A brief history of mathematics education in Turkey: K-12 mathematics curricula

In this study, we survey the history of mathematics education in Turkey starting with its historical roots in the foundation of the republic. The changes in mathematics education in Turkey over the last century are investigated through an analysis of changes in curricular documents for K-12 schools. We consider the factors and reasons affecting curriculum developments, changes in philosophy and structure in terms of standards, objective and instructions. This article utilizes archival research techniques by examining original sources and illustrates the nature of the changes benefiting from a historical perspective. As a result of such analysis of the aforesaid sources, we have seen that the main reasons for changing mathematics curricula are: to build up a modern civilization in Turkey; the reports of John Dewey and the recommendations of Kate Wofford, William C. Varaceus and Watson Dickerman; the desire to become a member of the European Union; international factors and political situations.     

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.     

Teachers learning the registers of mathematics and mathematics education in another language: an exploratory study

Many mathematics teachers around the world teach in a language different from the one in which they studied or completed their teacher education. Often these teachers must learn both the registers of mathematics and of mathematics education to teach in the additional language. This paper examines the factors that help teachers to learn these registers in Māori, the Indigenous language of New Zealand. Many of these teachers are second-language learners of the Māori language and attended English-medium schools and teacher-education programmes. After a brief discussion about the key role of language in teaching mathematics, this paper examines data from teachers at two Māori-immersion schools and a professional development facilitator. The analysis provides initial understanding of the factors that support or hinder their learning of the mathematics registers. Finally, a research agenda is suggested for further investigation of this issue.     

Reverse mathematics and well-ordering principles: A pilot study

The larger project broached here is to look at the generally sentence “if X is well-ordered then f(X) is well-ordered”, where f is a standard proof-theoretic function from ordinals to ordinals. It has turned out that a statement of this form is often equivalent to the existence of countable coded ω-models for a particular theory T_f whose consistency can be proved by means of a cut elimination theorem in infinitary logic which crucially involves the function f. To illustrate this theme, we prove in this paper that the statement “if X is well-ordered then ε_X is well-ordered” is equivalent to . This was first proved by Marcone and Montalban [Alberto Marcone, Antonio Montalbán, The epsilon function for computability theorists, draft, 2007] using recursion-theoretic and combinatorial methods. The proof given here is principally proof-theoretic, the main techniques being Schütte’s method of proof search (deduction chains) [Kurt Schütte, Proof Theory, Springer-Verlag, Berlin, Heidelberg, 1977] and cut elimination for a (small) fragment of .     

Theories in practice: mathematics teaching and mathematics teacher education

Whilst research on the teaching of mathematics and the preparation of teachers of mathematics has been of major concern in our field for some decades, one can see a proliferation of such studies and of theories in relation to that work in recent years. This article is a reaction to the other papers in this special issue but I attempt, at the same time, to offer a different perspective. I examine first the theories of learning that are either explicitly or implicitly presented, noting the need for such theories in relation to teacher learning, separating them into: socio-cultural theories; Piagetian theory; and learning from practice. I go on to discuss the role of social and individual perspectives in authors’ approach. In the final section I consider the nature of the knowledge labelled as mathematical knowledge for teaching (MKT). I suggest that there is an implied telos about ‘good teaching’ in much of our research and that perhaps the challenge is to study what happens in practice and offer multiple stories of that practice in the spirit of “wild profusion” (Lather in Getting lost: Feminist efforts towards a double(d) science. SUNY Press, New York, 2007).     

Chromatic-Index-Critical Graphs of Orders 11 and 12

A chromatic-index-critical graphGonnvertices is non-trivial if it has at most Δ edges. We prove that there is no chromatic-index-critical graph of order 12, and that there are precisely two non-trivial chromatic-index-critical graphs on 11 vertices. Together with known results this implies that there are precisely three non-trivial chromatic-index-critical graphs of order ≤12.     

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.     

