Showing papers in "for the learning of mathematics in 1999"

Supporting the Participation of English Language Learners in Mathematical Discussions.

Abstract: The aim of this paper is to explore how teachers can support English language learners in learning mathematics and not only English. The focus of the analysis will be on one important aspect of learning mathematics: participation in mathematical discussions. I use a discourse perspective (Gee, 1996) on learning mathematics to address two pairs of questions central to mathematics instruction for students who are learning English (as well as for students who are native English speakers):

Beyond Being Told Not to Tell

Abstract: For the past several years, we have been developing and studying teaching practices through our own efforts to teach school mathematics. Ball's work has been at the elementary level, in third grade, and Chazan's at the secondary level, grade ten and above, in Algebra I. In our teaching, we have been attempting, among other things, to create opportunities for classroom discussions of the kinds envisioned in the US National Council for Teachers of Mathematics Standards (NCTM, 1989, 1991). At the same time, we have been exploring the complexities of such practice. By using our teaching as a site for research into, and as a source for formulating a critique of, what it takes to teach in the ways reformers promote, we have access to a particular 'insider' sense of the teacher's purposes and reasoning, beyond that which a researcher might have. [1] This article originated with frustration at current math education discourse about the teacher's role in discussion-

Arbitrary and necessary: part 1, a way of viewing the mathematics curriculum

Abstract: This paper was published in the journal For the Learning of Mathematics: an international journal of mathematics education.

Why Is Intuition So Important to Mathematicians but Missing from Mathematics Education

Abstract: In my recent Communication (Burton, 1999), I outlined some features of a interview-based study of mathematician's views and research practices. 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 'inmition'.

Semiotic Mediation: from History to Mathematics Classroom

Abstract: We carried out an exploratory study with expert university students in order to validate this hypothesis, exploring how students could reseal the rupture and restore a sense of unity between the figural and conceptual components In particular, what kinds of geometrical conceptions, if any, were these students able to mobilise? Mter reporting the findings of the study, we shift to the didactic plane and suggest certain tools of semiotic mediation (Vygotsky, 1978) which could be introduced in order to enable the students to achieve the conceptual oversight that is possibly lacking

Genre Analysis as a Way of Understanding Pedagogy in Mathematics Education.

Abstract: In this article, I argue that gerue analysis is a form of discourse analysis which can provide useful and sometimes surprising new perspectives on understanding teaching and learning An analysis of the written and spoken 'texts' of schooling, drawing on linguistic and literary analytic methods, can highlight relationships between educational genres and other culturally recognizable forms .. The study of such related gentes may allow researchers to uncover hidden cultmal meanings, assumptions and intentions inherent in the generic forms of schooling I will use two examples to illustrate the application of genre analysis in mathematics education. The first draws on a study of mathematical word problems (see Gerofsky, 1999), analyzing their geme features and suggesting a relationship between word problems, parables and riddles The second focuses on the language of initial calculus lectures at a university, uncovering generic similarities with the language of the conjurer, the salesperson or nurse Both studies question the messages, intentional and unintentional, canied by generic forms employed in mathematics education.

Whither Relevance? Mathematics Teachers' Discussion of the Use of 'Real-Life' Contexts in School Mathematics.

Abstract: As a mathematics teacher, I have been concerned about students' access to school mathematics. In post-apartheid South Africa, there has been a commitment to make school mathematics not only more accessible, but also more relevant for students. To that effect, a new curriculum, Curriculum 2005, which seeks to make learning more integrated and more 'relevant and connected to real-life situations' (DoE, 1997, p. 7) is currently being implemented. This article sets out to discuss emerging issues of how mathematics teachers, as critical agents of change at this important historical time in South Africa, understand the notion of 'relevance' and what they perceive to be some of its intricacies. South Africa's commitment to access and relevance in school mathematics is shared by many educationalists elsewhere (e.g. Schoenfeld, 1988; Volmink, 1994; Bishop, 1985). With that commitment has come a new impetus to develop a better and more critical understanding of the issue of 'relevance' or 'real-life contexts' in school mathematics. Work done in this sphere includes Boaler's (1993) argument that many school mathematics tasks are covered by 'real' life contexts, where:

