WitCH 60: Pythagorean Construction

Yesterday, David de Carvalho, the CEO of ACARA, gave a speech at The Age Schools Summit. de Carvalho used his speech to set the stage for ACARA’s imminent launch of the draft of the revised Australian Curriculum. (See here.)

We shall take a more careful look at de Carvalho’s speech in the near future. For now, we’ll settle with a WitCH, an excerpt from de Carvalho’s speech:

“So the ability to recall facts from memory is not necessarily evidence of having genuine understanding.

A student might, for example, memorise the formula for calculating the length of the hypotenuse if given the length of the other two sides of a right-angle triangle, but do they understand why that formula, known as Pythagoras’ theorem, works every time? The process of discovering that for themselves, with the assistance of the teacher, is what makes learning exciting. And it’s what make teaching exciting. Seeing the look of excitement on the face of the student when they experience that ‘aha!’ moment.”

And when we understand a topic, it is easier to recall the facts because they are no longer just random bits of information but are organised into intelligible ideas. Not only do we know where the dots are, but we know why they are there and how to join them.”

UPDATE (29/04/21)

Just a reminder, this is a WitCH. So, what specifically is crap about de Carvalho’s Pythagoras suggestion?

To guide the discussion, below is (arguably) the simplest, algebra-free proof of Pythagoras, which, undeniably, all students should see. Where does/should this proof fit in with the teaching of Pythagoras, and where does de Carvalho’s suggestion fit in with either of these?

UPDATE (03/05/21)

It is extremely helpful of De Carvalho to have selected such a fundamentally flawed example. If he had chosen more judiciously it would be more work to counter, to make the case here against inquiry-based learning. De Carvalho having chosen almost an anti-example, however, makes clear that ACARA’s inquiry push is nothing like a reasoned best-choice approach in given situations, and much more a religious fundamentalism: inquiry is, simply by being inquiry, the preferred method.

De Carvalho contrasts the WHAT of Pythagoras — the equation — with the WHY, the proof (or proof-like evidence for) that equation. Putting aside for the moment De Carvalho’s atrocious suggestion for getting to the WHY, we’ll first note that De Carvalho has failed to ask two fundamental questions:

  • HOW does Pythagoras’s theorem work?
  • WHY2 do we teach Pythagoras’s theorem?

The first question is about the HOW of the mechanics of dealing with the equation \boldsymbol{a^2 + b^2 = c^2}: manipulating for the unknown, taking roots and so on. This HOW is not glamorous, and not intrinsically difficult, but it is fundamental. Of course the application of Pythagoras’s theorem, including the mechanics, is in the draft curriculum (beginning in Year 8), but De Carvalho’s casual airbrushing of the HOW is a tell.

(As  a side note, we’re pleased to see that the single stupidest line in the Australian Curriculum is still there, in the Pythagoras elaborations.)

And now, the second question: WHY2 teach Pythagoras’s theorem? There are two strong answers to this question, both of which, in different ways, demonstrate that De Carvalho has no understanding of his example.

The first reason, the WHAT to teach Pythagoras is because it is so important: it is the fundamental formula for distance in Cartesian (and Euclidean) geometry.

The second reason, to teach the WHY of Pythagoras’s theorem is because it is a historical icon and because it is so beautiful. The proof illustrated above is gorgeous, it can easily be learned by a primary school student, and it should be learned, by all students, as one learns a poem. (One can of course extend this to be a third reason, to teach other proofs and the nature of geometric proof.)

Here is why these reasons show De Carvalho’s example to be so empty:

  • The two reasons for teaching Pythagoras are almost totally disconnected.
  • Hunting for a proof is the absolute worst way to appreciate the beauty of Pythagoras.

Arguing for the WHY, De Carvalho notes,

And when we understand a topic, it is easier to recall the facts …

And of course, as a general point, De Carvalho is correct. In regards to Pythagoras, however, De Carvalho is simply wrong. Pythagoras is one of the easiest equations to learn, and students simply don’t need the WHY to know WHAT it is and HOW it works. Secondly, the WHY doesn’t help with the recall whatsoever. Pythagoras is a theorem about areas, and its application in school is always to distances. The idea that a longer proof about areas will help students recall and understand the use of a simple formula about distances is utterly ridiculous.

