It’s happened again: another would-be educational revolutionary has decried the memorisation of times table facts. The message Jo Boaler puts out there as a “Stanford professor” is old, tired and terrifying. In her analysis she is right enough, so that her argument has merit, but she is terribly wrong in her conclusion.
The reasonable but misguided belief that times table memorisation was a Victorian invention, no longer valid in a modern education, is a myth that she and others, such as Sugata Mitra and Conrad Wolfram, continue to perpetuate with impunity; the latter two being little more than “laypersons with an opinion”, with not a teaching qualification between them.
The fact is, we already have moved away from times table memorisation in schools, and the results have been catastrophic; those of us in secondary education bear witness to this every day.
Here’s how the argument goes: timed tests can cause anxiety; true. Anxiety can trigger a release of the hormone cortisol, which can inhibit working memory; true. Therefore, we should no longer ask that children memorise their times tables; false. And no one needs to memorise their times tables anyway; false.
The key to this debate lies in understanding how working memory (WM) functions. WM is the site of our conscious awareness; we need it both for processing thoughts and holding on to information that we are thinking about. It is very limited in capacity - think three to four “slots” - and importantly the capacity can vary from person to person, with no hope that its size will ever change; we’re stuck with what we’ve got.
Boaler argues that she doesn’t need to know her times tables because she has processes for calculating the results quickly, as needed. For example, one can calculate the value of 7 x 8 by learning that multiplication by 8 involves “doubling, doubling and doubling again” until you arrive at 14, 28, 56. The advantage of this approach is that it works for all numbers; you could calculate 120 x 8 this way as well. This is true; I didn’t really know my times tables either, before becoming a mathematics teacher, and I also used strategies like that to get by. Yet, of course, it is only people who are fortunate enough to have larger working memory capacities, such as Stanford professors, who can hope to be successful this way.
For a child with smaller than average working memory capacity, trying to process any kind of mid-level arithmetic or basic algebra becomes a hellish nightmare, as they are still left painfully trying to process basics that should have been embedded from a very young age. On the other hand, if you simply know that “seven eights are 56” then recalling that fact from our infinitely-capacious long-term memory uses no WM at all; it liberates a person’s WM to instead focus on the more demanding, higher-level ideas.
The call to demonise times table memorisation further misses an important and fundamental component of mathematical development; the idea of relational understanding. Coined by Richard Skemp in the early 70s, this term refers to seeing the many related connections between mathematical facts, concepts and processes. This includes relationships between numbers. If you had to recalculate the value of 7 x 8 every time you needed it, you would never appreciate that there is a relationship between the numbers 7, 8 and 56.
The damage this does is evident every time we have to watch a child struggle in pain to do something as straight forward as simplifying the fraction 49/56, or factorising 28x + 56. For someone who knows their times table facts, this process is simple because the numbers 28, 49 and 56 are all intimately connected to the number 7; it’s painless, even obvious, that 49/56 should become “something over 8”, since 56, 7 and 8 are connected as a related triad of numbers. Being able to “notice” this renders such operations facile; being unable to notice this makes such operations painful and frustrating.
Ironically, it is being abandoned to this mental abyss that imprisons a person in the very state of anxiety that Boaler would have us avoid. In refusing to help children memorise times table facts, we are casting them out into an unforgiving cognitive oblivion.
What is missing in Boaler’s analysis, in her conclusion, is the realisation that it might simply be the methods at fault, rather than the objective. Indeed, there are many ways of helping children memorise times table facts that induce little or no anxiety at all, and in the best cases, promote real excitement! The greatest example I have seen of this is a programme called Times Table Rockstars, created by Bruno Reddy.
The rhetoric of “down with times tables” is an old one; the yoke of whose destructive legacy I felt, I hoped, we were finally beginning to cast off.
As we slowly move towards a system that embraces times tables memorisation - as supported in schools minister Nick Gibb’s letter to teachers visiting Shanghai, and by innovators such as Reddy - it is important that we now challenge this dead idea wherever it makes its effort to become the reanimated living-dead.
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