ࡱ> qsp @M`bjbj >d؝؝7`jjjjjjj~8>D4~DL:HHHjClClClClClClC$+FR}HCEj[||[[CjjHH4C!!![jHjHjC![jC!!6&?jjAH Z[ @CdC0D"@QI QI8A~~jjjjAQIjA(.|!dMCC~~D D X~~ DEVELOPING THE UNDERSTANDING BY MEANS OF A DIDACTIC MODEL THAT FAVORS THE MATHEMATICAL CREATIVITY AIDA MARA TORRES ALFONSO AND DMASA MARTNEZ MARTNEZ Abstract: The proposed work, is the result of an investigation carried out in the Central University of The Villas, defends the idea that it is contributed to develop the understanding in the students that study the first year of the career of the students of Degree in Mathematical if one conceives the mathematical understanding as capacity to develop by means of a didactic model with systemic focus and with bonds of different subjects from the beginning of the course that interrelates the diagnosis, the learning strategies and the formative evaluation as subsystems of the pattern with the objective of developing the mathematical thought in an active way. Key words: Creative Process, Diagnose of understanding, Levels of understanding, Didactic System INTRODUCTION In the different levels of the educational system the learning of the mathematics faces daily, serious problems that are related with the motivation, the understanding, the linking of what memorizes with the individual's necessities and the creativity. In the university classrooms, so that the youths carry out and understand the mathematical, the educational ones should propitiate in the students the process of development of ideas, respecting their own intellectual characteristics and stimulating in them the desire to learn for their professional preparation, that is to say, creating conditions for the development of the motivation and the mathematical creation. In the work we intend to determine the levels of understanding with which arrive each one from our students to the higher education, as starting point to keep in mind for the professors for the design of the teaching content. Another idea to discuss would be if in fact the development of the mathematical capacities and the values in each student traffic for a process that is necessary to have in consideration on the part of the educational ones. The learning centered in the students, respecting the disposition diversity before the professional preparation. The proposed pattern foments the creativity in the mathematical activity of the university students leaving of the principle that the professor values in more degree the process of the knowledge that the knowledge in if, propitiating a great variety of reflexive activities, the techniques in the group, the use of the technologies in a course of Mathematical Analysis and the fact of keeping in mind the individual necessities of the students; as well as the social ones, in function of completing the formative objectives that intends the community of professors that give classes in the first year university student. THE MATHEMATICAL UNDERSTANDING AS CAPACITY TO DEVELOP IN UNIVERSITY STUDENTS The formation of capacities is a process closely interrelated with that of acquisition of knowledge and abilities. In fact, it constitutes a result of the appropriately structured teaching process. Although the capacities depend certainly on the knowledge and abilities, doesn't only decrease to them, because the capacities are more stable qualities of the personality that are not formed with the speed with which the knowledge and abilities are acquired. On the other hand, the quality and the speed in the assimilation of the knowledge and the development of abilities in the students depend in turn on their capacities. After consulting different definitions of capacities we mention for their interest that of Rubinstein (1979:50): The capacities are a complex formation, a group of psychic properties that make suitable to the man for certain socially useful professional activity." When assuming this posture, in the one which the development of the mathematical understanding, evolves in movement form in hairspring, being necessary the possibilities that offers the capacity at a level given for the development of capacities at a superior level to take advantage. And the mathematical understanding