Are Our Students Thinking Too Much?

"It is a profoundly erroneous truism that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilisation advances by extending the number of operations we can perform without thinking about them." 

-Alfred North Whitehead, An Introduction to Mathematics

How many times in your life as a maths teacher have you said to your students, "think about what you're doing!"?

And how many times in your life as a maths teacher have you thought to yourself, "how do you still need to use your fingers to know 4 times 5 is 20?"

I know that I have these two conflicting belief systems. I believe that knowing times table facts, number bonds, the values of sine, cosine and tangent for common angles, the quadratic formula, and probably loads more things are very very useful. I would go so far as to say important. Maybe even so far as to say, your exam success depends on it.

But at the same time, I believe that rather than relying on memory, we should have a solid understanding of maths and think about what we are doing and why at each stage of a process. This is why I am conflicted.

A good example is long division. Students in UK Primary schools are generally taught chunking as their primary method of long division. On the one hand, I'm in favour of this. Chunking is my preferred mental method, and I have even started doing it as my main written method. However, students (especially those who do not know their times tables very well) are slow with chunking. It also takes up a lot of space. In secondary schools, I think most teachers teach the traditional bus stop method of long division. Students find this difficult at first but after much practice they can use it efficiently and accurately. But when they are used to this method, they do it without thinking about it. Is this a good thing or a bad thing?

Does it make sense to "save your brain" for the more challenging bits and rely on your subconscious to do the easy bits? Or should we be mindful at all times?

Should we memorise the quadratic formula or derive it by completing the square every time we want to use it? Should we memorise that cos(30) is root three over two, or draw a 30/60/90 triangle every time? Does it free up space in our memory if we don't memorise these? Do we free up space in our working memory or unburdon our central executive if we do memorise them? Which is more beneficial?

I think that in the UK, we place much much less emphasis on arithmetic fluency compared to other countries. I'm frequently told by friends and colleagues who went to school in Asia or Europe that you simply did not move on to the next grade or year group until you had mastered addition, subtraction, multiplication and division. And mastered means you can do these things efficiently and accurately. Over here, we move on to algebra before students know their number bonds. We teach fractions and percentages before they know their times tables. We say, it's OK if you're no good at arithmetic, you might be really good at shape and space. Is it the lack of arithmetic fluency that is holding UK students back? When it comes to maths, are our students thinking too much?

[Is it just me, or is that last sentence totally Carrie Bradshaw-esque?]

The GCSE for maths now has a much larger emphasis on functional problem solving. Teachers say this will be a challenge because our students are bad at problem solving. I think they're wrong. I think our students are brilliant at problem solving. But they do badly on these questions because the arithmetic involved in these problems poses a much bigger problem to our students than, say, Korean students. If you know how to do arithmetic automatically, without thinking, you only have the problem to think about. If you also have to think long and hard about the arithmetic, it interrupts your thinking process and tackling the problem becomes much harder. It also means you won't as easily spot neat tricks or patterns that simplify the problem.

For example, there is a Foundation GCSE question with a rectangular patio 3m by 3.6m. Henry has 32 square paving slabs with side length 60cm. Does he have enough slabs to cover his patio? If you are good at arithmetic you may spot that 60cm goes into 3m and 3.6m nicely. If you don't know your six times table, you won't notice this. That means you may choose a longer, more complicated method, like finding the area of the patio and the area of the slab and doing a division. This was a non-calculator paper so now students will be having to calculate 3 x 3.6 and 0.6 x 0.6, then use long division. It is much, much simpler to see that you need 5 rows of 6 slabs, which makes 30.

So although I'm a big fan of exploring, discovering, reasoning, and understanding, I'm starting to think that a bit of rote learning of efficient, accurate methods, and drilling of times tables (and their related quotients) would actually improve students' ability to solve problems a lot more than simply getting students to practise problem solving.

What do you think?

Emma x x x