Modeling the Variations in an Individual Student's Understandings of Fraction

• In Event: 35.069 - Fractions Learning: One Subject, Multiple Perspectives

Abstract

In my approach to researching fractions learning, I model students’ fractions understandings as dynamic reorganizations of mental operations (e.g., unitizing, disembedding, partitioning) first established as students develop understandings of whole numbers (Steffe, 2001). The products of the organizations of such operations are progressively more powerful fractions schemes. For example, the partitive unit fractions scheme, by which students know 1/n as an amount that can be iterated n times to form a whole, involves sequential partitioning and iterating. The form of the resulting structure (Piaget, 1970) is related to the concept of a whole number n consisting of 1s, as they both involve coordination of one unit within another unit. More advanced fractions knowledge involves coordination of 3 levels of units, e.g., for understanding improper fractions or the results of fractions as operators (Hackenberg, 2010; Steffe & Olive, 2010). From this perspective, a child’s learning of fractions involves her (not necessarily in awareness) reflection on the results of her activities coordinating units with both whole numbers and fractions.
To understand the dynamics of students’ units coordination, I analyze video-recordings of teaching experiments (Steffe & Thompson, 2000) conducted by researchers focused on students’ constructions of fractions schemes. In such teaching experiments, the teacher has two intertwined goals. First, the teacher is trying to understand his students’ thinking. Second, the teacher is trying to influence his students’ thinking, so that the student might experience perturbation of her current ways of operating whose resolution might result in accommodation (von Glasersfeld, 1995). The teaching experiments I analyze therefore provide an environment in which both a researcher and a student can experience constraints, stability, and variability in the student’s units coordinating activities across the changing contexts within a teaching experiment.
In my modeling approach, I consider the context of units coordinating activity in two ways. First, there is the context of the task environment. Are students manipulating physical materials? What representations and types of numbers are involved? Second, there is the context of the interaction. Are students reflecting on a direct query or challenge from their teacher? Do the students seem to find their current ways of operating adequate, or does it seem that they are challenged or perturbed by the results of their activities?
I attend to the stability in inferences of students’ units coordinating activity across such contexts using the construct of propensity to coordinate units (Boyce, 2014). Modeling changes in the propensity to coordinate units allows me to make conjectures regarding the relationships between the qualities of the contexts of sequences of observations and students’ units coordinating activity. One such conjecture is the utility of engaging sixth-grade students in activities with whole numbers immediately prior to engaging with fractions. Another is that introducing randomness in units coordination demands, realized through the use of games to select tasks, is useful for students’ maintaining productive units coordinating activity.

Author

• Steven Boyce, Virginia Polytechnic Institute and State University