Philosophy of Mathematics in School Curricula of Mathematics Education. A Case Study of Flanders (Belgium) mathematics curriculum

Monday 28 February 2005 by François, Karen

During my research into the mathematics curriculum of Flanders secondary education (age 12-18), I discovered that there is small scope for an explicit philosophy of mathematics. Nevertheless there are some initial concepts formulated in the general objectives which tended to a more absolutist view of mathematics. However, in formulating the new curriculum, 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 as absolute and certain, connected with some humanistic values. In this paper I shall present the data of the case study.

Philosophy of Mathematics in School Curricula of Mathematics Education
A Case Study of Flanders (Belgium) mathematics curriculum

Karen Fran?ois

Centre for Logic and Philosophy of Science
Free University Brussels
Karen.Francois@vub.ac.be

Abstract

Introduction

The research on the implicit and explicit philosophy of mathematics in school curricula of mathematics education takes place in a broader research project, in which we are looking for the relations between sciences, society, politics and the democratic constitutional state. Within this project, one of the key questions is: ’what is the place of mathematics in (or indeed above?) sciences’ and broader, ’what is the place of mathematics in society’? By narrowing this question we shall elaborate on the research questions how mathematical knowledge is reproduced in our society, how mathematics is handed down from generation to generation. Obviously, education is an important, if not, one of the most important ways to reproduce knowledge in our society.
In a fist stage of the research, I was looking for the question ’if there is room for a philosophy of mathematics in school curriculum’. At a later point of view, this question seems a rather simplistic one. And even so the question ’Why should we implement a philosophy of mathematics in the curriculum of education?’ because there still exists a philosophy of mathematics in the current curriculum. However, it has not been made explicit yet, it remains hidden. Moreover mathematics is strongly directed towards the performance of techniques and has nothing to do with the study of mathematics as a historical and cultural product nor with the underlying cultural values. If we make the difference between an implicit and an explicit philosophy than we can propose the question ’if there is room for an explicit philosophy of mathematics’. The existence of an implicit philosophy is obvious. The absence of an explicit philosophy of mathematics tells us something about the implicit values of the curriculum. To discover them, we need to implement an explicit philosophy. I shall present the main findings on the screening of the curricula of mathematics.

Curriculum screening

The research question is if there are targets - beside the technical oriented targets - integrated in the curricula concerning philosophical issues in the broad meaning of the term (e.g. (strictly) philosophical, cultural, historical, …).

The method used to analyse the curriculum was a qualitative screening:
? at the level of the curriculum as developed by the community (as strictly enforced by law) and (section 1 and 2)
? at the level of the authorities of the differing school systems (section 3).

In Flanders there are three differing school systems namely public schools, subsidized private schools and subsidized community schools. They have to integrate the attainment targets into their own-developed curricula.
Before we present the kind of philosophical items in the curriculum we need to point out that the curriculum comes in two parts where philosophical issues can be found. Part one is the view on mathematics in education and some general objectives (section 1); part two are the attainment targets (section 2). It is understandable that teachers are focused on part two, because the attainment targets are the criteria for the evaluation of pupils.

Secondary education has four forms: general, technical, art and vocational secondary education. The four forms of education are not organised separately in the first stage. From the second stage, they are organised separately. In the first grade, there is an A class which gives access to the general, technical and art secondary education. There is the B class, which only gives access to vocational secondary education.

In the following tables, you will find the listing of retained paragraphs on philosophy (in the broad meaning of the term) and the non-technical oriented targets.

1. The curriculum as developed by the community at the level of the view

In a first general overview (table 1), one can see that there is no room for philosophy in vocational education and that there exist a real gap between general and vocational education.

Table 1: "Philosophy" of mathematics at the level of the view - overview

Grade Type of education
I A typegeneral B typevocational
II general technical art vocational
III general technical art vocational

Now, we want to go in more detail into the philosophical issues. Tabel 2 presents the listing of all issues referring to non-technical goals into the curriculum. We differentiate by grade (I, II and III which correspond respectively with the age of I = 12-14; II = 14-16; III= 16-18) and by type of education (general, technical and art) if necessary.

Table 2: "Philosophy" of mathematics at the level of the view - detail

Grade I: A type (general education)
Ontological proposition: The proposition that mathematics is abstract and formal and that mathematics has no connection with reality, is up to a certain degree.Appreciation: Pupils must be encouraged to see the beauty and the perfection of a geometric figure, the clarity of a well reasoned argument and the elegance of a formula. The cultural and dynamic meanings of mathematics:The pupils should have an experience that mathematics has a practical use, and that it has an educative and aesthetic value. The history of mathematics helps pupils to understand that mathematics is an important aspect and component of culture, both in the past and the present.Mathematics in the past developed via many cultures. Due to the emphasis of this development, pupils will gain the knowledge that mathematics is a dynamic process.The fundamental goals are:Pupils will have the experience of mathematics as a dynamic sciencePupils will have the experience of mathematics as an important cultural component.
Grade II: general, technical and art education
The ontological proposition: is absentAppreciation:In addition: when the commission determined the selection of the goals, they took into account, the effect of the development of a relationship with mathematics.The cultural and dynamic meanings of mathematics: (more abstract)The pupils should have an experience that mathematics has a practical use, and that it has an educative and aesthetic value. Attention to the development of mathematics helps pupils to understand that mathematics is an important aspect and component of culture, both in the past and the present. In this manner pupils will gain the knowledge that mathematics is a dynamic process.The fundamental goals are:Pupils will have the experience of mathematics as a dynamic sciencePupils will have the experience of mathematics as an important cultural component.
Grade III: general education
Idem Grade II
Grade III: technical and art educationThe text marked in grey is dropped at this level.
Idem Grade IIThe cultural and dynamic meanings of mathematics: (a partial interpretation)The pupils should have an experience that mathematics has a practical use, and that it has an educative and aesthetic value. Attention to the development of mathematics helps pupils to understand that mathematics is an important aspect and component of culture, both in the past and the present. In this manner pupils will gain the knowledge that mathematics is a dynamic process.The fundamental goals are: (one goal has been dropped)Pupils will have the experience of mathematics as a dynamic sciencePupils will have the experience of mathematics as an important cultural component.

