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Proceedings of the Discussing Group 9 :
Promoting Creativity for All Students in Mathematics Education
The 11th International Congress on Mathematical Education
Monterrey, Mexico, July 6-13, 2008
FUNCTIONAL MODELING –
A CREATIVE WAY IN MODELING
HANS-STEFAN SILLER
Abstract: The purpose of this paper is to show that modeling is a
very important part in mathematics education. Because of the
increasing importance of interdisciplinary teaching it is necessary
to look at subjects where modeling can be done interdisciplinary.
With the help of this view, certain problems can be solved through
a functional sight. This leads to the way of Functional Modeling,
which can be done in particular in Mathematics and Informatics.
The importance of this view should be discussed in this discussion
group.
Key words: Creative Personality, Creative Process, Creative
Environment, Creative Mathematical Product, Creative Thinking,
Creative Working.
INTRODUCTION
To enforce mathematical creativity it is necessary to have a look beyond one’s
own nose. Especially in mathematics education it is possible to take up innermathematical ideas like the pivotal idea of the function and to apply it to other
related subjects. Therefore it is necessary that the mathematical concepts and
ideas are realized and comprehended in detail by the learners. After the process
of understanding these fundamental ideas and concepts can be implemented with
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and in other subjects. One creative possibility to enforce the fundamental idea of
the function with the help of other subjects, in particular with Informatics, is the
interdisciplinary way of Functional Modeling.
Modeling is a well known method for working in a creative and sophisticated
way with pupils and students. It is also well known that Modeling is a pivotal
idea as well as in Mathematics and Informatics (H.-St.Siller, 2007). Hence it is
obvious that this pivotal idea in both subjects is a fundamental idea for the
interdisciplinary aspect of Mathematics and Informatics. Through narrowing the
broad perception of modeling it is possible to focus on interesting topics like
functions and functional sight. With the help of this view it is possible that pupils
or students work creative both at the sheet of paper and, if they have the chance,
at a computer or programmable graphical calculator. For advancing creativity in
this issue it is almost essential to work with a personal digital assistant. Thus
Functional Modeling accompanies strongly to Functional Programming.
Functional programming can be done in several ways in school. A lot of
computer programs allow doing it. Therefore it is very important to think about
the ways which are offered to go and the programs which are concepted
functionally. For students and pupils it is possible to do Functional Modeling
with different programs. Some of them have already been discussed; some of
them should be discussed in detail. Students and pupils shall focus on:




EXCEL,
Computer Algebra Systems (CAS),
Graphical calculators,
Functional programming languages (Haskell, Scheme, ...).
Different scientists like Hubwieser, P. or Schneider, M. have thought about the
ways of functional modeling in education and they have developed adequate
concepts to do it effectively. There are a lot of possibilities (because of the
enormous variety in computer programs) for students and pupils to create
programs in a functional way. This is very important because only a wide
spectrum of applications can lead to an efficient education in Functional
Programming. How this can be done in EXCEL is told by P.Hubwieser (2007)
'Didaktik der Informatik' impressively. How Functional Modeling can be done
with a calculator Fuchs/Siller/Vásárhelyi have shown in their book 'Basics in
Functional Modeling' (K.Fuchs, H.-St.Siller, E.Vásárhelyi, 2008). Another
possibility for functional programming is the usage of (programming-) languages
like Scheme or Haskell additionally. Important facts, why functional
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Functional Modeling - A Creative Way in Modeling
programming could be done with such a language are:

instructor need not be a mathematician,

needn't have experiences in the use of CAS or graphical calculators,

needn't make investments for expensive program solutions (like EXCEL).
Another very important fact, for using a language like Haskell is that the 'BlackBox'-principle is eliminated by creating little programs for specific problems.
The students must understand the mathematical background to realize it in a
functional way. On the one hand it is possible to advance the effective use of
'White-Box'-principles (B.Buchberger, 1992) by this approach because it has to
be an important aim for instructors. On the other hand interdisciplinary teaching
gets more and more important. This is a fact which is included in several
curricula and the students are motivated for informatical/mathematical working
on computers through a strong interdisciplinary teaching in certain topics.
Another important argument for teaching functional modeling are recursive
functions as Fuchs, K. J. told in his presentation 'Functional Thinking - A
fundamental idea in teaching Computer Algebra Systems' (K.Fuchs, 2007). With
the help of recursive functions it is possible to focus on recursive definitions of
the binomial-coefficient, shown in (K.Fuchs, H.-St.Siller, E.Vásárhelyi, 2008),
the binomial-equations, shown in (K.Fuchs, H.-St.Siller, E.Vásárhelyi, 2008) or
some special sequences, like sequence of Naryana (Ars electronica 93, 1993)
which are calculated with computers. With an informatical sight on these
subjects, they are of increasing importance again and it is possible to show
interesting aspects for such things beyond mathematics.
AN EXAMPLE FOR WORKING IN A CREATIVE WAY WITH
PROGRAPH DIAGRAMS
Very interesting and ideal topics for creative interdisciplinary teaching are
number systems. In this subject it is possible to connect easy basic mathematical
knowledge with an informatical sight. Students should have knowledge
 in the division with residues,
 about different number systems, like the decimal system and the dual
system,
 in functions and functional representation, like recursive description.
If they are common in these things it is possible to look at the conversion of a
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natural number to a binary number in an interdisciplinary mathematical and
informatical way. The conceptual formulation could amount like the following:
Think about a functional way to derive binary
numbers out of natural numbers. Before the
implementation with a Computer Algebra System
draw an appropriate diagram to show the
coherences between the used functions.
PROGRAPH diagrams (S.Matwin, T.Pietrzykowski, 1985) (Diagram 1) are an
adequate illustration for Functional Modeling. Of course it is necessary that
students get an introduction to the different symbols in these diagrams. A short
summary of all the symbols can be found in [4]. If they are familiar with these
symbols the diagrams are very intuitive and easy to draw.
Diagram 1.
It is obvious to see that the conversion of a natural number to a binary
number can be done very easy through the help of

conditions,

basic mathematical functions (Modulo-function, Append-function,
Division-function, Integer-function, recursive-functions),

guiles.
326 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 4
Functional Modeling - A Creative Way in Modeling
If all these mathematical facts are known the description of functional
models with the help of this diagram type is very advantageous. The advantages
for using PROGRAPH diagrams are:

