Geometry Autobiography.
Walter Whiteley Updated September 2006
Like many of you, I took my last 'formal' geometry course back in high school. (In my case,
Ontario Grade 13, Analytic Geometry in the early 60's. Perhaps you took Algebra and Geometry
in OAC?) I also had a formal 'Euclidean Geometry' course in grade 10, complete with 'ruler and
compass' constructions and proofs. Such courses are not as common these days.
As an undergraduate math major (physics minor) I saw a few pictures (in calculus, analysis,
topology ... ) but there were no courses in 'geometry'. An indication of the 'algebraic' slant of my
undergraduate training is that text for the year long linear algebra course had exactly one picture
in it! As a result of this training (including how the high school geometry course was taught) I
had lots of questions about logic, about proofs etc., but did not even think about geometry, or
visual reasoning.
I went to graduate school to study logic - and took many logic courses (maybe ten?) and no
geometry courses. I had some questions - like: Why was `projective geometry' considered a
'dead' subject? What makes one part of mathematics important or something to be left behind?
However, after three years of graduate work, I lost the questions I had about 'proofs' (the
professors did not want to talk about my issues) and met a professor who was interested in
almost everything. In particular, he had some interesting questions about the form and the
content of theorems in analytic geometry. (He also had reasons why he thought these questions
were important.) I did my thesis on aspects of the foundations of analytic geometry -- and had to
buy a Schaum's Outline of Projective Geometry to get started! (Of course, my thesis was on a
branch of mathematics that the sociologists were then describing as the example of dead
mathematics! They were wrong, as predictors of the future, but right as people describing the
previous four decades.)
After graduation I knew I wanted to teach, but I was not sure I wanted to do research. (I still had
lots of questions - but not ones many other people shared. Who can keep working on questions
for which nobody else will care about the answers?) So I went to teach in a Quebec CEGEP
(equivalent to final year high school and first-year university in Ontario). For 20 years, with
occasional breaks to 'visit at universities and teach first-year courses', I taught at the college: precalculus, calculus, linear algebra ... . I found I was using more and more pictures: images to
visualize and explain how I thought about the math. Often my head was filled with an 'animated
movie' that was difficult to show to my students. The effort to ‘think visually’ and communicate
this has been with me ever since.
I also met an architectural engineer from the Université de Montreal who had lots of interesting
geometry questions. I could answer a few of these, from the work I had done for my thesis - and
I joined a group to work on some of the other questions. Essentially the questions concerned the
relationship between shapes in 3-space which are flexible when you expect them to be shaky and pictures of four dimensional objects. Three things made this experience important:
1 there were other people to talk to, to share the questions, and the excitement of conjectures and
answers;
2 there were tangible 'applications' - I could build actual models, and explore the ideas in
examples in my hands and eyes;
3 there was lots of pretty geometry: I used everything I knew and had to learn lots more.
I still work with these people, as well as a number of other people around the world whom I
met through this geometry. (My son has observed: do mathematics and travel the world!)
I started to do geometry research again - and have continued ever since. After a while, there
were Quebec government grants which reduced my teaching hours and gave me more time for
research. I broadened my research into some other areas of 'discrete applied geometry': areas
related to Computer Aided Design, Geometric Modeling, Robotics, etc.. I also worked with
people in Montreal who are doing research of spatial visualization, pedagogy of geometry, etc.
(including the same architectural engineer who taught 'descriptive geometry' for over 30 years).
Just over fourteen years ago, I was offered to chance to come to York, to teach mathematics
(including geometry - for my first time) and to continue my research. I have taught this course,
and variations on it, a number of times now. I have lots of mathematics I would like to explore
with you, as well as some pedagogical ideas I am working with. I hope to learn more geometry
(and more pedagogy) this year and anticipate you will have ideas and questions that I will learn
from.
In my research, I have learned lots of geometry. In my travels doing research, I have met lots
of people outside of mathematics who have questions about geometry - but know even less
than I do. [People in engineering, computer science, design, robotics, biology, philosophy,
medicine, psychology, biology, etc.] I still have lots of geometry questions of my own (more
questions every year). We will explore this – and try to develop your ability to ask and refine
your own questions. This is key to ‘thinking mathematically’.
In my research, I have used the computer for geometry in a variety of ways. I used the
standard `drawing' programs (on the Mac) to illustrate my ideas and to help my thinking as I
presented the constructions. I learned some of the program Maple to help with the algebraic
manipulation (analytic geometry) and in the process learned some of its plotting features. More
recently, I have been using programs such as The Geometer's Sketchpad to investigate
geometry problems, and to explore some geometry with undergraduate mathematics students.
