Conference Paper

The Impact of an Authentic Intervention on Students’Proportional Reasoning Skills

  • Kütahya Dumlupınar University
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Work on learning from a situative point of view suggests that to understand how children's thinking about a domain such as fractions develops, research needs to take into account the socially organized ways of thinking in which children participate during instruction. The aim of the study reported in this article was to explore children's fraction learning in a 1st-grade classroom in which the teacher elicited and built on children's informal knowledge of fractions. Instruction revolved around children's solutions and discussions of equal-sharing tasks. The study provides an account of children's learning that examines the relation between classroom talk and children's evolving fraction concepts, with a focus on the analysis of several key classroom interactions that resulted in cognitive change. Pretests and posttests indicated that children's understanding of fractions changed in important ways. The results suggest that how children think about fractions is influenced not only by how their own knowledge is structured but, perhaps more profoundly, by how the context for thinking about and discussing fractions is structured.
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This article deals with the role that so-called emergent models can play in the process of constituting formal mathematics. The underlying philosophy is that formal mathematics is something that is, or should be, constituted by the students themselves. In the instructional design theory for realistic mathematics education, models always have been employed to foster a process in which formal mathematics is reinvented by the students themselves. This article describes how the use of models became more and more explicated over time and developed into the notion of emergent models. The design of an instructional sequence, which deals with flexible mental computation strategies for addition and subtraction up to 100, is taken as an instance for elaborating what is meant by emergent models and what role they play in fostering the constitution of formal mathematics. The analysis shows that there are 3 interrelated processes. First. at a more holistic level, there is a global transition in which "the model" initially emerges as a model of informal mathematical activity and then gradually develops into a model for more formal mathematical reasoning. Second, the transition from "model of" to "model for" involves the constitution of anew mathematical reality that can be denoted formal in relation to the original starting points of the students. Third, in the series of instructional activities, there is not 1 model, but the model actually is shaped as a series of signs, in which each new sign comes to signify activity with a previous sign in a chain of signification.
Twenty-four sixth-grade children participated in clinical interviews on ratio and proportion before they had received any instruction in the domain. A framework involving problems of four semantic types was used to develop the interview questions, and student thinking was analyzed within the semantic types in terms of mathematical components critical to proportional reasoning. Two components, relative thinking and unitizing, were consistently related to higher levels of sophistication in a student's overall problem-solving ability within a semantic type. Part-part-whole problems failed to elicit any proportional reasoning because they could be solved using less sophisticated methods. Stretcher/shrinker problems were the most difficult because students failed to recognize the multiplicative nature of the problem situations. Student thinking was most sophisticated in the case of associated sets when problems were presented in a concrete pictorial mode.
Two related problems have to be solved before we can have a clearer picture of cognitive development:(i) Is development hierarchical, leading to higher-order systems controlling lower-order subsytems? (ii) If so, what are the mechanisms involved in a process of development? These two problems will be studied here taking as example a concept which finds its achievement only in late adolesence: the concept of proportion. Part I of this article is devoted to the first problem. It will bear on the experiment which was undertaken and the analysis of results leading to a differentiation of stages of development. These stages will be illustrated by typical protocols of each stage. Part II will be devoted to problem solving strategies at each stage, and finally to a second order analysis leading to an attempt to interpret the passage from one stage to the next in terms of increasing equilibration or adaptive restructuring of the strategies put to use to solve problems.
Proportional reasoning is one form of mathematical reasoning. Many aspects of our world operate according to proportional rules. In the science classroom, proportionality surfaces when density is explored, when balance beams are used, and when any two equivalent rates are compared. ln the mathematics classroom, proportionality surfaces when properties of similar triangles are examined, when scaling problems are investigated, and when trigonometric functions are defined. The importance of proportional reasoning is stressed in the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989, 82).
Previous studies showed that 6-year-old children can make proportional judgements when the first-order relations are accessible to them and that children use the half boundary in order to make such judgements. In these studies “half” proved to be an important boundary in children's initial understanding of proportion. These studies involved non-numerical tasks (continuous quantities) in which proportional judgements were established on the basis of part-part relations (ratio). This raises a question about ratio comparisons that are numerically represented (discontinuous quantities). Would children again use the half boundary in ratio comparisons between continuous and discontinuous quantities? In this experiment 6- to 8-year olds were given two tasks: In the sliced task, the standard and the choices were discontinuous quantities; in the non-sliced task, the standard was a continuous quantity and the choices were discontinuous quantities. Children's justifications reflected different levels of understanding of proportions and characteristics of the tasks. The results revealed that “dhalf” plays a crucial role in children's proportional reasoning even when discontinuous quantities are involved.
The study deals with qualitative features of proportional reasoning on dimensional intensive variables by eleven and thirteen year-old students. These features include types of comparison and types of strategies employed. Four proportional problems, varying in numerical and referential content were administered to 116 eleven year-olds and 137 thirteen year-olds. No significant age effect was found. The relative frequencies with which the type of comparison and various strategies were used was greatly affected by context, numerical content of the problem, and the immediately preceeding task. No type of comparison appeared to be more natural than another, as suggested in previous research.
Make your own paint chart: a realistic context for developing proportional reasoning with ratios
  • K Beswick
K. Beswick, "Make your own paint chart: a realistic context for developing proportional reasoning with ratios," Australian Mathematics Teacher, vol. 67, no. 1, pp. 6-11, 2011.
Groundworks: Algebraic puzzles and problems
  • C Greenes
  • C Fendell
C. Greenes and C. Fendell, Groundworks: Algebraic puzzles and problems, Chicago, IL: Creative Publications, 2000.
A proposed constructive itinerary from iterating composite units to ratio and proportion concepts
  • M T Battista
  • C Van Auken
  • Borrow
M. T. Battista and C. Van Auken Borrow, "A proposed constructive itinerary from iterating composite units to ratio and proportion concepts," presented at the annual meeting of the North American Chapter of the International Group for Psychology of Mathematics Education, Columbus, OH, October, 1995.
Rational number, ratio and proportion
  • M Behr
  • G Harel
  • T Post
  • R Lesh
M. Behr, G. Harel, T. Post, and R. Lesh. "Rational number, ratio and proportion," in Handbook on research of teaching and learning, D. Grouws Ed. New York, NY: McMillan. 1992, pp. 296-333.