I’d be grateful if you’d help me in a little experiment.
In the photo the letters A to J stand, in some order, for the digits 0 to 9. (As you’ll no doubt guess, a two-letter item stands for a number such as 23 rather than 2×3.)
The first question I’d ask you is roughly how long you needed to solve the whole set of statements and discover the unique solution. Did you see everything straight away? Five minutes? Ten? Were there any blind alleys, and fresh starts?
Secondly, you’ll want to reflect on the mathematical and reasoning skills you needed to call upon.
Thirdly, who might you see as suitable pupils for the problem?
For what it’s worth, I think it probably took me about 15 minutes, perhaps 20; I found four possibilities and had to explore each of them. It wasn’t we were working through it together that I realised there was a much better approach that avoided multiple possibilities and allows you to home in smoothly on the unambiguous solution.
And, as you’ve realised, “we” means the remarkable Amy and her partner Paddy; I’ve written before a couple of times (March 2017) about her unusually highly developed reasoning abilities. Now 55 years of teaching give me a pretty solid feeling that this isn’t a problem you dish out to your average 11-year-old (actually Amy was ten when we first me, but like me she’s had a birthday since then).
But I’ve had plenty of pupils who’ve been able to tackle this problem, and the biggest reason they can handle it is motivation. It’s one of many challenges they meet in Anita Straker’s “Martello Tower” adventure game, and by the time they’ve invested several weeks of effort they’re not going to let one more problem put them off.
Usually, however, I offer a clue or two, and my contribution with Amy and Paddy was much more limited. I did write out each of the statements onto card so they could sequence them as they wished and write on them to keep track, but otherwise
my contributions were restricted to comments like “What does that tell you?” and “What could you do next?” On completion I congratulated them, and Paddy said something to the effect that he didn’t know what I was making a fuss about, it had all seemed pretty easy!
This was our final session together, and eight weeks of working with them has reminded me yet again just how localised children’s abilities can be. Their performance in some other arithmetical problems was nothing like as advanced. In the adventure they need to identify a four-digit number using ‘more than’ / ‘less than’ clues and neither of them were great at that, and Amy was worse than Paddy. In Martello Tower they repeatedly need to use triangular numbers and neither of them ever really reached the stage where they could find TN16 without working from and earlier one like TN10 or TN12. My other pupils have almost invariably called upon the streamlined method long before the end of the adventure.
It’s not simply that some children are good at number and less so at spatial stuff, and vice versa. Amy and Paddy are able to operate a very high level in some number work, and much more mundanely at other activities even in related areas, and the difference in maturity can be quite dramatic.
I’m reminded that one year I was asked to lead the national evaluation of pupil performance in the Key Stage 2 national tests. One thing jumped out at me: for virtually every one of the hardest (level 5) questions something like 10% of the correct answers were given by children whose overall achievement was graded at below average level 3. And of course it was a different 10% each time; clearly there are a lot of Amys around, with a very jagged profile of skills across different areas of the curriculum.
Potentially this has huge consequences for the way we group and teach children, and I thought it was so important that we should be shouting it from the rooftops. But no-one else seemed at all interested, and rather to my relief the curriculum authority decided to keep the process in-house and never invited me to do the job again.