Wednesday, 18 March 2015

Discovering Pythagoras' Theorem

Discovery learning gets a bad press. For instance, in the Sutton Trust report: What Makes Great Teaching (2014), discovery learning is listed under ineffective practices:
"Enthusiasm for ‘discovery learning’ is not supported by research evidence, which broadly favours direct instruction (Kirschner et al, 2006). Although learners do need to build new understanding on what they already know, if teachers want them to learn new ideas, knowledge or methods they need to teach them directly." (p23)
Direct instruction it is then. But how do we square this with the psychological evidence cited in the same report:
“Basically, any time that you, as a learner, look up an answer, or have somebody tell or show you something that you could, drawing on current cues and your past knowledge, generate instead, you rob yourself of a powerful learning opportunity.” (from Bjork and Bjork, 2011, p61)
Perhaps I am missing some subtleties in the definition of discovery learning - I have not studied it in great detail, though I have the sense that the term brings with it certain amounts of baggage. It conjures up for me mental images of frustrated students failing to see what the teacher has been hoping for.

However, I do not for a second believe that discovery is worth giving up on completely as a learning experience, and neither do I believe that discovery learning should mean students working aimlessly. A student generating knowledge through their own exploration has a number of benefits:

1. As stated in the second quote, it is a more powerful learning experience.
2. Students come to understand necessary 'must-be' maths facts for what they are, rather than as arbitrary knowledge (see my previous post about Dave Hewitt's definitions of arbitrary and necessary maths knowledge.)
3. It fosters further curiosity and perhaps even wonderment in the learner.
4. It empowers students to feel ownership of their mathematics. In other words that they are included in the community of Mathematics, rather than being an external onlooker trying to make sense of it.

Discovering Pythagoras' Theorem

When introducing Pythagoras' Theorem with Year 8, I used a discovery approach. The lesson I followed, and the *free* resources are detailed here on mathspad.

Firstly, we tackled the matter of finding areas of 'tilted squares': (whiteboard tool, worksheet)


This is a rich and engaging enough activity by itself (so much so in fact that nrich devote a whole task to it.)

We then started putting those squares onto the sides of right-angled triangles: (worksheet)


...and before long, students have started to notice the two smaller squares sum to the larger, and thus Pythagoras' Theorem was discovered.

This realisation prompted various reactions from students. Some questioned whether it would be true for all triangles, other than just the ones they had been given. (Excellent question! Try your own triangles!) A few students seemed truly inspired by this discovery - fascinated that it was true for all right-angled triangles, but not for any others.

One student really wanted to know 'why' it worked - and after looking at a couple of proofs, the student's curiosity was still not satisfied and they continued to pester me (over a number of days, actually) with the same query... sir, you have proved that it works, but not why it works. After some reflection, I realised I could offer no satisfactory answer as to why it worked - it simply does. This is a necessary, intrinsic property of right-angled triangles - a 'must-be' maths fact. And so, eventually, my response was simply to start to share the student's sense of wonder: I can't answer why it works because it 'just does' - amazing, isn't it that the universe works this way?

Of course, not all students responded this way - some appeared unimpressed! Importantly though, they had come to know this 'must-be' maths fact not by having it presented to them as if it were arbitrary knowledge, but by exploring and seeing it for themselves, behaving mathematically in the process. This was certainly a powerful learning experience for students, they could later relate the idea into the algebraic form without difficulty and demonstrated a deep and sustained understanding of the idea.

How much this approach resonates with 'discovery learning' is up for debate. The students had clear guidance on what to do the whole time. Indeed, students were primarily engaged with one mathematical task - finding the areas of squares, when they spotted a pattern emerging (a mathematical instinct?) and thus 'discovered' Pythagoras' Theorem.

To me, distinguishing between arbitrary and necessary mathematical facts makes discovery learning meaningful. There is little point in students trying to discover arbitrary, conventional knowledge. Conjecturing, investigating and discovering necessary 'must-be' mathematics knowledge on the other hand, is what mathematics is truly about and - with the right guidance and framework - can constitute a powerful learning experience.

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