ࡱ> BDAJ`gbjbjss Bp3      NNN8DT q*^:"""ppppppp$shXvq j*""j*j*q  q@1@1@1j* " p@1j*p@1@1j7j$ r o BN.k2_pq0q%l8w08wdoo8w oH" $v@1 &,L'"""qq0^"""qj*j*j*j* N N         FIRST LITTLE STEPS ON A GOD-KNOWS-HOW LONG ROUTE DACE BONKA, AGNIS AND}NS Abstract: The experience of Latvia fostering creativity of students/ teachers in mathematical activities of various levels is described. Key words: problem solving, concept building, cognition, ways of creativity in mathematics MATHEMATICAL INTRODUCTION We consider as creative any activity not following the rules, formulated by others. Therefore, by definition, creative approaches cant be taught; they can be demonstrated. It should be clarified whether creative approaches are necessary at all. Since ancient times the scientists dream has been to solve all problems once and for all. The phenomena of undecidability, algorithmic unsolvability and independence have ruined most nave versions of these hopes. More realistic approaches appeared in 60-ies of last century within the framework of theoretical computer science. It was observed that the truth almost always is approached in the limit. We set hypotheses, experiment with them, eventually change them; but we hope that sometimes we will come to the right hypothesis which will never be changed, although we will never be sure that we have already reached it. A common situation is computer programs scanning. It was proved in early 70-ies (Barzdin et al.), that algorithms for such inference in the limit are impossible, except of most trivial cases. Moreover, they remain impossible if we consider weaker concepts of algorithms. It follows that the concept of algorithm can not serve as the basis of cognition. Therefore the creativity, as described above, is not only philosophical or ethical, but also exact necessity. PHILOSOPHICAL INTRODUCTION So far mankind has invented four principal ways of cognition: rational cognition in Cartesian sense (mathematics is the principal representative), empirical cognition as Roger Bacon described it (natural sciences is the principal representative), emotional cognition (literature and religion being the principal representatives), computer modeling (informatics being the principal representative). A student who doesnt acquire all these ways can be compared with a horse with one or even two lame legs. As mathematics is the main representative of rational cognition and plays a crucial role for empirical cognition and modelling, the progress of the society depends very much on its mathematical intelligence. A VERY SHORT HISTORICAL INTRODUCTION Latvia is a small country with app. 2300000 inhabitants. During its 900 years long written history it has been politically independent only for 40 years. It has no rich natural resources. So almost only its resource to restore the country is well-educated people. With such a small population each child is of great importance. Therefore constant efforts are made to develop the abilities of each pupil. What follows is what we have tested during a period of almost 40 years and found to be effective and important not only for geniuses, but for broad public also. MOST IMPORTANT AREAS FOR CREATIVITY We consider following mathematical activities; at our opinion, the percentage of children with natural abilities to proceed creatively is decreasing along the list from appr. 30% for A till appr. 1% for E: solving problems set by others, forming concepts and models, developing own approaches and tools, setting and solving own problems, building a structure of mathematical facts, methods and approaches. Nevertheless, these abilities can be developed. Creativity in the area A is often demonstrated by children on the basis of general development only; students who demonstrate it often become excellent in physics, computer science, arts/ literature etc. Creativity in the areas B E is an indicator of excellence in mathematics. EFFECTIVE TOOLS FOR FOSTERING MATHEMATICAL CREATIVITY 1. At our opinion, the most important tool is positive emotions. Almost nobody is interested in what, e.g., Fredholms second equation is; but a lot of people will become interested if they will hear how the solution of this equation can help to irrigate the desert and to save millions from starving. E.g., [1] is an excellent example. To avoid children skipping mathematics before they have tested themselves in it, the following approach should be followed: the motivation must be as emotional and humanitarian as only possible, the implementation must be as precise and exact as only possible. 