ࡱ> EGD| @pbjbj >v؝؝/B-B-B-8z-D-&|V.p.Z/Z/Z/000QSSSSSS$Rwio800o8o8wZ/Z/4>>>o8LZ/Z/Q>o8Q>>ITZ/J. `cB-;N:m0&ׄ ȕ =Hȕtȕ0(2>D4<5000ww(B-Q>XB- The experience of development of pupils creativity in Latvia ANDREJS CIBULIS, GUNTA LCE Abstract: methods of fostering creativity in mathematics of all children in the classroom and of exceptional children through serious research are described. Key words: creativity, curriculum, pentomino twins, polyominoes, problem solving, pupils research. INTRODUCTION The Latvian dictionary of pedagogical and psychological terms tells as that creativity is an ability to produce new mental values or to arrange ideas and facts of real life in specific structures, to formulate original and interesting ideas/ approaches (see [1]). It doesnt specify the meaning of the words in italics. Math teachers in Latvia have had a lot of discussions on a topic can all pupils be creative in mathematics, and should they be such? Conceptually it is recognized that creativity can be developed. In order to do this the pupil must be urged not only to reproduce the acquired knowledge but also use it to discover something new at least for him, if not for the whole world. At our opinion, each pupil is able to do this up to some extent. Some examples from the Latvian math curriculum that encourage creativity Increasing attention is paid to the development of creativity of all students in mathematics in Latvia. In December 2006 a new curriculum for primary education in mathematics was approved by government. From that moment the standard has strength of law. The standard states that the mandatory content of mathematics subject consists of three parts: 1. Acquiring mathematical tools. 2. Applying mathematics to the analysis of processes taking place in nature and society. 3. Creating and investigating mathematical models. So there is a law in Latvia, which says that all students in mathematics lessons have to learn to explore with typical mathematical methods. Standard asks the students to master: Accurate mathematical language; Making and analyzing mathematical models: Defining the informally settled problem with a more accurate definition, formulating it using a suitable mathematical model; Solving mathematical model and interpreting the solution. Predicting the results of the research, checking and proving the proposition. It must be mentioned that until the Grade 9 students must prove the propositions in separate cases only; nevertheless, the best students are always urged to do this. Starting with Grade 10, all students acquire diverse math argumentation forms; they understand the need to argument propositions and always try to do that. The standard also defines the order of the students learning attainments. In Latvia students skills and knowledge are valued in scale from 1 to 10, where 1 is the worst, but 10 is the best possible attainment. This scale also anticipates a chance to value students creative job. If the student is able to use his knowledge only in situations, which have been examined in the classroom, his attainment doesnt surpass an 8. Students learning attainments are valued with 9 and 10 only, if he can use his knowledge in non-standard situations. This attainment system helps also to find students very talented in mathematics. Those students are stimulated to attend additional in lessons mathematics, to prepare for and to take part in mathematics olympiads, where each exercise requires a creative approach. These students often make research which is only remotely connected with topics considered in the classroom. Some examples of problems that we use to recognize and assess our students mathematical creativity Problem 1 In a final