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This paper presents an example of how paper folding can be used in a geometry class to support conceptual understanding. Specifically, it explains an activity that constructs a rhombus and explores its attributes by using paper folding. The steps of constructing a rhombus are described and some discussion questions are given to consolidate understanding. © 2017 Informa UK Limited, trading as Taylor & Francis Group.

Sandor Szab6 (sszabo7@hotmail.com) was born and grew up in Hungary and graduated from E6tv6s University in Budapest. He has taught at the Technical University in Budapest, has held a variety of visiting positions, and currently is on the faculty of the University of Pecs in Hungary. His main mathematical interests are in algebra and its applications. He has written two monographs, Algebra and Tilingq with S. K. Stein, and Topics in Factorization of Abelian Groups.

In April, 1991 the NSF/SSMA conference entitled “A Network for Integrated Science and Mathematics Teaching and Learning” was held at the Wingspread conference facility in Racine, Wisconsin. Five plenary papers were presented offering very diverse perspectives related to the integration of science and mathematics education. This publication summarizes the significant points advanced in each plenary paper. The full text of these papers is being published as part of the SSMA Monograph Series.

(1993). Stay in Touch, Won't You? Using the One-Minute Paper. College Teaching: Vol. 41, No. 2, pp. 60-63.

Innovative methods can change the paradigm of teaching mathematics and inspire teachers to espouse new ideas and gain new experiences. The flipped classroom (FC) is currently an innovative pedagogical approach that has high potential to transform the teaching of mathematics. In the case study described in this paper, we investigated one mathematics teacher’s transformation of teaching in two mathematics classrooms through implementing interventions based on FC methods; furthermore, we identified several key points of FC design as well as challenges and opportunities afforded by teaching mathematics in FCs. The results of the study showed that the tasks posed by the teacher, the implemented discourse, teacher feedback and scaffolding, and the teaching–learning environment were changed in FCs, although the approaches used by the teacher to analyze the tasks and students’ learning were similar to those used in non-FCs, which points out the strengths of traditional teaching approaches. The study indicates that although teaching mathematics in FCs created some difficulties for teaching, well-designed FCs offered a great opportunity to promote students’ mathematical thinking and understanding. Overall, the results highlight that through FC, teachers can develop students’ mathematical potential with FCs.

In this study, we investigated which teacher characteristics influence student proficiency development regarding fractions in Grade 5 of Dutch primary education. At least three domains of research (i.e. perspectives) on effective teaching can be distinguished: studies focusing on teachers’ background characteristics, on their knowledge and conceptions regarding the subject they are teaching and on the domain-specific and general pedagogical characteristics of their teaching. In this study, effects of the three perspectives on student fraction proficiency were examined simultaneously using multilevel analyses. Findings revealed that teachers’ age and experience in the upper grades, their pedagogical content knowledge and the degree of student participation in their lessons had positive effects. Their subject matter knowledge, quality of their concept maps and the richness of the mathematics in their lessons had negative effects. Thus, effects were found pertaining to all three perspectives on teachers and their teaching we included in our study.

It is commonplace in the educational literature on mathematical practice to argue for a general conclusion from isolated quotations from famous mathematicians. In this paper, we supply a critique of this mode of inference. We review empirical results that show the diversity and instability of mathematicians’ opinions on mathematical practice. Next, we compare mathematicians’ diverse and conflicting testimony on the nature and purpose of proof. We lay especial emphasis on the diverse responses mathematicians give to the challenges that digital technologies present to older conceptions of mathematical practice. We examine the career of one much cited and anthologised paper, WP Thurston’s ‘On Proof and Progress in Mathematics’ (1994). This paper has been multiply anthologised and cited hundreds of times in educational and philosophical argument. We contrast this paper with the views of other, equally distinguished mathematicians whose use of digital technology in mathematics paints a very different picture of mathematical practice. The interesting question is not whether mathematicians disagree—they are human so of course they do. The question is how homogenous is their mathematical practice. If there are deep differences in practice between mathematicians, then it makes little sense to use isolated quotations as indicators of how mathematics is uniformly or usually done. The paper ends with reflections on the usefulness of quotations from research mathematicians for mathematical education.

