ࡱ> uwt`6bjbjss Mi+   8P<, |H|LGGGGGGG$Ih`L!H l"  !H 6H 0  G GnB E `B9C GdLH0|HC@MQ@M@E@M EL)!H!H-X|H     d     LEARNING MATHEMATICS, DOING MATHEMATICS: CREATIVITY IN CLASSROOM? NGEL HOMERO FLORES SAMANIEGO Abstract. We all are born with a creative capability that can be or not stimulated. As all human capabilities, creativity can be developed and improved ( HYPERLINK "http://www.psicologia-positiva.com/ creatividad.html" http://www.psicologia-positiva.com/ creatividad.html). In this paper I present a student centered teaching model called Learning Mathematics, Doing Mathematics. In this model we create a Teaching Environment in which it is possible to foster students mathematical thinking and problem solving skills with the aid of technology, specially Dynamic Geometry software. After this I rise the question for the discussion: can we foster creativity in such Teaching Environment? Key words: Teaching model; Mathematical thinking, Human values; The Geometers Sketchpad. MATH CLASSROOM AS A WORKSHOP One of the main pedagogical goals in Basic School (from Kinder Garden to High School, 5 to 17 years) is the education of citizens on a reflective and critical thinking. In the teaching model Learning Mathematics, Doing Mathematics is possible to develop this kind of thought in our students through the promotion of a mathematical thinking, in particular a deductive reasoning, that would be useful in decision making, and the ability to solve problems; all this with the use of the technology, specially educational software (such as Dynamic Geometry Software, and CAS calculators) and Internet; it is also possible to foster a set of human values that, in turn, will govern students attitudes; namely, cooperation, tolerance, respect and responsibility. In Learning Mathematics, Doing Mathematics we take for granted that mathematical problem solving, with emphasis in validation and justification of results, can contribute significantly to the development of a deductive reasoning. At the same time, a deductive reasoning, along with mathematical modeling, can contribute to successfully face problems that come from fields other than mathematics. All of this can be made easier with the use of technology. In this teaching model, teachers must be able to develop a Teaching Environment (TE) that foster knowledge and allow the acquisition of human values. The TE is constituted by all things that influence the learning process in a classroom: from furniture and its arrangement to learning materials as textbooks and the kind of activities students do in a classroom, and attitudes of teacher toward students, students toward teacher and among students. Within a classroom a student must feel confident and sure of what he is doing. There must be respect and tolerance toward what he is doing, toward his teacher and his classmates. In a TE in which the principal character is the teacher, the sole fact of being a teacher confers him the maximum authority and the right to decide if a student has learned a topic and deserve to pass to a higher level. Traditionally, school subjects, specially mathematics, have a high level of failure, this results in a high level of school desertion; therefore school have served as a filter that excludes many students from reach a good school education. So, in many instances, math teachers are the maximum authority that decides about the educational future of a student. SUITABLE TEACHING ENVIRONMENT In Learning Mathematics, Doing Mathematics, it is proposed a TE where students meet the right conditions to work and do mathematics with the aid of their teacher and classmates. The idea is to strip away the fears and stresses a student bring with him when he enters a math classroom. The student knows that he is coming to his math class to learn math, and that he is going to work collaboratively with his classmates; in this way, the math class becomes a workshop for doing math. According to Brousseau (1997), in a classroom, teacher transfers or delegates to students the responsibility to solve the teaching activity he gives them, and students take this responsibility as theirs, because they know that in doing them, they are going to learn the mathematics they need to learn. In solving a given problematic situation or doing an activity, students put into play three patterns: An action pattern that arises when students know what to do to solve the problem. A communication pattern that arises when one student knows what to do and communicates his solution path to others. And a validation pattern that comes out when two or more students have different solution paths to the same situation and they try to convince each other of the validity of their proposals. In the TE proposed, students work in pairs, but the communication among pairs is not forbidden. Besides, students are allow to move freely in the classroom if they want to, and communicate with other students and the teacher. In turn, teacher is always monitoring the activity of students and giving clues for the solution if that is the case. At a given moment, teacher could call students attention and make some commentaries to all the class or encourage some group discussions. The general idea is that students feel comfortable inside the math classroom and confident about asking questions and giving argumentations. So they can learn and foster other students learning. Now, the fostering of the aforementioned values is quite difficult in a TE where the focus is on teacher; in this kind of environments is the teacher who transmit knowledge to her students who, in turn, are merely passive recipients of that knowledge (a primary teacher used to say: students in a classroom are like stones with eyes). The