From lessons to lectures: NCEA mathematics results and first-year mathematics performance

Given the recent radical overhaul of secondary school qualifications in New Zealand, similar in style to those in the UK, there has been a distinct change in the tertiary entrant profile. In order to gain insight into this new situation that university institutions are faced with, we investigate some of the ways in which these recent changes have impacted upon tertiary level mathematics in New Zealand. To this end, we analyse the relationship between the final secondary school qualifications in Mathematics with calculus of incoming students and their results in the core first-year mathematics papers at Canterbury since 2005, when students entered the University of Canterbury with these new reformed school qualifications for the first time. These findings are used to investigate the suitability of this new qualification as a preparation for tertiary mathematics and to revise and update entrance recommendations for students wishing to succeed in their first-year mathematics study.     

Studying advanced mathematics in England: findings from a survey of student choices and attitudes

The UK Government has set a goal that the “vast majority” of students in England will be studying mathematics to the age of 18 by the end of the decade. The policy levers for achieving this goal include new Core Maths qualifications, designed for over 200,000 students who have achieved good grades at the age of 16 but then opt out of advanced or A level mathematics. This article reports findings from a cluster-sampled survey of over 10,000 17-year-olds in England in 2015. Participants’ views on post-16 mathematics are presented and discussed. The main finding is that they are strongly opposed to the idea of compulsory mathematical study, but are less antithetical to being encouraged to study mathematics beyond 16. We consider how attitudes vary by gender, prior attainment, study patterns and future aspirations. The article considers the implications of these findings in the current policy landscape.     

New roles for mathematics in multi-disciplinary, upper secondary school projects

A new concept, compulsory multi-disciplinary courses, was introduced in upper secondary school curriculum as a central part of a recent reform. This paper reports from a case study of such a triple/four-disciplinary project in mathematics, physics, chemistry and ‘general study preparation’ performed under the reform by a team of experienced teachers. The aim of the case study was to inquire how the teachers met the demands of the introduction of this new concept and, to look for signs of new relations established by the students between mathematics and other subjects, as a result of the multi-disciplinary teaching. The study revealed examples of good practice in planning and teaching. In addition, it served to illuminate interesting aspects of how students perceived the school subject mathematics and its relations to other subjects and to common sense.     

Teacher listening: The role of knowledge of content and students

In this research report we consider the kinds of knowledge needed by a mathematician as she implemented an inquiry-oriented abstract algebra curriculum. Specifically, we will explore instances in which the teacher was unable to make sense of students’ mathematical struggles in the moment. After describing each episode we will examine the instructor's efforts to listen to the students and the way that these efforts were supported or constrained by her mathematical knowledge for teaching. In particular, we will argue that in each case the instructor was ultimately constrained by her knowledge of how students were thinking about the mathematics.     

METHODS IN GHANAIAN PRIMARY MATHEMATICS TEXTBOOKS AND TEACHERS’ CLASSROOM PRACTICE

The official school mathematics curriculum has a powerful influence on classroom practice in a developing country like Ghana, where many teachers with low teaching qualifications hardly ever have access to other sources of information and activity for their teaching. The study reported in this paper investigated the congruence between the teaching methods presented in the official curriculum materials and teachers’ classroom practice. The study involved a range of methods: classroom observation and content and discourse analyses of lessons in mathematics. The study found that, though there was rhetoric in the introduction of the curriculum materials on the use of discovery teaching methods, few learning/teaching activities that would encourage the use of such discovery methods were included in the materials. It was observed that both the official curriculum and the teachers who implement it emphasise expository teaching methods.     

Problem solving and Working Mathematically: an Australian perspective

This article reviews “problem solving” in mathematics in Australia and how it has evolved in recent years. In particular, problem solving is examined from the perspectives of research, curricula and instructional practice, and assessment. We identify three key themes underlying observed changes in the research agenda in Australia in relation to problem solving: Obliteration, Maturation and Generalisation. Within state mathematics curricula in Australia, changes in the language and construction of the curriculum and in related policy documents have subsumed problem solving within the broader category of Working Mathematically. In relation to assessment, research in Australia has demonstrated the need for alignment of curriculum, instruction and assessment, particularly in the case of complex performances such as mathematical problem solving. Within the category of Working Mathematically, recent Australian curriculum documents appear to accept an obligation to provide both standards for mathematical problem solving and student work samples that illustrate such complex performances and how they might be assessed.     

