Mathematics Mastery

One of the biggest ideas to come from Singapore is the concept of teaching mathematics for mastery. In simple terms it means spending enough time on a topic in order to comprehend it thoroughly. Teaching for mastery also means that children spend a consolidated amount of time on a topic to understand how it relates to concrete experiences. This is based on a whole body of research by educational psychologists and specialists, which manifests itself in a carefully selected sequence of lessons that children go through (I can go into more detail on this in another post). It took the Singapore education system decades to perfect the idea of maths mastery and it is still undergoing improvements.

When talking to teachers and department heads, I often hear that Year 5 and 6 pupils aren’t really equipped to problem solve. This is a widespread problem – it doesn’t exist only in the UK. The 1983 Cockcroft report said we need to put problem solving at the heart of the mathematics education, but how do we do it? The Maths mastery that children have acquired using the Singapore approach gives them the fluency they need in number sense, recognising patterns and seeing connections, and visualisation – the three key components to problem solving.

On one of my school visits this week, we took an example from a Year 6 workbook from the My Pals are Here series and went over how children can solve this problem, particularly if they have benefited from having acquired mathematics mastery.

Question from My Pals are Here Year 6 workbook

Mrs Singh bought a set of furniture for $1760. This was 20% less than the usual price. Later, Mrs Singh sold the same set of furniture for 5% more than the usual price. How much did Mrs Singh sell the set of furniture for?

So we know that visualisation is key and it is something pupils will have learned from early on, therefore we are going to draw this out in a bar model that looks something like this.

The next thing to represent in this model is this idea that $1760 represents the purchase price once 20% has been taken off the normal price.
Two key things need to happen here:

1. pupils need to be able to accurate depict the 20% off in the diagram
2. pupils need to be able to represent percentages as fractions in order to depict the information correctly on the diagram.

Now, this is where a strong number sense becomes crucial. A pupil should be able to recognise that 20% is the same as a one-fifth. By recognising this fact they will realise that 1760 is four-fifths of the final price.

So now we have an opportunity to work out some numbers in our head with some mental strategies that have been reinforced in previous years. Here are two possible approaches:

1. Divide by two and divide by two again

2. Simplify the numbers to make it easy to divide

So now we add these facts to the model and work out the ‘normal price’.

By persistently reinforcing the value of the mastered skills, these ideas stick. Being able to understand that 20% is the same as one-fifth, for example, is valuable information to solve this problem, but pupils will only remember and use this meaningfully if it was taught with comprehension in mind – not as a series of procedural steps to memorise.

This is why teaching for mastery allows students to solve problems. Teaching isolated procedures is easier, but it doesn’t give pupils the relational understanding they require to make decisions when solving problems. The ultimate goal of teaching mathematics is not to create students who can memorise procedures in isolation, but to create thinkers. Only then can they progress to solve problems.