Creativity and Resolution of Problems beyond the cl assroom Nuno Amaral nualroam@gmail Doctoral student at Institute of Education, Univers ity of Lisbon 1. Introduction Besides being a learning objective, the resolution of problems is a means by which students learn mathematics, helping them acquire ways of th inking, persistent habits, curiosity and confidence with unknown situations that will be use ful outside of the mathematics class (NCTM, 2007, p. 57). Problem solving is a favoured activity for students to consolidate, broaden and deepen their mathematical knowledge, but is also essential for l earning concepts, representations and procedures (Ponte et al , 2007). Work with problems encourages reasoning an d justification, motivates communication, provides the use of different repres entations, allows the establishment of connections between mathematical content and turns mathematics into a useful discipline for everyday life (Boavida, Paiva, Cebola, Vale & Pimen tel, 2008). To many authors, problem solving should be the focal point of all mathematics educat ion as it promotes the development of high-level cognitive functions and ability to relate mathemati cs with the real world (Fernandes, 1994; NCTM, 2007; Ponte et al , 2007). In this perspective, the NCTM (2007) state s that problem solving should be an integral part of mathematics teaching and lea rning programs should be geared towards enabling students to: build new mathematical knowledge through problem so lving; solve problems that arise in mathematics and other contexts, apply and adapt a v ariety of appropriate strategies to solve problems, analyse and reflect on the process of sol ving mathematical problems (p. 57). Good problems, involving concepts and skills that s tudents should develop, are a special resource for learning mathematics (Carvalho, 2008); contributing to stimulate creativity and encouraging students to study mathematics, enabling them to broaden their horizons (Powell et al , 2009). According to Tall (1991), many researchers s upport that the resolution of problems is the most creative mathematical activity. Students benef it cognitively when solving problems and the gains are evident in the short term and intellectua lly important over time (Powell et al, 2009). The troubleshooting championships open new possibil ities for mathematics education (Borba, 2009) providing access to interesting and challengi ng problems, more than those who are solved regularly in the mathematics class (Jones & Simons, 1999; Jacinto, 2008), promoting the development of students autonomy, since they are a llowed to use their own strategies and different representations to communicate their resolutions (F reiman et al, 2009) 3 Creativity and Resolution of Problems beyond the cl assroom Nuno Amaral nualroam@gmail Doctoral student at Institute of Education, Univers ity of Lisbon 1. Introduction Besides being a learning objective, the resolution of problems is a means by which students learn mathematics, helping them acquire ways of th inking, persistent habits, curiosity and confidence with unknown situations that will be use ful outside of the mathematics class (NCTM, 2007, p. 57). Problem solving is a favoured activity for students to consolidate, broaden and deepen their mathematical knowledge, but is also essential for l earning concepts, representations and procedures (Ponte et al , 2007). Work with problems encourages reasoning an d justification, motivates communication, provides the use of different repres entations, allows the establishment of connections between mathematical content and turns mathematics into a useful discipline for everyday life (Boavida, Paiva, Cebola, Vale & Pimen tel, 2008). To many authors, problem solving should be the focal point of all mathematics educat ion as it promotes the development of high-level cognitive functions and ability to relate mathemati cs with the real world (Fernandes, 1994; NCTM, 2007; Ponte et al , 2007). In this perspective, the NCTM (2007) state s that problem solving should be an integral part of mathematics teaching and lea rning programs should be geared towards enabling students to: build new mathematical knowledge through problem so lving; solve problems that arise in mathematics and other contexts, apply and adapt a v ariety of appropriate strategies to solve problems, analyse and reflect on the process of sol ving mathematical problems (p. 57). Good problems, involving concepts and skills that s tudents should develop, are