THE IMPORTANCE OF USING MANIPULATIVES IN PRIMARY MATHEMATICS TEACHING

Written by Joan Clark
Primary Specialist at Modelex, Monaco.
The range of manipulatives for use in the teaching of mathematics in primary school is wide and varied; this includes any resources or materials which can be used in a practical, hands-on way in a classroom. In number, these may range from beads and counters, through cubes and number rods, dice and dominoes, to number lines, hundred squares and place value cards. Other areas of mathematics, similarly, have appropriate manipulatives, for example 3D shapes, teaching clocks, geometric shapes, pin boards. The list is long!

The rationale for, and positive impact of, use of manipulatives across all primary age ranges has been well researched through academic studies and endorsed by school inspection bodies such as OFSTED in the UK. To understand their importance, we need to consider both overall child cognitive development as well as what we value in effective teaching and learning.

Piaget's Theory of Child Development outlines four stages of cognitive maturity. The important aspect which impacts use of mathematical manipulatives is the recognition that until the age of twelve or so, children still think in concrete ways, with abstract thinking beginning to develop in the late primary years. Thus, the use of practical, hands-on resources is essential to support children to make sense of, and master, mathematical concepts, as this relates to their stage of cognitive development all through their primary education. This applies not only to arithmetical concepts but across the maths curriculum. A good example of this would be, for example, geometry/3D shape. Depending on their level of cognitive development, children are unable to visualise a 3D shape from a 2D picture on a page in the way that an adult can. Thus in learning about shape, they need to be able to handle, move and explore real 3D objects.

What is regarded as effective teaching has progressed from older, traditional methods of rote teaching of concepts where children are merely taught to follow a set procedure, to a model where understanding of the underlying maths is an essential part of the learning process. An example of this from my own experience would be subtraction where the strategy of 'borrow and pay back' (one to the top and one to the bottom which many people may recognise) was taught by rote pencil and paper, but without any understanding of why the strategy worked. It was only during my higher level maths studies that I actually understood the mathematics behind this.

Manipulatives, then, allow children to 'see' how the actual maths works and help them become true mathematical thinkers and problem solvers, not just followers of a rote-learned procedure. This has many advantages, among which are embedding deeper understanding and, therefore, supporting long term retention of concepts as well as reducing misconceptions in mathematical thinking. This firm foundation is especially important in maths because children need to continually build on previously learned concepts as they progress through their maths education. In addition, when children have regular opportunities to explain their learning, including using practical materials to support their explanations, we are encouraging deep learning and mastery. This clear and deeper understanding then supports the development of abstract thinking and, consequently, the key skills of actually using and applying what has been learned in unfamiliar contexts.

We now have at our fingertips a vast range of online, virtual resources to support mathematical development. While these have the power to motivate and greatly enhance children's maths learning, we also need to be aware of their limitations. Despite advanced graphics, these do not provide a tactile experience and do not allow children to physically handle the resources; they are 2D representations, much like a picture on paper. They are, perhaps, then best used after the introduction of physical resources as part of what is known as a Concrete - Pictorial - Abstract approach. In this way, although children are no longer actually handling physical resources, they still have the benefit of a visual representation to support their development into abstract thinking.

Children can also be supported easily at home with practical materials. For example in learning about fractions, cutting and sharing cakes, pizzas, coloured sweets, counters etc are ideal resources. Time can be a difficult concept, particularly for young children, therefore, making their own clocks, perhaps with only an hour hand in the first instance, then adding a minute hand can be effective.

The following website might be of interest to find out more about telling time with your child, it gives practical examples on how you can use many everyday materials and activities to tell the time: https://www.weareteachers.com/5-hands-on-ways-to-teach-telling-time

The variety is endless, but the important thing is to provide children with a range of practical resources and encourage them to use these as essential tools, supporting their mathematical journey, and to ensure that these are used across the primary years, consistent with our knowledge of child cognitive development, to support effective, deep, meaningful and engaging learning in maths.

