Stuart Rowlands

Where would formal, academic mathematics stand in a curriculum informed by ethnomathematics? A critical review of ethnomathematics

This paper is a critical review of the ethnomathematics literature and classifies ethnomathematics according to where it might stand in relation to the teaching of formal, academic mathematics. This paper investigates what it sees as four possibilities: ethnomathematics should replace academic mathematics, ethnomathematics should be a supplement to the mathematics curriculum, ethnomathematics should be used as a springboard for academic mathematics and ethnomathematics should be taken into consideration when preparing learning situations. We argue that it is only through the lens of formal, academic mathematics sensitive to cultural differences that the real value of the mathematics inherent in certain cultures and societies be understood and appreciated.

Can we Speak of Alternative Frameworks and Conceptual Change in Mechanics

This paper discusses the various conflicting trends in mechanics education that have appeared over the past two decades, and proposes the theory of schemata as a means to resolve the conflict that exists within the literature. The conflict has two causes: the prevailing relativism that exists within science education, and the mistaken view that student alternative ideas are concepts that are well defined. We argue that student alternative ideas can be best understood in terms of schema theory, and that schema theory can offer support to the Vygotskian idea of the teacher facilitating the construction of the Newtonian system within the students zone of proximal development. Within the context of schema theory we propose the category of idealised abstraction that has as its starting point the logical structure of Newtonian mechanics rather than the cognitive state of uninstructed students.

Turning Vygotsky on His Head: Vygotsky' `Scientifically Based Method' and the Socioculturalist's `Social Other'

Vygotsky has become an authority, but the authority has more to do with justifying a sociocultural relativism than it has with his Marxist objectivist approach to psychology and pedagogy. This paper is an attempt to understand Vygotskys perspective in relation to Marxist epistemology, and will critically examine the sociocultural interpretation of Vygotsky but within the light of his own perspective. It will be shown that the relativism of the sociocultural school not only takes Vygotskys zone of proximal development out of its social and historical context, but as a consequence downplays the zone of proximal development as a dynamic research methodology. As an extension of the discussion of the zone of proximal development, this paper will also examine the sociocultural interpretation of Vygotskys relation between scientific and everyday concepts, and the pedagogical consequences of such an interpretation.

Our response to Adam, Alangui and Barton's ``A Comment on Rowlands & Carson `Where would Formal, Academic Mathematics stand in a Curriculum informed by Ethnomathematics? A Critical Review'''

We would like to begin by expressing our gratitude to Adam et al. (2003) for the time and effort they took to produce their comments on our article, “Where would formal, academic mathematics stand in a curriculum informed by ethnomathematics?” (Rowlands and Carson, 2002). We do not believe that our aspirations for the education of children differ significantly from those of Adam et al. Both papers would seem to have clearly affirmed support for the dignity, uniqueness, and ingenuity of traditional and indigenous cultures, and to have expressed a clear intent to continue toward a rapprochement between the world’s standard canon of mathematics and the unique contributions one finds in the more localized forms of mathematics characteristic of traditional cultures. It is an encouraging sign of the times that our debate focuses on how best to serve the needs and interests of the children of traditional cultures and that traditional cultures are finally being understood as precious resources generally. Yet within that essentially noble set of understandings and purposes, there is room for reasonable and decent people to disagree. Our conversation does not involve simple problematics, by any means. In addition to purely mathematical issues, it involves questions of historical injury and of contemporary relationships between cultural groups whose core values are incommensurable. It is a topic that by its very nature can sometimes decay from patient reason to acrimony. Thus we wish to express our gratitude to Adam et al. for their informative and thought-provoking critique. We are disappointed with the analysis they gave, but we credit them with maintaining high standards of decorum in the process and for what we believe are honorable intentions. There is, perhaps, no more important or pressing social issue on the planet today than that of figuring out how peoples of varied cultural allegiances are to get along. Academics and educators have some part to play

Are ‘misconceptions’ or alternative frameworks of force and motion spontaneous or formed prior to instruction?

