Research on Undergraduate Mathematics Education:
A Way to Get Started

Ed Dubinsky
Georgia State University

Contents

• Introduction
• A brief history of RUMEC
• Focusing on the APOS theoretical perspective
• One way of growing
• RUMEC-II
• RUMEC-Mexico
• More information
• Readings

Introduction

RUME is an acronym for Research in Undergraduate Mathematics Education, an area which has been growing steadily over the past several years, but is still quite small. There seems to be a sizable group of mathematicians who want to move into it, as well as some recent Ph.Ds in mathematics education who specialize in post-secondary topics and feel the need for some additional preparation.

This phenomenon raises a number of questions: Why is RUME receiving this attention? What should be its relations with the communities of research in mathematics and research in K-12 mathematics education? What national organization should be its professional home? Why do some new Ph.Ds in mathematics education feel that their experience in graduate school needs to be augmented? I very much hope that such questions receive serious attention from mathematicians and mathematics educators, but it is not my purpose to deal with them here. I am not even going to discuss the best ways of helping people get started in RUME. Rather, my purpose is to discuss one particular approach that has a strong mentoring flavor.

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A brief history of RUMEC

A research organization started by happenstance at MAA's NSF-funded Project CLUME (Cooperative Learning in Undergraduate Mathematics), which conducts workshops and minicourses for helping mathematics faculty learn about cooperative learning. During a CLUME workshop in June, 1995, several participants asked for help in getting started in RUME.

The organizers of the workshop had also been involved in several research projects in which they had generated a large amount of data about students learning calculus, abstract algebra, and discrete mathematics. They were looking at many years of data analysis and interpretation to produce research reports. With this in mind, an evening meeting of the CLUME participants was called to see if anyone was interested in collaborating on the research that this data would support. The attendance was high, the reception to the idea was very strong, and so we decided to organize ourselves around the examination of this data. We called ourselves a Research in Undergraduate Mathematics Education Community or RUMEC.

The Exxon Educational Foundation provided funding to get us started and we invited fellows from MAA's Project NExT (an Exxon-funded program to help recent Ph.D.'s get started in the profession) to participate. A few others joined, including some already working in RUME. Our structure focused on projects and we formed small groups, mixing experienced researchers with novices on specific research projects connected with the data. The idea was that the beginners would be mentored by the experienced people who would be able to expand their work with the help of the novices.

We also decided to institute ``internal reviewing." Every research paper is circulated amongst the membership and discussed in detail at a general meeting. This discussion can be hot and heavy, but openness to criticism has been both a result of, and a contributor to, our feelings of being together as a community. Authors' responses to suggestions are completely voluntary. That is, the final decision of what is in the paper remains totally with the authors. Nevertheless, the members of RUMEC feel that the quality of papers is much improved by the process. The internal reviews also allow us to share our understandings of how to do this kind of research.

Our funding has allowed us to meet three or four times per year. Meetings combine plenary sessions for business, long sessions at which the internal reviews take place, and break-out sessions for individual research projects. At first, all of the meetings were in West Lafayette, or South Bend, Indiana which were geographically centered for most members, although people from all over the U.S. and Mexico attended. What began as a plan for regional meetings became a national and even international enterprise. Our January 1997 meeting was in Atlanta and our July 1997 meeting will be in Morelia, Mexico. We will return to South Bend in Fall 1997.

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Focusing on the APOS theoretical perspective

Almost all of RUMEC's projects initially used data that had been collected from instruments whose design was based a paradigm for education research associated with the APOS (action, process, object, schema) framework for analyzing concepts. (See the paper by Asiala et al in the bibliography at the end of this article.) We started off with everybody accepting this framework, temporarily, as our working paradigm, and we found that sticking to one point of view tends to sharpen our thinking. Now we are completing the analyses of the original data and more of our projects collect new material for analysis and interpretation, generally based on, or at least closely related to the APOS perspective. We feel that our structure and philosophy has worked well and we will continue with it as we develop new research projects.

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One way of growing

There are a number of mathematicians interested in hearing about and even engaging in post secondary mathematics education research. This is evident in the attendance at meetings where such matters are reported, in the increase in publications in this area beginning to appear in the literature, and in the number of people interested in joining RUMEC.

After a brief initial period of growth, RUMEC consisted of about 25-30 people actively involved in one or more research projects. There continued to be individuals contacting us with expressions of interest and Project NExT has new fellows each year who have been a source for the growth of RUMEC. The Exxon Educational Foundation continues to provide us with support and many of our members' institutions match the travel funds we can provide. We have felt a strong pressure to expand and, at the same time, have hesitated to lose our collegial spirit through increased size. Organizational challenges became substantial as membership increased from that first 25 to the 50-60 that we have now. Furthermore, the source of mentors, already small, is limited by our decision to stick to the APOS framework -- mentors are experienced researchers who work in frameworks of their own choosing and are not likely to switch to something else in order to work with novices. Thus, we have renamed ourselves, RUMEC-I and have begun to organize RUMEC-II and RUMEC-Mexico for new people.