The contribution focuses on issues related to the implementation of formative assessment methods into inquiry based teaching, by means of issues related to solving twelve multiple-step arithmetic word problems based on operations with natural and rational numbers. These word problems have multiple correct solution procedures and the presented qualitative exploratory empirical study investigates how varied and how usual might be correct solution procedures provided by diverse groups of solvers – future primary school teachers attending diverse university mathematics courses of diverse forms and/or time extent. According to written data collected from 149 solvers, six notions are introduced in the paper: majority, minority and even solution procedures, and majority, minority and mixed solvers. Issues regarding minority solvers are recognized as an important element for implementing formative assessment methods. All the six notions are illustrated in the paper by samples of solution procedures and diagrams of relative frequency. Implications are given for formative assessment within any kind of education involving multiple-step word problems, regardless of the extent of implemented inquiry.

The goal of this paper is to promote the use of R and Python as high-level, free, open-source programming environments that can be used as a computational and visualization tool for teaching differential equations. Both R and Python also allow for creating reproducible dynamic documents using Markdown, which combines live code, plain text, and expressions to generate different formats (pdf, html, Word), including the numerical and graphical output from the code, along with the code itself, properly formatted with minimum effort. This technology can be used for creating interactive teaching documents, randomized exams, homework and project reports, and even publication quality papers. We illustrate this technology with several examples implemented in R and Python. This approach has been classroom-tested, with some promising results. This article has online supplemental material posted on the PRIMUS website.

The paper investigates how the order of numerical data in word problems on division of a whole into unequal parts affects achievement and reasoning of 14-16-year old pupils. The variable was altered in two word problems, in one of which also the context was changed (psychological variable) and in the latter “if-clause” is or is not used (linguistic variable). 182 pupils were involved in the experiment. The solutions were analysed quantitatively using Item Response Theory as well as qualitatively. The data suggest that pupils’ success is affected by the order of numerical data in the statement in an unfamiliar context. The presence of “if-clause” in the statement was studied in a two-level problem. The order of numerical data played its role in case of formulations without “if”. The results of the experiment are of interest for mathematics education as well as for construction of tests. The paper is an extended version of the paper by Novotná (2018).

The paper presents and explores the experience of maths students studying in a context shaped by a core concept maths curriculum. The three vignettes that illuminate experience are drawn from a larger research project that worked with five teachers and 630 learners aged 16&ndash, 18 in Further Education classrooms in England. Analysis involved distinguishing different understandings of being good at maths, different views of a good maths curriculum and identifying enablers and barriers to being a &lsquo, successful&rsquo, maths student. Dewey&rsquo, s ideas about focused experience and collateral learning were used to deepen this analysis. The paper reports a surprising finding. In some cases, students recognize the positive impact learning maths had on developing their wider human capabilities. Maths teachers in England, working in the context of &lsquo, new public management&rsquo, may find reasons to take heart from the accounts of teaching and learning presented. For the international reader who is grappling with the challenge of reengaging maths students, the accounts of what matters to students could spur a reconsideration of priorities and practice.

In this survey paper, we describe the state of the field of research on teaching mathematics with technology with an emphasis on the secondary school phase. We synthesize themes, questions, results and perspectives emphasized in the articles that appear in this issue alongside the relevant foundations of these ideas within the key journal articles, handbooks and conference papers. Our aim is to give an overview of the field that provides opportunities for readers to gain deeper insights into theoretical, methodological, practical and societal challenges that concern teaching mathematics with technology in its broadest sense. Although this collection of articles was developed prior to the global coronavirus pandemic, we have taken the opportunity to survey the contributing authors to provide some country perspectives on the impact the pandemic has had on mathematics teaching with technology in the period January–July 2020. We conclude the survey paper by identifying some areas for future research in this increasingly relevant topic.

This paper begins with a brief overview of literature indicating that, although there have been significant advances in the field’s capacity to conduct both formative and summative assessments over the past decades, those advances have not been matched by comparable impact. The bulk of the paper is devoted to a series of examples from the Mathematics Assessment Project that illustrate issues of methods, and the unrealized potential for advances.

This paper uses a post-qualitative philosophical perspective to find new ways of understanding teaching and learning. The paper presents a series of examples that were used in a longitudinal study,...

