Understanding proportional reasoning for teaching

Proportional reasoning is an important cornerstone in children's mathematical development (Lamon, 1993). This sort of reasoning has been shown to develop across the early years of schooling (ages 8 to 10) through the middle years (ages 11-14) (Van Dooren, De Bock and Verschaffel, 2010). In the early years, children tend to use additive reasoning to generate solutions to problems, while later comparisons rely on multiplicative strategies including the use of ratios, fractions, percents and proportions. In part this is a result of the support provided by their teachers. Yet children in the middle years, while expected to understand and be able to use ratios and percents (Dole, 2000), often over-generalise and apply missing value proportions strategies to problems that require only additive reasoning (see Figure 1, problem 4). To support development teachers can use a variety of tools including ratio tables (Dole, 2008) and contextual activities (Beswick, 2011) the basis of which is an understanding of learners' reasoning. But understanding such reasoning and deciding which direction to take can be quite challenging. Dole, Clark, Wright, Hilton, and Roche (2008) found that teachers responded to student work containing evidence of proportional reasoning by "repeating the strategy of the child and evaluating it in terms of appropriateness" (p. 167). Teachers were challenged when asked to craft instructional interventions and develop productive lines of questioning to support students' proportional reasoning (Watson, Callingham and Donne, 2008) as revealed in their responses to student work. Given the challenges faced in the middle years' curriculum and the expertise the teachers have at solving proportional problems themselves, students' approaches may not be the first thinking teachers draw on as they plan for instruction. In this report we contrast teacher reasoning and student reasoning in work samples. We highlight some computational and reasoning approaches 11 to 14 year old students take on a comparison problem and suggest that teachers' understanding and use of student approaches is a fruitful basis for instruction.