Ruthi Barkai

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.

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.

Prospective Primary Teachers’ Beliefs Regarding the Roles of Explanations in the Classroom

This study classifies and discusses the views of 23 prospective primary teachers in Israel regarding the roles of explanations in the mathematics classroom, explanations given by teachers and those given by students. Results indicated that prospective teachers perceive explanations as playing various roles although greater emphasis is placed on building content knowledge than on developing a mathematical disposition. Results also hinted that perspectives of explanations may reflect on teachers’ beliefs regarding mathematics and their beliefs regarding the teaching and learning of mathematics.

Working with Parents to Promote Preschool Children’s Numeracy: Teachers’ Attitudes and Beliefs

This chapter describes the beliefs and attitudes of five preschool teachers towards involving families in promoting children’s numerical competencies, such as saying number words in a sequence to ten. The backgrounds of the children in each class, along with the teachers’ educational and social backgrounds, form the context of the study and are important variables when analysing the ways in which each teacher decides to involve families. In addition, the chapter describes various ways that teachers encouraged families to take part in their children’s mathematical growth (such as giving the children and parents mathematics homework) and to experience mathematics with their children (such as taking part in play with a mathematical theme). Dilemmas for preschool teacher educators are raised and discussed such as if and how teacher educators may act as mediators between preschool teachers and children’s families when promoting early mathematical growth.