Easy to learn because there are less basic concepts. Especially there are no
allocations, no loops or no skips.

Higher efficiency because the source code which should be implemented
is very short compared to an imperative program code.

Higher trustiness because considerations or proofs of the correctness of the
source code is easier because of the mathematical background.
Like Nassi-Shneiderman (I. Nassi, B. Shneidermann, 1973) diagrams for
imperative modeling PROGRAPH diagrams return a very bright picture of the
processes which should be described. The structure of the drawn diagrams can be
implemented 1:1 in a CAS. It doesn’t depend on the system.
Another very important point which has to be articulated is that students or
pupils are highly motivated and challenged by Functional Modeling also
problems can be very difficult in description.
The conversion of the PROGRAPH-diagram above can be done in several
different systems told before. One of the shortest implementations can be done in
the language Haskell, which is a typical functional programming language.
Therefore it is obvious that the source code is very short. The code for the
implementation is a general one. So it is easy to expand the above headed
diagram to a general form for converting decimal numbers into binary numbers
(Diagram 2):
Diagram 2
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Through the combining of diagram construction and programming languages or
Computer Algebra Systems the teaching with the help of computers or graphical
calculators gets more exciting again. For students it is more efficient because
mathematical facts will be understood through informatical handling. The
interdisciplinary aspect of Mathematics and Informatics gets starched and new
forms of education in a creative way are possible.
Another important point which should be mentioned is the partition of the
modeling process into a Black-Box and a White-Box part. With this
classification it is possible to get a clear, intuitive and on the first sight
understanding description of this process which is meaningful and detailed. In
the first part, the Black-Box-Part, the questions `Which information is for which
component?’ and ‘Which information is necessary for another part of the
process?’ can be answered. The inner structure of the used components is not
answered yet. This is an exercise for the second part, the White-Box-Part, where
the functions are implemented as a source-code. The special use of PROGRAPH
diagrams is very efficient in the first part because the structure can be seen on the
first sight. The functional character of the involved parts appears because it is
easy to see that each function is fed through a unique assignment of input
parameters.
CONCLUSIONS AND FUTURE WORK
Through engaging students or pupils with problems that can be solved through
Functional Modeling it is possible to expand the learner’s perception. A lot of
mathematical problems which have been solved through an imperative sight of
the learner can be solved with the help of a functional ansatz. Enforcing this
view to problems leads to a more understandable and accessible approach to
functions and functional thinking.
Lots of problems discussed in schools can be solved through the described
functional sight. So the students’ or pupils’ mathematical creativity gets
stimulated through solving problems with this view. Demonstrations for such
problems as well mathematical problems as interdisciplinary problems will be
given in different presentations of me during the year 2008 at different
conferences. Another possibility to find a lot of typical mathematical problems
that can be discussed in a functional way can be found in [4], which will be
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Functional Modeling - A Creative Way in Modeling
released in March 2008.
Through an interdisciplinary attendance Functional Modeling gets more
important and (mathematical) creativity can get strengthened. Therefore it is
necessary to attract attention to this interdisciplinary mathematical topic.
REFERENCES
Ars electronica 93 (1993) - Festival fuer Kunst, Technologie und Gesellschaft:
Festival Katalog 1993 - Genetische Kunst - Kuenstliches Leben,
http://www.aec.at/de/archiv les/19931/1993 278.pdf, Linz
Buchberger, B. (1992): Teaching Math by Math software: The White Box / Black
Box Principle, Paper of the RISC-Institute of the Johannes Kepler
University Linz, Linz
Fuchs, K.J. (2007): Functional Thinking - A fundamental idea in teaching
Computer Algebra Systems, ICT-Conference, Boston
Fuchs, K.J.; Siller, H.-St., Vásárhelyi, E. (2008): Basics in Functional Modeling,
Casio Europe GmbH, Budapest
Hubwieser, P. (2007): Didaktik der Informatik, Springer, Berlin
Matwin, S. & Pietrzykowski, T. (1985): The Programming Language
PROGRAPH: A Preliminary Report. In: Computer Languages, 10:2, pp.
91 – 125.
Nassi, I. and B. Shneidermann (1973): "Flowchart techniques for structured
programming", ACM SIGPLAN Notices, vol. 8, pp. 12-26, Aug.
Siller, H.-St. (2007): Modeling - a pivotal idea of interdisciplinary teaching in
mathematics and computer science, CADGME-Conference, Pecs, 2007
ABOUT THE AUTHOR
Univ.-Ass. Mag. Dr. Hans-Stefan Siller
IFFB - Department of Mathematics and Informatics Education
University of Salzburg
Hellbrunnerstr. 34
5020 Salzburg
Austria
Phone: +43 662 8044 5330
E-mails: [email protected] [email protected]
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