Stimulating Creativity
Hartwig Meissner
Westfaelische Wilhelms-Universitaet
Muenster - Germany
Abstract: Stimulating creativity is a process of interaction. Challenges ("Darstellungen") must provoke a powerful and flexible thinking ("Vorstellungen") to produce creative answers ("Darstellungen"). We will describe these aspects in more detail and give examples.
The following picture describes our view on the process of learning and understanding mathematics (MEISSNER 2002):
mathematical
concepts
Darstellungen
(external
representations)
more "concrete":
a 2 b2
ab
„a fraction is ...“
Vorstellungen
(concept images)
E
I
N
S
T
E
L
L
U
N
G
E
N
Which now is the place to stimulate creativity? "Creativity" is a highly complex phenomenon.
Many experts from different disciplines give various descriptions. Summarizing, two complementary modes of thinking are necessary (BISHOP 1981) and the thinking processes must be flexible
(KIESSWETTER 1983). Thus, stimulating creativity is a process of "communication". Via Darstellungen we can provoke creative ideas (Vorstellungen) and powerful Vorstellungen may lead to creative Darstellungen. We will analyze these aspects in more detail.
Darstellungen basicly are written or oral (external) representations, often combined with (observable) activities or manipulations or actions or … Darstellungen provoke or lead to Vorstellungen in
the head of the learner, that means Darstellungen are the origin to build up concept images, scripts,
or micro worlds. The process of building up Vorstellungen very much is influenced by affective
components (Einstellungen). The learning athmosphere, the type of social interactions, emotions,
"beliefs", etc. form the frame in which Vorstellungen develop. For more details see GOLDIN (2003)
or SCHLOEGELMANN (2002).
1. The Role of Darstellungen
Let us start with the example "wysiwyg". This sequence of symbols is a Darstellung. Which is the
Vorstellung you get by this Darstellung? For some of us this only is a strange sequence of letters,
from the alphabet, in a crazy order, no special meaning, … Others might remember a big step forward in the design of word processing software packages and they remember all the difficulties in
using computers to type difficult formulae or special symbols 1. That means the same Darstellung
"wysiwyg" can generate quite different Vorstellungen and of course this is true especially for many
"learners" of mathematical Darstellungen.
Thus "stimulating creativity" is a process which is determined by two components:
 We need stimulating Darstellungen, i.e. we need Challenges.
 These Darstellungen or Challenges must find a "resonance" in the domain of Vorstellungen,
i.e. we must develop or recall powerful and flexible Vorstellungen.
To fulfill these demands we shortly will analyze the process of learning mathematics. We distinguish two modes of working in mathematics, working syntactically and working semantically. Syntactical activities are given when we work like a machine or like a calculator or a computer. We
work on a sequence of symbols along given algorithms, we work as an "operator" or a "function
machine" or press the correct sequence of buttons to get a wanted display like a sequence of digits,
a graph, a table, etc ... Only algorithms or lists of mnemonic codes must be learnt. A very limited
Vorstellung, only a "mechanical" or "instrumental" understanding is sufficient to get a correct solution2.
Working semantically however means that the problem itself is in the foreground, the meaning of
the situation. A network of related problems, topics, or experiences should be recalled, consciously
or unconsciously. Formulae, algorithms or computers just are tools or computational aids to manage
the burden of sophisticated calculations or to get drawings to visualize relations or to experience
properties or to test assumptions. Working semantically needs a conceptual understanding.
Challenges to stimulate creativity at least must initiate semantical activities. They must recall or
stimulate or produce, consciously or unconsciously, a powerful and flexible network of related
knowledge and experiences. And we should avoid Darstellungen which only induce syntactical
stimulus response activities. We will give some examples.
1.1 surprising Darstellungen
A classical or traditional view gets destroyed (or expanded) by a surprising question or a provoking
keyword or an unexpected problem, i. e.3:
(a) Construct the development
(= net) of the given solid where all its edges
are visible. (The solid is not transparent.)
