Friday 31 January 2020

Rules First, Exceptions Later

I know that we as modern, innovative maths teachers like to say that maths isn't about rules and that it's about relationships and reasoning and all that stuff. But there are some mathematical processes that students have to be able to carry out in a fairly standard way and for the sake of this discussion, we're going to call them rules.

Just so you know what I'm talking about it, here are a few examples of processes that involve rules:
-adding fractions
-finding the area of a sector
-finding the volume of a prism
-solving equations of the form 5(x+3)=7.

When we teach these processes, I think the most obvious way to do it is to start off with easy examples that students can work out without having to use a formal rule, and then extrapolate their reasoning to apply to more complex situations, which is used to develop the rule that is then eventually used. But it's occurred to me that this intuitive approach may actually be inefficient and even detrimental to long term ability to recall these processes.

Let's take adding fractions as our first example. I'm guessing you probably teach this in the same order I've always taught it in:
1) adding fractions that have the same denominators
2) adding fractions with similar denominators (like 5/8 +3/4)
3) adding fractions with denominators that share common factors (like 5/6 + 3/8)
4) adding fractions with denominators that are coprime (like 3/8 + 5/7)

The logic behind this sequencing is obvious: number 1) requires a one-step process, number 2) requires a two- or three-step process, and number 3) requires a three- or four-step process, so this suggests a kind of hierarchy. Number 4) involves applying a rule that you have to use when no other option is available, so it makes sense it goes last.

Students often intuitively understand the rule for number 1), so they can master it within seconds. They are then pretty accepting of the rule for number 2), as it's easy to see that writing 3/4 as 6/8 will allow you to apply the rule from number 1). This then logically leads to a rule for number 3). Number 4) is technically the same rule as number 3) because you are still finding a common multiple, but in practice your approach to these questions is slightly different.

In summary, the rules students need to learn are:
1) Since the denominators are the same, just add the numerators.
2) The denominators are kind of the same, so change one fraction so that it matches the other, and then apply rule 1).
3) The denominators are similar, so identify what the lowest common multiple (or any common multiple) of the two denominators is, and then change both fractions so that the denominators match, and then apply rule 1).
4) Spend a while trying to find a common multiple using your times tables facts, realise you can't think of one, multiply the two numbers together to create a common multiple, then change both fractions to make that denominator. Then apply rule 1).

This is quite a lot of rules to remember just for one process.

Now consider an alternative (still referring to the original four cases):
1) Since the denominators are the same, just add the numerators.
2) Multiply the numerator and denominator of each fraction by the opposite denominator to make the denominators the same. Then apply rule 1).
3) Multiply the numerator and denominator of each fraction by the opposite denominator to make the denominators the same. Then apply rule 1).
4) Multiply the numerator and denominator of each fraction by the opposite denominator to make the denominators the same. Then apply rule 1).

In this version, there's only two rules to remember. Doesn't that sound so much nicer? The standard rule that works for case 4) works for cases 2) and 3) too, so why not just use this rule for all three?

Now I know a lot of you are feeling very upset about the fact that I am endorsing the idea of students adding 3/4 and 5/8 by making an unnecessarily large denominator of 32. You may also point out that this way would involve a lot more simplifying at the end. But hear me out!

I'm suggesting that we treat fractions with dissimilar denominators as the rule and the fractions with similar denominators as the exceptions. We teach the students the rule, we make sure they're super-duper confident with this process, let them feel really successful in doing this, give them lots of drills in this so that they're well practised and then teach them the exceptions. Say to the class, "of course with question e) there is a slightly easier way to do it, can anyone spot it?" and then discuss this and identify a few more exceptions like this and then do some practice with these. Of course, it's not really an exception because the other rule will still work, but it's exceptional in that there is another method you could use.

Let me give you another example: finding the area of a sector. I've always started out by finding the area of a semicircle, because students know straight away that this involves halving the area of the whole circle. They'll also get how to do quarter circles straight away. Then I would throw in a 120 degree angle or a 60 degree angle, and with a bit of thought they'd be able to tell me that you need to divide the area by 3 or 6. And then I'll put up a 24 degree angle and watch them struggle. They might eventually do 360 divided by 24 to get 15 and then tell me we need to divide by 15. And then finally I'll go to a 37 degree angle and they wouldn't be able to do it. At that point I would tell them, well the fraction that's shaded is 37 out of 360, so we do blah blah blah.

Now consider an alternative: start off with the 37 degree angle. Ask the class what fraction of the circle is shaded. Eventually they'll get that it's 37 out of 360. Then simply show them the correct calculation. And that's it. No faff. If they come across a question with a 24 degree angle, they can still do the same calculation, there's absolutely no need to work out that you can divide the circle area by 15. And even if you have a semicircle, the same calculation will work, and it won't exactly take much longer that halving (especially if it's a calculator question). Feel free to teach semicircles and quarter circles and even third circles as exceptions afterwards, as a little bonus shortcut that students will grasp easily. But I don't really see how starting with that will help students understand the process of finding the area of the 37 degree sector.

Another example: finding the volume of a prism. Isn't it weird that we learn how to find the volume of a cuboid separately from finding the volume of a prism, despite a cuboid being a prism and the prism rule working perfectly well for a cuboid? Why not just teach how to find the volume of a prism, and throw in cuboids, triangular prisms, trapezoidal prisms and compound prisms altogether (assuming they are all fluent in finding the area of rectangles, triangles, trapezia and compound shapes)? I would love to add cylinders to that list too but I found out embarrassingly recently that a cylinder is not, as I always believed it to be, a prism. Who knew?

One final example: solving equations that look like 4(x+2) = 8. Now there are two obvious methods to solving this equation. You could divide both sides by 4 or you could expand the bracket first. Dividing by 4 is a nicer method in this case and gets you the answer faster. However, what about the equation 4(x+2)=7? In this case, it would be unwise to divide by 4 first, so you should really expand the bracket. But there are no examples where expanding the bracket first doesn't work, so why don't we just teach our students to always expand first? (I have a feeling a lot of you do teach them this actually!) There are a lot more numbers that don't divide nicely by 4 than numbers that do (although Hilbert might disagree...) so most of the time the dividing method won't work, but the expanding method always will. It's nice to show the dividing shortcut as a little aside, once the students are really secure in the process of solving other equations with brackets. But I believe fluency in the rule should take precedence over the ability to identify exceptions.

I hope I've managed to explain my point well enough in this post. It would have been much easier if I could scribble stuff onto a whiteboard at times! Maybe that's the push I need to finally start the NotJustSums YouTube channel...

Oh and I should mention that this post was inspired by things I heard in Craig Barton's podcast which is excellent. My favourite episodes have been his interviews with Dani Quinn and with Tom Francome, and I strongly recommend that you check them out.

Emma x x x

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