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Task Design Prompts Fourth Grade Students to Use Multiple Forms of Reasoning

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Task Design Prompts Fourth Grade Students to Use Multiple Forms of Reasoning

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doi:10.7282/T3ZK5JF0

Abstract

This analytic demonstrates various forms of reasoning used by fourth grade students as they try to build a model with Cuisenaire rods to solve a fractions task for which there is no solution in the set. 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 Carolyn Maher, 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 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, but were formally introduced in grade 5.

This analytic demonstrates the structure of students’ arguments and forms of reasoning which were used as they worked on the following problem: "If I call the blue rod one, what rod will I call one half?" The students employed various forms of reasoning to explain the meaning of one half and to prove that there is no rod in the set that is one half as long as the blue rod. 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. These arguments employed the following forms of reasoning: upper and lower bounds, reasoning by cases, and generic.

The analytic begins with David’s explanation of why he thinks that no rod exists which can be called one half if the blue rod is called one. His indirect argument employed an upper and lower bounds form of reasoning. Jessica attempted to refute David’s argument using direct reasoning, but David implied that her argument led to a contradiction. Erik then proposed an alternate definition of one half, but his claim of finding halves was countered by Alan who used indirect reasoning to explain that two halves have to be of equal length. Andrew, too, revealed the contradiction inherent in Erik’s argument using indirect reasoning. Alan then used indirect reasoning to explain that the task cannot be solved. David concluded with reasoning by cases and generic reasoning to generalize the properties of the "odd" 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).

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.

This analytic demonstrates the structure of students’ arguments and forms of reasoning which were used as they worked on the following problem: "If I call the blue rod one, what rod will I call one half?" The students employed various forms of reasoning to explain the meaning of one half and to prove that there is no rod in the set that is one half as long as the blue rod. 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. These arguments employed the following forms of reasoning: upper and lower bounds, reasoning by cases, and generic.

The analytic begins with David’s explanation of why he thinks that no rod exists which can be called one half if the blue rod is called one. His indirect argument employed an upper and lower bounds form of reasoning. Jessica attempted to refute David’s argument using direct reasoning, but David implied that her argument led to a contradiction. Erik then proposed an alternate definition of one half, but his claim of finding halves was countered by Alan who used indirect reasoning to explain that two halves have to be of equal length. Andrew, too, revealed the contradiction inherent in Erik’s argument using indirect reasoning. Alan then used indirect reasoning to explain that the task cannot be solved. David concluded with reasoning by cases and generic reasoning to generalize the properties of the "odd" 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).

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.

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Reasoning

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Esther Winter

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2012-10-10T17:26:12-0400

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2015-06-18T13:52:51-0400

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2015-06-18T14:00:10-0400

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Esther Winter

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2014-08-20T15:35:33-0400

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