Fourth graders build towards proportional reasoning

This analytic highlights students’ reasoning as they worked on five tasks which built towards the development of proportional reasoning. The events in this analytic are selected from a study of fourth grade students from Colts Neck, a suburban New Jersey district (Maher, Martino, & Davis, 1994). The three sessions depicted by this analytic were facilitated by Researchers Carolyn Maher and Amy Martino and took place consecutively in September of 1993 as part of a research intervention aimed to study how students build ideas about fraction equivalence and operations with fractions prior to their introduction through the school’s curriculum (Yankelewitz, 2009). The first of the three sessions was the second in a sequence of twenty-five sessions that were conducted over the course of the intervention. Of the twenty-five sessions, seventeen were focused primarily on building basic fraction concepts including fractions used as operators, fractions as numbers, equivalence of fractions, comparison of fractions, and operations with fractions. Although students, prior to the fourth grade, were introduced to strong ideas related to fraction as operator, in this school district fraction operations were not a part of the 4th grade curriculum at that time. Rather, fraction operations were formally introduced in grade 5. The students in this class session investigated these ideas about fractions through a series of open-ended problem tasks. In this analytic, the students built models for solutions to the tasks using Cuisenaire rods, attending to the attribute of length. Although at first the students evidenced the use of additive reasoning, as the students worked on the given tasks, they intuitively began to use direct proportional reasoning. According to Yankelewitz (2009), proportional reasoning is a common form of clarified analogical reasoning that “has a sound mathematical basis and can be used to validate results due to the laws of numbers that are the source of the similarity of relations” (p. 88). According to Polya (1954), clarified analogical reasoning is useful in mathematics as a means of determining similarity of structure or relation between two mathematical propositions, functions, or operations when “the relations are governed by the same laws" (p. 29). In the first event of the analytic, Kimberly used additive reasoning as she incorrectly extended a solution to a previous problem. As illustrated in events 2 and 3, Alan challenged her solution, and Kimberly revised her solution with the use of proportional reasoning. During the next session (events 4 through 7), the students worked on a pair of tasks to name the red rod when a train of yellow and light green rods is given, first, the number name one and then the number name two. These tasks elicited direct proportional reasoning when students acknowledged that when the train is given the number name two, the number name assigned to the red rod is doubled compared to when the train is given the number name one. In the following session (events 8 and 9), the students then worked on other related tasks. They were asked to give the number name for the white rod both when the orange rod is given the number name ten and when it is given the number name fifty. This set of tasks also elicited direct proportional reasoning as students articulated that the number name assigned to the white rod must be multiplied by ten to equal the value of the orange rod. Thus, these sets of tasks encouraged the students to use direct proportional reasoning as they assigned number names to specific rods . References Maher, C. A., Martino, A. M., & Davis, R. B. (1994). Children’s different ways of thinking about fractions. In Proc. 18th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 3, pp. 208-215). Polya, G. (1954). Mathematics and plausible reasoning (Vols. 1 & 2). Princeton: Princeton University Press. Yankelewitz, D. (2009). The development of mathematical reasoning in elementary school students’ exploration of fraction ideas. Unpublished doctoral dissertation, Rutgers, The State University of New Jersey