Because of my research, I visited Berkeley California in 1966 and worked for three weeks with
the developers of The Geometer's Sketchpad (and some Spherical models I will show you next
class). I am very interested in the use of such tools, and manipulatives (which I use in all my
geometry courses), so that people can explore the ideas, make conjectures, and 'own' the
material with {\it their} questions, rather than just mine. (However I don't leave my own
questions at the classroom door.)
I am pleased that I managed to anticipate changes in geometry teaching - as Geometers
Sketchpad is now required for all grade 9 classes (and encouraged for all years) in the new
Ontario Curriculum. We can have a class discussion about the good and bad ways to use such
software. To me, these programs give an opportunity to change what we 'see' in geometry (and
in other mathematics) in important ways. If you are interested in more of my thoughts on this,
there are some related papers on my main web page that you can download. We will talk more
about 'seeing mathematically' during the course.
In the last few years, I have also been involved in consultations with the Ministry of
Education around the Ontario Curriculum. I would say that the geometry in high school is very
weak and disconnected (nothing left now in Grade 11, for example). There is some interesting
pieces of geometry in k-8, but it does not build to a coherent reasoning whole, and does not
attach to what little is done in high school. In the first two years of university, there is also
very little or no geometry. We can do a lot better, and I would be interested in talking further
about this.
I am also involved in a funded project looking at the preparation of elementary teachers in
visual and spatial reasoning. Since we will also be working in this area during the course, it is
likely that we will adapt some of the activities for this research for use in our class. One reason
for this is that I am convinced that: Spatial Reasoning is essential to many areas of problem
solving; and these are skills that can be ‘lost’ during the years of k-12.
Throughout the years, dating back to my high school days I have wondered about 'proofs', how
we decide which mathematics is 'true' and which is 'important'? Over time, I have concluded
that ‘formal proof’, while important, is over-rated. I have continued to read on the 'philosophy
of mathematics' and reflect on how I (and others around me) do mathematics. I am interested
in an `inquiry' approach to this course because I think it will be good for us, and because I
should practice the approach I ‘preach' for teaching high school students. In particular, I have
lots of ideas and questions about the potential for diagrams, objects and pictures inside
mathematical proofs.
I believe that mathematics, and mathematics education, are undergoing a 'paradigm shift'
about the essential role of diagrams in and visual tools in learning, doing and communicating
mathematics. At some point, we should have a discussion (debate) about what makes a good,
convincing 'proof' in geometry, in algebra etc. Again, I talk some about this in the papers on
my web site - talks I gave to audiences of mathematics educators. I will also hand out readings
related to this.
Since I insist you offer some of your geometry questions, let me close with a few of my
questions.
Here are some examples of questions that I have been working on. (I have been thinking about
it for about 15 years, and worked on this with undergraduate students over three summers. The
Federal government does give research grants for such work through NSERC!)
Remember how you check whether two triangles are congruent?
SSS: three sides equal;
SAS: two sides and the enclosed angle equal;
ASA: two angles and one side equal;
SSA: two sides and the non enclosed angle equal - was ambiguous - but, for example, if both
triangles were acute this was enough;
We never needed four facts - only three, and we never used three angles. (Three angles only
guarantees similarity - it is the same as having only two angles equal.)
Question: What data should you use to check that two sets of four points (quadrilaterals?) are
congruent?
How many facts? Which patterns of angles and lengths? What about a thousand points? This
problem has applications in computer science, and in control theory in electrical engineering,
as well as CAD/CAM.
Here is a second, easier geometry problem I play with:
Question: How many fingers do you need on a 'Robot hand' in order to be able to grip and
control a solid object, if you assume the fingers are like 'claws' and there is no friction?
Note that five is the wrong answer!
Here is a third question that I worked on over the last five years, funded by the US National
Institutes of Health. The answers are useful in studying drug docking and other issues of
disease.
Question: What portions of a protein are rigid, and how does it flex? Does a mathematical
model of a protein as a structural framework explain how it folds and how it functions? (E.g.
Mad Cow Disease is caused by a protein - prion - unfolding and folding into a different shape
which becomes a rigid plaque.)
Question: How much of my own reasoning in mathematics is based on my senses (visual and
kinesthetic) and my experiences with these senses? How do people learn to work with 3-D
symmetry? Do we need to work with 3-D objects in order to later understand computer
animations and then to reason 'in our minds eye'?
If you are interested in learning more about the kinds of applied geometry I do, talk to me.