exam on Equilateral and isosceles triangles. The locus of points for 7th grade the following problem was posed: Draw a map to the scale 1cm:400m that satisfies the following constraints! It is known that sports club is located at point S, but culture centre is located at point K. The distance between sports club and culture centre is 1.5 km. You don't have to write down details about the construction. It is known that home of Artis is equidistant from both sports club and culture centre. Is it possible to mark the home of Artis on the map? Justify your answer! Mark all the places on the map where the home of Artis could be. It is known that Evija lives 1200m from both sports club and culture centre. Is it possible to mark the home of Evija on the map? Justify your answer! Mark all the places on the map where the home of Evija could be. The home of Guna is right midway between sports club and culture centre. Mark all the places where it could be located! Justify your answer! The problem was included in the test with an aim to find out, whether the students can choose a mathematical model that fits each given situation and to use it correctly to justify their opinion. There is a good chance to develop the students creativity, if in the mathematical classes they look at problems that are simple enough, so that all students can explore them, and at the same time it is possible to generalize them. That opens the road for the students independent research. There are a lot of such problems in combinatory and in the combinatorial geometry. Problem 2 In how many ways is it possible to illuminate a room, if there are five bulbs of different colors in it? Two ways are considered different if and only if at least one bulb is switched differently in both configurations. Experience shows, that the students like this problem. Because 5 isnt any big number, they usually try to enumerate all the possible variants. A part of the students try to systemize the counting procedure. It allows to generalize the task problem. Problem 3 In how many ways is it possible to illuminate a room, if there are n bulbs of different colours in it? Two ways are considered different if and only if at least one bulb is switched differently in both configurations. Usually when trying to solve this problem, the students offer one of two approaches: The number of ways is  EMBED Equation.DSMT4  (examining each bulb separately); The number of ways is  EMBED Equation.3  (examining each possible number of switched-on bulbs separately). Because the velue of the sum doesnt depend of the counting type, you can understand that  EMBED Equation.3 . Its possible to prove also other combinatorial identities using the fact that the result of the counting doesnt depend on the way of counting. Such students evidences have been brought forward in the Latvian new scientists conference numerous times. Wide possibilities for the students to work creatively are provided by mathematical games and toys, such as pentomino. Experience shows that students like to stay during the breaks in the classrooms, where mathematical games are available. The National Conference for Young Scientists During classes of mathematics the problems requiring creative approach are considered frequently. They are included both in the every day lessons and in the tests. The solution usually is not new to the world however it is new to the solver. Dealing with such problems encourages students to discover something really new. They write explorative papers and take part in conferences, which are first held in every school, and then in regional conference. In the larger part of Latvian schools everybody learning in secondary school (Grade 10 