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.

Age-related expectations: who decides?

The Expected Standard

The Ofsted guidance on assessment contains multiple references such as

pupils achieve the expected standards by the end of year

Unfortunately, there isn't any such thing as 'the expected standard' any more, except for Years 2 and 6. The 2014 National Curriculum has no attainment targets in it, and no levels. 
The DfE only says:

The curriculum must include an assessment system which enables schools to check what pupils have learned and whether they are on track to meet expectations at the end of the key stage.

So who decides what the 'expected standard' is for each year group?

The National Curriculum

It should be easy for Maths. After all the new National Curriculum sets out the suggested teaching objectives for each year group. Surely these make up 'the expected standard'!

But do we expect a child to be able to demonstrate every statement? What about 50%? Or 90%? Maybe some of those teaching objectives are more important than others?

We don't really know what 'the expected standard' is for Year 6 yet either. The abandoned 'Performance Descriptors' talked about 'the majority' of the elements, so that implies 50% of the objectives.  The new tests will cover the teaching from Years 3-6 and have an 'expected score' (which will be scaled to 100); we don't know what this will be yet (it used to change year on year anyway).

Assessment Systems

There seem to be two schools of thought on assessment systems. Some take all the statements and turn the yearly programmes of study into a checklist. Rising Stars has a system like this, exemplifying each statement at three levels for 20 pages. Some stick a set of key statements, often based around the work done by the NAHT.
We still have to decide whether children need all the 'Key Performance Indicators', or a given percentage of the whole lot.

What's the point of Assessment?

We've spent a lot of this year talking about assessment without levels in school. Some of that discussion revolves around why we are assessing children and who the assessments are for.

Ofsted have some helpful ideas (how often do you get to write that?) in their note on the use of assessment information. In the covering letter to schools, Michael Wilshaw writes about 3 purposes for assessment:

1. Raise achievement

The primary reason teachers assess is to make their teaching better. They need to know which children are achieving learning intentions, which children are on the way, and which children need support. With the expectation in maths that most children access the same curriculum at the same pace, it becomes really important that we  know what impact our teaching is having.
There's no point assessing the children's attainment in our classrooms if that information just goes 'up' to leaders, parents and governors. It has to go down too, and be shared with children and impact on planning. The quicker that happens, the more impact the assessment process has.
Good day to day assessment involves having clear objectives, making judgements about how secure the children are and acting on it rapidly to raise the attainment of all, but particularly the children who struggle the most.

2. Identify pupils who are falling behind

As a maths leader, I need to know which children are falling behind the expected standard, and to do something about it. This might be working with the teacher to improve provision in the classroom. It might be working with the intervention leader to plan a focused support. It might be working with the parents or carers to support learning at home.
For me, this involves having a core set of expectations in maths for each year group that are essential by the end of the year. Teachers can then talk about pupils who are missing key skills (often from the previous year), that prevent them making progress.

3. Report to parents

Schools need to let parents know how their children are doing. Ofsted talks about pupils who 'achieve the expected standard', and pupils who 'need to catch up'. That's very black and white: the sheep and the goats.
To be fair, we've been reporting to parents in this way for a while - 'Jane is working at 3b, and that's the expected standard for a year 4'. But it means schools defining 'the expected standard', and how much of it you have to be working at (50%? 90%?)

What does it look like?

I think this gives a sensible model of assessment practice:

Day to day assessment

Teachers plan well-defined objectives for their blocks of lessons, linked to the curriculum, and make judgements about which children are meeting them. They allow enough time for all children in the class to meet the objectives, and intervene to support those who don't.
 This doesn't involve any evidence, tests or external involvement, other than TAs.

Progress meetings

Once or twice a term, teachers make judgements about progress on a set of core objectives (the really important ones) and come to a pupil progress meeting with the subject leader, where they discuss which children need further support. Ideally, the good day to day assessment practice should provide enough support to make these judgements.

Reporting

Lastly, there is an 'end of year' standard which pupils are judged on. Again, if this is based on the 'core objectives', it should not involved any extra testing. It does involve standardisation and moderation, because (like a level judgement) there will be plenty of children on the 'border' between judgements. This standard can be used to judge progress (of children and cohorts) and to report to parents.