With a focus on resources for learning through writing, this paper compares the interdisciplinary framings of two teacher education programmes. What are the implications of these framings and resources for students’ possibilities when they write independent projects/bachelor theses? The paper presents a case study with data (interviews, texts, observations) from two different teacher education programmes. The conceptual framework stems from the Academic Literacies approach. The analysis shows that one of the programmes shapes interdisciplinarity by a “block approach” that allows students to work independently, while the other is shaped as a “bridging approach” with tight collaboration between students and tutors. The two educational programmes apply the same national regulating documents in different ways, thus creating different possibilities for students to learn and write as “independent” or “collaborative” students, respectively. The results contribute to a discussion of higher education structure and the role of writing within it.

In nineteenth-century America, students buried their mathematics books. This practice consistently celebrated the milestone of passing through collegiate mathematics, yet it changed due to national events. This article considers the case of Bowdoin College, where students buried their books differently before and after the Civil War. Antebellum, they observed a complex “Burial of Calculus” with songs, parades, and mock prayers. Postbellum, students personified their books as a woman, placing stones marked “Anna” on the textbooks’ graves. Using archival investigations of students' pamphlets and textbooks, this paper argues that these changes resulted from the war's effects on education as well as changing attitudes toward death. Both the antebellum and postbellum rituals communicated understandings of mathematics and academic achievement, as connected through a mock funeral ritual. Through investigating these connections, this paper asserts the importance of student practices for our understanding of Civil War era education.

In this paper we report on the outcomes of two surveys carried out in higher education institutions of Ireland; one of students attending first-year undergraduate non-specialist mathematics modules and another of their lecturers. The surveys aimed to identify the topics that these students found difficult, whether they had most difficulty with the concepts or procedures involved in the topics, and the resources they used to overcome these difficulties. In this paper we focus on the mathematical concepts and procedures that students found most difficult. While there was agreement between students and lecturers on certain problematic topics, this was not uniform across all topics, and students rated their conceptual understanding higher than their ability to do questions, in contrast to lecturers’ opinions.

This paper attends to teacher intellectual risk-taking when attached to expression of positive emotions, in order to explore some of the reasons why teacher risk-taking may not appear in mathematics lessons. We know that risk-taking can be beneficial, but research has not really examined what form this might take in a classroom. In recent research, I investigated how positive emotions are discussed and used by experienced mathematics teachers. In particular how to examine the ‘in-the-moment’ emotions of the teacher, and what the modelling of experienced teachers tells us about the role of affect in mathematics teaching. This paper examines some affect episodes for elements of teacher risk-taking. The evidence suggests that teacher risk-taking enables the use of emotions, and vice versa, is integral to ‘good’ teaching, and that, in Bandura’s Social Learning Theory terms, modelling such behaviours appears beneficial to student learning and should be encouraged.

[This paper is part of the Focused Collection on Quantitative Methods in PER: A Critical Examination.] A relevant aim of research in education is to find and study the reasoning lines that students deploy when dealing with problematic situations. This can be done through an analysis of the answers students give to a questionnaire. In this paper, we discuss some methodological aspects involved in the quantitative analysis of a questionnaire by means of two different clustering methods, a hierarchical one and a nonhierarchical one. We start from the coding procedures needed to obtain analyzable data from the questionnaire and from a definition of a correlation coefficient suitable for measuring student similarity in the case of binary coding of student answers. Then, criteria to choose the optimal number of clusters are discussed, and for the same purpose a new coefficient is introduced that measures the total amount of information we can obtain from a clustering solution. We show that each cluster can be characterized by its centroid that summarizes the answers most frequently given by the cluster students to the questionnaire. Finally, an example of the application of these procedures to a student sample is given, and a comparison between the two clustering methods is discussed.

This paper reports on a study that aims to address the challenges of UN Sustainable Development Goal 4 to ensure inclusive and equitable quality education for all. The study focuses on school mathematics in particular. With regard to ensuring equitable access to quality education, it is argued that there is a need to consider the epistemic quality of what students come to know, make sense of and be able to do in school mathematics. Accordingly, the aim is to maximize the chances that all pupils will have epistemic access to school mathematics of high epistemic quality. The study is based on the theoretical framework of Joint Action Theory in Didactics (JATD). Associated research questions focus on the quality of teacher-student(s) joint action and on the epistemic quality of the content. The paper draws on empirical research findings of the Developing Mathematical Thinking in the Primary Classroom (DMTPC) project (2010–12) and also on the findings of a parallel study of mathematics teachers’ assessment practices in Ghana. One teacher’s action research project is used as an exemplar to illustrate how mathematics can become more accessible and inclusive thus leading to an evolution in mathematical thinking and high-quality epistemic access for all.