2. Although a lot of teaching aids are available today, the main of them still remains the Textbook. Textbooks developed along the following guidelines have appeared stimulating the creative approach: The book must be a multi - level one, with all levels in the same volume ( EMBED Equation.DSMT4 ); we never know at which moment an interest will awake. The course must be considered as the part of natural sciences at the beginning, and the deductive elements must be introduced step-by-step. The level of rigorousity must increase gradually; the proof is considered as "the argument that makes you to believe", and during the learning process the individual demand for more and more rigorous proofs must increase in each student. There must be a lot of problems of different nature and different level of difficulty. An approach there will never be too much examples! should be used. As an example of such a book, see [2]. 3. There are five principal parts of mathematics on school level: algebra, geometry, number theory, combinatorics, algorithmics. All of them should be reflected in school curricula. There are different tastes; a child can be initially very interested into only one of these areas. Unfortunately, official school curricula become more and more algebraic; so we face a thread to loose many children with other preferences. At our opinion, especially geometry should be paid much more attention at least for following reasons: geometry is fine; its results provide great esthetical feeling, geometry is the largest and, in fact, the only part of school mathematics curricula which is developed in a deductive manner, geometry is an appropriate language for teaching natural sciences. 4. Unfortunately, not all the teachers nor all the schools are on the same level. On the other side, creative students and even geniuses are born more or less uniformly. Therefore three types of educational activities play extremely important role in fostering creativity among children living far from big cities: correspondence courses for students organized by universities and scientific institutions, also by means of internet, summer camps/schools carried out in a rather informal manner, lecture courses/ seminars given to students and teachers in regional centers of rural area. At our experience, for a) a schedule including 4-5 sessions during academic year, each consisting of theory studies and problem solving, is appropriate. For b), 5 days with 6-8 hours of academic activities each is OK; for c), one session in a month is a usual frequency. The topics are selected accordingly to abovementioned principles. 5. When the main attention of official education system is paid to the drill of formal skills, a wide system of mathematical competitions is a powerful tool for providing high-level mathematical knowledge/ skills and possibilities for creative approach. Of course, math olympiads play the main role here. Nevertheless, there are many students who need more than some 4-5 hours (usually allowed during math olympiads) to go deep enough into the problem. For such children a system of correspondence contests is much more appropriate; moreover, it gives more possibilities for deep and profound thinking which is characteristic for creative approach. It appears good to have various correspondence contests for different age and skill groups. In Latvia we have three of them: Club of Professor Littledigit for students up to the 9th Grade. There are 6 rounds each year, each containing 10 problems. Problems