Deep intuition as a level in the development of the concept image

To explicate certain phenomena, e.g., the possibility of deduction without definition, we hypothesize that an individual is able to understand and appreciate reasoning with a due feeling of its necessity when the concept image of each concept involved in the reasoning has reached a certain level of development; we then speak of deep intuition. This conception is presented (with a variety of examples) in the framework of D. Tall’s theory of three worlds of mathematics (‘conceptual-embodied’, ‘proceptual-symbolic’, and ‘formal-axiomatic’).

Mathematical enculturation from the students’ perspective: shifts in problem-solving beliefs and behaviour during the bachelor programme

This study investigates the changes in mathematical problem-solving beliefs and behaviour of mathematics students during the years after entering university. Novice bachelor students fill in a questionnaire about their problem-solving beliefs and behaviour. At the end of their bachelor programme, as experienced bachelor students, they again fill in the questionnaire. As an educational exercise in academic reflection, they have to explain their individual shifts in beliefs, if any. Significant shifts for the group as a whole are reported, such as the growth of attention to metacognitive aspects in problem-solving or the growth of the belief that problem-solving is not only routine but has many productive aspects. On the one hand, the changes in beliefs and behaviour are mostly towards their teachers’ beliefs and behaviour, which were measured using the same questionnaire. On the other hand, students show aspects of the development of an individual problem-solving style. The students explain the shifts mainly by the specific nature of the mathematics problems encountered at university compared to secondary school mathematics problems. This study was carried out in the theoretical framework of learning as enculturation. Apparently, secondary mathematics education does not quite succeed in showing an authentic image of the culture of mathematics concerning problem-solving. This aspect partly explains the low number of students choosing to study mathematics.

James J. Kaput (1942–2005) imagineer and futurologist of mathematics education

Jim Kaput lived a full life in mathematics education and we have many reasons to be grateful to him, not only for his vision of the use of technology in mathematics, but also for his fundamental humanity. This paper considers the origins of his ‘big ideas’ as he lived through the most amazing innovations in technology that have changed our lives more in a generation than in many centuries before. His vision continues as is exemplified by the collected papers in this tribute to his life and work.

School mathematics and its everyday other? Revisiting Lave’s ‘Cognition in Practice’

In the last three decades there have been a variety of studies of what is often referred to as ‘everyday’ or ‘street’ mathematics. These studies have documented a rich variety of arithmetic practices involved in activities such as tailoring, carpet laying, dieting, or grocery shopping. More importantly, these studies have helped to rectify outmoded models of rationality, cognition, and (school) instruction. Despite these important achievements, doubts can be raised about the ways in which theoretical conclusions have been drawn from empirical materials. Furthermore, while these studies rightly criticised prevalent theories of rationality and cognition as too simplistic to account for everyday activities, it seems that some of the proposed alternatives suffer from similar flaws (i.e., are straightforward inversions of the to-be-opposed theories, rather than more nuanced views on complicated issues). In this article we illustrate our sceptical view by discussing four case studies in Jean Lave’s pioneering and influential ‘Cognition in Practice’ (1988). By looking at the case studies in detail, we investigate how Lave’s conclusions relate to the empirical materials and offer alternative characterisations. In particular, we question whether the empirical studies demonstrate the existence of two different kinds of mathematics (‘everyday’ and ‘school,’ or ‘formal’ and ‘informal’) and whether school instruction tries to replace the former with the latter.

Participation,financial support and the marginal student

This paper examines differences between the decision-making of marginal and nonmarginal students about participation in higher education (HE). We distinguish between two kinds of marginality: being ‘borderline’ on account of prior achievements in school and being ‘unsure’ after taking prior achievement into account. We identify a significant minority of students in their final year of schooling who are unsure about participation in higher education even though they have prior school achievements typical of entrants to HE. Being ‘unsure’ is found to be unrelated to socioeconomic background, but it is associated with significantly different attitudes towards the potential benefits and risks of participation, different levels of knowledge about financial support that is available and different approaches to information search about participation in HE.

From the Didactic to the Dialectic—Intercultural Exchanges in Contemporary Counsellor Education

Within the context of the growing development of intercultural counsellor education, the question of how different cultures reconceptualize and transform Western counselling theories for their own context is an important one. In this intercultural exchange in education, concepts such as ‘globalization’, ‘indigenization’ and ‘universalization’ have an impact on the education process. This paper briefly explores these terms and movements and then highlights findings from a research project that took place with a group of counselling graduates in Hong Kong examining how they undertook the processes of reconceptualization and transformation. The dialogical process involved in that enabled participants and researcher to co-explore the process of knowledge development within the counselling education field. Implications for counsellor education are highlighted.

