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Файл:Integration of Natural Science and Maths in Scientific Thought and Education. The materials of Russian - German Seminar in Moscow - Cologne, 2014
.pdf
Moscow State Pedagogical University
Institute of Physics, Technology and
Information Systems
Faculty of Physics and Information
Technologies
University of Cologne
Faculty of Mathematics and Natural
Sciences
Mathematical Institute
Integration of Natural Science and Maths in
Scientific Thought and Education
The materials of Russian–German Seminar
in Moscow – Cologne, 2014
Moscow - 2015
И851
Integration of Natural Science and Maths in Scientific Thought and Education. The
materials of Russian –German Seminar in Moscow – Cologne, 2014.- Moscow:
MSPU, 2015. - 64 р.
УДК 37:001:330:06.
ББК 74+72+65
И851
ISBN 978-5-4263-0290-7
The collection includes materials of the Russian–German seminar which took place
from September, 15, to September, 23, 2014 in Moscow (Russia) at the Faculty of
Physics and Information Technologies of Moscow State Pedagogical University and
from November, 27, to December, 5, 2014 in Cologne (Germany) at the
Mathematical Institute of University of Cologne.
Edited by Tassilo Kuepper, Natalia Purysheva and Dmitry Isaev.
Typeset by Anastasia Pautova
УДК 37:001:330:06.
ББК 74+72+65
ISBN 978-5-4263-0290-7 © Moscow State Pedagogical University (MSPU), 2015
© University of Cologne, 2015
2
Contents
Tassilo Kuepper
Intuition – A Helpful or a Treacherous Scout?
Seen from a mathematical point of view
4
Natalia S. Purysheva, Alexandra Zhelyabovskaya, Nika Rybalka
The Education in the Russian Federation
19
Dmitry A. Isaev
Integrated Science course for the middle level students
26
Ekaterina Andrianova
The simultaneous use of natural subjects in school education
30
Felix Beschorner
Pisa, Competences and Mathematics
31
Svetlana Biriukova
Integration of Mathematics and Natural sciences in education
38
Arseniia Burkova
Logarithms and Astronomy
44
Florian Lange
Brachistochrone problem
46
Tatiana Ozharovskaia, Margarita Rodionova
Integration of mathematics and biology in education (at school)
50
Anastasia Pautova
Physical tasks in mathematics and natural-science subjects
By the example of State Graduation Exam (SGE-11 and SGE-9)
54
Alexandra Pavlova
Laboratory workshop with using of IT-technologies as a method to form
science thinking of students of pedagogical specialties
57
Ekaterina Zakhodyakina
Using natural scientific and historical material on the physics lessons
59
3
INTUITION – A HELPFUL OR A TREACHEROUS SCOUT?
SEEN FROM A MATHEMATICAL POINT OF VIEW
Tassilo Kuepper
Mathematisches Institut Universität zu Köln
Contribution for joint Russian-German Seminar „Integration of Mathematics and
Natural Sciences in Education“ in Moscow and Yaroslawl on September 19, 2014
1. Introduction
Following the statement due to Albert Einstein
„The intuitive mind is a sacred gift and the rational mind is a faithful servant“
scientific thinking is caracterized by two important ingredients. While the approach
based on rational thinking and arguments serves as a basic tool intuition refers to a
more artistic skill and relies on abilities beyond rational capacities. Both are needed
to gain progress in scientific development and for that reason they should be
considered in education. In this contribution the focus will be on the role played by
intuition and in particular be adressed from a mathematical point of view.
In common language the notion of intuition is used to describe processes or activities
which are not based on a purely rational argument but rather on insights gained
without explicitly available knowledge. Expressions like „ decision based on gut
feeling“ or the ability to judge character refer to such procedures. Of course, due to
complexity of the situation or unsufficient information other more sophisticated
approaches are quite often not at hand so one has to rely on intuition.
Frequently such spontaneous decisions lacking a rational base appear retrospectively
to turn out right although it might be questioned if that impression really captures
the facts truly. The impression might as well have been materialized in an unbiased
and neutral way glorifying true predictions as a strong ability to develop true intuition
while in the contrary case a failing prediction is shrugged off simply as
misjudgement. A further caracteristic element of intuition is given by somekind of
unconscious recourse to hidden knowledge or abilities which appear as mysterious.
