Ole Skovsmose

The End of Innocence: A Critique of 'Ethnomathematics'?

Ethnomathematics originated in the former colonies, in response to the Eurocentrism of the history of mathematics, mathematics itself and mathematics education. It has also found expression in several other contexts. It is a part of the broader framework that elaborates the social and political dimensions of mathematics and mathematics education but especially, the dimension of culture. This focus on culture examined in the unique context of South Africa makes visible both conceptual difficulties in its formulation and also difficulties with respect to its interpretation into educational practice. This paper explores a critique of ethnomathematics using the South African situation and conceptual tools of a critical mathematrics education.

Critical Mathematics Education

Critical Mathematics education is described in terms of ‘concerns’ which cover the following issues: a) Citizenship identifies schooling as including the preparation of students to be an active part of political life. b) Mathematics may serve as a tool for identifying and analysing critical features of society, which may be global as well as having to do with the local environment of students. c) The students’ interest emphasises that the main focus of education cannot be the transformation of (pure) knowledge; instead educational practice must be understood in terms of acting persons. d) Culture and conflicts raise basic questions about discrimination. Does mathematics education reproduce inequalities which might be established by factors outside education but, nevertheless, are reinforced by educational practice? e) Mathematics itself might be problematic because of the function of mathematics as part of modem technology, which no longer can be reviewed with optimism. Mathematics is not only a tool for critique but also an object of critique. f) Critical mathematics education concentrates on life in the classroom to the extent that the communication between teacher and students can reflect power relations.

Towards a Critical Mathematics Education

To illustrate aspects of critical mathematics education a project involving 14–15 years old students is described. Mathematics education can be organized so as to develop different types of knowing: mathematical knowing, which can be associated with skills developed in traditional teaching; technological knowing, which can be associated with a competence in mathematical model building; and reflective knowing, which can be seen as a competence in evaluating applications of mathematics. The thesis discussed says that if mathemacy should be developed as a competence of importance in a critical education, it must integrate mathematical, technological as well as reflective knowing. Via the description of the project, a possible educational meaning is given to this thesis. Especially, it is discussed what it could mean to involve students in reflections about mathematics as a tool for technological design.

Landscapes of Investigation

According to many observations, traditional mathematics education falls within the exercise paradigm. This paradigm is contrasted with landscapes of investigation serving as invitations for students to be involved in processes of exploration and explanation. The distinction between the exercise paradigm and landscapes of investigation is combined with a distinction between three different types of reference which might provide mathematical concepts and classroom activities with meaning: references to mathematics; references to a semi-reality, and references to a real-life situation. The six possible learning milieus are illustrated by examples.

Mathematical Education Versus Critical Education.

This paper concentrates upon the relationship between mathematical education (ME) and critical education (CE) connected with the Frankfurt School and Critical Theory. To make the discussion as precise as possible a distinction is made between three alternatives in ME: Structuralism, pragmatism, and process-orientation. These alternatives are related to the key terms of CE in order to show the extent to which ME and CE contradict each other. The conclusion is that there does not exist any integration — nor even any close relationship-between ME and CE.Finally, this result is discussed in the light of the following two theses:(A)It is necessary to increase the interaction between ME and CE, if ME is not to degenerate into one of the most important ways of socializing students into the technological society and at the same time destroying the possibilities for developing a critical attitude towards precisely this technological society.(B)It is important for CE to interact with subjects from the technical sciences, and among these ME, if CE is not to be taken over by the technological development and fade away into an unimportant and uncritical educational theory.

Linking Mathematics Education and Democracy: Citizenship,Mathematical Archaeology, Mathemacy and Deliberative Interaction

The relationship between mathematics education and democracy is discussed in terms of citizenship, mathematical archaeology, mathemacy and deliberative interaction. The first issue concentrates on the learner as a member of society, the second on the social functions of mathematics and on how to get to grips with mathematics in use, the third refers to an integrated kind of competence including different forms of reflection (mathematics-oriented, model-oriented, context-oriented and lifeworld-oriented reflections), the fourth issue considers the classroom as a micro-society and deals with the nature of the teaching-learning process. These four issues are discussed with reference to an example of educational practice. “Our Community”, carried out among sixteen-year-old students as an interdisciplinary project including a one-week trainee service. Finally, it is indicated that a discussion of mathematics education and democracy is essential to a further development of social theory, as the notions of citizenship, mathematical archaeology, mathemacy and deliberative interaction become part of the discussion about modemity, reflexive modemity and other constructs from recent social theory.

