What are children learning?

 Ofsted's Coordinating mathematical success: the mathematics subject report came out in 2023 and had a lot to say about Teaching for Mastery, whilst carefully avoiding the phrase "Teaching for Mastery".

As well as highlighting recent improvements in mathematics teaching in England, they had some concerns. One of these was a lack of ambition:

An ambitious curriculum is one that maximises the mathematics that pupils learn. In some schools, teachers move on before ensuring pupils have learned important knowledge and committed that knowledge to long term memory. In schools where this is common, leaders focus on what pupils study, rather than on what pupils learn. Moving on when pupils are not mathematically ready gives the illusion of progress but creates ever greater gaps that will take more time to address in the future. 

Leaders should focus on what pupils learn, not on what pupils study.

 In some schools, teachers teach the curriculum that is put in front of them, then give the children a summative assessment a few times a year to see if they are 'working at age-related expectations'. Not surprisingly, Ofsted also criticise this approach:

Some leaders gain false assurance about the effectiveness of their curriculum design and practice through internal assessments that closely align with ‘expected performance thresholds’ of external assessments. This approach often leads to an acceptance of pupils moving through the mathematics curriculum with significant gaps in their knowledge and leaders failing to make necessary adjustments to their curriculums. In these schools, some pupils would be better served by studying less, but securely learning more.

A more ambitious curriculum would focus on what pupils learn

An ambitious curriculum is one that aims for the most number of pupils to learn the most that they can. Rather than making the expectations more challenging, our ambition should be for more children to securely learn less stuff.

A successful school would focus on their key performance indicators for each year group, and ensure that the children have mastered these before they move on. And if they haven't mastered them, they don't move on. Teachers don't cover the curriculum for their year, they cover the curriculum for the children in front of them. 

What does this look like in practice?

Assess for learning

Start by assessing the pre-requisite content. For example, before starting a year 3 unit on multiplication and division, assess that the children are secure in the relevant key objectives from year 2. You can use the DfE Guidance for this - each Ready to Progress Criteria has a set of assessment questions that cover the key learning and put it into context.

If the children are all secure then you can start to plan. If they are not all secure, then start by covering the key ideas again. Don't worry about the children who are secure at this point - they should be encouraged to think more deeply, explain to their peers and ensure that their understanding is deep enough for what is about to come. Don't be afraid to practice pre-requisite skills first.

Teach for the whole class

Think about the journey from this starting point to the key objectives for your year group. What are the big ideas along the way? The NCETM Professional Development materials provide a journey toward each key objective.

Teach it one step at a time. Give time for enough practice after each step. Don't worry that it seems too easy.

Intervene rapidly

If children aren't successful during the lesson, give the necessary additional support as soon as possible. Work with them directly in the lesson if you can to provide additional, temporary scaffolding. Refer back to the models and language used earlier in the lesson. Give small groups of children short bursts of support after the maths lesson or before the next one.

These rapid pre-, post- or in-lesson interventions are in order that children 'keep up, not catch up'. They are distinct from any ongoing 'gap filling' interventions, that are normally focused on filling gaps from previous years that children lack fluency in.

Two visions of Mastery

My main confusion about the NCETM 'Teaching for Mastery' programme was the name. Not the word 'teaching' - that's fairly self-explanatory. The 'for' is also understandable, in the same way that the 'Department for Education' uses the preposition to demonstrate that it isn't actually against education (not all of it, anyway). No, it's the word 'Mastery' that causes the dilemma.

If you look on the Education Endowment Foundation page for Mathematics, Mastery is at the top of the list. Look:

 

However, the definition of Masterylearning provided includes:

Students must demonstrate a high level of success on tests, typically at about the 80% level, before progressing to new content. Mastery learning can be contrasted with other approaches which require pupils to move through the curriculum at a pre-determined pace. Teachers seek to avoid unnecessary repetition by regularly assessing knowledge and skills. Those who do not reach the required level are provided with additional tuition, peer support, small group discussions, or homework so that they can reach the expected level.

Is this the only vision of what Mastery is?

