What tools and resources are available for mastery teaching?
Mastery Tools and Strategies
There are some key strategies, informed by proven theories, that are fundamental to the mastery approach.
Concrete, Pictorial, Abstract (CPA) is a highly effective approach to teaching that develops a deep and sustainable understanding of maths in pupils. Often referred to as the concrete, representational, abstract framework, CPA was developed by American psychologist Jerome Bruner. It is an essential technique within the Singapore method of teaching maths for mastery.
Concrete is the “doing” stage. During this stage, students use concrete objects to model problems. For example, if a problem involves adding pieces of fruit, children can first handle actual fruit. From there, they can progress to handling abstract counters or cubes which represent the fruit.
Pictorial is the “seeing” stage. Here, visual representations of concrete objects are used to model problems. This stage encourages children to make a mental connection between the physical object they just handled and the abstract pictures, diagrams or models that represent the objects from the problem.
Abstract is the “symbolic” stage, where children use abstract symbols to model problems. Students will not progress to this stage until they have demonstrated that they have a solid understanding of the concrete and pictorial stages of the problem. The abstract stage involves the teacher introducing abstract concepts. Children are introduced to the concept at a symbolic level, using only numbers, notation, and mathematical symbols (for example, +, −, ×, ÷) to indicate addition, subtraction, multiplication, or division.
How to put Bruner’s key theories into teaching practice
The Concrete, Pictorial, Abstract approach lies at the heart of the Maths — No Problem! approach. It enables a natural and supportive progression for learners as they develop their understanding and skills in maths.
For example, to explore equivalent fractions with pupils, the teacher could begin with multi-link cubes, progress to the pictorial problems in the textbook and then on to questions with no visuals except numbers and the necessary mathematical symbols.
This will result in a fully embedded understanding of the concepts throughout the classroom.
Throughout our lives, we all experience this process of learning, revising, and revisiting as we return to old concepts and make sense of new one’s.
Bruner termed this spiralling; the process of coming back to an area of learning to build upon it in light of new discoveries.
Spiralling may also mean we take steps backwards before progressing forward — something to consider when a pupil is not moving forward as we might anticipate.
Bar Modeling is an essential maths mastery strategy. A Singapore-style of maths model, bar modelling allows pupils to draw and visualize mathematical concepts to solve problems.
The bar method is primarily pictorial. Pupils will naturally advance from handling concrete objects to drawing pictorial representations to creating abstract rectangles to illustrate a problem.
With time and practice, pupils will no longer need to draw individual boxes or units. Instead, they’ll label one long rectangle or bar with a number.
Bar modelling provides pupils with a powerful tool for solving word problems. However, the lasting power of bar modelling is that once pupils master the approach, they can easily use bar models year after year across many maths topics.
Bar modelling is an excellent technique — though not the only one — for tackling ratio problems, volume problems, fractions, and more.
How to use part–whole bar models in your classroom
All bar modelling starts with the part–whole model, an essential maths mastery strategy that helps learners visualise the relationships between numbers.
These simple diagrams help learners make the jump from relying on concrete resources to thinking pictorially. Part-whole models are a great way for learners to make sense of a problem and decide which operation they need to use and why.
The part–whole model, sometimes called the part–part–whole model, is a simple pictorial representation of a problem that helps learners see the relationships between numbers. A horizontal bar shows the ‘whole’ amount. Beneath that, an identical bar is divided into pieces to show the ‘parts’ of the whole.
Part–whole bar models illustrate the relationship between what’s known and what still needs to be calculated. The bar can be split into as many parts as necessary. If the total is unknown, you might see the ‘whole’ bar replaced with brackets and a question mark.
Problem solving is always going to be difficult for primary learners. Applying mathematical knowledge can be hard when learners don’t understand which operation they need to use. A part–whole model shows learners the problem in an accessible way.
Part–whole bar models are ideal for:
- Missing numbers
They’re particularly useful for time and measurement problems as children often find these problems challenging.
Number Bonds show how numbers are split or combined. An essential strategy of Singapore maths, number bonds reflect the ‘part-part-whole’ relationship of numbers.
Number bonds let students split numbers in useful ways. They show how numbers join together, and how they break down into component parts.
When used in Year 1, number bonds forge the number sense needed for early primary students to move to addition and subtraction. As students progress, number bonds become an essential mental problem-solving strategy.
Division: Equal Sharing or Equal Grouping?
There are two ways you can approach teaching division: equal sharing and equal grouping.
Children should experience both concepts, but here’s the interesting bit: research suggests that teaching equal grouping before equal sharing will help your pupils develop a deeper understanding of division.
The difference between equal sharing and equal grouping boils down to what the quotient represents. When sharing, the quotient represents the quantity of shared objects in each group. When grouping, the quotient represents the amount of groups within the shared quantity.
At the start of teaching division, teachers often focus on sharing, but not always on grouping. At this stage, it’s important to give equal, or more, emphasis on grouping when pupils are first getting their heads around multiplicative structures.
One rationale for this belief is that sharing is more intuitive and it’s likely that pupils already have a notion of sharing — although not necessarily a very deep one.
Grouping is conceptually more difficult and it’s less likely that pupils will have a strong notion of it. Letting them struggle early on with grouping before moving on to the simpler notion of sharing is more likely to create the necessary relational depth.
How important is number sense? Turns out, very.
Number sense is an important construct that separates surface level understanding from subject mastery. So, what is number sense and why is it important for learners to develop this skill?
The construct of number sense refers to fluidity and flexibility with numbers. It helps children understand what numbers mean, improving their performance of mental mathematics and giving them the tools to look at maths in the outside world and make comparisons.
Number sense helps children understand how our number system works, and how numbers relate to each other. Children who develop number sense have a range of mathematical strategies at their disposal. They know when to use them and how to adapt them to meet different situations.
Good number sense helps children manipulate numbers to make calculations easier and gives them the confidence to be flexible in their approach to solving problems.
Children who develop number sense can assess how reasonable an answer is, and routinely estimate answers before calculating. They look for connections and readily spot patterns in numbers, which helps them predict future outcomes. They have several approaches to calculating and problem solving and can use and adapt these for new situations.
Children will still see Roman numerals in lots of real-life contexts as they are a rich part of our cultural heritage, they teach basic maths facts, and they can be fun to learn!
Why do children need to learn these cumbersome relics from the past?
Firstly, it’s one of the statutory requirements to ‘read Roman numerals to 100 (I to C)’ in the Year 4 maths programme of study for Number. But, learning Roman numerals is also a great way to help children increase their number sense, better understand how numbers work, and is particularly good for increasing mental maths skills.
Understanding fractions is essential to understanding maths generally. But common misconceptions around fractions can leave both teachers and learners feeling more unsure than they should.
Knowing what common misconceptions look like and how to address them can make a significant difference in your learners’ understanding.
For example, why is 1/4 smaller than 1/2? Children learn that 4 is greater than 2. And this is true. So does it also hold true that 1/4 is greater than 1/2?
If you understand the role of the denominator, then this fraction misconception should not exist. However, when you do see this misconception, there are steps you can take:
- Make the fraction using square or rectangular paper
- Cut out the pieces
- Compare the pieces to the written fractions
Can a learner make the connection between the number of pieces it takes to make one whole, and the size of the pieces? Once that connection is made, and the learner has an understanding of what the denominator is telling us, the learner will see that one quarter is smaller than one half.
We could go on to generalise and say the smaller the part is, the more parts will be needed to make one whole.