Fork in the Road

Abstract: We met in Recife, Brazil at a Psychology of Mathematics Education (PME) conference. It was July 1995 and DR was about to take up a post as a mathematics teacher educator in Canada after completing a Ph.D., while LB was an experienced teacher of mathematics teachers in the UK, whose background was teaching secondary mathematics. The following two extracts illustrate how we started telling each other stories about our practice over email. Extracts from e-mail 1: Raising an issue - right/wrong LB: When interviewing people for the Post-Graduate Certificate in Education course (our one year, postdegree programme), I ask about how the interviewee's images of mathematics have changed through their own education and schooling. Often this reveals in a particular person an association of mathematics with the security of getting 'ticks' - "it's a subject where it's possible to know that you're right" - and this is seen as a good feeling. When they start university, however, everything goes so fast that the only response is to rote learn the material. They do not feel in control.

Ethnomathematics: A Political Plaything.

Abstract: Anyone with an interest in both mathematics and linguistics must grin with rueful fascination as they watch the way that the term 'ethnomathematics' is thrown about both in the public media and in the mathematics education literature. It is a versatile toy, fulfilling several functions in an on-going game of catch. Since the 1984 ICME address by Ubiratan D'Ambrosio (1985) which is widely regarded as establishing the field in its contemporary form this word has been used in the political games of curriculum development in many countries. It has also:

Are Theories in Mathematics Education of any Use to Practice

Abstract: l. The gap between theory and practice Some years ago, I took part in arranging a seminar on activity forms and student engagement in Danish mathematics instruction in the 10th-12th grade (gymnasium). The organising committee intended the participants to leave the course with an outcome which they could connect to the problems they experienced in their daily teaching practice. So we started by asking the participants to write down what they found to be the greatest problems in connection with students' involvement and activity forms. One participant wrote:

The Stench of Perception and the Cacophony of Mediation.

Abstract: than his intellect. A great heritage of tradition and experience, the sum total of man's [sic] knowledge of nature at that era, had to be administered, employed, and even more, passed on. A vast and dense system of experiences, observations, instincts, and habits of investigation was slowly and hazily laid bare to the boy. Scarcely any of it was put into concepts. Virtually all of it had to be grasped, learned, tested with the senses. (Hermann Hesse, 1969, p. 42)

A Journey through Geometry: Sketches and Reflections on Learning.

Abstract: When I was 11, my mathematics teacher gave me a copy of Prelude to Mathematics by W. W Sawyer Much of it I did not understand, but I became fascinated by a chapter on weird geometries in which people got smaller as they moved about, and infinity was the circumference of a circle 'Non-Euclidean geometry' did not mean very much to me, but the images of 'straight lines' being curved and the angles in triangles not adding up to 180° appealed to my schoolboy sense of anarchy Breaking the rules of geometry seemed to produce new rules that had their own coherence and logic Nearly a decade later, I became deeply interested in differential geometry Until then, mathematics seemed to consist of a series of discrete and disconnected topics and techniques. Differential geometry changed all that I began to understand what mathematicians called elegance economy, coherence and structures that linked together what appeared to be unrelated areas. It was also powerful, providing a language to describe the large-scale structme of the universe, and exotic objects such as black holes. Differential geometry drew together my intellect, imagination and interest, allowing me to speculate in a precise way about "life, the universe and everything" Turtle geometry entered the picture when I got my first computer .. It liberated me from BASIC programming and opened out the possibility of an experimental and exploratory approach to mathematics Reading Turtle Geometry by Abelson and diSessa (1980), which starts with drawing polygons and finishes with a simulator for General Relativity, fired my interest in finding a turtle version of the models that I had met as a schoolboy. Iluough a series of sketches, I am going to describe the journey that brought together these tluee geometries in creating a turtle-based exploratory tool for non-Euclidean geometry My aim is to reflect on some aspects of learning mathematics that emerged along the way

Sounds of Canons at Our Gates: Post-Modernism Comes to Mathematics Education.

