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

Abstract

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.

Beyond Plug and Chug: an Analysis of Calculus I Homework

“…We draw on Herbst and Chazan's (2012) elaboration of the instructional triangle (Cohen et al 2003) to understand the nature of homework within the broader context of instruction. This theoretical tool employs the concepts of didactical contract and milieu (Brousseau 1997) to describe the interactions between teachers, students, and knowledge at stake through instruction (see Fig.…”

“…1 by the arrow from the teacher's component to the milieu within the student subsystem. Herbst and Chazan (2012) argue that successful instruction occurs when all aspects of this system work together. They employ the notion of didactical contract (Brousseau 1997) to understand the ways in which teachers, students, and the knowledge at stake (content) are bound to each other and the environment through predominantly implicit responsibilities.…”

“…Our study contributes to this literature by exploring the nature of homework assignments at selected universities that were identified as having a successful Calculus I program. We ground this exploration within the perspective of the instructional triangle as adapted by Herbst and Chazan (2012), which allows us to understand and focus on relationships between the aspects of homework as part of a system involving content, students, and the teacher. …”

Revisiting the didactic triangle: from the particular to the general

“…' Schoenfeld (2012) also provides a thoroughgoing account and illustration of the application of the didactic triangle and argues that the didactic triangle, ''as typically construed in the English-speaking literature (contrasting particularly with the French speaking), is too narrow'' because it does not represent (directly) classrooms as cultural systems and consider the cultural forces that shape them. This idea is also present in the paper by Herbst and Chazan (2012) who focus on the teacher as an agent in a complex system of interrelated agents, and consider the professional obligations of teachers. Herbst and Chazan observe that for teachers ''individual choice is possible but not cost free.…”

“…He presents a conceptual framework, a sociocultural lens to examine mathematically productive classrooms. In a similar vein, Herbst and Chazan (2012) elaborate the notion of the didactic triangle by philosophizing on ''rationality'' as a basis under which interactions can be justified. Their focus is on the interactions between the person and the larger system under which their actions are regulated.…”