Combining theories in research in mathematics teacher education

In this paper, we describe how the combination of two theories, each embedded in a different realm, may contribute to evaluating teachers’ knowledge. One is Shulman’s theory, embedded in general, teacher education, and the other is Fischbein’s theory, addressing learners’ mathematical conceptions and misconceptions. We first briefly describe each of the two theories and our suggestions for combining them, formulating the Shulman–Fischbein framework. Then, we present two research segments that illustrate the potential of the implementation of the Shulman–Fischbein framework to the study of mathematics teachers’ ways of thinking. We conclude with general comments on possible contributions of combining theories that were developed in mathematics education and in other domains to mathematics teacher education.     

Identifying authority structures in mathematics classroom discourse: a case of a teacher’s early experience in a new context

We explore a conceptual frame for analyzing mathematics classroom discourse to understand the way authority is at work. This case study of a teacher moving from a school where he is known to a new setting offers us the opportunity to explore the use of the conceptual frame as a tool for understanding how language practice and authority relate in a mathematics classroom. This case study illuminates the challenges of establishing disciplinary authority in a new context while also developing the students’ sense of authority within the discipline. To analyze the communication in the teacher’s grade 12 class in the first school and grade 9 class early in the year at the new school, we use the four categories of positioning drawn from our earlier analysis of pervasive language patterns in mathematics classrooms—personal authority, discourse as authority, discursive inevitability, and personal latitude.     

Being a mathematics teacher in times of reform

Within research on mathematics teachers and/or their professional development, the concept of identity emerges as a critique of views of how teaching practice is related to teachers’ ‘internal states’ of knowledge and beliefs. Identity relates teachers’ professional lives to teaching practices and to the contexts in which the teaching and/or professional development occurs. However, what might count as the context still needs in-depth discussion. In order to contribute to the development of a theoretical framework for understanding mathematics teachers’ professional lives, we will draw on one remarkable teacher’s identity as a primary mathematics teacher in relation to one political, sociocultural, and pedagogical context. We use this teacher’s experience to discuss how education policies that create what Ball (2003) called ‘terrors of performativity’ tend to impede the formation of a balanced teacher identity.     

Teaching and learning of mathematics: what really matters to teachers and students?

The learner’s perspective study, motivated by a strong belief that the characterization of the practices of mathematics classrooms must attend to learner practice with at least the same priority as that accorded to teacher practice, is a comprehensive study that adopts a complementary accounts methodology to negotiate meanings in classrooms. In Singapore, three mathematics teachers recognized for their locally defined ‘teaching competence’ participated in the study. The comprehensive sets of data from the three classrooms have been used to explore several premises related to the teaching and learning of mathematics. In this paper the student interview data and the teacher interview data were examined to ascertain what do students attach importance to and what do teachers attach importance to in a mathematics lesson? The findings of the student interview data showed that they attached importance to several sub-aspects of the three main aspects, i.e., exposition, seatwork and review and feedback of their teachers’ pedagogical practices. The findings of the teacher interview data showed that they attached importance to student’s self assessment, teacher’s demonstration of procedures, review of prior knowledge and close monitoring of their student’s progress in learning and detailed feedback of their work. It was also found that teachers and students did attach importance to some common lesson events.     

Mathematics teaching expertise development approaches and practices: similarities and differences between Western and Eastern countries

This paper comments on the other papers in this issue related to how “mathematics teaching expertise” is conceptualized and the approaches employed to facilitate its development in Western and Eastern countries. Similarities and differences are found to exist in the conceptualization of mathematics teaching expertise and the development approaches employed. The papers in this issue share the similarity of exploring mathematics teaching expertise from the perspective of knowledge. Under the influence of this perspective, the approaches mentioned in the papers mainly focus on the development of teachers’ knowledge. A feature in common among teacher development approaches employed in Western countries is to let teachers attend some courses or training programs designed or organized by mathematics teacher educators at universities. In contrast, teacher development approaches employed in Eastern countries, particularly those employed in Mainland China, are relatively more practical in nature and directly related to teachers’ needs, like learning from observing exemplary teaching. This shows that the conception of mathematics teaching expertise and development approaches are culturally and contextually dependent. It is argued that a broader perspective of mathematics teaching expertise should be taken to explore mathematics teaching expertise and its development, and teacher expertise development should be conceptualized as a complex system rather than as some separated knowledge, skills and techniques.     

