It's a duck - but who cares?
Can you tell me how this game works for subtraction?" I asked the busy assistant."Certainly, you lay the cards out like this."
"So what does the 'minus' here mean to young children?" "Take away." "And the answer is?" That brought the shop to a halt.
"One," came the rather sharp reply. "One what?" I asked. Nobody knew. So I wrote to the managing director of the company that made the game. "Dear Sir, When your grandchild asks you 'one what?', what do you say?" The answer: "I have passed your enquiry on to my product development director."
Poor grandchild. Later I discovered a redesigned game with "two snails minus one teddy". The "minus" card has been "improved" to make the "take away" point more clearly. There is a picture of a mouse taking away some cheese. Ah.
Grandchildren change and so do their mathematical needs. I found a book in the local supermarket to help with those dreaded times tables. We had now moved on from ducks and teddy bears to multiplying apples. We had 4 apples x 2 apples, 4 carrots x 3 carrots, and something that looked like 8 feet x 5 toes! Perhaps this was advanced stuff, and you graduated to that from your fingers.
So what does "multiplication" mean? I looked it up in some dictionaries. Something like the following was almost universal: the process of finding the number or quantity (product) obtained by repeated additions of a specified number or quantity (multiplicand) a specified number of times (multiplier) symbolised in various ways (eg, 3 x 4 equals 12, which means 3 + 3 + 3 + 3 = 12, to add the number three together four times.) No mention of "cats" x "dogs". Neither was 3 x 4 called "three fours," it was "four threes".
Pages 82 and 83 from the National Numeracy Project's "lessons book", headed Multiplication Algorithms, offer a mathematics framework and sample interactive lessons: "In this lesson pupils develop their preferred ways of doing multiplication." However, there doesn't seem to be any element of choice offered. More to the point, no child says "I prefer this because...". After all this interaction, the aim didn't seem to have been met.
The suggestion is that the teacher should ask: "What do you say when you see 3 x 4? Do you say 'three times four'? Or 'three fours'? Or 'four threes'? Or 'three multiplied by four'?" This is a language issue. The text goes on: "Discuss the fact that it is possible to interpret 3 x 4 in all these ways and that although they look different, fortunately they all give the same answer." But now we are moving from images in the mind to getting answers. How did that happen? Tricky stuff. The Framework section "Understanding Multiplication" - in both the National Numeracy Project and its successor the National Numeracy Strategy - leads off with grouping real objects expressing 2 + 2 + 2 as "3 lots of 2". It suggests we "understand multiplication as repeated addition" where "5 added together 3 times is 5 + 5 + 5 or 3 lots of 5 or 3 times 5 or 5 x 3 (or 3 x 5)". Who makes the choice? Does it mean that if four children ea two cakes each that is the same as two children eating four cakes each because eight cakes are eaten? Is this giving you headaches? Well, if your doctor prescribes you three painkillers once a day for 21 days, don't take 21 tablets for three days.
I wrote to the project's director about my worries over pages 82 and 83. The reply - that the aim of the lesson on page 84 was not as I said - left me wondering if I had a numeracy or literacy problem. The brief answer to the 3 x 4 or 4 x 3 issue was that it didn't matter and that children need a lot of "things" to help them abstract these ideas. Things? The section's front page says "no equipment needed". Can you say 3 + 3 + 3 + 3 "is the same as" 2 x 6? Just fancy, four children eating three cakes each is the same as six children having two cakes each. I would like to be the one having 12 cakes - or is that another issue?
I did some research and discovered the book Teaching Mathematics in Grades K-8: Research Based Methods, edited by Tom Post in 1988. It gives a powerful case for distinguishing between 4 x 3 and 3 x 4, saying: "For children, three lots of four and four lots of three are fundamentally different. They think in concrete terms," and, "Even when considered in the absence of any concrete model the expressions 6 x 3 and 6 V 3 involve two numbers with different roles to play in the execution of the operation." There is even an exercise asking readers to give "some real-life examples of situations in which a multiplication product a x b (for example, 5 x 6) is not the same as b x a (6 x 5)". It matters, it really does.
Turning at last to the Numeracy Project's page 84, we go into the "area model". Just like that. No area problem to solve, no carpets or wallpaper or lawns. Just do it. The page shows a rectangle with sides of 37 squares by 6 squares and we are going to multiply. We are to measure lengths in square units? So perhaps you can multiply "things" by "things" after all! But what would the units be?
I wrote to the Minister for School Standards and asked for clarification: do we really have a choice about the meaning of 3 x 4? Do we, actually, measure lengths in square units? Was that lesson really "interactive"? What are we saying when we "chant" number sentences such as 3 x 4? After all, it means 3 + 3 + 3 + 3, so saying "three fours" may well be reinforcing misconceptions. Is all this really about raising standards? I need to know.
The reply informed me that the National Numeracy Project has been "offered widespread support" by the Teacher Training Agency, Qualifications and Curriculum Authority and the Office for Standards in Education. The official stamp of approval has been given.
So where does this leave us? Back to the card game, and my answer is clearly: "A duck" - because the problem has to be something like 'how many more ducks have I got than teddy bears?'" As for the rest, who cares? The "rote learning" of it all seems to detach children from reality. Never mind ducks, we risk producing more parrots than anything else.
John Marshall was formerly a mathematics inspector for Staffordshire.