Concrete Resources Explained: How Use Them With Children To Develop Their Mastery Of Maths
If you’re the parent or teacher of a primary school aged child, you are likely to have encountered concrete resources at some point during the child’s maths education. Here Emma Johnson, an experienced primary maths leader explains what they are, and why you should know about them and how to use them.
As you probably know by now the relatively recent (since 2015) maths curriculum is based on the Singapore/Shanghai approach to teaching maths, with ‘mastery’ being the ultimate objective for all children. A key feature of this mastery maths teaching method is the concrete pictorial abstract (CPA) approach, where children are initially introduced to new concepts through the use of concrete resources.
- What are concrete resources?
- Using the Concrete-Pictorial-Abstract (CPA) approach
- Why these teaching aids are a crucial part of mastery teaching
- How to support your child with maths at home using concrete resources
- Place value
- Written calculation methods
- Telling the time
- Properties of shape
- Why you should use concrete resources in your home
What are concrete resources?
Concrete resources (also referred to as manipulatives) are objects or physical resources that children can handle and manipulate to aid their understanding of different maths concepts. A mastery teaching approach encourages children of all ages to keep using these concrete resources, in Key Stage 1 (KS1) as well as Key Stage 2 (KS2). While the abstract nature of maths can be confusing for children, through the use of these concrete, practical resources, they are able to ‘see’ the maths and make sense of what is actually happening.
Using the Concrete-Pictorial-Abstract (CPA) approach
Once children are confident with a concept using concrete resources, they progress to drawing pictorial representations or quick sketches of the objects. By doing this, they are no longer manipulating the physical resources, but are still benefiting from the visual support the resources provide.
Once children have a secure understanding of the concept through the use of concrete resources and visual images, they are then able to move on to the abstract.
Taking short division as an example, the formal method is difficult to understand. Although children can be taught the method without understanding the maths behind it, this can lead to confusion errors and difficulties with retaining or remembering the skills or knowledge required to complete division calculations successfully.
If children are taught to use place value counters (concrete) they can make sense of the method.
Once confident with this, they can move on to recording the place value counters pictorially and eventually move on to the abstract formal written method.
Why these teaching aids are a crucial part of mastery teaching
In the past, children were taught procedures, but not why or how the procedure worked. In other words, children learnt the methods to get to an answer, without any understanding of the maths behind each method or procedure.
Some of you may be thinking, ‘That worked for me, why do children need to understand the maths now?’
While there are children who are able to access the maths through just learning a procedure by rote, many others have great difficulty coping with the abstract nature of it.
Ultimate Guide Concrete Resources and Maths Manipulatives
These are the 15 most essential concrete maths resources and how they work
Teaching methods without meaning leads to misconceptions, errors and difficulties in retaining the methods. Once children can actually ‘see’ the maths, they are much more likely to understand and accurately remember the methods… So, although they should not be exclusive to, concrete resources are particularly important for those children who find maths difficult, or lack confidence in maths.
How to support your child with maths at home using concrete resources
It isn’t just at school that children can benefit from accessing concrete resources in maths. They are equally important at home. Lack of resources could be seen as a barrier for supporting your child this way at home, but it doesn’t have to be.
In this blog, I will point you in the direction of the best resources you can buy to teach the key topics, but I will also share ideas for sourcing/creating your own resources at home, to mimic those used in school.
I’ll be looking at some of the key topics in maths and identifying the best concrete resources for supporting children (both bought mathematical resources and free alternatives you can access easily at home).
Look out also in this blog for the videos made with my daughter Amber at home showing how the resources help to explain some of the trickier concepts.
There are of course many topics that you could cover when helping your child with maths at home, but as we can’t cover them all in one blog here are the ones we feel will be the best use of your time!
The following is information about how you can help your child to learn about place value at home.
School Mathematical resources – Dienes equipment, Numicon and place value counters
Children begin to learn about place value by representing numbers using Numicon and Dienes equipment.
These are excellent resources for bringing to life the relationship between units, tens and hundreds. Their relative sizes really enable children to visualise the numbers and see the relationship between them.
Once children understand place value using Numicon / Dienes equipment, they progress to using place value counters. These are used in a similar way, with counters representing ones, tens and hundreds, but with the key difference of all being the same size.
This is useful for bridging the gap from the concrete to abstract nature of place value.
Home alternatives for learning about place value
Straws, lollipop sticks or anything else that can be bundled together are a fantastic alternative to Dienes equipment. For example, individual straws can represent the ones and bundled together into tens to represent the tens.
Lego is another good alternative, using individual bricks for the ones and joining together to make tens. The children are able to visually see the link between the size of the individual straws/bricks and the bundles of ten straws/bricks.
Downloading and printing images of Numicon onto card is a free alternative to buying a set of Numicon!
Place value counters can easily be made using plain counters and a marker pen.
Alternatively, you use money, with 1p representing the ones, 10p then tens and £1 the hundreds.
Written calculation methods
The following is information about how you can help your child to learn about written calculation methods at home.
School Mathematical resources – Dienes equipment, Numicon and place value counters
Once children are secure with place value, they are able to use the Dienes equipment or Numicon as a first step in understanding the written methods for column addition and subtraction, grid method multiplication and bus stop/short division.
The equipment is set out in columns on a baseboard as a concrete representation of the formal written method. Children use it to add/subtract the ones first and then the tens. If the calculation requires exchanging, they are able to physically exchange the ten ones for a ten and vice versa.
As with place value, children progress to replacing Dienes equipment or Numicon with place value counters to solve calculations involving the four operations.