A case study of pedagogy of mathematics support tutors without a background in mathematics education

This study investigates the pedagogical skills and knowledge of three tertiary-level mathematics support tutors in a large group classroom setting. This is achieved through the use of video analysis and a theoretical framework comprising Rowland's Knowledge Quartet and general pedagogical knowledge. The study reports on the findings in relation to these tutors’ provision of mathematics support to first and second year undergraduate engineering students and second year undergraduate science students. It was found that tutors are lacking in various pedagogical skills which are needed for high-quality learning amongst service mathematics students (e.g. engineering/science/technology students), a demographic which have low levels of mathematics upon entering university. Tutors teach their support classes in a very fast didactic way with minimal opportunities for students to ask questions or to attempt problems. It was also found that this teaching method is even more so exaggerated in mandatory departmental mathematics tutorials that students take as part of their mathematics studies at tertiary level. The implications of the findings on mathematics tutor training at tertiary level are also discussed.     

New Roses: Simple Symmetric Venn Diagrams with 11 and 13 Curves

A symmetric \(n\) -Venn diagram is one that is invariant under \(n\) -fold rotation, up to a relabeling of curves. A simple \(n\) -Venn diagram is an \(n\) -Venn diagram in which at most two curves intersect at any point. In this paper, we introduce a new property of Venn diagrams called crosscut symmetry, which is related to dihedral symmetry. Utilizing a computer search restricted to diagrams with crosscut symmetry, we found many simple symmetric Venn diagrams with 11 curves. The question of the existence of a simple 11-Venn diagram has been open since the 1960s. The technique used to find the 11-Venn diagram is extended and a symmetric 13-Venn diagram is also demonstrated.     

Differentiation of teaching and learning mathematics: an action research study in tertiary education

Diversity and differentiation within our classrooms, at all levels of education, is nowadays a fact. It has been one of the biggest challenges for educators to respond to the needs of all students in such a mixed-ability classroom. Teachers’ inability to deal with students with different levels of readiness in a different way leads to school failure and all the negative outcomes that come with it. Differentiation of teaching and learning helps addressing this problem by respecting the different levels that exist in the classroom, and by responding to the needs of each learner. This article presents an action research study where a team of mathematics instructors and an expert in curriculum development developed and implemented a differentiated instruction learning environment in a first-year engineering calculus class at a university in Cyprus. This study provides evidence that differentiated instruction has a positive effect on student engagement and motivation and improves students’ understanding of difficult calculus concepts.     

Spatial visualization and measurement of area: A case study in spatialized mathematics instruction

The purpose of this study was to explore the influence of spatial visualization skills when students solve area tasks. Spatial visualization is closely related to mathematics achievement, but little is known about how these skills link to task success. We examined middle school students’ representations and solutions to area problems (both non-metric and metric) through qualitative and quantitative task analysis. Task solutions were analyzed as a function of spatial visualization skills and links were made between student solutions on tasks with different goals (i.e., non-metric and metric). Findings suggest that strong spatial visualizers solved the tasks with relative ease, with evidence for conceptual and procedural understanding. By contrast, Low and Average Spatial students more frequently produced errors due to failure to correctly determine linear measurements or apply appropriate formula, despite adequate procedural knowledge. A novel finding was the facilitating role of spatial skills in the link between metric task representation and success in determining a solution. From a teaching and learning perspective, these results highlight the need to connect emergent spatial skills with mathematical content and support students to develop conceptual understanding in parallel with procedural competence.     

The MACSI summer school: a case study in outreach in mathematics

To encourage the study of mathematics in Ireland, the Mathematics Applications Consortium for Science and Industry (MACSI) organizes a summer school once a year. The different aspects of this summer school are presented. Students are selected depending on their motivation, academic abilities, gender and geographical origins. Instruction and supervision is provided by academics, post-doctoral fellows and post-graduate students. The teaching programme evolves every year and reflects the interests of the people involved. Feedback from participants has been almost uniformly positive. Students favour interactive sessions and enjoy the residential aspect of the summer school. Food and accommodation are however the most costly aspects of this summer school. In this respect the support of Science Foundation Ireland has been invaluable.     

Educational philosophies and their influence on mathematics education— An ethnographic study in English and German mathematics classrooms

In the first part of the paper different educational philosophies, developed in, England and Germany in the last centuries, are described. In the second part results of an ethnographical study in English and German mathematics classrooms are presented. The study indicates the influence of this different educational philosophies on the educational systems of both countries and on the understanding of mathematics teaching and the teaching practice developed there