Revealing hidden mathematics curriculum to pre-teachers using technology: the case of partitions

Motivated by work done with pre-teachers of mathematics in a problem-solving course, this paper shows how computing technologies, including a spreadsheet and Maple, facilitate an informal journey into a hidden aspect of the formal content of the pre-college curriculum dealing with the arithmetic of partitions. By using three problems from different grade levels within a state curriculum as an example, the paper suggests that a deeper perspective on seemingly disconnected problem-solving contexts may serve as a powerful didactical tool in helping teachers to appreciate mathematics and its pedagogy as an integrated whole. The connection of the hidden aspect of the curriculum to the concept of mathematical play is also explored.     

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.     

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.     

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.     

Contextual approach in teaching mathematics: an example using the sum of series of positive integers

In this paper, a contextual approach to teaching Mathematics at the pre-university level is recommended, and an example is illustrated. A context in the form of a real common mathematical problem is presented to the students. Different approaches to tackle the problem (from topics within and outside the syllabus) can be elicited from students. The insight obtained from the various methods of solving the problem can be used to deepen students’ learnt concepts and to enhance concepts to be learnt later in the curriculum.     

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).     

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.     

Researching the enacted mathematics curriculum: learning from various perspectives on enactment

This issue of ZDM focuses on research related to the enacted curriculum from various perspectives within the context of the US educational system. In this editorial, we describe the broad view of curriculum enactment taken in this issue, highlighting that we mean more than just how instruction plays out within a classroom. For instance, enactment can occur at a national level as educational goals are enacted into a set of national objectives or standards. Enactment can occur as goals or standards are embedded into written curriculum materials or textbooks, both in terms of teacher guides and materials for students. Enactment can occur as teachers make decisions about how to use their written curriculum materials. Finally, enactment can occur as teachers and students engage and interact with written materials during classroom instruction. We elaborate briefly on these views and then outline the structure of this ZDM issue.     

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.     

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.     

Renewing textbooks to align with reformed curriculum in former colonies: Ugandan school mathematics textbooks

Several nations have reformed both their mathematics pedagogy and curriculum. The remaining challenge is to review teaching and learning resources to support the renewed pedagogy and curriculum. This paper responds to the following question: what pedagogy and curriculum are depicted in textbooks used in Uganda? Ugandan textbooks were analyzed in terms of mathematics content structure and genre, and presentation of written and non-written voice and looks (appearance). Whereas certain Ugandan mathematics textbooks used the narrative form and others chose to eliminate the use of extensive text, these textbooks include common characteristics such as spiral coverage of mathematics content. A few strides toward reform pedagogy, such as use of contexts familiar to learners in development of rules and concepts, were evinced among selected Uganda textbooks. More strides are needed in revising a majority of textbooks to align with the renewed curricula on certain aspects including integrating learning tools—digital and non-digital—within the textbook resources. A critical reflection on curriculum renewals adopted from other countries is needed when designing textbooks to match these renewals.     

Towards an authentic teaching of mathematics: Hans-Georg Steiner’s contribution to the reform of mathematics teaching

Hans-Georg Steiner was the “motor of the reform” of mathematics education in Germany. His main concern was to promote authentic teaching. His suggestions for teaching mathematical structures stimulated the process of reform, but were criticised as well. Two controversies are studied in this paper. The controversy with Detlef Laugwitz in 1965 was about the dichotomy “axiomatics vs. constructiveness”. Another controversy with Alexander Wittenberg in 1964 was about the problem of “elementary”. The following considerations can show the need for fundamental didactical analyses in mathematics education, as they were initiated by Hans-Georg Steiner.     

Reflecting mathematics: an approach to achieve mathematical literacy

Mathematics plays a dominant role in today's world. Although not everyone will become a mathematical expert, from an educational point of view, it is key for everyone to acquire a certain level of mathematical literacy, which allows reflecting and assessing mathematical processes important in every day live. Therefore the goal has to be to open perspectives and experiences beyond a mechanical and tight appearance of the subject. In this article a framework for the integration of reflection and assessment in the teaching practice is developed. An illustration through concrete examples is given.     

The potential of recreational mathematics to support the development of mathematical learning

ABSTRACT

A literature review establishes a working definition of recreational mathematics: a type of play which is enjoyable and requires mathematical thinking or skills to engage with. Typically, it is accessible to a wide range of people and can be effectively used to motivate engagement with and develop understanding of mathematical ideas or concepts. Recreational mathematics can be used in education for engagement and to develop mathematical skills, to maintain interest during procedural practice and to challenge and stretch students. It can also make cross-curricular links, including to history of mathematics. In undergraduate study, it can be used for engagement within standard curricula and for extra-curricular interest. Beyond this, there are opportunities to develop important graduate-level skills in problem-solving and communication. The development of a module ‘Game Theory and Recreational Mathematics’ is discussed. This provides an opportunity for fun and play, while developing graduate skills. It teaches some combinatorics, graph theory, game theory and algorithms/complexity, as well as scaffolding a Pólya-style problem-solving process. Assessment of problem-solving as a process via examination is outlined. Student feedback gives some indication that students appreciate the aims of the module, benefit from the explicit focus on problem-solving and understand the active nature of the learning.     

Handheld technology for mathematics education: flashback into the future

In the 1990s, handheld technology allowed overcoming infrastructural limitations that had hindered until then the integration of ICT in mathematics education. In this paper, we reflect on this integration of handheld technology from a personal perspective, as well as on the lessons to be learnt from it. The main lesson in our opinion concerns the growing awareness that students’ mathematical thinking is deeply affected by their work with technology in a complex and subtle way. Theories on instrumentation and orchestration make explicit this subtlety and help to design and realise technology-rich mathematics education. As a conclusion, extrapolation of these lessons to a future with mobile multi-functional handheld technology leads to the issues of connectivity and in- and out-of-school collaborative work as major issues for future research.