Students encountering obstacles using a CAS

The paper describes a pilot study on the use of computer algebra at upper secondary level. A symbolic calculator was introduced in a pre-examination class studying for advanced pre-university mathematics. With the theoretical framework of Realistic Mathematics Education and Developmental Research as a background, the study focused on the identification of obstacles that students encountered while using computer algebra. Five obstacles were identified that have both a technical and a mathematical character. It is the author's belief that taking these barriers seriously is important in developing useful pedagogical strategies.This revised version was published online in September 2005 with corrections to the Cover Date.     

Paradox,programming, and learning probability: A case study in a connected mathematics framework

Formal methods abound in the teaching of probability and statistics. In the Connected Probability project, we explore ways for learners to develop their intuitive conceptions of core probabilistic concepts. This article presents a case study of a learner engaged with a probability paradox. Through engaging with this paradoxical problem, she develops stronger intuitions about notions of randomness and distribution and the connections between them. The case illustrates a Connected Mathematics approach: that primary obstacles to learning probability are conceptual and epistemological; that engagement with paradox can be a powerful means of motivating learners to overcome these obstacles; that overcoming these obstacles involves learners making mathematics—not learning a “received” mathematics and that, through programming computational models, learners can more powerfully express and refine their mathematical understandings.     

Exploring main obstacles of inflexibility in mathematics teachers’ behaviour in accepting new ideas: the case of equivalence between infinite sets

This article seeks to explore the most important obstacles of inflexibility in mathematics teachers’ behaviour in accepting new ideas, especially, while encountering a new situation. In so doing, the concept of equivalence between infinite sets, which is one of the challenging issues in mathematics teaching and learning, was selected as the subject matter of the study. The participants of the study were 64 high-school mathematics teachers. To collect the data, they were exposed to a challenging new situation involving equivalence between infinite sets. Then they were interviewed regarding the decisions they had made. The analysis of the data revealed that the main obstacles to accepting new ideas by mathematics teachers were preference to well-known procedures and tendency to work with methods with a more general extensive domain of the function. In addition, while the spirit of the article seems to imply that teachers are not flexible, in terms of accepting a new method, the data on teacher choices highlighted the positive results of teacher flexibility.     

Optimal regularity in the obstacle problem for Kolmogorov operators related to American Asian options

In this paper we prove optimal interior regularity for solutions to the obstacle problem for a class of second order differential operators of Kolmogorov type. We treat smooth obstacles as well as non-smooth obstacles. All our proofs follow the same line of thought and are based on blow-ups, compactness, barriers and arguments by contradiction. The problem considered arises in financial mathematics, when considering path-dependent derivative contracts with early exercise feature.     

一个声波散射区域的重建方法与唯一性定理

一个声波散射区域的重建方法与唯一性定理@王连堂...     

Students,mathematics, and technology: assessing the present – challenging the future

This paper is concerned with identifying and assessing the impacts of various technologically enriched approaches to mathematics learning. Its purpose is to address the usefulness of emerging knowledge to enhance practice, and to contribute to theorizing about technology-based learning. Hence the main drivers are intentions to identify and elaborate on obstacles, errors, and unresolved problems on the one hand, and positives and insights on the other. The paper tells a cautionary tale about expectations of technology-enhanced learning, while simultaneously uncovering a rich base from which to theorize and test new appreciations of what is involved when students, technology, and mathematics connect in learning settings. A summary assessment is that this area still very much represents work in progress, but there is now a heightened realization, at least among those not transfixed by technological blindness, that the search for ultimate answers will require much greater exploration of machine–mathematics–learner relationships.     

Modelling in Mathematics Classrooms: reflections on past developments and the future

This paper describes the development of mathematical modelling as an element in school mathematics curricula and assessments. After an account of what has been achieved over the last forty years, illustrated by the experiences of two mathematician-modellers who were involved, I discuss the implications for the future—for what remains to be done to enable modelling to make its essential contribution to the «functional mathematics», the mathematical literacy, of future citizens and professionals. What changes in curriculum are likely to be needed? What do we know about achieving these changes, and what more do we need to know? What resources will be needed? How far have they already been developed? How can mathematics teachers be enabled to handle this challenge which, scandalously, is new to most of them? These are the overall questions addressed. The lessons from past experience on the challenges of large-scale of implementation of profound changes, such as teaching modelling in school mathematics, are discussed. Though there are major obstacles still to overcome, the situation is encouraging.     

