Developing a reform mathematics curriculum program in Sweden: relating international research and the local context

This paper reports on a research-based mathematics curriculum program development project in Sweden, whose educational context is currently characterized by multiple reform initiatives. Current reforms include a repositioning of the teacher as central for students’ learning, but also a trend toward initiatives and teacher resources that are more directive than has been the case in the past 30 years. Collecting data from multiple sources, such as teacher log books, lesson observations and feedback meetings, we build on input from 11 elementary school teachers trying out our materials, including student texts and a teachers’ guide, during four trial rounds. We analyze how international research about curriculum programs and teachers’ use of these programs are interpreted and operationalized within the Swedish context. In particular, the two research questions guiding the study are: (1) “How do Swedish teachers interact with and reason about the reform-based classroom practices promoted by the curriculum program?” and (2) “How do Swedish teachers interact with and reason about their use of a teachers’ guide?” From our experiences in the Swedish educational context, we suggest the following contextual aspects to take into account when designing a curriculum program whose design is grounded in international research literature: characteristics of current classroom practices, teachers’ role in classrooms, the level of explicit/implicit support teachers are used to receiving, and teachers’ experiences using a teachers’ guide.     

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.     

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.     

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.     

Conceptualizing the curriculum enactment process in mathematics education

This article explores the following question: What does it mean to enact curriculum? In order to do so, it offers a conceptualization of the enacted curriculum and situates it within a curriculum policy, design, and enactment system. The system depicts the formal and operational domains in which curricular aims and objectives are developed and curriculum plans formulated and enacted. The authors situate the enacted mathematics curriculum in the operational part of the system and define it as the interactions between teachers and students around mathematical tasks of a lesson and collection of lessons, but argue that understanding what it means to enact curriculum involves examining the many places within the system that curricular elements are translated and transformed. The authors describe each of the articles in this special issue with respect to the framework.     

Computer programming in the UK undergraduate mathematics curriculum

This paper reports a study which investigated the extent to which undergraduate mathematics students in the United Kingdom are currently taught to programme a computer as a core part of their mathematics degree programme. We undertook an online survey, with significant follow-up correspondence, to gather data on current curricula and received replies from 46 (63%) of the departments who teach a BSc mathematics degree. We found that 78% of BSc degree courses in mathematics included computer programming in a compulsory module but 11% of mathematics degree programmes do not teach programming to all their undergraduate mathematics students. In 2016, programming is most commonly taught to undergraduate mathematics students through imperative languages, notably MATLAB, using numerical analysis as the underlying (or parallel) mathematical subject matter. Statistics is a very popular choice in optional courses, using the package R. Computer algebra systems appear to be significantly less popular for compulsory first-year courses than a decade ago, and there was no mention of logic programming, functional programming or automatic theorem proving software. The modal form of assessment of computing modules is entirely by coursework (i.e. no examination).     

Visage—Visualization of algorithms in discrete mathematics

Which route should the garbage collectors' truck take? Just a simple question, but also the starting point of an exciting mathematics class. The only “hardware” you need is a city map, given on a sheet of paper or on the computer screen. Then lively discussions will take place in the classroom on how to find an optimal routing for the truck. The aim of this activity is to develop an algorithm that constructs Eulerian tours in graphs and to learn about graphs and their properties. This teaching sequence, and those stemming from discrete mathematics, in particular combinatorial optimization, are ideal for training problem solving skills and modeling—general competencies that, influenced by the German National Standards, are finding their way into curricula. In this article, we investigate how computers can help in providing individual teaching tools for students. Within the Visage project we focus on electronic activities that can enhance explorations with graphs and guide studients even if the teacher is not available—without taking away freedom and creativity. The software package is embedded into a standard DGS, and it offers many pre-built and teacher-customizable tools in the area of graph algorithms. Until now, there are no complete didactical concepts for teaching graph algorithms, in particular using new media. We see a huge potential in our methods, and the topic is highly requested on part of the teachers, as it introduces a modern and highly relevant part of mathematics into the curriculum.     