It has often been assumed that misconceptions of force and motion are part of an alternative framework and that conceptual change takes place when that framework is challenged and replaced with the Newtonian framework. There have also been variations of this theme, such as this structure is not coherent and conceptual change does not involve the replacement of concepts, conceptions or ideas but consists of the development of scientific ideas that can exist alongside ideas of the everyday. This article argues that misconceptions (or preconceptions, intuitive ideas, synthetic models, p-prims etc.) may not be formed until the learner considers force and motion within the learning situation and reports on a classroom observation (that is replicated with similar results) that suggest misconceptions arise, not because of prior experience, but spontaneously in the attempt at making sense of the terms of the discourse. The implications are that misconceptions may not be preformed, that research ought to consider the possible spontaneity in the students’ reasoning and then, if possible, attempt to discern any preformed elements or antecedents, and that we ought to reconsider what is meant by ‘conceptual change’. The classroom observation also suggests gravity as a particular stumbling-block for students. The implications for further research are discussed.

Using Computer Software in the Teaching of Mechanics

The paper looks at ways of using computer software in the teaching of mechanics. The various reasons for using software are discussed to justify the use of software. A number of examples are then considered to show how different types of software can be used. Examples shown are taken from very specific types of software, more general simulation software and mathematical software. The paper discusses using software to explore mechanics, to challenge ‘misconceptions’, to make links between mathematical representations and motion and to solve non-standard problems. The paper also stresses the need for structured approaches to the use of software.

Special Issue on History and Philosophy of Mathematics in Mathematics Education

Although Science & Education has published papers specifically concerning mathematics over the years—some quite significant for education—mathematics has never received the spotlight. Indeed, when the journal was established it bore the subtitle, Contributions from History, Philosophy and Sociology of Science and Mathematics, but so few papers on mathematics were received in those years that the word ‘‘mathematics’’ was duly dropped. Yet within the community of science and mathematics educators, there is no lack of interest at least in the contributions of history and philosophy of mathematics to mathematics education, particularly the history of mathematics. Besides publications such as those we have listed at the end of this introduction, one can point to the activities of the International Study Group on the Relations between the History and Pedagogy of Mathematics (HPM) which dates back to 1972, and which almost from the start became an affiliated study group of the International Commission on Mathematical Instruction (ICMI), the largest and oldest international organization for mathematics education. International conferences of HPM have been held in Europe, Canada, Australia, Taiwan, Mexico, and Korea in association with the quadrennial International Congress on Mathematical Education (ICME). The European Summer University on History and Epistemology in Mathematics Education (ESU), now held every 4 years as well, is another regular venue for meetings dealing with history in mathematics education, as is the working group on History in Mathematics Education at the biennial Congress of European

Identifying stumbling blocks in the development of student understanding of moments of forces

Student intuitive ideas in mechanics, especially concerning the relationship between force and motion, have been the subject of much research. Many students find aspects of the topic of moments of forces to be stumbling blocks, yet there has been little or no research in this area. This paper reports on a small‐scale investigation of student understanding of moments of forces to provide some indication as to the nature of intuitive ideas in this area, and to provide some suggestions as to the appropriate teaching strategy. The results of the investigation suggests three stumbling blocks in the conceptual understanding of moments of forces. The first stumbling block seems to contain problems where the forces applied are still acting vertically, but the points of application of the forces are not at the same horizontal level. The second stumbling block seems to either contain problems where the forces applied are vertical but there is no obvious sense of symmetry, or problems that represent a conceptual lin...

MISCONCEPTIONS OF FORCE: SPONTANEOUS REASONING OR WELL-FORMED IDEAS PRIOR TO INSTRUCTION?

Throughout its forty year history, the conceptual change literature assumed student misconceptions of force are formed prior to instruction. We argue that it may well be the case that misconceptions are not formed until the student considers force and motion in a scientific context for the first time. This has obvious implications for research methods. We are in the early stages of developing a research method for investigating conceptual change in mechanics. To illustrate this method, we have taken examples from one-to-one Socratic tutoring. We conclude by outlining the next step of the research, which is to build a model that will enable the Socratic method to reveal the characteristics of misconceptions.

VYGOTSKY AND THE ZPD: HAVE WE GOT IT RIGHT?

There is a disparity between the historical Vygotsky and the diversity of ‘our Vygotsky’. This disparity is complex because it not only involves the interpretation of text but also mistranslation, construction of meaning and the legitimisation of current trends. This article attempts to throw light on the historical Vygotsky by unpacking this disparity with the ZPD in particular and argues that the ZPD should be seen as part and parcel of a scientific method in the quest to change psychology into an abstract theoretical framework, similar in structure to Marxs Capital In relation to Vygotsky, the issues of interpretation, translation, the logic of Capital and the ‘scientifically correct method’ are discussed. In particular, the ZPD is discussed in the context of teaching mathematics as a formal, academic discipline.