ࡱ> dfc`8bjbjss B>F5  844 {P|H:NNNOOOOOOO$Qh_TO9 JJO 45P%%%  O%O%%RJ M< ?0?#$nK"rOdKP0{PK?U$?UDM?U MN^%0L|NNNOO{%XNNN{P D     CREATIVITY AND ELEMENTARY MATHEMATICAL THEORIES Grupo MUSA.E1 INTRODUCTION In an important article [Y1], I. Yaglon proposed the following difinition: Elementary mathematics is that which can be constructed and developed working together with children and teachers of schools and high schools. If this definition is adopted it implies the following aspects: Elementary mathematics is an experimental academic discipline, because in order to establish whether a certain mathematical topic is elementary or not, it is necessary to work it in teams that include students at basic level and medium, who must not only understand the topic in question but also be creative during the activity; that is, that they are capable of formulating conjectures, construct examples and counterexamples, argue, recognize and accept facts and data, recognize theorems and proofs, etc. Naturally, an elementary mathematical topic suppose a minimum quantity of previous knowledge because the must be within the reach of boy and girls. We are not saying that elementary mathematics is the same as school mathematics. In the latter, there are other interests, specially the practical interest, that is, applicability. In the area of the elemental, what is important is the construction of the mathematical knowledge even if this does not shows an immediate utility as is shown in the two example we present in this paper. In elementary mathematical, the meaning must be found in knowledge itself and not necessarily in its possible relations with other type of activities. We also not saying that children alone construct mathematical knowledge; creation is collective and there the presence of experts is essential: they are the ones who organize or plan the different activities, the ones who guide the work and offer the assistance that may be required when the students need it. The MUSA.E1 group (Matemticas Universidad Sergio Arboleda, Educacin uno) has been carrying experiments with group of children who wish to join the Project Semicircle of Universidad Sergio Arboleda [SEM], working in theories who requirements are not very numerous and thus accessible to the children with whom which we work. These theories, that result elementary if the experiments are successful, have been constructed using well-known facts, but in the process of creation, the children that take part obtain new results working with these theories. In what follows we present two examples. A PYTHAGOREAN THEORY In this theory we look for properties of plane figures in a special manner. These figures are constructed using, according to a basic ideas of Pythagoreans, regular polygons of the same size and shape, for example squares, to which the same ythagorean name is given: alphas or calculus. The figures are constructed according to the following axioms or rules: (R1) Each alpha is a figure that has four vertices, four edges, two regions and area 1. (R2) If one has a figure F, one can construct a new figure F appending an alpha A in such a way that at least one of the sides of A coincides with one of the sides of some alpha in F and so that there is no overlapping. In each case, it is explained how to count vertices, edges and regions. The area of F will be the area of F plus 1. Among the various theorems that can be formulated and proved in this elementary theory, one, which various students always find, is Eulers theorem: N + R L = 2 In none of the experiments carried out has any student constructed a proof of this result; however, when one discusses with them a proof, the majority understands it. The following theorem, conjectured by an 11 year-old boy, shows the creative possibilities of this kind of work. Theorem. A figure is simply connected if and only if R + L N = 2 (Area) Naturally, this is not the formulation constructed by the mentioned boy: it was given by the coordinator of the group. A GAUSSIAN THEORY In this theory, initially, the