Report to the ICMI Study on Geometry:

**Perspectives on the Teaching of Geometry for the 21st Century**

Professional Aspects

by David W. Henderson

Department of Mathematics

Cornell University

Ithaca, NY 14853-7901, USA

e-mail: dwh2@cornell.edu

I am writing this based upon my 30 years of experiences at the university, observing, listening to, and working with students and teachers of geometry. I am focusing on geometry which is what I know most about, but perhaps some of my remarks are applicable also to other parts of mathematics.

I am also writing from my own experiences with geometry. I have observed the geometry of the world around me ever since, when I was 6 years old, I drew a picture of what a room looked like to my cat. However, I did not realize that the geometry which I loved was a part of mathematics. In school mathematics was almost entirely calculations and memorizations -- it seem dead. I especially did not like my high school geometry course with its formal two-column proofs. At the university I majored in physics and philosophy and took only those mathematics courses which were required for physics students. I became absorbed in geometric-based aspects of physics: mechanics, optics, electricity and magnetism, and relativity. My first mathematics research paper grew out of a philosophy course where I became interested in the geometry of Venn Diagrams for more than 4 classes. There were no mathematics courses at this university in geometry except for analytic geometry and linear algebra which only lighted touched on any thing geometric. It was only in my fourth year at the university that I realized that the geometry which I loved was also a part of mathematics. This is not an uncommon story among research geometers.

*What do the universities need most from school geometry teaching?*

Answer: Universities need HUMAN BEINGS --

- All kinds of human beings.

- Human beings who are excited about geometry and who value their own deep experiences of geometry and the geometries of their cultures. Human beings who have developed and are confident with their own
*alive geometric reasoning.*

*What is alive geometric reasoning?*

I take the word "alive" from the famous mathematician David Hilbert who wrote:

In mathematics, as in any scientific research, we find two tendencies present. On the one hand, the tendency toward *abstraction* seeks to crystallize the *logical* relations inherent in the maze of material that is being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency toward* intuitive understanding* fosters a more immediate grasp of the objects one studies, a live *rapport* with them, so to speak, which stresses the concrete meaning of their relations.

- It is "living proofs", that is, convincing communications that answer -- Why?

- It is paying attention to meanings behind the formulas and words -- meanings based on intuition, imagination, and experiences of the world around us.

Do not just pay attention to the words;

Instead pay attention to meanings behind the words.

But, do not just pay attention to meanings behind the words;

Instead pay attention to your deep experience of those meanings.

- It is knowing that geometric definitions, assumptions, etc. vary with the context and with the point of view.

- It is using a variety of geometric contexts: 2- and 3-dimensional Euclidean geometry, geometry of surfaces (such as the sphere), transformation geometry, symmetries, graphs, analytic geometry, vector geometry, and so forth.

- It is combining geometry with algebra, number systems, probability, and calculus.
- It is applying geometry to the world of experiences.
- It is using physical models, drawings, images in the imagination.
- It is making conjectures, searching for counterexamples, and developing connections
- It is always asking -- Why?

*What will students with alive geometric reasoning find at the university?*

- Almost all geometry is taught in courses which are:

- Most professors know little geometry.

How is the situation of geometry in the university going to change?

It is going to be changed by human beings coming to the university who are excited by geometry and by alive geometric reasoning. If they can not be excited by geometry and alive geometric reasoning within mathematics then they will take their geometry elsewhere, for example to sciences, engineering, computer graphics, robotics, architecture, design, ... -- this is what I almost did. Look around in the universities that you know -- where is geometry most alive? In my experience, geometry is usually not most alive in the Department of Mathematics. There may be general exceptions to this in the universities of China and the former Soviet bloc where strong, alive geometric traditions have survived and flourished.

Elsewhere, geometry is gaining in importance in mathematics departments. For example, at my university (Cornell) there are now 8 undergraduate geometry courses and only one half of one of these is primarily based on formal axioms. In addition, the National Science Foundation (USA) is sponsoring week-long workshops at Cornell for mathematics professors to learn geometry and new ways of teaching it.

*Free geometry from the restrictions of formal deductive systems!*

Confining geometry within formal deductive systems is harmful because:

- Much interesting and useful geometry is either not taught at all or is presented in a way that is inaccessible to most students.

- Many important and useful questions are not asked.

- Students are being harmed.

- Mathematics is being harmed.

- Formal deductive systems usually do not gain for us the certainty that we strive for.

I plead with all who teach geometry:

Free geometry

from the restriction of

formal deductive systems,

replace these systems with

alive geometry reasoning

at all levels of schooling,

and let Geometry

LIVE!