As for appreciating the beauty of Pythagoras: if you are given a beautiful poem then you simply teach the poem. It is absurd to think that De Carvalho’s “discovering that for themselves” — which, anyway, will almost certainly be faked — will give students any proper appreciation of Pythagoras. All it can do, and inevitably what it will do, is obscure the simple beauty.

There are zillions of examples of the WHY being critical to understanding the WHAT, and there are even examples where the WHY should replace the WHAT altogether. There are examples where a limited form of inquiry is worthwhile in discovering the WHY and the WHAT. Pythagoras, however, is none of none of these. Pythagoras is only an example of ACARA’s constructivist dogmatism, and of De Carvalho’s ignorance.

35 Replies to “WitCH 60: Pythagorean Construction”

  1. “Some learning areas have only required some tidying – but others have required more focus. Maths for example has required greater improvement and updating.”

    Sounds like it’s done and dusted.

  2. I don’t think anyone disputes that being able to remember facts is the same thing as being able to understand why facts are facts in the first place.

    Again, the issue here is not that giving students a lovely little demonstration of how to “discover” the Pythagorean theorem is a terrible idea. It is fun and I like doing it.

    The issue as far as I see it is that there is a greater responsibility on the line. If we focus *only* on these demonstrations, and have *none* or very little of the basics — these include memorisation of facts that need to be recalled quickly, not recalculated every time — our curriculum will be worthless.

  3. I learned how to prove the Pythagorean theorem when I was 15. But I still didn’t know why it was true. I started thinking about why it was true when I was 40 and it took me ten years of thought. Now I know why it is true, but I wouldn’t have minded being told why it was true when I was 15.

    1. It is taught horribly. It can be taught easily to a primary school kid. de Carvalho’s suggestion would make it way, way worse.

  4. I think if we’re concentrating on explaining why things are true, I wouldn’t start with Pythagoras’ theorem. I’d start with getting students to know things like “why” \frac{8}{3} is “2.67”.

    I’ve noticed that brighter Year 10 students at my school have memorised some of the more common arithmetic results from seeing them on their calculators but still don’t know what fractions mean and don’t seem to know how to do division into decimals. I think school has let them down. These are the brighter students too. It’s not because of any fault of the student.

    1. But \displaystyle \frac{8}{3} is NOT 2.67.
      2.67 is a decimal approximation of \displaystyle \frac{8}{3}, correct to two decimal places.

      Students should be taught fractions in primary school. I know teachers who do this and do it well. I have also seen it done badly. Unfortunately, teachers of students in higher grades must hope that the teachers of students in lower grades are doing their job properly. Where things break down is when you get students in Yr 10 and Yr 11 (and even Year 12) who cannot perform basic arithmetic calculations such as 2/3 – 3/7. Some would argue that this doesn’t matter because calculators can do it … Show me a kid that cannot calculate 2/3 – 3/7 and I’ll show you the kid who has no chance with algebra. What’s not understood by many of these self-styled educational experts is that being able to handle algebraic abstraction requires being able to handle the concrete. (Some would argue that being able to do algebra doesn’t matter either because a CAS can do it. This is where the real trouble begins).

      A real bug-bear of mine is students who come into Specialist Maths 3/4 and have never seen or heard of a function that crosses its horizontal asymptote. Probe deeper and you’ll soon discover that the fundamental difference between vertical and horizontal asymptotes has not been taught in Methods 1/2. This is something that should be getting taught in Methods Unit 1, but it’s not. I don’t know whether it’s because of incompetence, laziness, lack of time, indifference or a slavish devotion to the Study Design and textbooks.

      Re: Pythagoras’ Theorem and de Cretin’s speech:

      I think the word he’s groping for is proof. I would be bitterly disappointed if a proof of Pythagoras’ Theorem is not already ‘taught’ in Yr 9 (or indeed Yr 8). I read somewhere that Pythagoras’ Theorem has the most number of different proofs of any theorem in mathematics – over 100 different proofs. That to me is an amazing piece of information that I’m sure would fascinate many students. And on this topic, I never see Pythagoras’ Theorem stated as

      “A triangle is a right-triangle if and only if \displaystyle a^2 + b^2 = c^2.

