Conceptual Learning - an example using times tables

My agenda in this article is to challenge the widespread culture and beliefs that have such a powerful influence on teaching early maths, using times table facts as an illustration and drawing in my experience working with the outliers, the pupils who struggle with learning maths.

I have written the article in three parts.

Part 1. A look back over some 40 years of experience with teaching, writing, teacher training around the world and researching into difficulties in learning basic maths.

Part 2. A critical look at some of the comments and ideas for interventions from a wide variety of sources.

Part 3. The role of developmental, conceptual teaching.

Part 1. Reflections

Straight out of 14 years of mainstream teaching, primarily physics, but some chemistry and maths as well, I became Head of a school for dyslexics. I met my class of twelve, very dyslexic 13-year olds as their new maths teacher. Back then you could find books and programmes for teaching language to dyslexics. Nobody warned me that some might have co-occurring difficulties with maths. My ignorance was not bliss for long, certainly not for my students. It didn’t take me long to find out that they did not ‘know’ their times tables, well not beyond the 2x, 5x, 10x and 1x facts. (I later came to learn that this was an extremely common situation.)

In my ignorance and in my belief that I could deal with this swiftly, we chanted times table facts at the start of every maths lesson. The only, slight, exoneration I give myself is that I didn’t get them singing the facts to some inane tune.

At the end of the week, when I once again said in that happy, positive voice we teachers sometimes use, ‘It’s times table time again!’ One of my students got up from his desk and banged his head against the wall. Fortuitously, my teacher training had given me some basic awareness of non-verbal behaviours and I recognised this as a sign of stress.

That image remains with me forty years later. I still feel the guilt.

There were two immediate lessons for me:

Never start lessons with something a significant percentage of your students cannot do. I include mental arithmetic, unless empathetically delivered, in this category.

Rote learning is not always the best strategy for teaching maths. In fact, very few strategies qualify for ‘always’ in learning. Our student population is heterogeneous and mathematical competence is a constellation of abilities.

So that was the start of my journey (got to have a journey) to find better ways of dealing with these facts and then to attempt to break down the beliefs that are so pervasive in maths education around this small collection of facts. The most worrying belief is that rote is THE way to deal with these facts. I wrote my first book about this in 1996, ‘What to do when you can’t learn the times tables.’ I don’t think it has, as yet, successfully challenged the entrenched belief in rote. See Part 2.

Part 2. A selection from the very many resources and comments on the web about learning the times table facts. Sometimes with my comments added.

Let’s start with a factor that demotivates many learners, speed of retrieving these facts (and often the general pressure to do any maths quickly).

(From a UK maths website quoting a Headteacher) ‘Children need to be able to recall any times tables answer within two or three seconds - preferably in one second. That leaves no time for counting the way up to the answer from 2x, 3x, 4x etc - the answer has to pop out of memory pretty much instantly.’

From Pinterest there is a scheme that is: ‘Designed for high speed so that you get lots of practice.’

(There are 608 Pins on math multiplication [30/11/2020] which suggests that there is no one way that works for everyone, but 608 for 121 facts is, maybe, excessive).

You could argue that quick retrieval reduces the load on the working memory needed to do, for example, mental arithmetic, but substantial research tells us that anxiety about having to answer quickly reduces working memory capacity.

Maths anxiety lowers motivation more often than it enhances it. There are many books and research papers on this issue.

Then there is this pseudo-philosophical, and maybe over-optimistic advice that I found on the web:

‘We memorize daily, but do you realize that your brain is like a computer? Whatever you put into it will most probably remain there for a long period of time. You can choose to fill it with unimportant things, or you can fill it with skills that can be a benefit to you your entire life. Like multiplication. You will use this skill for as long as you live. Using and teaching your brain to memorize is probably the most important skill of all.’

Way back in 1925, Buswell and Judd wrote a monograph which recognised the dominance of the first experience of learning something new incorrectly and its re-emergence even when it looks as though the correct information now dominates. Recent interest on the role of inhibition on learning supports the theory that initial learning needs to be correct.

Then there is the belief in the power of being able to remember these facts, which may be true, but only if that power resides in the majority of pupils: For example, from the tes, 'The fact is, learning times tables does pupils a world of good' My question is, ‘How many pupils achieve this world of good?’ And if they don’t, does it become a world of bad?

And then there’s this comment about the English Government’s proposed tests of these facts for ten-year old pupils, ‘Teachers should welcome the proposed tests with open arms, says one headteacher, as he argues that knowing times tables off by heart sets children up for life.’ This is not encouraging news for those pupils who, despite hours of trying to learn, don’t learn. That could be as many as 70%. My statistic is from the many teachers I have surveyed during my lectures. It can be revealing looking at ‘What if?’, but then following this with ‘What if not?’

Ofsted, the schools watchdog joins in supporting the rote learning myth, highlighting in their perception the importance of reciting learning times tables:

“Pupils should be able to recite their times tables up to 12 x 12 by the age of nine.

Primary schools that fail to teach times tables by heart are condemning children to a lifetime struggling with numbers, inspectors have warned.’ It doesn’t have to be that harsh.

