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.

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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. Thus,
Vietnamese educators and teachers should remember that mathematics in class should be taught with
applications, but it should be taught as empirical theory and should not support a naive believe system
of mathematics.
Applications in mathematics education? Yes, of course – but with caution
61
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