A teacher’s take on visualising multiplication…
I saw this blog post a while ago on reflectivemaths and after commenting, put it aside. However, recently working on some early multiplication concepts with my own students has brought it back to mind.
So, does 6×4 mean 6 lots of 4 or 4 lots of 6?
The reply by the above-mentioned blogger was “bearing in mind the answers the same of course. I’d say you start with 6 and then multiply by 4. So 4 lots of 6”
That makes a lot of sense. Though, my initial response is that I visualise 6×4 as “6 fours”. “x” essentially meaning “lots of”, rather than “times”. As in, 6×4 is 6 lots of 4, or 6 times 4 things (6 packets of 4 pens) rather than 6 things times 4 (4 packets of 6 pens). But yes, regardless, we still get 24 pens either way!
My school has been working with a Maths coach on childrens’ misconceptions, and multiplication is always a confusing topic. To try and stop confusing our students, we talk multiplication in terms of “6 fours” rather than “6 times 4” or “6 lots of 4”. This aims to take the confusion out of the “x”. Further, “6 fours” acknowledges 4 as it’s own whole. Whereas, the 4 in “6 times 4” means children are seeing it as “4 ones”. Basically, why can’t 4 be it’s own being (first version)? Why should 4 only ever be seen as a quantity of ones (second version)? In saying that though, the 4 pens I mentioned above still labels poor 4 as 4 ones.
Again though, the answer is still 24, no matter which way you look at it.
Thinking in terms of ‘repeat addition’ – the concept I was working on with my 7 year olds – do you see 6×4 as essentially 4+4+4+4+4+4 or 6+6+6+6?
What about those good old “times tables”? Going through the 2s, for example, do you start with “1×2=2, 2×2 =4, 3×2=6, 4×2=8”? Or “2×1=2, 2×2=4, 2×3=6, 2×4=8”? Again, even though the answers are of course the same, I think the way you say it makes a huge difference to how the concept is visualised. Hence why, again to try and stop misconceptions, we do 2s as doubles now. That is, “double 1 is 2, double 2 is 4, double 3 is 6, double 4 is 8”.
Maybe the best solution is just to present young children with all possibilities, so that at the end of the day, they can ‘connect’ with whichever way they prefer. 6×4 might be “6 times 4”, “6 lots of 4”, “6 fours”, “6+6+6+6”, “4+4+4+4+4+4”, arrays, maybe even “double double 6”. If the answer is the same, does it matter how we get there?
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