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RUMEC-II

The organization of RUMEC-II is quite different from that of RUMEC-I. RUMEC-II consists of about 20 individuals who are serious about working in RUME, and seven experienced researchers who will mentor the novices, either by advising them in some detail or by joining with them on collaborative projects. This group will focus on individual project work and will de-emphasize full group meetings. The idea of internal reviewing will be kept, but we will learn how to do it electronically, rather than face-to-face. The basic structure of RUMEC-II is to make connections between novices and mentors, based on common interests. We will maintain only a loose communication within the whole group, mainly to keep each other informed about progress in a general way and conduct internal reviews. There is no underlying philosophy or research paradigm necessarily shared by all members.

The development of RUMEC -II proceeded as follows. The mentors selected a set of readings that would be useful to anyone working in the field. Everyone being mentored was asked to begin with a specific set of these readings to complete over a three month period and to report to their mentor about what he/she had learned. Novices were then expected to continue their reading on a more gradual basis until they completed the list. The mentors worked with their mentees to move towards establishing specific research projects. The assignment of mentors was made on the basis of a very brief statement of interest and goals by the novices and by their selection of initial readings.

All of this began in October 1996 and some participants completed the initial readings by the AMS/MAA meeting in San Diego last January. We held a breakfast meeting in San Diego attended by 21 mentors and mentees and it seems that a good start has been made. A retreat is planned for the Spring. Of course it will be some months before we can ask how successful this approach is in helping people get started with research in this area.

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RUMEC-Mexico

Three RUMEC-I members are from Mexico and will organize RUMEC-Mexico. They will maintain close communication with RUMEC-I. At the moment, this group is receiving financial support from the Mexican Foundation, CONACYT.

This July, RUMEC-I will meet in Morelia, Mexico, July 18-20, intersecting for a day the July 14-18 meeting of the Latin American Committee on Mathematics Education (CLAME). It and RUMEC-I will hold a joint session on July 18, which we expect to stimulate membership in RUMEC-Mexico.

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More information

• RUMEC-I will sponsor a Research Conference in Collegiate Mathematics Education, September 4-7, 1997 at Central Michigan University, Mt. Pleasant, Michigan. This conference will have research reports, expository talks on research, its applications and effects, and panels on issues related to research and teaching practice. For more information, visit the conference web page.

• Articles on RUME can be found in journals such as Educational Studies in Mathematics, Journal for Research in Mathematics Education, Journal of Mathematical Behavior, and the volumes, Research in Collegiate Mathematics Education I, II published by the American Mathematical Society for the Conference Board of Mathematical Sciences. Also, expository articles on education research are beginning to appear in MAA publications.
• There will be a contributed paper sessions sponsored by the AMS/MAA Joint Committee on Research in Undergraduate Mathematics Education at the AMS/MAA Winter Meetings in Baltimore next January.

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Readings

What follows is a list of background readings used by RUMEC-II.

INTRODUCTORY

Becker, J.R. & B. Pence (1994). The Teaching and Learning of College Mathematics: Current Status and Future Directions, in Research Issues in Undergraduate Mathematics Learning, MAA Notes #33, 5-14.

Bouniaev, M.M.(1996). Stage-by-Stage Development of Mental Actions and Computer Based Instruction, Technology and Teacher Education Annual, 947-951.

Finkel, D. L., and G. S. Monk (1983). Teachers and Learning Groups: Dissolution of the Atlas Complex, in Learning in Groups, Jossey-Bass, 83-97.

Schoenfeld, A. (1991). On Pure and Applied Research in Mathematics Education, Journal of Mathematical Behavior, 10, 263-276.

Schoenfeld, A. (1993). Confessions of an Accidental Theorist,For the Learning of Mathematics, 8(1).

Selden, J. & Selden A. (1990). Constructivism in mathematics education: A view of how people learn, UME Trends, 2 (2), 8.

Selden, Annie, & Selden, John. (1993). Collegiate mathematics education research: What would that be like? College Mathematics Journal, 24(5), 431-445.

PHILOSOPHICAL, THEORETICAL AND FOUNDATIONAL WORKS

Bereiter, C. (1994). Constructivism, Socioculturalism, and Popper's World. Educational Researcher, 23, 7, 21-23.

Cobb, P. (1994). An Exchange: Constructivism in mathematics and Science Education, Educational Researcher, 23, 7, 4.

Cobb, P. (1994). Where is the Mind? Constructivist and Sociocultural Perspectives on Mathematical Development, Educational Researcher, 23, 7, 13-20.

Driver, Asoko, Leach, Mortimer & Scott (1994). Constructing Scientific Knowledge in the Classroom, Educational Researcher, 23, 7, 5-12.

