# How can I perk up Pythagoras for my students?

10 January 2023

## Question

Nigel, who's a teacher, says he is looking for some tips on how he can get Pythagoras' theorem and bring it to life a bit for his students. Any tips? Can you help?

Ems - Oh wow. Pythagoras happens to be one of the subjects that I studied at uni. I got fascinated by all the different ways that you can actually prove Pythagoras' theorem. There's a lot.

Chris - Just for those not in the know, you have a triangle that's got a right angle on one side, and if you add the square of one side plus the second side, then they add up together to the square of the long side. Okay, so that's what he's saying.

Ems - Yeah. That's great because we've got that procedural fluency; we know the rule. But the thing is, do we actually understand what it means? And I think there's a huge difference there between being able to recite it and understand it. The teaching is focusing on the understanding. Now there was this great geometry teacher and what he did is he drew the triangle on the board, the right angle triangle, and then on each side of it he drew a square. Now imagine you've got those squares and you've made them out of gold. So you've got three squares of different size, you've got one along the hypotenuse and you've got squares off the other two sides. Now, if you want to engage your students, why not offer them which one they want? Do you want that single square or do you want the two smaller squares? Now, if the class has really understood what Pythagoras is about, they're going to know it's no contest.

Chris - It must be the same.

Ems - But what happens when you do that with a class? They tend to go for the larger square. Now part of that might just be, "I want the biggest, it's mine", put my arms around it. But it's also saying something about the understanding. So I think choosing activities like that where it goes deeper into the understanding, the conceptual understanding, rather than just focusing on the fluency. Yes, we need to know the facts, but there's no point in knowing them if you can't apply them. You get builders nowadays, they'll talk about having a 3, 4, 5 to get your roofs correct at the angles. And they may not associate that with Pythagoras, which is a shame. Here we have a system at the moment where people are saying they don't understand why they do maths at school and somebody's going, "I want to go off and be a builder. Why do I need Pythagoras?" And then they'll go and use a 3, 4, 5 to try and get the roof right.

Chris - You're so right though, because I can remember, I must have been about eight, nine years old and my maths teacher was trying to introduce the concept of area. And when you are that age when someone has centimetres or metews squared, it just doesn't really mean anything to you until she said, "I'll tell you the story of a monk who used to walk around the courtyard at the monastery and all four sides of the cloister." And the cloister was paved in one metre wide flagstones. And he found out that he could work out exactly how many there were and the area of them if he walked and counted them, or he could do it the quick way and times that side by the number along that side. And that vivid picture that it created in a seven, eight year old brain meant I engaged with it. And rather than it being foreign language, "metre squared, centimetre squared, what on earth does that mean? Oh, now I actually can understand what this is about."

Ems - And you've got a visual there. And visuals are so important in mathematics. We go throughout mathematical history and when we've got the visual, we've often made the breakthrough. A really famous example is Florence Nightingale. There she was in the Crimea, lots of soldiers injured, but many more were dying of their wounds and she was trying to campaign for better conditions for them. And so she collected the data, had all the lists, and you think, as a mathematician, you've got it sussed: you've got the data, you present it and you win the argument. But they were just lists of numbers. And until you bring it alive and do a visual, which is what she did with an area map, you suddenly realise the huge number of casualties. So it's another example, if you do a visual, either someone walking around a courtyard, do you want the gold on the larger square, or is it the area of the soldiers and the ones who died from wounds? All of these visualisations are really important. So many people just think of maths and instantly think of algebra and X. And I think that's the case in many subjects, you start studying one thing and then, in order to achieve your goals, you move into something else. I started off studying mathematics and then I moved into mathematics education when I realised in the UK only 6% of professors of mathematicians are female. What's happening to all those others? It's our most popular A level subject but we're not converting it into female mathematicians. Think of the talent pool we're losing! So I think once you've got a problem or a challenge, you try to address it, you get those skills and then you can move into those things.