Michal Tabach

The Influence and Shaping of Digital Technologies on the Learning – and Learning Trajectories – of Mathematical Concepts

The significant development and use of digital technologies has opened up diverse routes for learners to construct and comprehend mathematical knowledge and to solve problems. This implies a revision of the pedagogical landscape in terms of the ways in which students engage in learning, and how understandings emerge. In this chapter we consider how the availability of digital technologies has allowed intended learning trajectories to be structured in particular forms and how these, coupled with the affordances of engaging mathematical tasks through digital pedagogical media, might shape the actual learning trajectories. The evolution of hypothetical learning trajectories is examined, while the transitions learners make when traversing these pathways are also considered. Particular instances are illustrated with examples in several settings.

A Mathematics Teacher's Practice in a Technological Environment: A Case Study Analysis Using Two Complementary Theories

Integrating technology in school mathematics has become more and more common. The teacher is a key person in integrating technology into everyday practice. To understand teacher practice in a technological environment, this study proposes using two theoretical perspectives: the theory of technological pedagogical content knowledge to analyze teachers’ knowledge, and instrumental orchestration to analyze teachers’ actions. Applying this dual perspective to one teacher’s practice can shed light on the complexities faced by a teacher who integrates technology in her practice.

Principles of Task Design for Conjecturing and Proving

Principles of task design should have both the fundamental function of a clear relation to the learner’s rules, learning powers or hypothetical learning trajectories and the practical function of easy evaluation of many similar tasks. Drawing on some theories and practical tasks in the literature, we developed a total of 11 principles of task design for learning mathematical conjecturing (4), transiting between conjecturing and proving (2), and proving (5). To further validate the functioning of those principles, more empirical research is encouraged.

Understanding Equivalence of Symbolic Expressions in a Spreadsheet-Based Environment

Use of spreadsheets in a beginning algebra course was investigated mainly with regard to their potential to promote generalization of patterns. Less is known about their use in promoting understanding and learning of transformational activities. The overall purpose of this paper is to consider the conceptual aspects of learning a transformational skill (use of the distributive law to produce equivalent algebraic expressions) in a learning sequence composed of both spreadsheets and paper-and-pencil activities. We conducted a sequence of classroom activities in several classes, and analyzed the students’ work on a spreadsheet activity and on an assessment activity by both qualitative and quantitative methods. The findings indicate both encouraging benefits and some potential sources of difficulties caused by the use of spreadsheets at initial stages of learning symbolic transformations.

Employing the CAMTE Framework: Focusing on Preschool Teachers’ Knowledge and Self-efficacy Related to Students’ Conceptions

This chapter presents the Cognitive Affective Mathematics Teacher Education (CAMTE) framework, a framework used in planning and implementing professional development for teachers. The CAMTE framework takes into consideration teachers’ knowledge as well as self-efficacy beliefs to teach mathematics. The context of counting and enumeration is used to illustrate how the framework can be used to investigate preschool teachers’ knowledge and self-efficacy related to children’s conceptions. Different aspects of teachers’ knowledge, such as knowledge of students and knowledge of tasks are discussed. Ways of promoting teachers’ self-efficacy are also presented. Finally, the case of one preschool teacher is described in detail, showing how the teacher began to adopt a constructivist approach to instruction.

Preschool Teachers’ Knowledge and Self-Efficacy Needed for Teaching Geometry: Are They Related?

This chapter focuses on methodological issues related to investigating preschool teachers’ self-efficacy for teaching geometry. The first issue discussed is the specificity, as opposed to the generality, of self-efficacy and the need to design instruments which are sensitive to this aspect of self-efficacy. Specificity may be related to content, in this case geometry and the specific figures under investigation. In other words, self-efficacy for teaching triangles may differ from self-efficacy for teaching pentagons. Self-efficacy may also be related to the specific action being performed, such as designing tasks for promoting knowledge versus designing tasks for evaluating knowledge. The chapter also investigates the relationship between preschool teachers’ knowledge and self-efficacy for identifying geometrical figures, presenting a method for studying this relationship but also raising questions related to this method.

The Camte Framework

This chapter is concerned with developing teachers’ knowledge for teaching mathematics in preschool. Like Alan Schoenfeld, we are concerned with teachers, in this case preschool teachers, knowing school mathematics in depth and in breadth. Like Gunter Torner, one of the founders of theMAVI (Mathematical Views) conference, we are concerned with the affective side of teacher education. The framework we present in this chapter combines both cognitive and affective aspects related to facilitating proficient mathematics teaching in preschool.

Instances of Promoting Creativity with Procedural Tasks

The learning of algebraic procedures in middle-school algebra is usually perceived as an algorithmic activity, achieved by performing sequences of short drill-and-practice tasks, which have little to do with conceptual learning or with creative mathematical thinking. The goal of this chapter is to explore possible ways by which all middle-grade students can be encouraged to apply higher-order thinking in the context of tasks that integrate procedural work, conceptual understanding and creative thinking. Each of the five instances presented in this chapter was intended to promote creative thinking in the context of procedural tasks. An a-priori task analysis and data collected in some of our previous studies indicate the presence of many learning competencies and high levels of mathematical creativity in the participating students’ work. Thus, we conclude that certain procedural tasks have a strong potential to promote higher-order, and creative thinking.

An organizer of mathematical statements for teachers: the six-cell matrix

Policy documents and researchers agree that proofs and proving should become common mathematical practice in school mathematics. Towards this end, teachers are encouraged to implement proving activities in their classrooms. This article suggests a tool that may help teachers to integrate proofs and proving in their practice – the six-cell matrix. In addition to presenting the tool, we demonstrate how the tool may be used by teachers in four phases of their practice: planning a sequence of lessons, during a lesson, as an assessment tool, and as a way of making sense of learning materials.

Uses of Technology in K–12 Mathematics Education: Concluding Remarks

The aim of this closing chapter is to reflect on the content of this book and on its overall focus on the development of mathematical proficiencies through the design and use of digital technology and of teaching and learning with and through these tools. As such, rather than making an attempt to provide an overview of the field as a whole, or trying to define overarching theoretical approaches, we chose to follow a bottom-up approach in which the chapters in this monograph form the point of departure. To do so, we reflect on the book’s content from four different perspectives. First, we describe a taxonomy of the use of digital tools in mathematics education, and set up an inventory of the different book chapters in terms of these types of educational use. Second, we address the learning of mathematics with and through technology. Third, the way in which the assessment of mathematics with and through digital technology is present in this monograph is reflected upon. Fourth, the topic of teachers teaching with technology is briefly addressed. We conclude with some final reflections, including suggestions for a future research agenda.