In this paper, we examine the intersection of two previously recognised dimensions of students' interpretations of symbolic notations within graphs of functions. One dimension distinguishes location-thinking, where notations refer only to a point's location on a graph, from value-thinking, where such a point is treated as a multiplicative object. The other dimension distinguishes a nominal interpretation of expressions, where expressions refer to positions in the plane, from cardinal and magnitude interpretations, where expressions describe a discrete count of units, or measure of length, respectively. Taken together these dimensions provide six distinct ways students interpret expressions, especially those involving function notation, on graphs. Further, the frameworks offer suggestions about the ways of thinking underlying each interpretation. Each case reveals new meanings and affordances indicated by the interplay between the two dimensions. We provide both a theoretical account and empirical example of each case.

Students’ difficulties with argumentation, proving, and the role of counterexamples in proving are well documented. Students in this study experienced an intervention for improving their argumentation and proving practices. The intervention included the eliminating counterexamples (ECE) framework as a means of constructing and critiquing viable arguments for a general claim. This framework involves constructing descriptions of all possible counterexamples to a conditional claim and determining whether or not a direct argument eliminates the possibility of counterexamples. This case study investigates U.S. eighth-grade (age 13) mathematics students’ conceptions about the validity of a direct argument after the students received instruction on the ECE framework. We describe student activities in response to the intervention, and we identify students’ conceptions that are inconsistent with canonical notions of mathematical proving and appear to be barriers to using the ECE framework.

We offer criteria that an observer can use to determine whether an argument that uses an example to argue for a general claim appeals to that example generically. We review existing literature on generic example and note the strengths of each contribution, as well as inconsistencies among uses of the term. We offer several examples from the literature and our own data to develop and illustrate criteria for assessing whether an example is used generically.

These fifth graders engaged in key mathematical practices by explaining and illustrating central mathematical ideas.

This paper describes a theoretical model for systemic change as it concerns the learning and teaching of mathematics in K – 12 schools, with particular attention being paid to the rural context. Systemic change is the active process of establishing change in the community through lasting, long-term relationships, practices, and procedures (Adelman & Taylor, 2003). Our purpose is to describe the mechanics of such change provided by the strategic, continuous, and monitored support of all three of the constituents: Teachers, administrators and community, and externally supported by a temporary catalyst. Systemic change is achieved when the removal of the external catalyst does not affect the rest of the model. Evidence to support this claim has been derived from our case studies.