ā¢ Defining a āmasterā
ā¢ Maths is a verb
ā¢ What does the National Curriculum say?
ā¢ Questions, questions
ā¢ Asian thinking
ā¢ Rounding up
ā¢ What maths can you see?
Masters of the classroom
Our book begins with consideration of what āmastery in mathematicsā actually is and what āmasteryā may look like in the classroom. We discuss key principles of mastery and how the teacher can develop mastery in their pupils.
Over the past few years, many educators, educational thinkers and politicians have promoted the use of the phrase āmasteryā in relation to mathematics. However, before talking about the value of any concept, we need to have at least some idea of what it is. Our, perhaps contentious, interpretation is that there are many commentators who use the term mastery, without knowing what it really is ā¦ but it does sound good!
Defining a āmasterā
It may surprise you that we choose to begin this book on mathematics with a little English grammar. However, in order for you to gain an understanding of the term āmasteryā, it is useful to consider our everyday use of the term, as the word āmasterā can be a noun, an adjective or a verb. Exploring these different uses may give us some insight into what the phrase means in terms of teaching mathematics.
As a noun, āmasterā could be used to describe such diverse things as:
ā¢ a person who owns a dog or other animal
ā¢ a person who captains a merchant ship
ā¢ a male teacher
ā¢ a young man.
These different uses of the word āmasterā lead us to a number of other descriptions including: mastery; mastermind; masterclass; master of ceremonies; grandmaster; chess master; masterpiece; masterstroke; master of disguise; old Master; headmaster; bandmaster; station master; postmaster.
Equally, āmasterā can be used as an adjective to describe many roles or objects, such as master chef, master baker, master bedroom, masterās degree, master tape. When used as a verb, one may talk about āmastering a skillā or even to āmastermind a crimeā.
The Oxford English Dictionary describes the verb āto masterā as, gaining complete knowledge of, or skill in a subject, technique, and the word āmasteryā as, complete knowledge or command of a subject or skills.
These definitions may lead you to ask, What is a complete knowledge?
Perhaps the greatest of all mathematical thinkers was Albert Einstein, who once said:
Wow! This sounds hard doesnāt it? The term āmasterā seems to imply being some sort of expert or genius ā or even a wizard!
Given these varied perspectives on what mastery may be, it is worth considering how this can possibly apply to primary mathematics. Can a child actually be a master of mathematics? And if so, what would this look like?
It is our view that a child can indeed be a master of mathematics; the art of the mathematics teacher is to ensure that children are presented with the opportunities to enable this to happen.
Maths is a verb
It is possible to find a book that will show you where you should place your fingers on a guitar to play a D or an A minor chord. Alternatively, someone may show you where to place your fingers on a piano keyboard to play a C chord. Does that mean you can learn to play the guitar or piano by just reading about it? Similarly, we have watched many cookery programmes that show us how to bake bread or make a soufflƩ. Does that mean therefore that after watching these programmes we can bake a tasty loaf or know that our soufflƩ will rise? Like many things, the knowledge of what to do is important, but to become proficient at any of these activities, practice is required. The only way to become proficient at a musical instrument or at cooking is to keep practising. The more we engage with the activity, the better at it we become. It is the same with mathematics.
Mathematics has a body of knowledge that requires acquisition. However, that knowledge alone is insufficient for us to become proficient in it. We need to practise using this new knowledge. First, perhaps by relating it to existing knowledge. Then, this knowledge needs to be applied, in a variety of situations, moving from the familiar to the unfamiliar. In this sense, we see mathematics as a verb ā a ādoingā word. Deep learning comes from doing mathematics. The Chinese philosopher Confucius famously said:
This sums up the way we would like children to learn mathematics.
Just do it
It may have been easy for Confucius to say we should get children to ādoā mathematics. But what does that mean? For some teachers, in days gone by, that may have meant finding a worksheet with enough questions to reduce the chance that a child could actually complete their work within the lesson. In such lessons, differentiation of work can become using the same questions with harder numbers.
Many American school textbooks have perhaps over a hundred questions on each topic. The evenly numbered questions have the answers at the back and are completed in the lesson. Only the Teacherās Guide has the answers to the odd numbers, so they become the homework.
Mastery moves completely away from this idea. Maths mastery is not about children completing the most questions, nor children completing similar questions with bigger numbers.
Reflection
Suppose you are planning to teach long multiplication. How would you know if each child in your class has learned long multiplication?
How many long multiplication questions would you expect children to do by themselves in order for you to be assured that they have understood the concept?
What does the National Curriculum say?
You may think that a good place to start considering any questions about maths mastery is the mathematics National Curriculum. Does it surprise you therefore that the words āmasterā, āmasteryā and āmasteringā do not appear anywhere in the mathematics content of the Primary National Curriculum? The words you do see in the mathematics National Curriculum include: fluency; conceptual understanding; mathematical reasoning; problem solving; enjoyment; curiosity and application. These are what we would like to define as some of the principles of mastery. It is perhaps by thinking of a child who can evidence some of these principles that we can move towards an image of mastery.
How to master maths
The primary mathematics curriculum is clear that children need to challenged by being offered
You may be able to recall lessons from your own mathematics education where the teacher finished a topic before you had really understood it and then quickly moved on to something different. This is what we call aquaplanic learning as only the surface of each topic is considered. Children can āaquaplaneā from one topic to another, making no connections whatsoever between any of them. Such approaches result in learning not being secure and may lead to a child learning only what they cannot do. A mastery approach seeks to eliminate aquaplanic learning.
We see āmasteryā as a pedagogical approach to teaching and learning within mathematics. Any aspect of the mathematics curriculum can be developed to mastery level: by this, we mean that new learning is assimilated into each childās existing knowledge and understanding. Children need to be able to apply new knowledge in a wide variety of contexts to fully demonstrate deep understanding of each new topic. By this token, a child ...