Academic journal article Australian Primary Mathematics Classroom

Using Quotitive Division Problems to Promote Place-Value Understanding

Academic journal article Australian Primary Mathematics Classroom

Using Quotitive Division Problems to Promote Place-Value Understanding

Article excerpt

A robust understanding of place value is essential. Using a problem-based approach set within meaningful contexts, students' attention may be drawn to the multiplicative structure of place value. By using quotitive division problems through a concrete-representational-abstract lesson structure, this study showed a powerful strengthening of Year 3 students' conceptual understanding of place value.

Place value is commonly introduced to young children using 'bundling into tens' activities. In this article, we share a problem-based approach and a lesson structure implemented with a class of seven-year-olds that focused on the multiplicative structure of place value. Children were given a variety of quotitive division problems set in meaningful contexts and encouraged to use multiple representations to support their learning. Quotitive division problems involving groups of ten provided opportunities for students to make connections between the concrete and symbolic representations for two- and three-digit numbers.


Part-whole thinking, or partitioning, is a key component of mathematical reasoning reflected in the content of current curriculum documents. It is a "big idea" identified by Baroody (2004, p. 199) and is foundational for understanding concepts such as place value. In order to develop the concept of place value, a student must have an understanding of part-whole relationships and four key properties: (1) positional (the quantity is represented by the position of a digit within a multi-digit number); (2) base-ten (numbers increase in powers of ten from right to left); (3) multiplicative (the value of each digit is its place value multiplied by its face value); and (4) additive (the total is represented by the sum of the values of the individual digits) (Ross, 1989). Children's early reasoning relies primarily on additive properties but it becomes important that they develop understanding of part-whole relationships using multiplicative structures and ideas about grouping, equal composing, and decomposing into equal-sized parts (Baroody, 2004).

Introducing multiplication and division

Multiplication and division can be introduced to students from a younger age than is often the case in practice. Children are provided with many opportunities at pre-school and early in their schooling to work with units of one, and to partition numbers or decompose quantities into parts, and to solve addition and subtraction problems using counting strategies. However, a focus on multiplication and division problem-solving provides an opportunity to move from counting by ones to concepts of the group, equivalence, and the unit (Sophian, 2012). For example, Sophian explains that children may see a collection of six shoes as six discrete items or as three pairs (groups of two as composite units). She argues that although the multiplicative relationship is more complex, "young children are capable of reasoning both additively and multiplicatively about relations between quantities" (p. 170). In order to understand multiplication, children need to appreciate how the whole can be composed from, or decomposed into, equal-sized parts (Baroody, 2004). Likewise, decomposing a whole into equal-size groups (equal partitioning) is the conceptual basis of division. Multiplicative thinking is also important for developing mathematical understanding in the domains of proportio and ratio, measurement, and place value.

A clear implication from the literature is that children's mathematics learning can be supported by the introduction of multiplication and division early in their schooling. There is substantial evidence that children prior to school age are able to work with equal-group multiplication and fair-share division (see, for example, Baroody, Lai, & Mix, 2006; Wright et al., 2014). Hence, it makes sense to capitalise on that prior knowledge in the mathematics classroom. It is also reasonable to expect that the experience of working with units greater than one may help children to develop part-whole thinking sooner than may be otherwise the case. …

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