Investigating the transformation of a secondary teacher’s knowledge of trigonometric functions

Shulman (1987) defined pedagogical content knowledge as the knowledge required to transform subject-matter knowledge into curricular material and pedagogical representations. This paper presents the results of an exploratory case study that examined a secondary teacher’s knowledge of sine and cosine values in both clinical and professional settings to discern the characteristics of mathematical schemes that facilitate their transformation into learning artifacts and experiences for students. My analysis revealed that the teacher’s knowledge of sine and cosine values consisted of uncoordinated quantitative and arithmetic schemes and that he was cognizant only of the behavioral proficiencies these schemes enable, not the mental actions and conceptual operations they entail. Based on these findings, I hypothesize that the extent to which a teacher is consciously aware of the mental activity that comprises their mathematical conceptions influences their capacity to transform their mathematical knowledge into curricular material and pedagogical representations to effectively support students’ conceptual learning.     

Aporism: Uncertainty about mathematics

Neither absolutism nor aposteriorism have questioned the progressive elements associated with the applications and the social functions of mathematical knowledge. Aporism raises this question by discussing the thesis of the formatting power of mathematics. This thesis unites linguistic relativism applied to mathematics and the idea that technology is a structuring principle in society. We are no longer surrounded by “nature”, instead we live in a techno-nature. Mathematical abstractions can be projected outside the sphere of mathematics, and in this way they modulate and eventually constitute fundamental categories of techno-nature. The Vico paradox expresses the difficulties of specifying the nature and function of technological actions. We are not even able to grasp and to understand what we have ourselves constructed. A critique cannot be guaranteed by scientific (or mathematical) thinking itself. Critique becomes a much more complex activity including reflections on technological actions. A crifique includes ethical considerations, and therefore a critique of mathematics is also ethical.     

Exploiting the potential of primary historical sources in primary school: a focus on teacher’s actions

Integrating history of mathematics in classes could be a hard task with young pupils. Indeed, original historical sources have a language that is far from the modern one. Such texts represent cultural artefacts that can give access to mathematical knowledge. The teacher can exploit such potential acting as a mediator between the mathematical signs of the source and those signs that are accessible to students. Through a case study, we investigate the role of the teacher in the process of semiotic mediation during a collective discussion. The analysed intervention is made of two phases: firstly, students work collaboratively and secondly, the teacher mediates a discussion aimed at institutionalizing the knowledge. During the discussion, working on a text from Tartaglia’s translation of Euclid’s Elements, a group of fifth graders constructs a definition of prime numbers. Referring to the Theory of Semiotic Mediation, we analyse the role of the teacher in building up semiotic chains linking students’ productions to an institutionalized knowledge emerging from the collective discussion. We highlight how teacher’s focalization on students’ words allows the progress of the discussion: the potential of the historical text is exploited fostering a definition that is close to culturally shared mathematics.     

A framework and rubric for guiding the training of mathematics tutors in third-level education

This paper uses a framework established by one of the author's in a previous study and refines it to develop a rubric for mathematics tutor training in third-level education. The paper further opens the discussion on the necessity of, and practical methods for, providing high quality, efficient tutor training at third-level. Models of mathematics teacher knowledge are unanimous in stating that solely possessing strong subject matter knowledge (SMK) is not sufficient for one to be deemed to be a good teacher. Despite the necessity of high quality tutoring at third-level, past research indicates that though tutors’ SMK is usually of high quality, their ability to transform their knowledge so that students will understand the content, as well as their general teaching skills (e.g. board work, speech), often has room to improve. This paper discusses the development of the rubric and its use in a mathematics tutor training programme at third-level. The tutor training programme was found to lead to improvements in multiple elements of the tutors’ teaching.     

