**Research on Undergraduate Mathematics Education:**

A Way to
Get Started

**Ed Dubinsky**

Georgia State University

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 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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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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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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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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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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- 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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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.

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