Characterizing Teacher Knowledge Tests and Their Use in the Mathematics Education Literature

From a traditional perspective, induction and deduction have been discussed as key ways to generate new knowledge. On the other hand, Charles Sanders Peirce introduced the notion of abduction, distinct from deduction and induction, to avoid the pitfalls of empiricism and rationalism. Abduction is the process of forming an explanatory hypothesis on an observed surprising result (C.P. 5.171). Peirce emphasized that abductive reasoning is the only way of creating new ideas, and both epistemologists and educational researchers have attempted to tackle this notorious problem, the so-called learning paradox, from a Peircean perspective on knowledge generation (Prawat, 1999). With mathematics educators’ recent interests in semiotic approaches, there has been growing attention to the importance of investigating abductive reasoning in mathematics education research. Studies have attempted to clarify the forms and uses of abductive reasoning in students’ mathematical inquiries in order to identify how students generate new mathematical ideas. Researchers also consider that investigating students’ abductive reasoning may help to interpret and understand what occurs in mathematics classrooms. Thus, exploring abductive reasoning in mathematics education may provide a more helpful cornerstone in understanding how mathematics teaching and learning progresses. This special issue of EURASIA Journal of Mathematics, Science and Technology Education aims to share current and future issues on abductive reasoning in mathematics education. In inviting the contributions for this special issue, we intend to offer the reader, original elements of reflection from a wide range of issues on abductive reasoning in mathematics education. David Reid gives an overview of the discussion of abductive reasoning in mathematics education researches based on a meta-analysis of the state-of-the-art literature. He presents the origins of the concept of abductive reasoning and identifies the most significant approaches in mathematics education literature that refers to abductive reasoning. He then proposes a framework in which the different approaches taken in the research literature can be placed and compared. Michael Hoffmann presents very fundamental issues regarding abductive reasoning. Hoffmann tackles two crucial questions related to knowledge creation from a Peircean perspective: Can diagrammatic reasoning indeed be conceived as a foundation of abductive creativity? What is the relationship between abduction and diagrammatic reasoning? To answer these questions, he clarifies the Peircean concept of diagram and diagrammatic reasoning and analyzes the significance of a consistent system of representation for diagrammatic reasoning. He then examines how diagrammatic reasoning and creation of abduction are related. Three further contributions address how abduction is related to various contexts of mathematics learning. Bettina Pedemonte presents the role of abduction in the proving process of students solving a geometrical problem. She focuses on two types of rules in problem solving: strategic rules and definitory rules. She then compares two types of abductions that are related to these two rules, and analyzes the relationship between these two types of abductions and the deductive proof. Ferdinand Rivera examines elementary students’ pattern generalization. He focuses on identifying multiple abductive actions in their pattern generalization, and shows how multiple abductions can be coordinated and how this coordination is related to pattern generalization. He also analyzes the relationship between elementary children’s structural incipient generalizations and the natural emergence of their understanding of functions, especially the central role of abduction in such an understanding. JinHyeong Park and Kyeong-Hwa Lee investigate the abductive nature of mathematical modeling and the characteristics of mathematical inquiries triggered by mathematical modeling. They identify four characteristics of mathematical inquiries triggered by mathematical modeling based on an analysis of didactical and historical cases: abductive, recursive, analogical, and context-dependent. Michael Meyer presents various task-design options that can be used to support students in discovering mathematical properties that refer to abductive reasoning. These task-design options are identified using intense scrutiny of the processes of discovering and verifying mathematical properties from a Peircean perspective on learning and knowledge creation. He also presents concrete examples of task design and empirical findings on the implementation of tasks. From theoretical issues to more practical issues, these contributions in this special issue present state-of-the-art issues in abductive reasoning in the mathematics education research community. We hope that these offer an informative insight into the lively research on abductive reasoning in mathematics education.

Using literature in mathematics education is not a novel idea. Many books and articles have been written on how and why to introduce children's literature in elementary mathematics education over the past ten years, but little has been done at the middle school and high school levels.

