Applications in mathematics education? Yes, of course – But with caution. About the dangers of a pure empirical belief system

55 HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1075.2017-0175 Educational Sciences, 2017, Vol. 62, Iss. 12, pp. 55-61 This paper is available online at APPLICATIONS IN MATHEMATICS EDUCATION? YES, OF COURSE – BUT WITH CAUTION. ABOUT THE DANGERS OF A PURE EMPIRICAL BELIEF SYSTEM Eduard Krause 1 , Ingo Witzke 1 and Nguyen Phuong Chi 2 1 University of Siegen, 2 Hanoi National University of Education Abstract. There is no doubt that using applications in mathematics lessons can help to provide a contribution to the life and to career-preparation and also to contribute more general knowledge. However, if teaching mathematics focuses on only applications and empirical activities, students may learn a naive- empiric theory that always prove the hypothesis by empirical observations not by logical reasoning. It is dangerous for students if they do not use logical reasoning in proving mathematical problems because mathematics often relies on logical and deductive reasoning. Thus, if mathematics in class is taught with applications, it should be taught as empirical theory and should not support a naive believe system of mathematics. Keywords: Applications in mathematics lessons, empirical theory, belief system, mathematics education, empirical belief system. 1. Introduction In Vietnamese secondary schools, mathematics knowledge is often introduced in an abstract way without any connection to its application in real life. That is the reason why students lack of experience and confidence to apply mathematics to out of school problems. Consequently, they leave school without a real awareness of the power and relevance that mathematics has in the modern society. Recognizing this problem, the Vietnamese government established a project of fundamental and global innovation for the Vietnamese education which emphasized that it is essential to develop for students the practical competence and the competence of applying learned knowledge to the reality [1]. Because of this orientation, many researchers, educators and teachers in Vietnam are concentrated on studying how to design a mathematics lesson which includes various applications, empirical activities and real life situations. There is no doubt that applications are important in mathematics lessons. However, in this paper we want to show that besides the advantages, teaching mathematics on applications and empirical objects may leads to a danger: While teachers want to provide an abstract-formal mathematical theory cleverly illustrated, the students may learn an naive-empiric theory on the illustration materials. 2. Content 2.1. Application in the mathematics lesson as a contribution to the general knowledge Received: May 21, 2017. Revised: August 5, 2017. Accepted: August 10, 2017. Contact: Nguyen Phuong Chi, e-mail address: chinp@hnue.edu.vn Eduard Krause, Ingo Witzke and Nguyen Phuong Chi 56 The goals of mathematics education are set differently by different take holders. Besides teaching the rudimentary mathematical abilities also typical skills should be taught. The aim is certainly to provide a contribution to the life and to career-preparation and also to contribute more general knowledge. H. W. Heymann puts out general education on the following points [7]: Life Preparation: The students should learn in mathematics classes how mathematics can be used to cope with life. It is about the areas already affecting students in their environment, but also to living conditions that face them as adults later. Foundation of cultural coherence: Mathematics is an essential part of the cultural property. Mathematics in class must try to develop a reflected cultural identity that recognizes differences to other cultures and stresses cultural achievements. World orientation: The construction of a differentiated personal world view is made possible by addressing global key problems. For this purpose it is necessary annul the subject-borders and to understand the contribution of mathematics in the interaction of many subjects. Instructions for critical use of reason: Education for autonomy, emancipation and empowerment for critical use of reason is based on the assumption that man is a rational thinking and rational self- determination capable being and that this property is not handed on a plate. Especially mathematics helps one to understand things in a prudent perspective and to reflect critically. Development of willingness to take responsibility: Just the critical use of reason should enable to act responsibly. Acting responsibly means that the consequences of one’s action (or omission) for himself and others are considered and that one admitted for them. Exercise in understanding and cooperation: An important goal of education is to educate students to a cooperative coexistence. The basis of any cooperation is communication. Just the math requires a specific type of communication. Students should master this and get to work together, as many problems can only be solved jointly. Strengthening the student's ego: A healthy self-esteem is a prerequisite for achieving