This paper describes efforts by the author to engage and motivate students in undergraduate research by giving them a large audience and engaging the media in disseminating their work. I provide an introduction to public relations from the point of view of a mathematics professor, and describe some lessons I have learned in my own attempts to engage newsprint, radio, and television sources in stories about undergraduate research in mathematics. After describing some partially successful early attempts, the paper discusses a recent event in which undergraduate research in mathematics became, briefly, a “hot news item” in Central Washington. The paper concludes with some thoughts about the benefits of this type of work to our students and our profession.

Proportional reasoning, the ability to use ratios in situations involving comparison of quantities, is essential for mathematical competence, especially in the middle school years, and is an important determinant of success beyond school. Research shows students find proportional reasoning and its foundational concepts difficult. Proportional reasoning does not always develop naturally, however some research suggests that with targeted teaching, its development can be promoted. This paper reports on a large Australian study involving over 130 teachers and their students. A major goal of the study was to investigate the efficacy of ongoing teacher professional development for promoting middle years students’ proportional reasoning. A series of professional development workshops was designed to enhance the teachers’ understanding of proportional reasoning and to extend their repertoire of teaching strategies to promote their students’ proportional reasoning skills. The workshop design was informed by research literature on proportional reasoning teaching and learning as well as the results of a diagnostic instrument administered to over 2500 middle years students prior to the professional development. Between workshops, the teachers implemented a variety of targeted teaching activities. This paper reports on pre- and post- instrument student data collected at the beginning and end of the first year of the project (i.e., after completion of half of the workshops). The findings suggest that targeted professional development and explicit teaching can make a difference to students’ proportional reasoning.

This paper uses a sociocultural conceptual framework to provide an integrated view of academic literacy in mathematics for English Learners. The proposed definition of academic literacy in mathematics includes three integrated components: mathematical proficiency, mathematical practices, and mathematical discourse. The paper uses an analysis of a classroom discussion to illustrate how the three components of academic literacy in mathematics are intertwined, how academic literacy in mathematics is situated, and how participants engaged in academic literacy in mathematics use hybrid resources. The paper closes by describing the implications of this integrated view of academic literacy in mathematics for mathematics instruction for English Learners, arguing that it is important that the three components not be separated when designing instruction in general, and it is essential that mathematics instruction for English Learners address these three components simultaneously.

The impact of online math programs depends on its implementation, especially in vulnerable populations from developing countries. An existing online platform was adapted, at the request of the Chilean Ministry of Education, to exclusively include exercises previously designed and tested by a paper-based government program for elementary school. We carried out a cluster-randomized controlled trial (RCT) with 50 fourth grade classrooms. Treatment classrooms used the platform in a weekly 90-min math session. Due to a social instability outbreak in the country, a large unexpected disruption with huge absenteeism occurred in the second half of the semester, which turned this study into a unique opportunity to explore the robustness of the platform&rsquo, s effects on students&rsquo, learning. Using multiple imputation and multilevel models, we found a statistically significant effect size of 0.13, which corresponds to two extra months of learning. This effect is meaningful for four reasons. First, it has double the effect of the paper-based version. Second, it was achieved during one semester only. Third, is half that obtained with the platform for a complete year with its own set of exercises and with two sessions per week instead of one. Fourth, it was attained in a semester with a lot of absenteeism.

