Multiple-solution tasks: from a teacher education course to teacher practice

While development of a teacher’s expertise includes continuous incorporation of innovations throughout his/her career, teachers are often reluctant to adopt and implement new practices when challenged by innovative teaching approaches. This paper presents an analysis of the development of teachers’ expertise in relation to the implementation of novel (for them) instructional material. The study examines the ways in which teachers implement multiple-solution tasks (MSTs) (as an example of instructional tools new to the teacher) in their classes, following a professional development course in which they participated. The analysis focuses on the nature of MSTs implemented by the teachers and of the subsequent class discussion. The nature of MSTs is analyzed focusing on the goals with which MSTs were implemented, mathematical connections embedded in the MSTs, scaffolding provided to the learners and the learning settings. This analysis has led to the identification of four main implementation styles: straightforward, simple, adaptive and inventive. Concluding discussions are examined with respect to elevating and framing elements. Two lessons by mathematics teachers are described in the paper to explain how lessons were analyzed, and to exemplify adaptive and inventive implementation styles.     

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.     

The role of framing in productive classroom discussions: A case for teacher learning

In this paper, we contrast two mathematical arguments that occurred during an algebra lesson to illustrate the importance of relevant framings in the ensuing arguments. The lesson is taken from a graduate course for elementary teachers who are enrolled in a mathematics specialist program. We use constructs associated with enthnography of argumentation to characterize the framings for warrants and backings that supported the ensuing arguments. Our findings suggest that teachers fully participated in argumentations that were framed by problem situations that were familiar to them, ones that were couched in elementary, fundamental mathematical ideas, and that these types of argumentations were arguably more productive in terms of opportunities for learning.     

Improving teacher expertise through master teacher work stations: a case study

Professional development is emphasized in China not only for junior teachers, but also for experienced teachers. In this study, we investigated a master teacher??s work station that has been established at a provincial level for training experienced ??key?? teachers across the whole province. Three types of data were collected to examine the focuses and approaches used in this master teacher??s work station, teacher participants?? learning experiences and feedback, and selected teachers?? expertise improvement. The results show that the master teacher emphasized teachers?? development of a deeper understanding of elementary mathematics content and its structure through intensive studies of a textbook series. His training led to very positive feedback from teacher participants regarding what they have learned, especially in elementary mathematics content and content connections as presented in the textbooks. The detailed analysis of two teachers?? knowledge learning revealed dramatic changes in their thinking and instruction about specific content topics. Teacher expertise improvement through this master teacher??s work station is then summarized and discussed.     

Understanding and supporting teacher horizon knowledge around limits: a framework for evaluating textbooks for teachers

As part of recent scrutiny of teacher capacity, the question of teachers’ content knowledge of higher level mathematics emerges as important to the field of mathematics education. Elementary teachers in North America and some other countries tend to be subject generalists, yet it appears that some higher level mathematics background may be appropriate for teachers. An initial examination of a small sample of textbooks for teachers suggested the existence of a wide array of treatments and depth and quality of mathematics coverage. Based on the literature, a new framework was created to assess the mathematical quality of treatments for both specialized knowledge and horizon knowledge in mathematics textbooks for teachers. The framework was tested on a sample topic of the circle area formula derivation, chosen because it draws heavily on both specialized and horizon knowledge. The framework may contribute to similar analyses of other topics in a broader range of resources, in the overall quest to better describe the details of what constitutes appropriate mathematics horizon knowledge for teachers.     

Benefits of a teacher and coach collaboration: A case study

This paper describes a case study of a math teacher working with a math coach and the effects of their interaction. A guiding question was whether the coaching intervention had affected the teacher's classroom practices and, if so, in what way. The study utilized data from teacher/coach planning sessions, classroom lessons, follow-up debriefing sessions, and interviews with the teacher, coach and school principal. These data enabled the author to study the impact, if any, of the coaching on teacher beliefs and practices.     

Effectively coaching middle school teachers: A case for teacher and student learning

In this paper we report findings from a two-year, large-scale research project that describes the work of middle school mathematics specialists (also referred to as mathematics coaches or instructional coaches) who served in 10 school districts. We use mixed methods to describe how mathematics specialists spent their time supporting teachers and how these supports contributed to meaningful changes that teachers made in their instructional practices. We also report results that correlate student achievement scores with whether or not teachers were highly engaged with the mathematics specialists. We coordinate these quantitative results with findings from several case studies to illustrate the qualitatively different ways that mathematics specialists supported teachers’ ongoing work with their students. We also account for why some teachers participated more fully than others. Importantly, because mathematics specialists’ work was situated in different school settings with different demands, resources and administrative supports, these constraints and affordances contributed in part to how they could effectively support teachers’ work with their students.     

