ࡱ> ~ @t[bjbj i؝؝T2   " fff8T<" [:(b:0>00 ZZZZZZZ$]R-`[i 4l0"044[[;;;4d Z;4Z;;VT RY. P ^f98WZd[0[Wa9a4RY" " RYa fY,01;J2t2000[[" " Df:^" " f 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. 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  HYPERLINK "http://www.libraries.psu.edu/tas/locke/mss/c1684.html" \l "m0020" 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 self-determination); 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. Praxiological reflexion reflexion on the knowledge application, self-learning through objects of the own skills and abilities in an activity product (self-learning through and with ones 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) The basic characteristics in the considered approach consists in engaging the first 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 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 childrens 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 10th 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 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 6={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 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. 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,  HYPERLINK "http://www.humanmetrics.com/ebooks/probprognosis.asp" 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 -mails: natalia_1@abv.bg     Reflexive Approach and Creativity Nataliya Ton$%78An y  +~klI_Pg,TUyjjj[[yhgJhgJCJaJmHsHjhgJhgJCJUaJhgJhgJ;CJaJhgJhgJ5hgJhgJCJaJmH sH hgJhgJ6CJaJhgJhgJ56CJaJhgJhgJCJaJhgJhgJ5CJaJ'hgJhgJ;CJ$OJQJaJ$mH sH  hgJhgJOJQJaJmH sH h kOJQJaJmH sH "$%78 n  +,m~I $d\$a$gdgJ $d\$a$gdgJd\$gdgJ$mhd\$]m^ha$gdgJ $d\$a$gdgJd\$gdgJ?Ys[IP,-U _ "$ 8d\$a$gdgJ $d\$a$gdgJ $d\$a$gdgJd\$gdgJ$ & F d0xd\$^`0a$gdgJUVrswx &(01:;=>ABJKNORS\]_`cdklwxz{~>@n~hgJCJaJhgJhgJCJaJmHsHhgJhgJCJaJ0jhgJhgJCJUaJmHnHsHtHuN~egw(y()),,,,-------////$/%/&/'/+/,///0/3/ں枒" j-hgJhgJCJaJmHsHhgJhgJ;CJaJhgJhgJ;CJaJmH sH hgJhgJCJH*aJhgJhgJ5CJaJmH sH hgJhgJ5CJaJmHsHhgJhgJ5CJaJhgJhgJCJaJmHsHhgJhgJCJaJ2";""""$k'^(_)U+,,--/$/~0011N2295$ d\$a$gdgJ$ d\$a$gdgJ$ 8d\$a$gdgJ $d\$a$gdgJ3/4/000 0000 0,0006080:0>0N0R0z000000000000000011111112222 3!3%3&3'3(3/303:3<3|55556hgJhgJ5CJaJhgJhgJCJaJmH sH hgJhgJCJH*aJ" j-hgJhgJCJaJmHsHhgJhgJCJH*aJmHsHhgJhgJCJaJhgJhgJCJaJmHsH;95|556K7L7h78u9):+:7:::"<====D>>$a$gdgJ 0^`0gdgJ$0^`0a$gdgJ $d\$a$gdgJd\$gdgJd\$gdgJ $d\$a$gdgJ66666666"6#6&6'60616666K7h7):+:,:7:n::::::::::;; ; ;;;;;4;N<<<<<&=*=ƽեƥ䙑hgJh.CJaJh.CJaJjh.CJUaJhgJhgJ6CJaJhgJhgJ5CJaJhgJ5CJaJhgJhgJCJaJmH sH hgJhgJCJaJmHsHhgJhgJCJaJhgJhgJ5CJaJmHsH0*=,=.=====T?X?b?f?????????????????wj^HDh k*hgJh k5;CJ$OJQJaJ$mH sH hgJhgJ5OJQJhd5OJQJmH sH hd5OJQJhojhoUhgJh kCJaJ hgJhgJOJQJaJmH sH hgJhgJOJQJaJhgJhgJ5CJaJhgJhgJCJaJh.CJaJhr>h.0JCJaJjh.CJUaJ#jhr>h.CJUaJ>>>??T?????????????XXXXy&#$+D,gdY$a$$a$gdgJ $d\$a$gdgJ$a$gdgJ$a$gdgJcheva PAGE 192 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 2 PAGE 193 ICME 11, Mexico, 2008 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 PAGE 186 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 2 Introducing a new school material Rationalized memorizing of the interpretation of the example. Ith and II th signal system IIth signal system Mechanical memorizing of a particular example (attempt, image, geometry object, etc.) Consciously reproduction of the material Scheme 1. 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