Experimental mathematics in the curriculum (part 2)

In part 1 [1] of this work we showed how modern mathematicalresearch could, with a suitably chosen problem, be includedin the first year curriculum of undergraduate mathematicians.With the use of Computer Algebra Systems, even the average undergraduatemathematician can aspire to discover interesting yet still unexplainedbehaviour in many areas of mathematics. Of course, interestingresults still need a true expert to furnish proofs. This articlecontinues the exploration of the so-called Buffon puzzle anddemonstrates how it can be made accessible to undergraduates.Part 1 dealt with material delivered in lectures 1–12.In part 2, we describe work that can be carried out in lectures13–24.     

Re-sourcing curriculum materials: in search of appropriate frameworks for researching the enacted mathematics curriculum

This article provides a commentary to the eight papers of this issue of ZDM entitled “Researching the enacted mathematics curriculum.” It is structured around three main questions concerning (1) the layers of the curriculum addressed in the eight papers; (2) an identification of the main theoretical framework used, and an appreciation of this as compared to another European framework; and (3) challenges for future research on the enacted mathematics curriculum. The author outlines her views derived from a particular European perspective.     

Facing the change and meeting the challenge: mathematics curriculum of Tongwen Guan in China in the second half of the nineteenth century

It is commonly recognized that the second wave of transmission of Western science and mathematics into China took place in the second half of the nineteenth century, thereby starting a process of modernization which has become synonymous with ??westernization?? of the country. In this context mathematics education in China also went through a process of modernization. During this period of great change reformers faced serious challenge and went through a ??mental struggle??. This paper discusses this challenge and mental struggle by taking a close look at the controversy over the establishment of Tianwen Suanxue Guan (School of Astronomy and Mathematics) in Tongwen Guan (School of Combined Learning), one of the schools of Western learning at the time. The attempt to integrate Western mathematics with Chinese traditional mathematics is also discussed through the study of some questions sampled from examinations and students?? homework assignments of this school.     

How a standards-based mathematics curriculum differs from a traditional curriculum: with a focus on intended treatments of the ideas of variable

Analyzing the important features of different curricula is critical to understand their effects on students’ learning of algebra. Since the concept of variable is fundamental in algebra, this article compares the intended treatments of variable in an NSF-funded standards-based middle school curriculum (CMP) and a more traditionally based curriculum (Glencoe Mathematics). We found that CMP introduces variables as quantities that change or vary, and then it uses them to represent relationships. Glencoe Mathematics, on the other hand, treats variables predominantly as placeholders or unknowns, and then it uses them primarily to represent unknowns in equations. We found strong connections among variables, equation solving, and linear functions in CMP. Glencoe Mathematics, in contrast, emphasizes less on the connections between variables and functions or between algebraic equations and functions, but it does have a strong emphasis on the relation between variables and equation solving.     

On construction of parallel algorithms in problems of computational mathematics

It is shown by using the example of solution methods for systems of linear algebraic equations that algorithms possessing the important properties of being parallel and asynchronous can be constructed by reducing the problem under consideration to computing a path integral. Earlier, it was shown by the author and his colleagues that, with increasing the dimension n of the system, parallel and asynchronous Monte Carlo algorithms may become better than the corresponding iterative methods. A qualitative explanation of this phenomenon is suggested. The solution of a system of linear algebraic equations of the form X = AX + F admits a simple representation in form of a path integral only under the condition λ₁(A) < 1, where λ₁(A) is an eigenvalue of A with maximum absolute value. Otherwise, a recursive solution procedure with (coarse) parallelism properties can be constructed. However, in this case, additional conditions for synchronizing the algorithm are needed. Finally, it is shown how to obtain efficient analogues of stochastic algorithms (algorithms of the quasi-Monte Carlo method), which exhibit better rate of convergence than those in the Monte Carlo method, on the basis of results obtained by the author and Wagner in [10]. Similar approaches apply to a large class of problems of mathematical and theoretical physics in which integral representations of solutions are known.     

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.     

Integrating design problems in mathematics curriculum: an architecture college case study

This paper reports an attempt to improve results in the mathematics course in one of the architecture colleges in Israel through practise in applications. The effect of integrating structure design problems in the calculus curriculum on students' achievements and attitudes was examined. The applied topics in the curriculum were connected to calculus topics and studied through problembased learning activities. The integrated curriculum was implemented and the learning results in experimental and control groups were assessed by means of achievement tests, attitude questionnaires and student interviews. The learning achievements in the experimental group proved to be significantly higher than in the control group. The positive impact of learning applications on motivation, understanding, creativity and interest in mathematics is indicated.