Karen François

Philosophical Dimensions in Mathematics Education

Prelude.- Prelude.- Interlude.- The Untouchable and Frightening Status of Mathematics.- Interlude.- Philosophical Reflections in Mathematics Classrooms.- Interlude.- Integrating the Philosophy of Mathematics in Teacher Training Courses.- Interlude.- Learning Concepts Through the History of Mathematics.- Interlude.- The Meaning and Understanding of Mathematics.- Interlude.- The Formalist Mathematical Tradition as an Obstacle to Stochastical Reasoning.- Interlude.- Logic and Intuition in Mathematics and Mathematical Education.- Interlude.- A Place for Education in the Contemporary Philosophy of Mathematics.- Interlude.- Ethnomathematics in Practice.- Postlude.

Ethnomathematics in Practice

In this paper we elaborate on the difference Bishop made between mathematics with a capital M and mathematics with a small m. The relation between M and m is far from clear and, confronted with the task of teaching mathematics, the problem only sharpens. In the first section (Ethnomathematics) we give a brief presentation of the critical role of mathematics. We make clear that we should be conscious of the institutional aspects of mathematical learning and teaching. Mathematics education always takes place in an institutional context, e.g. the school context. The point is that mathematical learning or thinking is contextual in any living culture; it lives and develops and is used in a particular cultural context. We make a plea to consciously and explicitly seize and actively practice these different world views, to have pupils attain a level of comprehension and sophistication. This practice should be an ingrained part, implicitly or unconsciously, of the curriculum of mathematics. In the second section (Empirical Facts) we offer some suggestions about the practical use of ethnomathematics in the classroom, taking as a starting point examples of field research among the Navajo Indians (in the U.S.) and among the Turkish ethnic minority (in Belgium).

The Untouchable and Frightening Status of Mathematics

During my research into the mathematics curriculum of Flanders secondary education (age 12-18), I first discovered that there is small scope for an explicit philosophy of mathematics. Nevertheless, there are some initial concepts formulated in the general objectives which tend to a more absolutist view of mathematics. In formulating the new curriculum, however, there was some attention paid to the inclusion of humanistic values. The mainstream of the implicit philosophy of mathematics is still a rather absolutist one, viewing mathematical truth∈dex as absolute and certain, connected with some humanistic values. Second, I discovered a large gap between general and vocational education. On the one hand, we can say that mathematics in vocational education is completely embedded in a modular system, and that attention is paid to core skills. On the other hand, we must say that pupils are prepared for specific occupations, for personal and social functioning, and to survive in our society. Access to higher education is theoretically possible, but unlikely for the majority. Mathematics in general education is a separate and different course. General education provides a strong base for higher education. In this paper, I shall briefly present some findings of the case study in which I want to make the connection with the theoretical framework of Alan J. Bishop.Bishop believes in a difference between the small m and the large M of mathematics, where the small m stands for a set of mathematical basic competence (such as counting, designing, explaining, locating, measuring, and playing) and the large M stands for mathematics, as in the Western scientific discipline. I shall argue that pupils in vocational mathematics are taught the small m and pupils in general education are taught the large M. The more general the education is, the larger the M is, which gains higher respect in society. In line with this M-m distinction, I shall elaborate on the connection which exists between the view on mathematics education and the didactics used in classroom. International comparative research on the results of mathematics educations shows us—in the case of Flanders—the best results nearly all over the world. I shall, however, criticize the way in which mathematics in schools chooses between the smart and the not so smart and what the role of ethnomathematics should be in Western school curricula to overcome this social stratification. Finally, I want to go on to explore two central hidden values in mathematics education to demystify the untouchable and frightening status of mathematics.

On the Notion of a Phenomenological Constitution of Objectivity

In this paper I elaborate on the way in which Husserl analyzed the constitution of objectivity both of the ideal and of the material objects. A central question in The Origin of Geometry (1936) is how an internal, personal, psychological process of consciousness can evolve into the objectivity of objects. In line with the analysis of Husserl, I demonstrate the constitution of objectivity as a human practice with five layers which can be identified as: (1) the stage of “the self-evidence”, (2) the condition of “retention”, (3) the possibility of remembrance, (4) the inter-subjective stage of communication, and (5) the final stage of sedimentation. Throughout those five stages, we evolve from an intra-subjective through an inter-subjective into a final objective stage of an object, be it a real or an ideal object. With this phenomenological meaning of the concept of objectivity, both objectivity and subjectivity are not longer seen as the very opposite of each other. Instead, both concepts are indissolubly connected along a continuous line. Furthermore Husserl created a phenomenological foundation for both phenomena: mathematical objects and objects from the empirical sciences. Ever since, both objects are grounded in the original self-evidence which takes part at the Life-World.

