Fourth Graders Analyses of Equivalence: 1/5 or 2/10?
PurposesEffective teaching; Student model building; Reasoning; Representation
DescriptionMaher and Martino (1996) contend that when students are given sufficient time to work on a problem and are given the opportunity to discuss their solutions with their peers, they often express differences of opinion. These conflicts are not always resolved immediately; sometimes they are pushed off to a later class session, possibly being deferred over an extended period of time. This allows students to think about the problem and upon revisiting the identical or similar problem at a later date, students are able to build on their ideas and solidify their understanding. As noted by Francisco and Maher (2005), revisiting a concept in a related problem, “helps students build rich and durable forms of mathematical understanding of mathematical concepts” (p. 371).
This analytic highlights students’ foray into the world of fraction equivalence and shows how revisiting a task helps students build strong understanding of a fundamental mathematical concept. Their journey began in an earlier session where students showed uncertainty regarding whether or not two fractions are equivalent. This analytic portrays two subsequent sessions in which students revisit the concept of fraction equivalence and ultimately come to a clear consensus of the mathematical truth of equivalent fractions. The researcher, Carolyn Maher, then introduces the fraction notation used to demonstrate equivalence.
In an earlier session, the students had been asked to give a number name to two white rods if the orange rod was given the number name one. Mark and Andrew had offered the solution of one fifth and Meredith had countered that with justification for the solution of two tenths. The class did not reconcile the two explanations and the researcher Amy Martino therefore left the discussion for a later time. This analytic begins with a session in which the researcher, Carolyn Maher, continues the discussion and asks the students what they remembers about the problem “Is 1/5 = 2/10?” In revisiting this problem, Meredith builds on the work of a previous session and shows that the white rods are called tenths and that the red rods are called one fifth. She then demonstrates the equivalence of two white rods and one red rod and concludes that one fifth equals two tenths. The students once again revisit and strengthen their understanding of this concept in a later session when working on the task “Which is larger, one half or two thirds, and by how much.” Many students come up with models to show the solution as one sixth. Meredith demonstrates that the solution can be two twelfths as well, enabling students to further solidify their understanding of fraction equivalence. The researcher then records the students’ ideas in mathematical notation to show that the two solutions are equivalent.
References
Francisco, J. M., and Maher, C. A. (2005). Conditions for promoting reasoning in problem solving: Insights from a longitudinal study. The Journal of Mathematical Behavior, 24(3), 361-372.
Maher, C. A., and Martino, A. M. (1996). The development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 27(2), 194–214.
