Orit Zaslavsky

Functions, Graphs, and Graphing: Tasks, Learning, and Teaching

This review of the introductory instructional substance of functions and graphs analyzes research on the interpretation and construction tasks associated with functions and some of their representations: algebraic, tabular, and graphical. The review also analyzes the nature of learning in terms of intuitions and misconceptions, and the plausible approaches to teaching through sequences, explanations, and examples. The topic is significant because of (a) the increased recognition of the organizing power of the concept of functions from middle school mathematics through more advanced topics in high school and college, and (b) the symbolic connections that represent potentials for increased understanding between graphical and algebraic worlds. This is a review of a specific part of the mathematics subject mailer and how it is learned and may be taught; this specificity reflects the issues raised by recent theoretical research concerning how specific context and content contribute to learning and meaning.

Inhibiting Factors in Generating Examples by Mathematics Teachers and Student Teachers: The Case of Binary Operation.

The main objectives of this study were to identify difficulties encountered by mathematics teachers and student teachers associated with the concept of binary operation regarding the associative and commutative properties and to reveal possible sources for them. Thirty-six in-service mathematics teachers and 67 preservice mathematics teachers participated in the study. All participants were presented with a task calling for the generation of a counterexample, namely, a binary operation that is commutative but not associative. Responses to the task were analyzed according to four categories: correctness, productiveness, mathematical content, and underlying difficulties. The findings point to similarities and differences between the two groups. Both groups exhibited a weak concept by failing to produce a correct example and by using a limited content search-space. These findings suggest two main inhibiting factors: one related to the overgeneralization of the properties of basic binary operations and the other related to pseudo-similarities attributed to these properties, which seem to be created by the recurring theme of order. Teachers were superior to student teachers on the categories of correctness and productiveness. This study is part of a larger study the aim of which was to investigate ways in which mathematics teachers and student teachers generate counterexamples in mathematics. Two main goals of the study were, first, to identify difficulties encountered in generating examples related to a number of mathematical topics that teachers and student teachers are expected to teach and, second, to reveal possible sources of difficulty for them. The possible sources of difficulty in generating such examples were presumed to include the following: incomplete knowledge, inability to process existing knowledge, misconceptions, and insufficient logical knowledge. The case of binary operation represents a case in which the relevant knowledge is assumed to be available to secondary mathematics teachers and student teachers; however, that knowledge requires processing in order to produce such examples. The current work is designed to probe for factors inhibiting the processing involved in generating counterexamples, factors that in turn may shed light on the limited binary operation concept. The search for these limiting factors was carried out by an extensive analysis of the processes and difficulties involved in generating the required examples. As indicated by Bratina (1986), the task of generating an example is considered powerful in terms of revealing strengths and weaknesses. The notion of a binary operation is dealt with in various stages and courses at different levels and contexts throughout the mathematics education that prospective mathematics teachers experience. However, in many cases prospective teachers do not manage to integrate their various encounters into one comprehensive, abstract

Being sloppy about slope: The effect of changing the scale

What is the slope of a(linear) function? Due to the ubiquitous use of mathematical software, this seemingly simple question is shown to lead to some subtle issues that are not usually addressed in the school curriculum. In particular, we present evidence that there exists much confusion regarding the connection between the algebraic and geometric aspects of slope, scale and angle. The confusion arises when some common but undeclared default assumptions, concerning the isomorphism between the algebraic and geometric systems, are undermined. The participants in the study were 11th-grade students, prospective and in-service secondary mathematics teachers, mathematics educators and mathematicians — a total of 124 people. All participants responded to a simple but non-standard task, concerning the behavior of slope under a non-homogeneous change of scale. Analysis of the responses reveals two main approaches, which we have termed ‘analytic’ and ‘visual’, as well as some combinations of the two.

The many facets of a definition: The case of periodicity

Abstract This paper was triggered by an authentic conversation between two mathematics teacher educators who debated whether a constant function is a periodic function, within the framework of a professional development program for secondary mathematics teachers. Their initial conversation led to deep mathematical and pedagogical musing surrounding mathematical definitions. In this paper, we present various aspects of a mathematical definition, including the role and nature of definitions in school mathematics, critical versus preferable features of a definition, and the arbitrariness underlying the choice of definition. We discuss the interplay between logical and pedagogical considerations with respect to definitions, drawing on the definition of a periodic function as an example.

