Dreamworks Heroes Action Cards

Are your kids collecting the Dreamworks Heroes Action Cards from Countdown? Here’s a little game that you could play with them to practise multiplication and division by 6.

Part 1: Get them to fish out all the cards with numbers that are multiples of 6, i.e. the numbers in the 6 “times table”. Next, turn these cards over and look at the die/dice patterns on the back. Do they notice anything interesting?

Part 2: With the cards that are left, take a card and show them the number on the front of the card. Ask them to divide that number by 6 and calculate the remainder. Then turn the card over and see if their answer matches the die pattern on the back of the card!

In playing this game, it would be great if children progressed from skip counting in 6s to jumping straight to the required multiple of 6. So if the card number was 20, a beginner might say “Six, twelve, eighteen…I need TWO more to make 20”, while a seasoned player might say “Six goes into 20 three times with TWO left over”.

Parents can try this at home, or teachers (who haven’t banned the cards!) can try this in the classroom. Have fun!

Dr Audrey Tan, Mathmo Consulting

Automaticity and why it’s important to learn your ‘times tables’

Recent press articles, both here and in Canada, suggest that the debate over teaching mathematics in primary schools today boils down to one tiny matter: whether or not we get students to memorise their ‘times tables’. It’s kind of sad, since there is so much more to learning maths than just knowing your multiplication facts, but nevertheless, it is an important and necessary step to becoming numerically fluent. I’ll explain why later.

In February this year, The New Zealand Listener had a few weeks of amusing banter. In the cover story “The superstar learner”, Professor John Hattie, the highly acclaimed New Zealand educationist and co-author of “Visible Learning and the Science of How We Learn” (which discusses the importance of instant retrieval of number combinations), said

“I’m a great fan of teaching kids the times table by rote. Why would you want to understand why six times nine is 54? It is, just accept it. Then you can use it.”

Gasp, did he say rote? I’ll come back to that. He went on to call the Numeracy Project “a debacle” that “pooh-poohed” automaticity.

Predictably, one of the lead researchers of the Numeracy Development Projects rose to the bait. Associate Professor Jenny Young-Loveridge disputed the NP’s “pooh-poohing” of automaticity, but did say

“Current mathematics education reform emphasises thinking and reasoning mathematically, rather than mindless recall.”

Er, who said anything about mindless recall?? In all fairness, I find Jenny Young-Loveridge to be one of the more honest maths education researchers in this country. See her final longitudinal study paper “A Decade of Reform in Mathematics Education: Results for 2009 and Earlier Years“, in which she concedes that the results are disappointing and “we have underestimated just how difficult [changing the way mathematics is taught] is.” (So, why did the Ministry of Education at that time not ask some tough questions about the return on their $70 million investment?)

The icing on the cake was the prize-winning letter in the following week’s issue. David Prior-Williams, a (retired?) high school maths teacher, wrote about Little Doreen who “was never going to set the mathematical world on fire”, but she knew her ‘times tables’. He doubted that she would have understood an explanation of the sort proposed by Jenny Young-Loveridge, who of course provided not one, but two strategies to explain why six times nine is 54! (If you’re curious, you’ll find them at http://nzmaths.co.nz/resource/easy-nines.)

“[Little Doreen] would have lost interest in 10 seconds and been none the wiser. She didn’t need to know why; but behind the counter in the dress shop a few years later, when a customer wanted six buttons at 9c each, she would have been on top of the situation immediately.”

The rest of his letter were the words of a wise teacher who learned, through pure experience, that trying to explain why certain things were true was “not only too difficult, but unnecessary. …In fact, in many cases, the penny will drop as a consequence of repeated use.”

Out of the three points of view, I find Mr Prior-Williams’ to be the most compelling. He describes someone who understands multiplication well enough to apply it quickly and correctly, in context. In other words, Little Doreen is efficient and effective in the real world. I rather like the sound of that. Incidentally, I wonder how many shop assistants these days would be able to cope with a cash transaction in the absence of a calculator, built in to the till or otherwise?

Over to Canada now. Both Manitoba and, more recently, Alberta have inserted into their curricula a specific requirement for children to memorise their times tables. This has sparked a huge public debate, including one secondary math teacher in Ontario who asked “Knowing Your Times Tables Is Nice. But Is It Necessary for Success?