Home alternatives for learning about written calculation methods
As with place value, you can use anything else that can be bundled together as an alternative to Dienes equipment.
Place value counters are easy and cheap to make and coins are another great way for showing hundreds, tens and ones for written calculations.
The following is information about how you can help your child to learn about fractions at home.
School resources: Fraction cubes, fraction circles, Numicon, multi-link cubes, counters
Fractions are an area of maths children can find particularly difficult, due to their abstract nature. For this reason, it is essential they have practical resources to hold and manipulate or play with.
Fraction cubes or circles and Numicon are very useful for aiding understanding of the fraction of a whole, add to that using place value counters, multi-link cubes, skittles or smarties and you can demonstrate a real range of fraction concepts: from the basics of what a fraction is; to recognising equivalent fractions; understanding what happens when a fraction is greater than one and for adding/subtracting fractions.
Finding fractions of amounts is another concept children can have difficulty grasping, particularly when the numerator is greater than one. Using resources, such as cubes or counters – or beans, skittles, smarties or pasta pieces – and physically sharing them out really helps children understand.
Here is a video showing fractions of amounts using the pictorial example of the bar model to explain how it can be used to help children understand.
Home alternatives for learning about fractions
There are lots of cheap/free alternatives to buying fraction resources.
Fraction circles can be made out of paper plates, or downloaded off the internet and printed onto coloured card.
Lego and printable fraction strips are a good alternative to buying fraction cubes.
For fractions of amounts, anything that can be shared out can be used. Food works particularly well for this. For example: raisins, grapes, sweets etc…
The following is information about how you can help your child to learn about decimals at home.
School resources: tens frames, hundred squares, decimal place value counters, decimal cubes.
When children are first introduced to decimals, tens frames are useful for helping them to understand tenths, whilst hundred squares support children with recognising hundredths.
Tens frames are a particularly useful resource for enabling children to see the link between decimals and fractions, whilst also helping children to understand concepts such as rounding decimals.
Place value counters positioned on a place value grid are useful for helping children to understand place value of decimal numbers. Children can visually see how many tenths and/or hundredths a number has and they can help them understand how to compare and order decimals.
Home alternatives for learning about decimals at home
All these resources can be found/created at home. Tens frames, hundred squares and decimal strips can be printed off the internet whilst decimal place value counters can be made by writing 0.1 on one set of counters and 0.01 on another.
The following is information about how you can help your child to learn about percentages at home.
School resources: Hundred squares, percentage cubes
Hundred squares and number squares are useful for supporting children’s understanding of the concept of percentage being ‘out of 100’. Hundred squares can be shaded in, or place counters on each square, to represent different percentages.
Percentage cubes are also useful in helping children to understand percentages and enable children to see the link between fractions, decimals and percentages.
Home alternatives for learning about percentages at home
As with decimals, hundred squares and percentage strips can be printed off the internet for free and lego is a great concrete resource for supporting children with percentages.
Telling the time
The following is information about how you can help your child to learn how to tell the time at home.
School resources: Clock faces
Analogue clocks, which can be manipulated by moving the hours and minutes hand around, help children to grasp the concept of time.
Ideally these clocks will have removable hands, to enable children to look at the hours and minutes hands independently, before looking at both together on the same clock.
Children can struggle to understand the concept of time when presented with both hands together. It is much better to teach children to understand the hours hand by itself,
then the minutes hand,
before bringing both hands together onto the same clock.
Home alternatives for learning how to tell the time
Clocks can be made using card.
Here’s an ingenious solution to stop your young children becoming organic alarm clocks… That’s right, no waking up the grown-ups until 7am at the earliest!
Make two separate clocks to represent the hours and the minutes separately before looking at two hands together on one clock.
Counters or lego cubes can be used when making a clock showing the individual minutes.
In this video, we used blue counters to show minutes past and red counters to show minutes to; we look at both the hour and minutes hands on separate clocks, before looking at the two hands together on one clock…
Properties of shape
The following is information about how you can help your child to learn about the properties of shape at home.
School resources: 2D and 3D shapes, polydron geometry set
Do you remember these plastic polydron puzzle pieces from your school days?
Children are much more confident identifying properties of shapes when they are able to physically hold them. It is important children understand that the 2D shapes aren’t strictly 2D, but are 2D templates, which can be drawn round to create 2D shapes.
3D shapes the children can hold are much better than 2D drawings of 3D shapes for looking at properties, such as number faces/edges…
Polydron geometry sets are great enabling children to explore nets and constructing 3D shapes.
Home alternatives for learning the properties of shape
There are many opportunities both at home and when you’re out and about to investigate shapes and their properties.
Children can look around the home and outside – going to the shops, on the way to school, eye-spy in the car, wherever – and identify any 2D / 3D shapes they can see. This can generate lots of discussion round their properties.
Children can make 3D shapes using play-doh or lego. Alternatively, they could get creative using marshmallows and toothpicks – or bamboo skewers and masking tape – to construct a range of 3D shapes.
When making 3D shapes from nets, a whole range of nets can be downloaded for free from the internet, which is a great alternative to the plastic polydron geometry sets.
Why you should use concrete resources in your home
If you haven’t tried using concrete resources to support your child at home before, then give it a go!
This is by no means an exhaustive list of topics and resources.
Hopefully these suggestions have given you some ideas and a starting point, whilst providing an insight into how useful concrete resources can be in helping children to understand maths.
It doesn’t matter what age your child is, concrete maths manipulatives are a fantastic way to help your child grow their confidence so what are you waiting for?
Read more: What Is Part Whole Model?
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