The Learning Triangle

I’ve been reading What Counts? by Brian Butterworth. It’s a very worthwhile read for anyone interested in the psychology of learning. The book’s subtitle “How Every Brain is Hardwired for Math*” is appropriate, but not quite complete. I think Mr Butterworth’s book, although aimed at understanding the mathematical brain, goes a long way to explain learning in general.

The concept I’ve latched on to recently is that of the three critical arms of learning: Facts, Procedures and Understanding. Without any one of these, learning will be at best incomplete, and at worst, non-existent.

Case studies are given which show that children who learn their addition and multiplication facts by rote will often still make “obvious” conceptual errors when working out a problem, because they lack the understanding of why 3+5=8. They’ll be able to recite that sum (or tell you “eight” when you ask “What’s three plus five?”) but when presented with 13 + 35, they won’t necessarily apply the fact, or if they do, they may apply it in the wrong way. In fact, children who understand what addition is but don’t know their facts were shown to have been MORE successful later on, because they were able to work out the answer to a problem even if they didn’t know the answer to a smaller, “easier” one, such as 3 + 8.

Similarly, children (and adults, I should add) may know a procedure for multiplying two-digit numbers (carrying, and all that), but may apply the steps in the wrong order if they don’t understand why the procedure works.

Even understanding isn’t enough by itself; if you don’t have the facts and procedures in the first place, you may be able to work out simpler problems, but more complex ones will be too difficult. It may be that the facts, the memorisation of certain key details that seem to come up often, will happen as a result of pracitcing a concept (understanding), and that procedures may even reveal themselves to the learner over time. But then that just suggests that these two elements are important after all, at least in terms of efficient calculation.

The answer, then, is to ensure children learn with all three principles in mind.

* The copy I have of this book has clearly been marketed to the US, but there is strong evidence that Mr Butterworth’s original audience, or at the very least his own experience, is from the UK, specifically England.