Mixing methodologies for information systems development and strategy: A higher education case study

Although informed approaches from an interpretivist perspective have been widely promoted over the last two decades information systems development is largely informed by a functionalist perspective, into which human-centred or interpretivist issues are incorporated as they are seen to be required. By treating such developments as critically informed social systems, this paper offers an alternative viewpoint. Firstly, a theoretical justification for such an approach is offered, drawing on developments in Critical Management Science, themselves based on a strand of Critical Social Theory. Secondly, an action research-based intervention is detailed, demonstrating the practical applicability of such an approach, an applicability that has hitherto been questioned. The outcome is a synthesised approach to operational and strategic information systems development, within a critical framework, which, it is argued, offers a greater potential for success than the currently promoted approaches.     

The development of mathematics textbooks: historical reflections from a personal perspective

The paper begins with a reflective, historical overview of the role and development of textbooks and research into their use. Since much of the writing is based on personal experience, there is an obvious UK bias, although experiences there were not dissimilar to what was happening in many other countries at that time. Particular attention is then paid to comparative studies: the difficulties they pose and what can be usefully gained from them. To help in this latter aspect, an attempt is made to distinguish what would appear to be key features of textbooks (or, indeed, other educational materials) which might be assessed in such a study and, in particular, the vital role of the exercises to be found in them.     

Traditional instruction in advanced mathematics courses: a case study of one professor’s lectures and proofs in an introductory real analysis course

It is widely accepted by mathematics educators and mathematicians that most proof-oriented university mathematics courses are taught in a “definition-theorem-proof” format. However, there are relatively few empirical studies on what takes place during this instruction, why this instruction is used, and how it affects students’ learning. In this paper, I investigate these issues by examining a case study of one professor using this type of instruction in an introductory real analysis course. I first describe the professor’s actions in the classroom and argue that these actions are the result of the professor’s beliefs about mathematics, students, and education, as well as his knowledge of the material being covered. I then illustrate how the professor’s teaching style influenced the way that his students attempted to learn the material. Finally, I discuss the implications that the reported data have on mathematics education research.     

Using comics-based representations of teaching, and technology, to bring practice to teacher education courses

This article situates comic-based representations of teaching in the long history of tensions between theory and practice in teacher education. The article argues that comics can be semiotic resources in learning to teach and suggests how information technologies can support experiences with comics in university mathematics methods courses that (a) help learners see the mathematical work of teaching in lessons they observe, (b) allow candidates to explore tactical decision-making in teaching, and (c) support preservice teachers in rehearsing classroom interactions.     

Maximal distance for robotic simulation: The convex case

Given two hyper-rectangles inE ⁿ with sides having surface normals in the directions of the axes, each containing a set that touches all 2n sides of its containing hyper-rectangle, it is important to have an easily calculated upper bound on the distance between the sets, for use in a branch and bound algorithm applicable in collision avoidance in robotic simulation. In a previous paper, such a bound was given under the hypothesis that the sets are connected. Here, we consider the case where the sets are convex.The work of the second author was supported in part by the Natural Sciences and Engineering Research Council of Canada. The drawings were prepared by K. Stewart.     

Estimating Iraqi deaths: a case study with implications for mathematics education

In this paper, I present an account of attempts to quantify deaths of Iraqis during the occupation by US and other forces since the invasion of March 2003, and of the reactions to these attempts. This story illuminates many aspects of current socio-political reality, particularly, but by no means exclusively, in the United States. Here, these aspects are selectively discussed in relation to the overarching themes of what the story illuminates about the uses of statistical information in society and about shortcomings in mathematics education.     

