What Makes Effective Maths Teaching?

Maths teaching is not about speed or high completion rates. Effective maths teaching develops conceptual understanding, fluency, and problem-solving ability simultaneously.

This was articulated clearly in the EEF Guidance Report for Mathematics (2017), which synthesised research from meta-analyses and randomised trials. The report identified five key teaching approaches that consistently improve attainment:

1. Concrete-Pictorial-Abstract (CPA) approach — Teaching concepts with objects, then pictures, then symbols
2. Mastery approach — Deep understanding of core concepts before moving on, not breadth without depth
3. Visual representation and bar models — Using diagrams to make abstract concepts concrete
4. Whole-class instruction with formative assessment — Teaching together as a class, checking understanding constantly
5. Intelligent practising — Varied practice that builds flexibility, not repetitive drills

These approaches align with Rosenshine's Principles of Instruction (2010), which emphasise scaffolding, guided practice, and varied problem types. They also align with Singapore's approach, which has elevated maths attainment in primary schools across Europe.

The Mastery Approach: Slow Down to Speed Up

What is Mastery?

Mastery is the philosophy that every learner should understand a concept deeply before moving to the next topic. It contrasts with "acceleration," where faster learners skip ahead and slower learners fall further behind.

The research foundation: When learners have shallow understanding of fundamentals, they struggle with advanced concepts. For example, if a child doesn't understand place value, they will struggle with subtraction across 10s. Rushing past place value to progress "quicker" guarantees problems later.

What mastery looks like in practice:

• Year 1: Spend 3 weeks on number bonds to 5. Use objects, count repeatedly, explore arrangements. Move on only when 90% of the class secure.
• Year 3: Spend 2 weeks on multiplication as repeated addition before introducing the × symbol. Use concrete examples.
• Year 5: Spend 2 weeks on fractions as parts of a whole, using strips and diagrams, before symbolic notation.

The cost-benefit: Slower initial progress (maybe 10% slower in term 1), but stronger foundations. By term 3, mastery classes often overtake accelerated classes because learners understand deeply and learn faster.

Differentiation in a Mastery Model

Don't confuse mastery with "everyone does the same thing." Differentiation still exists, but it happens through depth, not acceleration.

What happens:

• Secure learners explore deeper variations: "If number bonds to 5 are {0,5}, {1,4}, {2,3}, what patterns do you notice? How would they change for number bonds to 6?"
• Developing learners get additional guided practice: work with manipulatives longer, use simpler contexts
• All move forward together once secure, avoiding permanent "low group" tracking

Why this works better than acceleration: When a faster learner solves 20 problems while a slower learner solves 5, the gap widens. When a faster learner deepens understanding while a slower learner builds fluency with the same concept, gaps narrow.

Key Approaches and Pedagogical Tools

Singapore Maths and the CPA Approach

Singapore Maths refers to a curriculum and pedagogy that has consistently outperformed other countries in international assessments (TIMSS). The core is the Concrete-Pictorial-Abstract (CPA) approach.

How it works:

• Concrete: Learners manipulate physical objects (blocks, counters, rods) to understand a concept
• Pictorial: Learners draw or interact with pictures/diagrams representing the same concept
• Abstract: Learners work with symbols and numerals, understanding the connection to concrete and pictorial

Example (Year 2, addition with regrouping):

• Concrete: "I have 15 Dienes rods. I add 7 more. Let me combine and regroup into 2 tens and 2 ones."
• Pictorial: "Draw 15 as 1 long and 5 ones. Draw 7 ones. Combine and regroup." (Visually shows the regrouping)
• Abstract: "15 + 7 = 22. 10 + 5 + 7 = 10 + 12 = 10 + 10 + 2 = 20 + 2 = 22."

Why it works: Learners see the "why" behind the algorithm, not just the "how." They build conceptual understanding that transfers to new problems.

Bar Models and Visual Representation

The bar model (also called tape diagram) is a visual representation tool that makes abstract word problems concrete. It is central to Singapore Maths and has transformed how many UK schools teach problem-solving.

How it works: Learners draw rectangles (bars) to represent known and unknown quantities, then use the visual to identify the operation needed.

Example (Year 3 word problem):

"Sam has 25 marbles. He buys 18 more. How many does he have now?"

Instead of guessing the operation, learners draw:

• Bar 1: "25 marbles"
• Bar 2: "18 marbles"
• Combined bar: "? total"
• Visual makes it obvious: combine the bars = addition. 25 + 18 = 43.

Why it works: Word problems require understanding the structure of the problem. Bar models make structure visible. Learners can solve the problem correctly even if they don't know the procedure, using visual reasoning instead.

EEF Evidence: The EEF found visual representation and bar models are among the most effective strategies for improving maths attainment.

Whole-Class Instruction with Formative Assessment

The mastery approach is typically delivered via whole-class instruction—not independent worksheets. The teacher leads a lesson where all learners engage with the same concept simultaneously, using mini whiteboards and cold calling to check understanding frequently.

What happens:

• "Today we're exploring number bonds to 10. I show 10 counters arranged in two groups. I ask, 'What are the two groups? What are the different ways to split 10?'"
• Learners draw arrangements on mini whiteboards and show responses
• Teacher scans responses: "Some of you show 7 + 3. Who can explain why that works?"
• Learners share and the teacher builds understanding through discussion

Why it works: Whole-class teaching with formative assessment (checking understanding constantly) prevents learners from practising misconceptions. It's also more efficient—the teacher addresses confusion immediately rather than finding errors in marked books days later.