They might be based on skills to read hardly visible signals as they are for example
still present in some „primitive“ cultures gained by long tradition.
Which are the reasons supporting the popular faith in the existence of an ability to
own the right intuition? For the german philosopher Kant intuition or to put it in his
terms „Anschauung“ forms an essential part of any perception:
„So fängt denn alle menschliche Erkenntnis mit Anschauungen an, geht von da zu
Begriffen und endigt mit Ideen.“
(Kritik der reinen Vernunft, B 730)
4
Another statement due to the french mathematician Poincaré examplifies the role of
intuition in Mathematics:
„C`est par la logique que nous prouvons, c`est par l`intuition que nous inventons.“
(„Mit Logik kann man Beweise führen, aber keine neuen Erkenntnisse gewinnen,
dazu gehört Intuition.“)
Many philosophers in history have dealt with intuition as a matter of the process
gaining insight. The list of names ranges from Platon, Epikur, Descartes, Spinoza,
Bergson, Husserl, Scheler, Hartmann up to present philosophers such as Dummett
and Kripke, for an overview see: ( http:/fr.wikipedia.org/wiki/Intuition)
In this contribution attention will be restricted to mathematical aspects related to
intuition. This special point of view might appear as a surprise for those who
consider intuition as a purely philosophical notion but there are interesting
connections to Mathematics as well ranging from geometrical perceptions up to the
foundations of modern Mathematics, in particular to the basic construction of set
theory. In fact it has been Kant who used mathematical examples to illustrate the
notion of intuition. While Kant in his epos „Kritik der reinen Vernunft“ presented
our notion of the surrounding space or of time as a perfect example illustrating the
role of intuition we nowadays know that the former understanding of space as
Euklidean does not hold anymore – hence intuition turned out as treacherous in that
case.
Another splendid illustration of intuition is given by the development of the notion
of a derivative due to Newton and Leibniz as a basic idea of „Calculus“ via
geometrical intuition. While Newton`s approach seems to be based on his
understanding of the physical process Leibniz is stimulated by geometrical arguments
approximating curves by tangents. In his famous book „Principia“ Newton is able to
derive the motion of a falling apple or the the traces of planets out of the abstract
axioms of motion and the law of gravity without explicit use of a formal notion of
derivative. The central idea is given by the link between position, velocity and
acceleration of a particle following a path. Interpreting the change of velocity as the
change of the ascent of the tangent when moving along the curve leads to the
formulation of the term „derivative“. It is worthwhile to note that this nowadays
familiar concept of a derivative has not been the only approach but the most
successful due to its convincing straightness. Less natural approaches such as
„subtangents“ introduced for example by Florimond de Beaune (1601-1652) did not
succeed.
It is remarkable that the existence of tangents at continuous curves obviously has
never been questioned at that time; following geometrical or physical intuiution
obviously neither Newton nor Leibniz took non-differentiable curves into
consideration at all.
5
It happened much later that Weierstrass (1861) proved the existence of a continuous
curve which is nowhere differentiable, hence contrary to intuition does not possess a
tangent at all – a fact which has been instrumentalizd in the critics of a naive use of
intuition. Before we proceed with a systematic discussion it is helpful to present a
few mathematical examples known for a strong appeal to intuition.
2. Examples illustrating treacherous aspects of intuition
A) The goat problem
This problem is popular in quizz shows. A player may choose among three doors.
While behind one door there is a prize there are blanks behind the others illustrated
by goats. When the player has decided which door he has chosen the quizz master
opens another door showing a goat. The player is now offered to change his decision.
Should he do so?
Contrary to intuition switching brings advantage since changing does only not
improve if the player had picked the „right“ door already at the beginning, hence
changing is successful with probability 2/3 compared to 1/3 in case of not changing.
Hence, using additional information pays off.
B) The Efron dices
This is an „unfair“ game with four dices showing the following points:
0 5 2 3
4 0 4 1 1 1 2 2 2 3 3 3
4 5 6 3
4 5 6 3
yellow blue white red
The game consists of a competion between two dices. The chosen dices are thrown
10 times in a row; each time it is counted which dice shows the higher number of
points, hence wins. The one who succeeeds more often is the winner. Which dice
should one choose? Is there a favourite?