Vision 1: Mastery is personalised learning

There is one vision of Mastery in which students work at their own pace through material, repeating it as necessary until they are sufficiently confident to move on. This approach works particularly well with a narrow curriculum, e.g. one focused entirely on numbers and calculations.
A student is ready to move on when they attain a particular score on a (timed) test. Then they move onto the next unit, and repeat until they have also 'mastered' that one. It could be seen as differentiation taken to the point of a personalised curriculum. Some teachers belief that through artificial intelligence and computer technology it may be possible to achieve this.
This approach may promote confidence in procedures and accuracy in test situations over a deeper understanding of the material. It can be very successful in developing fluency and confidence, but it may not meet the demands of the National Curriculum, in particular regarding conceptual fluency, mathematical reasoning, problem-solving and keeping the class together.
This version of Mastery has been made popular by the Kumon Maths Programme and Sal Khan's Mastery System.

Vision 2: Mastery is whole class

 The second vision of Mastery is one in which all students progress together at the same rate. In this version, the focus is on deeper understanding of the material to 'take up the slack' between differing ability students. It relies on the teacher to assess the whole class regularly and control the pace of learning so that no-one gets too far ahead or behind. When students do start to struggle, they need more and extra support to keep up. This approach focuses on differentation by support and very clear teaching and modelling to keep children working together.
This version of Mastery is heavily influenced by practice in education systems such as Shanghai and Singapore, where very carefully structured and represented material is sequenced to keep children together. Versions of this approach have been promoted by the Ark Schools Mathematics Mastery programme and the NCETM Teaching for Mastery . The NCETM defines Mastery as

At any one point in a pupil’s journey through school, achieving mastery is taken to mean acquiring a solid enough understanding of the maths that’s been taught to enable him/her move on to more advanced material

What is the difference?

In a personalised curriculum, children may progress at different paces. Some children may take a long time to master some ideas. Some children may make accelerated progress as result of grasping (or appearing to grasp) ideas more rapidly. This can be attractive to some parents.
In an Asian-style 'whole class' curriculum, children progress together. Some children may need more Maths teaching to keep up. All children develop understanding at a slower, but deeper pace. This approach should diminish the apparent gaps in attainment between groups of children. This can be attractive to schools (and governments).

What is not Mastery?

If these two visions seem like polar opposites it's worth considering what they have in common. Both contrast to a view of education in which, to quote the EEF report above, "pupils.. move through the curriculum at a pre-determined pace". This approach, which was commonly found in schools following the National Numeracy Strategy and the schemes and textbooks based upon it, focused on coverage of the curriculum rather than Mastery of it.
The NNS focused on whole class teaching, with differentiated tasks to allow different groups of children to achieve different outcomes. The sequence of lessons continued regardless of the competence of the children, but employed a 'spiral curriculum' model where each topic was revisited every term. Children who struggled with a concept were expected to improve next time they reached it. As a consequence the gaps between the majority of children who mastered the material and the minority who didn't grew over time, resulting in more differentiation, resulting in widing gaps between children who could and could not do Maths..

What is Mastery?

Whichever model you subscribe to, there are some key ideas behind any vision of Mastery.

1. Mastery is for everyone. It is based on a belief that all children will achieve.
2. Assessment is important. Teachers need objective evidence of what children have achieved and can do.
3. Teachers need to move at the pace of the child, and not 'cover' the curriculum. I would argue teachers should not even consider 'delivering' the material for their year group, but instead focus on the current understanding of their children.
4. Teachers need to start where the child is, and progress through a series of clearly defined points, checking for understanding as they go.
5. Mastery needs to be seen as a whole school journey, not as the progress from the start to the end of a particular year group.

What is this thing called Mastery?

You've probably heard a lot about 'Mastery' by now. It's been in the TES. Every catalog that arrives in your pigeonhole has 'Perfect for Mastery' stamped on the front. But what is it all about?

Why Mastery?

You might be forgiven for assuming that primary schools in England are not very good at teaching maths. There was a time when that might have been true (probably just before I became a teacher). But our position in international comparisons has been rising steadily for 20 years.

