Do you remember learning the multiplication table as a child? As a reminder of how difficult it can be for a child to learn these facts for the first time, try memorizing the following nonsensical arithmetic “facts”:
Seven and five are ninety-three
Eight and seven are thirty-five
Five and three are twenty-nine
Three and nine are fifty-eight
Difficult, right? Now think how hard it must be for children to memorize the 36 multiplication facts in the times table, as they are often asked to do in elementary school.
“Learning the multiplication table is one of the biggest challenges that children must overcome using rote learning.”
Learning the multiplication table is one of the biggest challenges that children must overcome using rote learning – the memorization of facts through repetition. Granted, conceptual learning is important, because the multiplication table has its inner logic – for example, multiplying larger numbers yields a larger result. To become proficient in arithmetic and algebra, however, children must be able to recall the multiplication table up to 10 x 10 immediately, without relying on time-consuming strategies. A calculator or smartphone may be useful for solving a single multiplication problem, but it becomes a burden if that is merely a single step in solving a larger, multi-step mathematical problem.
One thing that makes it hard to memorize the multiplication table is similarity between facts – as in the case of 8 x 8 = 64 and 8 x 6 = 48. Unfortunately, many memory tasks become harder if the items to be remembered are similar. For example, it is harder to repeat a list of words if the words resemble one another. Similar items interfere with each other in our memory because they have overlapping representations in the brain, and our brains are not very good at coping with this “cognitive overlap”.
“One thing that makes it hard to memorize the multiplication table is similarity between facts – as in the case of 8 x 8 = 64 and 8 x 6 = 48.”
Overcoming the similarity problem
Although there are many similarities in the 36 multiplication facts, there are also many individual facts that are dissimilar, like 6 x 4 = 24 and 9 x 9 = 81. In my lab, we reasoned that children would find it easier to learn if a given lesson focused only on dissimilar facts, as similarity-induced interference would be minimal. We conducted an experiment in which we split the times table into several groups of dissimilar multiplication facts. We taught young children, who had not started learning the multiplication table, a group of these facts in each lesson. The children learnt them quite well. When we tried teaching similar facts in a single lesson, the children found it difficult, getting the answers wrong about 50% more often.
Our low-similarity method was very effective for memorizing the multiplication table. What’s more, the method seemed to be especially effective for an individual with a learning disorder who is particularly sensitive to similarity-induced interference.
This low-similarity learning method is very different from the method currently used in many schools to teach the multiplication table. Typically, children learn the multiplication facts in a column-by-column manner: first the products of 2, then the products of 3, and so on. In each lesson, children learn multiplication facts that are highly similar to one another. This may make the memorization challenge harder than it needs to be, especially for children who are highly sensitive to similarity-induced interference.
“Both low-similarity learning and column-by-column learning seem to be potentially useful.”
Combining learning strategies
Should schools drop the column-by-column method and switch to the low-similarity method? Not necessarily. The column-by-column method has its benefits – for example, it makes the relationship between multiplication and addition more transparent. This may help children understand multiplication at the conceptual level. It may also enable them to develop calculation-based backup strategies for solving multiplication problems. For example, if you forget how much 7 x 6 is, you can compute 7 x 3 x 2 instead.
Both low-similarity learning and column-by-column learning seem to be potentially useful, with each strategy probably serving a different goal. Column-by-column learning may be most helpful during the initial stages of learning, which focus on multiplication at the conceptual level. Low-similarity learning may be more beneficial during later learning stages, which emphasize rote memorization. If this is true, the best way to teach multiplication may be to start with the column-by-column approach at school, and then to switch to the low-similarity method in practice situations – in school, doing homework, or when playing multiplication computer games that aim to automatize knowledge of the times table.