Dubinsky, E. (1991). Reflective Abstraction in Advanced Mathematical Thinking, in D. Tall (Ed.), Advanced Mathematical Thinking, Dordrecht: Kluwer, 95-126.

Glasersfeld, E. von (1985). Learning as a Constructive Activity, in C. Janvier (ed.), Problems of Representation in the Teaching and Learning of Mathematics, Hillsdale, NJ: Erlbaum, 3-17.

Leontiev A.N, (1979). The Problem of Activity in Psychology, in V. Wertsch (ed.), The Concept of Activity in Soviet Psychology, 37-72.

McLeod, D. B. (1992). Research on Affect in Mathematics Education: A Reconceptualization, in Douglas A. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, 575-596.

Piaget, J. & Garcia, R. (1989).Psychology and the History of Science, New York: Columbia University Press.

Tall, D. (1992). The Transition to Advanced Mathematical Thinking: Functions, Limits, Infinity and Proof, in Douglas A. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, 495-511.

Vygotsky, L. (1978). Mind and Society: The development of higher psychological processes, Cambridge, MA: Harvard University Press.

METHODOLOGY

Asiala, Brown, DeVries, Dubinsky, Mathews & Thomas (1996). A Framework for Research and Curriculum Development in Undergraduate Mathematics Education, Research in Collegiate Mathematics Education II, 1-32.

Eisenhart, Margaret A. (1988). The ethnographic research tradition and mathematics education research. Journal for Research in Mathematics Education, 19(2), 99-114 ,

Romberg, T. (1992). Perspectives on Scholarship and Research Methods, in Douglas A. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York.

Schoenfeld, A. (1994). Some Notes on the Enterprise (Research in Collegiate Mathematics Education, That Is). Research in Collegiate Mathematics Education I, 1-19.

(SMALL) LARGELY EMPIRICAL STUDIES

Dubinsky, E. and Yiparaki, O. (1996). Predicate Calculus and the Mathematical Thinking of Students, Paper presented at the DIMACS Symposium on Teaching Logic and Reasoning, Rutgers University, July 25-26, 1996. Available online.

Gray, E. & Tall, D. O. (1993). Success and failure in mathematics: the flexible meaning of symbols as process and concept, Mathematics Teaching, 142, 6-10.

Monk, S. (1992). Students' Understanding of a Function Given by a Physical Model. In Harel, G. and Dubinsky, E. (eds.), The Concept of Function: Aspects of Epistemology and Pedagogy, MAA Notes 25, 175-193.

Schoenfeld, A. (1989). Explorations of Students' Mathematical Beliefs and Behavior, Journal for Research in Mathematics Education, 20, 4, 338-355.

Selden J., Mason A. & Selden, A. (1994). Even Good Calculus Students Can't Solve Nonroutine Problems, in J. Kaput and E. Dubinsky (eds.), Research Issues in Undergraduate Mathematics Learning: Preliminary Analyses and Results, MAA Notes 33, 19-26.

Schoenfeld, A. (1988). When Good Teaching Leads to Bad Results: The Disasters of ``Well-Taught" Mathematics Courses, Educational Psychologist, 23(1), 145-166.

Tall, D. & Vinner, S. (1981). Concept Image and Concept Definition with Particular Reference to Limits and Continuity, Educational Studies in Mathematics, 12 (2), 151-169.

Tall, David (1992). Students' Difficulties in Calculus. Plenary presentation in Working Group 3, ICME-7, Quebec.

LARGER STUDIES

Breidenbach, D., Dubinsky, E., Hawks, J., and Nichols, D. (1992). Development of the Process Conception of Function, Educational Studies in Mathematics, 23, 247-285.

Heid, K. (1988). Resequencing Skills and Concepts in Applied Calculus Using the Computer as a Tool, Journal for Research in Mathematics Education, 19 (1), 3-25.

Schoenfeld, A. (1985). Mathematical Problem Solving, New York: Academic Press.

RELATIONS TO K-12

Fennema, E., & Loef, M. (1992). Teachers' knowledge and its impact, in D. A. Grouws (ed.), Handbook of research on mathematics teaching and learning, New York: Macmillan, 147-164.

Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding, in D. C. Grouws (ed.), Handbook of research on mathematics teaching and learning, New York: Macmillan, 65-97.

Martin, W. G. & Harel, G. (1989). Proof frames of Preservice Elementary Teachers, Journal for Research in Mathematics Education, 20 (1), 41-51.

Romberg, T. A., & Carpenter, T. P. (1986). Research on Teaching and Learning Mathematics: Two Disciplines of Scientific Inquiry, in M. C. Wittrock (Ed.), Handbook of research on teaching, New York: Macmillan Publishing Company, 850 - 873.

Skemp, R. (1976). Relational understanding and instrumental understanding, Mathematics Teaching, 77, 20-26.