## November 01, 2010

### Do We Really Need Formulae?

Ok, so I was at my PP1 school today and I observed four exemplary lessons with three excellent teachers. What I'm about to blog about should not be seen as a criticism of the teachers, but of maths as a subject.

The first lesson was a year twelve lesson about coordinate geometry: finding gradients, midpoints, lengths and equations of straight lines. The pupils were using this formula: y-y' = m(x-x'). If you have the gradient and a point, you can work out the equation, and if you have two points but no gradient, you can also work out the equation. This is not a formula I ever used in school, although I do remember being told it. When I was working out equations of straight lines, I sketched them first, and then just sort of worked it out. I'd find the gradient first (change in y over change in x, a sort of formula but not as rigid and equationy as the above) and then if it was, say, 3 I'd write  y = 3x + c , substitute my point in and rearrange to get c. This all felt quite intuitive to me after I was told it once, and I never forgot how to do it. I observed some very talented maths students asking me: is it y on the left or x? Is it y-y' or y'-y? What if I don't have two ys and two xs? The formula is emphasised in text books and such because it's supposed to be easy to remember and execute. But wouldn't a more relational understanding of coordinate geometry be better? Pupils were asking me, what's the formula for a perpendicular bisector? Well I have no idea if there is one, but if there were, it would have to be pretty complicated. But if you understand how points and gradients and lengths work, you can work out absolutely anything, given enough information. I think what we need are relationships (thought of as as sentences, pictures, or even just feelings) rather than formulae. Pupils doing A level maths should have the cognitive ability to think relationally rather than instrumentally.

The next two lessons were entirely different year groups but partly involved the same concept: formulae for areas and perimeters. I tried to think carefully about my thought processes when it comes to finding the perimeter of a rectangle. Do I think of a formula and substitute in the length and width? After metacognising on this, I realised I don't really: I see the four lengths of the rectangle in my head, and add them up in whatever way is convenient: sometimes I double both and add them, sometimes I add them and then double. The formula 2l + 2w never enters into it. In the year 7 class, the teacher asked pupils for their ways of working out the perimeter. They came up with: l + w + l +w, (l+w)x2, and l+l+w+w. The teacher then said, although they're all really good methods, the one that is most recognised is 2l+2w, and that they should learn that. I sort of got the feeling the teacher resented having to say this. The pupils had already done several questions on finding perimeters of rectangles, and were then told to do some more, using the proper formula and writing it out in a specific way. For some reason, the pupils who, ten minutes ago, could find the answers, were suddenly getting wrong answers and saying they were stuck! I don't doubt that the pupils will eventually get the hang of it and will be successful with the formula in the future, but it does still feel like a step backwards in terms of learning.

The lessons didn't mention trapeziums, but I'm going to reflect on my way of working out the area of them. For some reason, I could never ever ever remember the formula for the area of a trapezium. It was one I always had to look up (even during my degree, how embarrassing!) Just now, I've been drawing pictures on a scrap of paper, trying to see why the formula is 1/2(a+b)x h. I remembered Charlie telling me a few weeks ago that it was the average of the two parallel sides times the height. I was trying to draw a picture that shows why it's the average. To be honest, I still can't! There are just too many types of trapeziums (proper ones, parallelogrammy ones, wonky ones...) for me to work it out. This is probably why I could never remember the formula. You see, the area of a triangle fornula makes perfect sense to me, it's half a rectangle. The trapezium one just seems odd to me. However, as long as I can sketch it, I can work out the area of any trapezium just by breaking it into triangles and rectangles. My relational understanding of polygons allows me to do this. So what's the point in having a formula?

If you're interested in this sort of thing and would like to know what I'm on about when I say instrumental/relational understanding, read Skemp's classic paper (google: relational instrumental Skemp). And if anyone knows why the formula for the area of a trapezium is as it is, answers on a postcard to the "C" pigeon hole in the Avon building please!

Emma x x x

### 2 comments by 2 or more people

1. #### Jennifer Ingram

I can never remember the formula for the area of a trapezium either, but I have lots of pictures that help me work it out. Same is true of A level trig identities :-) I much prefer being able to work out a formula than remembering it in general, it gives me an understanding of how and why the formula works…

04 Nov 2010, 22:17

2. Ha Emma, I’m getting frustrated by this too. “Miss what’s the formula for ….” then when they’re given it they just mindlessly plug numbers in, often making outrageously obvious mistakes. Really shows their lack of understanding. On a similar note, I’ve found pupils like to avoid drawing pictures, because smart people do things in their head and little pictures are for babies. #

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Break’s over, I’ll finish this comment tonight!

08 Nov 2010, 11:23

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