Designing the Secondary Math Curriculum: What form should a geometry course for prospective secondary teachers take?

From: Lou Talman
Subject: Geometry Course for Prospective Secondary Teachers
Date: 16 May 1995 16:25:56 -0400

I am currently developing a new geometry course aimed at fulfilling the State of Colorado's certification requirement (for mathematics in secondary ed.) of a three-hour course in geometry. (Of course, there're a number of other requirements it should also meet -- like being useful to practicing teachers later in their careers...) I'm doing this under the auspices of the Rocky Mountain Teacher Education Collaborative, an NSF-funded consortium comprising Colorado State University, the University of Northern Colorado, and my own institution, Metropolitan State College of Denver. Its goal is the improvement of teacher education across the board.

My course is supposed to incorporate current knowledge of the way students learn, including collaborative learning, and to use current technology such as Geometer's Sketchpad or Cabri. The course is to be offered at the junior level; I'm to offer it for the first time in the fall of this year.

• What, in the opinion of the members of this forum, belongs in such a course?
  What doesn't?

• What do practicing secondary teachers think of the geometry courses they took (should have taken)? What did those courses do right? Wrong? What did they supply? What should they have supplied?

• What do practicing teacher educators say about the same things?

• What do practicing geometers say about the same things?

• What do practicing mathematicians who are not particularly interested in geometry as mathematics say about the same things?

• What are the (de-)merits of Sketchpad vs. Cabri? Are there particularly striking (or even moderately important) things one can do with one but not the other?

• What other technology is available?

• What forms of collaborative learning do Forum members have (un-)favorable experience with? What else do we know about the ways in which students learn that might be of interest in building a working solution to this problem?

• What other relevant things have I forgotten to ask? What are the answers to those unasked questions?

From: Walter Whiteley
Subject: Re: Geometry Course for Prospective Secondary Teachers
Date: 16 May 1995 18:40:23 -0400

In response to Lou's query:

A Geometry Course for High School Teacher Candidates

I just finished teaching a year-long course for a group of current high school mathematics teachers. For the first half of the course, I had an additional group of teacher candidates (Concurrent Education) and Math Majors who are considering a further program in Education. By the way - I use a large definition of geometry - including some topology of surfaces, Euler's formula, many visualization issues. [Where else will they see this?]

I have a couple of strong suggestions - and some other possibilities:

• I would definitely use one of Sketchpad or Cabri. (I used some Sketchpad, but not enough). I think the ideal setting would include a number of weeks in which one class would be held in a computer lab (two per machine) with items to explore.

• If the equipment will support it, I would get them onto the internet in general (and the Geometry Forum in particular). About half my class got into this, but others had inadequate access.

• I would use lots of physical manipulatives and devices - lots of emphasis on visualizing, playing with objects, asking 'what if ... ? etc.. Among others, I used Polydron (for making polyhedra and simple plane tilings), spheres (for spherical geometry - see below), sticks, string, clay, paper to cut and fold, other materials for 3-d examples. [One reason for me to do this is to model this behaviour, so they do this in their classrooms.]

• For part of the course, I used a preprint version of a book that will be published this summer: Experiencing Geometry, by David Henderson. It was primarily about spherical geometry - with the explicit exploration of which plane properties apply to the sphere, and which change. Lots of emphasis on writing, playing with physical models (what is a 'straight line on a sphere? Why? Which properties, local, global symmetry, etc. are being used?) I would use it again - it re- explores SAS etc. (including AAA). I would probably supplement it with more expamples of 'plane proofs' and what generalizes. This time, I took the Forum discussion on 'symmetries of a quadrilateral' and asked them to apply this to the sphere. [I could send you a version of this assignment if you wish.]

• For this you must have physical spheres. Key Curriculum Press is working on some nice plastic models. (You might contact them for information - they let people 'borrow' a trunk of them for several weeks at a time this year.) I made reasonable ones out of craft store 'xmas decorations' - clear spheres (cut off the hanging tabs) and clear tori as bases. A cut-off large yogurt container fit over a hemisphere as a 'spherical ruler' for drawing with overhead transparencey pens. Alternatives some students used included elastics on tennis balls.

• I used projects - presented to the class and written up for me (after the presentation). This allowed a variety of interesting topics to be introduced [4-D and the hypercube, knot theory, plane symmetries, icosahedral symmetries, visualization and geometry, quasi-crystals, the 4-colour problem, finite geometries, axioms for projective geometry, rigidity and projective geometry, projections of spheres and map making, ... ]. This experience will assist them in asking their students to do projects.

• I emphasized asking questions about geometry and asking geometric questions. Every assignment ended with their questions, their responses to my responses to their questions, etc.. This dialogue was time consuming - but generated MOST of the topics for projects, as well as interesting issues related to teaching geometry, the role of proofs, etc., and some reflections on their own struggles with geometry.

