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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
REFLEXIVE APPROACH AND CREATIVITY
NATALIYA TONCHEVA
Abstract: The mathematics education has wide variety of possibilities to
develop qualities, useful not only in working knowledge but in students’
self-learning process. The aim of this paper is to show examples which can
help students to learn more about their own accomplishments and to follow
their creativity by using “reflexive” mathematical problems.
Interesting methods that can increase the students’ creativity can be used
in different parts of mathematics. The probability theory in school is a
successful example. In the paper is shown a possibility of reflexive
approach in teaching theory of probabilities. Some types of reflection are
pointed out. Some examples for different aged students are shown. There
are offered some problems which can be used to develop the reflexive
abilities of students and to foster their creativity.
The main thesis of this work is to show that students want to show their
reflexive groping when they feel heuristic elements and intuition in their
mathematical solutions. They are looking for the way to the concrete
heuristic, purposefully. The students feel better when they can speak to the
other students and to the teacher about their intuitive arguments. In the
shown approach some suitable problems, examples and reflexive talks can
increase the mathematical creativity of students.
Key words: Mathematical Creativity, Self-learning Process, Reflexive
Approach, Heuristic Elements, Association, Intuition, Reflexive Talks,
Theory of Probabilities.
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INTRODUCTION
Reflexive approach can increase the students’ creativity in different parts of
mathematics. The probability theory is a successful example. One of the gaps in
teaching the probability theory is that the specificity of the considered subjects
has not been used completely, the sole stress being laid on the particular school
contents neglecting the useful and not so often elements of heuristic reasoning
developing the students’ creativity very successfully.
By tradition the subject from the main body of “Probability and statistics”
follows the historical road of science development. This approach has proved its
effectiveness but it will be nice to report the psychological aspects when the
school material is rendered. Reflexion appears to be useful tool along these lines.
REFLEXIVE APPROACH AND CREATIVITY IN TEACHING
PROBABILITY THEORY
REFLEXION NATURE
It is considered that John Locke and Gotfrid Leibnitz were the forefathers of
reflexion. The concept “reflexion” was published in 1690 for the first time in “An
Essay Concerning Human Understanding” by John Locke. His is the view:
“Under reflexion in the following exposition I understand the observation on
which the mind subjects its activity and the ways of its manifestation as a result
of which the ideas for this activity arise in the mind (Locke, 1985, p.155).
We accept the basic conditions for the different types of reflexion which
according to (Vasilev, Dimova, Kolarova-Kancheva, 2005, p.51) under a
synthesized form are:

Intellectual reflexion – „reflexion on the own cognitive activity”.

Personality reflexion – it manifests in the progress of the following
processes: self-determination (with an emphasis on the personality selfdetermination); self-improvement motivation; purpose formation (with an
emphasis on the inner purpose formation); planning; anticipation;
experience.