a special resource for learning mathematics (Carvalho, 2008); contributing to stimulate creativity and encouraging students to study mathematics, enabling them to broaden their horizons (Powell et al , 2009). According to Tall (1991), many researchers s upport that the resolution of problems is the most creative mathematical activity. Students benef it cognitively when solving problems and the gains are evident in the short term and intellectua lly important over time (Powell et al, 2009). The troubleshooting championships open new possibil ities for mathematics education (Borba, 2009) providing access to interesting and challengi ng problems, more than those who are solved regularly in the mathematics class (Jones & Simons, 1999; Jacinto, 2008), promoting the development of students autonomy, since they are a llowed to use their own strategies and different representations to communicate their resolutions (F reiman et al, 2009). 2. Research Problem Many are the students who take part in extracurricu lar activities for learning mathematics to develop their understanding in this area of knowled ge (Kenderov et al , 2009). The activities outside school, such as problem solving championships, as l ong as challenging, interesting and fun, can be key to a deep understanding and widening of mathema tical knowledge, providing opportunities for students to strengthen their mathematical thinking (Losada, Yup, Gjone & Pourkazemi, 2009). They are environments that have the ability to prepare s tudents for testing and deploying their mathematical ideas (Moyer, Niezgoda & Stanley, 2005 ). These projects originate authentic contexts of learning, looking ahead to develop the knowledge and skills that can be transferred to new problematic situations successfully (Wells, 2007). The importance of investigating the processes that students develop in the business of solving pr oblems within and outside the classroom, 4 particularly when problems are related to mathemati cal concepts, is highlighted by several authors (English, Lesh & Fennewald, 2008; Barbeau & Taylor, 2009). The main objective of this research project is to s tudy the role of creativity in problem solving by t he participants of the Mathematics championship SUB12 final, through the representations used to communicate the resolutions, focusing on three main guiding questions: 1. How is creativity manifested in the students’ forms of representation, in the resolution of mathematical problems of the SUB12? 1.1. Do the representations used express powerful mathem atical ideas? 1.2. Do the representations reveal specific aspects of t he mathematical reasoning? 1.3. Do the representations include aesthetic elements, relevant to the communication of 4 particularly when problems are related to mathemati cal concepts, is highlighted by several authors (English, Lesh & Fennewald, 2008; Barbeau & Taylor, 2009). The main objective of this research project is to s tudy the role of creativity in problem solving by t he participants of the Mathematics championship SUB12 final, through the representations used to communicate the resolutions, focusing on three main guiding questions: 1. How is creativity manifested in the students’ forms of representation, in the resolution of mathematical problems of the SUB12? 1.1. Do the representations used express powerful mathem atical ideas? 1.2. Do the representations reveal specific aspects of t he mathematical reasoning? 1.3. Do the representations include aesthetic elements, relevant to the communication of mathematical ideas? 2. How is creativity manifested in the strategies used by the students in the resolution of the SUB12 problems? 2.1. What characterises the student’s powerful and origi nal strategies? 2.2. What distinctive elements can be found in creative strategies (mathematical connections, establishment of relations, identification of objec tives ...)? 3. What can be transposed about the creativity shown b y the SUB12 students in the resolution of problems to the process of teaching and learning of mathematics? 3.1. What do the talented students think about the oppor tunity to develop creativity within mathematics? 3.2. What do the Mathematics Teachers think about the op portunity to develop creativity within mathematics? 3.3. How does the creativity of talented students relate to the feel of problem solving within mathematics? 