It is easy for us to identify what is different against an invariant background. It could be a red flower against a deep blue sky, a single item moved on an otherwise untouched shelf, a new sound amid a symphony of common city noises.

Our mind focuses in on that difference. And the inverse is also true - we notice what stays the same when everything else around it is changing.

The Swedish educational psychologist Ference Marton recognised this and was the first to articulate that we learn what varies against a background of invariance - that’s the only way we can perceive things.

So if we want a child to learn something specific, we need to design teaching and tasks that help that child to become aware of that item through comparing differences.

This is variation theory. In maths, this theory is gaining prominence as a method of task design. It was once rarely spoken of in our schools, but that is changing.

And yet, it is not always being used in a manner faithful to the original research: in fact, myths are arising that suggest it may have been unhelpfully oversimplified - in some cases, the resources are moving so far away from the original design intentions that they may be doing little good.

So for maths teachers - and, indeed, any other teacher - how do you do it properly?

Variation: the spice of life?

There are some task designers and educators who have been interested in variation theory for decades now, and it’s good to see it gaining prominence in mathematics education in the UK. It helps teachers to focus on what they want students to learn, and which tasks and sequences might help them learn it.

It is also true that variation theory is often incorporated into practice without the teacher recognising it.

It is normal to teach the linear function, for example, by seeing what happens when the constant term varies (the graph is translated up or down) and what happens when the coefficient of x varies (the gradient of the graph changes).

As far back as the 1970s, the primary series of books Maths in Schools used sequences of tasks, examples and questions in similar ways, but didn’t call it “variation theory”.

A typical task would be to compare varied ways of doing the same calculation, thus helping students to understand the rules of arithmetic. For example:

Imagine that some children are finding the area of a field on a map and have to calculate 23cm x 47cm

* Winston’s method: 23 x 50 = half of (23 x 100) = 1150; 3 x 23 = 69; 1150 - 69 = 1081

* Milly’s method: 2 × 47 = 94; 10 ×94 =940; 3 × 47 = 141; 141 + 940 = 1081

* Nick’s method: 3 × 7 = 21; 3 × 40 = 120; 20 × 7 = 140; 20 × 40 = 800; 800 + 140 + 120 + 21 = 1081

* Use Milly’s method to calculate 35 × 72

* Use Nick’s method to calculate 35 × 72

* Can you explain why Winston’s method works?

In a similar way, the investigative and exploratory methods used widely in the 1980s and 1990s depended on generating carefully varied cases and special cases of a mathematical situation to expose its structure.

For instance, the question “How close can you get to 700 by writing only the digits 5, 6 and 9 (as many times as you like) and a decimal point?” might take some time and the generation and comparison of several similar examples.

This variation has happened without the instruction of an underlying theory because teachers have naturally come to the conclusion that tasks designed in this way work. But it has never been explicit or as widespread as it could be. And it has often not been used as effectively as it should be.

What we know is that when variation theory is used properly, there is very strong evidence that it is effective.

Hamsa Venkat and Mike Askew, working in Johannesburg, have found that incorporating variation theory can make mathematics teaching more coherent and connected (for example, see Ekdhal et al, 2018). Likewise, Angela Barlow and her colleagues in Tennessee have found improvements in early algebra when variation principles are used to plan and teach (see Barlow et al, Teaching and Learning Mathematics Through Variation, Huang and Li, 2017). And it seems to work in Shanghai if success in international tests are taken as evidence.

Some teachers have taken note. They are beginning to use variation theory more consciously in their task design, and they are using it well.

Unfortunately, like many good ideas that become “the next best thing” in education, variation theory might have been simplified beyond recognition at times and it is subject to some myths that need dispelling.

So before those teachers who are using it without enough understanding get carried away, and before those new to it start replicating the myths, here’s a brief guide to how to use it properly.

At a time when curriculum and task design is at the forefront of thinking, it may be useful for teachers of all subjects, too.

