On the instructional triangle and sources of justification for actions in mathematics teaching

We elaborate on the notion of the instructional triangle, to address the question of how the nature of instructional activity can help justify actions in mathematics teaching. We propose a practical rationality of mathematics teaching composed of norms for the relationships between elements of the instructional system and obligations that a person in the position of the mathematics teacher needs to satisfy. We propose such constructs as articulations of a rationality that can help explain the instructional actions a teacher takes in promoting and recognizing learning, supporting work, and making decisions.The ideas reported in this paper have been developed in part with the support of National Science Foundation grants ESI-0353285 and DRL-0918425 to the authors.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/91281/1/On_the_instructional_triangle_PH&DC_May2012.pdf-

La production de faits en didactique des mathématiques

Théorie des situations, Brousseau. Situation adidactique, structuration du milie

Accounting for student perspectives in task design

This chapter aims to provide insights into students’ perspectives about the meanings and purposes of mathematical tasks and to understand how appropriate task design might help minimize any gaps between teacher intentions and student mathematical activity. Throughout the chapter, we explore accounts of how students understand the meaning and purpose of the mathematical activity they undertake, as well as how task design might take account of what we know about these perspectives. For instance, we discuss research that indicates ways in which the perceptions of students may differ from the intentions of teachers and task designers and attempt to articulate the nature of those differences to raise both theoretical and methodological challenges concerning how an observer can appreciate the student’s point of view. We also discuss ways in which task design that takes account of students’ responses might reduce the discrepancies between the intentions of designers and/or teachers and students’ perceptions of their activity and achievements

How to Teach and Assess Whole Number Arithmetic: A Commentary on Chapter 11

International audienceTeachers are accountable for classroom interactions and pupils’ work assessment. However, those visible actions are only a part of teachers’ work, and include also. (1) Planning not only a single lesson but also a sequence of lessons and more gener-ally thinking about and designing the entire mathematical theme (for instance, WNA as a whole). This usually depends on use of resources that are selected by the teacher. (2) Selecting the physical objects to be used during the lesson (or the sequence of lessons), the textbook which can be used by the pupils and/or by the teacher as a source of inspiration, the tasks that might be designed by others and are available (by sharing with colleagues, by browsing the Internet, etc.), the items for the assessment, etc. These aspects of teaching require teacher knowledge, which is not easily observed, since it is accessible only by means of what the teacher might say about her activity, which is always a reconstruction on her part, and what the teacher is doing in the classroom, which is subject to diverse interpretations.Various aspects of teacher knowledge have been considered within different frameworks that all take into account pedagogical content knowledge (Shulman 1986 ). This model has been refined by Deborah Ball and her colleagues (Ball et al. 2008 ), who have examined the impact mathematical knowledge for teaching on the quality of instruction (Hill et al. 2008 ). It is thus my purpose to highlight some aspects of whole number arithmetic knowledge for teaching which seem important to promote interest-dense situations (Bikner-Ahsbahs et al. 2014 ) and thus the development of pupils’ metacognitive strategies. The whole book containing this chapter is freely available on Springer web site (see DOI)