Embedded Files
One of the first things you will notice in a DJS Maths lesson is that there are lot of objects available to children. This is because we follow a Concrete- Pictorial-Abstract approach to maths (CPA). Below is an overview of these different stages and what they look like
Concrete
Concrete
At the concrete stage, children use physical objects to represent numbers.
This allows handling, movement and manipulation of the number. These items are referred to as manipulatives or manipulables.
While this is a very powerful tool when first learning about number, it is also incredibly effective for building understanding of concepts such as multiplication, division, fractions, decimals and ratio. Being able to use materials in this way allows a much more meaningful understanding of number and operations, and provides an essential tool towards problem solving.
How many pegs are on the line? Does the colour have significance? Are all the pegs worth the same? Does it matter that they are upside down?
How many are there? How could we count this many marbles effectively? How could we group them? Are they all the same?
Pictorial
Pictorial
At this stage, children start to use representations of the physical objects. This could involve drawing the items, or drawing a representation of the item. An intermediate step is where we use more standardised objects (such as counters or multi-link) to represent another item (eg apples).
Constructing models of problems allows for children to visualise what the problem is and start to calculate more efficiently. This process can be shown with drawings, sketches or by use of bars. Using bars to represent a problem in known as bar modelling.
Here, apples can be represented as counters. While still a physical object, this is a step towards a pictorial representation.
Drawing apples as circles is a lot simpler than drawing realistic apples.
Now the drawings of apples have been replaced with bars- this still shows us the same: red apples are double the quantity of green apples.
This visually represents the same as lining the apples up physically, or drawing them as objects.
Abstract
Abstract
The abstract stage introduces symbols to represent numbers. This is the most efficient approach, as it requires less writing down, drawing or collection of resources.
Abstract symbols are a powerful shortcut as we have an agreed understanding of what each one represents. They are however, totally abstract. Without knowing and truly understanding what they represent, this meaning is empty.
Numerals are abstract representations of number- there is nothing in the numeral 2 that shows it represents two.
(Unlike the Roman numeral, which has 2 lines).
This means we have to learn and experience what these symbols represent to have any level of understanding of what they show us.
Mathematical operations each have an abstract symbol showing us is a simple way what a calculation requires us to do. Signs for addition, subtraction multiplication and division are well known shorthand. We also need to explore an understanding of equals and the inequalities <> to show size of amounts. This is something we explore in a concrete and pictorial manner first to cement understanding of it.
There are some fantastic websites offering further information about the both the CPA approach, and the mastery approach that we follow at DJS .
These buttons will take you to third party websites, the content of which DJS is not responsible for.
Page updated
Report abuse