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.     

Gender and Mathematics: recent development from a Swedish perspective

A fairly large study of attitudes towards mathematics among Swedish students at secondary level was conducted during 2001–2004. A newly developed instrument was used that was designed to capture gender stereotyped attitudes among students related to various aspects of mathematics in education and future life. The new scale and its development are described with reference to the original Australian studies. The scale builds on the Fennema–Sherman attitude scale “Mathematics as a male domain” but allows mathematics to be viewed as female, male or gender neutral, reflecting a different societal and educational situation than in the seventies when attitudes towards mathematics as a male domain were first investigated. The Swedish study shows that mathematics is perceived as gendered, mostly as a male domain, by large minorities of students at secondary level. However, the results are complex, with clear differences in responses from female and male students. Furthermore, mathematics is also viewed as female in some aspects. A comparison with Australian data shows that Swedish students are less inclined to view mathematics as a female domain than Australian students of the same age. The relevance of the study is related to the lack of equity in mathematics in education and as a professional field in the Swedish society, documented by earlier research.     

Mathematics professional development as design for boundary encounters

This theoretical paper examines a process for researchers and teachers to exchange knowledge. We use the concepts of communities of practice, boundary encounters, and boundary objects to conceptualize this process within mathematics professional development (MPD). We also use the ideas from design research to discuss how mathematics professional development researchers can make professional development the focus of their research. In particular, we examine the question: How can MPD be conceptualized and designed around research-based knowledge in ways that promote knowledge exchange about students’ mathematics and mathematics learning among researchers and teachers to improve the practices of both the research and the teaching communities? We propose that MPD is a premier space for researchers and teachers to exchange knowledge from their communities, impacting both researchers’ and teachers’ practices without reducing the importance of either.     

Mathematics for engineers and scientists: some teaching approaches

The paper discusses various approaches that are being adopted at Edinburgh University for the teaching of mathematics to students studying engineering or science. The teaching organization and the role of lectures, practical and tutorial sessions are considered. Also highlighted is the use of support material such as tape‐booklet sequences, video and various experimental schemes.     

Mathematics teaching before and after the Meiji Restoration

This paper outlines mathematical education before the Meiji Restoration, and how it changed as a result. The Meiji Restoration in 1868 completely changed the social structure of Japan. In the Edo period (1600?C1868) Japan was divided into domains (han) governed by local lords (daimyo). Tokugawa Shogunate supervised local lords and governed Japan indirectly. In the Edo period there were no wars for more than two centuries and many people participated in cultural activities. Japanese mathematics developed in its own way under the influence of old Chinese mathematics. Japan also had a good education system so that the literacy rate was quite high. Each domain had its own school for samurai but mainly education was provided privately. Private schools for elementary education were called terakoya, in which mainly reading and writing and often arithmetic by the soroban (Japanese abacus) were taught. In the Edo period the soroban (abacus) was the only tool for computation and Arabic numerals were not used. The Meiji government was eager to establish a modern centralized state in which education played a key role. In 1872 the Ministry of Education declared the Education Order, whereby in elementary schools only western mathematics should be taught and the soroban should not be used. But almost all teachers only knew Japanese traditional mathematics ??wasan?? so they insisted on using the soroban. This was the starting point of a long dispute on the soroban in elementary education in Japan. Two Japanese mathematicians, KIKUCHI Dairoku and FUJISAWA Rikitaro, played a central role in the modernization of mathematical education in Japan. KIKUCHI studied mathematics in England and brought back English synthetic geometry to Japan. FUJISAWA was a student of KIKUCHI at the Imperial University and studied mathematics in Germany. He was the first Japanese mathematician to make a contribution to original research in the modern sense. He published a book on mathematical education in elementary school, which built the foundation of mathematical education in Japan.     