will be valued like a capacity in development as we can validate that is to say by means of the actings that go showing the students in the dynamics of the process of teaching learning, the easiness, speed and depth with which they acquire the mathematical knowledge during the whole school course. To determine the levels of understanding for those that the students will traffic in the first year of the university, we assume the classification that presents the pedagogy of the understanding of the school of Harvard developed by Perkins and Gardner (Perkins, 1999), in which the understanding is an ability that should be approached from an interdisciplinary perspective. And also that to understand a mathematical object consists on being able to recognize their characteristics, properties and representations; to relate it with other mathematical objects and to use it in all the variety of didactic situations that you/they are proposed by the professor, as well as to express it openly, with arguments that demonstrate that their thought has evolved after a productive intellectual effort (Godino and Batanero, 1994). Will it be possible to develop the mathematical understanding as a capacity in the university students? The capacities are originated, formed and developed in the activity and for the activity, that is to say, they are acquired during the life, without denying the paper that carry out in the development of the capacities the biological factors, starting from the aptitudes that being necessary conditions for the corresponding capacity doesn't predetermine it. For example, a student with the best aptitudes toward the mathematical one, without carrying out the corresponding activity, that is to say, if doesnt study the mathematical one, doesnt discover their mysteries, their history and their narrow knots with the social and economic development, it will never be neither a successful mathematician, neither a professional that can use it in any other field of the knowledge. On the other hand, a student that arrives to our universities without the good conditions of aptitudes toward the mathematical one, in a process of teaching surely longer learning could be formed, on the base of the realization of activities conceived for achievement of this objective, the corresponding capacities. The development of capacities doesn't have limit, it is certain historical - socially, while more evolves the society, but possibilities have of being developed the capacities For example, the use of the new technologies in educational activities in the university where the students can contrast different solutions to problems outlined by the professor, the analysis of theoretical foundations of diverse topics of the Mathematical Analysis, the search of against examples and the possibility of deepening in the history and the development of Mathematical until the present time, opens an interesting field where you can develop the mathematical understanding as capacity. The qualitative study of the capacities to develop in an university student is emphasized, leaving of each individual's potentialities, using the methods of the observation and the experimentation mainly. For example the development of the mathematical understanding in university students, it should be conceived leaving of determining the levels of understanding with which arrive each one from our students to the university classrooms, it should constitute a starting point to keep in mind for the professors when designing the teaching content. To diagnose the levels of mathematical understanding with those that the students arrive to the universities: starting point to develop the mathematical creativity. In the approach of the authors: the valuation of the initial diagnosis should go consenting to the progressive diagnosis: formative evaluation, what implies to determine the characteristics of the students and the changes that experience in their understanding, like part of the didactic pattern to develop the understanding that intends. As the depth of the mathematical understanding it can vary for content it becomes necessary to distinguish in the students, weak actings of others