Teaching for Mastery: Deeper thinking not harder sums!

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Depth NOT acceleration

TC 1The old national curriculum, measured in terms of levels, encouraged undue pace. Children were accelerated onto more complex concepts before really mastering earlier ones. Imagine the cubes to the left represent the building blocks of maths. As you move on swiftly from one topic to the next the tower gets taller and taller, until eventually it comes crashing down. By moving on before a child is ready, before they fully understand the concept, you are helping them build a tall tower, which at some point will become unstable.

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The new national curriculum encourages the study of fewer skills in greater depth in order to achieve mastery. By taking the same cubes and arranging them differently, the tower remains standing.

So by ensuring that the foundations are solid before moving on, children develop a more stable platform from which to build.


Curriculum: Sprial vs Linear

Many primary schools are still following a spiral curriculum where topics get revisited each term. The nature of this is that you have to move on quite quickly from one topic to the next, often spending only one week on something and not really having time to ensure that the children have properly and deeply undersood the concepts being taught. This suited the old national curriculum as it was content heavy and didn’t require any depth of understanding.

An alternative to this approach is following a linear curriculum in which topics are blocked together and covered only once in the year. Doing so allows more time for children to gain a deeper understanding of the key topics as there is less pressure to move on before they are ready. Evidence shows that children who have a deeper understanding of number and an enhanced ability to reason mathematically can progress through other topics more quickly.

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This is an example of how a linear curriculum might be structured. Note the emphasis that is placed on number in the first two terms. It is important to note that just because the curriculum is linear, it does not mean that the skills learnt in one block are not revisited again throughout the year as research clearly shows that interleaving ideas and revisting them aids long term retention. For example in the scheme above, during the geometry in the spring term you might look at perimeter and in doing so revisit addition and subtraction.

It is important at this stage in the year that schools consider whether the scheme they are following really provides their children with the most solid foundations.

There are many schemes availble on the market to suppot teaching for mastery (two of which we review in Textbooks: Compare and contrast). In addition to these, there is a FREE scheme that has been created by the White Rose Maths Hub, which provides an excellent starting point for any primary school wishing to adopt a new approach. We are currently working with the White Rose Maths Hub to further enhance these resources, details of what we are adding and expected timelines can be found here.

Depth is achieved through variation

There are three elements that are critical on the journey towards mastery in maths and that we need to develop in our children. Fluency, reasoning and problem solving. Without one, the next cannot follow and it is only by developing these three skills that children can move towards mastery.

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This flow of learning was first seen in Shanghai by the teachers who went on the exchange trip. Children must first gain fluency in whatever topic they are studying before they move on to reason mathematically and then begin to solve a variety of different problems which probe and challenge their depth of understanding.

Here’s a few examples to help illustrate how this development might look.

Example 1:


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Problem Solving:

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Example 2:


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Problem Solving:

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Can you see how each task builds on the previous one? The skills required are the same but the depth of understanding required is very different. When these activities are put in the context of whole-class, mixed-ability teaching where the aim is for all children to develop the skills of fluency, reasoning and problem solving at the same time, the slow progression through these three elements is even more critical. More will be written on this blog shortly regarding whole-class, mixed-ability teaching and how it can and does work.

Intelligent Practice

Variation is a phrase which has been banded about all too often since the first teachers returned from the Shanghai exchange visit. It think it has taken us a very long time to fully appreciate what is meany by variation…

Variation is NOT:

  • more of the same thing but a bit harder.
  • the same as variety!

When using intelligent practice, all tasks are selected and sequenced carefully with purpose, offering appropriate variation so that when viewed together they reveal something about the underlying mathematical structure, concept or process. Put simply, variation reveals concepts.

Variation is an approach to teaching. It is the art of sequencing similar but increasingly complex problems to “generate disturbance of some sort for the learner” Festinger (1957)

Consider the following example:

23 + 10 = [   ]

23 + 11 = [   ]

23 + 12 = [   ]


23 + 9 = [   ]

23 + 8 = [   ]

23 + 7 = [   ]

By changing just one small element at a time children are given the chance to develop their understanding of a concept rather than just learning a process to get the right answers. However, it is not as simple as just finding patterns. Care must be taken to ensure the structural concept is understood; not just a superficial procedure found, as it is this deep conceptual understanding that leads to mastery.

If you just rearrange the questions a little you can quickly see how big a difference the order the questions are in can make to the development of a child’s understanding.

23 + 9 = [   ]

23 + 12 = [   ]

10 + 23 = [   ]

23 + 7 = [   ]

23 + 11 = [   ]

8 + 23 = [   ]

It is this intelligent practice, the appropriate selection and sequencing of questions, that can make a fundamental difference to a child’s conceptual understanding.

Conceptual understanding leads to mastery

It is critical that children are taught maths in a way the develops a deep conceptual understanding as this is the only way of securing solid foundations. One way in which we can help children to do this is by using the concrete-pictorial-abstract (CPA) approach. This is based on research by psychologist Jerome Bruner, which suggests that there are three steps (or representations) necessary for pupils to develop understanding of a concept. Reinforcement is achieved by going back and forth between these representations.


In this stage a student is first introduced to an idea or a skill by acting it out with real objects. In division, for example, this might be done by separating balls into groups of red ones and green ones or by sharing 10 biscuits among 5 children. This is a ‘hands on’ component using real objects and it is the foundation for conceptual understanding.


In this stage a student has sufficiently understood the hands-on experiences performed and can now relate them to representations, such as a diagram or picture of the problem. In the case of a division exercise this could be the action of circling objects.


In this symbolic stage a student is now capable of representing problems by using mathematical notation, for example: 10 ÷ 2 = 5 Students only use abstract numbers and figures when they have enough context to understand what they mean This is the ‘final’ and most challenging of the three stages.


The following example demonstrates the principles of CPA. It starts with a pictorial representation of the problem using a clear diagram (the use of counters is important for any child still at the concrete stage), but it then builds the abstract alongside, so that the child can make that leap between the pictorial and the abstract as and when they are comfortable in doing so.

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In this second example the concrete is used as a starting point for the question. Two oranges are divided up into quarters, three of them are circled from each which leaves six quarters which are then put back together to make one and a half oranges. The pictorial stage sees the introduction of circles instead of oranges and then the final stage is the introduction of the abstract, the numerical calculation. By putting all three alongside each other students can see the progression and when they are ready, they will naturally make the transition from concrete to pictorial to abstract.

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In Summary…

  1. Teach fewer concepts in greater depth
  2. Intelligent practice gives children a rich appreciation of the concepts through variation
  3. If you help children unlock this depth of learning then, over time, whole class, mixed ability teaching will become even more successful.


Many of the ideas in this post have been adapted from ideas seen in Tom Collin’s session at the spring conference. Click here to download the powerpoint Session 1 – Teaching for Mastery

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