A micro-curricular analysis of unified mathematics curricula in Turkey

This article aims to give a detailed micro-level curricular analysis of the extent to which the intended mathematics curriculum matches the potentially implemented curriculum using the case of Turkey. The article makes inferences about what it means to have a match or mismatch between these two types of curricula. As a result, it is clear that even though there is a close match between the intended and the potentially implemented mathematics curricula, such a match does not seem to be enough to help students to have a solid understanding of targeted mathematical concepts outlined in the overall Turkish curricular standards.     

Cross-national comparisons of mathematics curriculum materials: what might we learn?

In most of the world, there is a broad public interest in cross-national comparisons. Despite the myriad difficulties inherent in such comparisons, and despite the fact that most studies explicitly acknowledge limited generalizability, news media representatives, policy makers, politicians and educators often take the findings at face value as providing solid scientific evidence of the achievement of students in the countries studied. This often leads in turn to superficial commentaries and unhelpful recommendations. Our task as scholars of mathematics education is to demonstrate the nuanced complexity of such comparisons. The papers in this issue of ZDM succeed in this regard. Using a range of scholarly perspectives (mathematical, psychological and socio-cultural), the authors offer many insightful observations regarding curriculum and curriculum materials in East Asian countries (especially China) and the USA, with careful attention to the central role of curriculum in the teaching and learning of mathematics in schools. In this commentary study I discuss a few of these insights and develop one illustrative example of how cross-national comparative analyses such as these could be used to sensibly inform policy discussions related to changes in mathematics curriculum, teaching, or teacher preparation.     

Exploring the potential role of visual reasoning tasks among inexperienced solvers

The collective case study described herein explores solution approaches to a task requiring visual reasoning by students and teachers unfamiliar with such tasks. The context of this study is the teaching and learning of calculus in the Palestinian educational system. In the Palestinian mathematics curriculum the roles of visual displays rarely go beyond the illustrative and supplementary, while tasks which demand visual reasoning are absent. In the study, ten teachers and twelve secondary and first year university students were presented with a calculus problem, selected in an attempt to explore visual reasoning on the notions of function and its derivative and how it interrelates with conceptual reasoning. A construct named “visual inferential conceptual reasoning” was developed and implemented in order to analyze the responses. In addition, subjects’ reflections on the task, as well as their attitudes about possible uses of visual reasoning tasks in general, were collected and analyzed. Most participants faced initial difficulties of different kinds while solving the problem; however, in their solution processes various approaches were developed. Reflecting on these processes, subjects tended to agree that such tasks can promote and enhance conceptual understanding, and thus their incorporation in the curriculum would be beneficial.     

SCHOOL OR COLLEGE MATHEMATICS AND WORKPLACE PRACTICE: AN ACTIVITY THEORY PERSPECTIVE

This paper reports on a case study in which we detail how a college mathematics and chemistry student struggles to make sense of the graphical output of an experiment in an industrial chemistry laboratory. The student's attempts to interpret unfamiliar graphical conventions are described and contrasted with those of college mathematics. Our analysis of this draws on activity theory to assist in understanding the position of the student in both the college and the workplace. This highlights the limitations of the experience of the student at college and we question how the mathematics curriculum might be adapted to assist students in making sense of workplace graphical output.     

Gender justice,human rights and measurement in the mathematics classroom

In line with international trends, the new South African mathematics curriculum implores mathematics educators to realize a pedagogy in their classrooms that is more practical, activity-oriented, and connected to their learners' lives. Drawing on data from a larger study that explores theory–practice relations in mathematics education, this paper shows how such progressive practices, when interpreted with respect to the teaching of measurement which required learners to use different measuring instruments for measuring the school grounds in learning about length and perimeter, were found to be deeply gendered. In two different contexts of an ‘African' township school and a predominantly ‘Indian' suburban school, girls in a grade 6 mathematics classroom faced direct sexism as they struggled to take the opportunity to participate in the activity and learn how to measure – an important mathematical competence and everyday knowledge and skill. The article analyses the data with reference to the human rights imperatives of the new national curricula and approaches to addressing disadvantage and discrimination for girls in mathematics classrooms.     

