Title: Inquiry-based approaches to mathematics learning, teaching, and mathematics education research
Abstract: Florensa, Bosch, & Gascón) deal explicitly with supporting mathematics teachers to move from teaching approaches characterized as traditional toward approaches based on inquiry.This focus foregrounds concern for pedagogy but in each case teachers' knowledge, conceptualized in different ways, is also a consideration.For Weiland, Orrill, Nagar, Brown, and Burke, the knowledge upon which middle school teachers can draw in their teaching of proportional reasoning and that constitutes robust understanding of that concept is the central focus.Nevertheless, concern for pedagogy, particularly aspects of pedagogical knowledge, is also apparent in their report.As a collection, the articles offer diverse conceptualizations of mathematical knowledge and raise questions about that knowledge, what it means to do mathematics and the broad social purposes of that activity, the intersection of research and teaching, and the relationship between teachers and researchers.All of the studies reported involved small numbers of participants and detailed analyses of qualitative data and hence questions as to the scalability of the approaches suggested also arise.Andrews-Larson et al. examined how two groups of undergraduate mathematics instructors engaged in pedagogical reasoning as part of a workshop on inquiry-oriented instruction.Each group engaged with a mathematics task in either abstract algebra or linear algebra firstly in the role of doers of mathematics and then as mathematics instructors as they viewed video of students working on the same task.Andrews-Larson et al. found that the more deeply the mathematicians engaged with the mathematics inherent in the task, the greater was their engagement with the evidence of student mathematical reasoning evident in the video.The group that engaged more deeply with the mathematics and student reasoning were more likely to focus on supporting students in their mathematical work and maintaining the students' ownership of the ways in which they represented the mathematics.This was in contrast to the group that engaged less deeply with the mathematics and hence with the evidence of student thinking, that focussed more on describing the representational choices that the students made and on evaluating the students' contributions.Andrews-Larson et al. note subtle differences in the ways in which the facilitators of the two groups oriented the participants to their task and speculate that this may account for the difference in the foci of the activity of the two groups.The authors further venture that