Ivy Kidron

The Need for Proof and Proving: Mathematical and Pedagogical Perspectives

This chapter first examines why mathematics educators need to teach proof, as reflected in the needs that propelled proof to develop historically. We analyse the interconnections between the functions of proof within the discipline of mathematics and the needs for proof. We then take a learner’s perspective and discuss learners’ difficulties in understanding and appreciating proof, as well as a number of intellectual needs that may drive learners to prove (for certitude, for causality, for quantification, for communication, and for structure and connection). We conclude by examining pedagogical issues involved in teachers’ attempts to foster necessity-based learning that motivates the need to prove, in particular the use of tasks and activities that elicit uncertainty, cognitive conflict and inquiry-based learning.

Interacting Parallel Constructions of Knowledge in a CAS Context

We consider the influence of a CAS context on a learner’s process of constructing a justification for the bifurcations in a logistic dynamical process. We describe how instrumentation led to cognitive constructions and how the roles of the learner and the CAS intertwine, especially close to the branching and combining of constructing actions. The CAS has a major influence on parallel constructions after branching and it facilitates combining. Hence, the CAS has the upper hand near branching points but the learner has the upper hand near combining points.

CERME7 Working Group 16: Different theoretical perspectives and approaches in research in mathematics education

CERME7 WG16 papers provide evidence of progress made since the central term of networking emerged at CERME4. WG16 aims to understand how theories can be connected successfully while respecting their underlying conceptual and methodological assumptions, a process we call ‘networking theories’. At CERME7 the manner in which a theory conceptualises, and accounts for, the emergence and nature of mathematical objects was often related to the question of the theory’s underlying conceptual assumptions. Font et al. explore this question by synthesising the onto-semiotic approach (OSA), APOS theory and the cognitive science of mathematics, as regards their use of the concept ‘mathematical object’. OSA extends the other theories by addressing the role of semiotic representations. Sollervall uses Peirce’s semiotic triangle and Duval’s theory of registers to negotiate disciplinary and individual perspectives on the notion of meaning in mathematics. In Santi’s paper, two semiotic approaches are coordinated, providing complementary views of Duval’s and Radford’s theories which can only be linked in a diachronic way due to strong differences in theoretical principles. LaCroix compares Cultural Historical Activity Theory, Engestrom’s interpretation and Radford’s theory of knowledge objectification. Some papers explore cognitive and social dimensions. Jay offers a reconciliation of perspectives with a semiotic approach which appears to propose a neutral arena for negotiation of definitions of crucial terms like concept, understanding and learning. In Kidron, Bikner-Ahsbahs and Dreyfus, ‘networking theories’ focus on two theoretical concepts: the need for a new construct, and interest. A new phenomenon emerges that enriches both views: General Epistemic Need, which appears as the driving force that makes students progress in their learning processes. Craig deals with the social context of research, and offers a way of exploring patterns of research collaboration within mathematics education research. Networking theories were also used to analyse mathematical classroom discourses. Ligozat, Wickman and Hamza look at classroom activities from institutional and participant perspectives, and combine two theories which focus on social

Advancing Research by Means of the Networking of Theories

Networking different theories is a rather new and promising way of doing research. This chapter presents the concept of the networking of different theories and its methodology, including networking strategies like research heuristics and cross-methodologies. The variety of networking is outlined by illustrating examples, and methodological reflections on the difficulties and benefits that accompany the networking are described.

Undergraduate engineering—a comparative study of first year performance in single gender campuses

The Jerusalem College of Technology is an institution for higher education in Israel, where the majority of the students study towards an undergraduate degree in Engineering (Electronics, Applied Optics, Computers, etc.). The studies are held on three different campuses, one campus for men and two for women. We describe the organization of the Foundation Year (i.e. the basic first year courses), and discuss the similarities and differences between the male and female populations, with respect to their learning skills and performances.

A Cross-Methodology for the Networking of Theories: The General Epistemic Need (GEN) as a New Concept at the Boundary of Two Theories

This example illustrates how research including the networking of two epistemic actions models from different theoretical perspectives is conducted and yields a new concept at the boundary of the two theoretical approaches. It illustrates a cross-methodology and the networking strategies described in the previous chapter of this part. The cross-methodology comprises five cross-over stages that systematically link the research process from the two perspectives in every methodical step and reveal an in-depth comprehension of the new concept from the two perspectives.

Starting Points for Dealing with the Diversity of Theories

This chapter presents the main ideas and constructs of the book and uses the triplet (system of principles, methodologies, set of paradigmatic questions) for describing the theories involved. In Part II (Chaps. 3, 4, 5, 6, and 7), the diversity of five theoretical approaches is presented; these approaches are compared and systematically put into a dialogue throughout the book. In Part III (Chaps. 9, 10, 11, and 12), four case studies of networking practices between these approaches show how this dialogue can take place. Chapter 8 and Part IV (Chaps. 13, 14, 15, 16, and 17) provide methodological discussions and reflections on the presented networking practices.

Introduction to Abstraction in Context (AiC)

The chapter briefly introduces the theoretical framework of Abstraction in Context (AiC) by referring to the data from Chap. 2. AiC provides a model of nested epistemic actions for investigating, at a micro-analytic level, learning processes which lead to new (to the learner) constructs (concepts, strategies, …). AiC posits three phases: the need for a new construct, the emergence of the new construct, and its consolidation.

Networking Different Theoretical Perspectives

During the last decade researchers in mathematics education have devoted efforts to understanding how theories can be connected successfully while respecting their underlying conceptual and methodological assumptions, a process called ‘networking theories’. Both authors of this chapter have had the privilege of collaborating with Michele Artigue and colleagues where we explored ways of handling the diversity of theories in mathematics education. In this chapter, we describe this collaboration and explain the reasons for networking theories as well as the expected difficulties of the networking process. We characterise different cases of networking and describe methodological reflections on the difficulties and benefits that accompany the networking. We refer to Michele Artigue’s contribution to the research on “networking theories” in general and in methodologies of the networking of theories in particular.

Constructing knowledge about the trigonometric functions and their geometric meaning on the unit circle

ABSTRACT Processes of knowledge construction are investigated. A learner is constructing knowledge about the trigonometric functions and their geometric meaning on the unit circle. The analysis is based on the dynamically nested epistemic action model for abstraction in context. Different tasks are offered to the learner. In his effort to perform the different tasks, he has the opportunity to understand the process used to create unit circle representations of trigonometric expressions. The theoretical framework of abstraction in context is used to analyse the evolution of the learners construction of knowledge in the transition from ‘triangle’ trigonometry to ‘circle’ trigonometry.