      The theorem is always used from triangle to formula – I never see the theorem used from formula to triangle …

      And de Cretin is silent on the basic skills required in order to successfully apply Pythagoras’ Theorem. And how those basic skills are acquired. That’s the real problem. It’s only easier to understand the ‘facts’ and “join the dots” when the facts that those facts are built upon are mastered:
      Great fleas have little fleas upon their backs to bite ’em,
      And little fleas have lesser fleas, and so ad infinitum.
      And the great fleas themselves, in turn, have greater fleas to go on;
      While these again have greater still, and greater still, and so on

      And how is that mastery achieved …? The answer to this question is what makes these self-styled educational experts squirm and prevaricate.

            1. I don’t think you were confusing. It’s just your example caught another learn-nothing aspect of primary mathematics.

      1. JF, I think you are wrong: I do not believe de Carvalho is struggling to find the word “proof”. His purpose here is entirely different.

        1. I got something else wrong, too. Re-reading de Cretin’s excerpt, he only talks about finding the hypotenuse. So all the algebraic skills required for finding one of the other two sides are moot in his book. He would probably want three different statements of Pythagoras’ Theorem, just like what teachers do with Ohm’s Law in science classes.

        1. Thanks, Greg (but not for the pun). The link works: I don’t know if you wanted something smarter than that from WordPress.

  5. There is no evidence that Pythagoras knew how to prove Pythagoras’ theorem! It appears near the end of Book 1 in Euclid’s “Elements”. This is not to say that Euclid was the first to prove it. “Elements” is one of the great works in mathematics, yet rarely read even though it could be read with profit by secondary students. I have taught it several times at universities. It’s beautiful. Reading “Elements” changed the way that I write out proofs.

    1. In a way, it is actually two of the propositions in Book 1: 47 and 48.

      47 demonstrates that all right angle triangles obey the equation a^2+b^2=c^2 and proposition 48 demonstrates that this is unique to right angle triangles.

      There is plenty of evidence that Indian and Chinese mathematicians also knew the idea and some authors have suggested that Pythagoras actually learned the idea while studying in Egypt…

      1. The “if and only if” aspect is what JF was referring to.

        I don’t think there’s clear evidence that the Egyptians knew of the theorem, much less that Pythagoras (or his followers) learned it from them. The Chinese knew of it pretty early, but there is no evidence I’ve seen of an early Chinese proof. I’m not sure when the Indians learned of it, with or without proof, but I haven’t heard of it being pre-Pythagorean.

        There’s a tendency to see a right-angled triangle on same ancient civilisation’s wall, and then the cheer squad for that civilisation takes it as proof of a proof.

        1. Sure. I’m not pretending to be an expert on any of this, but I do think it is entirely possible that different cultures discovered this idea independently of each other at different times.

          The evidence (which is a bit of a strong word in this case) for the Egyptians is their use of the 3-4-5 ratios in design, which is a well known Pythagorean triple/triad/call-it-what-you-want.

          We do know that Pythagoras travelled a lot as a young man and did some study in temples in Egypt – perhaps a few people have, over time joined a few dots and it becomes folklore of sorts…

          1. Definitely different cultures independently discovered the theorem. I’m not sure there’s any evidence of Pre-Pythagorean proofs. I’m also not sure there’s any evidence the Egyptians used 3-4-5 to create right-angled triangles.

            1. There is evidence if you start with the assumption that it is true and then actively look for 3-4-5 triangles in designs: 12 knots in a rope, equally spaced, will create a right-angled triangle. Whether or not civilisations *knew* this would be a right-angle… I don’t think it really matters as much as that we *now know* it does.