Mathematics Curriculum Evaluation

Differentes recherches ont etabli l'influence des enseignants sur la qualite de l'apprentissage des enfants. La question est de savoir ce qui rend un enseignant plus performant qu'un autre. Quatre indicateurs, etudies par les AA., peuvent etre utilises dans ce but : les acquisitions pedagogiques des enseignants, leurs participations dans les programmes d'etudes, leurs connaissances des matieres enseignees et leurs strategies pedagogiques

Mathematics is the discipline that a significant majority of most incoming researchers in mathematics education have prior qualifications and experience in. Upon entry into the field of mathematics education research, these newcomers–often students on a postgraduate programme in mathematics education–need a broadened understanding on how to read, converse, write and conduct research in the largely unfamiliar territory of mathematics education. The intervention into the practices of post-graduate teaching and supervision in the field of mathematics education that I describe here aims at fostering this broadened understanding and thus facilitating newcomers’ participation in the practices of the mathematics education research community. Here I outline the theoretical underpinnings of the intervention and exemplify one of its parts (an Activity Set designed to facilitate incoming students’ engagement with the mathematics education research literature). I supplement the discussion of the intervention with comments sampled from student interview and student written evaluation data as well as observations of the activities’ implementation. The main themes touched upon include: learning how to identify appropriate mathematics education literature; reading increasingly more complex writings in mathematics education; coping with the complexity of literate mathematics education discourse; working towards a contextualised understanding of literate mathematics education discourse. I conclude with indicating the directions that the intervention, and its evaluation, is currently taking and a brief discussion of broader implications, theoretical as well as concerning the supervision and teaching of post-graduate students in mathematics education.

This chapter discusses research carried out by the Working Group Mathematics Education in Middle School (6th, 7th, 8th, and 9th grades), WG02, of the Brazilian Society of Mathematics Education, SBEM. The objective of WG02 is to collect, map, and present research on mathematics education in the middle school, to disseminate the results obtained to the community of mathematics teachers and educators, and to identify the main requirements concerning teaching and learning mathematics in these school grades. Amongst the investigations examined were studies addressing public policies as well as curricular requirements and methodological approaches associated with these school grades. Teachers’ thoughts and beliefs and the influence of their conceptual frameworks on the development of their work as teachers were also examined.

This article has two purposes: firstly to introduce this special issue on scaffolding and dialogic teaching in mathematics education and secondly to review the recent literature on these topics as well as the articles in this special issue. First we define and characterise scaffolding and dialogic teaching and provide a brief historical overview of the scaffolding metaphor. Then we present a review study of the recent scaffolding literature in mathematics education (2010–2015) based on 21 publications that fulfilled our criteria and 14 articles in this special issue that have scaffolding as a central focus. This is complemented with a brief review of the recent literature on dialogic teaching. We critically discuss some of the issues emerging from these reviews and provide some recommendations. We argue that scaffolding has the potential to be a useful integrative concept within mathematics education, especially when taking advantage of the insights from the dialogic teaching literature.

This book, Approaches to Qualitative Research in Mathematics Education: Examples of Methodology and Methods, edited by Angelika Bikner-Ahsbahs, Christine Knipping, and Norma Presmeg, is a timely and valuable addition to the research literature in mathematics education. Across 18 chapters written by more than 30 prominent researchers, the book offers a range of insight and detail that goes beyond what is normally possible within individual articles in mathematics education research journals. The book complements other efforts to capture the many advances in the mathematics education research field over recent years. By capturing the important distinction between ‘method’ (the ‘how’) and ‘methodology’ (the ‘why’) through facilitating the various authors in probing the underlying assumptions of their approaches, the book is to be strongly recommended. While it is not a straightforward read, the various sections of the book offer unmatched insight into the conduct of important research in mathematics education.