Direct comparision for this stochastic experiment shows the following outcome:
blue is „worse“ than yellow, yellow „worse“ than red, red „worse“ than white, but
white „worse“ than blue!
Hence this is an example for a cyclic, non-transitive process, usually unfamiliar at
first glance and working against naive intuition.
C) ½ + ¼ + 1/8 + 1/16 + … = 1
6
A simple geometrical proof by partition of the unit square shows that this series
Probability of Leading Digit
1
30,1 % 2 17,6%
3
12,5 % 4 9,7%
5
7,9 % 6 6,7%
7
5,8 % 8 5,1%
9
4,6%
converges to a finite limit contrary to the divergent harmonic series
1 + ½ + 1/3 + …
D) Objects with constant diameter
What is the characteristic feature of a circle? A naive approach would rely on a
constant diameter but the situation is more complicated. Are there other objects
(wheels) of constant diameter? The existence of „Gleichdicks“ offers new insight and
such fascinating applications such as drills used to drill (almost) square holes.
E) Fermat conjecture
In Diophant`s book Arithmetica there is a famous note on the margin by Pierre
Fermat (1640) claiming:
For all n larger than 2 there are no integers x, y, z different from zero such that
x**n + y**n = z**n
together with the additional remark:„I know a proof, but there is not enough space on
the margin to write it down!“
„Cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non
caparet.“
(See S. Singh for illustration.)
For centuries this claim has challenged leading mathemacians but only Andrew Wiles
was the first one to succeed in 1995 with publication in the Annals of Mathematics.
With regard to the complexity this problem has exhibited over the centuries one
might speculate if Fermat really was aware of a proof or if he had simply been
captured by intuition.
F) Benford´s Law
Benford`s law describes the situation that in many empirical data sets the distribution
of the first digit is given as in the following table:
7
At a first glance this is a surprising phenomenon which has been first detected by the
american physicist Newcomb in 1881. Using logarithmic tables for computations as
common at that time Newcomb noticed that in his books of tables the first pages
showed signs of much more intense use than the following ones. This observation led
him to the conclusion that numbers with low first digit occured more frequently in
computations than those starting with higher digits. A systematic check confirmed
that impression. Partly forgotten this result was reinvented by Benford in 1938;
nowadays this fact is used as an easy tool to check data sets with regard to validity
and falsification. Although true in many situations this law seems to be counter
intuitive at first.
G) A Game Theoretic Example
In a group of people everybody is asked to write secretely a number between zero
and 100 on a piece of paper. Then the arithmetic average of these numbers is taken.
The one wins the game whose number is closest to twothird of the average.
Is there a winning strategy? Which number should one choose to win?
A successful strategy requires both mathematical and psychological qualities, hence
some kind of intuition.
In a naive approach to these examples intuition suggests a response which in some
cases turns out to be misleading; a subtle analysis will provide the correct solution,
sometimes relying on new and deeper insight for example into properties of systems
without transitivity.
3. Formal Definition of „Intuition“
For a scientific approach to the meaning of intuition more abstract criteria seem to be
adequate. A typical characteristic of intuition is given by statements for which there
is no further argument at hand, hence statements which appear as evident,
concerning something which in german is denoted as „Ding für sich“.
In a formal definition (www.philosophie-woerterbuch.de/online-woerterbuch)“
Intuition (or in German: Anschauung) is defined as the act of capturing imminently
given facts, hence statements which are evident and neither can be proven nor need a
proof; that is statements which are „eines Beweises weder fähig noch bedürftig“.
Due to Kant our imagination of time or the surrounding space was used to serve as a
typical example : „Der Raum ist kein diskursiver, oder wie man sagt, allgemeiner
Begriff von Verhältnissen der Dinge überhaupt, sondern eine reine
Anschauung.“(Kant, Kritik der r. Vernunft, B39). Modern Physics though proved that
this imagination needs an extension.
8
4. Geometry as perfect counter-intuitive example
Related to skill and experience with direct application in astronomy or survey
Mathematics was seen in babylonian or egyptian times as the science of measuring
and counting and for that reason was considered as an empirical discipline
producing verifiable results (Behnke-Bachmann-Fladt-Süss §1).
