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:

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Interesting post. I also devised a number of mental arithmetic methods when I was young. When I say devised, these approaches seemed obvious to me and still serve me well.

What interests me also is that another “maths blogger” who considers themselves to be quite a big deal has commented that neither you nor your student has used the “best” method. You make the point that maths teachers often ask a student to stick to the method they have taught them.

Your student seemed very successful with his method. Surely the “best” method is the one that the student finds the best and is most successful in using despite being pressured to use the one that the teacher thinks is “best”.

Maybe this is a maths teacher thing

Well, I haven’t seen the response you mention, but I imagine you and I are very much on the same page here. What I want from a “best” method is that it’s pretty efficient and it’s highly desirable that it’s based on understanding. You’d have to be very hard-hearted to say that Tim’s method fails either criterion; his image and language may be unconventional, but they’re built on a perfectly sound analysis of the situation.

What the best method will be depends on both the individual learner and the specific situation. Would we want anyone to use the same “best” method for each of 61-2, 61-58, 61-43, 61-39, 61-21, …?

I recall posting a piece in March 2014 (The Best Question) which makes a similar point.

With the government making strong demands that children should use traditional algorithms this is a real hot potato, and I’ve a couple more pieces on children’s methods in the pipeline, so do stick around. Incidentally, I discover that – perhaps like you – Tim has a number of other interesting strategies that he calls upon. I’ve a few more sessions with him to come, so who knows what we may discover?