TEACHING APPLIED MATHEMATICS AS A BRIDGE FROM PHILOSOPHY OF SCIENCE TO PHILOSOPHY OF MATHEMATICS EDUCATION
Peter Collignon
University of Erfurt, Germany
peter.collignon @ uni-erfurt.de
Abstract
The following considerations focus on the possibilities of independent and creative use of mathematics with a view to an enlightened perception of science. Looking at the example of calculus applied on disciplines beyond the natural sciences several connections between philosophy of science and common postulates of mathematics education are pointed out.
Introduction
In spite of (or actually due to) the fact that we can observe a co-evolution of calculus and natural sciences, especially physics, it became apparent that using calculus in social sciences, particularly economics, provides a promising approach to insights on how to implement mathematics in so-called exact sciences. Students, but also teachers, often show a naively realistic understanding of science. Within teacher education, a widening of student’s horizon can be achieved by using a certain disassociation effect caused by the discrepancy between the mathematical subject and the field of applications. This may enable students to adopt an adequate epistemological perspective. In this context the modelling aspect is included as well as a constructivist view on the subjects under (mathematical) investigation (cf. Ernest, 2004). Among other things this approach facilitates an unconventional access to Freudenthal’s method of reinvention (cf. Freudenthal, 1983).
An extended didactical triangle
The well-known didactical triangle shows the relationships between students, instructors and the subject to be taught. In some cases an extension can be meaningful. Here the vertex “Philosophy“ is added and we get an didactical tetrahedron. All vertices are connected in pairs.
Figure 1. An extended didactical triangle
If one unfolds the tetrahedron on the left, you get one of its nets. This opens up the possibility of diversifying its top and indicating three important aspects:
· Philosophy as a leading science,
· Philosophy of Mathematics Education*),
· Philosophy of Science in general**).
Within teacher education, an expansion of scientific understanding can be achieved that promotes an adequate epistemological view and modifies naive-realistic ideas, for instance by emphasizing a rather constructivist view. For this purpose, covering mathematical economics using methods of calculus is a promising approach. According to Fischer & Malle (1985, p. 107), the absence of a law-of-nature-character is even necessary to allow learners the free use of mathematics describing the so-called reality. In this sense the modelling perspective demonstrates a human distance to reality.
Jablonka (1996) states, that this view assumes an understanding of the underlying mathematical concepts separate from the context. This consideration can be modified by discussing the usage of a certain mathematical concept (calculus) for modelling (economic) circumstances, which originally did not contribute to its genesis. However, this approach should go along with an adequate awareness of the evolution and the conditions of the mathematics involved. In case of calculus students should have an awareness for the co-evolution with natural sciences, especially physics (cf. Boyer, 1959).
The use of calculus in economics
Physicists had good reasons to develop mathematical concepts regarding the continuum, for space and time are usually experienced as continuous. This is also applicable to change within space and time, the motion. The quantification of time-dependent processes and geometric phenomena involving curvature, torsion and the like required some kind of infinitesimal calculus. Aristotle already carried out such calculations, e.g. with respect to the parabola. But he did not yet use our “modern” concept of a function that made its way nearly 2000 years later, promoted by Euler, Weierstrass and others.
Using real functions makes it easy to define such concepts like monotony, (local) extrema and other terms connected with curve sketching. This opens up the possibility of involving economic categories, which are often connected with concerns of optimizing. The application of calculus to economics assumes that there are quantities, one independent and one dependent at least, such that economic circumstances are represented in a proper way.
What could those quantities be? The fundamental economic quantities are mainly of discrete type, such as numbers of items and currency units. In the first instance there is no reason of applying infinitesimal methods that require continuity or deal with infinity. But for what reason do we use calculus in economic contexts? Today’s well-established form of calculus provides a “calculus” – understood as a collection of methods for certain calculations – which is tempting because of its algorithmic character and is taught in classrooms. Many economic facts are adjusted and modified, so that they meet the conditions for a continuous perspective. There are several possibilities: Firstly, one may interpret geometrical quantities in terms of economics. For instance, certain areas or volumes are regarded to be proportional to costs. Among other things, this is motivated by the problem of economical packaging. Furthermore, one can define continuous economic quantities in an artificial way, so as there is a continuous mathematical model of the initial discrete situation. In this case, the continuous version is interpreted as an approximation of the “real” issue.
Mathematical model, reality and perception of science
For a self-contained application of mathematics – we call it mathematical modelling – there are challenges and opportunities at the same time:
· Social and economic conceptions are no “laws of nature“.
· There are no canonical models.
· It is not clear from the beginning how far economic issues are a subject suitable for mathematization.
· In physics there are theories – models – that are looked upon as the (supposed) “truth“ (cf. Cartwright, 2002).
· In social sciences and economics there are well-established models too, but they are considered as less authoritative.
· Social and economic circumstances can be used as a starting point for giving an idea about how quantitative science ‘works‘.
· This offers an opportunity to encourage students to model self-contained.
· Ideally, the discussion of these considerations helps students to get an adequate perception of science.
As mentioned above, the genesis of calculus was strongly influenced by the natural sciences, particularly physics. For instance, there have been “models” developed with the purpose of a description of planetary movements or to understand ballistic phenomena. Calculus uses the real numbers as an indispensable basis. This was (and is) assumed to be adequately in several modelling situations. Contemporary modelling has extended the scope of applications. There are mathematical models using calculus e.g. in social sciences and economy. These subjects deal with quantities that are not originally discrete in every case. This raises the question of whether infinitesimal concepts are a suitable tool for describing the “reality” in those subjects. There are different possible answers; they depend on the perspective, the targets under investigation and the perception of science.
In educational contexts there are several additional aspects to consider. Students will receive deeper insight in the process of modelling and achieve an appropriate concept of science by considering the factors mentioned above. Infinitesimal calculus is both – a standard tool in mathematical modelling as well as an example of a concept that is often used without critical reflection. It provides a suitable occasion to discuss the diversity of modelling, which ultimately results in considerations about science paradigms. For this, applying mathematics beyond the natural sciences is a promising thematic anchor.
References
Cartwright, Nancy (2002): How the laws of physics lie. Oxford, New York: Oxford University Press.
Boyer, Carl B. (1959): The History of the Calculus and Its Conceptual Development. New York: Dover Publications Inc.
Ernest, Paul (2004): What is the Philosophy of Mathematics Education? Philosophy of Mathematics Education Journal, 18, 17pp. http://www.ex.ac.uk/~PErnest/pome18/contents.htm.
Fischer, Roland & Malle, Günther (1985): Mensch und Mathematik. Eine Einführung in didaktisches Denken und Handeln. Mannheim; Wien; Zürich: Bibliographisches Institut.
Freudenthal, Hans (1983): Didactical Phenomenology of Mathematical Structures. Dordrecht: D. Reidel Publishing Company.
Jablonka, Eva (1996): Meta-Analyse von Zugängen zur mathematischen Modellbildung und Konsequenzen für den Unterricht. Berlin: Transparent H. & E. Preuß.