## Monday, September 28, 2015

### Still In Love After All These Years

One of the things that keeps me loving Maths is its ability to surprise me. A couple of weeks ago I posted this puzzle that I had created on Twitter:
I had a solution which I knew was correct but I wasn't satisfied with my approach (which was heavily algebraic). I wondered if anyone in the Twitterverse could provide a more visual approach. I am indebted to Carlos Luna (@el_luna) who shared with me his solution which I've shown at the bottom of this post (SPOILER: it has the answer!). He said he used Carpets Theorem to solve it.
Carpets Theorem? Now I maybe accused of being a touch naive but I have never heard of it. But a quick check online had me up to speed with Carpets Theorem and also faced with this most beautiful result of it:

What can you say about the areas A and B in this trapezoid?

All these years that I've drawn the diagonals on trapezia and seen these triangles but never once considered how their areas relate. Well, Carpets Theorem says that the areas are identical! And the proof of it is such a thing of beauty I cannot share it with you because that would deny you the enjoyment.

I was gobsmacked, totally gobsmacked, and in the best of ways.

So why do I bring up this personal story? Imagine it from a student perspective: making a Mathematical conjecture and then suddenly seeing that it is true all the time. I was reminded of the importance of this this morning when a colleague tweeted the following:

Can you imagine the look on Noah's face when he clicks that 8+2 is the same as 2+8? It is still the same feeling that I got when I discovered that the two triangles are the same area in the example above. The teacher has deliberately created a task that will guide Noah into making this discovery himself. It is not always an easy thing to do but my belief is that the more we give students opportunities like these, the more they will become better mathematicians.

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Here is @el_luna 's solution:

1. Hey Mike. Nice post. Even after taking the Geometry course through the MMT at UWaterloo, I am not sure I recognize Carpet's Theorem haha.

I love the activity your colleague created to show the commutativity of addition facts for 10 (now that your colleague has this picture, challenge them to find the picture for associativity of addition!). I agree that students certainly get very excited when they get that "Aha!" moment through a well-designed discovery lesson. However, I am skeptic (of course, feel free to disagree!) that this method will lead to "better" mathematicians. I suppose it might boil down to what we believe "better" means, but I think that whether a student internalizes a fact (say 9+1=1+9) utilizing a more guided approach, or a more direct approach, it is still the same fact - and it would be difficult to say that one approach is necessarily "better" at creating this memory connection. Anyway, just my thoughts!

1. Thanks for the comments Bryan. I think by 'better' I mean that such activities are essential in helping them think mathematically: in this case, making a generalisation. But it needs to be backed up with other activities that will help the internalizing of the fact which, in turn, will lead to more fluent recall. For me, the 'discovery' and the 'practice' both need to be place to produce better mathematicians. I hope this clarifies the post; thanks for the nudge.