Efficacy: ‘the ability to produce a desired or intended result.’
Efficiency: ‘achieving maximum productivity with minimum wasted effort or expense.’
In this post I’ll try and explore an aspect of mastery in mathematics that I’ve been thinking about: the fundamental (and often implicit) understandings that underpin effective and efficient strategies.
There are many ways to solve mathematics problems. To take a previously used example, consider the area of a triangle. The GCSE formula booklet gives it as: 1/2 x b x h. Yet in the past I have usually begin by teaching it as ‘b x h divided by two’. Why?
- It doesn’t require pre-knowledge of fraction multiplication
- It explicitly encourages students to multiply the lengths first, before halving; this avoids error-prone situations where students start by halving an odd-numbered length and then have to attempt to multiply this by another integer. (Of course, the commutativity of multiplication means the GCSE formula doesn’t necessarily lead you to these errors; nonetheless, such an understanding is implicit rather than explicit within the formula, compared to the formula I teach.)
For these two reasons, I think my version of the formula minimizes cognitive load (though that might not always be a good thing) and is a good place to start. In other words, I think this formula has greater efficacy than the alternative: it is more likely to get my students to the right answer, and less likely to lead them into making mistakes.
Yet this strategy isn’t always very efficient; and I think this reveals something important in teaching for mastery.
Take this triangle:
My preferred strategy would work here. Many of my students would have to do a bit of long multiplication and then the bus stop method, but I’d be confident they could do that. Again, it’s effective.
However there’s a much quicker way if you simply do 1/2 x 18 first, allowing you to work out the area by doing 9 x 7 – an automatic times table fact (or at least should be). This method is more efficient: you get the answer much more quickly. Then again, it has its limitations: if a student only knew how to do this method, then they could quite possibly be stumped by a triangle with a long odd-numbered perpendicular height (say, of 19cm) and an odd-numbered base, since they’d have to do a bit of decimal multiplication.
This contrast in methods comes up a lot. They both have clear advantages and certain drawbacks.
But it’s not always as clear cut a choice between efficacy and efficiency. Another example is in equation solving:
Assuming the balance method, there’s (at least) two ways to do this:
- Minus 16 from both sides, and then divide/multiply both sides by a negative number
- Add 3x to both sides, thus giving you a positive coefficient on x again.
At first I only showed one method to my year 10s – the former. This was again due to considerations of cognitive load: they are used to cancelling out constants, then cancelling out the coefficient on the unknown; the only new piece of information is to pay attention to negative coefficients. They happily went ahead with this and showed good success in such questions.
Yet a few lessons later, I’m teaching inequalities:
The previous method works, of course, except there’s a big new caveat – don’t forget to flip the inequality when you divide by a negative number! Yet by this stage I’m not so happy with my previous choice. ‘Flipping the inequality when you multiply/divide by a negative number’ is an isolated fact, completely unrelated to anything else my students know or have experienced, and so I’m not optimistic about its retention. On the other hand, if in the first place I had taught them to +3x to both sides, this method would have extended to inequalities without similar problems of learning new, hard-to-remember facts.
A working definition might be helpful here. In general, I think an effective method is:
- easy to remember due to being strongly linked with previous, well-understood concepts;
- can get you the right answer in all/most cases of the problem
Whereas an efficient method:
- enables one to find answers quickly
- requires a deeper understanding of less obvious fundamental concepts
- but doesn’t always generalize well to all problems, leading to potential misconceptions
As I’ve thought more about these two contrasts, there isn’t a clear distinction between the two, as my equation/inequality dilemma shows. My initial choice of method for equation solving was based on considerations of efficacy; I wanted to link solving such equations with previously-encountered methods to solve equations, so as to minimize differences and thus bolster memory. Yet in the context of inequalities it was no longer a particularly effective method.
It seems to me that I have to make multiple decisions like this every day, given the proliferation of methods for maths problems. I teach 4-5 different topics a day and the scheme of work means I have to cover a lot. As a result, the efficacy consideration usually wins out. (A spiral curriculum means that in one fortnight with one class, I’m meant to have covered area, perimeter, surface area, volume, and Pythagoras). But to be crystal clear, by efficacy I really mean short-term efficacy – what method helps most immediately with that specific topic of the day. This teaching for short-term efficacy can lead to a lack of efficacy in related topics further down the road, just as in my equation/inequality case.