3. Creativity and Resolution of Problems beyond the classroom From Silver’s point of view (1997), creativity is a n evidence of mathematical knowledge. For the author, although creativity is often associated with ingenuity or exceptional abilities, it can be widely promoted in the scholar population in genera l. The students with mathematical potential show creat ivity, high concentration, intuition, originality, stability and flexibility; solve probl ems differently from other students, enjoy specific activities since it allows them to create something new and to be autonomous in their approaches to problems; stand out for their superior capabilities to communicate and explain their symbolic resolutions (Applebaum, Freiman & Leikin, 2008). Th ey are creative students who demonstrate an evolved mathematical thinking, able to combine know ledge, imagination and inspiration (Leikin, 2007). Various authors (Ervynck, 1991; Piirto, 1999 ; Silver, 1997, Sheffield, 2003, referred by Applebaum, Freiman & Leikin, 2008) consider that th ese students should be offered the opportunity to show their mathematical knowledge, creativity, c uriosity, detail and imagination. Creativity is evident when students have the opportunity to find and use their own solving methods (Pehkonen, 1997). In that sense, the tasks should be especiall y challenging (Applebaum & Leikin, 2007; Sheffield 2003; Freiman, 2006, referred by Applebau m, Freiman & Leikin, 2008). Karp & Leikin (2009) believe that the challenging mathematic acti vities free from routine, research-based and rich in problem solving, can lead students to discover a nd realise their talent. Leikin (2007) defends the resolution of problems as an effective tool to promote and explore mathematical reasoning, the production of mathemati cal connections and creativity. For this author, 5 problem solving promotes advanced mathematical thin king, since it requires strategic thinking, insight, perseverance, creativity and skill. Meanin gful involvement with mathematical problems, over time, allows students to build, find and defin e effective schemes and strategies important in resolution (Powell et al, 2009). A study by Ching ( 1997) revealed that by having the opportunity to think for themselves, some students find interestin g ways to solve problems. According to the author, creativity and mathematical talent emerge w hen students are autonomous. On the other hand, Hashimoto (1997) argues that students mathem atical creativity emerges when they are able to combine different ways of thinking about a problem. 4. The resolution of problems and ways of represent ation The representations are an important vehicle for le arning and communicating, providing vital tools to register, analyse, resolve and communicate mathematical data, problems and ideas (Preston & Garner, 2003). The representation of mathematical ideas arises fro m processes occurring in the individuals’ minds (Boavida et al , 2008) and how to represent them is critical to be used and understood (NCTM, 2007; Ponte & Serrazina, 2000). To understan d the representations and to be able to represent mathematically, is a skill that increases the ability to think mathematically (NCTM, 2007; Boavida et al, 2008; Ponte and Serrazina, 2000), he lping to make learning more meaningful (Clements, 2004). According to NCTM (2007): Representations should be treated as essential elem ents in supporting the students understanding of the concepts and the mathematical relations, commun ication approaches, arguments, and mathematical knowledge for themselves and others, t o identify connections between inter-related mathematical concepts, and application of mathemati cs to realistic problems through modelling (p. 75). The representations designate concepts and always r efer to something, an object or situation (Ponte & Serrazina, 2000). They constitute themselv es through signs that are used when connected to a meaning (Saraiva, 2008; and Matos e Serrazina, 1996) and can take different forms, for example, symbolic, iconic and active (Boavida et al , 2008). The symbolic representations consist of the transl ation of experience in terms of symbolic language (Boavida et al , 2008, p. 71). The symbols are especially useful for solving problem since they allow accurate expression of mathematica l ideas and in a condensed manner (Ponte, Matos & White, 2008). They are usually used with th e objective to facilitate communication regarding the concepts (Matos & Serrazina, 1996). The iconic representations are based on visual org anisation, in the use of figures, pictures, schemes, diagrams or drawings to