to Grade 12) has to perform at least one research. The student can choose the field. Usually the students who choose mathematics have marks mainly 9 or 10 and have successfully competed in mathematics olympiads. There is approximately 1 such student out of 10. Exceptions are specialized mathematics classes. Because the average level of these classes is high, the students often write explorative papers. They confess that it encourages them to make deeper research in mathematics. The best from every region take part in The National Conference for Young Scientists, which is organized annually in Latvia. The winner is asked to defend his/her research in elimination contest. If the person meets the requirements of the jury successfully, he/she is nominated to participate in the European Union Contest for Young Scientists. Students` research Generally speaking the quality of students` papers depends strongly on their own qualities (serendipity, patience, working capacity, preliminary training and background, etc.), but topics and tutor have a significant role as well. The papers that are presented in The National Conference for Young Scientists can be split into three groups. Compilative papers. Papers related to teaching mathematics: tests, games which can be used in the teaching process, etc. Papers in which some unexplored mathematical question is investigated. To gain some idea on the content of the pupils research let us mention some of the topics in 2006-2007: Compatibility Problem for Polyhexes; Invariants and Quazi-invariants of a Function; Methods of Solving Problems of Latvian Mathematical Olympiads (2000-2005); Partitioning of the Rectangle 6(10 in Pentomino Twins; Analysis of Disentanglement Puzzles; Application of Graph Theory to the Tram Network Optimization in Riga. We consider in more details the contest paper Pentomino Twins (two equal shapes) by Inga Sakn+te and Madars Virza (Valmiera State Gymnasium, Grade 11, 2006). They investigated the Pentomino twins problem for chess board, namely: find all pentomino twins fitted on a chess board 8(8. Twins problem for the rectangles 4(15 and 5(12 has been solved in [2]. The problem of pentomino twins for chess board is attractive, very difficult and as far as we know had not been solved earlier. Elaborating a computer programme for this problem is, in no case, an easy task. However, M.Virza was able to create computer programmes by means of which one can find all p-twins. The first algorithm found all the ways how to divide the chess-board into two equal parts. The number of such partitions (92263) was known earlier [3]. After that the second algorithm examined both equal parts trying to cover them with 12 different pentominoes. Moreover, additional efforts were made to select twins who would be good for creating mathematical toys. As particularly interesting ones we have found pentominoes twins which can be covered with all 12 distinct pentominoes in only one way. There exist 549 twins altogether like these. The result has been used for invention of a new mathematical toy. An example of a mathematical toy Try to cover the board in Fig.1 consisting of 2 equal parts and 4 empty squares using all 12 pentominoes! This task can be solved in only one way (see Fig.2) Fig. 1Fig. 2 CONCLUSIONS Creativity as the ability to produce something subjectively new is inherent in all children. It can be developed, e.g. , in lessons of mathematics. Some children (up to 10%) are able to perform a research in the exact sense, providing us with objectively new information. Along with careful tutoring, the existence of formal infrastructure for such activities is welcome. A success on math Olympiads is a serious indicator of being capable to reach this higher level of creativity. REFERENCES Skujina V. at al. Dictionary of pedagogical and psychological terms. Zvaigzne ABC, Riga, 2000. Cibulis A. Pentominoes. Part II , Riga, LU, 2001, pp.106 (in Latvian).  