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- Careful selection of exercises can turn purposeful practice in maths into a task that also develops conceptual understanding.
- Poor, off-the-cuff or random selection of exercises can create barriers, feed misconceptions and at best miss out on opportunities for better learning.
- Using a framework of variation theory can help teachers examine and improve their practices and tasks, preferably collaboratively

If students can learn a spurious rule for answering questions rather than the desired concept, they will grab it with both hands. In my class a student worked out that if the scenario involves faults in tyres, it must be a binomial distribution. Another rule was that if the question gives a mean and standard deviation, then it must relate to a normal distribution. These rules often work in that they help students get correct answers in assessments, but they can hinder learning. When I became aware of this in my online resources, I made sure to vary the contexts and the ways that scenarios are introduced in order to go against any spurious rules.

Variation is inevitable, so we need to let it work FOR us. In creating experiments, as statisticians, we learn about variation, and we use control groups to limit the variation as much as possible to what we are focussing on. Similarly when we are making up exercises for students, we need to consider carefully what we wish to vary and what we wish to control.

Variation theory in the teaching of maths is a framework to examine our tasks and teaching sequences. I have found it fascinating to examine the examples I give when teaching and see the variation that I miss if I do not take time to prepare well. Kullberg et al state: ” The variation theory of learning emphasizes variation as a necessary condition for learners to be able to discern new aspects of an object of learning.”

The following quote from Watson and Mason speaks to me:

“We claim that if the teacher offers data that systematically expose mathematical structure, the empiricism of modeling can give way to the dance of exemplification, generalization and conceptualization that characterizes formal mathematics.”

For a fuller summary, leading to some more great articles, see the Cambridge Espresso by Lucy Rycroft-Smith

Recently I have been working with the Theorem of Pythagoras. I wonder if the adults I teach have been shown non-examples of right-triangles and seen that the theorem does not hold there? Is it really clear that the rule only applies to right-triangles? Do we have examples where we cannot use it – there is no right-angle, so will need other tools?

When we are teaching about binomial distribution, we need to give instances where the scenario is not suitable to model as a binomial distribution.

We examine a pair of box plots for evidence of difference between two groups. One example of a pair of boxplots gives little indication as to what matters and what does not. We need to see multiple examples of pairs of box plots, with varying degrees of overlap, differences in location and spread, different contexts and even different orientations, in order to see the commonalities. And coming back to the contrast – students need also to see pairs of boxplots that do NOT show evidence of difference between the groups.

In order to be able to provide a good range of examples and exercises, we need to think carefully about the potential aspects for variation.

My adult students often find it difficult to multiply by powers of ten. I have explored the different variations that are possible, and came up with the following list, which may or may not be exhaustive:

- Different powers of 10 (10, 100, 1000)
- Other number has trailing place value zeroes. (eg 340, 7300)
- Digits after the decimal point. (eg 5.7, 29.33)
- Number needing to have zeroes added vs just change in position (eg 5.5 × 10, 6.5 × 100)
- Zeroes inside the number (eg 805 × 10)
- Staying less than 1 (eg 0.0072 × 10)

This is invaluable when I am creating a learning task made up of a number of exercises. I can make sure that I include all the variations in a way that helps conceptual development.

I call this the binomial faulty tyre effect as one of my students told me that if the question is about tyres (tires for US audience) then the distribution must be binomial. We can unknowingly set up patterns that create false learning. Using certain contexts for different distributions can provide a hook on which to hang the distribution, but students can entrench unimportant or coincidental effects. Sometimes use the same context for different distributions. Here is an example of using the same context for two different probability distributions.

- About 50% of all calls to an emergency centre are false alarms. Out of twenty calls to an emergency call centre, how likely is it that there will be fewer than five false alarms?
- Emergency calls occur at the rate of thirty in one hour. What is the probability that there will be a call in the next five minutes?

This is talking about combinations of factors. For the boxplot example, we make sure that we look at different overlaps, different spreads and different combinations of both.

This does seem like a lot of work, but once a set of tasks is developed, it can be used and refined by many teachers over many lessons. This is a benefit of the Japanese “lesson study” approach, where several teachers focus on one specific lesson and work together to create a teaching sequence, observe the teaching and learning and develop the plan further.

This is my initial take on variation theory. If you found this interesting I would suggest following up the references on the Cambridge Espresso. There is also an interesting podcast with Craig Barton talking with Watson and Mason about variation and task development. Craig Barton has a whole website around variation theory. He is clearly a fan!

I find the ideas discussed here around variation and structured exercises compelling. I have long been concerned that problems made up in response to children’s input may well lack the required variation to develop full understanding of a desired concept. Conversely I am also concerned that questions made up “on the fly” by a teacher who is not confident in mathematics may lead to confusion over concepts for which the teacher has not yet got a strong understanding. I would be interested to see how variation theory would fit in with teaching mathematics through student inquiry.

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