We are very pleased to write this introductory paper to the special issue of ZDM that provides readers with a glimpse of mathematics education in Korea. In July 2012 (7th–15th), the 12th International Congress on Mathematical Education (ICME-12) takes place at COEX in the center of Seoul, Korea. The ICME is the most important congress in the mathematics education community. It is held once every 4 years under the auspices of the International Commission of Mathematical Instruction (ICMI). The impetus for having this special issue during ICME-12 arose from the scarcity of literature on Korean mathematics education despite Korean students’ high achievement on international comparison studies. Korean students have consistently performed well in recent international assessments of mathematics achievement such as the Trends in International Mathematics and Science Study (TIMSS), the OECD Program for International Student Assessment (PISA), and the International Mathematics Olympiad. In addition, a recent study titled Mathematics Teaching in the 21st Century (MT21) compared prospective teachers’ knowledge and beliefs about teaching and learning in six participant countries, reporting that Korean prospective secondary mathematics teachers were better prepared than those in other countries (Schmidt et al., 2007; Schmidt, Blomeke, Tatto, 2011). Despite such outstanding performances of Korean students and prospective teachers, very little literature is available on factors that contribute toward it. In addition relatively little is also written about mathematics education in Korea. Many scholars, particularly in the West, are curious about Korea’s excellence in teacher quality and student achievement (e.g. Akiba, LeTendre, & Scribner, 2007; Kang & Hong, 2008). There are, however, a relatively small set of systematic studies that examine various aspects of Korean mathematics education substantively and carefully (Kwon, Park, & Park, 2011). Further, some studies on Asian education have even mistakenly assumed that mathematics education in Korea is almost the same as that in other East Asian countries (e.g., Leung, 2011). Considering teaching and learning as cultural practices (e.g., Alexander, 2000; Fernandez & Yoshida, 2004; Ma, 1999; Stigler & Hiebert, 1999), this could mislead readers to ignore the uniqueness of mathematics education in Korea. As a historical–cultural construct, educational practice of mathematics is situated in a specific cultural context of South Korean society, where there are a range of unique cognitive and normative assumptions around the functions and purposes of education, the kinds of knowledge pursued in school and society, the roles taken by teachers and students. Thus, this special issue not only overviews major aspects of Korean mathematics education, it also addresses the historical and cultural foundations of Korean mathematics education in order to foster an understanding of its practices. This issue, a thematic collection of studies, consists of nine papers that present and discuss emerging issues and challenges of Korean mathematics education with respect to various aspects of mathematics education such as curriculum, teaching practices, student learning, textbooks, assessment, and teacher education. It makes an important contribution by Korean mathematics educators to the international community. O. N. Kwon (&) S. J. Cho Department of Mathematics Education, College of Education, Seoul National University, Gwanak-ro 1, Gwanak-gu, Seoul 151-748, Korea e-mail: onkwon@snu.ac.kr

In this chapter, contrasting views on the contributions that teachers as researchers are making to the field of mathematics education are used as a springboard to examine developments over the past decade. On the one hand, there is a growing movement for more teachers to become involved in a critical exploration of their practice through such methods as critical reflection, action research, and lesson studies. The contrasting position makes the claim that these activities have done little to add to the body of knowledge on mathematics education. The chapter starts by scanning the literature in mathematics education and then takes a step back and explores the broad roots and issues of the teacher-researcher movement. A particular example of teachers studying for a higher degree in teaching provides an opportunity to explore the interface between the two contrasting positions. The chapter closes with an appeal to academics in mathematics education to seek ways to listen to and engage with the informed voices of teachers.

In this paper, we expand our prior work on mathematics education in contexts of language diversity by elaborating on the three perspectives on language described by Ruiz (NABE J 8(2):15–34, 1984): language-as-right, language-as-resource, and language-as-problem. We illustrate our arguments with data taken from research contexts in Catalonia-Spain and South Africa. In these two parts of the world, the language policy in education has long been an issue, with a monolingual orientation that values one language (i.e., Catalan in Catalonia and English in South Africa) over others. Throughout the introduction of specific examples of policy documents, classroom practices, and participants’ reports, our main point is that the right of using the students’ languages makes sense because it is itself more than an intrinsic human right; it is an option that potentially benefits the creation of mathematics learning opportunities. Especially for the instances of classroom practices, our examples can be considered as representative in that they point to a common situation in our data: despite the fact of the language of learning and teaching being fixed, there is room for the learners and the teacher to take or react to a decision on what language to use, with whom, and how in concrete moments of the interaction. However, on the basis of our studies and drawing on the literature in mathematics education and language diversity, we argue that language rights are not sufficiently connected to language as a pedagogical resource. The enactment of these rights is still contributing in many ways to the social and political construction of problems concerning the role of certain languages in classroom interaction. We conclude the paper by discussing some possibilities for framing language as a resource that provide effective support to all students’ learning of mathematics.

