Mathematics education uncovered and recovered

Times tables again

After a short break, the Department against Education launched a new attack on children. This time the attack takes the form of a new compulsory national test for children in Year 4, the times tables test.

The obsession with memorising times tables has always puzzled me. Weekly times tables speed tests, the practice of chanting the tables during the lunch breaks, and times tables competitions give the impression that learning them by heart is considered the most important part of school maths education.

In “Seven Myths about Education”, Daisy Christodoulou emphasisesthe need for memorising times tables:

“Just learning that 4 × 4 is 16 will be of limited use. But learning all of the 12 times tables, and learning them all so securely that we can hardly not think of the answer when the problem is presented, is the basis of mathematical understanding. If we want pupils to have good conceptual understanding, they need more facts, not fewer”

The phrase “all of the 12 times tables” makes me smile. Numbers up to 12 is not all there is, so why stop at 12? Would memorising the facts up to 27 x 27 be even better? Back to serious mode, I should note that there is hardly any practical use for remembering times tables beyond 10 x 10, unless you are going to compete in “Countdown”.

It might be a good idea to base conceptual understanding on facts, but times tables facts on their own do little to contribute to that understanding. What matters are patterns in the sequences of answers that reveal the commutative, associative, and distributive laws of multiplication. Without recognising these laws, the pure knowledge of times tables facts is useless.

If a child is lucky to have spotted the regularities in the times tables and is encouraged to use these patterns, then they would grasp the concept of multiplication and apply it successfully. However, focusing on memorising the answers is often too stressful and distracts from exploring the patterns, thus doing more harm than good.

Knowing times tables by heart (be it “all the 12 times tables” or just those up to 10 x 10) is neither necessary nor sufficient for good conceptual understanding of multiplication. I have met many children who had learnt their times tables brilliantly, but still did not realise that, for example, 5 x 7 is half of 10 x 7, or failed to use their knowledge of 12 x 12 to work out 13 x 12. I have also met children who were deemed to be the bottom of the class in terms of recalling their times tables, but who demonstrated an excellent grasp of the properties of multiplication and division, and could apply them successfully for solving problems.

Given that this unhealthy obsession with times tables is ubiquitous, I cannot see what the government’s intention to reinforce it by introducing the tests would achieve, apart from extra pressure on children and an opportunity for the ministers to issue pathetic statements about improvements in education. The idea is a lasting legacy of Gove’s damaging reforms and a response to a consultation on primary assessment.

The DaE paper says: “There is strong evidence to show that being able to recall multiplication tables with fluency plays a crucial role in being able to solve more complex mathematical problems involving division, algebra, fractions and proportional reasoning”.

The evidence they mention is not provided, so it is impossible to discuss its credibility. However, whatever research – if any – was the basis of the decision, it probably shows no more than correlation between remembering the times tables and ability to solve mathematical problems. The proponents of times tables testing might know their own times tables, but they seem to have forgotten another principle taught in school that is much more important for policy-making: correlation is not causation. Both good knowledge of times tables and ability to solve mathematical problems are affected by various factors such as interest in dealing with numbers or memorisation skills not to mention the quality of teaching, so the correlation is not surprising. However, it is wrong to assume that demanding rote learning of times tables is the best way to improve children’s ability to solve problems. After all, the causation, if any, might be the other way round: solving mathematical problems could require a frequent use of times tables and results in learning times tables by heart.

As for the times tables being “critical for everyday life”, I can honestly say that my excellent knowledge of times tables is of no use in my everyday life. Seriously, how often would one need to recall that 7 x 7 = 49, that is,  7 + 7 + 7 + 7 + 7 + 7 + 7 = 10 + 10 + 10 + 10 + 9?  And if the special occasion requiring the knowledge of this fact does arise, it will not be difficult to work it out or reach for a calculator.

Knowing that 7 x 7 =49 and understanding the distributive law would help me to work out something more complicated like 17 x 7 = 119, but then again I would hardly need it in the first place. At the same time, the laws of multiplication go well beyond practicalities of everyday life of an average person. They open a door to a world of abstract mathematics,  and introducing children into that world, showing its beauties and peculiarities, should be the true purpose of mathematical education. Sadly, with initiatives that prioritise rote learning of pointless facts, the door to this world remains shut.



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