I suppose the biggest thing I’ve learned is that you learn a lot and you keep learning. A Support Group of teachers from twenty schools unanimously agreed that One-To-One working tells you so much about the way children learn that all teachers – or at the very least all NQTs – should experience some as part of their professional development. Quite possibly I learned as much in three years of tutoring as I did in my first three years of teaching.
1……All*** children can enjoy and benefit from an hour of intensive mathematics. Nothing is more important in mathematics learning than confidence and One-To-One gives them this, allowing them to play a full part in discussing the work, responding to challenge, and achieving success.
2……Every child has a unique profile of learning – their own individual set of misconceptions. I’ve met 11 and 12 year-olds with no understanding of odd and even, or who could do division but didn’t have a clue what multiplication involves. Yet all these children could somehow get around the glaring gaps in knowledge and function in mathematics so that superficially their attainment and understanding appeared perfectly normal within their group.
3……Learning is not neat and linear. We’ve known this for years, of course, but it’s a very convenient assumption for those who write schemes and text-books.
4……I don’t think I’ve had any pupil who didn’t understand the place value system, at least as far as tens and units are concerned. Each of them can add two-digit numbers successfully, on paper and usually mentally. That’s just about the only generalisation I’d like to make. Everything else – subtraction, multiplication, division, larger numbers, smaller numbers (fractions, decimals, negative numbers) – will be fully understood by some and a total mystery to others. As a rule of thumb, of course, it can mean that the other rules of subtraction, multiplication, and division may be better tackled by relating them to addition.
5……Many (most?) children have techniques they’ve taught themselves. For example, that you can add nine by increasing the tens digit by one and decreasing the units digit by one. Usually they see these as dirty and rather underhand. I reckon that one of the best things I can do is to explore these methods, help the child see why they work, and that they’re extensible (so the adding 9 technique can be adapted to add 8, or 19, or 99, ….). I want children to recognise they’ve done something of high quality that they can be proud of.
6……Conversely, many of the methods that we use may not be used by children. For me it’s obvious to say that since 4 times 6 is 24 then 8 times 6 will be twice as much. In my experience it’s very rare for children to use this or find it helpful when it’s pointed out.
7……Everyone enjoys playing games. For me, the advantages include (a) I get to see how children handle numbers naturally rather than when they’re presented in a sum, (b) games allow me to gradually increase the demand so they find themselves doing things they wouldn’t otherwise be comfortable with, (c) games put children into situations where they have to use a wide variety of thinking skills – to estimate, to plan ahead, to consider what-if situations.
8……Many children have hobbies and interests which have given them extensive stores of knowledge. Dinosaurs and sports stars are obvious examples, but another pupil told me about high diving, and someone else was fascinated by the building of a home extension. Another decided we’d finish our sessions by planning a picnic. All of these gave me the chance to tailor some mathematics to their own interests.
*** (I’m really talking about the 65% – 70% who aren’t at the extremes of the ability range, though actually I’m pretty sure everything I say is true for everyone)