Mastery: a concept that, when it first started gaining traction in the UK, probably felt like nothing more than yet another buzzword to many of us. Nowadays though, it’s quickly becoming so rooted in the principles of our education system that you’re likely leaning on at least some of its principles, even if you hadn’t realised it yourself.

A full and rigorous exploration of this new way of educating would be difficult to do justice here, and if you think you’ve already *mastered* the concept, then this guide may not be for you. But if you’re just starting your mastery journey, and you’re interested in how you can make your resources more appropriate for a mastery-focused world, then there’s hopefully something in this article for you.

Whilst the use of mastery has now become popular in many subjects, its origins—in the UK at least, from where I’m writing—were in Maths, and in particular at younger key stages. So that’s the perspective from which I’ll be writing, with a specific focus on published educational resources (again, this being the perspective from which I’m writing, as an author and editor of educational content). However, the beauty of mastery is that its principles are broad enough to be applicable to any subject at any level.

The idea of Maths mastery in the UK initially came from teaching methodologies used in Asia, particularly Singapore and Shanghai, where they regularly attain higher scores in PISA rankings than similarly developed Western economies. Some western-based educators and institutions started looking East to see how they could improve their own results, mostly to great success, and so the idea caught on.

So, what specifically sets mastery apart from the more traditional ways of writing and teaching educational content?

*Note that individual cities take part in PISA rankings in China, hence the specific mention of Shanghai. Note also that they don’t call it mastery… it’s simply the way they teach!*

## Same journey, different speeds

Our educational system, especially at secondary level, has generally worked on the principle that more-able students have the capacity to learn more content than their less-confident counterparts – one example being the foundation and higher equivalents of GCSE exams, in which the higher versions include topics that are not covered at all in foundation.

The mastery approach, however, works on the basis that everyone should be able to learn all of the same content, with the only difference being that less-confident students simply take longer to get there. The idea revolves around making sure you’ve understood all the ins and outs of a given topic *before* you move onto the next one, spending as much time as you need to get there. A few key principles of the mastery approach include (but aren’t necessarily limited to):

- Using language accurately and consistently, so that arguments can be clearly formed, explained, and understood.
- Understanding a topic in depth, using many different alternative representations of a problem to better understand how and why it works.
- Being able to fully understand why—and also specifically why
*not*—a particular solution, explanation, or broader principle works. - Using structured models (conceptual and physical) to explain concepts metaphorically.

Whilst allowing students to arrive at their own pace is logistically a difficult principle to apply in an education system that’s built for the reverse, there *are* still some simple ways of building these mastery principles into our educational resources, and so here I’ll describe 3 quick changes you can make to your published questions, explanations, or expositions, in order to make them ready for an increasingly mastery-friendly world:

## Tip 1: Less repetition, clearer differences

For many, it may feel like a rite of passage to have spent hours trudging through question after question, each with hundreds and hundreds of sub-parts, slowly losing the will to live as that seemingly impossible homework deadline loomed.

It turns out, though, that this style of rote learning is often not very helpful – no matter what your older relatives might tell you.

The mastery approach does away with endless repetition of what seem like randomly generated questions, and instead focuses on a smaller, more meaningful set of questions with clear and meaningful changes between each one. This way, the results of each individual change on the final answer can be identified more easily and committed to the student’s memory bank more efficiently.

### How to integrate this into your content – an example

**Instead of doing this…**

1) Solve these inequalities

a) *x* + 2 > 0

b) *x* + 3 > 0

c) 2*x* + 5 > 0

d) *x* + 2 < 0

e) 7 − 4 > 0

f) 4 − 6 < 0

g) 0 > 2*x* + 6

h) 6 + *x* > 0

i) 5 + 2*x* < 4

j) *x* + 3 > 10

k) 3 + *x* > 10

l) 13 + 2 < *x*

m) 100 + 1 > *x*

n) 100 − 7*x* < 0

o) 28 + 5*x* > 28

p) 100 > 20*x*

**Why isn’t it great?**

- No structure between questions, with largely varying numbers, so patterns aren’t easy to spot.
- Often multiple changes happen from one subpart to another, so the difficulty level doesn’t scale nicely.
- Way too many sub-parts, so students will get bored and lose focus.
- Numbers and iterations seem to be applied randomly.