teacher is in charge to keep discipline in the classroom and attract students attention. In order to fulfill this, teacher imposes a series of rules that produce a silent class where students are taking notes, give opinions only when they are allowed to do it, reproduce the exercise that they are asked to reproduce and seldom communicate with other classmates. Often, discipline is imposed under the threat of a punishment if it is broken or relaxed. In an environment like this there is no cooperation, respect, tolerance and responsibility, fear is the only value we can foster. Consequently, it is necessary to change focus from teacher to students. They must be an active element in the acquisition of their knowledge. Teacher should recognize students own individuality and give them the chance to practice it it is absurd to expect that all individuals in a class grasp a concept and understand the same things with only one speech given by the teacher. Students must do mathematics and learn from it; they should solve problem situations with cooperation of their classmates and teachers guidance. Students should have the opportunity to discuss those situations with their classmates and proceed accordingly. Math classroom should be the place where students Learn Mathematics, Doing Mathematics. It should be the ideal place in which our students learn to live together in harmony, tolerance, respect and love. In a Learning Mathematics, Doing Mathematics Teaching Environment, students are faced with problem situations in which they have the opportunity to question and challenge their selves or to another classmates; explore ideas, make conjectures and try to prove them; reflect on ideas, situations, actions and results; communicate proceedings and outcomes. All these features of students activities and actions promote motivation and creativity (see for instance Her Majesty's Inspectorate of Education: Emerging Good Practice in Promoting Creativity, 2006). We understand creativity as the production of an idea, concept or discovery that is new, original and useful for the individual that produce it as well for the community where this individual is working, at least for a given period. And all of these are increased with the use of technology, specially the Dynamic Geometry software The Geometers Sketchpad. Using Sketchpad makes easier not only construct and explore Geometry, but also algebra and most of school mathematics. Let us see an example: In a 10th grade class students were working in pairs on the next problem: Construct a square and its diagonals; inside the square there are four triangles, how are these triangles compared to each other? Explain your answer. We were studying congruence criteria and the instructor hoped that students use them to explain their result. They were at a computer lab and working with Sketchpad. All students jump into the conclusion that these four triangles are congruent to each other; and most of the explanations were in terms of measures these students were using what we say is an empirical argumentative scheme (Flores, 2007) they measured triangles angles and sides. Five, out of 20 couples, try to explain their results using square properties and congruence criteria; and a couple explain its result in the following terms: If you construct a square and its diagonals, and draw a circle with center at the diagonal intersection and passing through a vertex, you will notice that the circle passes through the other three vertices as well. This means that the four triangles are isosceles triangles and shear common sides of the same length, because they are radius of the same circle. Besides, the bases of these triangle have the same length because they are sides of the same square, consequently, the four triangles are of the same measure and congruent to each other. They came out with this explanation because, from an earlier session, their instructor told them that if they wanted to explain something in a mathematical way, they should look for reasons out of the software. So they explained later that they figured out how to measure the sides of the triangles without using Sketchpad. They said that they could use a compass instead of the software. This explanation is half way between an empirical scheme and an analytical one (Flores, 2007) and the process that led to the explanation is an unexpected one and, perhaps, it falls in the category of been creative. We have several other examples where students use a different and unexpected way of solving problems in Geometry and Algebra, but due to the lack of space it is impossible to mention them; but still the question remains and is open to discussion: It is possible to foster creativity in a teaching model centered on students in which they work in a suitable Teaching Environment? REFERENCES Brousseau, G., 1997, Theory of Didactical Situations in Mathematics, Mathematics Education Library, Kluwer Academic Publishers. HMIE, Promoting creativity in education, 2006,  HYPERLINK "http://www.hmie.gov.uk/documents/publication/hmiepcie.html" http://www.hmie.gov.uk/documents/publication/hmiepcie.html Flores, H., 2007, Esquemas de argumentacin en profesores de matemticas de Bachillerato, Educacin Matemtica, vol. 19-2. ABOUT THE AUTHOR ngel Homero Flores Samaniego Profesor de Carrera de Tiempo Completo, rea de Matemticas Colegio de Ciencias y Humanidades-UNAM Mxico, E_mails:  HYPERLINK "mailto:ahfs@servidor.unam.mx" ahfs@servidor.unam.mx ,  HYPERLINK "mailto:ahfs58@yahoo.com.mx" ahfs58@yahoo.com.mx  PAGE 170 PAGE 170 PAGE 170 PAGE 170 PAGE 170    ngel Homero Flores Samaniego Learning Mathematics, Doing Mathematics: Creativity in Classroom? 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