the aforementioned objectives, because "responsibility as an ethical, understanding as a social, critical use of reason as a personal and intellectual principle sponsor himself comprehending as subject, conscious acting, courage developing personality ahead which is able to commit to such values" [7, p. 117]. Achieving these goals is accompanied by an application-oriented mathematics. So the aspect of life preparation requires already directly applying mathematical theories to everyday life situations. A contribution to world orientation mathematics can only afford, when the cross links with other subjects are seen. Heinrich Winter formulated a shorter but not less pithy description of the goals in mathematic teaching [14]: The mathematics teaching is characterized as general education when he allows three basic experiences: (W1) "understand and perceive in a specific way phenomena of the world around us, natural, social and cultural, that all of us tackle or should tackle. (W2) get to know mathematical objects and facts, represented in language, symbols, images and formulas, as intellectual creations, as a deductive orderly world of its own kind (W3) acquire problem solving skills, that exceed mathematics (heuristic capabilities) by dealing with tasks. Applications in mathematics education? Yes, of course – but with caution 57 This formulation grants in W1 application a special importance in mathematics education. With the help of mathematics students should have a special look on phenomena in nature, society and culture and should understand these occurrences in a better way. Of course mathematics is not identified by application – and students should get to know that mathematics is a mental creation, which does not need to be applied in reality (W2) genuinely, but they should also learn, that it is often useful to apply mathematics on real-life situations or other subjects. But how is about the relationship between mathematics and reality? And how should this relation is taught in class? 2.2. The relationship between mathematics and reality “The most distinctive characteristic which differentiates mathematics from the various branches of empirical science, and which accounts for its fame as the queen of the sciences, is no doubt the peculiar certainty and necessity of its results” [6]. In this way Hempel described in 1945 the specific feature of mathematics. The causes of the certainty and necessity of its results he sees in the fact that since Hilbert mathematics is not genuinely bound on to logical to reality. In this way also Einstein describes mathematics saying [4]: "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality". Figure 1. Schematic illustration of naively empiricism Therefore, students in school should get to know mathematical objects and facts, represented in language, symbols, images and formulas, as intellectual creations, as a deductive orderly world of its own kind (W2), but also as a view on reality (W1) to describe and understand the world. Another point is that mathematical terms are very abstract. Learning seems to be easier when you deal with concrete things, according to the inactively-iconic-symbolic-principle of Bruner [2; 3]. But teaching mathematics on empirical object bears a danger: While teachers want to provide an abstract-formal mathematical theory cleverly illustrated, the pupils may learn an naive-empiric theory on the illustration materials [13;16]. An empiric-deductive understanding of mathematics in contrast to a naive-empiric understanding must not be deficiently at school. Historical analyses (cf. [15]) show that many mathematicians owned such a view in the history of mathematics: Examples are Moritz Pasch for the geometry and Leibniz and Jakob and Johann Bernoulli in the analysis. By this empiric Eduard Krause, Ingo Witzke and Nguyen Phuong Chi 58 character school mathematics shows epistemological parallels to physics: Questions are derived with empiric objects and the derived sentences are experienced in the reality. So the meta-structure of physics (Nature of Science) also can be discussed for school mathematics. Because of the empirical character, this kind of mathematics has many epistemological parallels to natural sciences like physics. How can the development of theory that refers to reality be described? A naively answer on this question could be that in empirical theories a law can be found through generalization of empirical findings. This imagination can be called naive-empirical and is illustrated in Fig. 1. This description can be criticized in at least two points: - An objectively observation of reality is not possible. According to the theory of Karl Popper every observer is influenced by a priori theory. The terms in a theory cannot be found directly empirically – they are man-created [11]. - In many theories deduction plays a crucial role. For example in physics or in (historical) mathematical theories logical reasoning makes an essential part. The scheme of Fig. 1 lacks this aspect of deduction. A more detailed meta-description how a theory can be related to reality can be found in Einsteins EJASE-Model. This is taken out a letter Einstein wrote to Solovine