Mathematics coaching and instructional reform: Individual and collective change

Mathematics coaching initiatives are being implemented in schools and districts across the country, guided by the notion that these initiatives will foster individual teacher’s learning and thereby support system-wide instructional improvement in mathematics. This paper explores the evolving roles that mathematics coaches played in a system-wide instructional improvement effort focused on elementary mathematics education in a medium-sized suburban school district. Using social network analysis and qualitative analysis of interviews, we argue that coaches facilitated teachers’ implementation of a new mathematics curriculum by acting as brokers, first as intermediaries between the district office and schools, then as catalysts for collective inquiry. Further, we show how coaches’ work was both enabled and constrained over time by various organizational dimensions at the school and district levels. Overall, our findings suggest that district and school leaders should think beyond the roles and responsibilities of individual coaches, and consider how to support coaches as participants in system-wide networks focused on continuous learning and instructional improvement.     

Teachers' Conceptions of Integrated Mathematics Curricula

In this qualitative research study, we sought to understand teachers' conceptions of integrated mathematics. The participants were teachers in the first year of implementation of a state‐mandated, high school integrated mathematics curriculum. The primary data sources for this study included focus group and individual interviews. Through our analysis, we found that the teachers had varied conceptions of what the term integrated meant in reference to mathematics curricula. These varied conceptions led to the development of the Conceptions of Integrated Mathematics Curricula Framework describing the different conceptions of integrated mathematics held by the teachers. The four conceptions—integration by strands, integration by topics, interdisciplinary integration, and contextual integration—refer to the different ideas teachers connect as well as the time frame over which these connections are emphasized. The results indicate that even when teachers use the same integrated mathematics curriculum, they may have varying conceptions of which ideas they are supposed to connect and how these connections can be emphasized. These varied conceptions of integration among teachers may lead students to experience the same adopted curriculum in very different ways.     

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.     

Rhetoric,Reality, and Possibilities: Interdisciplinary Teaching and Secondary Mathematics

The National Council of Teachers of Mathematics has proposed a broad core mathematics curriculum for all high school students. One emphasis in that core is on “mathematical connections” both among mathematical topics and between mathematics and other disciplines of study. It is suggested that mathematics should become a more integrated part of all students' high school education. In this article, working definitions for the terms curriculum, interdisciplinary, and integrated and a model of three categories of curriculum design based on the work of Harold Alberty are developed. This article then examines how a “connected” mathematics core curriculum might be situated within the different categories of curriculum organization. Examples from research on interdisciplinary education in high schools are presented. Issues arising from this study suggest the need for a greater emphasis on building and using models of curriculum integration both to frame and to give impetus to the work being done by teachers and administrators.     

Can Technology Help Us Make the Mathematics Curriculum Intellectually Stimulating and Socially Responsible?

In order to answer the question posed in the title of this paper, we must take a wide perspective and explore the goal societies have for maintaining educational systems, how curriculum contributes to the attainment of these goals, how mathematics in the curriculum contributes to effectiveness in attaining these goals and finally some of the ways in which appropriately crafted technology can help to make mathematics a more effective part of the curriculum. This revised version was published online in July 2006 with corrections to the Cover Date.     

Detecting Students' Experiences of Discontinuities Between Middle School and High School Mathematics Programs: Learning During Boundary Crossing

Transitions from middle school to high school mathematics programs can be problematic for students due to potential differences between instructional approaches and curriculum materials. Given the minimal research on how students experience such differences, we report on the experiences of two students as they moved out of an integrated, problem-based mathematics program in their middle school into a high school mathematics program that emphasized procedural fluency. We conducted an average of two interviews per year for two and a half years with participants and engaged in participant-observation at their high school. In this study, we illustrate an analytic process for detecting discontinuities between settings from participants' perspectives. We determined that students experienced a discontinuity if they reported meaningful differences between settings (frequent mention, in detail, with emphasis terms) and concurrently reported a change in attitude. Additionally, these students' experiences illustrate two opportunities to learn during boundary-crossing experiences: identification and reflection.     

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.     

Connections and confusion: teaching perimeter and area with a problem-solving oriented unit

Problem-solving-oriented mathematics curricula are viewed as important vehicles to help achieve K-12 mathematics education reform goals. Although mathematics curriculum projects are currently underway to develop such materials, little is known about how teachers actually use problem-solving-oriented curricula in their classrooms. This article profiles a middle-school mathematics teacher and examines her use of two problems from a pilot version of a sixth-grade unit developed by a mathematics curriculum project. The teacher's use of the two problems reveals that although problem-solving-oriented curricula can be used to yield rich opportunities for problem solving and making mathematical connections, such materials can also provide sites for student confusion and uncertainty. Examination of this variance suggests that further attention should be devoted to learning about teachers' use of problem-solving-oriented mathematics curricula. Such inquiry could inform the increasing development and use of problem-solving-oriented curricula.