• I introduced (and tried to return at regular intervals to) Klein's hierarchy of geometries. For each project, the LISTENERS were asked to report which level(s) of geometry were appropriate. I don't (yet) have good material for this - so it was only partially successful. However, I think it is a critical topic for modern users of geometry. Anyone using geometry in applied areas such as computational geometry, robotics, CAD, etc. should know the levels of geometry, and the value of moving up the hierarchy to simplify their problems. High school students who ask the right questions should have a teacher who can point them in the right direction.

• an interesting source for problems related to 'transformations' (Euclidean, similarity, affine, projective) is the three volume series Geometric Transformations, by Yoglom (MAA). These were written for Russian High School students - so they are tough, but the books include solutions. A number of them fit nicely with Sketchpad (Cabri) constructions and animations to 'see' what the solution looks like.

Walter Whiteley
York University
Toronto Ontario

From: Margaret Sinclair
Subject: Re: Geometry Course for Prospective Secondary Teachers
Date: 17 May 1995 21:05:49 -0400

Lou,

I was part of Walter Whiteley's geometry class for teachers. Everything he mentioned in his reply was of benefit to us -- the push to get us on the Internet, the Russian questions, his lectures on Klein's heirarchy, the introdution to Sketchpad, the projects and so forth. But two things stand out:

First, many (all?) math teachers think they're pretty good at math. They've forgotten how hard really new material can be. Attempting spherical geometry was a humbling experience. We had no frame of reference! We were so used to working from known theorems, it was shocking to be unsure. It's important, I believe, to get teachers exploring, and the questions Professor Whiteley asked us to consider made us think, but they also forced us to experiment, and this brings me to the other point.

Second, because we had to experiment we needed to use models. I had played around with cuisennaire rods, algebra tiles, and so forth at teacher's college and at school, but I had never worked with them as a person who didn't know the answer. It was eye-opening to all of us to see how much we needed the hands-on materials to discover the answers. They weren't window dressing; they were essential. My children couldn't believe that I was drawing great circles with magic markers on plastic spheres to discover how many quadrilaterals were formed and which symmetry groups they belonged to when the sides of one quadrilateral were extended.

Prof. Whiteley could have taught the course from a theoretical point of view, with blackboard diagrams and lots of formulas, but he chose to have us explore, and along with the spherical geometry we learned a great deal about what makes learning possible and enjoyable.

Margaret Sinclair

From: Anthony D Thrall
Subject: Re: Geometry Course for Prospective Secondary Teachers
Date: 18 May 1995 03:21:53 -0400

I am gratified that Lou has raised this question in an urgent context; I trust that he will elicit many deeply felt and thoughtful responses.

Related to Lou's requests for input are the broad issues of how much and what kind of geometry we should offer and encourage in high school. We have discussed "how much geometry?" in previous volleys, and Lou has set this aside for us since he's talking about a one-year course (at the junior level).

On the one hand I do not want to distract us from Lou's urgent requests; on the other hand I want to take this opportunity to make a plug for a longer-term, on-going discussion of the broader issues. In particular, we have an opportunity, through Annie Fetter's wonderful recording of our wrangling, to remember, resume, and build upon our discussions to a degree that was not convenient in the past.

I say this because I am dismayed by our institutional amnesia for previous discussions, e.g., Behnke et al (1960), or Tuller (1967).

The preceding plea for community memory is my major point, but I feel obliged to mention my personal inclinations about high school geometry curriculum, which are as follows. (Lou: I am an applied statistician - mathematician, as well as a recent graduate of the Stanford Teacher Education Program.)

I believe we must restore and update: (i) our notion of a liberal education; and (ii) the place of mathematics in such education. Certainly it is appropriate to marvel at and coo over the brilliant toddlers of human civilization, The Greeks. But it is unseemly for us now, perhaps in our pre-pubescence, to ape these toddlers.

Geometry as they practiced it was a contemplation on several related, and near-religious topics, such as the structure of the world and our place in it. A similar contemplation today must sooner or later reckon with the subsequent two and a half millenia of science, including Kepler, Newton, and Einstein.

The task before us is to present the important ideas, both in their technical substance and in their historical development, within the allotted time and attention span that our students have for us.

These considerations lead me to strongly endorse Klein's conception of geometry as pertaining to the invariants of specified transformations. In general, I think we need to elaborate a few simple themes, helping students to discover the power of these ideas through technical investigations.

At center stage is the notion of a group of transformations. Why not introduce this in high school geometry? The students already know how to compose and "undo" actions on the computer. I would count the year a success if we could get this idea across, along with some appreciation for the potential power of (future) elaborations of the idea.

Behnke et al (1960) discuss the technical difficulties of this and other proposals. Tuller (1967) considers and classifies the respective justifications for such proposals.

References:

Behnke, H., G. Choquet, J. Dieudonne, W. Fenchel, H. Freudenthal, G. Hajos, and G. Pickert (1960). Lectures on Modern Teaching of Geometry and Related Topics. Proceedings of the seminar held at Aarhus from 30-May-1960 to 2-June-1960 by the International Commission for Mathematics Instruction (ICMI). Aarhus, Denmark: Univeritet. Mathematics Institute.

Tuller, Annita (1967). A Modern Introduction to Geometries. Princeton, New Jersey: D. Van Nostrand Company, Inc.