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June 10, 2011

The best way to teach solving equations?

I think this is one of the things that maths teachers argue over the most.

There's the "move it to the other side" method, the "do it to both sides method" and (one I'd not heard of until recently) the "cover up method".

Method 1 is what I was taught and what I use myself. I knew that it was not a very good method as it does not explain why you're doing that and how it works. I defended this method forcefully, sure that my way of thinking must be the best way (I'm arrogant like that). But since teaching both methods at my placement school, and teaching different methods to my one-to-one tutees, I've come to realise that the second method is much less confusing, more intuitive, and more memorable.

Here is how I would write a model solution:

3x + 5 = 10

    - 5      -5

3x       = 5

3x/3    = 5/3 (written as fractions)

And then I'd cross out the two threes in the fraction on the left because they cancel each other out.

Writing the -5 underneath makes it really easy for pupils to see what they're doing and it's really hard to make a mistake.

Doing the dividing bit as fractions is good because it encourages pupils to leave things as fractions, it makes division easier (instead of actually dividing, just cancel down the fraction as far as you can in easy steps), and it means that the whole side will be divided and not just the last term.

That is the method I would recommend for higher-tier pupils.

The cover-up method is one I cam across recently and I think it's really good because it's the way a pupil would intuitively try to work out the answer.

In equations like 10 - x =7, most pupils can just "see" that the answer is 3. In equations like 3x + 4 = 10, it's harder to just seei it, but using the cover up method you can. Cover up the 3x, and say "what add 4 gives you 10?" They'll say 6. Then you uncover the 3x and say, "well then 3x must be 6. So what's one x?" And they'll say 2 (after painstakingly counting on their fingers, if they're anything like my pupils!)

I wouldn't teach higher tier pupils this because it's hard to apply it to more complicated equations (although it obviously still works). But for lower ability pupils it seems to be less confusing.

I'm glad I'm starting to realise what methods I'm going to be teaching because I want to be consistent, so that pupils don't get confused with lots of different methods.


April 12, 2011

No bad classes, just bad teachers

I've always thought that there ARE such thing as bad pupils and bad classes. Classes that are practically unteachable, classes that are impossible to manage. There's a year 9 class at my PP2 school that might have fallen into that category. But I witnessed a miracle recently.

Their class teacher is an NQT who was at Warwick last year. She is young and very calm, quiet and soft. She told me that she hated giving out punishments or even warnings, because she felt guilty. I suffer from this affliction too. Observing her has taught me so much about behaviour for learning. Let me tell you why...

This year 9 class was awful. Constant low-level disruption, everyone shouting out, little work going on, low motivation, etc etc. But this amazing teacher decided she'd had enough of that. She saw an opportunity and took it: the year nines were starting the GCSE course, so she decided that would be a great time for a reform.

Everything about this lesson (and the subsequent lessons, there's been 4 now I think) was meticulously planned and perfectly executed. Here is a blow-by-blow account:

> textbooks, exercise books and rulers had been set out before the pupils entered. The date and title were written on the board, along with the instruction to copy these down into their book.This was good because it let the pupils know that things were going to be different this lesson, and also it gave the lesson a nice, orderly and purposeful start.

>There was a new seating plan.This was good because pupils who are often silly together had been separated.

>She started the lesson by explaining that they were starting the GCSE course and hence were technically year 10s. She explained the importance of GCSEs and getting a C (this group is predicted Ds and Cs). This gave the pupils the impression that maths lessons had taken on a new importance. Calling them year 10s made them want to act more mature than normal.

>She then explained that as year 10s, the discipline will be stricter. She outlined her system of warnings and consequences. She emphasised that the class was a team, and they had to work together to achieve a good result. Outlining the consequences made sure the pupils were aware that they will be punished, and more strictly than before. Emphasising that their class is a team ensures that pupils pay less attention to the ones who disrupt.

>They started the work, and the pupils were impeccably behaved.

>The work was quite easy, so the pupils felt really motivated by the fact that they could do GCSE level work.

>She kept giving out positive comments to reinforce the good behaviour.

The transformation was amazing. What's even more amazing is that in the lessons that followed, they were even better.

Another good thing I saw:

>Before giving them a load of questions to do, she wrote on the board some grade boundaries for E, D and C. She told them to choose a target for the lesson. Then they did the work, and then marked it. Then they counted up the marks and see what grade they got for that lesson. This really motivated the pupils to work hard, and they were so happy to meet their targets.

What this has taught me is that "bad" classes can change. All it takes is a change of approach. Although this teacher had perhaps been too soft on them before, she was able to take back control. I've always been under the impression that if you're not strict enough from the start, the class will never respect your authority. I know now that that's not true.

We're starting a new term in two weeks. Why not use that as an opportunity to have a fresh start with your most difficult classes?

Emma x x x


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


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