Roza Leikin

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.

Solution Spaces of Multiple-Solution Connecting Tasks as a Mirror of the Development of Mathematics Teachers’ Knowledge

This study aims to deepen our understanding of the development of teacher knowledge in systematic (through learning) and craft (through teaching) modes. The main research tools of this study were multiple-solution connecting tasks. We used the notion of solution spaces to analyze the data and demonstrated that modifications of the teachers’ solution spaces were situated in their practices of varying types. We found that the implementation of multiple-solution connecting tasks in systematic mode meaningfully developed teachers’ problem-solving performance due not only to the reproduction of solutions offered during the course but to the production of new solutions. Furthermore, in creating opportunities for students to solve tasks with multiple solutions, teachers expanded their personal solution spaces. We conclude that the combining of systematic and craft modes is an optimal condition for the development of teachers’ knowledge.RésuméCette étude a pour but d’approfondir notre compréhension du développement des connaissances chez les enseignants en matière de modes systématiques (grâce à l’apprentissage et à l’enseignement). Les principaux outils de recherche de cette étude sont les tâches de connexion à solutions multiples. Nous avons exploité la notion d’espaces de solution pour analyser les données, et nous avons démontré que les modifications des espaces de solutions proposées par les enseignants sont présentes dans différents types de pratiques. Nos résultats montrent que la réalisation systématique de tâches de connexion à solutions multiples améliore de façon significative la performance des enseignants en termes de solution de problèmes, non seulement grâce à la reproduction de solutions offertes durant la formation, mais également grâce à la production de solutions nouvelles. De plus, en créant pour les étudiants des occasions de résoudre certaines tâches au moyen de solutions multiples, les enseignants ont élargi leurs espaces de solutions personnelles. En conclusion, nous estimons que la combinaison des modes systématique et artisanal est une condition optimale pour le développement des connaissances chez les enseignants.

Learning through Teaching: A Case Study on the Development of a Mathematics Teacher's Proficiency in Managing an Inquiry-Based Classroom

This study examined the development of a mathematics teacher’s proficiency in managing whole-class discussion in the context of an inquiry-based classroom. We analysed three lessons taught with the same class by a teacher-researcher. The first and second lessons were 10 months apart, the second and third lessons were 6 months apart. For each of the three lessons the analysis was carried out at two levels: macro-level analysis was applied to the general organisation of the inquiry-based lesson and micro-level analysis was applied to both the teacher’s structure of the discussion and to the quality of the discussion. Based on the two-level analysis we formulate criteria to define teacher proficiency and demonstrate the enhancement of proficiency through teaching.

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 Balance of Teacher Knowledge: Mathematics and Pedagogy

Teacher education and the professional development of practicing teachers need to provide a sound basis of knowledge for teaching, theoretically but also with strong ties to issues of practice. Although this seems like a common-sense statement, it is harder to make a reality than expected. At least three factors could account for this difficulty: the sheer complexity of the knowledge required for teaching, the interconnectedness of knowledge, and the fact that teachers’ knowledge comes from different and in certain cases even contradictory sources. Consequently, (a) there is still a lack of comprehensive and categorical descriptions that frame teachers’ knowledge, particularly for content-oriented viewpoints, and (b) there is apparently no broad consensus about the status of that knowledge—is it private knowledge, based on personal experience and only in the personal realm of thinking and acting, or is it knowledge coming from and staying in practice, or is it discursively generated, shared, and general knowledge? In this chapter we describe aspects of the research on the relationship between teachers’ content knowledge and pedagogical practices from various perspectives to address the question, “Is there evidence for a systematic interdependent relationship of content and pedagogy?” Such evidence comes from different sources. One can measure both knowledge facets by the means of questionnaires, by directly observing the teaching practice, and by case studies of selected teachers. Learning about both knowledge facets can occur within the teaching practice as such but also from prospective and practicing teacher education. Moreover, teachers learn from practice—from within the teacher’s own practice and from the practice of others as well as from student oral discourse and written productions in their classes.

On the content-dependence of prospective teachers’ knowledge: a case of exemplifying definitions

In this article, we demonstrate that prospective teachers’ content knowledge related to defining mathematical concepts is dependent on content area. We use the example of generation (a research tool we developed in a previous study) to investigate prospective teachers’ knowledge. We asked prospective secondary mathematics teachers to provide multiple examples of definitions of concepts from different areas of mathematics. We examined teacher-generated examples of concept definition and analysed individual and collective example spaces, focusing on their correctness and richness. We demonstrate differences in prospective teachers’ knowledge associated with defining mathematical concepts in geometry, algebra and calculus.

Qualities of professional dialogue: connecting graduate research on teaching and the undergraduate teachers’ program

Undergraduate and graduate mathematics teacher education frameworks must ensure professional development of two groups of practitioners within the community of mathematics educators, namely pre-service and in-service mathematics teachers. The relationship between them serves as an example of connection theory and practice in mathematics teacher education. The theory – in the form of models of teachers’ interactions and teachers’ flexibility – emerges from careful analysis of in-service teacher dialogue with their students, colleagues, and mentors. It is implemented in practice with pre-service teachers in order to raise the quality of their discourse about teaching. This implementation sheds new light on the role and function of the proposed models.

Challenging Mathematics with Multiple Solution Tasks and Mathematical Investigations in Geometry

In this chapter, I analyze multiple solution tasks (MSTs) and mathematical investigations (MIs) and the interplay between them. I argue that MSTs and MIs are effective instructional tools for balancing the level of mathematical challenge in the mathematics classroom and, thus, for realizing students’ mathematical potential at different levels. Additionally, these tasks lead to the development of mathematical knowledge, mental flexibility, and critical thinking. They also deepen mathematical understanding since they promote the design of mathematical connections of different types. I present several examples of MSTs and MIs and analyze these mathematical tasks from the perspective of their conventionality, the mathematical connections embedded in the tasks, and their potential for developing learners’ mathematical creativity. MIs will be presented in this paper in connection to MSTs. Particular emphasis is placed on analyzing the relationships between production of multiple solutions, mathematical investigations, and varying levels of mathematical challenge.

Teachers’ Opportunities to Learn Mathematics Through Teaching

In this chapter we discuss theoretical and empirical grounds for teachers’ learning through teaching (LTT). We review several theories on teachers’ knowledge and the potential changes in this knowledge, and then focus on learning mathematics. We consider several specific examples of teachers’ learning in a variety of instructional situations and identify the types of learning that have occurred. We further identify the sources of LTT, the types of knowledge acquired through teaching, and consider factors that support teachers’ learning.

Teacher Development and Mathematical Challenge

In this chapter we look at the issues around teacher professional development as they relate to teaching using mathematical challenges. We look at what mathematics is; discuss why challenging mathematics problems are important in school classrooms; give some examples of problems that can provide a challenge in a classroom situation; and suggest some barriers that might inhibit the use of challenging problems. We then look at the mathematics education research that is relevant to the theme of the chapter. This is followed by effective pedagogy and teacher preparation, which includes both theoretical and practical aspects and suggestions for someone who may want to design a professional development project using challenging mathematics.