Report to the ICMI Study on Geometry:
Perspectives on the Teaching of Geometry for the 21st Century
Geometry from School to University:
by David W. Henderson
Department of Mathematics
Ithaca, NY 14853-7901, USA
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 --
Including more women and more persons from racial and cultural groups that are now underrepresented in mathematics. There is research evidence that successful learning takes place for many women and underrepresented students when instruction builds upon personal experiences and provides for a diversity of ideas and perspectives. See, for example [Belenky et al, 1986], [Cheek, 1984], and [Valverde, 1984].
- All kinds of human beings.
In my experience these are the qualities in the students who are attracted to geometry and who are the most creative with it. These are also the students who can step back from their individual courses and see the underlying ideas and strands that run between the different parts of mathematics. These are the students who become the best mathematicians, teachers, and users of mathematics.
- 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 not formal 2-column proofs -- computers can now do formal proofs in geometry. What we need are alive human proofs which:
- It is "living proofs", that is, convincing communications that answer -- Why?
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 what to have explained. As an example, my shortest research paper has a very concise simple proof that anyone who understands the terms involved can easily follow logically step-by-step. But, I have received more questions from other mathematicians about that paper than about any of my other research papers and most of the questions were of the sort: "Why is it true?" "Where did it come from?" "How did you see it?" "What does it mean?" They accepted the proof logically but were not satisfied -- it was not alive for them.
It is not memorizing formulas, theorems, and proofs -- this is again something that computers can do. We, as human beings, can do more. As Tenzin Gyatso, the fourteenth Dalai Lama has said:
- 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 not definitions and assumptions that are fixed in a desire for consistency. It is an observable empirical fact that mathematicians and mathematics textbooks are not consistent with definitions and assumptions. We find this true even when the general context is the same. For example, I looked in the plane geometry textbooks in the Cornell library and found 9 different definitions of the term "angle". Also, calculus textbooks do agree on whether the function y = f(x) = 1/x is continuous are not continuous; and analysis textbooks have many different axioms for the real numbers that have different intuitive connections and necessitate different proofs.
- It is knowing that geometric definitions, assumptions, etc. vary with the context and with the point of view.
It is not Euclidean geometry as a single formal system. When a mathematician is constructing a proof that needs a geometric argument she/he is free to use whatever tools work best in the particular situation. Mathematicians do not limit themselves in this way. Also, those who use geometry in applications, do not feel restricted to a single formal system.
- 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?
axiomatic and formal -- Euclidean geometry, hyperbolic geometry, topology, or
- Almost all geometry is taught in courses which are:
mostly algebra -- linear algebra, algebraic geometry, or
mostly analysis -- differential geometry.
In most countries there are few university-level courses in geometry and very few mathematicians who research specialty is geometry. This seems to be a phenomenon only of this century. At the end of the last century mathematicians explicitly attempted to rigorize and formalize all of mathematics by weeding out reliances on geometry and geometry intuition.
- 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:
For example, spherical geometry (the geometry of the surface of our planet, the geometry of visual perception, the geometry of astronomical observations) is almost entirely absent from our courses and textbooks, apparently because it does not have a convenient formalization. Also, differential geometry (the geometry of curves and surfaces, the geometry of the configuration spaces of mechanical systems, the geometry of our physical space/time) has extremely difficult formalisms which make it inaccessible to most students and even, I suspect, most mathematicians are uncomfortable with the formalisms of differential geometry. When freed from the confines of formal systems it is possible to present spherical and differential geometry in ways that are based on geometric experiences and intuitions and that are accessible, but yet in ways that are based on alive proofs in the sense of convincing communications that answer -- Why?.
- Much interesting and useful geometry is either not taught at all or is presented in a way that is inaccessible to most students.
For example, the reliance on a formal Euclidean deductive system rarely allows for questions such as "What do we mean by straight?", "Why is Side-Angle-Side true on the plane (but not on the sphere)?", "What is the geometric meaning of tangency?", and "How do we experience the connections between linear algebra, transformations, symmetries, and Euclidean geometry?".
- Many important and useful questions are not asked.
When a student's experiences lead her/him to understand a piece of geometry in a way that is not contained in the formal system them the student is likely to lose confidence in their own thinking and understanding even when it is backed up by alive geometric reasoning. For example, almost all deductive systems for Euclidean geometry will not allow a student to understand (prove) that the opposite angles formed by two intersecting straight lines are congruent because half of a full revolution about the point of intersection will (because of the symmetries of a straight line) take one angle onto the other. Or, a student is not allowed by formal deductive systems to see the truth of the commutative law for arithmetic multiplication by observing the geometric relationship between an a-by-b array and an b-by-a array. Deductive systems do not allow for alive geometric reasoning which (in my experiences with students and teachers) is a natural human process and thus they serve to deaden human beings whose thinking and understandings are forced to reside in these systems. We now have machines that can do the computations and formal manipulations of deductive systems, what we need more of now is alive human reasoning.
- Students are being harmed.
Historically, most current-day mathematics was based on geometric explorations, geometric reasonings, and geometric understandings. The developers of our current deductive systems in algebra and analysis explicitly attempted to week out all references and reliances on geometry and the geometric intuitions on which the algebra and analysis was originally based. When we confine mathematics to these formal systems we teach the students to distrust geometry, not to value it, and not to use their intuitions in understanding mathematics. Many, many students who have a natural interest in mathematics are lost to mathematics by this process -- I almost was.
- Mathematics is being harmed.
Formal deductive systems are useful and powerful in some circumstances, for example, in deciding which propositions can be logically deduced from other propositions and whether certain processes or algorithms will always produce the expected result. But, these deductive systems only give us certainty that certain steps (that can in principle be mechanized) can be carried out. They usually do gain us certainty for the human questions of "Why?" or the human desire for experiencing meanings. (See the quotes above from Hilbert and Tenzin Gyatso, the discussion above of my shortest research paper, and the discussion of how students are harmed.)
- Formal deductive systems usually do not gain for us the certainty that we strive for.
I plead with all who teach geometry:
from the restriction of
formal deductive systems,
replace these systems with
alive geometry reasoning
at all levels of schooling,
and let Geometry