Olympification Versus Aesthetization: The Appeal of Mathematics Outside the Classroom

In this chapter we explore how mathematics education is caught by a meritocratic sense of the useful and how it could benefit from a more creative and experiential approach. The notion of olympification in mathematics education comes to the fore in the analysis of the differences between the measurements of PISA and TIMSS, further detailed by an example of Flanders (Belgium). Besides the observation of the olympification we consider the possibility of another perspective on mathematics education, looking at a way of bringing classroom mathematics in interaction with the material grounding of mathematics and with other experiences in life. Based on the content analysis of eight international journals concerning mathematical education we demonstrate the extent in which teachers and researchers take care of outside classroom experiences as possible input for a mathematical curiosity and understanding. Focusing on the relation between mathematics and art we will shortly explore different examples of mathematics within the arts. Finally we bring an example of how a mathematician can creatively bring mathematics outside the classroom.

Reforms and collaborations in Europe–China doctoral education

ABSTRACT This special issue focuses on the reforms and collaborations in Europe–China doctoral education. The articles in this special issue provide an insightful picture of the recent reforms in doctoral education in China and EU countries. Next to the structural reforms in Europe and China, the special issue papers have also specifically focused on EU–China cooperation in doctoral education, such as the current cooperation models, quality assurance in joint programmes and student experiences. In additions to an introduction of seven papers included in the special issue, the introduction provides necessary background information concerning Europe–China doctoral education as well as a reflection on the general issues, challenges and their implications.

Math and Music: Slow and Not For Profit

This chapter looks at the impact of recent societal approaches of knowledge and science from the perspectives of two rather distant educational domains, mathematics and music. Science’s attempt at ‘self-understanding’ has led to a set of control mechanisms, either generating ‘closure’—the scientists’ non-involvement in society—or ‘economisation’, producing patents and other lucrative benefits. While scientometrics became the tool and the rule for measuring the economic impact of science, counter movements, like the slow science movement, citizen science, empowering music-art initiatives and other critical approaches focus on intrinsic and ethical questions of education and knowledge. Thinking about knowledge and research in terms of quantifiable products impacts heavily upon the domains of science and arts, while the complexity of knowledge acquisition forces society to consider also other parameters like equality, personal development and participatory processes.

We’re (Not) Only in It for the Money (Frank Zappa): The Financial Structure of STEM and STEAM Research

The development of the philosophy of science in the twentieth century has created a framework where issues concerning funding dynamics can be easily accommodated. It combines the historical-philosophical approach of Thomas Kuhn (The structure of scientific revolutions (2nd ed., enlarged). The University of Chicago Press, Chicago, [1962] (1970)) with the sociological approach of Robert K. Merton (The normative structure of science. In: Storer NW (ed) The sociology of science. Theoretical and empirical investigations. The University of Chicago Press, Chicago, pp 267–278, [1942] (1974)), linking the ‘exact’ sciences to economy and politics. Out of this came a new domain, namely the study of scientific practices (Pickering P (ed): Science as practice and culture. The University of Chicago Press, Chicago, 1992). Given this broad theoretical framework, we will specify by looking at the case of STEM education and its variant Science, Technology, Engineering and Mathematics with Art (STEAM) education as an example par excellence. Without going into the technical details of the financial support of the projects, we prefer to open a philosophical debate on the way how policies on academic subjects influence a whole society and the personal life of both researchers and people/pupils involved in education.

The Concept of Culture in Critical Mathematics Education

A well-known critique in the research literature of critical mathematics education suggests that framing educational questions in cultural terms can encourage ethnic-cultural essentialism, obscure conflicts within cultures and promote an ethnographic or anthropological stance towards learners. Nevertheless, we believe that some of the obstacles to learning mathematics are cultural. ‘Stereotype threat’, for example, has a basis in culture. Consequently, the aims of critical mathematics education cannot be seriously pursued without including a cultural approach in educational research. We argue that an adequate conception of culture is available and should include normative/descriptive and material/ideal dyads as dialectical moments.

Perceptions of European and Chinese stakeholders on doctoral education in China and Europe

ABSTRACT This study investigates perceptions of European and Chinese stakeholders on doctoral education (DE) in China and Europe, particularly the cooperation between the two sides. Data were collected through online and paper survey from both European and Chinese stakeholders (N = 946). The results provide insights for policy-makers, university administrators, doctoral students and their supervisors when planning and engaging in DE in collaboration between European and Chinese higher education institutions (HEIs). Various cooperation models are currently implemented between Chinese and European HEIs. European respondents reported a higher awareness level of cooperation issues compared to the Chinese respondents. Significant differences were found between the Chinese and European respondents regarding their perceptions, perceived advantages and challenges, as well as expectations on cooperation in DE.