Esther Levenson

Using Theories to Build Kindergarten Teachers’ Mathematical Knowledge for Teaching

This chapter describes how a theory of Deborah Ball and her colleagues, embedded in the realm of teacher knowledge, was combined with a theory of David Tall and Shlomo Vinner, embedded in the realm of mathematics knowledge, to develop kindergarten teachers’ knowledge for teaching geometrical concepts. The chapter describes the separate theories and how they may be combined to build a more comprehensive and refined tool for building and evaluating mathematical knowledge for teaching. It also shows how kindergarten teachers used the combination of these theories to inform their practice. Finally, we discuss possibilities for the development and application of this combined-theories tool.

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.

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.

Engaging children with ABA patterns on a computer tablet: filling in the blanks

ABSTRACT Several studies have investigated children’s engagement with repeating pattern tasks, but few have related to patterns with ABA as the minimal unit of repeat. This study focuses on identifying intuitive characteristics of children’s work as they engage with repeating pattern tasks of different structures, focusing on the differences between ABA patterns and other patterns. The 11 children in this study are between the ages of four and seven years and are observed as they play with a repeating pattern application on computer tablet. A qualitative study of three children’s interactions with the application, including an analysis of gestures, found that completing an ABA pattern was non-intuitive.

Evaluating the potential of tasks to occasion mathematical creativity: definitions and measurements

ABSTRACT Creativity is often characterised by three components: fluency, flexibility, and originality. Specifically, in the mathematics classroom, in order to promote these aspects of creativity, educators recommend engaging students with multiple-solution tasks and open-ended tasks. In the past, various methods were used to measure fluency, flexibility and originality. This study raises questions and dilemmas regarding these methods and measures, and illustrates the complexity of these issues with data collected from fifth-grade students engaging with three such tasks. Some of the questions raised are related to differences between multiple-solution tasks and open-ended tasks, and between creative process and creative outcomes. In addition, the gap between the potential embedded within a task and its actual results is discussed.

Early Childhood Teachers’ Knowledge and Self-efficacy for Evaluating Solutions to Repeating Pattern Tasks

This study examines three aspects of early childhood teachers’ patterning knowledge: identifying features, errors and appropriate continuations of repeating patterns. Fifty-one practicing early childhood teachers’ self-efficacy is investigated in relation to performance on patterning tasks. Results indicated that teachers held high self-efficacy beliefs about solving patterning tasks correctly. Regarding performance, teachers were able to identify repeating patterns and errors in those patterns. However, when evaluating ways in which a repeating pattern may be continued, teachers found it more difficult to choose correct continuations for patterns that did not end with a complete unit of repeat than for those patterns that did. They tended to only choose continuations which would end the pattern with a complete unit of repeat. These results are discussed in light of findings from related previous studies.

Using Cases and Events in Teacher Education: Prospective Teachers’ Preferences

The use of cases as a pedagogical tool in teacher education is seen as one way of bringing practice closer to theory. This study describes the use of cases in a university course for secondary school prospective mathematics teachers. The study investigates participants’ views of cases taken from different sources and presented in different situations. In general, participants felt that the use of cases had an impact on their understanding of common mathematical errors but that cases based on mistakes they had themselves made during homework assignments were most meaningful.

Using Children’s Patterning Tasks During Professional Development for Preschool Teachers

Patterning activities in preschool are considered one way for enhancing young children’s appreciation for structure. Preschool teachers, however, are not always aware of the mathematics behind these activities. This paper describes one part of a professional development program that employs the use of tasks for children to promote preschool teachers’ knowledge for teaching patterns. Segments of the program reflect how the refined Cognitive Affective Mathematics Teacher Education framework helped to ensure that while engaging in pattern tasks for children, teachers enhanced their mathematics knowledge, knowledge of students, and knowledge of tasks.