Proofs and hypotheses

On the basis of an analysis of common features and differences between general statements in every day situations, in physics and in mathematics the paper proposes a didactical approach to proof. It is centred around the idea that inventing hypotheses and testing their consequences is more productive for the understanding of the epistemological nature of proof than forming elaborate chains of deductions. Inventing hypotheses is important within and outside of mathematics. In this approach proving and forming models get in close contact. The idea is exemplified by a teaching unit on the angle sum theorem in Euclidean geometry.     

Key Developmental Understandings in Mathematics: A Direction for Investigating and Establishing Learning Goals

Although mathematics educators seem to agree on the importance of teaching mathematics for understanding, what they mean by understanding varies greatly. In this article, I elaborate and exemplify the construct of key developmental understanding to emphasize a particular aspect of teaching for understanding and to offer a construct that could be used to frame the identification of conceptual learning goals in mathematics. The key developmental understanding construct is based on extant empirical and theoretical work. The construct can be used in the context of research and curriculum development. Using a classroom example involving fractions, I illustrate how focusing on key developmental understandings leads to particular, potentially useful types of pedagogical thinking and directions for inquiry.     

Key Developmental Understandings in Mathematics: A Direction for Investigating and Establishing Learning Goals

Although mathematics educators seem to agree on the importance of teaching mathematics for understanding, what they mean by understanding varies greatly. In this article, I elaborate and exemplify the construct of key developmental understanding to emphasize a particular aspect of teaching for understanding and to offer a construct that could be used to frame the identification of conceptual learning goals in mathematics. The key developmental understanding construct is based on extant empirical and theoretical work. The construct can be used in the context of research and curriculum development. Using a classroom example involving fractions, I illustrate how focusing on key developmental understandings leads to particular, potentially useful types of pedagogical thinking and directions for inquiry.     

Trends in researching the socioeconomic influences on mathematical achievement

We introduce the topic of socioeconomic influences on mathematical achievement through an overview of existing research reports and articles. International trends in the way the topic has emerged and become increasingly important in the international field of mathematics education research are outlined. From this review, there is a discussion about what appears to be neglected in previous work in this area and how the papers in this issue of ZDM provide information about some of these neglected areas. The main argument in this article is that socioeconomic influences on mathematical achievement should not be considered as a taken-for-granted fact that is accepted uncritically. Instead, it is suggested that the relationship between multiple socioeconomic influences and various understandings of mathematical achievement are historically contingent ways of understanding exclusions and inclusions in mathematics education practices. Research is not simply “evidencing” the facts of these relationships; research is also implicated in constructing the ways in which we think about these. Thus, mathematics education researchers could devise more nuanced approaches for understanding the social, political and historical constitution of these relationships.     

The role of the researcher’s epistemology in mathematics education: an essay on the case of proof

Is there a shared meaning of “mathematical proof” among researchers in mathematics education? Almost all researchers may agree on a formal definition of mathematical proof. But beyond this minimal agreement, what is the state of our field? After three decades of activity in this area, being familiar with the most influential pieces of work, I realize that the sharing of keywords hides important differences in the understanding. These differences could be obstacles to scientific progress in this area, if they are not made explicit and addressed as such. In this essay I take a sample of research projects which have impacted the teaching and learning of mathematical proof, in order to describe where the gaps are. Then I suggest a possible scientific programme which aspires to strengthen the research practice in this domain. Eventually, I make the additional claim that this programme could hold for other areas of research in mathematics education.     

Language and communication in mathematics education: an overview of research in the field

Within the field of mathematics education, the central role language plays in the learning, teaching, and doing of mathematics is increasingly recognised, but there is not agreement about what this role (or these roles) might be or even about what the term ‘language’ itself encompasses. In this issue of ZDM, we have compiled a collection of scholarship on language in mathematics education research, representing a range of approaches to the topic. In this survey paper, we outline a categorisation of ways of conceiving of language and its relevance to mathematics education, the theoretical resources drawn upon to systematise these conceptions, and the methodological approaches employed by researchers. We identify four broad areas of concern in mathematics education that are addressed by language-oriented research: analysis of the development of students’ mathematical knowledge; understanding the shaping of mathematical activity; understanding processes of teaching and learning in relation to other social interactions; and multilingual contexts. A further area of concern that has not yet received substantial attention within mathematics education research is the development of the linguistic competencies and knowledge required for participation in mathematical practices. We also discuss methodological issues raised by the dominance of English within the international research community and suggest some implications for researchers, editors and publishers.     

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.     