A Teacher’s Conception of Definition and Use of Examples When Doing and Teaching Mathematics

To contribute to an understanding of the nature of teachers’ mathematical knowledge and its role in teaching, the case study reported in this article investigated a teacher’s conception of a metamathematical concept, definition, and her use of examples in doing and teaching mathematics. Using an enactivist perspective on mathematical knowledge, the authors give an account of the case of Lily, a prospective, then beginning, teacher who conceived of mathematical definition as an object with particular form and function and engaged in purposeful, specialized use of examples when doing and teaching mathematics. Lily’s case illustrates how a teacher’s interpretation of examples (as exemplifications or single instances) and conception of the form and function of definitions can influence her doing and teaching mathematics. An implication is that teacher preparation should foster teachers’ abilities to use examples purposefully to provide students with rich opportunities to engage in mathematical processes such as defining.     

Transforming the mathematical practices of learners and teachers through digital technology*

ABSTRACT

This article argues that mathematical knowledge, and its related pedagogy, is inextricably linked to the tools in which the knowledge is expressed. The focus is on digital tools and the different roles they play in shaping mathematical meanings and in transforming the mathematical practices of learners and teachers. Six categories of digital tool-use that distinguish their differing potential are presented: (1) dynamic and graphical tools; (2) tools that outsource processing power; (3) tools that offer new representational infrastructures for mathematics; (4) tools that help to bridge the gap between school mathematics and the students’ world; (5) tools that exploit high-bandwidth connectivity to support mathematics learning; and (6) tools that offer intelligent support for the teacher when their students engage in exploratory learning with digital technologies. Following exemplification of each category, the article ends with some reflections on the progress of research in this area and identifies some remaining challenges.     

Teachers' beliefs on teacher training contents and related characteristics of implementation—the example of introducing the topic study method in mathematics classrooms

So-called “bottom-up” strategies for implementation based on mathematics teachers' own developmental activities are considered to be a powerful approach when encouraging teachers to introduce alternative instructional practices. For evaluational research of in-service teacher training programs using “bottom-up” implementation strategies, the way how teachers implement contents of the teacher training is at the centre of interest. As the teachers' active role in the implementation process is necessary, their individual beliefs on the contents of the teacher training and their expectancies might influence the teachers' implementational activities. These beliefs can be considered as components of professional knowledge and pedagogical contents knowledge (Shulman, 1986) in particular. For this reason, the study focuses on the development of beliefs on contents of a teacher training program throughout the training on the one hand and relationships with characteristics of implementation on the other hand. We consider the example of introducing a student-centred learning environment, the so-called topic study method, in the teachers' classrooms. The results indicate that there are interdependencies between beliefs on the teacher training contents and characteristics of implementation.     

Reflections on the promise and complexity of mathematics coaching

If students are to develop mathematical proficiency, then mathematics teaching must both change and improve. In an effort to provide site-based professional development addressing the mathematical content and pedagogical demands that teachers encounter in reality of public schooling, many school districts are turning to elementary mathematics coaches. Knowledgeable coaches can have a significant positive impact on teachers, yet this study documents substantial variance in the amount of coaching delivered and in the nature of activity that coaches undertake within schools. Coaches are frequently responsive to the needs of individual teachers. If this support is primarily marked by shared teaching or provision of instructional materials, it may not transform either instruction or teacher knowledge. Similarly if coaches assume duties that primarily address an administrator’s needs, they will have less time to enhance a school’s mathematics program. Coaches need to engage teachers in fundamental dialogue about mathematical content, mathematical learning, and student understanding. It may be that this dialogue and the effectiveness of a coach’s work with individual teachers would benefit from a coach’s concurrent work with grade-level teams. When a coach leads a grade-level team through discussion of targeted goals and approaches, the coach may facilitate individual teacher learning while building collective learning. When coupled with the support of a principal, this partnership may foster instructional change across a school.     

