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The fields of quantitative and covariational reasoning boast a wide range of powerful theoretical tools, which are described carefully in the literature. Less frequent and explicit attention, however, has been paid to writing down detailed, practical guidance for operationalizing these theoretical constructs. Some guidance is provided by covariational reasoning frameworks, but much is left unsaid concerning the inherent complexities and ambiguities involved in analyzing students' moment-by-moment behaviors and what these behaviors convey about their covariational reasoning. In an effort to more clearly link theory to analytic methodology, we share three lessons about analyzing students' covariational reasoning to make research more accessible to newcomers and better address what is often left unsaid in the covariational reasoning literature. [For the complete proceedings, see ED630060.]

We reflect on the limitations of our research group's prior methods for assessing covariational reasoning which primarily used graphical tasks found in extant literature. Graphical tasks dominate the literature on covariational reasoning, and through our use of these tasks we came to question the heavy reliance on them. Our concerns led us to ask the following: (1) What are the limitations of using tasks with graphs to assess covariational reasoning? (2) How can we improve assessment of covariational reasoning to accommodate students with nonnormative graphing schemes? We offer this piece as the beginning of a conversation to develop improved methodologies that attend to the ubiquity of students' nonnormative graphing schemes. [For the complete proceedings, see ED629884.]

This research report explores relationships between fractions' task representations (discrete, rectangular, or circular) and elementary prospective teachers' (PTs) fractions conceptions. Studies show PTs' conceptions of fractions are centered on a part-whole understanding, which may be problematic when teaching children about improper fractions. We studied PTs' conceptions of fractions using a task-based written assessment. The assessment also included PTs' rankings of task difficulty. We found that PTs' responses involving circles representations aligned best with the empirical trajectory of children's developing understandings of fractions. We discuss implications for supporting PTs in conceptualizing fractions as measures. [For complete proceedings, see ED581294.]

This proposal explores how the activity of subitizing--quickly apprehending the numerosity of a small set of items--changes with the development of number concepts. We describe how varying the orientations of items in teaching experiment sessions promoted one pre-schooler, Frank, to attend to subgroups of items and change his thinking about conjoining two numbers. The results illustrate how game-play oriented subitizing activities may support growth in number understandings. [For the complete proceedings, see ED583989.]

In this proposal, I introduce a method for modeling the dynamics of a sixth-grade student's accommodation of his fractions scheme to include a disembedding operation (Steffe & Olive, 2010). I will describe a three-part approach consisting of a constructivist teaching experiment, retrospective analysis, and stochastic modeling of the student's activity across ecologies within the teaching experiment (cf. Steffe & Thompson, 2000). As disembedding requires coordination of two levels of units--a disembedded part and an unadulterated whole (Steffe, 2001)--the results also serve as an initial model for describing the dynamics of an individual's units coordinating within and across contexts. [For the complete proceedings, see ED584443.]

An interview with a sixth-grade student illustrates how her number sense and understanding of variability relate to her ability and proclivity to apply a frequentist (statistical) approach to probability tasks. A general suggestion for teaching about mathematics of uncertainty through the gradual strengthening of estimation, as per the historical development, is also discussed. [For the complete proceedings, see ED585874.]