Mathematics Education as a Design Science

We propose re-conceptualizing the field of mathematics education research as that of a design science akin to engineering and other emerging interdisciplinary fields which involve the interaction of “subjects”, conceptual systems and technology influenced by social constraints and affordances. Numerous examples from the history and philosophy of science and mathematics and ongoing findings of M&M research are drawn to illustrate our notion of mathematics education research as a design sicence. Our ideas are intended as a framework and do not constitute a, “grand” theory (see Lester. 2005, this issue). That is, we provide a framework (a system of thinking together with accompanying concepts, language, methodologies, tools, and so on) that provides structure to help mathematics education researchers develop both models and theories, which encourage diversity and emphasize Darwinian processes such as: (a) selection (rigorous testing), (b) communication (so that productive ways of thinking spread throughout relevant communities), and (c) accumulation (so that productive ways of thinking are not lost and get integrated into future developments).     

Justification as a teaching and learning practice: Its (potential) multifacted role in middle grades mathematics classrooms

Justification is a core mathematics practice. Although the purposes of justification in the mathematician community have been studied extensively, we know relatively little about its role in K-12 classrooms. This paper documents the range of purposes identified by 12 middle grades teachers who were working actively to incorporate justification into their classrooms and compares this set of purposes with those documented in the research mathematician community. Results indicate that the teachers viewed justification as a powerful practice to accomplish a range of valued classroom teaching and learning functions. Some of these purposes overlapped with the purposes in the mathematician community; others were unique to the classroom community. Perhaps surprisingly, absent was the role of justification in verifying mathematical results. An analysis of the relationship between the purposes documented in the mathematics classroom community and the research mathematician community highlights how these differences may reflect the distinct goals and professional activities of the two communities. Implications for mathematics education and teacher development are discussed.     

Mathematics and power: an alliance in the foundations of mathematics and its teaching

Following a genealogic approach, this paper discusses how logic and calculation are linked to epistemology, spirituality and politics; how mathematics education can be understood as an institution for a mathematical enculturation; and how, therefore, mathematics education necessarily (re)produces techniques of power which privilege some children while disadvantaging others. This approach criticises other critical studies on social dimensions of mathematics education which argue that the social dimensions are to be found in the application or teaching of mathematics alone. Instead, mathematics itself has, since its very beginning, been a knowledge which allows power, represents a specific world view and serves the interests of certain groups in society.     

Capillary breakup of liquid threads: a singularity-free solution

The process of capillary breakup of a thread of Newtonian liquidis considered theoretically in the simplest case where the threadis surrounded by an inviscid, dynamically passive gas. The goalis to remove the singularities inherent in the known solutionsto the problem obtained in the framework of the standard modeland explain some puzzling qualitative features of the processobserved in experiments. The analysis is based on the idea that,since the known solutions indicate that the rate at which freshfree-surface area is created tends to infinity as breakup isapproached, one has that the surface tension, whose relaxationto equilibrium is always associated with a small but finiterelaxation time, is bound to deviate from its equilibrium valuein the process of breakup. This gives rise to a surface-tensiongradient which starts to pull the liquid thread apart (the flow-inducedMarangoni effect), whilst the role of the capillary pressure-drivensqueezing of the liquid out of the neck diminishes as the surfacetension in the minimal cross-section decreases. An earlier developedtheory incorporating the interface formation process is appliedwithout any ad hoc alterations and analysed in the frameworkof the slender-jet approximation. The resulting solution issingularity-free and allows one to describe some previouslyunexplained features of experiment by Kowalewski (1996, FluidDyn. Res., 17, 121).     

Commentary to Lesh and Sriraman: Mathematics education as a design science

In this commentary, the author discusses the strengths and weaknesses to Lesh & Sriraman's (2005) ambitious proposal of re-conceptualizing the field of mathematics education research as that of a design science.     

Numeracy task design: a case of changing mathematics teaching practice

Over the last 15 years, numeracy has become more and more prominent in curriculum initiatives around the world. Yet, the notion of numeracy is still not well defined, and as such, often not well understood by the teachers who are charged with the responsibility of helping our students to develop their numeracy skills. In this article I explore the work of a team of mathematics teachers brought together for the purpose of developing a set of numeracy tasks for use within district wide numeracy assessments. Results indicate that these teachers’ experience designing these tasks, and pilot testing them in their own classrooms, propelled them to make massive changes in their own mathematics teaching practice. Through a lens of Rapid and Profound Change (Journal of Mathematics Teacher Education 13:411–423, 2010) the mechanism and catalyst behind these changes are revealed.