Computer Manipulatives in an Ordinary Differential Equations Course: Development, Implementation, and Assessment

The d’Arbeloff Interactive Mathematics Project or d’AIMP is an initiative that seeks to enhance and ultimately transform the teaching and learning of introductory mathematics at the Massachusetts Institute of Technology. A result of this project is a suite of “mathlets,” a carefully developed set of dynamic computer applets for use in the university’s ordinary differential equations course. In this paper, we present the rationale for such computer innovations, the philosophy behind their design, as well as a discussion of their careful development and implementation. Survey results are reported which yielded positive student feedback and suggestions for improvement.

Prospective teachers’ reasoning and response to a student’s non-traditional strategy when dividing fractions

Recognizing meaning in students’ mathematical ideas is challenging, especially when such ideas are different from standard mathematics. This study examined, through a teaching-scenario task, the reasoning and responses of prospective elementary and secondary teachers to a student’s non-traditional strategy for dividing fractions. Six categories of reasoning were constructed, making a distinction between deep and surface layers. The connections between the participants’ reasoning, their teaching response, and their beliefs about mathematics teaching were investigated. We found that there were not only differences but also similarities between the prospective elementary and secondary teachers’ reasoning and responses. We also found that those who unpacked the mathematical underpinning of the student’s non-traditional strategy tended to use what we call “teacher-focused” responses, whereas those doing less analysis work tended to construct “student-focused” responses. These results and their implications are discussed in relation to the influential factors the participants themselves identified to explain their approach to the given teaching-scenario task.

‘Interdisciplinary strategies’ in U.S. research universities

In the context of increasing support for interdisciplinary modes of research, many in the policy, scientific, and academic communities propose that universities should change structurally to reduce the barriers to investigation that involves researchers from multiple disciplines. This paper examines ‘interdisciplinary strategies’ in U.S. research universities—deliberate efforts to spur collaborative research across traditional departmental and disciplinary boundaries, including the creation and adaptation of university policies, practices, and structures. It identifies and analyzes the use of incentive grants to initiate new interdisciplinary units, the establishment of ‘campus-wide institutes’ that steer campus investments in interdisciplinary areas, and new modes of faculty hiring and evaluation. Illustrative examples are provided, and the implications of these strategies are discussed.

Powering Up, Philanthropically

Faculty members have an important role to play in helping to fund programs and direct donor dollars at colleges and universities. This paper outlines the ‘golden age of philanthropy’ that the United States has currently entered, explaining the combination of material resources and spiritual inclinations that have led to increased giving. It then outlines several principles that faculty members should keep in mind in approaching prospective donors, including the reasons for giving, donors’ interest in ideas, and the importance of persistence, local focus, and collaboration. Finally it offers several practical steps that faculty members can take to marshal philanthropic support for their visions.

On semiotics and subjectivity: a response to Tony Brown’s “signifying ‘students’, ‘teachers’, and ‘mathematics’: a reading of a special issue”

In this response we address some of the significant issues that Tony Brown raised in his analysis and critique of the Special Issue of Educational Studies in Mathematics on “Semiotic perspectives in mathematics education” (Sáenz-Ludlow & Presmeg, Educational Studies in Mathematics 61(1–2), 2006). Among these issues are conceptualizations of subjectivity and the notion that particular readings of Peircean and Vygotskian semiotics may limit the ways that authors define key actors or elements in mathematics education, namely students, teachers and the nature of mathematics. To deepen the conversation, we comment on Brown’s approach and explore the theoretical apparatus of Jacques Lacan that informs Brown’s discourse. We show some of the intrinsic limitations of the Lacanian idea of subjectivity that permeates Brown’s insightful analysis and conclude with a suggestion about some possible lines of research in mathematics education.

Competence-Based Vocational Education and Training (VET): the Cases of England and France in a European Perspective

This paper examines the notion of ‘competence’ in the VET systems of France and England. While both countries have developed ‘competence-based’ approaches, underlying the similar terminology are distinct meanings, rooted in the countries’ institutional structures and labour processes. A key distinction is identified between a knowledge-based model in France and a skills-based model in England. Competence in the French sense is multi-dimensional and relies on the integration of practical and theoretical knowledge, as well as personal and social qualities within a broadly defined occupational field. By contrast, in England, competence refers to the performance of fragmented and narrowly defined tasks, with minimal underpinning knowledge. Thus, whereas ‘competence’ in the English VET system usually denotes functional employability for what may be relatively low-skilled employment, in France, it encapsulates the multi-dimensional development of the individual as a citizen as well as an employee.

Challenges in deepening prospective teachers’ understanding of multiplication through justification

This article reports challenges faced by prospective elementary teachers as they revisited whole number multiplication through a sequence of tasks that required them to develop and justify reasoning strategies for multiplication. Classroom episodes and student work are used both to illustrate these challenges, as well as to demonstrate growth over time. Implications for the design of mathematics courses for prospective teachers’ are discussed. Although the study is situated in the context of multiplication, it has implications for teachers reasoning and justification in other areas of mathematics.

