Enactment of school mathematics curriculum in Singapore: whither research!

The official curriculum for mathematics in Singapore schools is based on a framework that has mathematical problem solving as its primary goal. It is detailed and one may say that the gap between the designated curriculum and teacher intended curriculum is often very narrow. This is so as the main source of instructional materials is textbooks which are very closely aligned with the official national curriculum. There is a dearth of research on the enactment of the curriculum in Singapore schools, with the few research studies done so far appearing to cover only a narrow focus. The author’s view is that, even though only a few such studies have been published, schools have always been engaged in small-scale investigations, the findings of which are necessary to guide decisions on matters related to choice of textbooks and pedagogies for improved student learning. Considering all the published research and the investigative work undertaken by educators in Singapore, it may be said that the conceptual model proposed by Remillard and Heck is rigorous. In addition, the issues in this particular issue of ZDM offer educators, both classroom teachers and others, very good perspectives for research on the enactment of the school mathematics curriculum.     

Creativity and school mathematics: some modest observations

In spite of its importance, research into mathematical creativity has not received the same attention as research into student difficulties in learning mathematics. In this paper, I review some of the history, policies and research in this area from the last 40 years, and then critique the papers in this issue, noting progress that has been made in the field.     

Stepping beyond high school mathematics: a case study of high school women

The Summer Explorations and Research Collaborations for High School Girls (SEARCH) Program, held annually since 2004 at Mount Holyoke College in the US, was created for talented high school girls to explore mathematics beyond that taught in high school. Our study, which focuses on factors that facilitate or inhibit the pursuit of higher level mathematics by girls, is centered on the 2006 SEARCH Program. We present a combination of qualitative and quantitative data drawn from student journals written during SEARCH, program evaluations written at the end of SEARCH, post-program interviews, and comparisons with two peer group samples. From this data we point to important factors, such as developing a mathematical voice, gaining a broader view of advanced mathematics, being challenged in a supportive atmosphere, and having a positive stance toward risk-taking, that may help to maintain the interest of talented girls in advanced mathematical studies.     

Links between science and mathematics in school

There exist obvious links between the science and mathematics taught in schools. This paper discusses the responses of trainee teachers in science and mathematics to a number of questions relevant to both subjects and points to failures of understanding and of communication. It further seeks to highlight problems of liaison between subject teachers.     

The concept of invariance in school mathematics

In this paper, we highlight examples from school mathematics in which invariance did not receive the attention it deserves. We describe how problems related to invariance stimulated the interest of both teachers and students. In school mathematics, invariance is of particular relevance in teaching and learning geometry. When permitted change leaves some relationships or properties invariant, these properties prove to be inherently interesting to teachers and students.     

Pro-efficiency: Data speak more than technical efficiency

In this study, we demonstrate a new method of addressing efficiency in situations in which only the input and output data are available, while evaluating efficiency more accurately than is possible via the ordinary data envelopment analysis (DEA). Technical efficiency is important, but management always desires information regarding the profit aspects of performance. In practice, however, the precise price data are frequently unavailable. Is it possible to approximate profit efficiency in the absence of price information? We develop a simple and usable approach, a linear programming model, for the evaluation of profit efficiency. Our approach implies technical efficiency in DEA and gives rise to the upper bound of profit efficiency, referred to as pro-efficiency. We also report a successful application of our method to a securities company, in which a comparison of the actual profit data and the pro-efficiency measures of the company’s branches demonstrates a significant correlation.     

Ideal and reality: Cross-curriculum work in school mathematics in South Africa

Within various school mathematics dispensations in South Africa the intention for cross-curriculum work is expressed in the official documents describing the intended school mathematics curriculum. This paper traces this expressed intention from 1962 to the present. The view is adopted that textbook authors are the major interpreters of the intended curriculum and therefore the manifestations of the cross-curricular ideal in school textbooks for the various periods are described and commented on. Using the manifestation of the cross-curricular ideal in the South African situation as backdrop, the paper concludes by suggesting ways to deal with three issues that seemingly mitigate against the realisation of this ideal. It is argued that the applications of and modelling in mathematics should be treated as a distinet separate section in the school mathematics curriculum, that mathematics activities, should be designed so that learners with various levels of mathematical sophistication and expertise can deal with both the embedded context and the mathematics and that problems which use context only as a disguise for “pure” mathematics should not be summarily dismissed and written off as useless for the realisation of cross-curricular goals.     