The conceptualisation of mathematics competencies in the international teacher education study TEDS-M

The main aim of the international teacher education study Teacher Education and Development Study in Mathematics (TEDS-M), carried out under the auspices of the International Association for the Evaluation of Educational Achievement (IEA), was to understand how national policies and institutional practices influence the outcomes of mathematics teacher education. This paper reports on the definition of effective mathematics teacher education in TEDS-M, distinguishing between mathematics content knowledge and mathematics pedagogical content knowledge as essential cognitive components of mathematics teachers’ professional competencies. These competence facets were implemented as proficiency tests based on extensive coordination and validation processes by experts from all participating countries. International acceptance of the tests was accomplished whereas, by necessity, national specifications had to be left out, as is common in comparative large-scale assessments. In this paper, the nature of the TEDS-M tests for the primary study is analysed and commented on detail. The aims are to increase our understanding of mathematics content knowledge and mathematics pedagogical content knowledge, which are still fuzzy domains, to provide a substantive background for interpretations of the test results and to examine whether some educational traditions may be more accurately reflected in the test items than others. For this purpose, several items that have been released by the IEA are presented and elaborately analysed in order to substantiate the test design of TEDS-M. Our main conclusion is that the overall validity of the TEDS-M tests can be regarded as a given, but that readers have to be aware of limitations, amongst others from a continental European point of view.     

Learning from the Eastern and the Western debate: the case of mathematics teacher education

Comparative studies have gained significant influence in the last decades, and school systems of many countries have been revised referring to better results of other countries in international large-scale assessments. Authors of such studies commonly link their interpretations of the results to distinctions between “Eastern” and “Western” cultures, in particular with respect to the consistent and continuing outstanding performance of East Asian learners compared with their Western counterparts. One question is whether the same achievement pattern holds for future teachers and whether similar cultural differences may cause it. International Association for the Evaluation of Educational Achievement’s “Teacher Education and Development Study in Mathematics” (TEDS-M) was the first comparative study that focused on the outcomes of teacher education with standardised testing. In this paper—based on the TEDS-M results—commonalities and differences in the achievement of future teachers from Eastern and Western countries are explored and related to a cultural perspective. Cultural differences between Eastern and Western approaches concerning mathematics, mathematics education and mathematics teachers are analysed with respect to the achievement pattern. The paper closes with reflections on possible consequences concerning the development of teachers’ knowledge and teachers’ expertise in mathematics education.     

Investigating textbooks as crucial interfaces between culture, policy and teacher curricular practice: two contrasted case studies in France and Norway

This paper reports on an investigation of mathematics curriculum documents, commonly used textbooks and teacher ‘curricular practice’ with respect to educational traditions in France and Norway. The study has helped to develop a deeper understanding of (1) educational traditions in France and Norway; (2) the ways the educational traditions permeate the system, from policy documents through textbooks into the classroom; and (3) the connections between the worlds of policy, textbooks and teacher curricular practice in mathematics. Results show that French and Norwegian curricular documents and practices were influenced by egalitarian values, albeit differently interpreted and ‘lived’ in each country. In terms of mathematics, using a focus on grade 6 geometric transformations has shown that French textbooks and teacher curricular practices emphasized theoretical properties and mathematical reasoning, whereas the Norwegian counterparts stressed practical and inquiry-based activities. It is argued that mathematics textbooks can be regarded as a crucial interface between culture, policy and curricular practice, and as a pivotal resource in teachers’ resource system for curricular practice, even in times of digitization.     

Math MOOC UniTo : an Italian project on MOOCs for mathematics teacher education,and the development of a new theoretical framework

This paper focuses on MOOCs (massive open online courses), a fairly recent paradigm in e-learning educational projects. Despite the high dropout rate, and     

Conceptualizing and organizing content for teaching and learning in selected Chinese, Japanese and US mathematics textbooks: the case of fraction division

In this study, selected Chinese, Japanese and US mathematics textbooks were examined in terms of their ways of conceptualizing and organizing content for the teaching and learning of fraction division. Three Chinese mathematics textbook series, three Japanese textbook series, and four US textbook series were selected and examined to locate the content instruction of fraction division. Textbook organization of fraction division and other content topics were described. Further analyses were then conducted to specify how the content topic of fraction division was conceptualized and introduced. Specific attention was also given to the textbooks’ uses of content constructs including examples, representations, and exercise problems in order to show their approaches for the teaching and learning of fraction division. The results provide a glimpse of the metaphors of mathematics teaching and learning that have been employed in Chinese, Japanese, and US textbooks. In particular, the results from the textbook analyses demonstrate how conceptual underpinnings were developed while targeting procedures and operations. Implications of the study are then discussed.     

Designing a national blended learning program for “out-of-field” mathematics teacher professional development

“Out-of-field” teaching refers to the practice of assigning secondary school teachers to teach subjects that do not match their training or edu