Fourteen Papers by Leone Burton
Introduction: Educational systems have developed out of social demands and have tended to follow social changes rather than to lead them. This is partly because of their nature. Educational systems are staffed by persons whose formal education began from fifteen to fifty years earlier. They rely upon expensive capital equipment which often becomes outmoded before it is outlived. Inherent in these systems which elapses between a child's entry and leaving which, in itself, causes a minimum six- to ten-year pause before the need for and the consequences of new changes can be assessed. But not all rigidity is due to these restraints. There are many others that can be overcome once the need to remove them has been perceived. The organization and movement of the labour force in the educational system, the content of curricula, the methods used, and most of all the attitudes adopted towards change, are all subject to alteration.
Conclusions: The four areas which have been examined, universal primary education, literacy, technical education and university provision, are essential parts of the educational systems of the developed world. They are seen by the developing world as being pre-requisites for their achieving economic, political and social growth. If examined in terms of economic and social efficiency, they are all lacking in varying degree in the developing countries. In the developed world, not only does the basic infrastructure exist, but resources are available for the consistent research which makes it more efficient. The kinds of changing methods which have been described are of great relevance to the developing countries - they are being introduced and refined in the industrialized nations. The result is that the gap between developed and underdeveloped is widening. While considerable efforts are being made to make education in the former more relevant and more efficient, in the latter the provision is inadequate, often not relevant and often inefficient.
Abstract: It is the purpose of this article to report upon the early stages of a research project in Mathematical Education, currently in progress at the Polytechnic of the South Bank, London. The project, supported by funds from the Social Science Research Council, is entitled, 'The Skills and Procedures of Mathematical Problem Solving in pupils of 9-13 years' (S.P.M.P.S.) The article explores the background to the need for such a project, both from a mathematical and from a pedagogic point of view.
Conclusions: Once teachers see the pleasure, involvement, tenacity and satisfaction that problem solving can evoke in children, they will be encouraged to begin, themselves, to experiment and explore methods they use to teach mathematics; to make available to children an area of study which is thoughtful, provocative, beautiful, curious, a means invented by man to help him in his search better to understand and explain the world about him.
Introduction: The word "problem" is used generally to refer to a difficulty - "What is your problem?". In the mathematics classroom it has a particular usage referring to the set of exercises at the end of a chapter in the text. In North American classrooms, it has achieved the further particular meaning of "word problem" - the presentation of a mathematical situation in written language.
Conclusions: The thesis being advanced is that puzzle behaviour by teacher and pupils is not productive. However, the content of the puzzles does not have to be jettisoned. If the teacher is aware of the puzzle/problem distinction and its rationale, the emphasis will be changed from seeking a right answer to open search behaviour, from single solution to variation, comparison and evaluation of methods and resolutions, from meeting external requirements to becoming self-aware of one's own and consequently of others (different) thinking processes. Out of this environment springs motivation to learn mathematics. The key to making such a shift lies in the questions and expectations of teachers and pupils - the desire and interest to explore and understand. Both teaching and learning them become problem solving activities.
Abstract: This paper argues that mathematical thinking is not thinking about the subject matter of mathematics but a style of thinking that is a function of particular operations, processes, and dynamics recognizably mathematical. It further suggests that because mathematical thinking becomes confused with thinking about mathematics, there has been little success in separating process from content in the classroom presentation of the subject. A descriptive model of mathematical thinking is presented and then used to provide a practical response to the questions, Can mathematical thinking be taught? In what ways? The teacher is encouraged to recognize both what constitutes mathematical thinking, whether in the mathematics class or some other, and what conditions are necessary to foster it.
Conclusion: The key to recognizing and using mathematical thinking lies in creating an atmosphere that builds confidence to question, challenge, and reflect. Behind such behaviour is an acknowledgment of the need to (i) query assumptions, (ii) negotiate meanings, (iii) pose questions, (iv) make conjectures, (v) search for justifying and falsifying arguments that convince, (vi) check, modify, alter, (vii) be self-critical, (viii) be aware of different approaches, (ix) be willing to shift, renegotiate, change direction.
