The instructional triangle

Kelli Nipper and Paola Sztajn have written an article that was recently published in Journal of Mathematics Teacher Education. The article is entitled: Expanding the instructional triangle: conceptualizing mathematics teacher development.

Abstract As mathematics educators think about teaching that
promotes students’ opportunities to learn, attention must be given to
the conceptualization of the professional development of teachers and
those who teach teachers. In this article, we generalize and expand the
instructional triangle to consider different interactions in a variety
of teacher development contexts. We have done so by addressing issues
of language for models of teachers’ professional development at
different levels and by providing examples of situations in which these
models can be applied. Through the expansion of our understanding and
use of the instructional triangle we can further develop the concept of
mathematics teacher development.

Teachers are professionals with a rich knowledge that is both content specific and general. They shape instruction by the way they interpret and respond to students and materials (p. 2). The notion of "the instructional triangle" is based on the definition of instruction as (they refer to Cohen and Ball, 1999, p. 5 here): the interaction between teachers and students around educational material. These ideas are also shared by other researchers. One of them, Barbara Jaworski, created the teaching triad, consisting of:

• management of student learning
• sensitivity to students
• engagement in challenging mathematics

Nipper and Sztajn describe how they have tried to expand this instructional triangle to teacher education, and as a response to language issues, they suggest to replace the ordinary triangle: teacher - student - mathematics with the more general: organizer - participants - content. For a further elaboration of their analysis and theoretical suggestions, you should dig deeper into the article!