Monday, January 12, 2015

The Extended Family of Pythagorean Theorem Proofs

In 4th Grade today, I previewed a proof of the Pythagorean Theorem. Sort of.

This activity (TERC) has nothing to do with right triangles, but it has a lot to do with this proof of the Pythagorean Theorem. (Shell Center)

The deep structure of this proof of the Pythagorean Theorem is identical to that of the array problem above. The same area is to be described in two different ways. One of those ways is obtained by describing the area of the shape as a whole and the second comes the sum of its parts. An equation (an identity) is then derived by equating these two descriptions.

One reason why my 9th Graders have a hard time with proofs of the Pythagorean Theorem is because they aren't familiar with this type of proof. And it is a type of proof, one that shows up throughout mathematics. For instance, it shows up in the study of algebra when studying visual patterns. (Shell Center)

These types of proofs continue to show up throughout mathematics. The "Proofs Without Words" genre is littered with them. Consider the following, which is a proof that the a square is a sum of odd numbers. (Wikipedia)

Why does this matter? I have two takeaways, though I'm curious to know what you think.
  • The justification for teaching proofs of the Pythagorean Theorem might have nothing to do with understanding the Pythagorean Theorem. Instead, there is a genre of proof that shows up throughout mathematics that competent students need to be able to grasp.
  • I think that students find proofs of the Pythagorean Theorem difficult in the main because they do not understand how this genre of proof works. This goes against the typical analysis which would say that students have trouble thinking logically or that they lack persistence.  (Previously: Top 4 Reasons Students Struggle With Proof
The deep pedagogy of proofs of the Pythagorean Theorem differs from its surface appearance. What other areas of k-12 math are like this?

[This post follows up on this.]


  1. On looking at the diagram it is not at all evident that this has anything to do with Pythagoras. It is almost an accident that a^2 + b^2 = c^2. I devised a different cutting up of a square, for a different purpose, and then saw that it was in effect another proof of Pythagoras. I would be interested to know how the Penelope diagram arose in the first place.
    In passing, Penelope's arithmetic is so bad that it looks like "working backwards".

    1. Yeah, her arithmetic is bad. It's part of an activity that asks students to understand and then improve the proofs, including Penelope's.

  2. I think that students find proofs of the Pythagorean Theorem difficult in the main because they do not understand how this genre of proof works.

    I think that it is worse than that. I think young students cannot hold the multiple steps together as a whole in their brains.

    They don't understand how proofs work because they cannot follow one, even laid out, as a whole. To them it is a series of steps, each logical but which -- mysteriously -- are said to form a whole.

    Like the colour blind unable to see what is plainly there, they cannot see it, no matter how simple and "obvious" we make the red dots inside the green dots.

    They think perfectly logically, provided you do not ask them to assemble more than two steps by themselves towards an abstract goal. They have persistence, they basically don't see the target to which they are meant to be aiming.

    I have a saying "no matter what you want to be true, biology wins every time". Lots of well meaning people have tried to conquer our biological flaws, and all have lost.

    1. My read of the research literature (bolstered with some generous anecdotal experience) is that this just aint the case. Kids can reason logically, that's my opinion. I know that lots of teachers disagree with me, but I think my argument is well-supported. One place where I've argued this on the blog is here. (link)

    2. If you haven't done it, I heartily recommend an activity to directly engage with what the students consider to be a proof. This happened to involve the pythagorean theorem as well. Gave out 2 proofs of the theorem (einstein's scaling proof, a cut-and-paster), one proof of the converse (if a^2+b^2= c^2 then right triangle) and a table of illustrations (showing a, b, c, a^2, b^2, a^2+b^2 and c^2 in columns). Then, we had a discussion about what was:
      (a) easier to understand
      (b) more convincing
      (c) more elegant/beautiful

      This helped me get a much clearer view of the students' understanding and led to some very interesting discussions. One of the most interesting things, for me, was the students' confusion about mathematical argument and non-mathematical argument. To give a small taste, examples are often very compelling in non-mathematical argument, but carry very little weight toward a mathematical proof. Shifting between those ideas and modes is obvious and natural with practice, but was a big source of confusion for these students.

    3. Shifting between those ideas and modes is obvious and natural with practice, but was a big source of confusion for these students.

      Totally agreed. I think this is a great way into the idea that mathematical proof is different than everyday reasoning.