Abstract: For the last few years, the author has been part of a team developing a mathematics course for students most of whom are women, many of whom are black and all of whom are attempting to gain entry to a teacher-training course by successfully completing a one year re-entry course. It is a requirement that teacher-training students should have attained a suitable standard in mathematics. Further Education colleges where re-entry courses are sited have a sad history of student failure in mathematics. The theoretical environment in which the development of this course took place is described in order to place the course in context. In particular, attention is drawn to the re-definition of mathematics which encourages student enquiry and experimentation in order to establish a basis for understanding the subject, and to the teaching/learning model which creates an environment of respect and confidence. The roles of students and staff in a learning environment of this kind are discussed. Particular attention is paid to the attitudes and feelings of the students and the effects on their expectations.
Conclusions: In this course it was successfully demonstrated task-focussed, there is no opportunity nor any plausible reason for an individual, or a group, to dominate the class, nor to monopolise the teachers' attention, nor to indulge in any other of the tactics which have been described in the literature. Under these conditions, it would appear that all learners benefit. The course can be used by any group aiming at this level of mathematics learning. A small trial was conducted in a secondary school with pupils of 15-16 years but the degree to which the methods and the structure is transferable to schools has yet to be systematically investigated.
Abstract: Strategies used by ethnic minority women and men and white women to gain senior management positions within educational institutions were investigated using a life history approach. Participants were also asked to reflect upon institutional strategies that might operationalise equal opportunities policies. The report describes the experiences they recounted and identifies the themes, strategies and structures to which they drew attention.
Conclusions: The key themes identified in the literature were: (i) the conceptualisation of management as masculine, to which we would add 'white'; (ii) discrimination in promotion; (iii) women's career patterns; (iv) domestic responsibilities, to which we would add responsibilities to their communities; (v) mentoring, which is clearly neither singular nor simple; and (vi) management styles. While this study reinforced the importance of all these themes and added recruitment, staff development and institutional structures to the list, it also provided particular personal and institutional strategies which can successfully address under-representation by 'race' or sex. Yet again, as in other studies reported in the research and practice literature, attention is drawn to the failure of institutions to implement policies and strategies appropriate to social justice. The lack of adequate management theories which incorporate aspects of institutional structure pertinent to 'race' and gender issues is again underlined. Equal opportunities is not the responsibility of those aspiring to senior educational management but must be addressed, and required, by those already in positions of power. Change lies within their remit.
Introduction: This Special Issue of 'Educational Studies in Mathematics' reflects a concern in mathematics education which, as one of the papers points out overtly, and others imply, is not new but has come to dominate a lot of pupils, teachers' and politicians' time. The political climate in many countries, led by pressure from such economically powerful institutions as the World Bank, has been affected by the need to supply answers to questions about systemic performance in education. Such system-wide questions lead to the introduction of programmes which seek to establish if pupils are learning and what they are learning. These programmes, of which, perhaps the most internationally well-known are the International Mathematics Studies, the Third of which is imminent, attempt to provide performance data linked to syllabi. While such snapshots can be useful in some arenas, and are clearly valued by politicians, they do not address the nature of learning mathematics, the culture of the mathematics classroom, or the variations of learning style which are appropriate and necessary to success in the discipline. And, there is a clear difference between the emphasis that many outside of classrooms place on assessment, particularly testing, and the ongoing sense of monitoring progress which effective teachers and learners value. However, our inevitable cultural obsession that what is particular to our own society is generalisable across all others, is challenged by learning about expectations, practices and outcomes elsewhere.
Conclusions: Since this is a guest editorial, there are no conclusions.
Abstract: There is, now, an extensive critical literature on gender and the nature of science three aspects of which, philosophy, pedagogy and epistemology, seem to be pertinent to a discussion of gender and mathematics. Although untangling the inter-relationships between these three is no simple matter, they make effective starting points in order to ask similar questions of mathematics to those asked by our colleagues in science. In the process of asking such questions, a major difference between the empirical approach of the sciences, and the analytic nature of mathematics, is exposed and leads towards the definition of a new epistemological position in mathematics.
