Establishing norms and creating a mathematical community

This analytic describes the representations, reasoning, and justification used by students to express their understanding of fraction ideas while building solutions to a set of tasks. These tasks were introduced during the first session of a research intervention that was conducted over twenty-five sessions to study how students build fraction ideas prior to their introduction through the school’s curriculum. Of these 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. In addition to showcasing the foundational skill of building models to justify or reject claims made by students or the teacher, this analytic illustrates the importance of establishing socio-mathematical norms (Steencken, 2001; Yackel and Cobb, 1996) that establish what criteria establish the validity of a mathematical solution in the mathematics classroom. In this classroom, a norm was immediately established that clearly formed justifications were to be agreed upon by all members of the mathematical community. In addition, the analytic shows how a mathematical community was created by promoting group interaction and encouraging students to share their ideas and challenge one another. This analytic demonstrates the importance of collaboration and the ways that students can learn by constructing their own problems in an attempt to challenge their peers. The researchers, Carolyn Maher and Amy Martino, began by posing some simple tasks to the students. The tasks were punctuated with questions such as “What would you do to convince me?” The students were encouraged to justify their solutions, first to their partner and then to the researchers as well as the rest of the class. They demonstrated their solutions verbally in addition to building models to support their arguments at their desks before sharing their rod models using an overhead projector. At the start of the analytic, with Researcher Maher asked the class to demonstrate that the light green rod is half as long as the dark green rod. She then asked the students to name the light green rod if the dark green rod is given the number name one. After working other problems dealing with halves, the next task that was posed introduced a unit fraction other than one half. Researcher Maher asked the students if the red rod is one third as long as the dark green rod and what number name would be given to the red rod if the dark green rod was given the number name “one.” Next, she asked them if the light green rod is one third as long as the blue rod, and then asked what number name would be given to the light green rod if the blue rod was given the number name “one.” After working on the two tasks where the solution was a rod with length one third of a second given rod, the researcher asked how it could be that two rods of different length could be called one third. She then led a discussion on the fact that the number names of the rods can change but the color names remain the same, since number names would depend on which rod was given the number name “one”. She also emphasized that two different rods could not be called one in the same model. Researcher Maher then posed a task to the class to name the dark green rod if the red rod is given the number name, “one”. In this example the answer was three, a whole number. Researcher Maher then contrasted a problem where the length of the red rod was one fourth that of the length of the brown rod (brown was given the number name “one”) and where brown would be called four, if red were called “one”. She again emphasized that it must first be established what rod is named one. The session ended with pairs of students first posing a challenging question to the researchers and the rest of the class and then posing tasks to their partners. Steencken, E. (2001). Tracing the growth in understanding of fraction ideas: A fourth grade case study. Unpublished doctoral dissertation, Rutgers, The State University of New Jersey, New Brunswick. Steencken, E. P. & Maher, C. A. (2003). Tracing fourth graders’ learning of fractions: Episodes from a yearlong teaching experiment. The Journal of Mathematical Behavior, 22 (2), 113-132. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for research in mathematics education, 458-477