Intelligent Practising (Varied Practice)

Practice is essential in maths, but not all practice is equal. Varied practice (sometimes called "interleaving") is more effective than blocked practice for long-term retention and transfer.

What happens:

• Blocked practice (less effective): "Solve 10 doubling questions, then 10 halving questions." Learners solve them quickly but often forget the strategy after a week.
• Varied practice (more effective): "Solve this mix: some doubling, some halving, some with both. Work out which strategy applies to each." Learners must think about when to use each strategy, strengthening flexibility.

Research: Rohrer & Taylor (2007) found that interleaved practice is substantially more effective for learning problem-solving flexibility, even though it feels harder during learning.

Maths Tools and Digital Resources

MathsWatch

MathsWatch is a subscription platform providing short video explanations of maths concepts (typically 3–10 minutes each) plus linked worksheet tasks. Videos are indexed by topic and age group (primary and secondary).

When to use: Homework support (learners or parents watch a video if stuck), intervention (reteaching a concept a learner missed), and flipped classroom (learners watch video at home, practise in class).

Times Tables Rock Stars (TTRS)

TTRS is a gamified app for practising times tables through short gameplay sessions (3–5 minutes). Learners race against classmates and progress through levels as speed and accuracy improve.

When to use: Regular low-stakes retrieval practice (few minutes daily). TTRS is excellent for building fluency but should not replace conceptual teaching of multiplication.

Topmarks and Mathsframe

Free interactive maths tools covering topics from Early Years through KS3. Includes games, simulations, and visual demonstrations.

Examples: 100 square interactive grids, shape manipulatives, fraction bars, bar chart builders.

Hegarty Maths

Comprehensive video platform with explanations for all GCSE and A-Level topics. Similar to MathsWatch but more extensive secondary coverage.

Dynamo Maths (For SEND Learners)

Dynamo Maths is a multi-sensory intervention for learners with dyscalculia or maths anxiety. It uses colour-coded number lines, kinesthetic activities, and explicit, structured teaching.

When to use: Small group intervention for learners with persistent maths difficulties not responding to quality first teaching.

Maths for SEND Learners and Dyscalculia

Dyscalculia is a specific learning difficulty affecting number sense and calculation. It is not simple "maths anxiety" or low maths ability—it's a neurological difference in how the brain processes number.

Signs of dyscalculia: Difficulty with number bonds, counting, subitising (instantly recognising quantity without counting), and memorising times tables despite good teaching.

Effective approaches for dyscalculia:

• Multisensory teaching: Use colour, sound, movement, and tactile materials. Number lines with colours. Counting with rhythm.
• Concrete materials first: Stay at concrete stage longer than typical peers. Use number rods, counters, beads—physical objects learners can manipulate.
• Small group intervention: Intensive, explicit teaching in groups of 1–3, not whole-class. Programmes like Dynamo Maths or Numicon-based interventions.
• Overlearning: More repetitions, spaced over time, to embed learning. Mastery is slower but deeper.
• Accommodation: Number lines, number squares, or calculators to reduce working memory load. The goal is to build conceptual understanding, not always to compute mentally.

Maths Deep-Dive Questioning

Ofsted's deep-dive inspection framework specifically looks for evidence of "mastering" a maths topic. Teachers are expected to demonstrate:

• Knowledge of curriculum sequence and pedagogical progression
• Evidence that learners understand concepts deeply, not just procedurally
• Varied question types and formative assessment throughout teaching
• Monitoring of misconceptions and targeted intervention

Typical deep-dive questions:

• "Show me a maths lesson using bar models. Why did you choose bar models for this concept?"
• "How do you ensure learners develop conceptual understanding, not just procedural fluency?"
• "What's your approach to formative assessment? How do you use it to inform pacing?"
• "Tell me about a learner with a maths difficulty. What intervention are they receiving? How do you know it's working?"

Strong answers focus on pedagogy and evidence, not activities and pace through the curriculum.

Key Resources and Linked Articles

Key Research References

EEF (2017). Guidance Report: Mathematics. Based on meta-analyses and trials. Recommends CPA, mastery, visual representation, whole-class instruction, and varied practice as high-impact strategies.

Rosenshine, B. (2010). Principles of Instruction: Research-based Strategies that All Teachers Should Know. American Educator, 35(1), 12–20. Emphasises scaffolding, guided practice, and varied problem types.

Rohrer, D., & Taylor, K. (2007). The shuffling of mathematics problems improves learning. Instructional Science, 35(6), 481–498. Evidence for interleaved practice over blocked practice.

Key Takeaways

1. Mastery is about depth, not acceleration. Deep understanding of fundamentals prevents later learning gaps. Slower early progress (1–2 weeks longer per topic) pays dividends later.
2. Visual representation is foundational. Bar models, concrete materials, and pictorial diagrams make abstract concepts concrete and accessible to all learners, including those with dyscalculia.
3. Formative assessment should be constant. Check understanding before moving on. Use mini whiteboards and cold calling to keep all learners engaged and reveal misconceptions early.
4. Varied practice is more effective than blocked practice. Interleaving develops problem-solving flexibility. Learners who mix problem types learn to identify when to use each strategy.