Professional Development of Mathematics Educators: Trends and Tasks

In this chapter the professional growth of mathematics educators — including teachers, teacher educators, and educators of teacher educators — is presented as an ongoing lifelong process. So far as teachers are concerned, it occurs in various stages and contexts, beginning with their experiences as school students, followed by formal preservice preparation towards an academic qualification and teaching certificate. It continues in formal and informal inservice settings and, sometimes, in graduate studies. In this chapter we focus on trends in the thinking about, and practices within, inservice professional development programmes for mathematics educators — inservice teachers as well as inservice teacher educators. We begin by offering a unifying conceptual framework which takes into account interrelations between the different groups covered by the term ‘mathematics educators’. This framework, which acknowledges the central role of tasks and programmes in which participants engage, facilitates thinking about the complexities and underlying processes involved in professional development in mathematics education. We describe in detail the main types of programmes reported in the literature. Examples of tasks with special potential for enhancing the professional development of both teachers and teacher educators are discussed.

Learning through Teaching: The Case of Symmetry.

A study was carried out within the framework of an undergraduate course in teaching skills and strategies. An experimental part of the course was designed to provide an opportunity for the students to learn a mathematical topic through teaching it to eighth grade pupils in aLearning Through Teaching (LTT) environment. Symmetry was chosen as the focal mathematical topic for the experiment. This paper focuses on the development of the students’ understanding of line symmetry. The findings show that the implemented LTT environment served as a vehicle for the student teachers to learn mathematics, hi spite of the difficulties they encountered during the study, the students expressed positive dispositions towards symmetry and its role in mathematics.

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.

The Explanatory Power of Examples in Mathematics: Challenges for Teaching

I use the term ”instructional example,” to refer to an example offered by a teacher within the context of learning a particular topic. The important role of instructional examples in learning mathematics stems firstly from the central role that examples play in mathematics and mathematical thinking. Examples are an integral part of mathematics and a significant element of expert knowledge . In particular, examples are essential for generalization, abstraction, and analogical reasoning. Furthermore, from a teaching perspective, there are several pedagogical aspects of the use of instructional examples that highlight the significance and convey the complexity of this central element of teaching.

Example-Generation as Indicator and Catalyst of Mathematical and Pedagogical Understandings

This chapter examines the activities of example-generation and example-verification from both the teaching and learning perspectives. We closely examine how engaging learners in generating and verifying examples of a particular mathematical concept as a group activity serves both as an indicator of learners’ understandings and a catalyst for enhancing their understanding and expanding their example space that is associated with the particular concept. We present two cases that illustrate how the mathematics instruction may look when classroom activities and discussions build on example-generation and example-verification – the first case focuses on the concept of an irrational number and the second on the notion of a periodic function. The learners in these cases are in-service secondary mathematics teachers (MTLs), and the teacher is a mathematics teacher educator (MTE). We show how this kind of learning environment lends itself naturally to genuine opportunities for learners to engage in meaningful mathematics, to share and challenge their thinking, and to sense the need for unpacking mathematical subtleties regarding definitions and ideas. For practicing and prospective mathematics teachers, engaging in such activity and experiencing the potential learning opportunity that it offers is also likely to convince them to implement this approach in their classrooms.

Setting the Stage: A Conceptual Framework for Examining and Developing Tasks for Mathematics Teacher Education

This book is about tasks that teacher educators might use with prospective or practicing secondary mathematics teachers. There is a substantial literature that has established the critical role that tasks play in the teaching and learning process for school mathematics classes. Kilpatrick et al. (2001), for example, claim that the quality of teaching depends on whether teachers select cognitively demanding tasks, and whether these tasks unfold in the classroom in ways that allow the students to elaborate on the tasks and learn through those tasks. The basic argument is that it is through and around tasks that teachers and students communicate and learn mathematical ideas, so the tasks used by the teachers become the mediating tools. Christiansen and Walther (1986), drawing on the work of Leont’ev (1978), argued that the tasks set and the associated activity form the basis of the interaction between teaching and learning. Other authors who have similarly emphasized the critical role of tasks in creating learning opportunities for school students as well as the significant influences tasks have on what students actually learn include Stein and Lane (1996), Brousseau (1997), Hiebert and Wearne (1997), and Boaler (2002). This book is contributing to a related literature on the important role of tasks in teacher education.