Equity and Diversity, Teacher Education, Technology

Equity and Diversity, Teacher Education, Technology Annie Selden and John Selden UME Trends, Vol. 5, No. 6, January 1994, 8-9 These were the themes of the annual conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA) held at the Asilomar Conference Center on the Monterey Peninsula of California in October. The venue was reminiscent of summer camp with rustic-looking shingle buildings scattered amongst the dunes. Agreeable, sunny weather suggested long walks on the beach, but the excellent program lured one back indoors. The over two hundred participants were mainly researchers in mathematics education, but included a growing number of college mathematics faculty. The program often had eight parallel research papers concerned with topics such as epistemology and cognitive processes, teachers’ and students’ beliefs and attitudes, social and cultural factors affecting learning, problem solving, probability and statistics, functions and graphs, and modeling. There were also discussion groups, posters, and short oral presentations. Opportunities for informal chats occurred at meals and coffee breaks and an evening at the aquarium provided an excellent opportunity to watch a variety of sea life while getting to know colleagues better. The opening plenary panel consisted of Gilah Leder of Monash University in Australia, Walter Secada of University of Wisconsin-Madison, and Ubiratan D'Ambrosio of Brazil, who addressed equity and diversity. Leder mentioned several ways of viewing gender differences in mathematics education. The interventionist perspective assumes programs should be mounted to ensure more females engage in mathematics and related pursuits. The segregationist perspective sees current curricula as geared to males and seeks a different, often single-sex, education for females. The discipline perspective wants the mathematics taught in schools to be changed to be less authoritarian, more human, and more inclusive of girls. Leder linked these perspectives with several different feminist views -- the feminism of equality emphasizes intervention, the feminism of difference concentrates on women's unique ‘voice’ and often seeks segregation into single-sex schools, and radical feminism stresses the oppression of women socially, economically, and through the discipline. She noted that diversity and equity are often linked, but outside the school context, they do not necessarily go hand-in-hand. Walter Secada was critical of overly narrow uses of psychology and saw a need for scholarly inquiry into issues of mathematics-education equity. Much of the NCTM Standards concerns individual understanding, thought of as having coherence, as being constructed slowly and linked to pre-existing knowledge, and as sometimes involving re-organization. However, nothing is said of affect, context, or the social and cultural origins of understanding. He suggested an appropriate response to the kind of decontextualized mathematics currently taught in many schools may well be disengagement -- timeoff-task, absence from school, failure to persist, and dropping out. Ideas from ethnomathematics, situated cognition, social constructivism, and neo-Vygotsykian theories [see Research Sampler this issue] are promising avenues for research. D'Ambrosio, in responding, was intentionally broad and provocative. What keeps inequity going is what a certain group of society in power wants -- it's misguided to put so much emphasis on equity. We need an ethics of diversity which respects everyone's creativity; however, we should not expect society to change because of how we teach mathematics. Discussion groups focusing on equity and diversity were held the next day. In one of these, David Clarke of Australian Catholic University mentioned that he is involved with six ‘focus’ schools which seek strategies to achieve access and participation of groups ‘marginalized’ by culture. He asked