This research paper reports strategies used by Grade 6 learners in multiplying whole numbers in five selected primary schools in Kavango East and West regions. A total of 200 learners’ mathematics exercise books were analysed in order to identify the commonly used strategies by learners in multiplying whole numbers. A total of ten teachers teaching grade 6 mathematics were also requested to complete a questionnaire which required them to indicate the strategies that they employed in class when teaching multiplication of whole numbers. The teachers indicated that they used a variety of strategies including repeated addition, complete-number (Including doubling), partitioning and compensation to teach multiplication of whole numbers. The results also disclosed that the majority of the learners’ mathematics exercise books reflected the use of the traditional method of repeated addition contrary to the teachers’ claims. It was also found that a few of the learners used other strategies such as long method, short method and learner “invented” strategies. Additionally, the mathematics curriculum for upper primary learners (Grade 4-7 mathematics syllabus) requires learners to use paper and pencil algorithms to carry out multiplication of whole numbers without calculators (Ministry of Education, Arts & Culture [MoEAC], 2015, p. 2). However, at Grade 6, learners were expected to use paper and pencil algorithms to multiply numbers within the range 0-100000. Analysis of the learners’ exercise books indicated that the majority were not able to multiply a two digit by a single digit, a two digit by a two digit and a three digit by a two digit number.

After a long six-year lapse, the Curriculum and Assessment Policy Statement introduced in 2012 included geometry as part of the South African Grade 12 Mathematics Paper 2. The first cohort of matriculation students wrote Paper 2 in 2014. This article reports on the understanding of geometry terminology with which a group of 154 first-year mathematics education students entered a rural South African university in 2015; 126 volunteered to be part of the study. Responses to a 60-item multiple-choice questionnaire (30 verbally presented and 30 visually presented items) in geometry terminology provided the data for the study. A concept’s verbal description should be associated with its correct visual image. Van Hiele theory provided the lens for the study. An overall percentage mean score of 64% obtained in the test indicated that the majority of the students had a fairly good knowledge of basic geometry terminology. The students obtained a percentage mean score of 68% on visually presented items against that of 59% on verbally presented items implying a lower level thinking as per Van Hiele theory. The findings of this study imply a combination approach using visual and verbal representations to enhance conceptual understanding in geometry. This has to be complemented and supplemented through scaffolding to fill student teachers’ content gap.

Part of the international reflection on the use of history in mathematics teaching consists in a quest of frameworks and models suitable for empirical studies. Following this demand, this paper explores the way Balacheff’s cKȼ model, a model taken from the didactics of mathematics, can be used in the analysis of learning at student level. In the first part of this paper, Balacheff’s cKȼ model (conceptions, knowledge, ȼoncepts) is shortly presented, and in the second part, the relationship between the epistemological background of the model and the use of the history of mathematics is explored in order to show a possible suitableness. The third part addresses an example of a school activity (about ancient Indian geometry) in which the model is applied and the historical issues clarified. Questioning the role of problems both in the cKȼ model and in the use of history, the last part shows how a study at the students’ conception level enlightens the way in which historical elements can interact with contemporary mathematical learning.

Observation-based frameworks of instructional quality differ largely in the approach and the purposes of their development, their theoretical underpinnings, the instructional aspects covered, their operationalization and measurement, as well as the existing evidence on reliability and validity. The current paper summarizes and reflects on these differences by considering the 12 frameworks included in this special issue. By comparing the analysis of three focal mathematics lessons through the lens of each framework as presented in the preceding papers, this paper also examines the similarities, differences, and potential complementarities of these frameworks to describe and evaluate mathematics instruction. To do so, a common structure for comparing all frameworks is suggested and applied to the analyses of the three selected lessons. The paper concludes that although significant work has been pursued over the past years in exploring instructional quality through classroom observation frameworks, the field would benefit from establishing agreed-upon standards for understanding and studying instructional quality, as well as from more collaborative work. 50 3 535 553

In this paper, we extend a mixed method (MM) approach to lesson observation and analysis used in previous research in England, combining multiple structured observation instruments and qualitative field notes, to provide a framework for studying three videotaped lessons from 3rd-grade US mathematics classrooms. Two structured observation schedules are employed, one subject-specific and research-oriented and the other generic and inspection-oriented. Both instruments were previously developed based on evidence from the teacher effectiveness research (TER) knowledge base. Qualitative field notes, in addition to structured observation schedules, provide detailed narratives for each lesson video. Separate findings from each instrument and approach are presented, followed by an integrated analysis and synthesis of results. Although previous studies used similar methods to analyze teaching practice within broader research designs incorporating additional methods and perspectives (e.g. teacher interviews, pupil assessments, pupil questionnaires), this paper explicitly examines the strengths and limitations of the multi-instrument, mixed method approach to lesson observation. Using multiple observation instruments allows for triangulation as well as consideration of complementary foci (i.e. a content-specific instrument measures fine-grained aspects of practice not emphasized in a more generic instrument, and vice versa). Field notes facilitate rich descriptions and more thorough contextualization and illumination of teaching practice than structured observation ratings alone. Further, the MM approach allows for consideration of lesson features beyond those established in TER literature as sufficient to characterize ‘effective’ practice.

