Analogical reasoning has played an important role in the development of modern mathematics. However, there has been critique of analogies for the purpose of learning new mathematics. In this article, I counter that students can productively reason by analogy to learn new mathematics and even develop new mathematics themselves. I display examples…

In this paper we analyze common solutions that students often produce to isomorphic tasks involving proportional situations. We highlight some key distinctions across the tasks and between the different equations students write within each task to help elaborate the different interpretations of equivalence at play: numerical, transformational, and…

We reflect on the evolution of an instrument designed to gather data about student actions with dynamic textbooks in university mathematics in a large-scale project. We also discuss the evolution of our understanding of the mediating role of this instrument in gathering data remotely that allow us to access student use and reconstruct their…

Critical to constructing and interpreting graphs is an individual's understanding of the underlying coordinate systems, yet coordinate systems are often overlooked or taken-for-granted in both mathematics education research and curricula. In this paper, we foreground coordinate systems and present a distinction between two uses of coordinate…

While proof is often presented to mathematics undergraduates as a well-defined mathematical object, the proofs students encounter in different pedagogical contexts may bear salient differences. In this paper we draw on the work of Dawkins and Weber (2017) to explore variations in norms and values underlying a proof across different pedagogical…

This paper sets forth a construct that describes how many undergraduate students understand mathematical terms to refer to mathematical objects, namely that they only refer to those objects that satisfy the term. I call this students' pronominal sense of reference (PSR) because it means they treat terms as pronouns that point to objects, like…

Recognizing identity not only as an important educational outcome but also as being inter-related to students' knowledge and practice, this paper explores an affordance of proof scripts; exploring students' identities. Specifically, drawing on data from teaching experiments and the construct of perceptual ambiguity, this paper presents an analysis…

Prior learning provides the basis for new learning. Mathematics educators employ formative assessment to "elicit" and "use" student thinking as the foundation of their instruction. Yet, information can be elicited and used in a variety of ways, so not all formative assessment is equally "formative." This means that…

This paper demonstrates how questions of "provability" can help students engaged in reinvention of mathematical theory to understand the axiomatic game. While proof demonstrates how conclusions follow from assumptions, "provability" characterizes the dual relation that assumptions are "justified" when they afford…

In his "Proofs and Refutations," Lakatos identifies the "Primitive Conjecture" as the first stage in the pattern of mathematical discovery. In this article, I am interested in ways of reaching the "Primitive Conjecture" stage in an undergraduate classroom. I adapted Realistic Mathematics Education methods in an…

Many of the central ideas in an introductory undergraduate linear algebra course are closely tied to a set of interpretations of the matrix equation Ax = b (A is a matrix, x and b are vectors): linear combination interpretations, systems interpretations, and transformation interpretations. We consider graphic and symbolic representations for each,…

Starting from an interest in the teachers' use of diagrams and gestures during a traditional front lesson at tertiary level, this research takes a narratologic perspective to see a mathematical lesson as a story, and hence the students' notes as re-tellings of a mathematical story. The first minutes of a traditional mathematics lecture…

Weber and Alcock's (2004, 2009) syntactic/semantic framework provides a useful means of delineating two basic categories of proof-oriented activity. They define their dichotomy using Goldin's (1998) theory of representation systems. In this paper, I intend to clarify the framework by providing criteria for classifying student reasoning into…