whether the apprenticeship model of mathematics learning currently proposed by Jean Lave, et al., is appropriate for everyone. [E.g., Cognition and Practice: Mind, Mathematics and Culture in Everyday Life}, Cambridge U. Press, 1988.] For example, Australian aborigines consider it inappropriate to ask questions to which one knows the answers and this might interfere with apprenticeship. He suggested that metaphors beyond those of reflection, negotiation, and reflective abstraction need to be incorporated so as to clarify and account for both social and cognitive phenomena. Tom Cooney of University of Georgia, in his plenary address, emphasized that teachers' beliefs are formed very early and are hard to change. This has implications for implementation of the NCTM Standards. Using both Perry's personal world view and Belenky’s ‘voice’ schemes, he noted that many teacher education students are still at the stage of dualism or silence, respond to authority, and just want to be shown the “right way to do it.” Those at the stage of relativism, begin to listen -- those with permeable beliefs are more willing to change. Jere Confrey, whose research group at Cornell has developed Function Probe, gave the final plenary on diversity, tools, and knowledge. She sees a lack of diversity in the content of mathematics and advocates an “epistemology of multiple representations,” which speaks to a variety of students and protects the integrity of each representational form. For her, abstraction is implicated in the suppression of diversity. Tools mediate knowledge. She demonstrated how different tools can recast the way we think about mathematics -- first drawing an ellipse with two foci and a string, and then as South African carpenters do, using two fixed perpendicular cross-pieces and a moving stick, a method that allows one to see relative rates of change. Confrey encourages students to think of mathematics as a balance of systemic inquiry and ‘grounded activity.’ Taking the sine function as an example, Confrey noted there are conceptual hurdles for students in going from triangle calculations to a continuous understanding, that rate of change is hard to see here, and it is not clear that circular motion is being represented. She demonstrated a model Ferris wheel with a flashlight which she uses to help students conceptualize the following problem: You are riding on a Ferris wheel with a 12 foot radius. Each rotation takes 20 seconds. The lowest point of the Ferris wheel is 6 feet off the ground. Find a way to predict your height off the ground as a function of time. Students observe the Ferris wheel turning at a constant speed, but know when they ride one it goes slowly at the top and quickly down the sides. She sees such kinesthetic experiences as providing students with important “grounded activity” before turning to graphing using Function Probe. Students who use a visual approach typically recognize the periodicity and begin with y = sin x . First, they stretch the graph horizontally to get the period to 20 seconds. Then they stretch it vertically to get the 12 foot radius. Finally, they do a vertical translation of 18 feet to locate the function 6 feet above the ground. Among the many interesting research papers was the work of Bob Davis, Carolyn Maher, et al., of Rutgers, who reported a small portion of a longitudinal study looking at how children develop mathematical problem-solving and argumentation. Periodically since grade one, they have been working with and observing children now in sixth grade as they explore problem-solving situations in a group setting. As the children get older, they are asked to explain and justify their solutions orally and in writing. After revisiting the problem periodically in school and thinking about it at home, Milin, a fifth grader, invented the term “families” to keep track of different possibilities for two-colored towers of unifix cubes and came up with a situation-bound proof by mathematical induction, which another group member was able to explain to the class. Maher noted that children often begin by building their own solution ideas, as if to get a ‘feel’ for the problem space, and only subsequently consider the ideas of others to refine their own problem representations.

Martin Simon and Glendon Blume of Penn State described how prospective teachers negotiated standards for mathematical justification in a mathematics course which is part of a three-year NSFfunded project, Construction of Elementary Mathematics (CEM). Asked to find the area inside an irregular plane closed curve, nicknamed “the blob,” one group of students suggested, but could not agree amongst themselves, that one might lay a string around the outside, pull it into a rectangle or square, and calculate the resulting area. One girl suggested that using a string of 8 inches one could make both a 2 x 2 square and a 1 x 3 rectangle -- a counterexample convincing to both her and the instructor, but not to other students even after further discussion. The researchers hypothesized that these students did not think one exception would invalidate a rule, or they did not accept the abstract thought process whereby “the blob” and string were implicitly universally quantified. Clifford Konold, et al., of University of Massachusetts at Amherst have developed software that demonstrates various features of the law of large numbers about which people often have poor intuitions. While piloting their software in tutoring interviews with undergraduates, they found students quickly developed some understanding of why means of larger samples are less variable than those of smaller samples. However, rather than using a “representativeness” heuristic as previously suggested by Kahneman and Tversky, the majority of students giving incorrect answers explained their reasoning using a “more-is-more” argument. That is, large samples have more of everything -- scores, means, percentages, variability, etc. Formal discussion groups included topics such as teacher change, effective learning environments, classroom complexity, teacher preparation, cultural support for mathematical understanding, openended problems, and research in undergraduate teaching and learning. The latter, organized by members of the AMS-MAA Committee for Research in Undergraduate Mathematics Education (CRUME), was well attended. Steve Monk of the University of Washington posed the question, “What could researchers do for Professor X?” Ed Dubinsky of Purdue gave an update of recent CRUME activities, including the soon-to-appear first of a projected CBMS-sponsored series of annual volumes on research in undergraduate mathematics education. Karen Graham of the University of New Hampshire solicited opinions and help regarding a possible proposal for a collaborative conference to bring together mathematicians and mathematics education researchers. The next PME-NA conference will be November 5 - 8 on the LSU campus in Baton Rouge and have a technology focus, coordinated by Jim Kaput (e-mail: [email protected]). Proposals are due February 15. For further information, contact David Kirshner (e-mail: [email protected]).