Researchers from different fields have developed different observational instruments to capture instructional quality with a focus on generic versus content-specific dimensions or a combination of both. As this work is fast accumulating, the need to explore synergies and complementarities among existing work on instruction and its quality becomes imperative, given the complexity of instruction and the increasing realization that different frameworks illuminate certain instructional aspects but leave others less visible. This special issue makes a step toward exploring such synergies and complementarities, drawing on the analysis of the same 3 elementary-school lessons by 11 groups using 12 different frameworks. The purpose of the current paper is to provide an up-to-date overview of prior attempts made to work at the intersection of different observational frameworks. The paper also serves as the reference point for the other papers included in the special issue, by defining the goals and research questions driving the explorations presented in each paper, outlining the criteria for selecting the frameworks included in the special issue, describing the sampling approaches for the selected lessons, presenting the content of these lessons, and providing an overview of the structure of each paper. 50 3 355 366

International audience; This paper examines the existence and impact of theoretical mathematical foundations on the teaching and learning of whole numbers in elementary school in France. It shows that the study of the New Math reform –which was eventually itself replaced in the longer term – provides some keys to understanding the influence of mathematical theories on teaching and learning. The paper studies changes related to place value, a notion that was deeply impacted by the introduction of numeration bases other than ten in 1970, and their subsequent removal in the 1980s. What the author terms ‘numeration units’ (ones, tens, hundreds, thousands, etc.) and ‘powers-of-ten written in figures’ (1, 10, 100, 1000, etc.) are key tools for describing and understanding changes. The author identifies two theories that have formed the basis for place value teaching in the 20th century, and examines some aspects of their influence. The paper also addresses epistemological issues in the relation between academic mathematics and school mathematics, and highlights the role of units in the teaching of basic arithmetic.

Sir Michael Wilshaw, the head of the Office for Standards in Education (OfSTED), declared a 'new wave' of Local Area Under-performance Inspections (LAUI) of schools 'denying children the standard of education they deserve'. This paper examines how the threat of LAUI played out over three mathematics lessons taught by a teacher in her first year in the profession. A Foucauldian approach is mobilised with regard to disciplinary power and 'docile bodies'. The paper argues that, in the case in point, LAUI was a tool mediating performative conditions and, ultimately, the docile body. The paper will be of concern to policy sociologists, teachers, school leaders, and those interested in school inspection.

This paper documents the author’s adaptation of team-based learning (TBL), an active learning pedagogy developed by Larry Michaelsen and others, in the linear algebra classroom. The paper discusses the standard components of TBL and the necessary changes to those components for the needs of the course in question. There is also an empirically controlled analysis of the effects of TBL on the student learning experience in the first year of TBL use.

The purpose of this paper is to explore the relationship of subject areas to teachers’ technology integration. Educational technology research has often identified “culture clashes” to explain differences in technology use between subject areas. These clashes are frequently attributed to core features, values and beliefs held in the subject area cultures, but there has been little research analyzing the relationship between subject areas and integration. Using a validated path model as a conceptual framework, this paper presents an analysis of the relationship between three subject areas (English, Mathematics, Science) and known factors of teacher beliefs and readiness to use technology in teaching, which directly impact on secondary-level teachers’ technology integration, over three years. Findings show that time and subject areas are both associated with teachers’ readiness, but only subject areas are associated with teachers’ beliefs. Implications for practice and future research are discussed.

