Suppose you wanted students to learn about deductive proof, and the proof of this theorem in particular… how would you then proceed from this abductive approach? How would your students become aware of a proof of this theorem that was recognised as a valid mathematical proof by the mathematical community (if we can agree that such a thing exists)?

Thanks for the response, Danny.

I don’t disapprove of explanation in general. As lots of researchers have pointed out (the van Hieles come fastest to mind; Harel & Sowder also) deductive reasoning is weird and hard to learn. Bringing new learners into the community of mathematicians and helping them understand what constitutes a valid proof is also difficult work. I don’t dismiss the role of explanation or direct teacher modeling in helping students with that work.

But I don’t think explanation is an effective or interesting introduction to that work.

This post asks “how should we motivate proof?” How do we help students understand why mathematicians think a proof is necessary?

Suppose you wanted students to learn about deductive proof, and the proof of this theorem in particular… how would you then proceed from this abductive approach?

My next move here is to feign surprise at the slowly forming circle and ask students to please create a right triangle that doesn’t lie on the circle.

M Ruppel:

As to the previous commenter, Thales’ thm is not a particularly important piece of content in and of itself, but it’s one of my favorite proofs for students to build. It requires careful attention to definitions and previously-learned theorems as well as a bit of creativity (drawing that auxiliary line). Personally, my favorite part of the proof is that students don’t solve for a or b, and in fact have no knowledge of what a and b are, but they prove that a+b=90…different flavor than they are used to.

I appreciate this distinction. The math is less important here than the mathematical practices it lets us glimpse.

If, instead, the prompt started with “Drag Points A,B, and C to create three different triangles where the segment below is the longest side,” students would start by getting a random mess of points (nothing interesting here). A follow-up prompt asking students to then “Drag Points A,B, and C to create three different right triangles where the segment below is the hypotenuse,” students would see that it is the right triangles that make the points form a circle. I worry that if we go too far down the “cool teacher magic trick” road in Geometry, kids start to see it as just a cool picture that makes them do a proof afterwards.

As to the previous commenter, Thales’ thm is not a particularly important piece of content in and of itself, but it’s one of my favorite proofs for students to build. It requires careful attention to definitions and previously-learned theorems as well as a bit of creativity (drawing that auxiliary line). Personally, my favorite part of the proof is that students don’t solve for a or b, and in fact have no knowledge of what a and b are, but they prove that a+b=90…different flavor than they are used to.

I agree with Barb in thinking that the learning of content should be embedded in the larger goal of learning about the discipline of mathematics. Trying to ignite curiosity and wonder in the students is surely one step, among many, that would be necessary in that direction.

Or, if it is “Steps 1,2,3” Bury the important middle step(s), see if the kids can dig ’em up.

Or you can play Jeopardy! The answer is “90 degrees” and the kids give the questions.

Combining disparate ideas is fertile ground for inquiry: Can you find a way to do transformations with compass & ruler constructions? Is there a geometry task you normally do without coordinates? Try it with coordinates and see if you can find a pattern.

It takes time to incorporate inquiry into your class, but with patience, you can do it. Remember that good math theorems can take years to prove, and so can crafting good lessons/units.

Also, there’s some good stuff available from Judah Schwartz.
https://sites.google.com/site/mathmindhabits/

What I’d love is a resource for teachers that broke down the different theorems and formulas and definitions into explorations for our students to set them up for the need for the theorem or formula or definition.

You’re talking hundreds of theorems and formulas here. What you are asking for is a life’s work. Many of the most important are beyond our students anyway (e.g. finding the value of pi).

Most decent texts give one or two simple ones, because that’s all we have time for in our courses anyway. Such explorations are useful but they are, if you are going to do them properly, slow.

Also, your students won’t thank you for it, unless they are wildly different from mine. “When will I use this?” is a problem we face. But we also face “Do I need this for my SATs?”. We need to try and balance those things, and long deviations into the wilds of proofs doesn’t do that.

Here are my (current) thoughts on ‘teaching proof’:

(1) http://www.squeaktime.com/blog/what-is-proof-and-why-is-it-important, and

(2) http://www.squeaktime.com/blog/what-might-we-consider-when-teaching-proof

On reading these posts, you may see that my preferred pedagogy around proof is not (necessarily) explaining.

That said, I don’t think there is anything ‘wrong’ with explicit teaching of deductive proof(s), and I must admit I am not entirely sure how one would make students aware of deductive proofs without explaining or showing them one in some way.

What you present here is of course not a deductive proof, rather a prelude to one, an example of abductive reasoning that *may* be considered (by, say, Balacheff) to actually be an obstruction to learning about deductive proof. What are your thoughts on this?

Suppose you wanted students to learn about deductive proof, and the proof of this theorem in particular… how would you then proceed from this abductive approach? How would your students become aware of a proof of this theorem that was recognised as a valid mathematical proof by the mathematical community (if we can agree that such a thing exists)?

Danny

Can you provide the responses to the question “What would it look like if we laid all of our triangles on top of each other”? These responses would be a better measure of the engagement level.

Interestingly, a few treatment group participants asked whether the triangles were right triangles. Also interesting is that the control questions were, arguably, more sophisticated. Not using that as evidence for one approach or the other — just interesting.

Are the question and the reveal too cluttered? I have difficulty seeing order out of the chaos of three overlapping triangles. The control version is still too formal for my taste. In fact, one question was “what is an auxiliary line”.

Proving something is true often does not show why it is true. We have all had the misfortune of slogging through a proof, struggling to check that the next line follows from the previous, but not having any intuitive sense of why the darn thing is true. Does the control approach elicit a need for proof or a need for a reason “why does it make a circle”? I think the latter, and I am not sure how to address that in this instance.

Thanks!

The experimental version sets this up (A)…and then stops. The control version sets it up (A)…and then explains the solution (B). I find the Desmos-ified version much more engaging and trust that students will find it more engaging. Still, is it possible that control students asked fewer questions simply because the most likely question _was already answered_?