illustrate concept s, procedures, or a relation between them (Boavida et al , 2008, p. 71). Such representations have a very im portant role in the representation of mathematical ideas. For example, the construction o f figures can be a powerful learning experience for students, because they have to make explicit va rious aspects that are often assumed when the images are pre-established (Clements, 2004). The active representations are associated with act ion. The proper and direct manipulation of objects, (...), contributes to the construction of concepts (Boavida et al , 2008, p. 71). Students should be able to use given representation s and select from among various representations, which are most useful for a partic ular situation. It is also desirable to be able to create their own representations (Preston & Garner, 2003). The representations constructed by students are important to the understanding of math ematical ideas (Ponte & Serrazina, 2000), allowing recognition of its modes of interpretation and reasoning used (NCTM, 2007) and can be used to provoke discussion of mathematical ideas (C lements, 2004). Allowing students to build their representations has advantages in understandi ng and problem solving, provides the development of own solving methods (NCTM, 2007) and provides a starting point for consideration of alternative representations (Ponte & Serrazina, 2000). The ease of using multiple representations and the flexibility to switch between different rep resentations are critical components in the ability to solve mathematical problems (Heinze, Star, & Ver schaffel, 2009). Each form of representation takes place in the used thinking processes that und erlies the choice of strategy and tool for its communication (Preston & Garner, 2003). To Heinze, Star and Verschaffel (2009), the use of flexible and adaptable representations is part of a cognitive variability, which allows the solving of problems more quickly and accurately. The learning environments that confront students wi th multiple representations and which enable a fluent and flexible combination of differe nt representations can be effective in helping students understand and learn mathematical concepts by developing a genuine willingness to learn mathematics (Heinze, Star & Verschaffel , 2009). 5. The Research Design Since the objective of this research is to study th e phenomenon in the context in which it happens, we opted essentially for a qualitative met hodology. The intention is not to generalise, but to deepen the knowledge of a particular situation i n a given context. In this case, it is intended to describe and interpret the creativity within proble m solving, outside the classroom, through the representations used by the students to communicate resolutions. 5.1. Empirical Field The SUB12 is a Problem Solving championship in Math ematics, supported by the Department of Mathematics within the Institute of S cience and Technology at the University of Algarve, geared for students in the 5 th and 6 th grades within the Algarve and Alentejo regions in Portugal. It appeared in the 2005/2006 school year as a complement to mathematical activities developed in the classroom by teachers and it aims to tackle some of the limitations pointed out by teachers, including the difficulty in implementing a regular and solid task in the area of problem solving within the mathematics classroom. The championship is divided into two distinct phase s: the qualifying phase , which consists of 12 problems, held from January until June, and is h eld remotely via computer and the Internet ; the final phase , in which finalists students participate in a tour nament at the University of Algarve, consisting of five problems. 5.2. Study Participants This study will focus on the participants of the Ma thematics SUB12 Championship. It is not intended to standardise the results or generalise c onclusions, so the selection of participants will b e intentional (Ferreira and Carmo, 1998). The method of selection will be determined by scrutinising the resolutions produced by the participants and th eir interest in solving problems outside the classroom. 5.3. Data Collection Given the interpretative nature of the research, th e methodology to follow, will be based on the collection of qualitative data in order to desc ribe and interpret the phenomenon under study in the most complete and authentic manner possible. It will seek to diversify the sources and data collecting instruments in order to obtain a signifi cant set of data so that the descriptions and interpretations are well founded and valid. In this sense, the data will be gathered from docum ents produced by participants of the SUB12, from a diary of notes and reflections of the researcher, complemented by the application of semi-structured interviews. 