that are more advanced. For what we will characterize according to the theoretical mark of reference four levels of understanding: Informative understanding - reproductive (routine procedures) The actings of understanding are based on the intuitive knowledge, it is when the students don't show domain signs of what they know being shown few reflexive. Understanding in the practical level or of resolution of problems The actings of understanding in the practical level are mainly based on the traditional mechanisms of exercises, as procedures mechanical step for step. Theoretical understanding The actings of theoretical understanding demonstrate a flexible use of concepts or contents of the discipline. With the students' well structured help and the community of professors the actings in this level discover before the agents of the process of teaching learning, the relationship between disciplinary knowledge and the daily life, examining the opportunities and the consequences of using this knowledge. Understanding in the investigation level The actings of investigation understanding are mainly integrative, creative and critical. In this level, the students are able to move with flexibility and to validate the purposes of the investigation that have been proposed. The actings in this level demonstrate disciplinary understanding: they can reflect the critical conscience of the students about the construction of the knowledge. Therefore, as the main objective it is to diagnose the levels of understanding in the students that arrive to the university classrooms, it is demanded from a design of tasks, exercises and problems that propitiate a chain of acting of understanding of wide variety and growing complexity, when offering them the possibility to carry out different activities that require thought, for example: to explain, to find evidence, against examples, to generalize, to apply in new situations, to present analogies and to represent in different ways. On the other hand it will be demanded the students to show the acquired capacities in a way that it can be observed, making that their thought transforms evident before the professor, another student or the group.. Will the mathematical creativity be propitiated in the university students, with the realization of an initial diagnosis of its levels of understanding only? The authors sustain the idea that the capacities, when being been indirect of the learning process should be to be propitiated by means of a didactic model that links that diagnosis with the learning strategies directed to obtain that objective and a continuous evaluation that keeps in mind the unit of the affective thing and the cognitive thing in each student and during the whole process of teaching learning. The proposed model foments the creativity in the mathematical activity of the university students leaving of the principle that the professor values in more degree the process toward the learning that the knowledge in if, propitiating a great variety of reflexive activities, the techniques in groups, the use of the technologies in a course of Mathematical Analysis and the fact of keeping in mind the individual necessities of the students, his potentialities; as well as their interests and motivations, in function of completing the formative objectives that intends the community of professors that impart classes in that first year university student. CONCLUSIONS AND FUTURE WORK To achieve then, didactic proposals based on developing the mathematical creativity in the university students are considered as a challenge for the educational one almost as much as a necessity. Since it should achieve the student to be able to develop activities that imply something more than to reproduce information or to communicate a concept. It should be able to apply it to new situations, to link it to different contexts, to transfer it or to present it in a creative way. For that this is needed to verify which is each students understanding level, for of there to design the pedagogic practice that allows their evolution and development. The thematic one approached it possesses great importance and present time if one keeps in