I wrote this piece for the NCETM (National Centre For Excellence In The Teaching Of Mathematics) as part of their micro-site focusing upon One-To-One tuition. The site contains several other case studies and lots more information at: https://www.ncetm.org.uk/resources/30012

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Oliver was my first 1-2-1 pupil and was remarkable in that he functioned mathematically as a perfectly normal Y8 pupil without having any insight at all into the properties and significance of odd and even numbers. He was the first of many children with very basic gaps in their knowledge who still, by some process I don’t fully understand, manage to cope with what seems a major gap in their knowledge.

Oliver is in Y8. He’s always very chatty; we began the session with him switching on a computer and this led him into telling me how the computers in the classroom are numbered (I did think about taking the opportunity to use this to discuss counting processes but decided against it). The computer then gave a message about the amount of space Oliver has on the system, but again I decided against taking the opportunity to discuss the numbers involved – kilobytes, megabytes, etc – though I may think about the potential for a later session.

Oliver has areas of complete mastery (e.g. his understanding of place value in whole numbers) but there are other areas which are so fuzzy they’re almost non-existent – for example, while he recognises odd and even numbers he has no insight into what oddness and evenness mean nor how they combine. Number bonds are often worked out on his fingers. He knows most of his multiplication tables, but the facts are often isolated from each other and we had no success when I tried to use 3×8 to derive others like 6×8 or 9×3. He’s enthusiastic, but dives in and is more likely than not to get hold of wrong ideas and can get stuck into an incorrect idea which he can’t discard.

We’re currently working on the target “Strategies for mental manipulation of two- and three-digit numbers”, and in a previous session I’d found him very weak on doubling, and halving multiples of 10. In particular, though he’s OK halving even multiples of 10 he finds trouble with odd multiples, even 30.

I’d made a set of cards (e.g. with 60 on one side, 30 on the other). (A set of cards seems much less threatening than a list of examples, and in particular allows you to select which numbers to use and how many of them according to how things develop). As we went through we sorted them into those Oliver could do with ease and those that gave him trouble. He began to develop a strategy of partitioning (more likely, I suspect, to build upon a strategy he’d previously met but not mastered).

Oliver loves playing games, so we used the In The Box game. We’ve played a version before; the preliminary phase involves selecting resource numbers which are then combined to make target numbers. Oliver remembered this and chose his numbers strategically. There was discussion of why he chose them and why he liked particular numbers, e.g. 107, 13 630, 4 were all “nice looking numbers” – though I couldn’t find out why.

Initially we played a version involving doubling. He played with insight rather than randomly or simply looking for soft choices. We reached a stalemate stage where some numbers couldn’t be used, so we changed them into ones which could, which of course meant we had to decide what made some numbers unsuitable and the properties needed by their replacements. His strategy for doubling was consistent and involved partitioning (he never used simple addition or doubled a number like 19 by doubling 20 and correcting, even though we’d done something very similar last week). So 68 was doubled by splitting into 60 and 8, and thence to 120 and 16, and 136. I was struck by this and other examples – in the previous session he’d struggled to add numbers using the open number line image and needed either to use paper or make extensive use of fingers. I’d begun the session with the aim of improving his ability to double and halve multiples of ten, but here he was making a valiant attempt to double 13 630 mentally; he couldn’t quite manage this, but he produced a good approximation.

He often makes it easy to observe his thinking by muttering what he’s doing. “I’m talking to myself again” he said with a little embarrassment. I encouraged him not to see it as a weakness, but as a technique that suits his learning style.

When we played the halving version of the game he happily halved odd numbers (always to n.5, never to n½) e.g. he halved 43 to 21.5. When it suited his play, he gleefully halved 1 to 0.5. I, in turn, halved 0 to 0 and a discussion resulted.

During the halving version Oliver found he could tackle numbers by using the same partitioning strategy and exclaimed with some excitement “It works the other way round!”

During the games Oliver’s ability to double and half progressed by leaps and bounds. By the end of the session this has become a clear and mastered strategy, and he used it comfortably with three-digit numbers and not just tens. So he’d halve 130 by partitioning into 100, 20, 10. Other examples: 124 was halved by partitioning into 100, 20, and 4; 136 into 100, 30, 6 (this was particularly interesting, because he couldn’t halve 30 in the previous session or at the start of the session).

I’d expected simply to do a bit of practice on doubling and halving multiples of 10, and gently to explore hundreds, but Oliver progressed so far that it would have been silly to stop once we’d done what I’d planned. He would happily play games for the whole time, but we have two sessions to go and I may revisit the game if we get time – we could play a quartering version, or a multiplying by 4 version, and explore successive doubling / halving.

The planning had been to move into the next target (Division) early in the session, but there were only ten minutes to go, so I used an old favourite – arrange a set of 0 to 9 digit cards to make five pairs totalling 10. Oliver tried this and recognised and explained the impossibility, and I then asked him to make five pairs which do all total the same number. He couldn’t reach a solution, but he got a long way, and his method involved adding the ten digits together, so this will make a nice lead into the division target after all next time.

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