Within the greek philosophical school around Platon another approach was initiated
in form of a systematic and deductive access based on evident rules (axioms) as
starting point to derive theoretic results in a purely logical way. Euclid`s „elements“
are known as the most prominent piece of this direction.
Evidence of mathematical axioms means that their content is generally accepted and
can be characterized by the fact that they can neither be proven nor need any proof.
In his book „Elements“ Euclid derives a theory nowadays known as „Euclidian
Geometry“. For centuries that theory was considered as a protype of a mathematical
theory based on simple axioms, and in that sense it served as perfect example to
illustrate the „a priori“ character of Mathematics.
The basic foundation of „Euclidian Geometry“ is formed by several axioms most of
them reflecting truly evident facts. Following M. Lehn („Welche Geometrie gilt?“)
the first five of these axioms are formulated as:
(1) Any two points can be connected by a straight line.
(2) Each bounded straight line can be extended to a connected infinite straight
line.
(3) Around each point and radius one can draw a circle.
(4) All right angles are equal.
(5) Consider a straight line intersecting two other straight lines S and T. If the
angles on the same side of the first line and in between S and T are less than
two rights then S and T intersect on that side.
While the first four axioms are immediately obvious the fifth seems to be more
delicate, not just because of its cryptical formulation but conceptually. In fact it is
equivalent to the famous „axiom of paralleles“ which can be stated as follows:
For each straight line S and each point P not on S there is another straight line
through P not intesecting S. This line is called the parallel to S through P.
In addition the „Fifth Axiom“ is equivalent to the wellknown statement:
In any planar triangle the sum of angles amounts to 180°.
Verification of the „Axiom of Paralleles“ is difficult insofar as it requires statements
concerning areas far away, in fact at infinity which are beyond our access.
9
For centuries Euclid`s „Elements“ were considered as a basic opus in mathematics
forming the foundation for various parts of geometry which was not questioned for a
long time. Nevertheless, the „Fifth“ attracted attention, and there were various trials
to get rid of it and to derive it from the other axioms. In particular Giovanni Sacchieri
(1773) is known for such attempts. A crucial step forward was made independently
by the three scientists Carl Friederich Gauss (1777-1855), Nikolai Ivanovich
Lobatschewski (1792-1856) and Janos Bolyai (1802-1860) with the development of a
„Non-Euclidian Geometry“ establishing a geometry built up without the axiom of
paralleles. For a good review of the history see Donald O`Shea: „Poincare´s
Vermutung“ ( p. 92 ff)
Nikolai Ivanowitsch Lobatschewski started his studies in Kasan and became
Professor, Dean and finally rector of the university.
Through his efforts to reject statements of Kant known as „Transzendaler Idealismus“
claiming that our imagination of space and time is given a priori he was lead to the
creation of „Non-Euclidian Geometry“ first announced in 1826 and eventually after
previous rejection in Kasan and Saint Petersburg due to its revolutionary novelty
published in 1829. Later publication in 1840 in a book on german led with support by
Gauss to his admission into the Göttingen Academy of Science.
The father of Janos Bolyai, Farkas B., was acquainted with Gauss. Besides sceptical
warnings by his father Janos continued to work on replacements of the „Fifth“ and
presented his results concerning a „Hyperbolic Geometry“ around 1823; in 1832 his
results were published in Farkas` book „Tentamen“.
Although Gauß was familiar with these new developments of a „Non-Euclidean
Geometry“ he never published any results in this direction. Probably though he has
been the first to reckognize the imprtance of the new development. In a letter to
Torinus he wrote in 1824:
„Die Annahme, dass die Summe der drei Winkel kleiner sei als 180 Grad, führt auf
eine eigene, von der unsrigen (euklidischen) ganz verschiedene Geometrie, die in sich
selbst durchaus consequent ist, die ich für mich selbst ganz befriedigend ausgebildet
habe, …“ (s. O`Shea p. 98)
„The assumption that the sum of the three angles is less than 180° leads to a new,
completely different geometry which is consistent and which I have developed just
for myself to my satisfaction, …“ (see O`Shea p. 98)
Besides these notes these results have not been published by Gauss.
Until 1850 these developments were hardly known in Mathematics. This has been
changed by the famous inaugural lecture of Riemann in 1854 where he presented a
systematic approach:
„Über die Hypothesen, welche der Geometrie zugrunde liegen.“
10
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