Challenges to Learning Multiplication

Jonathan Firth's Memory & Metacognition Updates #81

If you look back on your school days, perhaps you remember spending time memorising multiplication tables (or ‘times tables’).

It’s widely recognised that accuracy and fluency in mentally calculating ‘number facts’ – sums such as 7x4 – are important for all learners. These underpin many types of problem from science to cooking, and therefore play a role in schools, careers, and everyday life.

But how should we best teach them? A reader of these updates from Penicuik High School in Scotland asked me if memory research could point to an effective strategy, and I’m happy to share some thoughts and evidence.

Memory

The teacher in question asked about memory tricks, and certainly, the challenge here is to get the number bonds so well learned that they are more or less automatic. Doing so will free up the limited space in working memory (see update #70).

However, memorisation is not easy! It also tends to be done badly.

For most people, the intuitive assumption is that to memorise something, you need to focus on repetition. My own experience of multiplication in school involved a lot of (short-term) repetition.

However, as discussed previously on these updates, desirable difficulties such as spacing, retrieval and variation lead to better outcomes. This could mean:

• Practising a multiplication after a delay (spacing)

• Varying the context. So, for example, rather than always asking about 4x6 straight after 4x5, students have to practise the number facts out of order (as they would in real problems).

• Focusing on retrieving number facts from memory, again ideally out of context (retrieval). This could be done via mini whiteboards.

I’d like to share with you a short paper that looked at using retrieval practice for the learning of number facts:

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Ophuis‐Cox, F. H., Catrysse, L., & Camp, G. (2023). The effect of retrieval practice on fluently retrieving multiplication facts in an authentic elementary school setting. Applied Cognitive Psychology, 37(6), 1463–1469.

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The authors carried out a classroom based study, using the 3x and 4x multiplication tables. Students practised one of these using flashcards for retrieval, and the other using traditional chanting out loud (a form of restudy, with no retrieval needed).

The authors found that:

“…retrieval practice led to a stronger short-term and long-term increase in the fluency of retrieving multiplication facts” (p. 1463).

The study is rare in being a classroom experiment on mathematics conducted with elementary students. It adds to our broader understanding: active retrieval is better than rote rehearsal (similar results have been found with the teaching of spelling).

Other approaches

Some researchers have looked at strategies that get away from traditional tables.

García-Orza et al. (2021) compared typical memorisation approaches with a ‘memory and rules’ (M&R) method. The latter involved memorisation of some number facts, together with rules to help with others. For example, students didn’t try to memorise 7x5, because this is the same as 5x7.

The researchers found that higher-attaining students did better with the M&R method…but lower-attaining students did worse! This is in some ways unsurprising. The M&R method involves more complex processing, and the application of rules is metacognitively demanding, which would be harder for weaker students.

However, the researchers rightly pointed out that better understanding of mathematical rules can help with transfer (e.g. using maths for problem solving in science). Therefore, the approach does have promise!

The study was done with 7–8 year olds, and in my view, this approach might be best as consolidation, after learning the traditional times tables (ideally using desirable difficulties to do so). If students are older and more experienced, more of them may gain from the rules approach.

Final thoughts

So, where does this leave the memorisation of times tables? To me, it is massively important that students gain fluency with basic number facts, but there are some caveats:

• Memorising based on rote rehearsal is ineffective.

• Desirable difficulties such as retrieval and variation can be used, individually or in combination.

• Educators should bear in mind the importance of understanding and transfer.

• Unfortunately, there is no real shortcut. However, as students gain fluency in number facts, practice of these will happen incidentally, as they do other tasks.

• Anything that undermines fluency in number facts, such as reliance on calculators, will be harmful in the long term.

Finally, it’s worth noting that applying desirable difficulties in the classroom will lead to students making more errors. There has been a general desire for error-free learning across education. However, it turns out that avoiding errors is unnecessary and even counterproductive, as it makes practice too easy and predictable (Metcalfe, 2017).

I hope that was interesting. And in a similar vein, I’ll write next week about the teaching of spelling!

Jonathan

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