are published in newspapers and on the INTERNET. Contest of young mathematicians for students up to 7th Grade, originally developed for weaker students than the participants of CPL. The problems are published in regional newspapers and on the INTERNET. So much or how much? contest for the students up to 4th Grade, organized jointly with colleagues from Lithuania. The problems are published in INTERNET. At the end of the school year an international correspondence competition between the students of two countries is organized. Contest problems serve also as a basis for number of teaching aids developed for a general education system, as problems given there are well-balanced accordingly to the following alternatives: discrete/continuous mathematics deductive/algorithmic mathematics algebra/combinatorics/geometry/number theory prove/calculate/decide whether/construct. For examples see [3]. It has been observed that in the younger grades the students are much more interested in constructive and algorithmic problems than in other types of them. All parts of algorithmics should be used to provide possibilities for creative approaches: the inference of an algorithm from examples; the analysis of a given algorithm; the development of an algorithm; the optimization of an algorithm; the proof of correctness of an algorithm; the proof of non-existence of an algorithm. New types of algorithms as probabilistic, nondeterministic, frequential etc. become important in theory and applications. These fresh and almost non-explored areas often give a chance for unexpected and effective approach even to younger students, see, e.g., [4]. 6. The following classes of problems have appeared to be most interesting for those students who afterwards have become scientists, university professors, inventors, project leaders etc.: problems with many possible solutions (see, e.g., [5]) open-ended problems; they are often given also in contests problems with a paradoxical question and/or solution (see, e.g., [6]) problems the solution of which combine ideas and approaches considered as not-related to each other in a traditional teaching/learning process. Mainly they are problems solvable by the method of interpretations (see, e.g., [7]) or those lying on the interface between discrete and continuous mathematics (see, e.g., [8]). 7. Broad and serious mathematical education is necessary for acting creatively in mathematics: an empty head doesnt think. The teachers who want to foster their students creativity have a possibility to study a lot of mathematics usually left outside both high school and university curricula. As well they can practice in conducting corresponding activities. One of the branches of master studies in mathematics at The University of Latvia is Modern elementary mathematics, mainly aimed at educating future teachers for work with pupils on higher level than usually. Such courses as General methods of elementary mathematics, Modern elementary geometry, Practice in olympiad mathematics, Mathematical models in natural sciences etc. are combined with involving students into organizing the abovementioned activities. CONCLUSIONS Creativity, as we understand it here, can be observed on various levels. It can be stimulated and developed. Emotional and psychological factors are important in this. An active participation of the students is an essential part in the considered approaches. A student demonstrating the creativity on the lower levels in mathematics often does the same in other areas; this is not the case for higher levels. REFERENCES Krner, T. The Pleasures of Counting. Cambridge University Press, 1996. And~ns, A., E.Falkenateina, A.Grava. Geometry, Parts 1-5 (in Latvian). R+ga, Zvaigzne, 1992-1996. http://nms.lu.lv And~ns, A. & G.Lce. What Probability Theory should be taught to bright high School Students? Proceedings 6th Int. Conf. APLIMAT. - SUT, Bratislava, 2007, pp. 251 - 256. And~ns, A., D.Bonka, G.Lce. There are many Roads to the only Truth. Integral, Nov. 2006, vol. 9, issue 6, pp. 10-14. 