Sunday, September 15, 2013

My (Current) Rules for Teaching Geometry

Presented mostly without comment, and with varying degrees of confidence:
  1. Rarely limit our study to just triangles. I'm looking at you, triangle congruence and triangle similarity. You think that AA is a shortcut for similarity? Look at this square; look at this rectangle. Why should triangles be different?
  2. Ask kids to prove things that they don't have the tools for proving. For example, ask kids to offer a convincing argument that all these triangles are of equal size.

    That'll create the need for similarity (or the side-splitter theorem, or parallel line theorems) in a jiffy.
  3. Vary the problems that we work on between the visually obvious and the visually unintuitive, because one of our major jobs is to understand the relationship between the way we see things and mathematical argument.
  4. Treat definitions like theorems. (Thanks, Justin + Lakatos!)
  5. Two-column proof, or axiomatic proof, is an abstraction of proof. It is not a natural or normal way to communicate mathematically. The few cases I know of mathematicians working from axioms are remarkable for this peculiarity. (Most often, mathematicians are studying the properties of these formal systems, rather than working within the formal system itself.) So care has to be taken to put formal proof in its proper context, lest kids walk out of the year thinking that mathematicians sit in a room and pump out theorems from axioms all day.
I'd love to hear your thinking about these, as Geometry is a course that I still feel slightly uncomfortable with. See you in the comments!


  1. I really like rules one and two. I haven't started teaching yet, but that definitely is something I would keep in mind if I were teaching. Having them try to prove something would be a good way to begin the lesson and show why people came up with and needed these principles.

  2. Good post and this would be a great discussion.

    Trying to think about my rules:
    1. Reasoning is on a continuum, make sense, make conjectures, make arguments, make proofs. Provide for prerequisite opportunities.
    2. The meat of geometry is noticing. Provide opportunities to notice and reasons to do so.
    3. Students need to communicate to each other as well as me, and have the chance to critique and question.

    Relevant to yours above:
    Whenever possible, definitions should only be introduced after arguing, to create community agreement. (Just had this experience with pentominoes.) Also okay to use definitions to provoke arguments, to make students work out a definition you know. (I do the polygon definition like this.)

    Thanks! - John

    1. (@RosieL52)

      Really like the idea of reasoning on a continuum. I used to think students were "at" a particular van Heile level, but now I think it is more subtle than that. Switch to a new geometric topic and we might be back at "level 0" again so starting with noticing and sense-making before progressing to anything like proof just seems right to me. Hands-on "play" is a really important piece of the sense-making step for me with my students. (Patty paper, pushpin & card stock-segment linkages, origami, etc.)

      I also think this mirrors my own mathematical experiences. I felt completely at home with axiomatic systems in my undergrad geometry course, but then I still had to "play with" the ideas in other branches of mathematics before I could get a real sense of the overall structures. I did take abstract algebra and real analysis only after finishing a calc sequence...

  3. Nice topic. Here are some general rules of mine:
    Rules for Teaching Geometry
    1. Students learn best by doing, discovering, explaining.
    2. Teaching geometry is more about processes than covering content.
    3. Research shows that 70% of all high school students enter geometry operating at level 0 or 1 on the van Hiele measure of geometric reasoning. Don't start your course assuming they are ready for level 3 deductive reasoning. Focus on proof as a means of "explaining why" rather than a way of convincing someone the given conjecture is true.
    4. Have fun. Each school year explore non-traditional topics (keep yourself excited!). Do projects. Play games. Solve puzzles.

    1. Focus on proof as a means of "explaining why" rather than a way of convincing someone the given conjecture is true.

      I think that this is really interesting. I've tended to start with proofs as convincing, because I thought that "prove to convince" was a more concrete context than "prove to explain," for kids who aren't really comfortable with proof.

      Your comment is making me wonder whether I'm really taking them anywhere with offering them chances to use proof to convince.

    2. Just to clarify: it's clear to me that "prove to convince" runs out pretty quickly. The question, for me, is how do I start out the year? Do I start with proving to convince (on the assumption that it's more available) and then abstract to prove to explain, or do we start with explaining and abstract to the level of careful proof?

    3. This is Rose (@RosieL52)...

      One problem with "prove to convince" is that often times students _are_ already convinced ("Of course those angles are equal/lines are parallel/triangles are congruent/etc. - anyone can see that!") and so writing a "convincing" argument seems pointless.

      If you haven't looked at it yet, I would recommend M deVillier's book for Sketchpad. This is an excerpt from the introduction that is well worth reading:

      I really want my students to understand the structure of a logical argument to help them organize their reasoning/ critique others' reasoning so I'm doing some work with games and puzzles (shikaku, nim, eleusis express, tic tac toe, animalogic, etc.) to frame it up. I'm also using non-mathematical examples to practice all this initially (ala Harold Jacobs).

      I've taught geometry a number of times, but I am really still evolving (that will probably never end, though!).