We describe a four-level spectrum of pedagogical awareness (SPA) that emerged from the analysis of six undergraduate mathematics tutors' (1) conceptualizations of their first-year students' difficulties; (2) descriptive accounts of their strategies for facilitating the overcoming of these difficulties; and (3) self-reflective accounts regarding their teaching practices. These were recorded in 45 semistructured interviews conducted during the 8-week Oxford University term and following minimally participant observation of their tutorials. We exemplify the four levels of SPA, across these three strands, in 12 characteristic examples where tutors discuss episodes from the observed tutorials. The design and analysis of the study reflect three underlying theoretical perspectives: constructivist, sociocultural, and enactivist. Finally, we draw on participants' comments to suggest that our methodology encouraged reflection, raised awareness, and demonstrated the pedagogical potential of collaboration between mathematicians and mathematics educators. We see such awareness and collaborative goodwill as prerequisites to conceptualizing and implementing reform of pedagogical practice.

This article deals with 12th-grade students' conceptions of a mathematical definition. Their conceptions of a definition were revealed through individual and group activities in which they were asked to consider a number of possible definitions of four mathematical concepts: two geometric and two analytic. Data consisted of written responses   to questionnaires and transcriptions of videotaped group discussions. The findings point to three types of students' arguments: mathematical, communicative, and figurative. In addition, two types of reasoning were identified surrounding the contemplation of   alternative definitions: for the geometric concepts, the dominant type of reasoning was a definition-based reasoning; for the analytic concepts, the dominant type was an example-based reasoning. Students' conceptions of a definition are described in terms   of the features and roles they attribute to a mathematical definition.

This article proposes a taxonomy of strategies for single-digit multiplication, then uses it to elucidate the nature of the learning tasks involved in multiplication. In preceding work, it has generally been assumed that much of children's strategy development is driven by changes in their general conceptual capabilities relating to number. In   contrast, we argue that, during the period in which single-digit multiplication is the focus of explicit classroom attention, changes in strategy use are primarily driven by the learning of number-specific computational resources. For this reason, we categorize   multiplication strategies based upon the number-specific resources that are employed in their execution. To support our conclusions, we draw from a corpus of interviews with third-grade students that were conducted before, during, and after   instruction in multiplication.