(b) Draw a polygon, and mark a point in its interior (exterior) from which one cannot see the whole
length of any side.
(c) Draw the set of points in a plane that have the same distance from the two given rays r1 and r2:
r1
r1
r2
r2
„wysiwyg“ means „what you see is what you get“ ( for example like in WORD, but not in TECH)
„instrumental understanding“ according to SKEMP (1976).
3
These challenges from FRANTISEC KURINA and more challenges from others can be found in MEISSNER e.a. (1999).
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2
We can expand traditional views and further the development of flexible Vorstellungen by confronting the learner with surprising or unexpected situations, combined with challenging keywords like
"describe"“, "discuss", "measure", "count", "compare", "imagine", ... :
(d) impossible?
(e) impossible?
(f) impossible?
These or similar pictures may stimulate developing own creative ideas. Additional nice examples
are given for example by ENZENSBERGER or PAPY or ESCHER In school also the interaction of the
two subjects art and mathematics might further the design of own new creations.
1.2 deep Darstellungen
With a given Darstellung in mathematics we often only recall a limited frame or Vorstellung (which
is an advantage for beginners because it is easier to start working in a familiar environment). Thus a
lot of mathematical Darstellungen look “familiar” though often there are powerful additional aspects and properties hidden. Working with such a kind of Darstellung suddenly opens new views,
new aspects arise and the horizon gets expanded. Hidden properties consciously appear and expand
the existing Vorstellungen. Though there is no change in the Darstellung the related Vorstellungen
have become more flexible and powerful. This type of Darstellungen therefore also might be helpful to "develop or recall powerful and flexible Vorstellungen". We will give some examples4.
(a) Multiplication. Select a path from A to B. Change the direction at each corner. Multiply the
numbers of each step you go. Find the path with the smallest product. You have 4 trials.
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Those who do not see the hidden properties immediately should work more intensively on the problem given.
(b) Solids. Given is a solid where the base and the upper face are parallel and congruent. Describe
the solid. Base and upper face are regular n-polygons, describe again. Assume the solid is not a
prism, but base and upper face still are regular n-polygons, describe. All side faces are regular triangles, describe now. Which is the name of the solid when also the base is a regular triangle?
(c) Can you tessellate with "Escher-Birds", with regular triangles, or with pythagorean triangles a
given square, a given triangle, a given stripe? Not seeing the “deep” relations many students start
with trials, though logical reasoning in most of the cases immediately would lead to an answer.
(d) Shadow: Take three letters (such as your initials). Can you make a ‘solid’ shape which casts
your three letters as shadows in three perpendicular directions? Can you make a shape which shows
your letters as light with a shadow surrounding them? (MASON in MEISSNER e.a. 1999, p. 188)
2. The Development of Vorstellungen
We pointed out that some kind of specific Darstellungen might help to develop or recall powerful
and flexible Vorstellungen (which are necessary to stimulate creativity). How else can we stimulate
the development of powerful and flexible Vorstellungen? Regarding this question in more detail we
should distinguish two types of Vorstellungen.
According to STRAUSS (1982) young children have a global non-differentiated concept of a certain
domain which is appropriate to solve operations or tasks adequately within that domain. The concept is biological in origin and refers to a "common sense knowledge". It is a spontaneous concept
in the sense of VYGOTSKY, based on an intuitive thinking (in the sense of BRUNER). But then
schooling starts and another concept develops - a "cultural" (STRAUSS) or "scientific" (VYGOTSKY)
concept - which is reflective and self-conscious, and which is based on analytic thinking.
These two types of Vorstellungen can interfere. Abilities relating to the common sense knowledge
decrease into a "chaos" while adequate abilities of a cultural knowledge have not yet developed.
The global view gets destroyed and the children suddenly cannot solve problems which they could
solve before5. But step by step the schooling builds up a new and more structured concept. Some of
the former abilities "reappear" more powerful than before, now based on a different view. Other
abilities are lost for ever.
