### Comparing 1/2 and 1/3: Confusion about the Unit

PurposesStudent collaboration; Student engagement; Student model building; Reasoning; Representation
DescriptionThe data for this RUanalytic come from Sessions 5 (September 29th) and 6 (October 1st) of the fourth grade fraction intervention and continues the students’ investigation about which fraction is larger, 1/2 or 1/3 (Steenken, 2001; Yankelewitz, 2009). The particular focus is on the students’ reasoning as they propose, defend, and argue about precisely how much bigger 1/2 is than 1/3. This RUanalytic is the second in a series of three related RUanalytics. The first RUanalytic in the series focuses on comparing 1/2 and 1/3 and the introduction of the candy bar metaphor (See “Which is larger, 1/2 or 1/3? An Introduction to Comparing Unit Fractions”). The third RUanalytic in this series focuses on comparing the models that students created to compare fractions and analyzing students’ justification for their choice of unit (See “Comparing Models and Justifying the Choice of Unit”).

In addition to Session 3, which provided an introduction to comparing fractions and introduced the problem that is the focus of this RUanalytic, the investigations during the sessions depicted in this RUanalytic build directly on the first two sessions of the intervention during which the students constructed Cuisenaire rod models to support their ideas about basic relationships among whole numbers and fractions. Determining number names for each of the ten Cuisenaire rods based on a specific rod or train of rods that was assigned as the “unit” and given the number name 1 was fundamental to their reasoning throughout these sessions. (See RUanalytics: “Establishing Norms and Creating a Mathematical Community,” “Task Design Prompts Fourth Grade Students to Use Multiple Forms of Reasoning,” “Fourth Graders Design a New Rod,” “Fourth Graders Reason by Cases as They Explore Fraction Ideas,” and “Fourth Graders Build Toward Proportional Reasoning.”)

Prior to the events in Sessions 5 and 6 captured in this RUanalytic, during Session 3 (September 24th), Researcher Carolyn Maher contextualized the exploration of comparing fractions by describing an actual situation in which she introduced chocolate candy bars as a metaphor for thinking about sharing and comparing particular fractions. Students discussed how half of a smaller candy bar is smaller than the corresponding half of a larger candy bar and they agreed about the necessity of using the “same candy bar” or the same unit when comparing fractions. This idea was revisited frequently as the students built Cuisenaire rod models to compare 1/2 and 1/3, and is a primary focus of this RUanalytic. See RUanalytic “Which is Larger, 1/2 or 1/3? An Introduction to Comparing Unit Fractions” for a detailed look at the initial construction.

After students explored fraction and whole number relationships in Sessions 1 and 2 and the establishment of the “candy bar metaphor” developed in Session 3, they began their comparison of 1/2 and 1/3, using rods to determine which was larger. As a result of this exploration, students claimed that 1/2 was larger than 1/3 and developed two distinct rod models to support their claim. In one model the train formed by an orange rod and a red rod was given the number name 1 and represented the unit, with the dark green rod being given the number name 1/2, and the purple rod, 1/3. The second model, built by a number of students, represented the unit, 1, with the dark green rod. The light green rod was given the number name 1/2, and the red rod, 1/3. In both models students justified their comparison by showing that the rod with the number name 1/2 is indeed longer than the rod with the number name 1/3 and that the difference could be represented by a third rod that is smaller than either of the two (See RUanalytic “Which is Larger, 1/2 or 1/3? An Introduction to Comparing Unit Fractions”).

The two models mentioned above form the basis for the discussion in this RUanalytic. The intent of this narrative is to present not only the students’ reasoning but also specific instances of confusion as they work together to name the rod that represents the difference between 1/2 and 1/3. Based on the two models that the students created, the events provide illustrations of students changing the values of rods, both within and between the two models. As various students share their reasoning about their own ideas and models and those presented by other students, the notion of a unit appears to stabilize within the group.

