Patricio G. Herbst
This article uses a classroom episode in which a teacher and her students undertake a task of proving a proposition about angles as a context for analyzing what is involved in the teacher's work of engaging students in producing a proof. The analysis invokes theoretical notions of didactical contract and double bind to uncover and explain conflicting demands that the practice of assigning two-column proofs imposes on high school teachers. Two aspects of the work of teaching—what teachers do to create a task in which students can produce a proof and what teachers do to get students to prove a proposition—are the focus of the analysis of the episode. This analysis supports the argument that the traditional custom of engaging students in doing formal, two-column proofs places contradictory demands on the teacher regarding how the ideas for a proof will be developed. Recognizing these contradictory demands clarifies why the teacher in the analyzed episode ends up suggesting the key ideas for the proof. The analysis, coupled with current recommendations about the role of proof in school mathematics, suggests that it is advantageous for teachers to avoid treating proof only as a formal process.
Patricio G. Herbst
Two questions are asked that concern the work of teaching high school geometry with problems and engaging students in building a reasoned conjecture: What kinds of negotiation are needed in order to engage students in such activity? How do those negotiations impact the mathematical activity in which students participate? A teacher's work is analyzed in two classes with an area problem designed to bring about and prove a conjecture about the relationship between the medians and area of a triangle. The article stresses that to understand the conditions of possibility to teach geometry with problems, questions of epistemological and instructional nature need to be asked—not only whether and how certain ideas can be conceived by students as they work on a problem but also whether and how the kind of activity that will allow such conception can be summoned by customary ways of transacting work for knowledge.
Inah Ko and Patricio Herbst
This study proposes task of teaching as an organizer of dimensionality in teachers’ subject matter knowledge for teaching (SMK) and investigates it in the context of measuring SMK for teaching high school geometry (SMK-G). We hypothesize that teachers use different SMK-G in different aspects of their teaching work and that such differences can be scaled and associated with key elements of instruction. Analyses of 602 high school teachers’ responses to two sets of items designed to measure the SMK-G used in two particular tasks of teaching—understanding students’ work (USW) and choosing givens for a problem (CGP)—suggested the two scales of SMK-G to be distinguishable and differently related to experience in teaching high school geometry.
Justin K. Dimmel and Patricio G. Herbst
We investigated how secondary mathematics teachers check student geometry proofs. From video records of geometry teachers checking proofs, we conjectured that teachers have different expectations for details that follow from written statements than for details that are conveyed by diagrams. To test our conjectures, we randomly assigned 44 secondary mathematics teachers to 1 of 3 experiment groups (n & 13, n & 15, n & 16) in which they viewed and rated representations of instructional practice. Participants in each group viewed treatment or control versions of instructional scenarios and rated the appropriateness of the teachers' work in different segments of each scenario. We compared participants' ratings across and within experiment groups. We found that participants rated lower instruction that deviated from what we hypothesized to be their expectations, confirming our hypotheses.
Talli Nachlieli, Patricio Herbst, and Gloriana González
This article reports on an investigation of how teachers of geometry perceived an episode of instruction presented to them as a case of engaging students in proving. Confirming what was hypothesized, participants found it remarkable that a teacher would allow a student to make an assumption while proving. But they perceived this episode in various ways, casting the episode as one of as many as 10 different stories. Those different castings of the episode make use of intellectual resources for professional practice that practitioners could use to negotiate the norms of a situation in which they had made a tactical but problematic move. This collection of stories attests to the effectiveness of the technique used for eliciting the rationality of mathematics teaching: By confronting practitioners with episodes of teaching in which some norms have been breached, one can learn about the rationality underlying the norms of customary teaching.
Justin K. Dimmel and Patricio G. Herbst
Geometry diagrams use the visual features of specific drawn objects to convey meaning about generic mathematical entities. We examine the semiotic structure of these visual features in two parts. One, we conduct a semiotic inquiry to conceptualize geometry diagrams as mathematical texts that comprise choices from different semiotic systems. Two, we use the semiotic catalog that results from this inquiry to analyze 2,300 diagrams from 22 high school geometry textbooks in which the dates of publication span the 20th century. In the first part of the article, we identify axes along which the features of geometry diagrams can vary, and in the second part of the article, we show the viability of using the semiotic framework to conduct empirical studies of diagrams in geometry textbooks.