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Article by Colin Foster# Mathematical Etudes

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Age 11 to 18

Published 2017 Revised 2023

The NRICH website is full of ** lovely rich tasks**, attractively packaged, all ready to be used. And yet often when I go into mathematics classrooms, I see students doing dull, repetitive exercises. Why does this happen, when all these lovely NRICH problems are freely available, just a click away?

It seems that there is a battle going on in mathematics classrooms between the kind of

Here's an example of what I mean by each:

Tedious Exercises |
Lovely Rich Tasks |
---|---|

What are the factors of 6? | Find some numbers with exactly 4 factors |

You could easily imagine a page in a textbook containing ** tedious exercises** like this:

- What are the factors of 6?
- What are the factors of 7?
- What are the factors of 20?

and so on.

I call these ** tedious exercises** because there is no bigger picture here. You work your way through these, but you could stop at any point and it wouldn't matter. The textbook author perhaps decides how many to put in according to how much space they have available on the page. A student could do every other one; or just do the last one. You just slog through them; the
answers don't matter - when you finish one you just move on to the next one until the time is up.

Contrast that with what I call

Find some numbers with exactly 4 factors.

I like this task very much.[1] It is wide open (there is certainly more than one number with exactly 4 factors), which means that different students can give different correct responses. It feels like a beginning rather than an end. As I accumulate numbers with 4 factors, I begin to ask, “*Why* do these numbers have 4 factors? What makes that happen?” If I
find numbers with other numbers of factors, that's almost as useful, really, because the 4 is clearly arbitrary. The real *question behind the question* is “What determines how many factors a number has?” Exploring the number of factors that numbers have is a lovely investigation, which I have written about elsewhere (Foster, 2017). Unlike the * tedious exercises*,
which might be too easy for some students, and consequently boring, this task can be mathematically interesting for almost anyone, and I have used it with students of all ages from 11 to 18, as well as with student teachers and experienced teachers. Everyone can get something out of it. Can you write down a number with exactly 30 factors? Can you find the

Clearly, both of these tasks (the

Mathematics teachers often think that

I recently carried out some research to explore whether this is true or not (Foster, under review). For some time, I have been working on devising what I call mathematical etudes (Foster, 2013, 2014). These are intended to be not just

You can see two examples of

Several schools recently very kindly trialled some of the

[1] I feel free to say how nice this task is because, although I've used it a lot, I certainly didn't invent it. I've no idea where I first came across it. Maybe it's as old as numbers? I'm pretty sure that no one could claim that anyone in particular “invented it”!

**References:**

Foster, C. (2013). Mathematical études: Embedding opportunities for developing procedural fluency within rich mathematical contexts. *International Journal of Mathematical Education in Science and Technology, 44*(5), 765-774.

Available open access at: https://doi.org/10.1080/0020739X.2013.770089

Foster, C. (2014). Mathematical fluency without drill and practice. *Mathematics Teaching, 240*, 5-7.

Foster, C. (2017). Thirty factors. In *Questions pupils ask* (pp. 127-132). Mathematical Association. Available from: http://members.m-a.org.uk/Shop/product/1114

Foster, C. (2017). Developing mathematical fluency: Comparing exercises and rich tasks. Educational Studies in Mathematics, online first.

Available open access at: https://doi.org/10.1007/s10649-017-9788-x

*Colin Foster is an Assistant Professor in mathematics education in the School of Education at the University of Nottingham. He has written many books and articles for mathematics teachers (see www.foster77.co.uk).
The Mathematical Etudes Project lives at www.mathematicaletudes.com.*