Just compare the learning of mathematics and the learning of our native tongue. Why can young
children learn their native tongue without almost any instructions or explanations? Why do they
learn a correct grammar also without instruction? We argue that the development of these intuitive
or unconscious Vorstellungen is the result of "communication". Interaction is necessary and adaptation. In the PIAGETian meaning the learner realizes (consciously or unconsciously) a conflict and
seeks "equilibrium".
Here, we think, we can learn how to stimulate the development of more powerful and flexible Vorstellungen also in mathematics education. We must concentrate more on the development of the intuitive type of Vorstellungen, i.e. on the development of a common sense knowledge like number
sense, function sense, proportional feeling, number relations, experiences with infinity, etc. We
must emphasize more "communication" and interaction to broaden the field of experiences for (unconscious) adaptations in the PIAGETian meaning. Thus in mathematics teaching we should not concentrate only on the deterministic or analytic style of thinking. We also should allow collecting
more spontaneous and intuitive experiences. We must allow trial and error behaviour, we must emphasize guess and test activities. These "interactive" activities are responsible for the development
of intuitive (and often unconscious) Vorstellungen.
There are two conditions to fulfill:
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examples see in MEISSNER (1986)


The mathematics teachers must allow and further guess and test activities.
We need useful and effective questions and activities to practice guess and test.
The latter easily is possible by the use of calculators and computers. Pressing buttons we get an
immediate feed back to see if our assumption was senseful or how we can find a better assumption.
We will give some examples.
(a) "Hit the Target". This calculator game trains the understanding of multiplicative structures: An
interval [A,B] is given and a number n. Find a second number s so that the product of n and s is
within the interval [A,B]. Our more than 1000 guess-and-test protocols (MEISSNER 1987) show that
the students after a certain training develop excellent estimation skills (guessing the starting number) and a very good proportional feeling (very often less than three guesses to find a correct solution).
(b) Calculator Games Using the "Constant Facility". Simple calculators often have a "constant facility", that means "operators" like "-253" or "47" can be stored. In the game "BIG ZERO" we hide a
subtraction operator and ask "Which is the input for getting 0 in the display?" In the game "BIG
ONE" we hide a division operator and ask "Which is the input for getting 1 in the display?" Discovering these hidden operators by guess-and-test develops intuitive Vorstellungen of additive respectively multiplicative structures.
(c) Percentages. There are calculators which work syntactically like we speak in our daily life:
"635 + 13 % = ..." needs the key stroke sequence 6 3 5 + 1 3 % =
We introduced the topic percentages in
100
more than 10 classes with such calculators.
100
There were no formulae or reverse functions.
We always used the same sequence of key
strokes, if necessary the problems were solved
by guess and test. Results see post test (6 pro50
blems, white bars). We also administered the
same test with about 500 students who got the
traditional course of teaching percentages (dark
50
bars).
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3
4
5
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(d) Interest, compound interest, growth and decay, …. A similar approach is possible. The students
develop intuitive Vorstellungen about the relations between the many variables.
(e) Functions. The traditional school curricuriculum has had not much success in developing a
deeper understanding between the gestalt of a graph and the related algebraic term. Using guess and
test with computers we developed that missing link for linear and quadratic functions: Our students
very easily could sketch the gestalt for a given term and write down a term for a given graph.
We urge our students to write protocols from their guess and test work because these protocols are
excellent Darstellungen from their (mainly intuitive) Vorstellungen. Discussions then can bring the
shift from an unconscious feeling to a conscious insight.
3. Summary
To stimulate creativity we need challenging Darstellungen and powerful and flexible Vorstellungen.
Here we distinguish two complementary modes, a mainly intuitive and unconscious common sense
knowledge and a reflective and self-conscious analytical thinking. To be creative, a flexible change
between both types of Vorstellungen is necessary and especially in mathematics education we
should concentrate more on the development of the intuitive type of Vorstellungen, for example by
allowing and furthering guess and test activities.
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