### The development of mathematical conceptual understanding

#### Definition of conceptual understanding

Hiebert and Lefevre provide a useful definition of conceptual understanding, describing it as “knowledge that is rich in relationships….. so that all pieces of information link to some network” (1986, p.3). For students with sound conceptual understanding, “physical objects, actions performed on them and abstractions are all interrelated in a strong mental structure” (Clements, 1999, p. 48).

In most countries around the world today, conceptual understanding is one of the core proficiencies that must be demonstrated across most mathematical strands and as such, is assessed and reported on. An example of conceptual understanding is shown below:

A Year/Grade 2 student with a solid conceptual understanding of number and place value might know, represent and explain how the number 32;

• is part of a count (31, 32, 33… or 12, 22, 32, 42…)
• can represent cardinality (the last number in a count represents the total number of objects in a group)
• be the result of an operation (such as 52 – 20 = 32)
• be a whole made up of a number of parts (32 = 10 + 16 + 6 )
• is conventionally partitioned as 3 tens and 2 ones, which is represented by the digits, 32
• but can be repartitioned in different ways; it is also 32 ones
• is greater than 22 but less than 42 and different to 320

#### How do primary students develop mathematical conceptual understanding?

Based on the findings of scholars such as Piaget and Bruner, educators have long believed that learners (particularly children) make a shift from sensory-concrete thinking where they must act upon and observe actual objects to make sense of an idea, to abstract thinking where conceptual understanding is developed and expressed via symbols and language (McNeil & Utall, 2009). Recent researchers have challenged some of these notions, with a number of them believing that early learners are capable of abstract thought. Others see that adults also use sensory experience and perceptual scaffolding to construct understandings (Mix, 2009). However, it is still generally accepted that direct interaction with physical objects can help students make sense of abstract concepts.

Most mathematical concepts encountered by students at school are abstract. Long division or subtraction with repartitioning cannot be directly represented; we interact with these concepts via words, symbols and numbers. (Mix, 2009). This lack of direct connection increases the probability that learners can attach the wrong meaning to a symbolic representation or use mathematical processes with little real understanding of what they are doing.

Thus, manipulatives have been created to physically represent such abstractions; to embody the concepts in a way that everyday objects do not and symbols can not (Mix, 2009).

#### The role of manipulatives

Clements states that a good manipulative “aids students in building, strengthening and connecting various representations of mathematical ideas”, leading to greater conceptual understanding (1999, p 50). Research suggests a good manipulative:

• creates links between multiple physical and symbolic representations of the same abstract concept (Clements, 1999)
• is consistent with the cognitive and mathematical structures and processes it represents (McNeill & Uttal, 2009)
• scaffolds and supports the learner as they construct new understandings (Mix, 2009)
• promotes mathematical precision and dialogue (Samara & Clements, 2009)

#### The limitations of manipulatives

It is important to note that a manipulative has no inherent mathematical meaning in itself. Dienes, the creator of base-10 blocks stressed, “one cannot over-emphasise that it is not the material itself which creates the true mathematical learning situation” (1971, p. 55). Teachers, particularly early years educators can over-rely on the power of manipulatives, sometimes thinking that their provision alone is enough to facilitate learning and transfer knowledge (Uttal, O’Doherty, Newland, Hand & DeLoache, 2009). They may also see them as “fun” add-ons to formal teaching programs and avoid them, particularly in the Upper Primary years (Moyer et al, 2002).