# How To Use Concrete, Pictorial and Abstract Resources in Maths

Written by Dan

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When teaching maths, it is essential to use various resources to help students understand concepts. Concrete, pictorial and abstract resources can all be used to support learning.

Each resource has its benefits and can be used in different ways. Here are some tips on using each type of resource in your teaching.

Related: For more, check out our article on The Importance Of Teaching Arithmetic here.

## What are concrete, pictorial and abstract resources in maths?

The Concrete Pictorial Abstract (CPA) methodology is a system that utilizes physical and visual tools to strengthen a child’s comprehension of abstract topics. Pupils are first exposed to the new mathematical concept with tangible materials, such as fruit or Dienes blocks.

Once they become comfortable solving problems with these substantial resources, they will be given puzzles featuring pictures – pictorial renditions of the tangible objects used in the earlier activities.

By having students solve problems using only abstracts, such as numbers or symbols, we can help them better comprehend the link between mathematics and reality.

This will, in turn, aid them in further securing their knowledge of the mathematical concept being taught. Constructing these tasks throughout a lesson plan is an effective way to do this.

Related: For more, check out our article on Tips For Improving Students Maths here.

## Where has it come from?

In primary mathematics education, it is impossible to avoid hearing about ‘maths’ and ‘mastery’ in one breath. It’s no shock; this Government has placed ‘mastery’ within its spotlight policy for improving maths skills, and millions are being poured into the Teaching for Mastery program that includes many schools across Britain.

Until 2015, the concept of ‘mastery’ was rarely known. East Asian countries such as Singapore and Shanghai have been touting their high-achieving Maths students through media coverage and a Teacher Exchange Programme.

As a result, teachers often assumed that mastery teaching originated from these nations — with much focus on the CPA approach furthering this misconception.

Tracing the beginnings of CPA, we again turn to American psychologist Jerome Bruner’s teaching strategies in the 1960s. He argued that abstract learning (especially math) can be perplexing for little ones and needs support via compelling visualizations and physical tools.

Hence, he introduced this approach as a scaffolding strategy to help children learn better.

His results showed that when students use the CPA approach to learn mathematics, they can progress through each stage with a deeper understanding of what is known. This allows them to retain more knowledge and build on their mathematical foundation.

Teachers often need to better perceive the CPA method as a technique imported from Singapore. However, in reality, the Singapore Maths curriculum is heavily based on Bruner’s learning theories and Cockcroft Report recommendations of 1982 that promote connecting computational skills with practical scenarios to solve problems.

## How To Teach The Concrete, Pictorial and Abstract Methods

It is a common mistake to think that when teaching the concrete-pictorial-abstract (CPA) model, one must teach each stage sequentially. However, it’s important to remember that these stages should be taught together at the same time whenever new material or ideas are introduced.

By concurrently employing all three steps in an activity – with tangible resources, pictographic representations and abstract recordings – pupils can make robust correlations between them for more efficient learning.

To help children understand column addition in the early stages, it is helpful to provide them with objects they are familiar with. For instance, straws or lollipop sticks can be bundled into groups of ten to represent tens and ones.

Once kids feel comfortable making two-digit numbers using these materials, they can organize them on a baseboard to show the two digits in an equation for column addition.

First, children compute calculations where the digits sum to no more than nine without exchanging or regrouping. To further support their growth in understanding numbers, they can utilize substantial resources and annotate them on a baseboard to better comprehend the digits used. Doing this exercise will also help create an association towards abstract formal methods.

Once children feel confident in manipulating physical resources, they can demonstrate their understanding by representing it visually and connecting the three stages via writing down digits.

Once students are comfortable with Dienes base ten equipment, teachers should introduce them to hundreds of column addition of 3-digit and 2-digit numbers. With consistent practice comes greater assurance when using these essential math tools!

Children should familiarize themselves with digit numbers and record the visual representations on baseboards. Once they are confident in understanding our tens-based numbering system and its values, Dienes equipment can be substituted for place value counters—which provide an easier transition to abstract concepts since all of them have the same size.

Introduce the place value counters to children similarly to other resources, beginning without regrouping and gradually progressing to calculations that call for such. Furthermore, allow them to document their findings alongside concrete sources before moving on to pictorial representations once they have mastered them.

Additionally, these tools are excellent for familiarizing youngsters with more significant numbers; use column addition of thousands and ten-thousands columns!

## Create Your Own

When trying to get children interested in math, it can be helpful to create your own resources. Substantial resources like objects or models representing a concept can help a child understand the idea better. You could construct a tower of blocks to define counting or draw pictures of fractions on paper.

Pictorial representations such as charts, diagrams and tables are also very effective in illustrating relationships between numbers. In contrast, abstract resources like number lines, algorithms and equations allow children to manipulate numbers to find patterns or solve problems.

With some creativity, you can create engaging and interactive materials that will help your child learn maths more effectively.

Three main types of resources can be used to help children learn mathematical concepts: concrete, pictorial and abstract. Each type of resource has its advantages and disadvantages, so it is essential to choose the right kind of resource depending on the concept you are trying to teach and your child’s learning style.

Using a combination of all three types of resources, you can create an effective maths learning environment at home or in the classroom to help your child understand math concepts more quickly.

Do you have other tips for using concrete, pictorial and abstract resources in maths education? Please share them with us in the comments below!

If you want your class to get 100% in the Maths Sats! Check out our how-to guide!

I'm Dan Higgins, one of the faces behind The Teaching Couple. With 15 years in the education sector and a decade as a teacher, I've witnessed the highs and lows of school life. Over the years, my passion for supporting fellow teachers and making school more bearable has grown. The Teaching Couple is my platform to share strategies, tips, and insights from my journey. Together, we can shape a better school experience for all.

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