5.3.1 Documents produced and written notes The documents produced by the participants will ena ble meaningful data to be obtained on their representations in the resolution of problems , obtained in the final phase of the 2005/2006, 2006/2007, 2007/2008, 2008/2009 and 2009/2010 editi ons. In the 2009/2010 edition the finalists’ resolutions will be analysed in the final qualifyin g and in the final, with the aim of confronting the strategies used with and without access to the tech nologies. The notes and reflections written will serve to des cribe what the researcher observes, feels and thinks during the data analysis (Bogdan & Biklen, 1 994), especially regarding the resolutions and the processes used by the participants in the champ ionship. 5.3.2. Interviews Interviews will take place in order to obtain more specific data about the participants (Tuckman, 2005). The interviews will take place after the final sta ge of the SUB12 championship, for ten of the finalists in attendance and respective teachers , with the intention to collect participant data, orally and through their own words (Bogdan & Biklen , 1994), for example: the participants’ feedback on the resolution of problems; the partici pants’ motivation to solve mathematics problems; the emotions that the participants feel during the problem solving; the usefulness and importance of solving problems for the participants and the teach ers; understanding what teachers think of the participants and about their level of talent and cr eativity, what do they have of difference, what distinguishes them, what makes them creative, how t hey react in the classroom, how they work... this way the researcher will be able to develop ide as on how the participants interpret key aspects o the problem under study (Bogdan & Biklen, 1994). 5.4. Data analysis The data analysis will primarily be descriptive and interpretative in order to obtain a characterisation as complete as possible of the phe nomenon under investigation and an understanding of it, with the objective to answer t he questions asked. The aim is not to establish the relations of cause and effect, but to understand th e participants’ views in this study, in a perspecti ve supported by the concepts discussed and analysed wi thin the theoretical framework guiding the study. 6. Bibliography Applebaum, M., Leikin, R. & Freiman, V. (2008). Views on Teaching Mathematically Promising Students . ICME 11 Conference. disponível em google.br/#hl=pt- BR&source=hp&q=VIEWS+ON+TEACHING+MATHEMATICALLY+PRO MISING+STUDEN TS&btnG=Pesquisa+Google&aq=f&aqi=&aql=&oq=&gs_rfai= &fp=c808b9ad7f0c0da3 Applebaum, M. & Leinkin, R. (2007). Teacher ́s Conce ptions of Mathematical Challenge in School Mathematics. In J. H. Woo, H. C. Lew, K. S. Park & D. Y. Seo (Eds.). Proceedings of the 31 Conference of the International Group for the Psyc hology of Mathematics Education (pp. 9-16) (Volume 2), Seoul: PME. Boavida, A. M., Paiva, A. L., Cebola, G., Vale, I. & Pimentel, T. (2008). A Experiência Matemática no Ensino Básico . Lisboa: Ministério da Educação. Bogdan, R. e Biklen, S. (1994). Investigação qualitativa em educação – uma introdução à teoria e aos métodos . Porto: Porto Editora. Borba, M. C. (2009). Potential scenarios for Intern et use in the mathematics classroom. ZDM- International Journal on Mathematics Education , Vol. 41(4), p. 453-465. Carmo, H. & Ferreira, M. M. (1998). Metodologia da investigação – Guia para a Auto-apre ndizagem . Lisboa: Universidade Aberta. Carvalho, A. (2008). Representações dos alunos. Que raciocínios revelam? Educação e Matemática , N.