## On motivating mathematics education

Here is a funny comic from the Fake Science blog.

The problem is, there is a kernel of truth in this satirical comic. Given most problems we will encounter in life, we would use a ruler to find the third side of a triangle. Obviously I think that there are good reasons to learn the Pythagorean theorem, but for most real life applications, one could draw a careful scale diagram (an incredibly useful skill in itself) and apply ratios to your measurements of your diagrams to find the missing length.

So why do we teach the Pythagorean theorem? Is it because of the power this abstract idea has? Are there other abstract ideas which have equal value? Could you imagine a mathematics curriculum which includes lots of rich abstract ideas, but happens to not include this theorem? How important is this theorem anyway?

• I think the last phrase could also be ” … so you don’t need to use a ruler”

• Here’s a real life example where we used the Pythagorean theorem recently. Our neighbour has an air conditioner which sits on the other side of our fence, a straight diagonal from our upstairs bedroom window. It is extremely loud. There is a city bylaw which states that the distance from the window to the air conditioner must be less than x feet (can’t remember what x is). Obviously we could not measure this diagonal distance very easily, or accurately, especially when we were trying to do this quickly so as not to be noticed. (We didn’t mention any of this to the neighbour.) So we measured the straight line distance from the bottom of our house, through the slat in our fence, to the air conditioner. We then measured the distance from our window to the ground, applied the Pythagorean theorem and got the diagonal distance. Moral of the story: Yes, her air conditioner is too close. Did we work up the nerve to ask her to have it professionally moved? No, but at least we got to use the Pythagorean theorem.

As for mathematical applications, a world without the Pythagorean theorem seems unimaginable. A nice way to illustrate the the square root of two exists is by considering the diagonal of a right angled triangle with legs of length 1. We also wouldn’t get very far with trigonometry without it (eg. sin^2x+cos^2x=1). The list goes on….

• David Wees wrote:

That’s a good example, Ahypatia. I’m sure we can find other examples when the Pythagorean theorem is extremely useful, and as you point out, there are applications in all sorts of other areas of mathematics. Our previous discussion about abstract reasoning is another good reason to include teaching it. One of the things I really like about the Pythagorean theorem is that it is easy to prove true in a variety of different ways, including using experiments. I’m not really advocating for abolishing the teaching of the Pythagorean theorem, but I am questioning some of what we teach. I can’t imagine that all of it is the most important mathematics to teach.

• Dan Pearcy wrote:

With David’s permission, I have replied to this blog post on my blog. (danpearcymaths.wordpress.com)

• David Wees wrote:

Great post by Dan, I recommend reading it.

• Dan Pearcy wrote:

Thanks David – it was an interesting thing you did taking such a fundamental theorem in maths and questionning its relevance. Just asking the question makes everyone think about it in so much more detail than usual.

• […] the reasoning behind teaching an abstract concept is indeed something to consider, and references David Wee’s blog which questions the need to teach this theorem (perhaps the most convincing reason/s are […]

• […] David Wee’s blog, 21st Century Educator, he recently reflected on whether we need to teach the Pythagoras Theorem. He argued that while […]