Following my recent posts on questions to ask your child’s primary school teacher (here and here), I had a request to expand on my comments about the teaching of mathematics. There are a few issues surrounding maths that I believe parents should know about but before I go into that, I wish to make two points. Firstly, a fundamentally misconceived maths program taught by dedicated, evaluative teachers will always be better than one that meets the highest standards of evidence but that is taught badly. In my school we have specialist mathematics teachers and I think this is far more important than the specific details of the program. Secondly, the intention of this post is to inform parents and not to have a go at primary school teachers. Some teachers were offended at my comment that time-tables songs were not the best way to memorise tables. They thought I was suggesting that this is what many primary school teachers do. No – I was just setting up two contrasting alternatives in order to explain my point.
1. Discovery learning
Discovery learning is ineffective and most people tend to recognise this. So you don’t see many schools advertising their programs as discovery learning (apart from in AITSL’s illustrations of their teaching standards, bizarrely). Yet if it looks like a duck and quacks like a duck then it probably is a duck. And there are two powerful fallacies that drive people towards discovery learning. The first is the idea that we understand something better if we discover it for ourselves. We don’t. Secondly, we tend to assume that by asking students to emulate the behaviour of experts then our students will themselves become experts. Experts in maths are research mathematicians who make new discoveries so we should get our students doing that. Yet this is also fallacious thinking.
In primary maths, discovery learning takes on the form of ‘multiple’, ‘alternative’ or ‘invented’ strategies. Students are intended to make-up their own ways of solving problems and to solve a single problem in several different ways. Explicitly teaching a standard approach, such as the standard algorithm for addition, is discouraged. Of course, many students don’t discover much and so they pick-up these strategies from others or are led toward them by the teacher.
2. Big to little or little to big?
The kinds of alternative strategies that the students ‘discover’ are generally variations of strategies that we might use for mental arithmetic. Imagine I wanted to add 25 and 49. I would probably first add the 20 and the 40 to make 60. Then I can add the 5 and the 9 to get 14. Sometimes, little sub-moves will be encouraged as part of this e.g. take 1 from the 5 and add it to the 9 to get another 10 so that we have 70, then add the remaining 8.
Notice how this proceeds from big to little. We add the tens first and then the units. But when we add the units we find that we have yet another ten so we have to loop back and add this to the tens that we already had. This is inefficient when we get to larger numbers and is the reason why the standard algorithms generally start with the units first, then tens and so on. Indeed, students who use the standard approach seem to have more success, particular with larger and more complex calculations.
The objection to standard algorithms seems to be that kids can learn them as a process without understanding how they work. Presumably, they have to understand procedures they’ve invented themselves? This may be true if they really have invented them but I suspect such genuine invention is rare, with most children latching on to the ideas of others. In this case, these alternative procedures could also be replicated as a process without understanding.
In my view, students should be taught the standard approach and this approach should be explained to them. This requires the teachers to also understand how these processes work.
3. Words and pictures or actual maths?
Given that alternative strategies are meant to be ad hoc and contingent, there is no formal way for expressing them. You may see it done with pictures or even in words. Contrast this with the standard algorithms – their universality means that they follow a tightly defined set of notation. One way that alternative strategies may therefore be promoted is by insisting that students ‘explain their reasoning’ or draw diagrams when answering questions on homework or assessments. This is basically a way of marginalising the standard algorithms.
For instance, imagine the following question:
“A lottery syndicate of 13 people wins a total of $3 250 000. If the money is shared equally then how much would each member receive?”
A simple use of the long division algorithm is sufficient to explain what the student is doing, why they are doing it and to determine the right answer. If the student has gone wrong then the error will be easy to find. An insistence on words or pictures would be redundant unless you wish to privilege alternative strategies.
4. Maths anxiety and motivation
Maths anxiety is real. Some people struggle with maths – perhaps because they were not taught very well – and develop a fear of maths tests and maths more generally. It is complex and the chain of cause and effect is not entirely clear. Evidence does seem to point to timed tests as being associated with anxiety but perhaps better test preparation and framing would mitigate this. However, as well as advising us against timed tests, a whole raft of things that look and smell a lot like alternative strategies and related ideas such as the use of ‘authentic’ problems are proposed as possible solutions.
Authentic, real-world problem are considered good because the idea is that they will motivate children and so the children will learn more. In fact, a lot of discussion centres around motivating and engaging students. I am sceptical that many of the activities that are suggested as motivational are actually motivating for students and evidence suggests that motivation works the other way around. Maths achievement predicts motivation but motivation does not predict achievement. In other words, teach them maths, increase their competence and then they will start to feel more motivated about maths.
5. Cognitive load
Finally, it is worth mentioning that a lot of fashionable strategies are at odds with what we know about human cognition. Children should know their maths facts because that means that they don’t have to work out 5 x 8 whilst attending to other aspects of a complex problem. Those people who dismiss times-tables as ‘rote’ learning fail to take account of this. And so do those who propose big, messy, open-ended, real-world problems. Such problems have many facets and often contain information that is irrelevant to finding a solution. All of this needlessly increases cognitive load and makes learning less efficient.