º 102, 2-10. Clement, L. (2004). A Model for Understanding, Usin g, and Connecting Representations. Journal of Teaching Children Mathematics , 11(2), 97-102Ching, T. P. (1997). An Experiment to Discover Mathematical Talent in a Primary School in Kampong Air . ZDM - International Journal on Mathematics Education , Vol. 29(3), p. 94-96. English, L., Lesh, R. & Fennewald, T. (2008). Future directions and perspectives for problem solv ing research and curriculum development . Paper presented at the 11th International Congres s on Mathematical Education (ICME11) – Topic Study Group 19: Research and development in problem solving in mathematics education. Consultado em htt p://tsg.icme11.org/document/get/458. Freiman, V., Kadijevich, D., Kuntz, G., Pozdniakov, S. e Stedoy, I. (2009). Technological Environments beyond the Cassroom. In E. J. Barbeau & P. J. Taylo r (Eds.), Challenging Mathematics In and Beyond the Classroom. The 16th ICMI Study , (pp. 97-131). New York, NY: Springer. Jacinto, H. (2008). A Internet e a Actividade Matemática no Caso do Sub 14. Dissertação de Mestrado (não publicada), Universidade Católica Portuguesa, Lisboa Jones, K. e Simons, H. (1999). Online Mathematics Enrichment: an independent exter nal evaluation of the NRICH project 1998-99 . Centre for Research in Mathematics Education. UK: University of Southampton Design & Print Centre. Hashimoti, Y. (1997). The Methods of Fostering Crat ivity through Mathematical Problem Solving . ZDMInternational Journal on Mathematics Education , Vol. 29(3), p. 86-87. Haylock, D. (1997). Recognising Mathematical Creati vity in Schoolchildren. ZDM - International Journal on Mathematics Education , Vol. 29(3), p. 68-74. Fernandes, D. M. (1994). Utilização da folha de cálculo no 4.º ano de escola ridade: Estudo de uma turma . Dissertação de Mestrado, Universidade do Minho, B raga. Heinze, A., Star, J. R. & Verschaffel, L. (2009). F lexible and adaptive use of strategies and representations in mathematics education. ZDM-International Journal on Mathematics Education , Vol. 41(5), p. 553-540. Jukes, I, e Dosaj, A. (2006). Understanding Digital Children (DKs): Teaching & Le arning in the New Digital Landscape . (Consultado em 09 de Novembro de 2009, disponível em ibo.org/ibap/conference/documents/IanJuk es-UnderstandingDigitalKids.pdf) Karp, A. & Leikin, R. (2009). Mathematical gift and promise: exploring and developing. In M. Tzekaki, M. Kaldrimidou & H. Sakonidis (Eds.), Proceedings of the 33rd Conference of the Internati onal Group for the Psychology of Mathematics Education (pp. 185- 186)(Vol. 1). PME 33. Greece:Thessaloniki. Kenderov, P., Rejali, A., Bussi, M. G. B., Pandelie va,V., Richter, K., Maschieto, M., Kadijevich, D & Taylor, P. (2009). Challenges Beyond the Classroom : Sources and Organizational Issues. In E. J. Barbeau & P. J. Taylor (Eds.), Challenging Mathematics In and Beyond the Classroom . The 16th ICMI Study , pp. 53-96. New York, NY: Springer. Losada, M. F., Yeap, B., Gjone, G. & Pourkazemi, M. H.(2009). Curriculum and Assessment that Provide Challeng in Mathematics. In In J. B. Edward & J. T. Peter (Eds), Challenging Mathematics In and Beyond the Classroom. The 16th ICMI Study , pp. 285-315. New York, NY: Springer. Matos, J. M. & Serrazina, M. L. (1996). Didáctica da Matemática . Lisboa: Universidade Aberta. Moyer, P. S., Niezgoda, D. & Stanley, J. (2005). Yo ung Children’s Use of Virtual Manipulatives and Other Forms of Mathematical Representations. In J. M. William & P. C. Elliot (Eds.), Technology- supported mathematics learning environments: Sixty- seventh yearbook (pp. 17-34). Reston: National Council of Teachers of Mathematics. NCTM (2007). Princípios e Normas para a Matemática Escolar . Lisboa: APM. (Tradução portuguesa de NCTM, 2000). NCTM (1991). Normas para o currículo e a avaliação em matemática escolar . Lisboa: APM e IIE. (Tradução portuguesa de Curriculum and Evaluation S tandards for Teaching Mathematics, 1989). Pehkonen, E. (1997). The State-of-Art in Mathematic al Creativity . ZDM - International Journal on Mathematics Education , Vol. 29(3), p. 63-67. Ponte, J.P. (1997). O Ensino da Matemática na Socie dade da Informação (Editorial). Educação Matemática , Nº 45, pp. 1-2. Ponte, J. P., Branco, N. & Matos, A. (2008). O simb olismo e o desenvolvimento do pensamento algébrico dos alunos. Educação e Matemática , N.º 100, 89-96. Ponte, J. P., Serrazina, L., Guimarães, H. M., Bred a, A., Guimarães, F., Sousa, H., Menezes, L., Marti ns, M. E. G. & Oliveira, P. A. (2007). Programa de Matemática do Ensino Básico . Ministério da Educação, Lisboa. Ponte, J. P. & Serrazina, M. L. (2000). Didáctica da matemática do 1.º ciclo . Lisboa: Universidade Aberta. Powell, A. B., Borge, I. C., Fioriti, G. I., Kondra tieva, M., Koublanova, E. & S Group for the Psychology of Mathematics Education (pp. 185- 186)(Vol. 1). PME 33. Greece:Thessaloniki. Kenderov, P., Rejali, A., Bussi, M. G. B., Pandelie va,V., Richter, K., Maschieto, M., Kadijevich, D & Taylor, P. (2009). Challenges Beyond the Classroom : Sources and Organizational Issues. In E. J. Barbeau & P. J. Taylor (Eds.), Challenging Mathematics In and Beyond the Classroom . The 16th ICMI Study , pp. 53-96. New York, NY: Springer. Losada, M. F., Yeap, B., Gjone, G. & Pourkazemi, M. H.