This research paper reports on phase one of an investigation of video recorded intensive one-to-one teaching interactions with 6–7-year-old students who were in their second year of schooling in Australia and identified by the their teacher as low attaining in early number. The two-phased study from which this paper emerges was originally conducted in 1998 as part of my Bachelor of Teaching Honours (Research) program at Southern Cross University Lismore, New South Wales. That study identified teaching interactions particularly suited to one-to-one teaching in the Maths Recovery Program, a program designed for these students who were at risk of failure in early number. Since that time a great deal has not changed with limited literature available that comprehensively reports on teaching interactions in intensive one-to-one settings. Revisiting the original study is considered timely given the increasing number of withdrawal and intensive programs now funded and adopted by schools and yet, rarely reported on in terms of the effectiveness of the teaching interactions that occur in such settings. This paper then presents a discussion of a preliminary series of teaching interactions that either positively and or negatively influence an intensive one-to-one teaching and learning setting.

Published version of an article in the journal: Educational Studies in Mathematics. Also available from the publisher at: http://dx.doi.org/10.1007/s10649-013-9489-z This paper reports a case study from a mathematics teaching developmental research project. The theoretical foundation for the research comprises communities of inquiry and critical alignment, with which the developmental methodology has a particular synergy. This synergy is the main focus of the paper. The paper elaborates theoretical and methodological antecedents of the project and traces these through a case study of developments in the practices of one upper secondary school team and a group of university didacticians (mathematics teacher educators and researchers) during the first year of the project. The case study reveals that critical alignment and inquiry (necessarily) bring uncertainty and risk, and foster tensions within the teachers’ practice and between the practices of teachers and didacticians. In exposing these uncertainties, risks and tensions, the paper points to their value for the learning and knowledge gained by participants.

In this paper, I comment on the set of papers in this special issue on mathematical problem posing. I offer some observations about the papers in relation to several key issues, and I suggest some productive directions for continued research inquiry on mathematical problem posing.

This paper proposes a reflection on the experience of a teaching intervention, in the format of higher education teaching training, in the Mathematics course of a university in Rio Grande do Sul, Brazil. The intervention was carried out in the teaching practice subject, which focused on the discussion of Brazilian National Curricular Parameters for Basic Education. This production is marked by moments, which, although separate are complementary and interrelated. The lessons given by the professional teacher were observed and analyzed as well as the activities employed by the teacher in pre-service training and these were followed by analysis of the undergraduate students’ registers, resulting from the reflections upon the activities carried out during the intervention. Contributions originated in this paper are mainly the links found by the undergraduate students among the fields, blocks of content and concepts of Mathematics in elementary school. The interrelations built reveal possibilities for the treatment of mathematical concepts in teacher education and project the trainees as elementary school mathematics teachers. Keywords: Teaching practice. Mathematics in Elementary School. Mathematics Instruction.

Background:This paper reports on one part of a large study which attempted to identify the linguistic and hierarchical characteristics of van Hiele theory amongst grade 10 learners.Materials and methods:The sample consisted of a total of 359 participants from five conveniently selected schools from Mthatha District in the Eastern Cape Province of South Africa. The performance of these learners on a written geometry test was analysed. On the basis of the analysis, 30 learners were selected to be the subjects for interviews.This paper focuses on the analysis of the interview transcripts to discern the van Hiele characteristics among learners’ geometrical tasks.The ethical requirements were met.Results:This paper reports particularly on linguistic and hierarchical characteristics of the van Hiele theory.Conclusions:The findings from the study affirmed the existence of the linguistic and hierarchic characteristics of the van Hiele theory.

The UK Government has set a goal that the ‘vast majority’ of students in England will be studying mathematics to 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 paper reports findings from a cluster-sampled survey of over ten thousand 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 paper considers the implications of these findings in the current policy landscape.

There is growing support for making the study of mathematics to age 18 compulsory for all young people in England. This paper aims to inform this debate through new insights into historic A-level Mathematics participation trends. We analyse full-year cohorts from the Department for Education’s National Pupil Database for age-16 students from 2004-2010, a total of just over 4.5 million young people. Using a cohort-tracking approach we aim to better understand the flow of young people through upper secondary mathematics education. Earlier work identified GCSE attainment as the strongest predictor of A-Level Mathematics participation. In this paper we show that the percentage of students progressing to A-Level by GCSE grade has not changed significantly over the period in question, with some exceptions. This implies that the increase in A-level Mathematics numbers is largely explained by the growing proportion of higher GCSE grades. We discuss the implications for policy that this raises, e.g. the possible impact of making GCSE mathematics more demanding.

