DescriptionThe students use Cuisenaire Rods to develop relational understanding (Skemp, 1978) of finding the difference of two fractions and explore, informally, comparison problems with fractions, developing an understanding of least common multiple. Students work on the following problems, “If we agree that the 8 students who received 1 and 1/4 pieces of candy got more than the 9 students who received 1 and 1/9 pieces, how much more did each of the students in the group of 8 receive? How much larger is 1/4 than 1/9?”. This analytic traces classroom and researcher questioning techniques that prompt students to build models with rods and use the models to demonstrate their solutions.
The roles of the researcher that this analytic focuses on are guiding students sense making from the models they build. Also, researchers initiate shifts in discussion by inviting students to clarify their ideas and provide backing for their ideas with evidence from their rod models. Guiding and initiating shifts are described in the Yackel (2002) article as a way to make, “what was previously done in action can become an explicit topic of conversation.” (Yackel, 2002, p.424). Clarifying and ensuring attention to significant concepts is described in the article as well as a way to focus students on certain pieces of information. The article describes backing as strategies, which according to Foreman, et al (1998) are successful when “taken-as-shared” (Yackel, 2002, p.425).
In the first event, Andrew and Jessica describe how they solved the Candy Bar Problem from the previous class to determine the number of pieces of candy bar each student will receive if a candy bar is scored into ten pieces. Researcher Maher, in prompting a new discussion topic, asked the students to compare their results. This is an illustration of initiating shift in the discussion. Since each group had a different number of students for the sharing, the candy bar was distributed differently in each group. The students were encouraged to compare which students received more candy, the students in Andrew’s group or the students in Jessica’s group, and by how much? This introduction related to the problem from the previous class and the solution would provide the students with valid information about this problem. This exemplifies a way to gain the interest of the students. This illustration of shift allowed the students to see the importance of the task at hand.
In the first event, Meredith conjectured that ¼ is larger than 1/9 by 1/5. In the second event, Researcher Maher states that if she has one-half and Amy has one-fourth that by Meredith’s reasoning, Researcher Maher would have one-half more than Amy. Researcher Maher explicates the problem at hand by providing the students with another example to foster their understanding. This is an illustration of attention to concept and backing by providing scaffolding for the students to see that there was no backing to the claim made and encourage the students to try to investigate further. Through this similar problem, the students develop an argument that their original conjecture is incorrect and identify their misconception without Researcher Maher telling them their initial conjecture is incorrect.
In the third event, Researcher Davis prompts James to identify the number names for each rod that he is using to represent the respective fraction. James, in identifying the number names for the rods, identifies an error he has made in his original conjecture that ¼ is bigger than 1/9 by 1/5. This is an illustration of guiding. Researcher Davis does not tell him or insinuate that there has been an error made. He only guides him through his thought process, making it possible for James to discover the error on his own.
In the fourth event, the students are unsure of the number name of the yellow rod. The students argue that the number name is both 5/36 and 1/5. Researcher Maher leads the students to fold back on the idea of sum of fractions parts equaling to a unit whole. She places five yellow rods down along the chain that represents one unit and states, “Would that be one fifth? Is that big enough to be one fifth?”. This illustrates a move of assuring attention to the concept. After creating a representation, the students are able to identify that five yellow rods would not create the unit.
In the fifth event, Michael states that Researcher Maher provided him with another example to explore in order to argue that it is possible to create a model that works for comparing two fractions, one with an odd and the other an even denominator. This is an illustration of eliciting backing, and it turned out to be an important discovery for Michael. By providing Michael with an example, he was able to determine that creating a model for which one can compare the unit fractions ¼ and 1/9 with respect to the unit determined by a train of rods, it was in fact possible to back his conjecture. Creating this representation allowed Michael to identify strategies to use for creating a model for comparing the fractions, one-fourth and one-ninth.
The researchers in this analytic invite the students to make their own connections, use similar problems to accept or decline a theory presented, and investigate whether or not their conjectures are valid. The researchers encourage the students to evaluate their own understanding and provide backing for their claims, thus applying the concepts they were investigating and building knowledge to apply to solving other problems. These components help the students to develop relational understanding, defined by R.R. Skemp as, “…knowing not only what method worked but why...” (Skemp, 1978, p.40). The students were able to come to their conclusions encouraged by the guided strategies provided by the researchers.
References:
Yackel, E. (2002). What we can learn from analyzing the teacher’s role in collective
argumentation. Journal of Mathematical Behavior, 21, 423-440
A32, Fraction problems: Sharing and Number Lines (front view), Grade 4, October 29, 1993, raw footage
[video]. Retrieved from http://dx.doi.org/doi:10.7282/T32N547Z
Skemp, R.R. (1978). Relational Understanding and Instrumental Understanding. Arithmetic Teacher, 37–
43.
Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics.
Washington, DC: National Governors Association Center for Best Practices and the Council of
Chief State School Officers.