Examining novice teacher leaders’ facilitation of mathematics professional development

This paper reports on novice teacher leaders’ efforts to enact mathematics PD through an analysis of their facilitation in workshops conducted at their schools. We consider the extent to which teacher leaders facilitated the Problem-Solving Cycle model of PD with integrity to its key characteristics. We examine the characteristics they enacted particularly well and those that were the most problematic to enact. Facilitators were generally successful with respect to workshop culture and selecting video clips for use in the PD workshops. They had more difficulty supporting discussions to foster aspects of mathematics teachers’ specialized content knowledge and pedagogical content knowledge. We suggest a number of activities that may help to better prepare novice PD leaders to hold effective workshops. Furthermore, we conjecture that leaders of mathematics PD draw from a construct we have labeled Mathematical Knowledge for Professional Development (MKPD), and we posit some domains that may comprise this construct.     

Research into teacher knowledge: a stimulus for development in mathematics teacher education practice

In this paper, we document some developments in teacher education practice at one university, brought about by reflection on research into mathematics teacher knowledge. The authors are three members of the Cambridge-based research team who developed the Knowledge Quartet (KQ), a theory of mathematics teacher knowledge, with a focus on classroom situations in which this knowledge is applied. At the same time as being researchers, the authors were elementary mathematics teacher education instructors. They found that the KQ research brought about new awareness of the importance of some components of mathematics didactics, as well as providing new tools for undertaking some aspects of their teacher educator role. The paper explores some of these awarenesses and tools in detail.     

Examining Secondary Mathematics Teachers' Opportunities to Develop Mathematically in Professional Learning Communities

To make progress toward ambitious and equitable goals for students’ mathematical development, teachers need opportunities to develop specialized ways of knowing mathematics such as mathematical knowledge for teaching (MKT) for their work with students in the classroom. Professional learning communities (PLCs) are a common model used to support focused teacher collaboration and, in turn, foster teacher development, instructional improvement, and student outcomes. However, there is a lack of specificity in what is known about teachers’ work in PLCs and what teachers can gain from those experiences, despite broad claims of their benefit. We discuss an investigation of the work of secondary mathematics teachers in PLCs at two high schools to describe and explicate possible opportunities for teachers to develop the mathematical knowledge needed for the work of teaching and the ways in which these opportunities may be pursued or hindered. The findings show that, without pointed focus on mathematical content, opportunities to develop MKT can be rare, even among mathematics teachers. Two detailed images of teacher discussion are shared to highlight these claims. This article contributes to the ongoing discussion about the affordances and limitations of PLCs for mathematics teachers, considerations for their use, and how they can be supported.     

How well prepared are the teachers of tomorrow? An examination of prospective mathematics teachers' pedagogical content knowledge

There is a growing emphasis in the teaching profession on pedagogical content knowledge (PCK) as an important knowledge component. The study reported in this article investigates Turkish prospective mathematics teachers’ mathematics teaching knowledge in the numbers content domain. A series of 10 open-ended scenario-type questions were adopted to challenge 83 prospective mathematics teachers’ knowledge of the learner and presentation of content in the context of PCK. The participants’ responses were analysed by means of rubrics and scoring guides developed by the researchers. The results showed that many of the future teachers performed well in determining what misconceptions students might express in the given scenarios. However, a majority of the participants performed poorly on presentation of content in terms of instructional strategies. In line with these results, the authors offer some suggestions for teacher training programmes.     

Good mathematics instruction in South Korea

There have been only a few studies of Korean mathematics instruction in international contexts. Given this, this paper describes in detail a sixth grade teacher’s mathematics instruction in order to investigate closely what may be counted as high-quality teaching and learning in Korea. This paper then discusses several key characteristics of good mathematics instruction along with some background information on Korean educational practice. This paper concludes with remarks that good mathematics instruction may be perceived differently with regard to underlying social and cultural norms.