“Success stories,” i.e., cases in which mathematical problems posed in a controlled setting are perceived by the problem posers or other individuals as interesting, cognitively demanding, or surprising, are essential for understanding the nature of problem posing. This paper analyzes two success stories that occurred with individuals of different mathematical backgrounds and experience in the context of a problem-posing task known from past research as the Billiard Task. The analysis focuses on understanding the ways the participants develop their initial ideas into problems they evaluate as interesting ones. Three common traits were inferred from the participants' problem-posing actions, despite individual differences. First, the participants relied on particular sets of prototypical problems, but strived to make new problems not too similar to the prototypes. Second, exploration and problem solving were involved in posing the most interesting problems. Third, the participants' problem posing involved similar stages: warming-up, searching for an interesting mathematical phenomenon, hiding the problem-posing process in the problem's formulation, and reviewing. The paper concludes with remarks about possible implications of the findings for research and practice.

This paper applies and extends Valsiner’s ‘zone theory’ (zones of free movement and promoted actions) through an examination of an intervention to establish inquiry communities in primary mathematics classrooms. Valsiner’s zone theory, in a classroom setting, views students’ freedom of choice of action and thought as mediated by the teacher. The extension of this theory adds that teachers’ freedom of choice of action and thought is mediated by ‘significant others’ and also by the actions of students with new freedoms. The paper presents necessary theoretical constructs and provides extracts from lessons and from teacher-teacher collaboration to illustrate the theoretical extension.

THIS PAPER FOCUSES ON VERY young students' ability to engage in repeating pattern tasks and identifying strategies that assist them to ascertain the structure of the pattern. It describes results of a study which is part of the Early Years Generalising Project (EYGP) and involves Australian students in Years 1 to 4 (ages 5 – 10). This paper reports on the results from the early years' cohort (Year 1 and 2 students). Clinical interviews were used to collect data concerning students' ability to determine elements in different positions when two units of a repeating pattern were shown. This meant that students were required to identify the multiplicative structure of the pattern. Results indicate there are particular strategies that assist students to predict these elements, and there appears to be a hierarchy of pattern activities that help students to understand the structure of repeating patterns.

The set of papers presented in this issue comprise a multiple-case study which attends to instructional resources-teacher knowledge and curriculum materials-to understand how they individually and jointly contribute to instructional quality. We approach this inquiry by comparing lessons taught by teachers with differing mathematical knowledge for teaching who were using either the same or different editions of a US Standards-based curriculum. This introductory paper situates the work reported in the next four case-study papers by outlining the analytic framework guiding the exploration and detailing the methods for addressing the research questions. © 2012 Charalambos Y. Charalambous and Heather C. Hill. 44 4 443 466 Cited By :24

This paper presents the main results of the international study of students’ mathematics and science achievement TIMSS 2011. The focus is on presentation and discussion of the results achieved by Serbian students. In Serbia, this study was conducted in the fourth grade of primary school on a nationally representative sample of students - a total of 4379 students from 156 primary schools were tested, i.e. 219 classes. The average achievement of students from Serbia was 516 points in mathematics and in science, which in both cases represents achievement that is significantly higher than the center point of the TIMSS 4th grade scale. The paper also presents results according to content domains, differences in achievement between boys and girls, as well as students’ distribution according to international benchmarks. The results are observed from the view point of overlapping of the Serbian curricula and the TIMSS test. Overall, these results can be seen as encouraging indicators of the Serbian education system for the first education cycle, particularly if we bear in mind relatively poor performance of students in higher grades in Serbia which had been shown in various previous studies. Since TIMSS findings should primarily be used in planning the development of education, the concluding part discusses the importance of regular participation of Serbia in international studies of this kind, as well as the importance of greater involvement of domestic researchers in analysis of available international databases. [Projekat Ministarstva nauke Republike Srbije, br. 47008: Vrednovanje postignuca ucenika u matematici i prirodnim naukama - TIMSS 2011, Unapređivanje kvaliteta i dostupnosti obrazovanja u procesima modernizacije Srbije i br. 179034: Od podsticanja inicijative, saradnje i stvaralastva u obrazovanju do novih uloga i identiteta u drustvu]