It's probably no coincidence that the vast majority of the increase takes place between 1999 and 2007, the years of the National Numeracy Strategies. In fact, our 2015 score places us firmly in the second highest performing group of countries, along with Russia, Ireland, Norway and Belgium.

That's a great place to be; but the gap between England and the next highest performing group of countries is a statistically massive 27 points. The highest performing group of countries are all East Asian - Singapore, Japan, Korea, etc.

What can we learn from East Asian countries?

It would be ridiculous to throw out all the improvements we have made in the last 20 years and try to import the 'Shanghai approach' or the 'Singapore approach' wholesale. But there may be things that we can learn from these countries:

1. Whole class teaching

Whole class teaching doesn't have to mean passive learning. It can incorporate pair work, group work and investigations, but the expectation is that the class will be learning the same thing at the same time. The National Curriculum(2014) has the expectation that the 'majority of pupils will move through the programmes of study at broadly the same pace', and it's very hard to achieve this if pupils have different learning objectives.

This style of teaching that goes back and forth between teacher led and group led is often called 'Ping Pong'. It's a world away from the 'input - independent - plenary' triad of the Numeracy Strategy.

2. Mixed-Ability classrooms

My hackles tend to rise at the word 'ability', because, while we all certainly have different abilities at different things, these are not fixed. Our abilities change, as we grow and learn. I was a 'low ability' artist at school. I'm an acceptable sketcher now. It just took me longer. I accept that children will grasp things at different rates, and good teaching takes account of this. I don't accept that the child's prior attainment dictates their future capabilities. 

In many high-performing countries, such as China, children are not put into sets, streams or groups for Maths in primary schools. This doesn't mean we should not differentiate, but we should base this on what the children can and can't do now, not on prior attainment data that labels them 'higher' or 'lower'. 

3. Small Steps

If the whole class are moving together, they all have to succeed. It's no good half the class achieving the learning intention, and the other half picking it up at some later point. So the learning has to be broken down into small steps, that everyone can understand, and sequenced carefully, so that learning builds on firm foundations. Shanghai teachers have a massive advantage here, in that they tend to teach maths across year groups, and so are more aware of the sequence of steps.

4. Challenge and depth

Learning isn't a passive process, at least according to constructivist models. Mastery lessons often start with a real-life scenario or a problem for children to explore and represent, leading into discussion and the examination of strategies to solve it.  They link into prior experiences. 
The use of variation also creates challenge throughout the lesson along with the expectation that children can explain a process rather than just get to an answer. It's common to include a challenge question to make children think deeper - but this is for the whole class, not just the 'higher attaining'.

4. Representation and Structure

English schools are aware of the value of 'Concrete-Pictorial-Abstract' and the constructivist work of Jerome Bruner. Singapore schools embed this firmly in their exploration of mathematical concepts. Shanghai schools are less wedded to concrete resources, but they use lots of structured representations of number, such as tens frames. The use of a structured representation makes it clear what the 

5. Variation

This seems to be quite unique to countries like China, where children are taught from very early on to search for patterns and to see connections. When planning a lesson, teachers use variation to make concepts clearer, such as showing different representations or making subtle changes to a process. When setting tasks, there is often a patterns or a link between the examples that draws out the structure of the mathematics.

6. Sentence Stems

There has been a lot of work done in the last ten years on 'Success Criteria' in English classrooms, largely based on the work of Shirley Clarke. A  sentence stem is a model for children to follow, that focuses on the key aspect of the lesson. This allows for chorusing and repetition: 'I say, you say, we all say', as well as a 'hook' to refer back to in subsequent lessons.

7. Textbooks

You can't help noticing that many of these countries use a single textbook, usually produced by a central agency. Whilst this is may be helpful, it doesn't teach the children - there is no substitute for a well-designed lesson and teacher subject (and pedagogical) knowledge. The DfE's attempt to produce a choice of textbooks that embody the above principles has been an shambles so far, but hopefully things will improve as publishers catch up. 

So what is a Mastery Curriculum?

