving a lot of attention as "constructivism" today. But what I find interesting is the difference between two types of procedural learning: "separate" and "connected."
In procedural learning the voice of reason begins to evaluate validity. The "separate" form recognizes the need for proof; it is characterized by such traits as logic and rigor. Using the "connected" form, one seeks to compare one's observations with those of others. It is characterized by such traits as intuition and creativity. Separate knowing is the means by which mathematicians present their results and teach, and more men are comfortable with this type. On the other hand, connected knowing is how mathematicians actually do their work. Women's studies in the 1970s recognized that women are more comfortable with connected knowing. Although connected knowing has now been incorporated into the NCTM standards, no credit for it has been given to women's studies.
Q. You seem to be suggesting that men and women are comfortable with different ways of knowing. How can this be explained?
J. There are several possibilities: biological causes, math anxiety, the perception of mathematics as a male domain, teachers' differential treatment of males and females, and two other factors called "causal attribution" and "autonomous learning behavior."
F. "Autonomous learning behavior" is something I pinpointed in 1985. It requires and develops one's ability to work independently in high-cognitive-level activities, and female participation in autonomous learning behavior is lower.
T. I have found "causal attribution" very prevalent in my research. When boys experience success, they attribute it to ability; but when they experience failure, they chalk it up to not enough effort. In contrast, girls attribute their successes to luck and feel that their failures result from not enough ability.
Q. What about teachers' differential treatment of males and females?
J. Even I was guilty of giving the boys more attention when I taught junior high school math.
F. The boys seem to have more interesting things to say - perhaps it's because of their greater risk taking that Joan alluded to - but it unwittingly captures our attention.
Q. You also mentioned biological differences. What evidence is there for this?
J. I would cite the difference in the way boys and girls feel comfortable with separate and connected knowing. But don't get me wrong: feminist pedagogy is not anti-male!
F. I think a great deal of this is personal belief. Research shows that, in general, the differences are greater within each sex than between the sexes, two exceptions being spatial visualization and aggressiveness. While girls with weak spatial skills have experienced difficulty solving problems, it is possible for appropriate curriculum redesign to compensate for these weak skills. I feel that men and women should be treated equally, as is legally required by Title IX. If anything, I feel that we have "nurtured" our females too much. Requiring independence of both female and male students is necessary to develop confidence.
T. I'm basically in agreement with Liz. I feel that males and females are equal, even indistinguishable intellectually, if left alone. That the two sexes exhibit different choices of courses in math and science means that an explanation in terms of external causes is needed. I feel it derives from the way the sexes are socialized, the internalization of the different sexes of their beliefs of society's expectations for them.
Q. Given the existence of a gender equity problem in math and science, what should we do about it?
F. One thing that's important, I feel, is that girls and boys have the same experiences - and we need to take into account that they don't enter our classrooms with the same experiences.
J. And we need to use those experiences to enhance students' learning. That's an important aspect of feminist pedagogy.
F. Rather than try to change student attitudes toward mathematics, we can also change mathematics (and the way it is taught) to provide equal outcomes for the sexes.
J. That's important in feminist pedagogy, too. For example, the nature of proof in mathematics needs to be changed. Consider the standard way of proving that the sum of two odd numbers is even - it's a classic example of separate knowing. It can just as easily be proved by joining together two egg cartons cut off to hold odd numbers of eggs in two rows: the two cartons joined together always come out even!
F. The egg carton approach is also more relevant to our students' lives, and that's important, too.
J. It can also be expressed in ordinary language rather than mathematicians' language. Students can easily describe a sine curve in terms of its period and amplitude, but I feel they understand it better if they tell me how close the cycles are to each other and how high and low the curve goes.
F. We should also foster cooperation rather than competition. Our math courses should be designed to help students learn rather than weed them out.
J. And teachers must monitor male-female interactions in cooperative learning groups to make sure that males don't dominate and that females don't end up being secretaries.
C. Something else that I find to be important is knowing where our students are coming from. I like to have my students write their math autobiographies.
F. I couldn't agree with you more.
T. Me, too. And the value of these autobiographies is directly related to how specific they are. By the way, another thing related to communication that distresses me is that the math-anxious never talk to anyone about the math they're trying to learn.
J. Written communication is an important aspect of learning, too. Women typically feel more comfortable using narrative writing.
Q. The changes you are recommending are designed to help females learn math better and enjoy it more. Will they work equally well with males?
F. What we have looked at so far suggests that boys can thrive in a cooperative learning environment as well as in a competitive one, but a teacher employing cooperative learning should enlist the support of parents in using it. Using cooperative learning teaches cooperative skills as well as content.
J. Feminist pedagogy is good for boys as well as for girls. It will also eliminate gender-unequal subtleties from the classroom. And this is even more valid for eliminating racial and socioeconomic inequities.
