PurposeReasoning

DescriptionThis analytic demonstrates the development and use of upper and lower bound arguments by fourth grade students as they worked on fraction comparison tasks using Cuisenaire rods to build models. 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 four sessions depicted by this analytic were facilitated by Researchers Carolyn Maher and Amy Martino and took place consecutively in October of 1993 as part of a research intervention aimed to study how students build fraction ideas prior to their introduction through the school’s curriculum (Yankelewitz, 2009). The first of the four sessions was the eighth in a sequence of twenty-five sessions that were conducted during 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.

Yankelewitz (2009) describes the use of upper and lower bound reasoning:

When reasoning by upper and lower bounds, a student defines the upper and lower boundaries or limits of a class of numbers or mathematical objects. For example, for the set of numbers 1 < x < 4, the upper bound of the set is 4 and the lower bound is 1, since all the numbers in the set are contained within the two bounds. After these bounds have been defined, the student reasons about the objects that are not contained between the bounds and draws conclusions based on this reasoning.

This analytic demonstrates how upper and lower bound arguments were developed by students, taken up by others, and reused in different problem contexts.

The analytic begins with Michael’s use of an upper bound argument while working to create a model to demonstrate the difference between the fractions, one half and three fourths. Although his argument was presented to Brian, there is no evidence that it was taken up by others in the class. The following day, however, Alan began to develop an upper bound argument as he and other students worked with rod models to compare the fractions, two thirds and three fourths. The next day they continued working on the problem by creating larger rod models to demonstrate the difference. Alan noticed that as the rod model grows larger, at some point no single rod can be used to represent one third. He articulated this concern to Erik, who then alerted Andrew, who was separately creating a large rod model. Alan’s argument, however, was unclear. Later that day, Alan restated his argument as he, Erik, David, and Meredith worked on the floor creating larger models. Alan’s argument was stated more clearly, but excluded the upper bound from the set. On the following day Alan explained his conjecture to the class using his upper bound argument and including the upper bound in the set. His reasoning was then adopted by other students in the class when later that day, David and Meredith used an upper bound argument similar to Alan’s to demonstrate that no single rod could be used to represent one third or one fourth for a model they built. Then, later in the session, Alan used an upper bound argument in the context of another problem, the comparison of one half and two fifths. Further evidence of this type of reasoning was displayed three days later, when Erik used a lower bound argument to justify a claim that they had found the smallest possible model to represent the difference between two thirds and three fourths.

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).

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 (2009) describes the use of upper and lower bound reasoning:

When reasoning by upper and lower bounds, a student defines the upper and lower boundaries or limits of a class of numbers or mathematical objects. For example, for the set of numbers 1 < x < 4, the upper bound of the set is 4 and the lower bound is 1, since all the numbers in the set are contained within the two bounds. After these bounds have been defined, the student reasons about the objects that are not contained between the bounds and draws conclusions based on this reasoning.

This analytic demonstrates how upper and lower bound arguments were developed by students, taken up by others, and reused in different problem contexts.

The analytic begins with Michael’s use of an upper bound argument while working to create a model to demonstrate the difference between the fractions, one half and three fourths. Although his argument was presented to Brian, there is no evidence that it was taken up by others in the class. The following day, however, Alan began to develop an upper bound argument as he and other students worked with rod models to compare the fractions, two thirds and three fourths. The next day they continued working on the problem by creating larger rod models to demonstrate the difference. Alan noticed that as the rod model grows larger, at some point no single rod can be used to represent one third. He articulated this concern to Erik, who then alerted Andrew, who was separately creating a large rod model. Alan’s argument, however, was unclear. Later that day, Alan restated his argument as he, Erik, David, and Meredith worked on the floor creating larger models. Alan’s argument was stated more clearly, but excluded the upper bound from the set. On the following day Alan explained his conjecture to the class using his upper bound argument and including the upper bound in the set. His reasoning was then adopted by other students in the class when later that day, David and Meredith used an upper bound argument similar to Alan’s to demonstrate that no single rod could be used to represent one third or one fourth for a model they built. Then, later in the session, Alan used an upper bound argument in the context of another problem, the comparison of one half and two fifths. Further evidence of this type of reasoning was displayed three days later, when Erik used a lower bound argument to justify a claim that they had found the smallest possible model to represent the difference between two thirds and three fourths.

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).

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.

Created on2014-02-17T14:15:29-0400

Published on2015-05-06T15:53:52-0400

Persistent URLhttp://dx.doi.org/doi:10.7282/T3ZS2Z8N