Alan Thanks for the detailed response. Yes your class sounds much better than the image of 'proof in Euclidean Geometry' would suggest, at first.
From the point of view of exploratory geometry, the book I use (including for some of our top mathematics majors - students headed for graduate school, sometimes even in geometry as a research area) is David Henderson: Experiencing Geometry on Plane and Sphere. I consider learning to ask the 'geometer's questions' to be more important than learning to give a formal proof. I do emphasize the use of the hands on and visual, in these university classes (and classes for in service high school teachers) as well as in my own research. OF course, one really wants a balanced approach, with multiple ways of understaning a problem and a topic, all intereconnected in a consistent whole. Unfortunately, the visual has been thoroughly discounted, treated as something you get, or don't get, akin to 'intuition' - not as something that needs to be taught and developed as carefully as, say, algebra. In the spirit of asking questions and making conjectures, I would strongly suggest a component of dynamic geometry software. These activities bring out the best in many students - sometimes turning over the usual 'ordering by ability' of students in a geometry class. The software also gives my an opportunity to make visible to the students some of the visual thinking which I do myself, letting them see more of what is behind the questions I ask, the ways I solve geometry problems, etc. It also gives them an opportunity to develop such work with dynamic visual images in their own minds, while not ingoring the importance of matching such thinking with clear communication about what those processes are, and how the results are convincing.
I would mention, again, the importance of transformations in any approach to geometry. While they were not a big theme of classical Euclidean Geometry, they are, of course, at the very roots of congruences and even measurements. However, since the time of Klein's Hierarchy of Geometries, they have assumed a clearer role as a key organizing theme of work in geometry. In my own experiences working on applications of geometry (robotics, structural engineering, CAD programming, protein modeling, ... ) the issue of 'which geometry' has been key to understanding how to attach problems. The answer to 'which geometry', in turn, comes from understanding how various transformations effect the problems in question.
I note that I did, originally, train in formal logic (about 10 courses at graduate school at places like MIT and Harvard). It has its place - but to often teaching organized around the goal of teaching students to do formal proofs seriously distracts from actually thinking about the questions and methods of geometry per se. So people neglect transformations because they are not as easy to write up as formal proofs (at least for typical high school students). The result, for good students, may be something like my experiences in school - I learned I liked geometry, and only a decade later ALSO learned that I liked geometry - when I finally got to see geometry!
Walter Whiteley York University.
Hi, thanks for taking time to respond. Let me respond. > I just wanted to day how saddened I am by the proposed constraints > within which a geometry book should be 'good'. Sorry for the term 'good'! I deliberately put this in quotes to show that this was a subjective opinion. I am looking for a book that stresses formal proof so this is 'good' for me. A person who was going to teach art students would probably want a different book (and so would I!)
> Formal proof IS a piece of mathematics - but only a piece. Yes this is quite true. There is some proof in the algebra I teach (the students explore and derive everything in class themselves) but I wanted to have proof presented in a formal mathematical way and in my opinion geometry is a good place to do this (less abstract than building algebra from the ground up)
> I hate to see a classroom that tastes like a medicine cabinet rather > that the life of mathematics. You should see my class. I am teaching an algebra 1 class for ages 11-13 where we meet once a week for 3 hours and the students are having a great time. One mother told me a story where her son told her about a nightmare he had where in his bad dream my math class had been replaced by soccer! The parents are estatic that math can be so fun. Half of the students are doing extra homework just because they love it (and the regular load of homework is around 160-200 problems a week and we are using a college text). I was sceptical that students between the ages of 11-13 could pay attention for 3 hours, but the energy in the class is fantastic and they are able to stay on top of it. We have a lot of fun. For example, we were discussing Cantors proof why the rational numbers are the same order of infinity as the positive integers and the students were abel to derive a symbolic formula of how to calculate the integer pairing for any square in the table. And they did this in about 30 minuets! Wow I was impressed. And I could go on and on. (By the way, this is not a special upper track math class, I have a wide range of student's)
>It is particularly sad when a rich > and important subject like geometry is pushed into a box of > formal proof. Probably means very little transformations > (probably the key part of applications of geometry) and very > little 3-D geometry - at least until the students have been > so saturated in plane games that they have lost touch with > the world they live in. A book is a spring board for a teacher like me. My whole class is taught socratically so the students are doing a lot of thinking out of the box all the time. We are constantly doing puzzlers and brain teaser that are woven into class. My point is that I start with the book and do a lot of enrichment way beyound the scope of the book which would be 3-D geometry, etc. I could go on about my philosphy of teaching, but what matters to me is what works for me as a teacher and produces results. When I say results, I mean that I am expanding the students minds to think creativly out of the box visavie math, have a firm understanding of why things work they way they do (so they are not just memorizing rules), they are getting excited about math and be prepaired to do problems (lets say problems from a physics textbook). I have found that the exploritory textbooks that for example Key Curriculum Press publishes, don't work for me and therefore don't work in my classroom. But when I use the 'medicine cabinet' books I get results as defined above. Remeber what works for me might not work for the next teacher. Teaching is an art, not a science and there is room form many different and divergent aproaches to math. And I firmly believe that there is no one right approach. And the proof is in the pudding! (pun intended)
> I recently saw a paper about testing students in grade 7, 9, 11 > on some pencil and paper exercises about 3-D mirror symmetry. > (The kinds of things that are central to say, stereo chemistry.) > Students got WORSE in these exercises as they proceeded through school! > Sad indeed when we teach understanding out of students. There are many > geometry courses which accomplish that sad task.
Ah yes, the public schools are really butchering math and the students at the same time. No wonder our universities are crammed with students for other countries. I have taught in the public schools but no longer. I love the freedom of teaching outside the system.