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.The work reported here has been done with the support of the U.S. National Science Foundation (NSF) grants ESI-0353285 and DRL- 0918425 to the authors. All opinions are those of the authors and do not necessarily represent the views of the Foundation.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/110693/1/Multimedia Scenarios and Professional Knowledge.pdf-1Description of Multimedia Scenarios and Professional Knowledge.pdf : Main Articl

Research on Practical Rationality: Studying the justification of actions in mathematics teaching

Building on our earlier work conceptualizing teaching as the management of instructional exchanges, we lay out a theory of the practical rationality of mathematics teaching—that is, a theory of the grounds upon which instructional actions specific to mathematics can be justified or rebuffed. We do that from a perspective informed by what experienced practitioners consider viable but also in ways that suggest operational avenues for the study of instructional improvement, in particular for improvements that enable students to do more authentic mathematical work. We show how different kinds of experiments can be used to engage in theory building and provide examples of initial work in building this theory

TEACHING THE SOLVING OF LINEAR EQUATIONS – WHAT IS AT STAKE?

To test a model which characterizes what is at stake in the situation of solving linear equations (Chazan & Lueke, 2009), we analyse talk of teachers who, stimulated by watching an animation of classroom interaction (Chazan & Herbst, in press) share with their colleagues how they teach their students how to solve linear equations. The teacher talk illustrates two key aspects of our model of the situation of solving linear equations. First, the teachers in the sample conceive of it as their responsibility to teach their students a method for solving this class of problems; applying the steps of the method successfully means knowing how to solve linear equations. Second, teaching the method of solving linear equations does not involve the presentation of mathematical arguments, but at the same time is not exactly justification-free; the teachers present students with similes that motivate the steps in the method.The research reported in this article is supported by NSF grant ESI-0353285, to Daniel Chazan, University of Maryland, and Patricio Herbst, University of Michigan. Opinions expressed here are the sole responsibility of the authors and do not reflect the views of the Foundation

Revisiting the functions of proof in mathematics classrooms: A view from a theory of instructional exchanges

We consider the functions of proof in mathematics from the perspective of the work of the mathematics teacher. The work of proving and the time spent on proving, what can a teacher account it to? How can he or she justify it? We frame that problem in a descriptive theory of teaching and place within that frame the work of scholars who have inquired on the function of proof in mathematics. We argue that the multiple functions that proof plays in mathematics are resources that a teacher could use to account for the work of proving. We describe how the functions of proof identified in the literature can assist the work of the teacher and illustrate the role these functions of proof can play using classroom scenarios that showcase the work of proving. Since the teacher is not only accountable to mathematics but also accountable to students’ learning of that mathematics some times work is valuable because it helps represent important mathematical knowledge, sometimes because it helps students acquire, or demonstrate they have, knowledge. The existence of these different sources of value is not only a resource for the teacher to value diverse work but also permits to anticipate management dilemmas concerning the different ways of accounting for the work of proving.Work reported in this paper has been done with the support of the US National Science Foundation (NSF), grant ESI-0353285 to the first and third authors. Opinions expressed are those of the authors and do not necessarily represent the views of the Foundation.http://deepblue.lib.umich.edu/bitstream/2027.42/78168/1/Herbst_etal-Functions_of_proof.pd

Framing, normativity, and serviceability in teachers’ decision-making during lessons

To operationalize the notion that lessons are the knowledge base of the teaching profession and enable the study of teacher decision making during lessons, we provide a conceptualization of lesson as part of a multiverse of mathematics teaching. Using the case of problem-based instruction and a particular example in teaching high school geometry, we characterize framing, normativity, and serviceability as three systems of choice within the practical rationality of mathematics teaching which are useful for a teacher in managing decision making in problem-based lessons.Work supported with a grant from the James S. McDonnell Foundation AWD 220020524, Teachers as Learners program.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/171072/1/LessonConception-Herbst et al8v2.pdfDescription of LessonConception-Herbst et al8v2.pdf : Main ArticleSEL

STUDYING THE PRACTICAL RATIONALITY OF MATHEMATICS TEACHING: WHAT GOES INTO “INSTALLING” A THEOREM IN GEOMETRY?