0A8;L52, . 8 4@. 0>G=K5 <0B5<0B8G5A:85 >;8<?804K. 0C:0,1981. D.Bonka, D. & A.And~ns. The Method of Interpretations: Possible Failures. Proc. Int. Conf. "Matematika ir matematikos destymas" - Kaunas, KTU, 2005. - pp. 5 - 9. And~ns, A. & I.BrziFa. Research Possibilities for middle and high School Students: Perspectives via Computer Science. Dept. Math. Rep. Series, vol.14 - University of eske Budejovice, 2006, pp. 52 - 56. ABOUT LNfh 3 4 N qSZ ;L%3cܻ~ocXooo~~o~~~~h"hh~"mH sH h"hh~"5mH sH h"hh~"CJaJmH sH h"hh~"5CJaJh"hh~"6CJaJh"hh~"56CJaJh"hh~"CJaJh"hh~"5CJ aJ h"hh~"OJQJaJmH sH h~"CJ$OJQJaJ$mH sH $h"hh~"CJ$OJQJaJ$mH sH  h~"h~"5B*CJ$aJ$ph"Nfh 3 4 N  <`gd~"d\$gd~" $d\$a$gd~"d\$gd~" $d\$a$gd~"cd\$]c^gd~"$cd\$]c^a$gd~"d\$gd~" $d\$a$gd~"\gg:5pq3-y$ & F S<d\$^`a$gd~"$ & F S<d\$^`a$gd~" $<<a$gd~" d\$`gd~"d\$gd~" $d\$a$gd~"$ & F S<d\$^`a$gd~"%G ^!c,r$ & F d\$^`a$gd~"$ & F <^`a$gd~" $<a$gd~"d\$gd~" $d\$a$gd~"$ & F Sd\$^`a$gd~"$ & F S<d\$^`a$gd~" cevw  !!$$''((w)y)...../u1w14446666777J8Կ{rhv92h~"CJh"hh~"CJaJmH sH h~"CJaJmH sH h"hh~"6CJaJh"hh~"CJH*aJ!jh"hh~"CJEHUaJ)jhK h~"CJUVmHnHsHtHh~"CJaJjh~"CJUaJh"hh~"CJaJh"hh~"5CJaJ(,XF  W !S"x$ & F S0d\$^`0a$gd~"$ & F S0<^`0a$gd~" $<a$gd~" $d\$a$gd~"$ & F h<d\$^`a$gd~"$ & F h0<d\$^`0a$gd~" S""#c#r$$'q(?)Y*+;+]+++z$ & Fd\$^`a$gd~"$ & F<^`a$gd~" $<a$gd~"$ & F <d\$^`a$gd~" $<d\$a$gd~" $d\$a$gd~"$ & F S0<^`0a$gd~"++,,-=-`---.{///30u14446d\$gd~" $d\$a$gd~"$ & F 0<d\$^`0a$gd~"$ & F <d\$^`a$gd~" $<a$gd~" $d\$a$gd~"666b7(8J89:;X<==XBXTYYZZ[ \A\\\gd~" $7`7a$gd~"$a$gd~" $d\$a$gd~"$0^`0a$gd~" $d\$a$gd~"J89"9&9P9T9t9999,:v::::::::::::::::;;;;;ǷǨǙǨǨljzjX"h"hh~"6CJ\aJmHsHh"hh~"CJ\aJmHsHh"hh~"CJaJmH&sH&h"hh~"6CJaJmHsHh"hh~"CJaJmHsHh"hh~"CJaJmHsHh"hh~"6CJaJmHsHh"hh~"CJaJmHsHh"hh~"6CJH*\aJh"hh~"6CJ\aJh"hh~"CJ\aJ;0<V<X<t<H=====>XBXYTYYYYZZZϳϚp`O`;O&jh~"h~"5CJUaJ jh~"h~"5CJUaJh~"h~"5CJaJmHsHh~"h~"5CJaJmH sH h~"h~"5CJaJUh"hh~"5CJaJhv92h~"CJh"hh~"5CJaJmHsHh"hh~"6CJaJmHsHh"hh~"CJ\aJh"hh~"CJaJmHsHh"hh~"CJ\aJmHsH"h"hh~"6CJ\aJmHsHTHE AUTHORS Dace Bonka, Mg.math. A.Liepa s Correspondence Mathematics School, Faculty of Physics and Mathematics, University of Latvia, 8 Ze<<u str., Riga LV1002, Latvia Cell phone: +371 29868289, fax: +371 67033701, E-mail:  HYPERLINK "mailto:dace.bonka@lu.lv" dace.bonka@lu.lv, Web page:  HYPERLINK "http://nms.lu.lv" http://nms.lu.lv Agnis And~ns, Prof., Dr.habil.math. A.Liepa s Correspondence Mathematics School, Faculty of Physics and Mathematics, University of Latvia, 8 Ze<<u str., Riga LV1002, Latvia Cell phone: +371 26566419, fax: +371 67033701, E-mail:  HYPERLINK "mailto:agnis.andzans@lu.lv" agnis.andzans@lu.lv , Web page:  HYPERLINK "http://nms.lu.lv" http://nms.lu.lv     First Little Steps on a God-Knows-How Ong Route Dace Bonka and Agnis ZZVZXZZZZZZZ[ \A\I\J\r\s\t\\\\\\\\\\\\\\\\\\\\̸̬̈tܬkc_c_c_c_h^ jh^ Uh~"h4#-aJ&jih~"h~"5CJUaJ&j~h~"h~"5CJUaJh~"h~"5CJaJmH sH h~"h~"5CJaJ&jh~"h~"5CJUaJh~"h~"5CJaJmHsH jh~"h~"5CJUaJ#h~"h~"0J5CJaJmHsH%\\\\\\\\\]] ]ddd.dddd e$a$gd+$'&#$+D,a$gdY$hh]h`ha$gd*$y&#$+D,gdYgd~"$a$gd~"$a$gd$a$gd~"\\\\\]] ]]]^ddddd"d$d*d,d.d8d@dDdddĸĶĭӘtgWCWg'h;hsc6CJOJQJaJmHsHh;hsc6CJOJQJaJhsc6CJOJQJaJ%hM0JCJOJQJaJmHnHu hAoIhsc0JCJOJQJaJ)jhAoIhsc0JCJOJQJUaJhhscaJUh~"CJOJQJaJh-h~"CJOJQJaJhsc h khsch05OJQJaJmH sH #h-h~"5OJQJaJmH sH And~ns PAGE 302 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 4 PAGE 297 ICME 11, Mexico, 2008 Proceedings of the Discussing Group 9 : Promoting Creativity for All Students in Mathematics Education The 11th International Congress on Mathematical Education Monterrey, Mexico, July 6-13, 2008 PAGE 296 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 4 dddddddddee"eeefffffffffffٵȧ|ٵj]Mh;hsc6CJOJQJaJhsc6CJOJQJaJ#h;hsc0J6CJOJQJaJh-Chsc56CJaJh-Chsc56CJ aJ h;hscCJOJQJaJhsc0JCJOJQJaJ%hM0JCJOJQJaJmHnHu hAoIhsc0JCJOJQJaJ)jhAoIhsc0JCJOJQJUaJhschM6CJOJQJaJ e"eteeelfffff|g~ggggggggg$a$gd+$ 9r  n!&#$+Dq]a$gd+y&#$+D,gdY$ 9r  ]a$gdN$  ]a$gdNffdgxgzg~gggggh~"h4#-aJhfJVhA3h^ hschM6CJOJQJaJhsc6CJOJQJaJh;hsc6CJOJQJaJ'h;hsc6CJOJQJaJmHsH gggggggd~"@ 00&P 1hP(:p0. 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