    4. Michael, I think you're honing in on exactly the right problem: kids don't struggle with all logical reasoning. They struggle with mathematical proof specifically.

      That's because mathematical proof ISN'T pure logic. It's a communicative and conceptual framework with its own conventions, vocabulary, history, and culture. It's a worthy object of study in its own right.

      Now, as much as I love your analysis and your collection of analogous problems, I'm not sure kids need so much scaffolding with different genres of proof. (I find that "the whole is the sum of its parts" essence of the Pythagorean proof goes down smooth if I give it about three minutes of clear, punchy direct instruction.)

      I do find, though, that they need an engaging introduction into the realm of proof itself.

      For this, I find the standby story of the Wiles-Fermat theorem works. (Guy claims he's "proved" something ~1650, but doesn't write it down. It takes 3+ centuries for someone to actually prove it, and when he does, it's a 200-page behemoth building on ideas that didn't even exist in 1650."

      Hearing that story covers a lot of bases:

      -Proof is the essence of a mathematician's work.
      -Just a few examples can't prove something; it needs to be shown in all possible cases.
      -Sometimes a proof is really hard.
      -Sometimes a proof is really long.
      -A proof often builds on previously-proved things.
      -A proof has to be very careful and airtight.
      -A proof needs to be clear and understandable to others, so they can see your reasoning.
      -Even something that seems obvious can be hard to prove.

      Anyway, looking forward to your continued taxonomization of proof strategies, if that's where you're headed. And if not, I quite like this family tree you've charted.

      -Ben Orlin

    5. But I accept that kids think logically, and I said so. Actually I find it odd that teachers think otherwise.

      But just because we can assemble logical structures mentally doesn't mean all proofs are available to us. I struggle to keep hold in my mind multi-stage proofs that use number theory. They just don't stick, because my brain is unable to hold them (whereas multi-stage geometric reasoning is a piece of cake, by the genetic lottery).

      Proofs, as other comments have noted, are not about thinking logically at all. Kids can do long, complicated logical thinking with hard numbers well before they can do them with abstract quantities.

      The issue with proofs is their abstractness, not the use of logic, and it requires a certain brain maturity to hold abstract thought. I don't think most students have that until 15 years old.

      It's why we teach proof through geometry. Because the ability to see an example of what is being shown makes things much easier. It reduces the abstract loading.

    6. Replying to Anonymous a.k.a. Ben Orlin:

      I'm not sure kids need so much scaffolding with different genres of proof.

      That's some solid skepticism. I can't really make the case that this sort of scaffolding is especially important -- did it help my students? would it help your students? who knows? -- but I think that's an OK state of affairs.

      First, I'm OK with this scaffolding being minimally helpful (though for me it was quite helpful). Second, because this might not be scaffolding as much as a way to build deeper connections for students. By making the deeper structure of proof explicit, maybe we can teach students some powerful paradigms for mathematical proof, instead of a smattering of interesting proofs.

      If we want students to understand the properties that polygons have, we'll likely start by studying triangles, studying quadrilaterals, and then generalizing to the general polygon case. Similarly, if we want students to understand things about proof ("a proof has to be very careful and airtight") then we might start with one category of proofs ("area-equivalence arguments") and then another category ("equivalence-relationship arguments") and then we might generalize to proof. My ambition here is to lay out the landscape of proof a bit so that we might better do all the things that we might want to do as teachers -- scaffold, generalize, connect, compare and reflect.

    7. Nice. I find that pretty persuasive.

      I was being a little sneaky with my comment: "need" is rarely the right standard for math teaching. They don't really "need" anything in particular from grades 8-12. But they benefit immensely from a lot if it, and that's what counts.

      Do you have an example of an equivalence-relationship argument? I'd be curious.

      "The landscape of proof" is a tricky thing. Lots of ways to map that landscape (e.g., I often characterize geometry as "we study triangles really carefully, then build everything out of triangles," which is a type of description that's orthogonal to the type you're seeking). That said, I love the idea of classifying proof strategies. There's only a few I've ever seen explicitly taught:

      -Proof by contradiction
      -Counterexample (as a way of disproving a general statement)

      Do you have a list you're generating? Is there an existing list out there?

      -Ben Orlin [sorry for the 'Anonymous' thing, but my accounts all have weirdo names]

    8. I'm generating a list, but right now my list is very very short. Understanding proof is a long-term project for me, but I still don't think that I really understand congruence/similarity proofs. That's what I'm really obsessed with at the moment.