HYPERLINK "http://www.research.att.com/~njas/sequences/" http://www.research.att.com/~njas/sequences/ ABOUT THE AUTHORS Andrejs Cibulis, Assoc. Prof. University of Latvia 29 Rainis boulevard Riga, LV-1459 Latvia Phone: ++371 7211421 -mail:  HYPERLINK FH   ` a b A I N Y h i ޹ugXIh$Eh?oCJaJmH sH h?oh?oCJaJmH sH h?oh?o5;CJaJh?o5;CJaJh?oh?o5CJaJh?oCJaJh?oh?o6CJaJh?oh?o56CJaJh?oh?oCJaJh?oh?o5CJ aJ h?o5;CJ$aJ$h?oh?o5;CJ$aJ$ h?oh?oOJQJaJmH sH  h?oh5B*CJaJphH a b o ' h #########w #w#$d@&\$]a$gd?od\$gd?o$d\$`a$gd?o$Pd\$]P^a$gd?o$d\$`a$gd?o$d\$`a$gd?o$d@&\$]`a$gd?o$d\$`a$gd?o $d\$a$gd?o eph p7q/####P#P########## $d\$a$gd?o $[$a$gd?o$ & F0<[$\$`0a$gd$E$ & F0<[$\$`0a$gd$E$[$\$a$gd?o $[$a$gd?o $d\$a$gd?oeg:;E#( o p ĸĬϏϏs\G)jh!'h!'CJEHUaJmH sH -j.L h!'CJUVaJmHnHsHtHh!'CJaJmH sH jh!'CJUaJmH sH h?oh?o6CJaJmH sH h?o5CJaJmH sH h?oh?o6CJaJh?oh?oCJH*aJh?oh?oCJaJh?oh?o5CJaJmH sH "h?oh?o5;CJaJmH sH h?oh?oCJaJmH sH o:;E#( Y !!"#################`~#> # #### $[$a$gd?o $[$a$gd?o$ & Fd*$\$`a$gd?o$ & Fd\$`a$gd?o x!y!!!!!!####෢vj[I9h?oh?o;CJaJmH sH "h?oh?o5;CJaJmH sH h?o5;CJaJmH sH hFCJaJmH sH )jQh?ohFCJEHUaJmH sH -jL hFCJUVaJmHnHsHtH)jh?oh?oCJEHUaJmH sH +jq6K h?oh?oCJUVaJmH sH %jh?oh?oCJUaJmH sH h?oh?oCJaJmH sH h?oh?oCJH*aJmH sH #$ ) ) )v***7+,2333K44######P##o#P######$ B d\$a$gd?o$ d\$a$gd?o$ d\$a$gd?o$ d\$a$gd?o$ & F 80d\$]^`0a$gd?o$d\$]a$gd?o$d\$]a$gd?o# ) ) )+++++2,3,<,?,K,L,\,],^,_,a,b,n,o,t,v,,,,--./F////////000022333444444)h?oh?oB* OJQJaJmH phsH  h?oh?oOJQJaJmH sH #h?oh?o5B*CJ\aJph33 jh?oh?oCJaJh?oh?o6CJaJh?oh?o5;CJaJh?o5;CJaJh?oh?oCJaJ64444444444444444444444444#Ff $d$&`#$/If\$a$gd?o$ B d\$`a$gd?o444444444444444444444444444Ffd$d$&`#$/If\$a$gd?o444444444455'5(5D5E5Y5Z5[555Z6e6J7U7g7777777777n8p8r888899:dFdЅЃU jh?oh?o5CJUaJh?oh?o0JCJaJ#jj`h?oh?oCJUaJjh?oh?oCJUaJh?oh?o6CJaJh?oh?o5CJaJh?oCJaJ h?oh?oOJQJaJmH sH h?oh?oCJaJ-444444444444444444444444444Ff $d$&`#$/If\$a$gd?o444444444444444444444444444Ff*$d$&`#$/If\$a$gd?o44444444444444444444444444Ff>Ff4$d$&`#$/If\$a$gd?o44555555555 5 5 5 5 555555555555FfcH$d$&`#$/If\$a$gd?o55555555 5!5"5#5$5%5&5'5(5)5*5+5,5-54555657585FfQ$d$&`#$/If\$a$gd?o8595:5;5<5=5D5E5F5G5H5I5J5K5L5M5N5O5P5Q5R5S5T5U5V5W5X5FfX$d$&`#$/If\$a$gd?oX5Y5Z5[5g56I7J7U777888##########$ & F0d\$]`0a$gd?o$ & F0d\$`0a$gd?o $d\$a$gd?o $d\$a$gd?o $d\$a$gd?od\$gd?o$d\$`a$gd?oFf\$d$&`#$/If\$a$gd?o 829\99999dddddd e2eee$$If]^a$gd?o$$If]^`a$gd?o$x$If]^`a$gd?o"mailto:Andrejs.Cibulis@mii.lu.lv" Andrejs.Cibulis@mii.lu.lv Gunta Lce, University of Latvia 8 Zellu Str. Riga Lv  1002 Latvia Phone: +37126673183 -mail:  HYPERLINK "mailto:gunta.lace@lu.lv" gunta.lace@lu.lv   PAGE 554 PAGE 554 PAGE 554 PAGE 558 PAGE 554    Andrejs Cibulis and Gunta Lce The Experience of Development of Pupils Creativity in Latvia PAGE 220 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 3 PAGE 219 ICME 11, MFdHdJd|d~dBeDeeeeeeeeeeeeeeeeeeeefffff f,f.f4f6f8ffBfDfHfJfxxxxxplplplph>=}jh>=}Uh>=}0JmHnHu h>=}0Jjh>=}0JUh?ohjMOJQJaJh?oh?oOJQJaJ&j~bh?oh?o5CJUaJh?oh?o5CJaJh?oh?o0J5CJaJ jh?oh?o5CJUaJ&jah?oh?o5CJUaJ*eeee:f=}f gg(gggg p p6pvpwpppp$ 9r  ]a$gdIi$ 9r  ]a$gd52$a$gd|*$'&#$+D,a$gd F:$ee]e`ea$gd|*y&#$+D,gd F:gd?o$d@&\$]`a$gd?oexico, 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 217 ICME 11, Mexico, 2008 ghp p pvpwpppppppppppppppp볢|xkh?ohjMOJQJaJh>=} hIihVhVCJOJQJaJ%hV0JCJOJQJaJmHnHu hAoIhV0JCJOJQJaJ)jhAoIhV0JCJOJQJUaJ h-ChVh-ChV56CJaJh-ChV56CJ aJ hVUh;hVCJOJQJaJpppppp $d\$a$gd?o$a$gd:'&#$+D,gd F:< 00&P 1hP:pG=. 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