A capital allocation based on a solvency exchange option

In this paper we propose a new capital allocation method based on an idea of [Sherris, M., 2006. Solvency, capital allocation and fair rate of return in insurance. J. Risk Insurance 73 (1), 71-96]. The proposed method explicitly accommodates the notion of limited liability of the shareholders. We show how the allocated capital can be decomposed, so that each stakeholder can have a clearer understanding of their contribution. We also challenge the no undercut principle, one of the widely accepted allocation axioms, and assert that this axiom is merely a property that certain allocation methods may or may not meet.     

Mathematical modelling in university education. An experiment at the University of Oldenburg

This paper suggests that mathematics teacher educators should listen carefully to what their students are saying. More specifically, it demonstrates how from one pre-teacher's non-traditional geometric representation of a unit fraction, a variety of learning environments that lead to the enrichment of mathematics for teaching can be developed. The paper shows how new knowledge may be generated through an attempt to validate an intuitive idea; in other words, how the quest for rigour may serve as a catalyst for the growth of mathematical concepts in the context of K-16 mathematics.     

On an assumption of geometric foundation of numbers

In line with the latest positions of Gottlob Frege, this article puts forward the hypothesis that the cognitive bases of mathematics are geometric in nature. Starting from the geometry axioms of the Elements of Euclid, we introduce a geometric theory of proportions along the lines of the one introduced by Grassmann in Ausdehnungslehre in 1844. Assuming as axioms, the cognitive contents of the theorems of Pappus and Desargues, through their configurations, in an Euclidean plane a natural field structure can be identified that reveals the purely geometric nature of complex numbers. Reasoning based on figures is becoming a growing interdisciplinary field in logic, philosophy and cognitive sciences, and is also of considerable interest in the field of education, moreover, recently, it has been emphasized that the mutual assistance that geometry and complex numbers give is poorly pointed out in teaching and that a unitary vision of geometrical aspects and calculation can be clarifying.     

Smoke and mirrors: Combinatorial properties of small cardinals equiconsistent with huge cardinals

Since the work of Godel and Cohen many questions in infinite combinatorics have been shown to be independent of the usual axioms for mathematics, Zermelo Frankel Set Theory with the Axiom of Choice (ZFC). Attempts to strengthen the axioms to settle these problems have converged on a system of principles collectively known as Large Cardinal Axioms.These principles are linearly ordered in terms of consistency strength. As far as is currently known, all natural independent combinatorial statements are equiconsistent with some large cardinal axiom. The standard techniques for showing this use forcing in one direction and inner model theory in the other direction.The conspicuous open problems that remain are suspected to involve combinatorial principles much stronger than the large cardinals for which there is a current fine-structural inner model theory for.The main results in this paper show that many standard constructions give objects with combinatorial properties that are, in turn, strong enough to show the existence of models with large cardinals are larger than any cardinal for which there is a standard inner model theory.     

Cotangent bundles for “matrix algebras converge to the sphere”

In the high-energy quantum-physics literature one finds statements such as “matrix algebras converge to the sphere”. Earlier I provided a general setting for understanding such statements, in which the matrix algebras are viewed as compact quantum metric spaces, and convergence is with respect to a quantum Gromov–Hausdorff-type distance. More recently I have dealt with corresponding statements in the literature about vector bundles on spheres and matrix algebras. But physicists want, even more, to treat structures on spheres (and other spaces) such as Dirac operators, Yang–Mills functionals, etc., and they want to approximate these by corresponding structures on matrix algebras. In preparation for understanding what the Dirac operators should be, we determine here what the corresponding “cotangent bundles” should be for the matrix algebras, since it is on them that a “Riemannian metric” must be defined, which is then the information needed to determine a Dirac operator. (In the physics literature there are at least 3 inequivalent suggestions for the Dirac operators.)     

Argumentation and participation in the primary mathematics classroom: Two episodes and related theoretical abductions

The main assumption of this article is that learning mathematics depends on the student's participation in processes of collective argumentation. On the empirical level, such processes will be analyzed with Toulmin's theory of argumentation and Goffman's idea of decomposition of the speaker's role. On the theoretical level, different statuses of participation in processes of argumentation will be considered. By means of the method of comparative analysis, different grades of autonomy according to the interactional contribution of a student can be reconstructed. The paper finishes with remarks about consequences for improving mathematics teaching in schools and mathematics teacher education at university level.     

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.     