Dialogue reflexion – it is realized in the “communication between the
subjects in teaching through empathy processes (co-experience) and
reflexive listening”.
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
Praxiological reflexion – “reflexion on the knowledge application”, “selflearning through objects of the own skills and abilities in an activity
product (self-learning through and with one’s own activities and works)”.
MODELS OF REFLEXION APPROACH IN TEACING
J.Feigenberg and I.Vajnberg (2000) discovered the ability of people to make a
probabilistic prognosis in their common activities. The problem of unoptimal use
of the natural facts for probability prognoses in mathematical education we
solved with the help of a developed scheme for application the reflexive approach
in acquiring the new school material (Scheme 1)
Introducing a new
school material
Ith and II th
signal
system
Mechanical memorizing
of a particular example
(attempt, image, geometry
object, etc.)
Rationalized memorizing
of the interpretation of
the example.
IIth
signal
system
Consciously
reproduction of
the material
Scheme 1. Scheme of study the new concepts, theorems and methods for the solving
problems from the main body of “Probability and statistics” in 10 grade.
The basic characteristics in the considered approach consists in engaging the first
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signal system, mnemonic treatment of the material and the revealing of the
logical connections in the second signal system on the basis of the images
received from the first signal system. The ability of the students to carry out a
similar teaching is innate and a similar process is completely natural for the man
(Feigenberg, Vajnberg, 2000, p.31). The scheme follows Herbartianian structure
of teaching (Andreev, 1980, p. 98), whose formal degrees are:
Clarity●Association●System●Method
The application of the teaching according to this scheme is experimented with
202 students from 10th grade (with age of 17) in Bulgarian schools in 2006/2007
and shows its good results.
It is possible to be used with others age groups.
EXAMPLE APPLICATIONS
It turns out that the students regardless of their age feel the necessity to express
their reflexive quests when they face a particular act of their intuition. They
purposefully look for the way to reach a particular heuristics.
In connection with an experiment reporting the behavior of probability thinking
the following “Game” was played with 50 children at pre-school age – in each
group was given a picture of a small bear with three hoops (small, medium and
large) and a ball (the diameters of the ball a little less than the diameter of the
smallest hoop and the medium hoop is more attractive than the others two). The
children are asked to help the Bear win a reward (honey) when they hit one of the
hoops with the ball. They have to choose which of the hoops they would live to
hit.
The most usual comments of the children on the given task were:
“ – The Bear will choose the largest hoop because the ball can go in it most
easily.”
“ – The Bear will choose the largest hoop because it is the biggest one.”
“– The Bear will choose the smallest hoop because it is as big as the ball” – when
asked additionally it turned out the children giving this answer had not
understood the question correctly and they had looked for the smallest hoop
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Reflexive Approach and Creativity
where the ball could go in.
Particularly interesting were the following statements:
“ – The Bear will choose the smallest hoop because it will be given more honey.
– Why will it be given more honey?
– Because it will push the ball with much difficulty.”
“ – The Bear will choose the hoop with the ribbon because it is the nicest one.” –
This answer is met very rarely in the described activity but when is played with
real hoops it is frequently met and this shows the possibility of the harmful
influence on the probability thinking.
It is interesting that the children were able to follow their own arguments very
easily and they could express them very precisely and clearly. Although that at
this age they have not reached the stage of the formal operations (12 years) they
were able to dissociate their intuitive conclusions with the help of the reflexive
thoughts. The manifestation of an empathy with unreal object (The Bear)
motivates the children to seek the best decision. They gave their answer twice –
once after the task is given and second – after everyone had spoken. The teachers
did not show their opinion by any means when the children spoke. The result of
the second vote improved considerably. In this “Game” the typical reflexive
listening was observed and despite the young age of the participants it can be
claimed that the children apply dialogue reflexion in a particular situation.
In the children’s statements it is evident that they are in the clear it is not certain
the Bear will win no matter with hoop choose and this can be accepted as a
display of creativity assisted by the intellectual and praxiological reflexion,
With the students from 10th grade the reflexion is on a quite different level. At
this age the students possess a certain inherent or acquired experience in similar
reasoning. This facilitates the introduction of the reflexive approach in school
training.
The availability of a number of primary concepts in school subjects from the
main body of “Probability and statistics” for 10 th grade is an extra reason for the
introduction of reflexion in school training. The differentiating of intuition and
the recognition of certain heuristic presents the training in mathematics in a quite
new light. An example for similar application of reflexion is the following talk
presenting the role of intuition in treating equally possible/not equally possible
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events.
According to the level and motivation of the students the talk can follow different
directions with a different degree of reflexivity. In the most common variant the
teacher can render examples (equally possible elementary; equally possible but
unelementary; not equally possible) and to leave the students to analyse their
reasoning by themselves.
General scenario of the talk:
„  Equally possible events – events which at a given attempt have equal chances
of realization. This is not a definition but a simply objective estimation of
possibilities according to the conditions of the given attempt.
 How will they be estimated?
 The judgment will be intuitive. For example with dices (beforehand is
indicated that if dices are “right” and the conditions of the attempt are “ideal”)
the elementary events will be equally possible. And according to you what are
events S={falls an even number on a dice} and T={falls an uneven number on a
dice}.
 …equally possible…
 What are the events Е6={falls 6 on a dice} and S (as above) for the same
attempt?
 …unequally possible…
 What are the events U={falls a big side when a match box is tossed} and
V={falls the smallest side when a match box is tossed}?
 … unequally possible…
 Think how you can determine which possibilities are equally possible and
which are not!
(The students describe their sensations and thoughts when they solve similar
problems).”
Unlike the children of pre-school age the 10th grade students are not so resolute in
their particular argumentation. They need a greater motivation. It is interesting
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that except the example with the match box, the students can draw the wrong
conclusion that all possible events at one attempt are equally possible. At such
situation the example stimulates the strong act of creativity assisted by different
types of reflexion simultaneously. A similar study of the own thinking is well
liked by the students and when applied regularly in practice it helps for the good
atmosphere in class and also stimulates the further active own opinion of the
students.
Example problems contributing to the creativity through reflexion:
Problem 1: Draw a “picture” (dice, an urn with balls, etc.). Make at least 3
problems for finding a probability on this picture!
Problem 2: Imagine you have forgotten the classical definition for probability.
You only remember that when a right dice is tossed, the probability one of sides
to appear is 1/6, and when a wrong dice is tossed this definition cannot be used.
Try to restore the definition grounding in the data from your memories. Note
what you use!
CONCLUSIONS AND FUTURE WORK
A number of examples of similar problems and fragments can be given in
teaching mathematics. As a disadvantage can be noted the time which such an
approach takes and a possible disturbance of the discipline in class. On the other
hand the reflexion increases the interest of students and the specific skills and the
creativity runs high.
The experiment proves that similar training improves the mathematical culture of
the students and helps the more effective application of the obtained
mathematical knowledge in practice.
An interest for future study presents the realization of non-standard
interdisciplinary connections on the basis of the applications of reflexive
approach in probability teaching.
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REFERENCES
Vasilev, V., Dimova, J., Kolarova-Kancheva, T. (2005). Reflection and Training
– I part (bg), Makros, Plovdiv.
Locke, J. (1985). Selected Works in 3 Volumes (rus), Volume 2. Misli, Moskow
Feigenberg, J., Vajnberg, I. (2000). The Obssesive Syndrome (rus), Independent
psychiatric journal, №IV, pp. 31–34
Feigenberg, J. (2000) Probabilistic Prognosis in Human Behavior,
www.humanmetrics.com/ebooks/probprognosis.asp
ABOUT THE AUTHOR
Nataliya Toncheva
Department of Didactics of Mathematics and Informatics
Faculty of Mathematics and Informatics
University of Shoumen
115 Universitetska str.
9712 Shoumen
Bulgaria
Cell phone: +359 899 333 798
Е-mails: [email protected]
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