Integrating algebra and proof in high school mathematics: An exploratory study

Frequently, in the US students’ work with proofs is largely concentrated to the domain of high school geometry, thus providing students with a distorted image of what proof entails, which is at odds with the central role that proof plays in mathematics. Despite the centrality of proof in mathematics, there is a lack of studies addressing how to integrate proof into other mathematical domains. In this paper, we discuss a teaching experiment designed to integrate algebra and proof in the high school curriculum. Algebraic proof was envisioned as the vehicle that would provide high school students the opportunity to learn not only about proof in a context other than geometry, but also about aspects of algebra. Results from the experiment indicate that students meaningfully learned about aspects of both algebra and proof in that they produced algebraic proofs involving multiple variables, based on conjectures they themselves generated.     

Discrete mathematics in the high school curriculum

In this paper we present some topics from the field of discrete mathematics which might be suitable for the high school curriculum. These topics yield both easy to understand challenging problems and important applications of discrete mathematics. We choose elements from number theory and various aspects of coding theory. Many examples and problems are included.     

Introducing computer algebra to school teachers of mathematics

Since the last decade, the use of computer algebra systems atthe Hong Kong school level is still very limited. Among variousreasons behind, the lack of exposure of this kind of softwareto local school teachers should be taken into account. In thisarticle, we describe how to introduce MAPLE in a BEd moduleof a local teacher-training programme. Reflection on the effectivenessof the module and barriers faced by the school teachers at presentare discussed.     

Selection of content in high school mathematics textbooks: an international comparison

As a component of the ongoing development of the mathematics curriculum in China, we compare the country’s high school mathematics textbooks with those of several other countries. We base our analysis on the assumption that textbooks, as primary printed teaching resources, are key tools for interpreting educational policy. In this paper, we compare what content is selected and how that content is presented in series of high school textbooks from China, France, Germany, Japan, Russia, the United Kingdom, and the United States according to four core domains: algebra, geometry, statistics and probability, and calculus. We then discuss the implications of the analysis for the reform of the high school mathematics curriculum in China, particularly as it applies to the development of textbooks within that process of reform. The comparative results provide us with the opportunity to recognise the distinguishing features of the content and presentation of the Chinese mathematics curriculum and lead to some suggestions for future curriculum development.     

Didactics of mathematics as a scientific discipline: A sketch of its development from a personal (autobiographic) point of view

At the 8th International Congress on Mathematical Education (ICME-8) held in Sevilla, Spain, July 1996, Working Group 25 was devoted to the theme “Didactics of Mathematics as a Scientific Discipline”. The proceedings of this activity have been published by Nicolina A. Malara (s. footnote 1). The plenary talk given by the author on “Basic Characteristics and Recent Trends in the Development of Didactics of Mathematics as a Scientific Discipline” as well as the round-table discussion under his chairmanship are documented in the proceedings. Furthermore the author has contributed to the proceedings an autobiographicly oriented paper which goes back to his own time as a schoolboy and refers to his professional development in connections with various national and international institutions and movements. The present slightly extended reproduction of that contribution is mediating aspects of developmental dynamics which can hardly be grasped by highly objectified presentations. As for a similar paper see H. G. Steiner: “Some notable anniversaries and some personal German-Italian memories in didactis of mathematics ZDM 26 (1994), 211–214; also in Italian: L’insegnamento della matematica e delle scienze integrate (Numero Speziale), vol. 5 N.17A-17B, Sept.–Oct. 1994, 425–437.     

A reform in undergraduate mathematics curriculum : more emphasis on social and pedagogical skills

The paper describes the changes that are being made in the mathematics teachers' subject studies in the Department of Mathematics at the University of Joensuu, in order to provide our mathematics students both with a sufficiently deep knowledge of mathematics and science, and with present-day expertise in their profession as teachers. While the formal structure of the mathematics curriculum remains structured and taught as courses with mostly traditional names like algebra, analysis, and linear algebra, there are also totally new ‘professionally oriented’ courses. Some of the old courses—with rather traditional and rigorous contents—have been changed in a more student-driven direction. In these ‘pedagogically oriented’ courses students are encouraged, and even forced, to study co-operatively in social interaction, for example to negotiate how to solve a problem decently, or how to build a formal definition for a concept with certain wanted attributes. As an ultimate example of a pedagogical experiment we describe in more detail an abstract algebra course, where co-operative learning is combined with intensive programming in a mathematically oriented computer environment.     

Approaches and practices in developing school mathematics textbooks in China

In this study, we aim to examine and discuss approaches and practices in developing mathematics textbooks in China, with a special focus on the development of secondary school mathematics textbook in the context of recent school mathematics reform. Textbook development in China has its own history. This study reveals some common practices and approaches developed and used in selecting, presenting and organizing content in mathematics textbooks over the years. With the recent curriculum reform taking place in China, we also discuss some new developments in compiling and publishing high school mathematics textbooks. Implications obtained from Chinese practices in textbook development are then discussed in a broad context.