Research School Network: Variation theory as a tool to reach struggling learners in maths Alison Ford discusses how variation theory helps students identify critical aspects of maths concepts

I teach a class of 20 Year 9 students. Although the school has lower levels of disadvantage and SEND than the national average, many students in this class have identified additional needs and experience barriers to consistent engagement, including attendance challenges, language processing and high levels of maths anxiety.

Barriers like these can be compounded by the curriculum experience. The Ofsted report Coordinating Mathematical Success (2023) found that students in lower-attaining groups are often ​‘rushed’ through content too quickly and experience a reduced curriculum with fewer opportunities for mathematical problem solving. Hallam and Parsons (2013) also found that students in lower-attaining classes are exposed to less subject-specific vocabulary than their peers in higher-attaining groups.

All students are entitled to full access to a high-quality mathematics curriculum, not a ​‘dumbed down’ version. The curriculum should remain ambitious while being designed with laser-sharp precision and accessibility in mind.

One way I’ve been focusing on that precision is through examples, explanations and tasks, all of which need to be crystal clear and concise. This is especially important when attendance interrupts the continuity of learning, meaning it is important not to assume secure prior knowledge, including content covered in previous lessons.

Examples, non-examples and variation theory

Taking Recommendation 6 of the EEF’s Improving Mathematics in Key Stages 2 and 3 guidance report as a starting point, this year I have learnt a lot about variation theory (Watson 2017, Marton and Pang 2013) and have been applying its principles to my examples, explanations and tasks.

Variation theory proposes that understanding develops as learners discern what is different and what is the same between examples. Through this, they build a general understanding of a new concept, fusing the features together and extrapolating beyond the examples given. In other words, carefully designed variation helps students identify the critical aspects of a concept and distinguish it from related but different concepts.

An example

Recently, I wanted to use the word ​‘variable’ with my class, so I designed the following series of examples to introduce the concept. In particular, I wanted students to understand that a variable is not simply ​“a letter in maths”, but a representation of an unknown or changeable quantity.

I deliberately started with two non-examples, drawing on teacher Kris Boulton’s work (2025) to give the students something to compare the new concept to. There is minimal variation between the second and third examples so that they can see the small change that moves us from a specific value to a variable.

Then there were further minimal changes so that they did not develop the misconception that only certain letters could be variables. The difference between the final two examples was vital to show how very similar looking situations (3a and 2m) do not belong to the same concepts.

They were presented one at a time with carefully scripted brief explanations to accompany each example. I then gave them a testing sequence of similar examples in a different order to check their understanding.

For learners whose prior knowledge is insecure or interrupted, this level of precision matters. It reduces reliance on remembered procedures or isolated definitions and instead builds understanding through discernment of critical aspects.

The impact has been clear. The class use the word variable now with confidence. It became particularly useful when teaching them complex skills such as multiplying through an equation with more than one variable in preparation for solving simultaneous equations. Additionally, retrieval of the concept has been straightforward thanks to the clarity with which they learnt it. Careful planning of how to highlight the discerning features using example and non-examples has led to long-lasting learning and increased confidence.

References

Boulton, K (2025) How to Teach Element 1 of 4: Categoricals

Hallam, S., & Parsons, S. (2013). The incidence and make up of ability grouped sets in the UK primary school. Research Papers in Education, 28(4), 393 – 420.

Marton, F., & Pang, M. F. (2013). Meanings are acquired from experiencing differences against a background of sameness, rather than from experiencing sameness against a background of difference: Putting a conjecture to test by embedding it into a pedagogical tool. Frontline Learning Research, 1(1), 24 – 41.

Watson, A. (2017) ​‘Pedagogy of variations: synthesis of various notions of variation pedagogy’, in Huang, R. and Li, Y. (eds) Teaching and learning mathematics through variation. Rotterdam: Brill Sense, pp. 85 – 103.