Conclusions: If the nature of knowing mathematics were to be confirmed as matching the description given in this paper, the scientism and technocentrism which dominate much thinking in and about mathematics, and constrain many mathematics classrooms, would no longer be sustainable. Mathematics could then be re-perceived as humane, responsive, negotiable and creative. One expected product of such a change would be in the constituency of learners who were attracted to study mathematics but I would also expect changes in the perception of what is mathematics and of how mathematics is studied and learned. That such a possibility, in schools, is not outside the realms of possibility is suggested in Boaler (1993). We can also learn from experiences in other disciplines. English, for example, attracts predominantly female constituencies of learners at the undergraduate level, many of whom have been successful in developing academic careers.
Abstract: The paper explores the background and implementation of a project in which school students researched the factors which affect the decision to remain at school or leave at 16. The two major concerns of the project were equity and access to higher education. It was the result of a collaboration between researchers from the University of Birmingham, school teachers and school students. A critique is offered of the method of students as researchers focusing upon five main issues, organisation, time, equity, research style and clashing cultures.
Conclusions: Working with Students as Researchers requires time. It became clear to us that aspects of the project which we had thought had been explored were, for the students, within an agenda which remained hidden. This was partly to do with trust, but also to do with time and the siting of the project mainly at the school, and not at the University. Consequently, there was restriction on the choice of research methods, and the support given to the students came mainly from the teachers who, themselves, were under time pressures. The researchers could not know what the students were doing, or demanding in the period between meeting them at the outset and towards the end of the project. Future research using this method should assure greater co-operation between the different participants. Such co-operation may be demanding in time and resource.
Abstract: In 1997, an interview-based study of 70 research mathematicians was undertaken with a focus on how they 'come to know' mathematics, i.e. their epistemologies. In this paper, I discuss how these mathematicians understand their practices, locating them in the communities of which they claim membership, identifying the style which dominates their organisation of research and looking at their lived contradictions. I examine how they talk about 'knowing' mathematics, the metaphors on which they draw, the empiricist connections central to the work of the applied mathematicians and statisticians, and the importance of connectivities to the construction of their mathematical Big Picture. I compare the stories of these research mathematicians with practices in mathematics classrooms and conclude with an appeal for teachers to pay attention to the practices of research mathematicians and their implications for coming to know mathematics.
Conclusions: Out of the interviews with the research mathematicians, I have a clear image of how impossible it is to speak about mathematics as if it is one thing, mathematical practices as if they are uniform and mathematicians as if they are discrete from both of these. Would it not be enlightening for more learners ... to discover that they, too, can practise mathematics in the many different ways that I have described for the many different reasons that my participants explain towards the many different outcomes that are possible? With a quick nod in the direction of the Cockcroft Report (1982), I will give the final word to a male Professor of pure mathematics: "To know is to understand and normally understanding is not as though you have things that you know and then you try to prove something in order to understand it."
Introduction: In 1997, I undertook an interview-based study of thirty-five female and thirty-five male career research mathematicians in twenty-two universities in England, Scotland, Northern Ireland and the Republic of Ireland. The purpose of the study was to explore how robust an epistemological model which I had developed was, as a way of describing how mathematicians come to know mathematics, and furthermore to ask what we, as mathematical educators, might learn from their research practices. The model has five components which describe knowing in mathematics. These are its person- and cultural/social-relatedness, the aesthetics of mathematical thinking invoked, intuition, styles of thinking and connectivities. ....
Conclusions: With respect to how mathematicians go about their work and how social and interactive it is, one of the biggest surprises of the study, for me, was that mathematics is no longer seen, certainly by most of those I interviewed, as something you do on your own in an eyrie away from human contact (what I think of as the Andrew Wiles effect). Of the seventy mathematicians, only three males and one female claimed never to work with others and most were extremely enthusiastic about the benefits of joint work. They identified two different forms of what they termed 'collaboration', one of which I am calling co-operation because the partners brought different disciplinary skills and knowledge to the team (for example, a statistician working with non-mathematicians). ... Indeed, I found it quite depressing how rare it was for these mathematicians to see any connection between what they did in their own researching and what they expected of, and received from, their students. This was noticeable when they talked about the role of intuition.
Introduction: I have been exploring how research mathematicians come to know the mathematics they develop, with a view to substantiating different learning strategies which consequently might inform practices when working with less sophisticated learners. This article follows up in more detail on the brief comments I made there about what my interviewees had to say about the topic of 'intuition'.