Implementing the Standards: Keys to Establishing Positive Professional Inertia in Preservice Mathematics Teachers

This paper summarizes and integrates the lessons learned from the last decade of professional development efforts based on the standards of the National Council of Teachers of Mathematics (1989 , 1991 , 1995 ). The fundamental challenges to such reform are identified, then the rest of the paper is dedicated to strategies that have been helpful in overcoming these obstacles. The challenges include both the teachers' views of mathematics and their image of teaching. The Immerse and Instill approach describes strategies that encourage teachers to implement standards‐based teaching upon entering the teaching field (Immerse) as well as instilling some of the professional habits necessary to keep teachers and their students actively engaged (Instill).     

Mathematics and Science Integration: Models and Characterizations

The squeeze on instructional time and other factors increasingly leads educators to consider mathematics and science integration in an effort to be more efficient and effective. Unfortunately, the need for common understandings for what it means to integrate these disciplines, as well as the need for improving disciplinary knowledge, appears to continue to be significant obstacles to an integrated approach to instruction. In this study we report the results of a survey containing six instructional scenarios administered to thirty-three middle grades science and math teachers. Analysis of teacher responses revealed that while teachers applied similar criteria in their reasoning, they did not possess common characterizations for integration. Furthermore, analysis suggested that content knowledge serves as a barrier to recognizing integrated examples. Implications for professional development planners include the need to develop and provide teachers with constructs and parameters for what constitutes mathematics and science integration. Continued emphasis on improving teacher content knowledge in both mathematics and science is also a prerequisite to enabling teachers to integrate content.     

Mathematics,religion, and Marxism in the Soviet Union in the 1930s

Mathematicians and traditional pure mathematics in the U.S.S.R. came under attack in the 1930s from the platforms of religion and nationalism in attempts to establish dialectical materialist mathematics. These platforms came to light at the 16th Congress of the All-Union Communist Party in 1930. A leader of the attacks was the mathematically inclined Marxist/Stalinist ideolog Ernst Kolman (1892–1979), in whose writings randomness and probability (more amenable to philosophical discourse than other areas of mathematics) are recurring issues. The recently rediscovered booklet Mathematics and Religion, written in 1933 under the influence of Kolman by the mathematician and political activist Mikhail Kh. Orlov (1900–1936), encompasses the Bolshevik position on mathematics and religion. A nucleus of vilification in Orlov's book for his religiously inclined pre-revolutionary writings is the mathematician Pavel Alekseevich Nekrasov (1853–1924), and by extension, the Moscow Mathematical Society, a focus of attack by Kolman and others. The attacks, and especially the probabilistic aspects are studied below under a sequence of headings: Mathematics and the 16th Party Congress; Moscow Mathematical Society; Nekrasov's Probability and Social Physics; Politically Correct Probability; Politically Correct Statistics?; Orlov, Religion, and Mathematics; Orlov and Kolman on Religion and Mathematics. There are biographical appendices on Kolman and Orlov. The outstanding mathematician/probabilists of the time, Andrei N. Kolmogorov (1903–1987) and Sergei N. Bernstein (1880–1968), were prominent among the obstacles to the “reforms” of Kolman (in Moscow) and Orlov (in Ukraine), respectively, and feature in this study.     

The influence of overcoming fixation in mathematics towards divergent thinking in open-ended mathematics problems on Japanese junior high school students

The purpose of this study is to find the influence of overcoming fixation in mathematical problem-solving towards divergent thinking in open-ended mathematics problems. This paper presents findings of investigation of Japanese junior high school students. Students who can overcome fixation in mathematics can contribute varied and original ideas in open-ended situations in mathematics.     

Olga Taussky,a torchbearer for mathematics and teacher of mathematicians

Olga Taussky-Todd's mathematical and personal life (1906-1995), her achievements and obstacles, her scientific reasoning and teaching all have served as inspiration to many mathematicians. We describe her role in the mathematics world of the previous century as a torchbearer for mathematics and mathematicians, bearing the “torch of scientific truth” that burns inside of mathematics and its applications. Besides her many deep math contributions – too many to elaborate – she excelled at distilling and presenting mathematical concepts and ideas in her work and gave us many visionary papers and math talks. By sharing her mathematical vision freely she has inspired many of us. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

数学建模对高中生构建数学底层思维的作用及其教学实施建议

数学底层思维即用数学的眼光观察世界、用数学的思维分析世界以及用数学的语言表达世界,是人们面对自然和社会中纷繁多样的现象和问题时,所展现的自发的、不依赖监督的、融汇数学学科核心素养的思维方式.作为国家高中新课程标准中数学六大核心素养之一的数学建模,是培养学生数学底层思维的良好载体,对人才培养和社会发展均起到良好的促进作用.本文主要阐述了数学建模对高中生构建数学底层思维的作用,并结合教学实例给出教学实施建议.     