C. Absolutely! Eliminating gender bias in math and science will serve to make these fields inclusive for all, especially for people of color.
Q. What kind of programs employing the strategies you have recommended are in the offing?
F. Megan Franke and Tom Carpenter have joined me in developing Cognitively Guided Instruction (CGI). The basis of this curriculum is problem solving, with the problems based on learners' understanding. Learners talk about their problem solutions, and multiple solution strategies are expected. For example, in listening to and understanding students' thinking, teachers will ask, "Did anybody solve it in a different way?"
In this way, all children are empowered to become problem solvers and to learn with understanding. They will develop a positive self image and feel valued as individuals. They will also learn with a culturally appropriate curriculum.
J. I'd also like to throw in my guidelines for female-friendly science: Teachers should set experiments in context, link physical science principles to the human body, and stress safety precautions rather than dangers. They should also aim for a balanced view of benefits and disadvantages of scientific developments, make aesthetically pleasing exhibits, and employ imaginative writing.
Introduction The Gender Equity Dialogue is to be read preceding this activity. In the activity four panelists are interviewed by a moderator. The activity is based on the process model of Hilda Taba. It consists of three parts (I) identification and classification of concepts and ideas from the dialogue, (II) use/development of a data retrieval chart, and (III) implementation of a problem solving activity. A facilitator is needed to involve participants in the three stages of the activities. The facilitator's questions are suggestions and should be modified according to the needs of the participants.
Participants have now seen and heard the dialogue among Tobias, Fennema, Jacobs, and Countryman.
Facilitator: Ask questions and record participant answers on a chalk board.
Facilitator: Involve the participants in classifying the information on the chalkboard. Encourage participants to classify in more ways than by the name of the panel member.
Facilitator: This part can be done in different ways. In the most open way, groups of participants could use the categories developed in Part I to develop a data retrieval chart. Individuals could engage in research (from dialogue or library sources) to complete the chart. Another method is to suggest the categories for the chart. In this case, copies should be available for participants. Again, the chart could be completed in one session using the existing dialogue or be expanded into a research activity.
Facilitator: Participants are instructed that they will receive a problem in the form of a new dialogue, an interview with Maxine Greene. As they listen to this dialogue, they should attempt to respond to the following questions:
Responses can be shared in small or large groups. They can be written or oral.
Maxine Greene has taught philosophy of education at Teachers College, Columbia University, for 25 years. Active in the Lincoln Center Institute for the Arts, she is past president of the American Educational Research Association, the Philosophy of Education Society, and the American Educational Studies Association. Dr. Greene has written five books - the latest of which is Dialectic of Freedom - as well as numerous articles.
Q. How did your career of teaching the philosophy of education begin?
A. In retrospect, you might say that it started by accident. I was already married and the mother of school children, and I was looking for a course to take. The only criterion was that it had to occur between the hours of 10 and 2. It happened to be a course on the history and philosophy of education at New York University.
Q. So it wasn't something that you charted from the time you were in high school?
A. No. You see, I never had the same opportunities as my brother. But I have always believed in the equality of the sexes. I think singling out differences is belittling. John Dewey argued that human rights were invented to counter the divine rights of kings. But those rights belong to women as well as to men.
Q. How were you able to combine your developing career in the philosophy of education with your family responsibilities?
A. Women have always had to juggle more than one track at a time, while men have needed to think about only one thing at a time. Yet women have been esteemed for their silence: what they have had to say has been deemed to be of low importance. What I sought in my youth was to be connected to society at large; and I guess I did pretty well - because I was once introduced as "someone who thinks as a man." But do you know what I did then?
A. I curtsied. Although I don't like singling out people for their differences, I don't believe in repressing them, either. I'm interested in "human possibilities" - for people to choose whom they become.
Q. How well do you feel individual differences are respected today?
A. Not very well. In spite of the emphasis given to multiculturalism today, the public sphere of affairs is very monolithic. It is characterized by rationality and efficiency and by a lack of compassion. It is really a male sphere. It also assumes that its males are all heterosexual. It has been carefully constructed over the years, with its differences buried, and women excluded.
The messy things like compassion have been relegated to the private sphere of affairs. There public norms like justice play no role at all. True equity would require reforming the community to allow the private sphere to merge with the public. Equity allows for differences from traditional norms. In the past such differences have been bullied. We have been asked to "leave our biases at home." It is important that our perspectives be validated and that their differences be respected.
Instead of perceiving the world as monolithic, with differences repressed, we need to be willing to let things get messy. Nor can we perceive of things as entirely complete, lest the possibilities of becoming be foreclosed. We are torn between the comforts of our traditions and the possibilities we can achieve by departing from them. Equity is a chance for all of us to be our own crazy selves.