Giving Professors Permission to Change Their Teaching

by

David W. Henderson

Department of Mathematics

Cornell University

Ithaca, NY 14853-7901, USA

dwh2@cornell.edu

There is much talk these days about changing university mathematics classes to include less of the traditional lecture/exam mode of teaching/learning and more of writing assignments, cooperative learning, problems-based curriculum, alternate forms of assessment, and, in general, to make the mathematics courses reflect the living activity of mathematicians -- conjecturing, problem solving, communicating, exploring ideas, and using multiple sources for methods. Most of us do not teach advanced mathematics courses in ways that reflect our own activity of doing mathematics; nor do we build classroom atmospheres which encourage students to be active in mathematics parallel to the ways that we are active in mathematics. To a very large extent, we seem to teach mathematics in the same way that we were taught mathematics. To change is not easy for any of us. How do we break this cycle?

I will give a brief description of several efforts that I am involved in that are attempting to help with different aspects of this issue. I do not see these as a solution to the problem but only as one way to start the exploration for what I expect will be numerous solutions. In what follows I am talking about university mathematics course at the levels above calculus -- courses that are taken by mathematics majors, by future teachers of mathematics, and by users of mathematics in engineering and the sciences. There has already been much talk about "reforming" the teaching and learning of calculus.

Giving Permission.

Most important of all. We all need to know that it is OK for us to explore and try out different teaching strategies. There has been many articles suggesting different approaches and some university and even governments are encouraging such changes, but words and decrees do not seem to work in many cases. To require change of any individual teacher can backfire and cause resentment and lead to only token involvement in the new strategy, with disappointing results. In addition, we must recognize that there are many successful strategies and that each person must find what works best for them. They must be comfortable with it and believe in it. So I believe that any attempt at change must allow the individual teacher to explore and change slowly by trying out and experiencing different methods. Much of this permission must come from the university or one's colleagues, but there are other sources that can help. These sources include changing views of what is the nature of mathematics, new texts that model different approaches to the content and pedagogy; and include workshops in which the faculty can experience different approaches.

Changing Views on the Nature of Mathematics and Proof.

Almost all of our upper-level university textbooks and courses in mathematics have followed the traditional definition/example/theorem/proof/exercise model. The mode is based on the Formalist view of mathematics that gained dominance in mathematics in the first half of this century. The formalist view of mathematics was codified in the Bourbaki volumes and in the 1960's and 1970's was instrumental in the worldwide push (by the ICMI and others) to base school mathematics instruction on the "New Math". More recently there have been many calls to change this model -- for example in the USA, William Thurston, "On Proof and Progress in Mathematics", Bull. Amer. Math. Soc. 30 (1994), 161-177 and Rueben Hersh, What is Mathematics, Really?, Oxford University Press, 1997. Now, I do not see any mathematicians (other than some working in logic, foundations, or computer science theory -- whose job it is to study formalizations of mathematics) advocating Formalism as an adequate description of mathematics and mathematics activity. But yet we still teach (for the most part) in the Formalist-based definition/example/theorem/proof/exercise mode.

Central to our view of what distinguishes mathematics from other human endeavors is the notion of "proof". In most texts and upper-level courses, I find that a Formalist view of proof still dominates even though it, I believe, does not describe how most mathematicians actually consider proofs in their every day work. In addition, the formal notions of proof have a deadening effect in the classroom. I find in my own teaching that the most important step to take is to reexamine what we mean by "proof". I propose the following definition as being closer to the way that mathematicians actually work and closer to how we want our students to work:

Proof is convincing communication that answer -- Why?

It is not formal 2-column proofs -- computers can now do formal proofs in mathematics. What we need are alive human proofs which:

are communications -- when we prove something we are not done until we can communicate it to others and the nature of this communication, of course, depends on the community to which one is communicating and is thus in part a social phenomenon.

are convincing -- a proof "works" when it convinces others. Of course some persons become convinced too easily so we are more confident in the proof if it convinces some one who was originally a skeptic. Also, a proof that convinces me may not convince you or my students.

answer -- Why? -- The proof should explain, especially it should explain something that the hearer of the proof wants to have explained. I think most people in mathematics have had the experience of logically following a proof step by step but are still dissatisfied because it did not answer questions were of the sort: "Why is it true?" "Where did it

come from?" "How did you see it?" "What does it mean?".

I find that using this definition of proof has changed my teaching and changed the ways in which my students learn and desire to do proofs and thus changes the ways in which my students learn mathematics.

New Texts that Model Different Approaches to the Curriculum.

With the support of a National Science Foundation (NSF) curriculum development grant, I have written two new texts, Experiencing Geometry on Plane and Sphere (1996) and Differential Geometry: A Geometric Introduction (1998), both published by Prentice-Hall. The first text is for a third year course for perspective teachers and mathematics majors and the second is for fourth year majors. In both texts the main development of the content is through the problems (some of them open-ended). The students are encouraged to experience the meanings of the content and then to find (with hints) their own proofs for the key results. In both texts the working notion of proof is "A convincing communication that answers -- why". The material is not presented in logical or axiomatic order but rather in an order dictated by basic intuitions and geometric understandings.

I have heard from many faculty that though they do not follow the text they have been encouraged to try their own problems and their own ways of encouraging their students to be active in the creation of their mathematical understanding instead of passively learning what the professor lectures.

Providing Experiences in Workshops.

Besides numerous short workshops of one and half hour to a full day in length. The National Science Foundation in the USA has supported Undergraduate Faculty Enhancement (UFE) workshops. I (with three others, Kelly Gaddis, Jane-Jane Lo, and Avery Solomon) have lead four of these week-long workshops for university faculty on the teaching of upper-level undergraduate mathematics. They have continued to be popular and to attract more applications from faculty in the USA than we can accommodate. In our UFE workshops we provide opportunities for university mathematics faculty to:

1. experience new teaching strategies and techniques emphasizing the use of writing to foster continuing dialogue between teacher and student, the use of cooperative learning, the use of hands-on exploration, and the use of challenging and open-ended problems and projects.

2. experience and/or see examples of non-test based assessment schemes; including portfolio assessment, and writing assignments which foster both understanding and creativity, and dissolve the line between instruction and assessment.

3. learn about and observe these and other examples of pedagogical techniques that have been successful in encouraging women and underrepresented minorities to excel in mathematics; in particular, how to encourage diverse ideas and build upon student experiences.

4. have the opportunity to interact and share with colleagues and experts who have common interests in the teaching of mathematics to future school teachers,

5. develop plans to incorporate both teaching techniques and new curricular materials into their own courses with the help of other participants and the project staff.

6. be committed to participate after the workshop in several of the following follow-up activities: use materials and techniques from the workshop in their own teaching, give a presentation/workshop on the new techniques and materials to faculty in their home institution, share their own classroom experiences and materials with the workshop staff and other participants via e-mail, and use the workshop staff as a resource to assist in the implementation and evaluation of the new materials and techniques.

Summary:

I consider the most important aspect of the above is that they give faculty the permission to explore and search for their own ways to make the teaching/learning of advanced mathematics more human and more in line with the way that mathematicians actually work. An important part of this is the modeling of the definition:

Proof is a convincing communication that answers -- why?