              1. That’s the kind of “evidence” that has people spotting the golden ratio in buildings, or constructivism in Singapore.

                1. My point exactly. PT is quite a rich theorem if you want to push it further from a number-theory perspective (such as proving that exactly one of a-b-c will be a multiple of 3, if a, b, c are integers then the product is always a multiple of 60… I could go on)

                  But these are all ideas best left until much further down the track and only for a specific type of student.

                  The obsession that “discovery learning” somehow leads to greater understanding than a succinct explanation of the idea followed by some well chosen examples and well chosen skills-practice is crap.

                  In fact, it is more than crap, because for some students the approach has the potential to do harm, rather than do no good and there is a significant difference.

        2. I stumbled on this a few years ago and kept the link because I like seeing things taught from a different point of view and in a different style.

          Click to access Exemp21.pdf

          It’s some kind of exemplar of how to teach about the Chinese proofs to Hong Kong students (KS3 = year 7-9 I think). It seems to show Chinese proofs. Have you seen those?

          1. Hi wst, I read a bit about it, for a (mot very good) talk I gave: Was Pythagoras Chinese? I concluded he wasn’t, which didn’t make the evening papers.

            1. Well, according to that (quite patriotic) lesson plan, if he’d been Chinese his proofs would have been more vigorous and intuitive. So yes, I guess not. 🙂

  6. Even though no one asked, here’s what I came up with after 10 years of thought: Draw a circle and one of its diameters.
    Pick a point on the diameter, separating the diameter into 2 segments. Let each segment be the diameter of a circle. We now have 3 circles and it is easy to see that the sum of the circumferences of the 2 smaller circles equals the circumference of the larger circle. We have 2 sets of similar objects, circles and line segments juxtaposed in equivalent ways. And we have the distributive law: pi times the sum of the 2 smaller segments equals pi times the larger segment.
    The Pythagorean theorem is understood with a similar idea. Drawing an altitude to the hypotenuse creates three similar right triangles with the areas of the 2 smaller ones equaling the area of the larger one. I like to call them Larry, Moe, and Curly Joe. Draw the square on each of their hypotenuses. Let c be the length of the largest hypotenuse and let h be the height of the altitude. 2c/h times the areas of each triangle will yield the areas of each of the three squares. That’s what I wish someone had shown me when I was 15.

    1. Thanks, Marc. Yes, that is a lovely proof, and I was thinking of that proof when I included “arguably” to describe the more familiar proof above.

      It raises a point about this blog. It’d probably be worthwhile going Dr. Jeckyll occasionally, and include some *good* mathematics. Still, there’s a reason why I stopped doing that kind of thing …

  7. OK… I rather like PT as a piece of mathematics, so I’ll have a go…

    PT to me is an equation: a^{2}+b^{2}=c^{2}

    But, equally importantly, to me it is a statement of a geometric fact.

    I show Year 8/9 students the Garfield (US president, not fictional cat) proof that uses a trapezium as I think they can handle it pretty easily – not that there is anything wrong with the pictures above, just personal preference.

    So… where is the crap?

    Well, firstly I don’t think the author really understands what PT is about – rather, making it out to be some amazing mystery that can only really be understood by hours of abstract thinking.

    Sure, it has some abstract applications (Euclidean distance formula) and some equally bizarre generalisations (Fermat’s last theorem), but all in all, it is perhaps the simplest (and therefore I like it the most) mathematical statement about something that happens with lengths.

    Secondly, the idea that students will somehow “discover” this theorem is crap. They might, if given a large helping of guidance, experience what some of my current and former colleagues have called “discovery learning” but, really, if you are using a measuring tape and squaring numbers with a calculator to get approximately equal numbers, you really haven’t discovered anything.

    Thirdly, and perhaps most importantly, the author seems to be over-obsessing with the PROCESS rather than the IDEA.

    a^{2}+b^{2}=c^{2} is an idea that anyone can understand once they know what it means to square something. The geometric interpretation then pretty much confirms the truth of it all – a bit of practice, a few 3D examples perhaps if we really want and… done. Move on.

    1. I agree — just wanted to comment that I don’t think it is particularly honest to say that Fermat’s last theorem is a generalisation of Pythagoras’. (I’d also not call it bizarre, but that’s personal taste. Unexpected maybe.)

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