This paper 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.The research reported in this article is supported by NSF, grant ESI-0353285 to Herbst and Chazan. Opinions expressed here are the sole responsibility of the authors and do not reflect the views of the Foundation.http://deepblue.lib.umich.edu/bitstream/2027.42/78023/1/ITH-PHTN&DC.pd

DIRECTING FOCUS AND ENABLING INQUIRY WITH REPRESENTATIONS OF PRACTICE: WRITTEN CASES, STORYBOARDS, AND TEACHER EDUCATION

We discuss affordances and liabilities of using a storyboard to depict a written case of a teacher’s dilemma that involves race, opportunity to learn, and student community. We rely on reflections by the teacher educator who authored the written case and later depicted it as a storyboard to use it with his preservice teachers (PSTs). The analysis involved, first, organizing the signifiers in each of the two representations of practice into what we call concentric spheres of stratification, and secondly, contrasting the various meanings attributed to signifiers by both the author and his PSTs. We suggest that the resources of storyboard allow for more inquiry and alternative narratives than is available from the single modality of text in the written case.Work reported here has been done with the support of NSF grant DRL- 1316241 to D. Chazan and P. Herbst. All opinions are those of the authors and do not necessarily represent the views of the foundation.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/143005/1/HerbstBoileauClarketalPMENA17.pdfDescription of HerbstBoileauClarketalPMENA17.pdf : Main Articl

Representations of mathematics teaching and their use in teacher education: What do we need in a pedagogy for the 21st century?

The introductory content of this paper is taken from the first author’s Pattishall Award Lecture at the UM School of Education in May 2010.First discussion document for PMENA’s RMT Working Group:“Facilitating sessions where teachers interact with and discuss representations of teaching”The work of project ThEMaT has been done with the support of NSF grants ESI-0353285 and DRL- 0918425. All opinions are those of the authors and do not necessarily represent the views of the foundation.http://deepblue.lib.umich.edu/bitstream/2027.42/78158/1/WG-RMT-Final.pd

Digital assessment and the “machine”

In this chapter, we explore assessment that is performed automatically by the digital environment, what might be called assessment “through technology.” Based on experience with the design of three innovative content-specific automatic assessment platforms, the main goal of the chapter is to exemplify design considerations to map the development of digital assessments. These digital platforms use various methods to assess mathematics through student’s interaction with digital learning resources. We address the fit between an assessment’s task design and its goals, the analysis of student work by the platform, and the report that is then produced. We want tasks to offer opportunities for students to express mathematical ideas, to take advantage of the opportunities provided by automatic assessment, as well as to meet the goals of assessments. In writing tasks, designers must also take into account two other design considerations: How can the digital assessment platform interpret and analyze student work in variable and flexible ways? How can the platform “make sense” of student work, so as to be able to generate feedback or to report on learning achievements? What are the ways in which insights from this analysis will be made accessible and to whom? Taken together, examination of the design of the tasks given to students to collect data, how that data is analyzed by the machine, and how the machine reports on that analysis allows us to map current digital assessment practices. We close by emphasizing the importance of continued engagement of the mathematics education community with the design of digital assessment platforms because mathematics education stakeholders bring with them a content-specific focus on higher-level thinking in mathematics and on students’ conceptions and misconceptions

Using comics-based representations of teaching, and technology, to bring practice to teacher education courses

This article situates comic-based representations of teaching in the long history of tensions between theory and practice in teacher education. The article argues that comics can be semiotic resources in learning to teach and suggests how information technologies can support experiences with comics in university mathematics methods courses that a) help learners see the mathematical work of teaching in lessons they observe, b) allow candidates to explore tactical decision making in teaching, and c) support pre-service teachers in rehearsing classroom interactions.Work described in this paper has been done with support of NSF grant ESI-0353285 to Herbst and Chazan. All opinions are those of the authors and do not represent the views of the Foundation.http://deepblue.lib.umich.edu/bitstream/2027.42/78017/4/PHetal-Comics_RoT_share.pd