Unrealistic expectations do not help learning.

There are many suggestions in the education world to ensure that these facts are memorised. They range from the basic to the bizarre. This following is about as basic as you can get: ‘Your life will be a lot easier when you can simply remember the multiplication tables. So ... train your memory!’ Education is not that simple.

Now, let me return to Pinterest for some slightly less pedagogical input.

It seems that using tunes and thus singing the times tables is a popular idea. Maybe some pupils agree. Here is one selection:

2x This old man

3x Happy birthday to you

4x Old MacDonald

6x Jingle Bells

7x Row, row, row your boat

8x Mary had a little lamb

9x Twinkle, twinkle

12x Itsy-bitsy spider

Perfect for the 17-year old who failed at their first attempt at GCSE.

There are many examples of more fact-limited ‘tricks.’ Some, at least, show patterns, for example, The sixes rule: ‘If you multiply 6 by an even number, it will end with the same digit, so for 6 x 6 the ones digit is 6 and the tens digit will be half the ones digit (3). So, 6 x 6 = 36.’ That’s not going to happen in that ‘preferably one second.’

Some tricks do not do patterns, for example, ‘Six times eight, so finish your plate. Six times eight is forty eight.’ ‘There is a monkey in a tree so nine times seven is sixty three.’ There’s a touch of cross-curricular content there. And maybe something that is missing from Sir David Attenborough’s commentaries.

Often such tricks are described as ‘fun’ and are accompanied by cartoons. I realise I might be sounding like Mr Grumpy here. Maybe I’m missing a grumpy chance for the 7x facts.

Often the tricks require the pupil to learn more to learn less. And rarely are these tricks conceptual. They usually demand more memory capacity. Sometimes they have dubious conceptual validity, as with, ‘10× is maybe the easiest of them all .... just put a zero after it.’ Desperate parents may not filter out such harmful stuff.

Part 3. The role of developmental, conceptual teaching.

Much of maths develops. It helps to know the history of a topic and its future. Then teachers will know if the foundations are secure and thus not teach something that will have to be untaught in the future, such as, ‘Take the little from the big.’

So, I propose that we expose our pedagogy to the question, ‘What else are you teaching?’ Rote learning may well set in place information that will help future development, but it is not the only way, especially with maths where there are many opportunities to use what you know to work out what you don’t know. There are often alternatives and often these can develop understanding. In maths the learner needs to know facts, but also how to use these facts to develop their access to other ‘facts’. Maths is a good subject for that philosophy.

When potential pupils, usually aged 11 or 12 years, came to my (specialist) school for an interview I would ask them, in as low stress a manner as I could muster, ‘How are you getting on with your times tables? Which ones do you reckon you really know well?’ Inevitably the reply was, ‘The 2s, 5s and 10s.’ Some would add, ‘and the 0s and 1s’. That’s enough, especially if they are automatically retrieved. I can work with that to develop a lot of maths. And that situation is by no means restricted to ‘special’ pupils.

When I taught physics in secondary schools, I used experiments and visual images to help develop concepts. I do the same with maths, always linking the images to the symbols.

There is not enough space to give much detail of interventions in this article, so I’ll just give a few examples. There is much more in my books and videos. The solution is not a piecemeal collection of unrelated explanations and cartoons. It should be a chance to develop an understanding of key concepts.

If we work from what the learner knows to take them to new knowledge and understanding, so, from my experience, this means using the key numbers. For example, break 3 into 2 + 1, 4 into 2 x 2, 6 into 5 + 1, 7 into 5 + 2, 8 into 10 - 2 and 9 into 10 – 1. This partitioning can be developed to many other examples, such as 15 = 10 + 5 (and using 5 as half of 10). This enhances number sense and the concept of breaking down numbers, partitioning.

Learners need to be able to link multiplication to repeated addition, for example, 7 x 8 = 8 + 8 + 8 + 8 + 8 + 8 + 8 and then into partial products, 5 x 8 + 2 x 8. The ability to link all four operations enhances understanding of each separate operation and thinking that is not rigid.

The ‘What else are you teaching’ continues with using 10 as an estimation for 9, each 9 being 1 less than each 10 and thus to refinement, for example, 8 lots of 9 equals 8 lots of 10 minus 8 lots of 1.

4x and 8x, can illustrate repeated multiplication.

Cuisenaire rods are useful images, but areas, y = ab, are developmental for ‘long’ multiplication and division and algebra and fractions.

In 2017 a joint discussion from the Association of Teachers of Mathematics and the Mathematical Association (MT.2109. 265. 11-13) goes some way to maximising the opportunities arising from teaching multiplication facts but lacks coherence and consistency.

In my recent book, ‘How to Teach Maths: Understanding Learners’ Needs’ (2020), I break down a typical maths curriculum to show the vertical progression of topics. Among the benefits of this is that it can be used to answer the key question for intervention, ‘How far back to I go?’

When memory fails, our pupils need an alternative. An over-dependence on teaching early maths with rote dominating denies them the chances to conceptualise and progress.

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