In the following paragraph we move on to the more important level of the attainment targets. While it is possible to ignore the view and the general objectives of a curriculum, teachers are supposed to taken into account the attainment targets. They have to focus on that part, because the attainment targets are the criteria for the evaluation of pupils and schoolbooks are build up based on this targets.

2. The curriculum as developed by the community at the level of the attainment targets

Looking at the overview of possible locations for philosophical issues (table 3), one will see there are no philosophical (historical or cultural) goals formulated, neither for the B type (vocational education), nor for the first grade. The philosophical issues are reserved only for the second and third grade of general education.

Table 3: "Philosophy" of mathematics at the level of the attainment targets - overview

Grade Type of education
I A typegeneral B typevocational
II general technical art vocational
III general technical art vocational

Tabel 4 presents the listing of the non-technical attainment targets of the curriculum. We only retained three different attainment targets over the six years of secondary education. Philosophical goals are completely skipped in vocational education and at the first grade (age 12-14). While there is some attention paid for a philosophy of mathematics at the level of the introduction of the curriculum, one can see that at the level of the content of the course, the real stuff that pupils have to gain, there is only very small room for philosophical reflection.

Table 4: "Philosophy" of mathematics at the level of the set attainment targets - detail

Grade II: general education
Pupils can give examples of the contribution of mathematics to art.
Grade II: technical and art education
Pupils will gain appreciation for mathematics (possibilities and limitations) in confrontation with the cultural, historical and scientific aspects of mathematics.
Grade III: general education
Idem II: general education
Grade III: technical and art education
Pupils can give examples of the application of mathematics in other courses and into society in general.

3. The curriculum as developed by the differing school organisations

It is now interesting to see what happens at the level of the self-developed curriculum of the differing school organisations. They have the freedom to add some targets, fill in the content of the attainment targets and of attitude. What did they do with this freedom? We shall not repeat those aspects which are integral to the self-developed curricula, we shall only pay attention to the new aspects which have been added.

Firstly, it can be said, they (unfortunately) do not appear to have done a great deal with this freedom. Only the curriculum of the Catholic schools displays an explicit ideological message which is directed at teachers of mathematics:

"A teacher of mathematics at a catholic school will teach the same mathematics as their colleagues in public and community schools.
However, they have a duty to refer to the ideological project, wherever they can. As a member of the Christian pedagogical project, they should be alert in order to seize any opportunity to emphasize a deeper and more intense dimensions. Also the courses in mathematics opens up other possibilities. The better teachers know their pupils personally, the better they can feel the moment when pupils have the openness to advance toward ontological and existential questions."

Looking at the level of the attainment targets we discovered the following three additional ’philosophical’ issues.

Public school: Grade II: general, technical and art education
Mathematics can contribute to expressive-creative education, specifically to architecture, art of painting and sculpture.

Catholic school: Grade II: general, technical and art education
The education of mathematics is bound with other disciplines and courses. Moreover even mathematics has during the past centuries been developed in a historical context with its specific ideas and problems. So, it is also important to pay attention to the historical context to assist the pupils to gain an understanding of mathematical problems.

Catholic school: Grade II: only for general education
For example, in geography, a teacher can pay attention to the contribution of mathematics in architecture, music, painting and sculpture. (Escher, Vasar?ly, Mondriaan, Roelofs, Le Corbusier, The Pantheon, …)

Conclusion

In a following step of our research we shall elaborate on the interpretation of these empirical data. Based on the theoretical framework of both, Bishop ([1991], 1997) and Ernest ([1991] (2003), we shall argue why we retained the following three findings: 1) the absolutist view on mathematics, 2) the gap between vocational and general education and 3) the related distinction between mathematics with a capital ’M’ and with a small ’m’.

References

Bishop, Alan. J. (1991) (1997) Mathematical Enculturation, A Cultural Perspective on Mathematics Education, Mathematics Education Library, Vol. 6, Kluwer Academic Publishers, Dordrecht / Boston / London.
Ernest, Paul (1991) (2003) The Philosophy of Mathematics Education, Studies in Mathematics Education, RoutledgeFalmer, London.
Fran?ois, Karen / van Bendegem, Jean Paul (2004) ’Philosophy of mathematics in the curriculum of mathematics. Questions and Problems Raised by a case study of secondary education in Flanders, Belgium’, Contribution for the 10th International Congress on Mathematics Education, Discussion Group 4, http://www.icme-organisers.dk/dg04/

The curricula

The extended list of the references of the curricula, developed by the community and the own-developed curricula by the differing school systems are available at the website.

The curricula, developed by the community:
http://www.ond.vlaanderen.be/dvo/secundair/index.htm

The own-developed curricula by the differing school systems:
In Flanders there are three differing school systems namely public schools, subsidized private schools and subsidized community schools. They have to integrate the attainment targets into their own-developed curricula.

Public schools:
http://www.rago.be/pbd/so/index.htm

Subsidized private schools (mostly catholic schools):
http://www.vvkso.be/

Subsidized community schools:
http://www.ovsg.be//default.asp?folder=663&foldername=Secundair+Onderwijs