# What we learn in school

February 23, 2009Link: What we learn in school

I recently took on a remarkably talented tutee. Seeing the natural ease with which he, at 12 years old, sees deeply and sharply into all kinds of mathematical problems is a pleasure. For me, making sure this kind of student gets proper guidance is essential. Part of the reason, I’m sure, is that I was accelerated in math when I was younger… and I’m not sure anything in my education gave me any kind of idea what math was about. My first real exposure—and one of the reason I ended up as a mathematician today—came when I went to math camp summer after my ninth grade year. It’s hard to communicate how profoundly my relationship to the subject changed as a result of that program.

But what does school offer? Here’s an article from London that gives a sense of what school math is like for gifted students (and one in particular, named Joe). I’ll post the last half of the article here:

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“In the National Numeracy Project’s Framework for Teaching Mathematics (DfEE, 1999) the suggestion is that there are three ways of making the content of mathematics lessons more challenging:

- by offering the same content within a shorter time (acceleration)
- by offering additional content outside the regular curriculum (broadening)
- or by offering increased cognitive challenge within the same curriculum area (deepening).

I would predict that if you talk to any group of maths teachers in any school you would find that acceleration is the predominant strategy to increase challenge. Broadening, after all, requires additional preparation by hard-pressed teachers so is unlikely to find universal favour. Non-specialist colleagues may lack the confidence (or competence) to ask deeper questions requiring higher-order levels of thinking, so acceleration by default becomes the most common strategy. Certainly for Joe it was the most frequent offer.

Of course, there are advantages for a teacher in accelerating pupils through the curriculum. There is no need to invent activities or acquire different resources as everything is already mapped out for the older pupils. It’s an easy option but can set up possible problems for the subsequent teacher and, as we saw from Joe’s experience, unless carefully planned can lead to repetition or, perhaps even worse, to gaps. For the student it may well offer increased self-esteem but it may also be lonely and seem like a random and disjointed set of experiences.

**Thinking like a mathematician**

For me, the saddest part of Joe’s story is that he is not inspired to continue his mathematical learning. He has no idea of what it is to behave mathematically because his perception is that maths is the acquisition of a set of skills, a set of exercises to be completed, a speed test. Essential though these skills are, they are just the starting point. I want able students to ‘think like mathematicians’ – to enquire, to generalise, to question and seek proof. I don’t want them to accept everything I offer or be able to regurgitate pre-digested algorithms. I want them to engage with the thinking and find some things difficult. I want them to know that mathematicians see things in lots of different ways: that maths is creative and although there may be accepted elegant methods of solving particular problems or calculations, there are other equally acceptable ways of doing the same. I want them to take time to deepen their understanding rather than skating through the curriculum at high speed.

Writers on the subject (Krutetskii, Kennard) say that these forms of behaviour (generalising, seeking proof and so on) are characteristic of able mathematicians. This is what they do naturally. The teacher’s role is to offer opportunities, activities, investigations for able students to display these characteristics, and to model such behaviour themselves and hence help students to refine their thinking. Spending time deepening understanding in this way is time well spent for able pupils, and can have knock-on effects on the rest of the class who are exposed to behaviour that they might otherwise never experience.

**I want able students to think like mathematicians– to enquire, to generalise, to question and seek proof.”**

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My new tutee is already accelerated. I’m going to try to offer the depth and breadth that, sadly, he may never get in school.