Commentary on Theories of Mathematics Education

<strong>Commentary</strong> on Understanding a Teacher’s Actions in the Classroom 425
chapter, a missing factor is the teacher’s views of the expectations from the videolesson.
It seems that this factor significantly influences her decision at the critical,
turning point of the lesson. It is, probably, possible to identify and describe these
expectations as beliefs (or as goals). But it seems that such definitions somehow
obscured the specific nature of expectations.
• The relationship between knowledge, beliefs and goals. The relationship between
knowledge and beliefs has been addressed in numerous writings in mathematics
education (see, for instance, Forgasz and Leder 2008; Leder et al. 2002;
Thompson 1992). Less attention has been paid to the relationship between knowledge
and goals and even less so to the relationship between beliefs and goals. This
chapter provides many insights to the latter issue.
• The relationship between professional development and teacher change. The
chapter emphasizes that the teacher that was video-taped attended, shortly before
she was video-taped, in-service training courses on the use of open tasks and
on the use of computers in mathematics instruction. The authors clarify that
She [the teacher] tried to adapt the imparted issues to the topic of linear functions. While
the lesson did not develop as desired, she shifted back to her hitherto established traditional
teaching repertoire. (p. XX)
Similar phenomena are described in the large and fundamentally important literature
that examines the various stages of teacher change (see, for instance, Tirosh
and Graeber 2003). The teacher’s behavior and her statements during the interview
suggest that she has not gained enough knowledge and confidence with the
technological tools that she intended to use and that she is eager to use these new
methods of instruction. Time is a critical factor in this change process. We suggest
that readers of this chapter will pause for a while and attempt to provide their
own response to the question: What methods are effective in assisting experience
teachers in the challenging, often frustrating change process?
• The appropriateness of the teacher’s decision. Törner, Rolka, Rösken and Sriraman
clarify, at the beginning and throughout the chapter, that their aim is neither
to assess the video-taped lesson nor to evaluate the teacher’s decision. Yet, near
the end of the chapter, we hear the voice of a teacher who participated in a discussion
on this lesson who “commented aptly the situation: When the house is on
fire, who will then worry about pedagogy? Then you can rely only on the systematic
nature of the content”. Törner, Rolka, Rösken and Sriraman add two, related
notes. The first “In a deliberately provocative formulation, subject matter related
goals and beliefs might be called hard and pedagogical content goals and beliefs
soft”. The second “Obviously, pedagogy then loses out in the game pedagogy
versus content”. These and other comments in the chapter regarding the teacher’s
decision could lead to a long-lasting discussion on the relationship between the
aims and the means in mathematics education.
It is possible to draw attention to other, significant issues that are implicitly or explicitly
discussed in the chapter. However, at this point we will suggest to read (or
re-read) the chapter, and to pay specific attention to various aspects that are related
to the interplay between the general and the specific, those that are mentioned in