Conclusions: Intuition, insight or instinct was seen by most of the seventy mathematicians whom I interviewed as a necessary component for developing knowing. Yet none of them offered any comments on whether, and how, they themselves had had their intuitions nurtured as part of their learning process. While many referred to the centrality of their intuitions to how they came to know within the research process, some were very dismissive of their students', and others of their own, power to bring intuition into play. More importantly, some considered intuition as something you either had or did not have. ... Further, those who do believe that intuition is an important feature can be challenged by this work to answer the following question: if it is so important, how could you set out to nurture it in your students? .... I would like to encourage mathematicians, indeed anyone who has responsibility for the learning of mathematics, to open mathematical activity to include the subjectivity of intuitions, to model their own intuitive processes, to create the conditions in which learners are encouraged to value and explore their own and their colleagues' intuitions and the means that they use to gather them. This seems to me to be a necessary step which provides a justification for, but is prior to, the search for convincing argument and, ultimately, proof.
Abstract: A theoretical model describing coming to know mathematics was matched against the descriptions gathered from interviews with 70 practising research mathematicians. The model proved to be a good framework for understanding their epistemological practices and a number of pedagogical implications of wider application are drawn from the results.
Conclusions: If we are honestly worried about the many students who turn away from mathematics as well as some of the dysfunctional experiences in schools and universities and within the mathematical community itself as reported in the literature and recounted by the mathematicians in the study, I believe we have a responsibility to make the learning of mathematics more akin to how mathematicians learn and to be less obsessed with the necessity to teach 'the basics' in the absence of any student's need to know. This is an epistemological question in the broadest sense rather than a narrow focus on the acquisition of knowledge objects. Learning mathematics could be a (re)search activity and 'it is characteristic of frontier research that it opens up new ways of seeing the world' while recognising that 'the most important form of learning is that which enables us to see something in the world in a different way'. As a research activity, the creativity, challenge, motivation and delight of which the mathematicians spoke could be available to those engaged in exploring the discipline as learners. But for this to happen requires a major shift in understanding the ways in which learning demands an incorporation of knowledge objects into the activity and the variety of styles of engaging with those objects such that 'insights into how knowledge is formed in various domains of knowledge' become 'parts of those domains'. It is a source of great sadness that we are currently in an educational climate that makes such shifts more difficult and, consequently, less likely. I believe that the potential outcomes for learners of mathematics are, as a result, all too predictable.
Abstract: Data from a study of the practices of research mathematicians are used to highlight an exploration of commonalities across and differences between learners in mathematics classrooms. Such commonalities and differences are, it is claimed, central to a teacher's understandings of how to address her role and responsibilities. Commonalities discussed are those between learners, across mathematics and between communities of practice. Differences between syllabus and practice, between school mathematics, mathematics in the world, and academic mathematics and, finally, between members of the different communities of students, teachers of mathematics, mathematics educators, and mathematicians are explored.
Conclusions: To build a climate of trust and respect within and across communities that we engage more fruitfully with the challenge of learning mathematics, is a non-trivial task. I think we have the direction and the directions for doing so here. To review what I have been identifying and recommending, I began with a set of questions that were derived from my theoretical perspective. These were: (i) How is the mathematical learning community constituted, how does it function and how is it sustained? (ii) How are the identities of pupils and teachers sustained and nurtured? (iii) How is active learning experienced? (iv) In that process, how is meaning recognised and made? (v) How is what is being learnt connected to what is known and familiar? (vi) In turn, how does what is being learnt inform and enhance knowledge and experience? These are practical questions which every teacher, and researcher, can ask of the classrooms in which they are working. My responses to these questions were built around the recognition of, indeed trust in, the commonalities between learners, across mathematics, and within different communities of practice. Together with the recognition of commonalities, I discussed the reconciling of, indeed respect for, differences between syllabus and practice, between school mathematics, mathematics in the world, and academic mathematics and, finally, between members of different communities such as students, teachers, mathematics educators, mathematicians, parents. I offered an epistemological model related to classroom practices to help structure the thinking
JOC/EFR November 2017
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