Proofs in Geometry – The Modification Continues…

Two statements of interest to me:

• I get more consistent daily hits on my blog for teaching geometry proofs than anything else. Shiver.
• Dan Meyer’s recent post on proofs in Geometry gets to the heart of what bothers me about teaching proofs at all. Double shiver.

These statements have made me think about my approach in doing proofs with students in my 9th grade course, which has previously been a geometry course, but is morphing into something slightly different in anticipation of our move to the IB program. I like the concept of teaching proofs because I force students to confront the idea that there’s a difference between things they know must be true, might be true, and will never be true. I started the unit asking the class the following questions:

• Will the sun rise tomorrow?
• Will student A always be older than her younger sister?
• Will the boys volleyball team win the tournament this weekend>

The clear difference between these questions was also clear to my students. The word ‘obviously’ came up at least once, as expected.

The idea of proving something that is obvious is certainly an exercise of questionable purpose, mostly because it confines student thinking in the mould of classroom mathematics. As geometry teachers, we do this as a scaffold to help students learn to write proofs of concepts that are not so obvious. The downside is the inherent lack of perplexity in this process, as Dan points out in his post. The rules of math that students routinely apply to solve textbook or routine problems already fit in this ‘obvious’ category either from tradition (‘I’ve done this since, like, forever’) or from obedience (‘My teacher/textbook says this is true, and that’s good enough for me.’)

I usually go to Geogebra to have students discover certain properties to be true, or give a quick numerical example showing why two angles supplementary to the same angle are congruent. They get this, but have a sense of detachment when I then ask them to prove it using the properties we reviewed in previous lessons. It seems to be very much related to what Kate Nowak pointed out in her comment to Dan’s post. Geometry software or numerical examples show something to be so obvious that proof isn’t necessary, so why circle back to then use the rules of mathematics to prove it to be true?

I had an idea this afternoon that I plan to try tomorrow to close this gap.
I wrote earlier about using spreadsheets with students to take some of the abstraction out of translating algebraic expressions. Making calculations with variables in the way a spreadsheet does shows very clearly the concept of variables, and also doing arithmetic with them. My idea here is to use a spreadsheet this way:

My students know that they should be able to change what is in the black cells, and enter formulas in the red cells so that they change based on what is in the black cells only. In doing this, they will be using their algebraic rules and geometric definitions to complete a formula. This hits the concrete examples I mentioned above – a 25 degree angle complementary to an angle will always be congruent to a 25 degree angle complementary to that same angle. It also uses the properties (definition of a complementary angle, subtraction property of equality, definition of congruence) to suggest the relationship between those angles using the language and structure of proof, which comes next in class.

Here is the spreadsheet file I’ve put together:
02 – SPR – Congruent Angles

I plan to have them complete the empty cells in this spreadsheet and then move on to filling in some reasons for steps of more formal proofs of these theorems afterwards, as I have done previously. I’d like to think that doing this will make it a little more clear how the observations students have relate to the properties they then use to prove the theorems.

I’d love you to hack away at my idea with feedback in the comments.