Identity research in mathematics education has become increasingly prominent over the past two decades. In the last few years, there have been several reviews of identity literature in the field of mathematics education generally, or specifically focused on mathematics learner identities or mathematics teacher identity. We begin our paper by summarizing the key findings of these reviews, pointing to various categorizations proposed by their authors, and the key critiques raised therein. We then report on our more recent review of identity literature in mathematics education published in the top twenty mathematics education journals over the past 5 years (47 articles in total). This review enables us firstly to speak descriptively about the extent to which the field is evolving (or not) in relation to earlier findings (dominant regions, focus of research, methods and perspectives used). Secondly, following deeper analysis of the papers, we speak to the way in which the field is addressing (or not) key critiques raised by reviews. Our analysis points to several theoretical, methodological and empirical absences and challenges requiring further engagement. We further highlight key absences in the research, particularly a lack of cross-national comparative studies and studies that connect learner and teacher identities. We then provide a summary discussion and analysis of the contributions of the present special issue papers and engage with how these take the field forward. Finally, we engage with the implications of our analysis and recommendations for future directions for research.

Mathematics education is arguably the most internationalised subject in higher education. This is reflected in the number of international conferences, exchanges and publications in the field. While a number of publications have addressed the internationalisation of mathematics education theory and research, the term globalisation is perhaps less familiar in the mathematics education literature. This chapter presents a theoretical construct towards the study of globalisation and internationalisation in mathematics education and identifies some key issues under each construct and that may require further discussion and research. These issues are illustrated by three case studies from Australia, East and Southeast Asia and Latin America. The chapter concludes by a discussion of global collaboration in mathematics education.

Although Science & Education has published papers specifically concerning mathematics over the years—some quite significant for education—mathematics has never received the spotlight. Indeed, when the journal was established it bore the subtitle, Contributions from History, Philosophy and Sociology of Science and Mathematics, but so few papers on mathematics were received in those years that the word ‘‘mathematics’’ was duly dropped. Yet within the community of science and mathematics educators, there is no lack of interest at least in the contributions of history and philosophy of mathematics to mathematics education, particularly the history of mathematics. Besides publications such as those we have listed at the end of this introduction, one can point to the activities of the International Study Group on the Relations between the History and Pedagogy of Mathematics (HPM) which dates back to 1972, and which almost from the start became an affiliated study group of the International Commission on Mathematical Instruction (ICMI), the largest and oldest international organization for mathematics education. International conferences of HPM have been held in Europe, Canada, Australia, Taiwan, Mexico, and Korea in association with the quadrennial International Congress on Mathematical Education (ICME). The European Summer University on History and Epistemology in Mathematics Education (ESU), now held every 4 years as well, is another regular venue for meetings dealing with history in mathematics education, as is the working group on History in Mathematics Education at the biennial Congress of European

In this chapter I raise a set of questions intended to make us reflect on our work as researchers, namely in the way we propagate and naturalise common assumptions or truths about mathematics education, as well as the mechanisms of power that makes it difficult for us to see beyond these well-accepted truths. I suggest that some of the forces that impact upon and restrict socially just outcomes for mathematics education are not just “external”, that is, originated outside the mathematics education community, but also, and perhaps more importantly for us, from the way research itself addresses the teaching and learning of mathematics in schools. Instead of positing ourselves as the beautiful souls of mathematics education, my invitation is for us to posit ourselves as part of the problem, and be willing to address some of our ideological assumptions before relegating to the social and political world the causes of our discontentment. For this purpose, I will rely on Foucault’s and Lacan’s works on the notion of truth, as a way to explore the role that contemporary mathematics education plays within capitalism.