in 1952. In the original version, which was reprinted and commended by Holton [8], Einstein describes the methodology of physics, but in his opinion it also can be generalized for all empirical theories. The essence of this model is shown schematically in Fig. 2. Figure 2. Schematic illustration of empirical theories The theory consists of concepts and axioms which can be regarded as intellectual creation. The way how the terms and axioms are created (the two round arrows in Fig.1 - Holton calls it the “Jump”) is not so easy describe (cf. [9]). Of those terms theorems can be concluded. What makes an empirical theory is that it has references to reality. That means, that the terms and axioms should be created in that way that the concluded theorems should be compatible with reality. That does not mean that a theorem is only true if it is verified in reality. The term “true” has different meaning in physics than in empirical mathematics, but both theories have in common that the sinfulness of a term depends on its empirical examination. So if mathematics in class is taught with applications, it should be taught as empirical theory and should not support a naive belief system of mathematics. The next chapter describes the dangers of pure empirical belief system. Applications in mathematics education? Yes, of course – but with caution 59 2.3. The dangers of a pure empirical belief system Alan Schoenfeld describes the importance of belief systems for the first time in his book “Mathematical problem solving” [12]. He understands the belief system as the conception of mathematics and the attitudes towards it. “One’s beliefs about mathematics [...] determine how one chooses to approach a problem, which techniques will be used or avoided, how long and how hard one will work on it, and so on. The belief system establishes the context within which we operate []” [2]. He demonstrates it on the following example, a geometry problem which students and a mathematician both had to solve: Two intersecting straight lines and a point P on one of the two lines are given. Show how a circle can be constructed using only a compass and a ruler in that way that the circle owns the two straight lines as tangents and P as a contact point (Fig.3). Figure 3. A geometric task to find out the belief-system The transcripts made Schoenfeld construct the archetype of students behaving like “pure empiricists” generating and verifying their Ideas and assumptions exclusively by drawing (Shoenfeld generates his category – system by analyzing transcripts of students and not using surveys like various other researches with a view on beliefs). A mathematician in contrast uses terms and definitions. The students’ review of a geometric hypothesis is solely done based on a drawing. Accuracy will increase its’ usefulness while the mathematician does not bother to draw his solution. Figure 4 e. g. shows that logical reasoning is not as important to students as a right-looking drawing because they are able to accept two different (logical inconsistent) solutions if both look appropriate. Overall the mathematician relies on logical reasoning while deductive reasoning is not important in the students’ process of problem solving. The different approaches can be explained by the different belief systems. The questions to determine the belief systems are: - How do the students / mathematicians come to hypotheses? - How to check the students / mathematician their hypotheses? - What role do logical deductions play? The belief-system of the students in the example calls Schoenfeld naive-empirical (naively science). It is named this way because it deals with real (empirical) objects’ (e. g. a drawing like in the example of Schoenfeld) and the hypothesis which is proven by empirical observation not by logical reasoning. This kind of mathematics describes a universe of discourse in the (physical) reality. The notion of truth relies on empirical facts gained through observation and experiments. Pure empirical mathematics do not need logical reasoning – it’s pure phenomenology. Eduard Krause, Ingo Witzke and Nguyen Phuong Chi 60 Figure 4. Students accept both solutions even though they are logically inconsistent, because the picture looks appropriate [2, p. 170] 3. Conclusion Using applications in mathematics lessons can help to provide a contribution to the life and to career-preparation and also to contribute more general knowledge including life preparation, foundation of cultural coherence, world orientation, instructions for critical use of reason, development of willingness to take responsibility, exercise in understanding and cooperation, and strengthen the student’s ego. However, it is dangerous if teaching mathematics focuses on only applications and empirical activities because it can make students learn a naive- empiric theory that always prove the hypothesis by empirical observations not by logical reasoning. If students do not use logical reasoning in proving mathematical problems, they will have difficulties when they face to an abstract- formal mathematical theory which relies on logical and deductive reasoning. 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