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Thoughts on Teaching Proof

This week's scheduled #geomchat on Twitter was about proof. There was lots of good stuff there. I would love to continue the conversation here.

I'm still processing the advice and thoughts of others on that chat, so for now I'm just going to post some of my own thoughts about proof here.

  1. I think it is a good idea to do lots of informal and guided proof work before introducing any Axiom/Postulate/Theorem-based structure. Euclid is all well and good, but most HS geo students aren't ready for Van Hiele Level 3 out of the gate (and many of them will really struggle to reach it with any consistency at all). This is a real problem in almost every classroom that follows a traditional text, since the vast majority of texts start formal proofs WAY too early. Exceptions I've seen include Discovering Geometry (which does a TON of work at VH Levels 1 and 2 early on and builds up to Level 3 quite slowly), CPM, and to a large extent CME, though I think it, too, rushes in a tad faster than I'd like. I personally don't use any of those book - I use this old thing (which is actually not NEARLY as bad as it looks at first glance) which starts Euclidean right out of the gate. So I teach it out of order; do the things that most texts treat as very algebraic (similarity, right triangles and trig, coordinate geometry and circle equation, area and volume) with a real emphasis on reasoning; derive area formulas and justify their working, use similarity to prove they Pythagorean theorem, do inductive work with patty paper / geogebra to decide AA~ and SAS~ guarantee similarity, etc. But no two columns. No list of theorems, even. A bit of flowcharting, in the similarity section. Then in the second semester, I jump back and do parallel line proofs into congruence into quadrilaterals.
  2. I think it's important that the students prove important things. That is, they should prove interesting theorems themselves. We prove Vertical Angles in teams, and come up with tSSS, SAS, ASA, AAS inductively. But then I have them write their own proofs for the Isosceles Triangle Theorem (both directions), the Angle Bisector theorem (both ways), HL Congruence, Perpendicular bisector theorem (both ways), a theorem I made up called the "Isosceles Triangle median/altitude/bisector" theorem which just says all four of them coincide in an isosceles triangle, and the three different converses (if a median is an altitude, the triangle is issoceles, etc.) I help them structure some of these proofs, occasionally help them get started as a class, and give them plenty of time to meet with me individually for advice, where I can scaffold appropriately; we use the theorems in non-proof problems without worrying about it.

What are your big proof ideas? Techniques that help? How much time do you spend doing it - are you a prove-all-year-longer, like some of these texts want us to be? Personally, I'd rather hang out at Van Hiele 2 all year and not stress about it (because my students need PLENTY of work on Van Hiele 2) but I don't have that luxury.

I want to jump on to this thread with another question, related but different from David's:

What activities that are not proofs do you do with your students to help them become better at proofs?

For example, last summer I wrote a few Connecting Representations (roughly, matching) activities for David Wees' project at New Visions. These activities asked students to math givens with proofs, diagrams with given/prove statements, along with other activities aimed at helping kids notice some of the hidden details of a proof.

What do you do?