Myth 1: Variation theory is just about the way exercises are ordered; for example, making tasks progressively harder

Structuring mathematical tasks, examples and exercises is one way to implement variation theory; but just plugging different numbers into the same type of question - getting more difficult by progressing through positive, negative, decimal and fractional values and tricky cases - isn’t.

If you are just doing that, it is an oversimplification of the theory and might not have the intended effect.

However, this does have its root in an important emphasis in the theory: that you choose your values carefully in every mathematical task, with a clear rationale for the effect it is likely to have on what learners pay attention to.

In the question above, Winston’s method works because one of the numbers is close to 50, so the next task might be to make up calculations for which Winston’s method works well and some for which Winston’s method does not work so well. This creates a need for a universal method as a backstop for when mental methods won’t help.

Myth 2: We have to ‘do’ variation theory now as part of mastery

Some have coupled variation theory with mastery in mathematics, and many teachers feel pressure to use it in some way as part of a political agenda.

However, variation theory doesn’t necessarily imply a certain type of teaching and can be used in conjunction with - or independently of - all sorts of other ideas.

It’s not a “part” of mastery and it’s not a box you can tick - it’s about exposing mathematical structure to learners as a way of enabling conceptual development, and can be applied in a wide variety of different ways. Mastery in mathematics - however you define it - may happen without explicit use of variation theory, and vice versa.

Myth 3: If I teach using variation theory, pupils will learn

We know that nothing in education is guaranteed, and this theory is about offering opportunities to learners, not forcing them to do anything (as if a theory could, anyway!). It may sound obvious, but using variation theory isn’t a sure-fire route to outcomes in learning or classroom achievement.

However, that isn’t to say that it can’t be immensely powerful. At its heart is the profound idea that learners are more likely to see mathematical structure if you pay attention to unfolding the mathematical structure through controlling variation, rather than asking learners to work through a variety of examples, possibly getting stuck in the details, and hoping learners will “get” the underlying structure all on their own. This idea is a simple yet powerful one, and the potential pay-offs are huge.

But variation theory is, by its nature, an idea that needs deep embedding. Changing a worksheet doesn’t do it, because helping pupils to see structure requires a holistic approach - it is about the task, the questioning AND the environment.

Myth 4: The only thing variation theory leads to is intelligent practice

The idea of intelligent practice is defined by the National Centre for Excellence in the Teaching of Mathematics (NCETM) as: “The precise designing of pupil activities and practice questions so that, rather than pupils repeating a mechanical activity, they are taken down a path where the thinking process is practised with increasing creativity.”

This can clearly be linked to variation theory, and it is broadly true to say that using variation theory is likely to lead to intelligent practice - but also to many other things, like fluency; being comfortable with a particular kind of discourse; seeing relationships; and making connections. It is also important to consider endpoints and outcomes - competence with a skill or algorithm isn’t the only purpose for using variation theory.

Myth 5: Variation theory is about step-by-step atomisation of mathematical content

By far the biggest worry is the way that variation theory may be used to justify an atomisation of content, a breaking down of mathematical ideas into smaller and smaller parts, which is a perception about variation theory that simply isn’t true. Rather it provides ways to look at, and build, complex ideas from various perspectives.

The grain size of the variation - how large the changes are - and also how many changes are made at once, is not prescribed by variation theory. Sometimes several things can be legitimately changed at once; sometimes the questions or tasks may combine several ideas at once; rarely is it about breaking down content into the tiniest, most digestible chunks possible, because mathematics is about connections and relationships, and disguising this from learners by neatly julienning it for them is at best unhelpful. Variation theory is about so much more than just “clever exercises”.

So, how should it be used?

Two constructs that are important to variation theory are: dimensions of possible variation and ranges of permissible change (Watson and Mason, 2006). These are helpful ideas when designing or analysing mathematical tasks from a learner perspective. If designing from scratch, you can ask:

* What is the key idea I am trying to teach?

* What are the mathematical variations around that key idea?

