Questions from the Oberwolfach conference on
New Trends in the Teaching and Learning of Mathematics,
27 November through 1 December, 1995

The following research, curricular, and pedagogical questions arose in response to the presentations given at this conference. They represent some of the important issues and problems that the participants jointly agree should be studied. The list is far from complete and should not be interpreted as an attempt to put forth a research program or agenda. Neither is it claimed that these questions are original or of equal importance. We only hope that they will serve to stimulate workers in the field to obtain new results and to improve the learning of mathematics by students throughout the world.

  1. What are appropriate methodologies for answering curricular and pedagogical questions?

  2. Are learning theories transferable across cultural and subject matter boundaries? Can they be applied to different topics and different groups of students in different countries?

  3. What are the different learning styles for mathematics that are prevalent among post-secondary students? How do these learning styles relate to various theories of learning? How immutable is the learning style of an individual student?

  4. What are the differences between how mathematics is learned by experts and by novices of different kinds?

  5. What do faculty and students mean by the word "understanding"? What is meant by "clarity"? What is the relationship between clarity and precision in the minds of students and faculty?

  6. Do the tools of technology change students' understanding of mathematics, and if so how? For example: some people argue that learning geometry with a software package does not promote the same understanding of geometry as learning in a paper and pencil environment. How can we transform this claim into a research question and what methodology can be developed to investigate this question?

  7. What are the student conceptions of the different notions of equality and approximate equality? How are these conceptions affected by technology?

  8. What are the difficulties that students have with formal mathematical language such as the use of "for all," "there exists," two-level quantifiers, and negation, and with the relationship of formal mathematical language to everyday language?

  9. In what ways is the concept of a solution to a differential equation difficult? What is the nature of that difficulty? In particular, what is the nature of the difficulties in understanding—symbolically, graphically, and visually—what it means to be a solution to a differential equation or initial value problem?

  10. What pedagogical strategies can be effective in helping students understand the systematic development of mathematical theories?

  11. How can we most effectively teach students to use definitions as a mathematician does, and in particular to turn a definition into "an operative form"?

  12. What is the relationship between time spent on mathematics outside of class and the level of student understanding? What pedagogical strategies are most effective in improving the quantity and quality of the time students spend on mathematics?

  13. What course designs and pedagogical strategies are most effective in taking into account the wide range of abilities and backgrounds of the students?

  14. What are the pedagogical advantages and disadvantages of the different ways in which technology can be used? Among these are visualization, the use of built-in mathematical tools, and programming.

  15. How does class size affect learning? How is this affected by technology and cooperative learning? What group sizes in cooperative learning best support learning?

  16. What are the advantages and disadvantages of using applications from both inside and outside mathematics and of using history? Do they improve the students' retention of the mathematics and/or the retention of the students in mathematics? What is their effect on understanding, and the appreciation of mathematics both for its internal beauty and its usefulness?

  17. What form or forms of proof are appropriate in different contexts for student learning and how should they be dealt with pedagogically?

  18. What algebra is appropriate as preparation for post-secondary work? How is the answer affected by subject? How is it affected by technology?

David Bressoud
Urs Kirchgraber
Ed Packel
Bill Barker
Ed Dubinsky
Werner Hartmann
Lisa Hefendehl-Hebeker
Wolfgang Henn
Reinhard Hoelzl
Deborah Hughes Hallett
Hans Niels Jahnke
Dan Kennedy
Heinz Klemenz
Colette Laborde
Hans-Christian Reichel
V. Frederick Rickey
Werner Schmidt
Inge Schwank
David Smith
Anita Solow
John Stillwell
David Tall
Bernd Wollring