Patricio Herbst

Engaging Students in Proving: A Double Bind on the Teacher

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 teachers 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.

Using Novel Tasks in Teaching Mathematics: Three Tensions Affecting the Work of the Teacher

Novel (as opposed to familiar) tasks can be contexts for students’ development of new knowledge. But managing such development is a complex activity for a teacher. The actions that a teacher took in managing the development of the mathematical concept of area in the context of a task comparing cardstock triangles are examined. The observation is made that some of the teacher’s actions shaped the mathematics at play in ways that seemed to counter the goals of the task. This article seeks to explain a possible rationality behind those contradictory actions. The hypothesis is presented that in managing task completion and knowledge development, a teacher has to cope with three subject-specific tensions related to direction of activity, representation of mathematical objects, and elicitation of students’ conceptual actions.

Studying the Practical Rationality of Mathematics Teaching: What Goes Into “Installing” a Theorem in Geometry?

This article presents a way of studying the rationality that mathematics teachers utilize in managing the teaching of theorems in high-school geometry. More generally, the study illustrates how to elicit the rationality that guides teachers in handling the demands of teaching practice. In particular, it illustrates how problematic classroom scenarios represented through animations of cartoon characters can facilitate thought experiments among groups of practitioners. Relying on video records from four study group sessions with experienced teachers of geometry, the study shows how these records can be parsed and inspected to identify categories of perception and appreciation with which experienced practitioners relate to an instance of an instructional situation. The study provides initial evidence that supports a theoretically derived hypothesis, namely that teachers of geometry as a group recognize as normative the expectation that a teacher will sanction or endorse those propositions that are to be remembered as theorems for later use. In interacting with a story in which students had proven a proposition that the teacher had not identified as a theorem, the study also shows the kind of tactical resources that teachers of geometry could use to make it feasible for students to reuse such a proposition.

Interactions with Diagrams and the Making of Reasoned Conjectures in Geometry

Four potential modes of interaction with diagrams in geometry are introduced. These are used to discuss how interaction with diagrams has supported the customary work of ‘doing proofs’ in American geometry classes and what interaction with diagrams might support the work of building reasoned conjectures. The extent to which the latter kind of interaction may induce tensions on the work of a teacher as she manages students’ mathematical work is illustrated.

Students' Conceptions of Congruency through the Use of Dynamic Geometry Software.

This paper describes students’ interactions with dynamic diagrams in the context of an American geometry class. Students used the dragging tool and the measuring tool in Cabri Geometry to make mathematical conjectures. The analysis, using the cK¢ model of conceptions, suggests that incorporating technology in mathematics classrooms enabled a measure-preserving conception of congruency with which students’ could shift focus from shapes to properties. Students also interacted with dynamic diagrams in a novel way, which we call the functional mode of interaction with diagrams, relating outputs and inputs that result when dragging a figure. Students’ participation in classroom interactions through discourse and through actions on diagrams provided evidence of learning using tools within dynamic geometry software.

''Doing Proofs'' in geometry classrooms

Proof in advanced mathematics classes: semantic and syntactic reasoning in the representation system of proof

Mathematical Knowledge for Teaching and its Specificity to High School Geometry Instruction

This chapter documents efforts to develop an instrument to measure mathematical knowledge for teaching high school geometry (MKT-G). We report on the process of developing and piloting questions that purported to measure various domains of MKT-G. Scores on a piloted set of items had no statistical relationship with total years of experience teaching, but all domain scores were found to have statistically significant correlations with years of experience teaching high school geometry. Other interesting relationships regarding teachers’ MKT-G scores are also reported. We use these results to propose a way of conceptualizing how instruction-specific considerations might matter in the design of MKT items. In particular, we propose that the instructional situations that are customary to a course of studies can be seen as units that organize much of the mathematical knowledge for teaching such a course.

Is the Role of Equations in the Doing of Word Problems in School Algebra Changing? Initial Indications From Teacher Study Groups

We illustrate a method, which is modeled on “breaching experiments,” for studying tacit norms that govern classroom interaction around particular mathematical content. Specifically, this study explores norms that govern teachers’ expectations for the doing of word problems in school algebra. Teacher study groups discussed representations of classroom interactions in which the class is doing a word problem and a student solves the word problem without writing an equation. Toulmins scheme for everyday argumentation was adapted to analyze the study group talk. This analysis suggests that, for this sample, word problem norms are in flux; it may no longer be necessary for a student to state and solve an equation to do an algebra word problem. We propose that further research will have to determine whether the emerging change in these norms is attributable to policy initiatives in local contexts or whether it represents a larger cultural shift in conceptions of algebraic symbols.

Studying professional knowledge use in practice using multimedia scenarios delivered online

We describe how multimedia scenarios delivered online can be used in instruments for the study of professional knowledge. Based on our work in the study of the knowledge and rationality involved in mathematics teaching, we describe how the study of professional knowledge writ large can benefit from the capacity to represent know-how using multimedia representations of practice and alternatives to it. These instruments can be used to study what professionals notice and decide to do in practice in ways that improve upon earlier uses of written representations of professional scenarios or videorecorded episodes. In particular, storyboards and animations of nondescript cartoon characters can be used to explore professional knowledge variables systematically while the multimodal representation of human activity in context ensures the face validity of questions.

An Analysis of Evaluative Comments in Teachers' Online Discussions of Representations of Practice.

It has been common to use video records of instruction in teacher professional development, but participants have rarely been encouraged to evaluate teachers and students’ actions in those records, allegedly because evaluation deters from the development of a professional discourse. In this study, we inspected teachers’ online discussions of animations of classroom episodes realized with cartoon characters, looking at the difference in the content of conversation turns when members made evaluative comments and when they did not make evaluative comments. We were interested in finding out whether making evaluative comments correlated with participants’ reflection on their professional practice and proposal of alternative teaching actions; for that purpose we used systemic functional linguistics (SFL) to develop a coding scheme that attended to evaluation, alternatives, and reflection in forum discussions. We found statistically significant evidence that the more the participants actively evaluated the teaching in the animations, the more they proposed alternative teaching actions and reflected on instructional practice. We relate these findings to the notion of social presence in online discussions.