Neo-Kantian foundations of geometry in the German Romantic period

While mathematics received relatively little attention in the idealistic systems of most of the German Romantics, it served as the foundation in the thought of the Neo-Kantian philosopher/mathematician Jakob Friedrich Fries (1773–1843). It fell to Fries to work out in detail the implications of Kant's declaration that all mathematical knowledge was synthetic a priori. In the process Fries called for a new science of the philosophy of mathematics, which he worked out in greatest detail in his Mathematische Naturphilosophie of 1822. In this work he analyzed the foundations of geometry with an eye to clearing up the historical controversy over Euclid's theory of parallels. Contrary to what might be expected, Fries' Kantian perspective provoked rather than inhibited a reexamination of Euclid's axioms. Fries' attempt to make explicit through axioms what was being implicitly assumed by Euclid while at the same time wishing to eliminate unnecessary axioms belies the claim that there was no concern to improve Euclid prior to the discovery of non-Euclidean geometry. Fries' work therefore serves as an important historical example of the difficulties facing those who wanted to provide geometry with a logically secure foundation in the era prior to the published work of Gauss, Bolyai, and others.     

Exemplary mathematics lessons: a view from the West

Cross-cultural comparative research provides a powerful means of achieving better understanding of one’s own practice. This article contrasts exemplary mathematics teaching, as highlighted in this special issue on its practice and development in East Asia, with the increasing emphasis on effective teaching in the West. It provides an overview of the papers in this issue and examines the similarities and differences between what is seen as exemplary practice and the ways in which its development is supported in the educational systems represented. It attempts to identify some of the cultural factors that influence our understanding of what constitutes exemplary practice and the possibilities for its development in the East and the West.     

计算机及数学软件在高职数学教学中的应用

阐述了计算机与数学软件运用于高等职业教育数学教学中的设想与实践探索．把数学实验作为传统教学、计算机和数学软件相结合的一种实践,贯穿于日常的教学过程,并列举了对应的课堂教学具体实例加以展示．最后指出,高职数学教学中应该加强这种教学模式的设计和开发．     

Time allocated to mathematics in post-primary schools in Ireland: are we in double trouble?

Mathematics educators and legislators worldwide have begun placing a greater emphasis on teaching mathematics for understanding and through the use of real-life applications. Revised curricula have led to the time allocated to mathematics in effected countries being scrutinised. This has resulted in policy-makers and educationalists worldwide calling for the inclusion of double class periods on the mathematics timetable. Research from the United States suggests that the introduction of double or block periods allow for the objectives of revised curricula to be realized. The aim of this study, which is set in the school context, is first to ascertain if schools in Ireland are scheduling double periods for mathematics at both lower post-primary level (Junior Cycle) and upper post-primary level (Senior Cycle). It also seeks to determine if there is a link between teachers’ levels of satisfaction with the time allocated to mathematics and the provision of double periods and to get insights from teachers in relation to their opinions on what can be achieved through the introduction of such classes. Questionnaires were sent to 400 post-primary schools (approximately 1600 teachers) which were selected using stratified sampling techniques. It was found that 8.7% of mathematics teachers reported the provision of double periods at Junior Cycle while 55% reported that double periods were included on their timetable at Senior Cycle. The study also identified a link between teachers’ levels of satisfaction with the time allocated to mathematics and the provision of double periods. Finally, teachers felt that double periods allowed for new teaching methodologies, which were promoted by the revised curricula, to be implemented and teaching for understanding was also more feasible. In essence, it was found that double periods have an influence on the mathematical experience of post-primary students as well as the teaching approaches employed.     

Hong Kong teachers’ views of effective mathematics teaching and learning

Twelve experienced mathematics teachers in Hong Kong were invited to face-to-face semi-structured interviews to express their views about mathematics, about mathematics learning and about the teacher and teaching. Mathematics was generally regarded as a subject that is practical, logical, useful and involves thinking. In view of the abstract nature of the subject, the teachers took abstract thinking as the goal of mathematics learning. They reflected that it is not just a matter of “how” and “when”, but one should build a path so that students can proceed from the concrete to the abstract. Their conceptions of mathematics understanding were tapped. Furthermore, the roles of memorisation, practices and concrete experiences were discussed, in relation with understanding. Teaching for understanding is unanimously supported and along this line, the characteristics of an effective mathematics lesson and of an effective mathematics teacher were discussed. Though many of the participants realize that there is no fixed rule for good practices, some of the indicators were put forth. To arrive at an effective mathematics lesson, good preparation, basic teaching skills and good relationship with the students are prerequisite.     

Reasoning and proof in geometry: effects of a learning environment based on heuristic worked-out examples

We analyze heuristic worked-out examples as a tool for learning argumentation and proof. Their use in the mathematics classroom was motivated by findings on traditional worked-out examples, which turned out to be efficient for learning algorithmic problem solving. The basic idea of heuristic worked-out examples is that they encourage explorative processes and thus reflect explicitly different phases while performing a proof. We tested the hypotheses that teaching with heuristic examples is more effective than usual classroom instruction in an experimental classroom study with 243 grade 8 students. The results suggest that heuristic worked-out examples were more effective than the usual mathematics instruction. In particular, students with an insufficient understanding of proof were able to benefit from this learning environment.