Prior to Event 1, three students reconstructed their earlier model with the unit represented by the train of orange and red rods and claimed that 1/2 is greater than 1/3. They used the overhead Cuisenaire rods to share their model and support their claim that the dark green rod, given the number name 1/2, is larger than the purple rod, given the number name 1/3, and that the difference can be represented by the red rod (See “Which is Larger, 1/2 or 1/3? An Introduction to Comparing Unit Fractions”). In Event 1, Researcher Maher asks the students to quantify how much bigger 1/2 is than 1/3 by giving a number name to the red rod. On the basis of their model, showing that three red rods are the same length as one dark green rod, the girls claim that 1/2 is 1/3 bigger than 1/2.

In Event 2, Brian challenges this claim. He argues that the red rod has the number name 1/6, not 1/3. In Event 3, Researcher Maher revoices Brian’s argument and asks whether the red rod can be named both 1/3 and 1/6 at the same time.

In Event 4, Jackie and Kelly present the second model that students constructed when they gave the dark green rod the number name 1. They use this rod to claim that 1/2 is greater than 1/3 by 1 white and refer to the white as 1. Jessica then recalls the candy bar metaphor established for determining units by observing that Kelly and Jackie are using a "different sized candy bar" by their selection of dark green rod as 1 rather than the train of orange and red rods as 1. Researcher Maher notes that the argument presented by Jackie and Kelly is developed within their model and asks them to share it. Her questions and the discussion in Events 4 and 5 focus on the issue confronted by naming both the dark green rod and the white rod 1. Erik argues that the students did not really mean that white should have the number name 1, but rather, it should have the number name 1/6. The students agree that it is not possible for the white rod to have the number name 1 within the same model where dark green has been given the number name 1, and that the difference in this model between 1/2 and 1/3 is indeed 1/6 since six white rods are the same length as the dark green rod.

In Event 6, Researcher Maher asks students to reconsider the model when the train of orange and red rods has the number name 1. Jessica, Laura, and Audra rebuild their model and again use it to reason that 1/2 is bigger than 1/3 by 1/3. Jessica claims 1/3 and 1/6 might both be answers. Researcher Maher asks the class to consider this claim. In Events 7 through 9, students discuss this claim. Brian, Erik, Michael, and Meredith challenge the claim, presenting a variety of arguments from their understanding of the rod models, all agreeing that the number name for the red rod should be 1/6.

Event 10 is taken from the sixth session on October 1st. In this session, the students are asked to revisit the comparing task. Jessica revisits the question, "How much bigger is 1/2 than 1/3?" She states that she no longer thinks that 1/2 is bigger than 1/3 by 1/3 and uses the model with the orange and red rod train as 1 to support her claim. She places six red rods beneath the train of orange and red rods, claiming that the number name for red should be 1/6. She is not able to complete the argument as to why the difference is 1/6 and asks for Erik to continue. When he finalizes the argument, Jessica agrees, with apparent relief, that this is her conclusion as well.

In his article, Discovery Learning and Constructivism, Robert B. Davis describes traditional school practice about ‘learning mathematics’ as usually learning, by rote, certain rules for writing, often meaningless, symbols on paper in some very specific ways. In contrast, he espoused and practiced a view of ‘learning mathematics’ as building up, in your mind, certain powerful symbol systems that allow you to represent particular situations, acquiring skill in creating such mental representations and in using them. (Davis, 1990). The events of this RUanalytic provide convincing evidence for the validity of Davis’s statement.

Following the events in this RUanalytic, the class revisits the idea that the students "used different candy bars." If viewers are interested in the details of this discussion, they are invited to view the RUanalytic, "Comparing Models and Justifying the Choice of Unit.”

References:
Davis, Robert B. (1990). Discovery learning and constructivism. In: R. Davis, C. Maher & N. Noddings (Eds.) Journal for Research in Mathematics Education. Monograph No. 4, Constructivist Views on the Teaching and Learning of Mathematics, 93-106.

Gerstein, M. (2015) Establishing norms and creating a mathematical community. Retrieved from: http://dx.doi.org/doi:10.7282/T30C4XH9.

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 Models and Justifying the Choice of Unit.

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