mind that the High Education in the world directs its objectives toward a much more personalized teaching where it is essential that the students are able to conquer the obstacles and to get ready professionally according to the requirements of the current society. The proposal developed in the work is directed to guarantee in the career of Licentiate in Mathematical, the responsibility that assumes the Cuban University of working to achieve permanency levels and of in agreement expenditure with the access levels. REFERENCES Gimnez Rodrguez, J. (1997). Evaluacin en Matemticas. Una integracin de perspectivas. Madrid: Sntesis, S. A. Godino, J. D. y Batanero, C. (1994). Significado institucional y personal de los objetos matemticos. Recherches en Didactique des Mathmatiques, Vol. 14, n 3: 325- 355. Perkins, D. (1999). Qu es la comprensin? In: WISKE. M.S. (Ed.) La Enseanza para la Comprensin- Vinculacin entre la investigacin y la prctica, 215-256. Barcelona, Espaa: Editorial Paids. Rubistein, S. L. (1979). El pensamiento en el desarrollo de la Psicologa. Principios y mtodos. La Habana: Pueblo y Educacin. ABOUT THE AUTHORS Lic. Aida Maria Torres Alfonso Department of Mathematical Faculty of Mathematical Physics and Calculation Universidad Central de Las Villas, Cuba -mails:  HYPERLINK "mailto:aida@uclv.edu.cu" aida@uclv.edu.cu ,  HYPERLINK "mailto:fresasjun22@yahoo.com" fresasjun22@yahoo.com Dra. Dmasa Martinez Martinez Department of Mathematical Faculty of Mathematical Physics and Calculation Universidad Central de Las Villas, Cuba -mails:  HYPERLINK "mailto:damasa@uclv.edu.cu" damasa@uclv.edu.cu ,  HYPERLINK "mailto:damasam58@yahoo.com" damasam58@yahoo.com     Developing the Understanding by Means of a Didactic Model that Favors the Mathematical Creativity Aida Mara Torres Alfonso and Dmasa Martnez Martnez PAGE 126 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 2 PAGE 125 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 120 DG 9: +,MNfg8 C Fopиsd\UdsLhx5CJaJ h#=hxh#=hx5h#=hxCJaJmH sH h#=hx5CJaJmH sH h#=hxCJaJh#=hx6CJaJh#=hx56CJaJh#=hx5CJaJmH sH hx5CJaJmH sH hx5CJ$\aJ$h#=hx5CJ$\aJ$#h#=hx5OJQJaJmH sH hx5OJQJaJmH sH ,Nfg8 I2 $d\$a$gdxd\$gdx$d\$`a$gdx$ Pd\$]P^a$gdx $d\$a$gdx $d@&\$a$gdx $d\$a$gdxAL`Fop@#!!#$%%&( $d\$a$gdx$ & F 0d\$`0a$gdx$ & F 0d\$`0a$gdx $d@&\$a$gdx $d\$a$gdx #d\$`#gdx $d\$a$gdxp@!%%&(=))$***~,,+1166 6;;;%;C;};;;;;;ا󐀐qaqah#=hx6CJaJmH sH h#=hxCJaJmH sH h#=hx6CJOJQJaJh#=hxCJOJQJaJhxCJaJhx5CJaJmH sH h#=hxCJaJmH sH h#=hx6CJaJhx5CJaJh#=hx5CJaJmH sH h#=hxCJaJh#=hx5CJaJ(=))$***~,,3.+11o366 6688:;;d\$gdx$#d\$`#a$gdx $d\$a$gdx $d\$a$gdx$ & F 0d\$`0a$gdx;%;;B<======= >Z>l?n????$a$gdY+$a$gdY+$a$gdx$d\$`a$gdx"$Td\$^T`a$gdx$07$8$H$^`0a$gdx"$0^`0a$gdx $d\$a$gdx;'<B<<<==e===== >Z>^>h>l>n>>>ϽϮzj]L]7L7)jh#=hY+OJQJUaJmH sH  h#=hY+OJQJaJmH sH h#=hY+OJQJaJh#=hY+5CJaJmH sH h#=hY+5CJaJh#=hx5CJaJhxCJOJQJaJh#=hx6CJOJQJaJh#=hxCJOJQJaJ"h#=hx6CJ]aJmH sH h#=hxCJaJmH sH h#=hx6CJaJmH sH "h#=hx6CJ]aJmH sH >>>>>:??h?j?l?n???D@@@@@@@@@ǯǟzmm\G\G)jh#=hxOJQJUaJmH sH  h#=hxOJQJaJmH sH h#=hY+OJQJaJh#=hx5CJaJmH sH hY+5CJaJh#=hx5CJaJhY+hY+5CJaJmH sH /jh#=hY+OJQJUaJmH sH  h#=hY+OJQJaJmH sH )jh#=hY+OJQJUaJmH sH $h#=hY+0JOJQJaJmH sH ?D@@AAAAAAAAAA BEBGBHBBBBB$hh]h`ha$gd*$y&#$+D,gdY $d\$a$gdx$a$$a$gdxgdx$a$gdx$a$gdx@ A"A(A*A,A|A~AAAAAAAAAAAAAAAƵuh\TPTPTPTPChY+CJOJQJ\aJhHQjhHQUhY+h k5mH sH hY+hxOJQJaJ$hY+hx0JOJQJaJmH sH /jhY+hxOJQJUaJmH sH )jhY+hxOJQJUaJmH sH  hY+hxOJQJaJmH sH #h#=hx5OJQJaJmH sH )jh#=hxOJQJUaJmH sH $h#=hx0JOJQJaJmH sH AABB B B B BBB)B*BDBEBGBHB~BBBBBBBBBBB¯‡vcvVFh;h k6CJOJQJaJh k6CJOJQJaJ%h>T0JCJOJQJaJmHnHu hAoIh k0JCJOJQJaJ)jhAoIh k0JCJOJQJUaJ$hxh kCJOJQJaJmH sH $hxhxCJOJQJaJmH sH h kh kh kCJOJQJaJhxCJOJQJ\aJhY+CJOJQJ\aJh#=hxCJOJQJ\aJBBBBBBBBBBBBCCpCqCCCCCCCCCCCCCʵtfXfʵF#h;h k0J6CJOJQJaJh-Ch k56CJaJh-Ch k56CJ aJ h;h kCJOJQJaJh k0JCJOJQJaJ%h>T0JCJOJQJaJmHnHu hAoIh k0JCJOJQJaJ)jhAoIh k0JCJOJQJUaJh kh k6CJOJQJaJh;h k6CJOJQJaJ'h;h k6CJOJQJaJmHsHBBBCC0CpCqCCCCCCJ`K`L`M`gdx$ 9r  n!&#$+Dq]a$gd+y&#$+D,gdY$ 9r  ]a$gdN$  ]a$gdN$a$gd+$'&#$+D,a$gdYPromoting Creativity for All Students in Mathematics Education, Section 2 C`>`I`K`L`M`hY+h k5mH sH hHQh kh k6CJOJQJaJh;h k6CJOJQJaJU< 00&P 1hPx:p>T. 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