Comparing Models and Justifying the Choice of Unit

PurposesStudent collaboration; Student engagement; Student model building; Reasoning; Representation
DescriptionThis RUanalytic is the third in a series of three RUanalytics that show students’ investigation of how to use the rods to compare 1/2 and 1/3. In the first RUanalytic, “Which is Larger, 1/2 or 1/3? An Introduction to Comparing Unit Fractions,” after establishing the candy bar metaphor, students examine the different rods, attempting to establish that one of them can serve as an appropriate unit, referred to as “1” or at times “the whole” or the “candy bar”, so that it is possible to compare the fractions 1/2 and 1/3. The challenge that students faced was to comply with the necessity of keeping the same unit that was established in the discussion about sharing candy bars. Specifically, the rods used to represent halves and thirds must be named with respect to the same unit rod. The students’ exploration resulted in two models for 1: one model where a train of an orange rod and a red rod represents the unit, 1, and a second model where a dark green rod represents the unit.

In the second RUanalytic, “Comparing 1/2 and 1/3: Confusion about the Unit,” students investigate how much greater 1/2 is than 1/3. During their exploration students are confronted by confusion stemming from their reference to two different units to name the red rod. In this RUanalytic, one group of students shifts the unit from the orange and red rod train to the dark green rod and calls the red rod 1/3. Another group claims that six red rods fit beneath the unit, so the red rod is 1/6.

In this third RUanalytic in the series, students analyze this confusion in more depth. The data come from Sessions 4 and 6 of the fourth grade fraction intervention and the focus is on the students’ reasoning as they revisit and reconsider the candy bar metaphor, in effect, analyzing both the importance and the complexity of the unit when working with fractions (Steencken, 2001; Yankelewitz, 2009).

The first three events are included in this RUanalytic to provide the viewer with the context for the discussion about the unit that occurs in the following events. Event 1 comes from Session 4 (Sept. 27th) and takes place as the students explore ways of using the Cuisenaire rods to determine which is greater, 1/2 or 1/3. Three students present their solutions to Researcher Martino. Their solutions include the two models that form the basis for the rest of the student discourse for this comparison task.

In the Session 5, which takes place between Event 1 (Session 4) and Event 2 (Session 6) of this RUanalytic, the class explores the comparison of 1/2 and 1/3. Some students claim that 1/2 is 1/3 bigger than 1/3 and some claim that 1/2 is 1/6 bigger than 1/3. Students use models to support different arguments, and not until Session 5 is the debate resolved when students agree that 1/2 is not greater than 1/3 by 1/3, but rather, 1/2 is greater than 1/3 by 1/6 (See “Comparing 1/2 and 1/3: Confusion about the Unit”). Event 2 of this RUanalytic shows the resolution of the confusion as two students together present an argument for why 1/2 is greater than 1/3 by 1/6 when the train of orange and red rods is given the number name 1. In Event 3, another student presents a succinct argument for why 1/2 is greater than 1/3 by 1/6 when the dark green rod is given the number name 1.

In the beginning of Event 4, Researcher Maher asks students whether they agree or disagree with Alan’s model that shows that 1/2 is greater than 1/3 by 1/6, in light of the previous argument where the train of the orange and red rods represents 1. Students begin their argumentation about whether both models are admissible. In Event 4, students present their disagreements and agreements.

At this point, students revisit the idea of the candy bar metaphor. Some students argue that when you change from the train of orange and red rods as the unit to the dark green rod as the unit, you "change candy bars" and that is "not fair." In Event 4, Michael points out that "it matters" because "if you wanted to give someone 1/6 of that candy bar and then you were going to give someone 1/6 of the other one, then the person with that size would get a smaller amount."

Other students argue that you can switch the candy bars. In Event 6, Andrew argues that it is ok as long as you do not switch the candy bar in the middle of the problem. He states, "Well, that’s right because … it’s just a different size candy bar. If you just gave half of that [one candy bar] to the person and the other half of that [the same candy bar] to another person you would still have the same size. You can’t switch the candy bars."

At the end of the RUanalytic, students confirm that the rods take on different number names when the unit changes. When the train of orange and red rods is given the number name 1, the red rod has the number name 1/6. When the dark green rod is given the number name 1, the white rod has the number name 1/6.

This RUanalytic can be viewed on its own; however, if viewers are interested in a more detailed depiction of the events leading up to this classroom discourse about the candy bar metaphor and the significance of the unit, they are invited to view the first and second RUanalytics in this three-analytic series: “Which is Larger, 1/2 or 1/3? An Introduction to Comparing Unit Fractions” and “Comparing 1/2 and 1/3: Confusion about the Unit.”


References:

Steencken, E. (2001). Tracing the growth in understanding of fraction ideas: A fourth grade case study (Doctoral dissertation).

Van Ness, C. K. and Alston, A. (2015). Comparing 1/2 and 1/3: Confusion about the Unit.

Van Ness, C. K. and Alston, A. (2015). Which is Larger, 1/2 or 1/3? An Introduction to Comparing Unit Fractions

Yankelewitz, D. (2009). The Development of Mathematical Reasoning in Elementary School Students’ Exploration of Fraction Ideas. (Doctoral dissertation, School of Education Rutgers, The State University of New Jersey).
Created on2015-06-30T16:32:05-0400
Published on2015-09-22T09:56:28-0400
Persistent URLhttp://dx.doi.org/doi:10.7282/T3XW4MQT