You bet it is! I am surprised that a secondary math teacher would think that “having the memorization of multiplication tables removed from the Ontario math curriculum is a step in the right direction…”. Where is the evidence that “through a ‘variety of’ strategies and tools,…the eventual memorization of multiplication tables will come”? [Update (Dec 2016): The author updated his article earlier in the year and has retracted these statements.]

In my experience, children who do not put some direct effort into memorising their ‘times tables’ never learn them, and it really holds them back. They are slow at multiplying, and even slower at dividing. They all but grind to a halt when simplifying fractions, finding least common multiples or highest common factors. Consequently, they struggle with elementary algebra, e.g. expanding/factorising algebraic expressions including quadratics, working with algebraic fractions. This is no surprise. Research by Robert Siegler et al. has shown that knowledge of fractions and division is a strong predictor of success in high school mathematics.

In case the previous paragraph isn’t clear, understanding multiplication in and of itself is not the ultimate goal. What is far more important is getting students to understand division and the inverse relationship between multiplication and division, the precursor to proportional reasoning. Not once did the secondary math teacher refer to division, except to quote an expectation from the Ontario curriculum. Instead, he focussed on multiplication strategies that exploit the distributive property of multiplication over addition, i.e. at the end of the day, students are still adding. Even our own educationists recognise that the transition from additive thinking to multiplicative thinking is important, and yet they condemn students to remain stuck in the world of addition by promoting these strategies! Don’t get me wrong, they are sometimes useful, (especially as a backstop when a basic multiplication fact has not yet been committed to memory,) but this is not where the emphasis should be. Furthermore, in my experience, students who rely on distributive strategies show no more aptitude for grasping the abstraction of these methods, e.g. expanding quadratics, than their memorisation peers when they reach high school. A common mistake is to think that (a + b)2 is the same as a2 + b2. So where is the deep understanding there?

Believe it or not, it is actually possible for children to memorise the multiplication tables and understand what multiplication means. Plenty of us have done it, despite what the educationists would have you believe. Sure, we all have anecdotes about children who rattle off their tables but don’t have a clue what it’s all about. Memorisation isn’t the problem here, failing to bring meaning to the concept of multiplication is the problem. That’s why good teaching is essential. (Actually, most children don’t have a problem with the early concept of multiplication as groupings. It’s the concept of division that needs more work; again, groupings is a very good place to start.)

As soon as we hear the word ‘rote’, we gasp in horror at the thought of mindless recall, lacking in engagement. But in the strictest sense, ‘rote’ simply means memorising by repetition. Does repetition necessarily mean mindless recall? Definitely not; good teaching should monitor for that and prevent it. In a broader context, especially if a student improves with repetition, one might call it practice.

Learning by rote may not necessarily promote understanding, but memorisation does promote understanding, as cognitive science tells us.

Daisy Christodoulou: “Facts are not opposed to understanding; they enable understanding. This is because of the way that our minds work. Our long-term memories are capable of storing a great deal of information whereas our working memories are limited. Therefore, it is very important that we do commit facts to long-term memory, as this allows us to ‘cheat’ the limitations of working memory. The facts we’ve committed to memory help us to understand the world and to solve problems.”

Marc Smith: “Memorising facts can build the foundations for higher thinking and problem solving. Constant recitation of times tables might not help children understand mathematical concepts but it may allow them to draw on what they have memorised in order to succeed in more complex mental arithmetic. Memorisation, therefore, produces a more efficient memory, taking it beyond its limitations of capacity and duration. …[There] exists a considerable body of evidence to suggest that a memory rife with facts learns better than one without.”

I am not a traditionalist who harks back to the days of old, when children used to chant their ‘times tables’ in the classroom every morning. I support the memorisation of multiplication tables because it truly helps. It doesn’t have to be done by mindless recall. There are plenty of creative resources out there to make the task more fun and engaging. Aside from the direct benefits, it helps to create ‘muscle memory’ in the brain. Quite simply, students who retain what they learn are able to make faster progress. Now who wouldn’t want that?

Dr Audrey Tan, Mathmo Consulting
May 2014

Further reading:

Thank you very much to the writers who choose to link to this post. Please let me know when you do. Let’s keep the discussion going!