Research School Network: Variation Theory in Maths: The Key to Better Maths Understanding How evidence-based strategies have driven substantial improvements in maths outcomes across the Trust.

Chris Whitehead is the Director of Secondary Maths at North East Learning Trust, a multi-academy trust comprising eight secondary schools. Schools that have joined the Trust in recent years have experienced a significant uplift in results. On average, these schools have seen a 25-percentage-point improvement at grade 4+ (bringing the Trust’s 2024 average to 79%, compared to the national average of 60%) and a 23-percentage-point increase at grade 5+ (with the Trust achieving 58% versus the national 42%).

In this blog, Chris shares how evidence-based strategies — high-quality teaching, variation theory, and effective modelling — have driven substantial improvements in maths outcomes across the Trust.


The Importance of a Strong Maths Education
The Education Endowment Foundation’s (EEF) Improving Mathematics in Key Stages 2& 3 states: ​‘Leaving school with a good GCSE in maths is a prerequisite for progressing into quality jobs, apprenticeships, and further education.’ This statement resonates with pupils, parents, teachers, and employers alike. Yet, too many students fall short of achieving the required grade, limiting their future opportunities.

At North East Learning Trust, we are committed to ensuring that every pupil — regardless of background — has the opportunity to excel in mathematics. Research consistently shows that excellent teaching is ​‘the most important lever schools have to improve pupil attainment.’ With this in mind, we have dedicated significant time and resources to curriculum design and teaching methodology, empowering teachers to deliver a high-quality, challenging curriculum effectively.

Through meticulous curriculum planning and a structured approach to problem-solving, we have achieved remarkable success in raising GCSE maths attainment across our schools.

Thoughtfully Designed Curriculum
Our approach is rooted in logical sequencing and step-by-step methodology, ensuring that every pupil builds a strong foundation. By carefully scaffolding new concepts onto prior knowledge, we create an inclusive learning environment where all students can thrive.

The Power of Variation Theory
Variation theory plays a central role in our teaching strategy, fostering deep conceptual understanding. Developed in the late 1990s, variation theory examines how students perceive the same phenomenon in different ways. In maths, we leverage this approach to expose pupils to multiple representations of the same concept. By doing so, we help students identify underlying mathematical principles rather than relying on surface-level patterns.

Small Steps, Big Success
Breaking mathematical concepts into the smallest possible steps ensures mastery at every stage. Our lesson structure follows a clear and effective sequence:

- Teacher Explanation: Introducing and contextualising the concept
- Worked Example: Demonstrating application through guided practice
- Independent Practice: Allowing students to apply their knowledge, with continuous formative assessment

Only when all pupils achieve success in these foundational steps do we introduce more complex challenges. This systematic approach ensures that learning is built on a rock-solid foundation — never on sand.

Here is an example of how we approach factorising.

Step 1- Definition and teacher explanation

Step 2- Guided Practice

Here, in our guided practice we start by factorising with all positive terms.

Step 3 – Independent practice

Applying the principles of variation theory, we use just factors of 24 and 36 to apply the rule and practise. Note the reflection questions – encouraging pupils to think about their previous worked examples – what has changed and what has stayed the same.

We assess pupil progress continuously during the independent practice – often using mini whiteboards to continuously check for understanding. Only when we are confident that pupils have got it, do we move on to factorising with a range of positive terms.

Step 4 – The same, but different.

This is followed by guided practice of negative terms.

As we move into some independent practice, again we use factors of 24 until pupils have mastered the application. Pupils are once again encouraged to think about what has changed and what has stayed the same.

Once we are sure that pupil have mastered this ( we check for understanding continuously), we move on to mixed independent practice, as shown below.

Practice Until Perfect
Repetition and practice are at the heart of our strategy. By focusing on mastery, pupils gain confidence and fluency in mathematical problem-solving. This systematic approach not only improves results but also fosters a genuine love for maths.

Conclusion
By embedding evidence-based teaching practices into our maths curriculum, we have transformed outcomes for our pupils, as well as improving their confidence and fluency in maths. Through logical sequencing, variation theory, and carefully structured problem-solving, we are equipping students with the skills and confidence they need to succeed — not just in their exams, but in life beyond the classroom.


References

Improving mathematics in key stages 2 and 3 EEF. Available at: https://educationendowmentfoundation.org.uk/education-evidence/guidance-reports/maths-ks‑2 – 3

High-quality Teaching EEF. Available at: https://educationendowmentfoundation.org.uk/support-for-schools/school-planning-support/1‑high-quality-teaching (Accessed: 01 February 2025).

The Variation Theory of learning. Available at: https://www.researchgate.net/publication/377353216_The_Variation_Theory_of_Learning (Accessed: 01 February 2025).