properties of the ring of the whole integers numbers are worked out and later those of the integers module n > 1, for different values of n. Among the activities that are proposed to students, one has the purpose of finding the following theorem to which some students arrive: Theorem. a integer m module n > 1, has a multiplicative inverse if and only m is different from zero and is relative prime with n. Naturally this assumes that the group of boys and girls have worked topics such as: Maximum common divisor of two whole numbers Minimum common multiple of two whole numbers Prime numbers Relative primes Several children in different groups have formulated the following theorem: Theorem: For n > 2, the sum of those integers module n that have multiplicative inverse is 0 module n. Naturally, with a theorem like this no one is going to win a Field medal; however, as it is work with children whose dates range between 9 and 11 years old, the central objective is that these persons construct mathematical knowledge at their level, and therefore elementary and hence practice academic values such as: construct and develop theories, formulate and solve problems, construct examples and counterexamples, argue and counter-argue, etc. and very specially, understand the meaning of the words theorem, proof, refutation, etc. REFERENCES [SEM] El semicrculo de la Universidad Sergio Arboleda  HYPERLINK "http://www.usa.edu.co/programas/matemticas/Semicrculo" www.usa.edu.co/programas/matemticas/Semicrculo [YI] Yaglom Issac et al. The Goemetric Vein, Springer Verlag, 1981, 253-269. ABOUT THE AUTHORS Grupo MUSA.E1, Coordinadores: Jess Hernando Prez, Reinaldo Nez Department of Mathematics Faculty of Engineering University Sergio Arboleda Calle 74 No. 14-14 Bogot Colombia Cell phone: 3 75 25 00 Ext. 2153 -mails:  HYPERLINK "mailto:jhppeluza@gmail.com" jhppeluza@gmail.com  HYPERLINK "mailto:reinaldo.nunez@usa.edu.co" reinaldo.nunez@usa.edu.co  PAGE 170 PAGE 170 PAGE 170 PAGE 170 PAGE 170    Grupo MUSA.E1 Coordinators: Jess Hernando Prez and Reinaldo Nez Creativity and Elementary Mathematical Theories PAGE 278 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 3 PAGE 277 ICME 11, Mexico, 2008 Proceedings of the Discussing Group 9 : Promoting Creativity for All Students in Mathematics Education 2O) i  esKL =>egļrhj0hj0CJaJmH sH hj0hj056CJaJhj0hj0OJQJ^Jhj0hj0aJhj0hj0CJaJhj0hj06CJaJhj0hj05hj0hj05CJaJhj0CJ$OJQJaJ$mH sH hj0hj0CJ$OJQJaJ$ hj0hj0OJQJaJmH sH +23ABO) i f J$ & F  d\$^`a$gdj0$ d\$a$gdj0$ & F  d\$^`a$gdj0 $d\$a$gdj0 $da$gdj0 $d\$a$gdj0 $d\$a$gdj0d\$gdj0F 8Jv ~esKL^ d\$gdM' $d\$a$gdj0 d\$`gdj0 $d\$a$gdj0$d\$`a$gdj0d\$gdj0 $d\$a$gdj0k/v4h$a$gdj0 $xa$gdj0 $d\$a$gdj0 $d\$a$gdj0$ & F<d\$^`a$gdj0 $<d\$a$gdj0BCDtuv4Zln񰡕|l\lO>O!jhj0hj0OJQJUaJhj0hj0OJQJaJhj0hj05CJaJmHsHhj0hj05CJaJmH sH hj05CJaJhj0hj05CJaJmH sH hj0hj05CJaJhj0hj0CJaJmHsHhj0hj0CJaJmH sH  hj0hj00JCJaJmH sH #jhj0hj0CJUaJjhj0hj0CJUaJhj0hj0CJaJmH sH ZF !!!gdj0$a$gdj0$a$gdM' $h^ha$gdM'$a$gdj0$a$gdj0( ) * C D E F G M N Q S Y Z ] _ e f i k q r u w } ~ ںڭچpf`fUf`fUf`fUf`fUf`fhr:U0JmHnHu hr:U0Jjhr:U0JU*jhj0hj0B*OJQJUaJph$hj0hj00JOJQJaJmH sH 'jhj0hj0OJQJUaJhj0hj0OJQJaJ hj0hj0OJQJaJmH sH hj0hj00JOJQJaJ!jhj0hj0OJQJUaJ'jGhj0hj0OJQJUaJ!~ !!!˷qqm_P_Im hFhM'h/05OJQJaJmH sH hIthM'5OJQJaJhM'hM'CJOJQJaJmH sH $hM'hM'CJOJQJaJmH sH h/0CJOJQJaJmH sH $hh/0CJOJQJaJmH sH 'hM'hM'5CJOJQJaJmH sH $hM'hM'CJOJQJaJmH sH hr:Ujhr:UUjhr:U0JUhr:U0JmHnHu!!!!!!!!!!\!g!i!j!p!q!t!u!v!}!!!!!8]8^8_8`8f8g8j8ٹمvhZXhQ h-ChM'Uh-ChM'56CJaJh-ChM'56CJ aJ h;hM'CJOJQJaJhM'CJOJQJaJhM''h;hM'6CJOJQJaJmHsHh;hM'6CJOJQJaJhM'6CJOJQJaJ%h/00JCJOJQJaJmHnHu hAoIhM'0JCJOJQJaJ)jhAoIhM'0JCJOJQJUaJ!!h!i!v!!!!!!;8^8_8l888$a$gd:'&#$+D,gd $ 9r  ]a$gdIi$ 9r  ]a$gd52$a$gd|*$'&#$+D,a$gd $ee]e`ea$gd|*y&#$+D,gd The 11th International Congress on Mathematical Education Monterrey, Mexico, July 6-13, 2008 PAGE 275 ICME 11, Mexico, 2008 j8k8l8s888888;hj0hj0OJQJaJhr:UhM' hIihM'h;hM'CJOJQJaJhM'CJOJQJaJ hAoIhM'0JCJOJQJaJ)jhAoIhM'0JCJOJQJUaJ888$a$gdj0@ 00&P 1hP:p/0. 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