remember the catchphrase ‘Ours is not to reason why. just invert and multiply”. AustralianHuman Computer Interaction researcher Dianne Siemon (2010. p. 3) argues that these ‘helpful rules’ not only lead to more fragile learning. but that they ‘serve to reinforce the view that school mathematics is about learning and applying fairly meaningless rules and procedures’. On the other side of the globe. French researchers complained that often students ‘must accept many things simply on the basis that the teacher says so. and in the long run [they havel no coherent foundation for the concepts’ (Brousseau. Brousseau. & Warfield. 2014. p_ 5). While in America. Susan Lamon (2012) links a more general fear of mathematics to the teaching of fractions: Many people have a feu of mathematics. In high school. they were reluctant to take more than the minimum required courses. They had feelings of -being lost’ or in the dark’ when it came to mathematics. For most of these people. their relationship with mathematics started downhill early in elementary school, right after they were introduced to fractions. They may have been able to pass courses — perhaps even get good grades — beyond the third and fourth grades by memorizing much 01 what they were expected to know. tart they can remember thc anxiety of not understanding what was going on in their mathematics classes.
Helping students develop the necessary conceptual understandings is better done with a curriculum that emphasises many representations — physical. pictorial. verbal. real world and symbolic — than with more traditional approaches that depend only on rote procedures (Lamon. 2012: Siegler & Fazio. 2010: Spangler. 2011: Way. 2011). Interacting with a range of representations and models with different attributes presents different challenges to students. ‘causing students to continuously rethink and ultimately generalize the concepts’ (Petit ct al.. 2010. p. 3). This approach will take longer as the concepts are developed, but thereafter less time will be spent reviewing and re-teaching (Spangler. 2011) and students will build proficiency more quickly (Lamon. 2012). The Australian Curriculum (see tables 12.1 and 12.2) recommends that during the early and middle years of primary education the focus should be on developing the meaning of fractions — as equal parts of a whole or equal parts of a set — and the investigation of equivalence of fractions. and the comparison and ordering of fractions using a range of models including number lines. In the upper years of primary schooL the focus shifts to solving problems with fractions. Each of these ideas is discussed in this section: however. if you are teaching in the upper primary you will find that many students are still struggling with understanding fractions and consequently trying to apply rules by rote to solve problems. These students will benefit from the opportunity to undertake activities. such as those described here, designed to develop their understanding of the meaning of fractions. Many materials, such as fraction bars and pattern blocks, are available to assist you in helping children develop concepts about fractions (see the Appendix C for paper versions of these commercial materials). Some of the most effective materials. such as construction paper and counters, are commonly found in the classroom. Mathematics literature can also be very useful in helping children develop concepts for fractions. For example. Mr Half Dar (Fisher & Sneed. 2008) develops the language of fractions. The Doorbell Rang (Hutchins. 1986) is excellent for introducing the division meaning of fractions.
Three meanings of fractions Their are three distinct meanings of fractions — part-whole. quotient and ratio — that are found in most primary mathematics programs. but the focus is usually on the part-whole meaning. with little development of the other two meanings. Ignoring these other meanings may be one source of students’ difficulty with fractions (Fuchs at al.. 2013).
Part-whole The part-whole meaning of a fraction indicates that a whole has been partitioned into equal parts — for example. the fraction ; indicates that a whole has been partitioned into five equal parts and that three of those parts are being considered. This fraction may be shown with a region model:
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