Further reading on manipulatives
If you want to find out more about using manipulatives with your primary aged child, the following websites might be of interest:
https://thirdspacelearning. com/blog/concrete-resources- cpa-explained/
https://www.weareteachers.com/ at-home-math-manipulatives/
https://www.ldatschool.ca/ manipulatives-support-math- learning/
https://www.parentcircle.com/ making-math-fun-using- manipulatives/article
https://www.scholastic.com/ parents/school-success/ homework-help/more-homework- help/math-manipulatives.html
THE IMPORTANCE OF USING MANIPULATIVES IN PRIMARY MATHEMATICS TEACHING
The range of manipulatives for use in the teaching of mathematics in primary school is wide and varied; this includes any resources or materials which can be used in a practical, hands-on way in a classroom. In number, these may range from beads and counters, through cubes and number rods, dice and dominoes, to number lines, hundred squares and place value cards. Other areas of mathematics, similarly, have appropriate manipulatives, for example 3D shapes, teaching clocks, geometric shapes, pin boards. The list is long!

The rationale for, and positive impact of, use of manipulatives across all primary age ranges has been well researched through academic studies and endorsed by school inspection bodies such as OFSTED in the UK. To understand their importance, we need to consider both overall child cognitive development as well as what we value in effective teaching and learning.

Piaget's Theory of Child Development outlines four stages of cognitive maturity. The important aspect which impacts use of mathematical manipulatives is the recognition that until the age of twelve or so, children still think in concrete ways, with abstract thinking beginning to develop in the late primary years. Thus, the use of practical, hands-on resources is essential to support children to make sense of, and master, mathematical concepts, as this relates to their stage of cognitive development all through their primary education. This applies not only to arithmetical concepts but across the maths curriculum. A good example of this would be, for example, geometry/3D shape. Depending on their level of cognitive development, children are unable to visualise a 3D shape from a 2D picture on a page in the way that an adult can. Thus in learning about shape, they need to be able to handle, move and explore real 3D objects.

What is regarded as effective teaching has progressed from older, traditional methods of rote teaching of concepts where children are merely taught to follow a set procedure, to a model where understanding of the underlying maths is an essential part of the learning process. An example of this from my own experience would be subtraction where the strategy of 'borrow and pay back' (one to the top and one to the bottom which many people may recognise) was taught by rote pencil and paper, but without any understanding of why the strategy worked. It was only during my higher level maths studies that I actually understood the mathematics behind this.

Manipulatives, then, allow children to 'see' how the actual maths works and help them become true mathematical thinkers and problem solvers, not just followers of a rote-learned procedure. This has many advantages, among which are embedding deeper understanding and, therefore, supporting long term retention of concepts as well as reducing misconceptions in mathematical thinking. This firm foundation is especially important in maths because children need to continually build on previously learned concepts as they progress through their maths education. In addition, when children have regular opportunities to explain their learning, including using practical materials to support their explanations, we are encouraging deep learning and mastery. This clear and deeper understanding then supports the development of abstract thinking and, consequently, the key skills of actually using and applying what has been learned in unfamiliar contexts.

We now have at our fingertips a vast range of online, virtual resources to support mathematical development. While these have the power to motivate and greatly enhance children's maths learning, we also need to be aware of their limitations. Despite advanced graphics, these do not provide a tactile experience and do not allow children to physically handle the resources; they are 2D representations, much like a picture on paper. They are, perhaps, then best used after the introduction of physical resources as part of what is known as a Concrete - Pictorial - Abstract approach. In this way, although children are no longer actually handling physical resources, they still have the benefit of a visual representation to support their development into abstract thinking.

Children can also be supported easily at home with practical materials. For example in learning about fractions, cutting and sharing cakes, pizzas, coloured sweets, counters etc are ideal resources. Time can be a difficult concept, particularly for young children, therefore, making their own clocks, perhaps with only an hour hand in the first instance, then adding a minute hand can be effective.

The following website might be of interest to find out more about telling time with your child, it gives practical examples on how you can use many everyday materials and activities to tell the time:https://www.weareteachers.com/5-hands-on-ways-to-teach-telling-time

The variety is endless, but the important thing is to provide children with a range of practical resources and encourage them to use these as essential tools, supporting their mathematical journey, and to ensure that these are used across the primary years, consistent with our knowledge of child cognitive development, to support effective, deep, meaningful and engaging learning in maths.

Further reading on manipulatives

If you want to find out more about using manipulatives with your primary aged child, the following websites might be of interest:
https://thirdspacelearning. com/blog/concrete-resources- cpa-explained/
https://www.weareteachers.com/ at-home-math-manipulatives/
https://www.ldatschool.ca/ manipulatives-support-math- learning/
https://www.parentcircle.com/ making-math-fun-using- manipulatives/article
https://www.scholastic.com/ parents/school-success/ homework-help/more-homework- help/math-manipulatives.html
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