We can define manipulatives as, ‘objects that can be handled and moved, and are used to develop understanding of a mathematical situation.’ (Making Numbers, 2017). The materials can be structured, such as Base 10 apparatus, Place Value Counters and Cuisenaire Number Rods. Research and practice show that by representing a maths concept with everyday objects, children can visualise, explore and manipulate them, giving easier access to mathematical knowledge and understanding. Therefore, as teachers, we need to think creatively about how and what we can use within our teaching to represent abstract concepts.
The purpose of manipulatives is to expose the abstract mathematical concept and allow children to communicate their ideas, which would otherwise remain abstract. Understanding and meaning are central to a child’s mathematical development, for without understanding no strategy is efficient or meaningful. We therefore need to ensure that children interpret and make sense of mathematics for themselves. This is when the most powerful learning happens, resulting in higher retention of the concept – and manipulatives support this process. Neglecting manipulatives and moving straight to the abstract often means that children have not had the opportunity to make sense of a concept for themselves, and therefore the child has not developed a deep and structural understanding of the mathematics.(Hoong et al, 2015).
Whilst manipulatives are tools for learning, they should not be seen as a crutch. Lio Moscardini (2009, cited in Black, 2013) argues that manipulatives will be crutches if used to model rote processes with little or no understanding of the structure of the mathematics. However, if children are given the opportunity to make sense of the mathematical concept, through the careful facilitation of the teacher and their skilled questioning, then the manipulative becomes an effective tool for understanding.
Fractions are generally seen as a challenging concept to teach and learn. However, Wendy Fortescue-Hubbard in her video “Hard to Teach – Starting off with Fractions” demonstrates clearly how manipulatives aid understanding by using strips of paper. She states that people have trouble with fractions because they have not had the opportunity to explore fractions in a hands-on, practical manner. Some of these strategies could also be employed with Bar Modelling resources or with manipulatives such as Fraction Action.
The Concrete-Pictorial-Abstract (CPA) approach underpins the teaching of mathematics in the Singapore curriculum. The Concrete, Pictorial and Abstract approach has its foundations in Bruner’s modes of representation. Previous understanding was that children move from Concrete to Pictorial and then to Abstract. However, recent thinking is that the three should be interlinked and presented together so that the child can see connections between the abstract and the concrete or pictorial. The three CPA elements can work powerfully together to enable learners of any age to make sense of concepts.
With thanks to Jill Todd from Bradford College and the West Yorkshire Maths Hub for writing this blog post.