Abstract: In a conversation with a classicist colleague, the Cambridge number theorist G H. Hardy once noted that from his perspective while the Romans had clearly been dead for millennia, the Greeks were rather like fellows from another college whom one just didn't happen to meet very often Given that he expressed these sentiments close to a century ago, it is rather surprising to note how 'Hardyish' the mathematical world has remained at its core Taking two recent examples, one need only think of the much-heralded, 'shoulders of giants' triumph of Andrew Wiles over the long-standing Fermat conjecture and the strangely Pythagorean life of Paul Erdos (given very accessible treatments, respectively, by Singh (1997) and Hoffman (1998)) From the viewpoint of many other disciplines, this continuing commitment to classical roots seems almost quaint, as the past two decades have seen extensive and often bitter debates leading, in some cases, to nearly complete capitulation on the part of scholars holding 'traditional' perspectives in several of the disciplines in the social sciences and humanities Many of the 'Visigoths' in this scenario have rallied under the banner of 'post-modernism' Several forests and much magnetism have been sacrificed to the consideration of just what this term entails, but agreement seems hard to find At a fundamental level, there is a rejection of orthodox conceptions of logic, linearity, value and hierarchy as they have been applied in literary, political, historical and sociological studies The generators of the traditional 'English' cuniculum, for instance, have, in the 'canon' wars at institutions like Starrford University, been subjected to tests of hue, penility and vitality and found to be all-too-largely pale, male and considerably-less-than-hale (hence the 'dead white male' descriptor) Collage and pastiche have become accepted forms of presentation not just in the visual and plastic arts, but also in films and novels. The puerile and the pop have become as much gtist for the academic mill as the abstruse and the esoteric Subgroups emerge; linguistic and historical grit generates the pearls ofpost-structrualism and post-colonialism Rationality wanes, deconstruction reigns, relativism rules. Yet again, as Marx had it: 'all that is solid melts into the air' The intellectual energy behind post-modernism has largely been French and German .. The francophone 'front four' of Foucault, Denida, Baudrillard and Lyotard have, in a barrage of baffling, cryptic, playful and frustrating texts provided a foundation, of sorts, for the post-modern perspective Working in the shadows of Heidegger and Nietzsche, the somewhat more sober Rhine-side scholars like Gadamer, Habermas and Schulze have presented us with interpretations of hermeneutic and critical theory. Deference is dead: long live 'differe(or a)nce'! In the kingdom of science, where logic, linearity and hierarchy are titular gods of very long standing, little attention has been paid to the canon wars. This is behaviour consistent with the accepted practice of aristocrats. From time to time, reports of skirmishes at the frontier (in this case, among the philosophy of science hill-tribes) filter back to the laboratory palaces Then, in 1996, a young and well-read physicist engineered a prank which catapulted science and post-modernism to the front pages of many journals Alan Sokal, of New York University, submitted a paper, 'Ttansgtessing the boundaries: toward a transformative hermeneutics of quantum gravity' to the fashionable journal, Social Text. There were, briefly, signs of satisfaction in post-modern ranks with this clear signal of advancing influence Once published, Sokal revealed in another journal that his paper was naught but meta-twaddle. Much discussion; Nobellists pontificate, philosophers equivocate; the ethics of it all, quelle scandale [I] In the far comer of the castle, the dowager empress, mathematics, seems restless Meanwhile, in the developing nation-state of mathematics education, some of the taller citizens have taken note of smoke coming from the language and society mountains across their border And while their most comprehensive documents from the early and middle parts of the decade (Grouws, 1992 and Bishop et al, 1996) mention things postmodern only in the briefest of passes, it has been rumoured that some of the more adventmous young have crossed the river and embarked on major scouting expeditions Enough The form, or lack thereof, especially for a novice, is quite intoxicating But properly-reared Canadian folk know that the siren of self-indulgence leads most often to excess and regtet, and hence with the merest flicker of disappointment at scuttling the possibility of \"Gullible's Travels\", we choose the Lewis Carroll-like wormhole that spins us out of the land of porno and back to terra more-orless firma, where not everything goes. I he three books in question that form the focus of this essay review are significant and substantial attempts to broaden and deepen the natrue of discourse about, and activities in, the field of mathematics education These publications of Brown, Dowling and Davis [2] share many attributes; there are significant differences between and