Mathematical Etudes

Stage: 3, 4 and 5
Article by Colin Foster

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 lovely rich tasks that you see on the NRICH site (and in lots of other places too, of course) and what I refer to as tedious exercises.

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:

  1. What are the factors of 6?
  2. What are the factors of 7?
  3. 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 lovely rich tasks, such as:

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 smallest such number? You can go on and on posing and solving related problems.

Clearly, both of these tasks (the tedious exercises and the lovely rich task) have something to do with factors, and both involve working out factors of numbers quite a bit, but could they be in any sense alternatives to each other? Could I simply throw out tedious exercises, which I don’t like, and replace them with lovely rich tasks? Will lovely rich tasks do the same job, or do we unfortunately need those tedious exercises? Do tedious exercises perhaps have a necessary place?

Mathematics teachers often think that lovely rich tasks are all very well, and in an ideal world you would spend lots of time on them, but in the real world we have assessments! And we all know that the easiest thing to test on an examination is procedural fluency on short, closed questions. So, since they dominate high-stakes assessments, tedious exercises are bound to dominate in the classroom. Unless we prepare our students well for those high-stakes assessments, we do them a disservice. And presumably the only way to get good at doing tedious questions is to spend a lot of time practising tedious questions. So, for many teachers, they seem to be a necessary evil.

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 lovely rich tasks, but lovely rich tasks that force students’ attention onto a particular, important mathematical procedure. So etudes don’t allow students a choice of method. Allowing students to choose what method they use is often a great idea, but if you want them to develop their skill at a specific technique, then you have to force them to use that technique, otherwise they can end up unwittingly avoiding their areas of weakness by (perhaps cleverly) solving the problem some other way than you intended. So, mathematical etudes are designed with a specific procedure in mind. But, rather than simply asking students to perform that procedure repeatedly (as with tedious exercises), in an etude there is some wider problem-solving goal – something a bit more interesting is going on at the same time!

You can see two examples of mathematical etudes on the NRICH website, Almost One and The Simple Life, and there are no doubt many more NRICH tasks (and, of course, in other places) which follow the same kind of design, and which could, if you like, be called etudes.

Etudes are intended to be much more interesting to do than tedious exercises, but they entail at least as much repetition of the essential technique. So students should gain as much (if not more) practice by doing them. But, at the same time, students have to think about what they are doing, perhaps unpick the procedure to achieve a desired goal, perhaps think backwards, perhaps modify what they do this time in response to what happened previously. All of this means that the etude can operate at multiple levels of challenge. The answers obtained to the procedure matter for some wider purpose, and so students are more likely to carry out the procedure carefully and check what they have done. Wrong answers are more likely to be challenged by a peer if they matter for some wider purpose. And having something else to focus on beyond the procedure should help to promote the desired automaticity of carrying it out.

Several schools recently very kindly trialled some of the etudes that I have been developing, using etudes with one class and traditional exercises with a parallel class. The data showed that the etudes were no worse than the exercises for developing students’ procedural fluency. Given that result, why would anyone use tedious exercises? One reason might be that they don’t have ready access to lots of nice etudes! I like to see that as a challenge for the mathematics education community: if we were to work together, could we develop a plentiful supply of etudes – and say goodbye to tedious exercises forever? 
 

[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
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.