Orly Buchbinder

What Can You Infer from This Example? Applications of Online, Rich-Media Tasks for Enhancing Pre-service Teachers’ Knowledge of the Roles of Examples in Proving

There is a consensus among mathematics educators that in order to provide students with rich learning opportunities to engage with reasoning and proving, prospective teachers must develop a strong knowledge base of mathematics, pedagogy and student epistemology. In this chapter we report on the design of a technology-based task “What can you infer from this example?” that addressed the content and pedagogical knowledge of the status of examples in proving of pre-service teachers (PSTs). The task, originally designed and implemented with high-school students, was modified for PSTs and expanded to involve multiple components, including scenarios of non-descript cartoon characters to represent student data. The task was administered through LessonSketch, an online interactive digital platform, to 4 cohorts of PSTs in Israel and the US, across 4 semesters. In this chapter we focus on theoretical and empirical considerations that guided our task design to provide rich learning opportunities for PSTs to enhance their content and pedagogical knowledge of the interplay between examples and proving, and address some of the challenges involved in the task implementation. We discuss the crucial role of technology in supporting PST learning and provide an emergent framework for developing instructional tasks that foster PSTs’ engagement with proving.

Systematic Exploration of Examples as Proof: Analysis with Four Theoretical Frameworks

This chapter offers a multi-layered analysis of one specific category of students’ example-based reasoning , which has received little attention in research literature so far: systematic exploration of examples. It involves dividing a conjecture’s domain into disjoint sub-domains and testing a single example in each sub-domain. I apply four theoretical frameworks to analyze student data: The Mathematical-logical framework for the interplay between examples and proof, Proof schemes framework, Transfer-in-pieces framework, and the Theory of instructional situations . Taken together, these frameworks allow to examine the data from mathematical, cognitive and social perspectives, thus broadening and deepening the insights into students thinking about the relationship between examples and proving. Implications for teaching and learning of proof in school mathematics are discussed.

Who is Right? – What Students’ and Prospective Teachers’ Responses to Scripted Dialog Reveal About Their Conceptions of Proof

This chapter explores the potential of using scripted student responses, embedded in a task titled Who is right?, as a tool to diagnose argumentation and proof-related conceptions of high-school students and pre-service mathematics teachers (PSTs). The data, collected in two separate studies, were examined for evidence of participants’ conceptions of the role of examples in proving and refuting universal statements. Additional analysis explored what types of criteria are used by the high-school students and the PSTs when evaluating scripted arguments, as well as whether participants were consistent in their evaluations across the collection of arguments. The data revealed that, when evaluating scripted arguments, high-school students used mainly mathematical criteria and strived to maintain consistency in their evaluations across the collection of arguments. On the contrary, PSTs applied both mathematical and pedagogical considerations in their evaluations, thus judging multiple, and even contradictory arguments as correct.

Representations of Practice in Teacher Education and Research—Spotlights on Different Approaches

Representations of practice provide an opportunity to refer to teachers’ professional environment both when designing tasks for teacher education or professional development, and when investigating aspects of teacher expertise. This volume amalgamates contributions by the members of the discussion group on representations of practice, which took place during ICME 13. The discussion group sought to collect experiences with different forms of representations of practice in pre-service and in-service teacher professional development settings, and of the use of representations of practice for researching into aspects of teacher expertise and its development. In this introductory chapter we provide an overview of different approaches to representing practice, and address key methodological issues that came up in the monograph’s chapters and in the discussion group’s meetings. We suggest four key questions along which such approaches can be discussed.

Examining the Mathematical Knowledge for Teaching of Proving in Scenarios Written by Pre-service Teachers

In this chapter, we examine what aspects of Mathematical Knowledge for Teaching of Proving (MKT-P) can be observed in written scenarios of classroom interactions, produced by pre-service teachers (PSTs) of mathematics. A group of 27 elementary and middle school PSTs completed an online interactive module, intended to trigger reflection on, and crystallization of their knowledge of the roles of examples in proving. To ground these processes in the context of teaching, the module engaged PSTs in analysis of several representations of practice such as a questionnaire about quadrilaterals with sample student work imbedded in it, and a classroom scenario in a storyboard format realized with cartoon characters. In addition, PSTs wrote a one-page continuation of that scenario describing how they would handle the situation if they were teaching the class. These scenarios proved to be a rich source of data on several aspects of MKT-P as well as general pedagogical knowledge.

Guided Discovery of the Nine-Point Circle Theorem and Its Proof.

ABSTRACT The nine-point circle theorem is one of the most beautiful and surprising theorems in Euclidean geometry. It establishes an existence of a circle passing through nine points, all of which are related to a single triangle. This paper describes a set of instructional activities that can help students discover the nine-point circle theorem through investigation in a dynamic geometry environment, and consequently prove it using a method of guided discovery. The paper concludes with a variety of suggestions for the ways in which the whole set of activities can be implemented in geometry classrooms.

Using Representations of Practice for Teacher Education and Research—Opportunities and Challenges

Representations of classroom practice offer the chance of referring to the teachers’ professional environment both when conceiving opportunities of professional development and when investigating aspects of teacher expertise. Representations of practice can stimulate teachers’ criteria-based analysis in environments that do not bring the full pressure and action constraints of the actual classroom. This discussion group aims at collecting experiences with different forms of use of representations of practice in pre-service and in-service teacher professional development activities and research into aspects of teacher expertise and its development. On this base, the discussion group plans to include an overview of different approaches to representing practice, address key issues of case-based learning, as well as methodological issues and questions related to validity of the construct(s) researchers or facilitators aim to address.