Mathematical education and democracy

Is it possible to develop the content and form of mathematical education in such a way that it may serve as a tool of democratization in both school and society? This question is related to two different arguments. The social argument of democratization states: (1) Mathematics has an extensive range of applications, (2) because of its applications mathematics has a “society-shaping” function, and (3) in order to carry out democratic obligations and rights it is necessary to be able to identify the main principles of the development of society. The pedagogical argument of democratization states: (1) Mathematical education has a “hidden curriculum”, (2) the “hidden curriculum” of mathematical education in a traditional form implants a servile attitude towards technological questions into a large number of students, and (3) we cannot expect any development of democratic competence in school unless the teaching-learning situation is based on a dialogue and unless the curriculum is not totally determined from outside the classroom.The social argument implies that we must aim at “empowering material” which could constitute a basis for reflective knowledge i.e. knowledge about how to evaluate and criticize a mathematical model, while the pedagogical argument implies that we must aim at “open material” leaving space for decisions to be taken in the classroom.Will it become possible to create materials at the same time open and empowering? To answer this question we have to analyse the concept ‘democratic competence’, which can be related to ‘reflective knowledge’ characterized by a specific object of knowledge and a specific way of knowledge production. The ultimate aim will be to unify these characteristics in an epistemological theory of mathematical education.This paper is a revised version of \ldDemocratization and Mathematical Education\rd, R. 88-33 Department of Mathematics and Computer Science, Aalborg University Centre.

Beyond the Tunnel Vision: Analysing the Relationship Between Mathematics, Society and Technology

Enthusiasm about the introduction of computers into mathematics education is widespread, the justifications for it however are very diverse: “Some call the effects of micro-computers on schools a revolution … Nothing before has so stirred schools into action. School systems, teachers, parents and children talk about computers as they never talked about programmed learning, educational television, open education nor raising the school leaving age, for that matter. Schools must have computers! No other educational technology has been thought to have such potential. People talk about how children are captivated by computer … while others stress computer-based jobs. Yet others urge affirmative action and remediation through computer-assisted-learning. Others point to the demands of technological culture when urging schools to use computers. On a different track some see a potential for more and better intellectual and social activities in schools, others stress self-image and self-expression. The range of possibilities is exceptional.”(Olsen 1988,p,9)

Project work in university mathematics education: A Danish experience : Aalborg University

This article discusses project work in university mathematics education. The practice perspective is obtained as students and teachers from Aalborg University share their experiences. A theoretical framework is introduced. It includes the following key-terms: Problem-centered studies, interdisciplinarity, participant-directed studies, and the exemplarity principle. The contrasting of this theoretical conception of project work with the practice shows that the original notion of project work has been modified as a consequence of its encounter with practice. The modification can be perceived as both a success and a failure. To discuss this, different perspectives on project work in mathematics are suggested.

Reflective Knowledge: Its Relation to the Mathematical Modelling Process

A distinction is made between three different types of knowledge related to a process of mathematical modelling: (a) mathematical knowledge itself; (b) technological knowledge, which in this context is knowledge about how to build and how to use a mathematical model; (c) reflective knowledge, to be interpreted as a more general conceptual framework, or metaknowledge, for discussing the nature of models and the criteria used in their constructions, applications and evaluations. Some types of problem relating to the modelling process—important to analyse in an attempt to develop reflective knowledge—are summarized. The first problem, accompanying a mathematization, is the phenomenon of disguising the complexity of the construction of the conceptual system, which constitutes the very foundation of the model. The second problem has to do with the confusion of the different possible guiding interests connected to a modelling. The third problem is caused by the nature of mathematical language, which makes it te...