(2009). Curriculum and Assessment that Provide Challeng in Mathematics. In In J. B. Edward & J. T. Peter (Eds), Challenging Mathematics In and Beyond the Classroom. The 16th ICMI Study , pp. 285-315. New York, NY: Springer. Matos, J. M. & Serrazina, M. L. (1996). Didáctica da Matemática . Lisboa: Universidade Aberta. Moyer, P. S., Niezgoda, D. & Stanley, J. (2005). Yo ung Children’s Use of Virtual Manipulatives and Other Forms of Mathematical Representations. In J. M. William & P. C. Elliot (Eds.), Technology- supported mathematics learning environments: Sixty- seventh yearbook (pp. 17-34). Reston: National Council of Teachers of Mathematics. NCTM (2007). Princípios e Normas para a Matemática Escolar . Lisboa: APM. (Tradução portuguesa de NCTM, 2000). NCTM (1991). Normas para o currículo e a avaliação em matemática escolar . Lisboa: APM e IIE. (Tradução portuguesa de Curriculum and Evaluation S tandards for Teaching Mathematics, 1989). Pehkonen, E. (1997). The State-of-Art in Mathematic al Creativity . ZDM - International Journal on Mathematics Education , Vol. 29(3), p. 63-67. Ponte, J.P. (1997). O Ensino da Matemática na Socie dade da Informação (Editorial). Educação Matemática , Nº 45, pp. 1-2. Ponte, J. P., Branco, N. & Matos, A. (2008). O simb olismo e o desenvolvimento do pensamento algébrico dos alunos. Educação e Matemática , N.º 100, 89-96. Ponte, J. P., Serrazina, L., Guimarães, H. M., Bred a, A., Guimarães, F., Sousa, H., Menezes, L., Marti ns, M. E. G. & Oliveira, P. A. (2007). Programa de Matemática do Ensino Básico . Ministério da Educação, Lisboa. Ponte, J. P. & Serrazina, M. L. (2000). Didáctica da matemática do 1.º ciclo . Lisboa: Universidade Aberta. Powell, A. B., Borge, I. C., Fioriti, G. I., Kondra tieva, M., Koublanova, E. & S Group for the Psychology of Mathematics Education (pp. 185- 186)(Vol. 1). PME 33. Greece:Thessaloniki. Kenderov, P., Rejali, A., Bussi, M. G. B., Pandelie va,V., Richter, K., Maschieto, M., Kadijevich, D & Taylor, P. (2009). Challenges Beyond the Classroom : Sources and Organizational Issues. In E. J. Barbeau & P. J. Taylor (Eds.), Challenging Mathematics In and Beyond the Classroom . The 16th ICMI Study , pp. 53-96. New York, NY: Springer. Losada, M. F., Yeap, B., Gjone, G. & Pourkazemi, M. H.(2009). Curriculum and Assessment that Provide Challeng in Mathematics. In In J. B. Edward & J. T. Peter (Eds), Challenging Mathematics In and Beyond the Classroom. The 16th ICMI Study , pp. 285-315. New York, NY: Springer. Matos, J. M. & Serrazina, M. L. (1996). Didáctica da Matemática . Lisboa: Universidade Aberta. Moyer, P. S., Niezgoda, D. & Stanley, J. (2005). Yo ung Children’s Use of Virtual Manipulatives and Other Forms of Mathematical Representations. In J. M. William & P. C. Elliot (Eds.), Technology- supported mathematics learning environments: Sixty- seventh yearbook (pp. 17-34). Reston: National Council of Teachers of Mathematics. NCTM (2007). Princípios e Normas para a Matemática Escolar . Lisboa: APM. (Tradução portuguesa de NCTM, 2000). NCTM (1991). Normas para o currículo e a avaliação em matemática escolar . Lisboa: APM e IIE. (Tradução portuguesa de Curriculum and Evaluation S tandards for Teaching Mathematics, 1989). Pehkonen, E. (1997). The State-of-Art in Mathematic al Creativity . ZDM - International Journal on Mathematics Education , Vol. 29(3), p. 63-67. Ponte, J.P. (1997). O Ensino da Matemática na Socie dade da Informação (Editorial). Educação Matemática , Nº 45, pp. 1-2. Ponte, J. P., Branco, N. & Matos, A. (2008). O simb olismo e o desenvolvimento do pensamento algébrico dos alunos. Educação e Matemática , N.º 100, 89-96. Ponte, J. P., Serrazina, L., Guimarães, H. M., Bred a, A., Guimarães, F., Sousa, H., Menezes, L., Marti ns, M. E. G. & Oliveira, P. A. (2007). Programa de Matemática do Ensino Básico . Ministério da Educação, Lisboa. Ponte, J. P. & Serrazina, M. L. (2000). Didáctica da matemática do 1.º ciclo . Lisboa: Universidade Aberta. Powell, A. B., Borge, I. C., Fioriti, G. I., Kondra tieva, M., Koublanova, E. & S Wells, M.G (2007). Collaborative online projects in a global community. In T. Townsend & R. Bates (Eds.), Handbook of Teacher Education: Globalization Standa rds and Profissionalism in Times of Change (pp. 657-674). The Netherlands: Springer.
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