partitioning whereas decimals can represent partitions of only tenths and powers of tenthsHuman Computer Interaction (hundredths. thousandths and so on). In this chapter. the initial concentration on fractions does not imply that the entire study of fractions pre-cedes the study of decimals. Teachers can introduce decimal notation after a beginning foundation has been built with fractions. In fact. many operations with decimals are easier than the corresponding operations with fractions and can be taught meaningfully before children learn all the operations with fractions. There is some controversy over the importance of studying fractions, since technology uses decimals almost exclusively: however, not only is an understanding of fractions required in order to undertake advanced mathematics study (Fuchs et al.. 2013: Lamon. 2012: Siegler & Fazio. 2010) but ‘it is virtually impossible to run a household or to understand magazine or newspaper articles’ (Lamon. 2012, p. 256) without an understanding of fractions. which provide the foundations of understandings of proportionality. Consequently. the Australian Curriculum continues to emphasise the development of sound conceptual understandings of fractions. Further expectations for children’s teaming of fractions and decimals from Year 3 to TABLE 12.2 Year 7 based on the Australian Curnculum proficiency strands
Year Proficiency strand
Year 3
Understanding: representrig unit fractions
Year 4
Understanding: extendrig piece value to decimals Fluency: ccomirlicatnq sequences of simple tractions Year 5 Understanding: using tractions to represent probabilities, comparing and ordering fractions and
dectrels and representing them In various ways Reasoning: continuing patterns InvoisAng tractions and decants
Year 6
Understanding: using tractions and decimals to describe probabilties, representrig fractions and decimals In various ways and describing connections between them. and making reasoratie estrnalions
Fluency: ccrrmiling between fractions and decimals, using operations WM fractions, decimals and percentage Problem solving: forrrulating and solving authentic problems using fractions and decimals
War 7 Understanding: recognising equivalences between tractions, decimals, percentages and ratios Fluency: representing fractions and decimals In various ways
Source: C Atioratun Curriculum. Assetitricnt Arai Rcportin! Authority. 2015.
12.1 Conceptual development of fractions
LEARNING OUTCOME 12.1 Companng three different meamngs of fractions and identifying the models of the part-whole meaning. The Australian Curriculum emphasises that students should be given the opportunity to develop concepts as well as number sense with fractions and decimals. This continues to be a challenging target. In an Australian study of 323 Year 6 students, barely half (50.5%) could correctly nominate which was larger when presented with the pair of fractions and 4 (Clarke & Roche, 2009). while only 30% of students were able to give the correct answer when asked to indicate ‘how much each person would get if three pizzas were shared between five people’ (Clarke, 2006). Many authors draw a distinction between conceptual and procedural knowledge. Siegler and Fazio (2010, p. 6) suggest that a conceptual knowledge of fractions is an understanding of the meaning of fractions. for example: their magnitudes and relations to physical quantities, an understanding of why arithmetic procedures with fractions me mathematically justified and why they yield the answers they do. such conceptual knowledge can be contrasted with procedural knowledge — the ability to execute a series of steps to solve a problem.
A range of researchers report that students who have been given the opportunity to develop stronger conceptual understandings of fractions are more successful remembering and applying procedures (Gabriel el al.. 2012: Petit. Laird. & Marsden. 2010: Siegler & Fazio, 2010: Vukovic et al.. 2014). Despite this. the teaching of fractions is an area where there has been a tendency to resort to teaching procedures: we all