**Do this**:

1) Solve these inequalities

a) *x* + 2 > 0

b) *x* + 2 < 0

c) 2*x* + 2 < 0

d) 2*x* + 2 < 4

e) *x* + 4 > 2

f) 4 + *x* > 10

g) 10 − *x* > 0

h) 4 − *x* > 10

**Why is this better?**

- Simple, similar numbers used, so the focus is on working with the key topic (inequalities) rather than manipulating numbers.
- Small, single differences between parts, so the result of each change on the solution is clear.
- Fewer parts, so easier to complete without losing focus.

### Tip 2: Ask why *not*, rather than why

*not*

It’s relatively common these days to include ‘flipped’ questions, where a student is given a solution and asked to explain why it works. The mastery approach, however, goes one step further – and also places focus on understanding why certain solutions do *not* work. This checks that the student doesn’t just understand why a given solution is accurate, but also specifically why an alternative solution would not work.

### How to integrate this into your content – an example

**Instead of doing this…**

1) A prime number is divisible by only itself and the number 1. A student claims that the only even prime number is therefore 2. Explain why the student is correct.

**Why isn’t it great?**

- The answer is somewhat given away by the question.
- The student is pointed straight towards the number they need to focus on (2), leaving little room to explore.

**Do this**:

1) A prime number is divisible by only itself and the number 1. A student claims that no even numbers can therefore be prime, because all even numbers are divisible by 2 as well as 1. Explain why the student is **not** correct.

**Why is this better?**

- The student is given more room to explore options (is the error in the first half of the statement, or the second?)
- It encourages efficient problem-solving: listing all even numbers will take forever, so let’s first think about
*where*the exceptions might lie, and start there. - It reinforces the idea that we should consider any exceptions when using mathematical rules.

## Tip 3: Use more modelling

Modelling mathematical problems is a big part of mastery teaching, so much so that practitioners often use the abbreviation ‘CPA’ to discuss the types of models that can be used:

**Concrete**involves using actual physical ‘manipulatives’ to represent problems and solutions, such as lining up different-length rods, each of which represents a given number, to investigate whether one number is divisible by another.**Pictorial**involves using diagrams to represent a concrete problem, such as shading in sections of a cup to represent what fraction of it is full of water.**Abstract**involves modelling problems with more abstract concepts or symbols. At younger key stages, this might involve using basic shapes to represent numbers of items, whilst at older key stages this will involve modelling real-world problems with algebraic formulae and equations.

### How to integrate this into your content – an example

A great way of including CPA modelling into your questions, if the style and tone of your product allows it, is to include problems that build on successive sub-parts, each of which presents a similar problem in an alternative form of modelling. For example:

**Starter class activity**: Build two ‘function machines’ out of cardboard boxes and give students plastic counters. When they place one or more counters in the first box, give them back twice as many. When they place counters in the second box, give them back the same amount plus three. Allow them to bring a small handful of counters to each function machine, in a mix of orders (first one then the other, and vice versa), and ask them to observe and record what happens.

**Question 1**: Draw two function machines next to each other. Label them ‘× 2’ and ‘+ 3’.

Using your function machines, work out what number you end up with if you put the following numbers into one machine and then the other:

a) 1, moving from left to right

b) 1, moving from right to left

c) 10, moving from left to right

d) 10, moving from right to left

e) −5, moving from left to right

f) −5, moving from right to left

g) −2, moving from left to right

h) −2, moving from right to left

**Question 2 (challenge)**: Create an equation to represent the effect of the two function machines in question 1:

a) From left to right

b) From right to left

Use *y* to represent the number that comes out at the end, and *x* to represent the number that you first put in.

*[Answer: a) y = 2x + 3 b) y = (x + 3) × 2 or y = 2(x + 3)]*

These are just a few of the ways in which you could integrate mastery into your products, and we’re really just scratching the surface here. We’d love to hear any thoughts, questions or advice that you might have from your journey into the brave new world of mastery, so please get in touch and let’s continue the conversation!

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