Mathematics university teachers' perception of pedagogical content knowledge (PCK)

Teaching mathematics in university levels is one of the most important fields of research in the area of mathematics education. Nevertheless, there is little information about teaching knowledge of mathematics university teachers. Pedagogical content knowledge (PCK) provides a suitable framework to study knowledge of teachers. The purpose of this paper is to make explicit the perception of mathematics university teachers about PCK. For this purpose, a phenomenological study was done. Data resources included semi-structured interviews with 10 mathematics university teachers who were in different places of the mathematics university teaching experience spectrum. Data analysis indicated a model consisting of four cognitive themes which are mathematics syntactic knowledge, knowledge about mathematics curriculum planning, knowledge about students' mathematics learning and knowledge about creating an influential mathematics teaching–learning environment. Besides, it was found out that three contextual themes influenced on PCK for teaching mathematics in university levels which were the nature of mathematics subjects, university teachers' features and terms of learning environment.     

Eliciting and Developing Teachers' Conceptions of Random Processes in a Probability and Statistics Course

The purpose of this study is to examine prospective mathematics specialists' engagement in an instructional sequence designed to elicit and develop their understandings of random processes. The study was conducted with two different sections of a probability and statistics course for K-8 teachers. Thirty-two teachers participated. Video analyses within a collaborative course design were used to support a teaching experiment about teachers' conceptions of random processes. In particular, teachers were asked whether the outcomes of Rock-Paper-Scissors (RPS) are generated randomly or not, were presented with a definition for random selection, and were asked to come to a conclusion about RPS. Teachers struggled to reconcile the equality of winning outcomes for each player with the potential for human interference in the process of generating outcomes. Ultimately, teachers concluded the outcomes were not generated randomly, but encountered a variety of unexpected obstacles along the way.     

An analysis of history of mathematics research literature in Turkey: the mathematics education perspective1

Research in history of mathematics gained momentum in the past two decades in Turkey. The present paper aims to describe the patterns in the history of mathematics research in Turkey and to analyse the research in Turkey using a mathematics education framework. The qualitative paradigm and a case study design are used in the study. The obtained data were analysed by using the document analysis technique with the help of a content analysis. The study group which is comprised of twenty-two postgraduate theses at master's or doctoral level were purposefully selected from the higher education council postgraduate theses database. Findings indicate a dearth of research in the area and that most of the theses are done in the area of mathematics education. Moreover, the focus, in general, was on attitudinal variables, and cognitive aspects seemed to be ignored.     

关于理工科高等数学研究型教学与大学生创新意识培养研究的构想

由于高等数学是高校理工科一门重要的基础课,因此进行高等数学研究型教学与大学生创新意识培养研究是当今高校理工科专业高等数学教学中进行素质教育的必然要求.本文论述了其研究的目的、理论依据和意义,同时阐述了研究型教学与创新意识的培养问题研究的现状和趋势.最后提出了理工科高等数学研究型教学与大学生创新意识培养研究的实施方案和计划、实施细则、效益分析和研究环境.     

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.     

数学学术资源开放获取文献类型及获取途径研究

基于互联网,对开放获取数学学术资源类型与布局进行调查,提出开放获取数学学术资源的方法与途径,并对网络开放数学资源应用对策提出几条建议,目的是推动开放获取数学资源在数学研究领域的普遍使用与有效传播.     

Textbook research in mathematics education: development status and directions

This paper presents a survey study aiming to systematically examine, analyse and review relevant research focusing on mathematics textbooks and hence identify future directions in this field of research. The literature surveyed is selected from different data sources, including mainly journal articles, research theses and conference proceedings. The survey revealed that important progress has been made over the last few decades in mathematics textbook research, though the major achievement has been concentrated in the areas of textbook analysis (including textbook comparison), and the use of textbooks in teaching and learning. It is overall no longer true that the textbook research in mathematics is “scattered, inconclusive, and often trivial” as described six decades ago; however, the development of research on mathematics textbooks has been unbalanced in different areas. Following the review and discussion, the paper proposes five needed directions for advancing the research in this field.