This point came home to me in an A-level class recently. First, back to the different methods for finding the area of a triangle – as discussed, efficiency here really requires understanding these four key concepts:
- The commutativity of multiplication
- The equivalence between dividing by x and multiplying by 1/x.
- The interchangeability in the order of division and multiplication
- Applying all of these insightfully when odd numbers are involved.
But a few weeks ago, I realised that these exact same mathematical understandings were hindering my C1 class. They were calculating the sum of arithmetic series using the formula (n/2) x (a+L). Pretty much all of them (A-A* at GCSE maths, of course) didn’t realise that it was entirely possible to divide (a+L) by 2, rather than n; since C1 is non calculator, they were spending lots of time doing calculations like 22.5 x 42, instead of spotting its equivalence with 45 x 21. No one had ever taught them key understandings 1 and 3; they were genuinely surprised when I told them that the ‘/2’ could simply be done to (a+L), and I had to convince a few of them why this was the case. While they were all successful maths students so far, they were lacking a pretty fundamental understanding that could have helped them be a little quicker in the future.
Off the top of my head, here are some other topics in maths which would benefit from making those 4 interrelated key concepts really explicit:
- Simplifying expressions such as (2a x 3b)/(15c) – many students struggle with simplifying the coefficient to 2/5 and writing it as 2ab/5cd
- Rationalising the denominator and then simplifying – similar to 1 – many students struggle with dealing with constants multiplied by surds
- Simplifying algebraic fractions when there’s some sort of factorisation involved (and knowing when further simplification isn’t possible)
- Equations with fractional coefficients, expressed in a variety of ways
- Area of trapeziums
- Volume of triangular prisms, pyramids, cones
Interestingly, the first four topics are ones that my students always find tricky to grasp, let alone master – yet I believe that at their core, the difficulties really revolve around the concepts of commutativity etc. mentioned above. What’s more: with these more advanced topics, it’s a struggle to teach effective methods, let alone efficient ones. While students can successfully find the area of a triangle without utilizing these key understandings, these key understandings are increasingly vital for questions such as simplifying (2a x 3b)/(5c x 3d). It’s harder to ‘paper over them’ in such topics.
So, how does this relate to planning, teaching, and mastery? The NCETM explains one aspect of mastery as ‘Concepts are often explored together to make mathematical relationships explicit and strengthen pupils’ understanding of mathematical connectivity.‘ I think the four key concepts above are a key example of such mathematical relationships and connectivity. Yet I have never explicitly devoted much time to teaching them particularly explicitly, or testing students on these concepts. Of course they should go beyond a lip-service hour on ‘commutativity’, and be interleaved throughout terms and years on various topics like area and then equations – and that takes some serious long-term planning.
I guess the first step is to recognize that these fundamental concepts are out there, and that maybe we don’t devote as much time to them as we should. The second is to begin identifying them – I’ve picked up on commutativity and interchangeable representations of multiplication and division. The third is then to incorporate them more explicitly into teaching and schemes of work. I don’t think a cursory hour long lesson on ‘commutativity’ is quite enough, but the exciting thing about these fundamental understandings is that they crop up everywhere and should be easy to interleave throughout various topics, if it’s first taught explicitly and then referenced and tested throughout a variety of contexts – say in year 7, you could weave it into lessons on multiplication, order of operations, area of triangles, area of trapeziums, volume….
I think another key batch of understandings revolve around the equals sign/inverse operations/the balance method, and I have some first thoughts of exciting ways to interweave that – for example, why don’t we explicitly teach mensuration as a subset of substitution, formulae, equations, and rearrangement, whilst introducing ‘reverse’ questions from a younger age? Currently, in my experience, the combination of algebra with mensuration doesn’t happen until GCSE, yet in theory they could be done a lot earlier, given the equation-solving skills that most KS3 students are taught.
Any other suggestions of similar concepts/understandings/curriculum sequencing ideas would be very welcome – please leave a comment.