A mastery curriculum is one designed for the children to move through at the same pace. It is carefully designed so each step lays the framework for the next and each concept is developed in enough detail that it becomes embedded. It provides enough challenge for the children who see the concept readily, and enough time for those who need it to become proficient.

I'm not saying for a moment that I think our current Mathematics National Curriculum comes anyway near this. Concepts are introduced too early (multiplication and division), some objectives are far too complex and some are repeated or missed out entirely in some year groups. A few make no sense. 

What it is and what it isn't.

A key idea when illustrating maths concepts to children is to show a non-example as well as examples (standard and non-standard). So:

Teaching for Mastery is:

• A set of principles.
• All children in the class (with very few exceptions) working together.
• High expectations for all children.
• Small steps that everyone can succeed in.
• Engaging and fun.

Teaching for Mastery is not:

• Passive learning.
• A published scheme of work.
• A scheme of work produced by a Maths Hub.
• A one-size-fits-all pedagogy.
• Something Ofsted are looking for.
• Something Ofsted are against.
• The end of differentiation.
• Turning the tables to face the front.
• Using a Powerpoint in lessons.
• Using a certain set of physical resources.
• Holding back high-attaining pupils.
• Something that we can start doing tomorrow.

It's not what you do, it's the way that you do it

So, those lovely people at the Education Endowment Fund have published a new guidance report, called Improving Mathematics in Key Stages 2&3. It comes with 8 main areas for improvement. Each of these headings has a series of recommendations underneath:

A lot of the language and thinking in the first half of the report ties in very closely with the big ideas in the NCETM Teaching for Mastery in Mathematics. Here are some of my highlights, with quotes from the report and my comments.

1. Use assessment to build on pupils’ existing knowledge and understanding

Giving Feedback

"Schools should be careful that their desire to provide effective feedback does not lead to onerous marking policies and a heavy teacher workload. Effective feedback can be given orally; it doesn’t have to be in the form of written marking."

We all know that feedback is best if it is timely, personal and relevant. The NCETM guidance on marking and feedback recommends "If interaction between teacher and pupils is good, then efficient marking strategies can be deployed."

Addressing misconceptions

"It is important that misconceptions are uncovered and addressed rather than side-stepped or ignored"

Part of planning the sequence of a lesson is identifying the misconceptions that children are most likely to make and how they will be addressed. Use of an anchor task or problem at the start of the lesson can be a way of uncovering children's ideas.

2. Use manipulatives and representations 

"A manipulative should enable a pupil to understand mathematics by illuminating the underlying general relationships"

The use of structured representations of number is one of the key themes of Teaching for Mastery. But it is important to see the concrete manipulative as only part of the journey. Once the representation has been introduced with the manipulative, it can be linked to the pictorial and abstract counterparts. The representation is always there for students to refer back to, but is not a tool to perform the calculations. 

"Teachers should purposefully select different representations of key mathematical ideas to discuss and compare with the aim of supporting pupils to develop more abstract, diagrammatic representations."

Teachers need to choose representations carefully to illustrate the key mathematical idea in the lesson. There is always a danger (maybe more so in KS1) of having too many representations on display or in the classroom so the children loose the links between them.

3. Teach strategies for solving problems

In the past we have tended to teach 'problem solving' as an add on to the curriculum. 

"OK, we've done a week of multiplication, now here are some multiplication problems to solve".

Teaching the skill of solving problems means giving children opportunities to get stuck.

"Help students to make use of appropriate diagrams and representations that provide insight into the structure of a problem and into its mathematical formulation."

The use of bar models to represent  the structure of a problem is a valuable way to develop a better understanding of the maths that is most likely to be useful.

4. Enable pupils to develop a rich network of mathematical knowledge

Ensure that pupils develop fluent recall of number facts

Children's automaticity of number facts allow them to concentrate.on the mathematical ideas without being distracted by trying to recall their number bonds to ten or their times table facts. 

Teach pupils to understand procedures

The connection between understanding and doing a procedure is a big bart of fluency. Procedural fluency is not just about performing an algorithm accurately and efficiently, but understanding when and why to use it. Algorithms without understanding ('just add a zero') tend to fail sooner or later.