* Which variations are most likely to help learners attend to that key idea?

* How could this task help to explore the dimensions of possible variation?

If you are analysing or choosing a task, you could ask:

* What are these dimensions of possible variation?

* How have they been explored in the task design?

* What are the learners being directed to pay attention to?

* Why might those specific examples have been chosen?

How students perceive a specific object of learning depends on what pattern of variation is provided, with different patterns of variation resulting in different types of learning. Most agree that there are four main patterns of variation:

* Contrast - identifying differences of an aspect

* Generalisation - seeing different ways in which the same aspect appears

* Separation - categorising aspects that can/do change from those that can’t

* Fusion - experiencing several critical aspects at the same time

However, Lai and Murray (2012) present a slightly different view by considering three different types of variation: one problem, multiple solution methods; one problem, multiple changes/ conditions; multiple problems, one solution method.

Either set of headings could be used to explore the different types of response from learners to varying different things in a task - in other words, have a play around, do your own research, and think about your skills of predicting how those in your classroom might respond to changes you make to what is in front of them.

You could also talk to other teachers about what you perceive to be the critical aspects of the task and why; see if they agree, and have a discussion about what you understand the term to mean.

For example, typically, when teaching about straight lines, we might want learners to understand the significance of the terms m and c in the equation of a straight line y = mx + c.

We may ask students to use a graph plotter to plot graphs of:

1. y = x - 3

2. y = x - 2

3. y = x - 1

4. y = x

5. y = x + 1

6. y = x + 2

7. y = x + 3

We may then ask, “What do you notice?” and then expect them to deduce that the “+c” determines where the line crosses the y - axis.

However, take a look at the resulting picture from the graph plotter. It seems clear that at least a number of pupils may associate the “+c” with where the line crosses the x - axis (at “- c”).

The key idea and aspect of mathematical structure that we want to focus on here is how we translate a function by “+c” in the y-direction. How might we develop our task to bring this to the fore?

What would be the benefits of reframing the task to explore plotting:

1. y = x + 1

2. y = 2x + 1

3. y = 3x + 1

4. y = x2 + 1

5. y = 2x2 + 1

And going on to ask them to find equations of lines and quadratics that pass through (0,2), (0,3), (0,-1) and so on?

Thinking about variation in this way might be something you can share with students, too - they could be encouraged to write these types of problem after encountering plenty of interesting examples.

You see, ultimately, variation theory is considering the journey for the learner - moving from early experiences and examples, symbols and system into structure, and exposing that mathematical structure in an inviting and accessible way.

What variation does the learner need to experience in order to move from what they know to what they don’t know?

And that is as applicable in maths as any other subject.

Anne Watson is professor emeritus of mathematics education at the University of Oxford and the author of many books and resources, including a chapter on “Pedagogy of Variations” in Teaching and Learning Mathematics Through Variation, edited by Huang, R, and Li, Y, and mentioned above. Geoff Wake is professor of mathematics education and convenor of the Centre for Research in Mathematics Education at the University of Nottingham. Lucy Rycroft-Smith works in communications and research at Cambridge Mathematics

Can variation theory work in other subjects

Mark Enser, head of geography and research lead at Heathfield Community College in East Sussex, ponders whether it could be applied to geography:

The idea of variation theory would seem to have some applications in geography; at least conceptually. 

A defining part of our subject is using contrasting places to identify common themes - such as the way rivers behave, leading to the Bradshaw model - or the truly unique, giving a sense of place. Variation theory could, therefore, be applied to understanding how changing one aspect of a place or event could have powerful ramifications, even when everything else stays the same. This could help students to understand the underlying principles.

For example, we could look at the disaster risk equation (risk is equal to magnitude, multiplied by vulnerability and then divided by the capacity to cope) for a particularly devastating natural disaster. We could ask students to alter just one aspect of the equation and see if this changes their analysis of the risk.

This would hopefully help them to understand which factors are most important in determining the level of risk.