Tim and his friend were both happy to spend their first few minutes with me chatting about themselves. The friend had a pretty good feeling about himself and said he felt he was good at mathematics. Tim, on the other hand, said “I’m pretty bad at maths” – so even though school feels they’re of pretty similar abilities they’ve clearly got very different self-images.
A few minutes later the calculation 74 – 46 came up. Tim gave it some thought and gave the correct answer, 28. He’d done it mentally, so I asked him to explain what he’d done. “6 minus 4 is 2” he said.
“Erm, aren’t you doing 4 minus 6?”, I asked.
“6 minus 4 is 2”, he repeated. “That tells me how many I’ve got to take away from 70. So I get 68, and then I take off the 40, and 28 is the answer.”
He did several more the same way and some three-digit subtractions as well, using the same method and getting each of them correct.
Now, for something like 70 years whenever I see something like 74 – 46 a little voice in my head says “4 minus 6 you can’t do, so you borrow a 10 ….” But the voice in Tim’s head is saying “4 minus 6 of course you can do, it’s negative 2”.
So Tim’s method is actually not a convenient trick, it’s actually better than the method I was taught. It’s based on clarity and understanding rather than confusing and misleading terms like “borrow” and “pay back”.
I’ve heard anecdotal accounts of children who’ve used this method before, but Tim is the first of my pupils who’s discovered it and articulated it to me. As you’d expect, I took some pleasure in telling Tim that that far from being bad at maths, his method represents much more insight and achievement than is needed by those pupils (like me in 1950 or thereabouts) who simply follow a rule given by the teacher.
(I recall with considerable embarrassment the first talk I ever gave at a parents’ evening. It was a new school and the hall was full with parents who wanted to know about about teaching. I demonstrated a subtraction example – and with 150 people watching I got in a total mess. Never in my life had I needed to think about what I was doing, I knew the rule and applied it automatically – until now. As I floundered around, I knew what every person in the audience was thinking, and I knew it wasn’t complimentary.)
I asked Tim when he’d devised his method – probably about 7, he thought. What did his teacher say about it, I asked. As I rather suspected, Tim had never previously disclosed it to any teacher, believing they wanted him to use the traditional written method – so Tim has always worked his subtractions mentally, and then writes out the sum with borrowing and carrying figures so the teacher won’t suspect he’s done anything unusual.
PS: I’ve just discovered this in my files. I can’t remember anything about Shelley – she may have been someone I met or someone who a colleague alerted me to – but clearly she’d used the same reasoning as Tim but recorded her working in a different way: