Fourth Graders Reason by Cases as They Explore Fraction Ideas

This analytic explores the various forms of reasoning employed by students as they share their solutions for the following task: "I’m going to call the orange and light green together one... Can you find a rod that has the number name one half?" This task, which does not have a solution, prompted the students to use multiple forms of reasoning to justify their claims; some of which are highlighted in this analytic (Yankelewitz, Mueller, & Maher, 2010). In particular, several students used reasoning by cases to justify their solutions. 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 session, facilitated by Researcher Amy Martino, took place in September of 1993, during the second session of a research intervention that was conducted over twenty-five sessions to study how students build ideas about fraction equivalence and operations with fractions prior to their introduction through the school’s curriculum (Yankelewitz, 2009). 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. Understanding the forms of reasoning used by children is important because reasoning is a fundamental component of mathematics learning, and by emphasizing reasoning and justification in mathematics education, children can naturally develop a foundation for formulating the idea of mathematical proof during their elementary school years (Yackel & Hanna, 2003). Yankelewitz (2009) analyzed the reasoning that students used as they worked on fraction tasks. She classified types of reasoning in two ways. First, she analyzed the structure of the argument as a whole by “pinpointing the data and the conclusion of the argument, as well as the function that the argument served” (p. 84) and classified the reasoning employed by these arguments as direct or indirect reasoning. Second, she analyzed the forms of reasoning “that were used to make up the direct or an indirect argument” (p. 85). These forms of reasoning included reasoning by cases, reasoning using upper and lower bounds, recursive reasoning, and reasoning using the generic example. The arguments posed by the students in this analytic were structured as both direct and indirect arguments. One form of reasoning is particularly highlighted by this analytic: exhaustive reasoning or reasoning by cases. According to Yankelewitz (2009): “Reasoning by cases, also known as the use of an argument by exhaustion… organizes the argument by considering a set of finite, distinct cases, and arrives at the same conclusion after consideration of each case. This form of reasoning requires a systematization of all possibilities into an organized set of cases that can be analyzed separately.” (pp. 85-86). During the group sharing session highlighted by this analytic, Andrew presented his solution first, using direct reasoning to show that there was no single rod that satisfied the condition. Meredith explained why no such rod exists using a partially flawed indirect argument. Brian then used exhaustive reasoning as he described the models he built in an attempt at finding all possible cases that could represent solutions for the task. Meredith, then, shared her solution to the problem, using direct reasoning to explain her model, with Erik elaborating on her presentation to explain why Meredith referred to the train of orange and green rods with number name, thirteen. The session ended with Erik presenting his solution of two groups of unequal halves, with an effort to be exhaustive to find an original solution to the task. 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). Yackel, E., & Hanna, G. (2003). Reasoning and proof. A research companion to Principles and Standards for School Mathematics, 227-236. Purpose(s) Reasoning 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. Yankelewitz, D., Mueller, M., & Maher, C. A. (2010). A task that elicits reasoning: A dual analysis. The Journal of Mathematical Behavior, 29(2), 76-85