Analysis of Korean students’ international mobility by 2-D model: driving force factor and directional factor

The purpose of this study is to understand the dynamics of Korean students’ international mobility to study abroad by using the 2-D Model. The first D, the driving force factor, explains how and what components of the dissatisfaction with domestic higher education perceived by Korean students drives students’ outward mobility to seek foreign higher education. The second D, the directional factor, describes the factors that influence the choice of destination country for students’ outward mobility, and is explained by the comparison of Korean students’ perceptions on the images of universities in the U.S., China, the U.K., and Australia and their expectations for higher education in each country (categorized as ‘academic’–‘environmental’). Two questionnaire surveys were conducted to analyze the two D factors and the research findings were integrated into suggestions for each country’s higher education institutions that can be incorporated into their recruitment strategies for international students.

The role of principal empowerment within a site-based management environment: empirical testing of a structural model

A model examining psychological empowerment, transformational leadership and innovation for school principals is proposed and tested within a Site (School)-Based Management environment. Developed from management literature, the model was adapted for state school principals experiencing increased decision-making powers and accountability. Empirical results provide a level of support for the model. ‘Role clarity’ and ‘access to resources’ were found to exert a positive influence on principals’ feelings of empowerment, though the value placed by principals on strategic information was found to have a slightly negative influence. The influence of empowerment on transformational leadership was not found to be as strong as anticipated. The paper includes a discussion of the implications of these findings for theory and policy development.

Socio-emotional orientations and teacher change

In this article we consider how elementary education students’ views of mathematics changed during their mathematics methods course. We focus on four female students: two started the course with mainly positive views of mathematics and a task orientation, two with negative views of the subject and an ego-defensive orientation. The biggest change observed was that the trainees’ views of teaching and learning mathematics became more positive. Moreover, what had been an ego-defensive orientation changed towards a social-dependence orientation. The crucial facilitators of change seemed to be (1) handling of and reflection on one’s experiences of learning and teaching mathematics, (2) exploring content with concrete materials, and (3) collaboration with a partner or working as a tutor of mathematics.

Student and early career mobility patterns among highly educated people in Germany,Finland, Italy,and the United Kingdom

This article addresses the question of how great are higher education students’ incentives to change study programs or institutions to improve one’s personal employability in the course of the higher education-to-work transition process. The posed question is addressed at a system level. Students’ mobility between programs and institutions is referred to as ‘student mobility’ and graduates’ mobility between jobs is referred to as ‘early career mobility.’ The relationship between these two separate components of mobility is discussed in three different institutional frames: German/Finnish, Italian, and British. When depicting the relationship between the two components of mobility, the article also considers parallel phenomena such as prolongation of the degree-earning process and participation in work-life with student status. Indicator level analysis concerning graduates of the year 2000 reveals important differences between the three institutional frames: in Germany and Finland, there is a high level of student mobility at the basic degree level combined with a low level of career mobility after graduation; in the UK the opposite transition logic than that of Germany and Finland occurs, and in Italy students demonstrate prolonged transitions with little student mobility and early career mobility.

Culture, food, and language: Perspectives from immigrant mothers in school science

The article explores the role of immigrant parents in middle school science as both teachers and learners as part of an urban middle school curriculum, the Linking in Food and the Environment (LiFE) program. The curriculum engaged parents as partners with science teachers to teach science through food. Over a 2-year period, parents attended a series of bilingual workshops, collaborated with classroom teachers, managed activities, guided student inquiry, and assisted in classroom management. The following study analyzes the role of culture, language, and identity as four mothers navigated their position as ‘insiders’ in a science classroom.

Tracing the development of preservice teachers’ efficacy beliefs in teaching mathematics during fieldwork

This paper focuses on efficacy beliefs with respect to teaching mathematics; it aims at testing and validating an existing scale for measuring efficacy beliefs in the domain of mathematics, examining the development of preservice teachers’ efficacy beliefs in mathematics during fieldwork, and identifying factors that contribute to the development of these beliefs. Exploratory factor analysis applied to longitudinal data collected through a questionnaire administered to almost 90 preservice teachers elicited a two-factor model, reflecting efficacy beliefs in mathematics instruction and classroom management. Using cluster analysis, we found four patterns in the development of preservice teachers’ efficacy beliefs regarding these two factors. Our analysis of semi-structured interviews with eight participants representing these four patterns suggested that preservice teachers’ efficacy beliefs were mainly informed by experimentation with teaching and interaction with mentors, tutors, peers, and pupils. Based on our findings, we draw implications for mathematics preservice teacher education and provide suggestions for future research.