24 SES 06 B, Issues of Mathematics Learning
Nowadays we are experiencing deep changes in different areas of society, in particular in mathematics education. So mathematics educators have a great challenge to face, mainly in developing higher order thinking skills in students such as posing and solving problems, reasoning and communication. It’s also important to analyze what features of teaching and learning are associated with better performance in mathematics by improving the quality of the main agents of change: teachers. These factors imply innovative strategies to improve teaching and learning, in particular tasks and resources that call for independent and critical problem solvers, curiosity and creativity, in order to develop mathematical knowledge and citizenship. Recognizing the central role of exploratory tasks in the development of mathematical knowledge and ability of students, it’s necessary to involve teachers in their selection and planning in order to acquire a deeper awareness of their effectiveness and educational value. Thus, great attention is necessary in supporting (future) teachers in the construction and refinement of these tasks. In fact, the tasks used in the classroom provide the starting point for the mathematical activity of students, since their nature influences the type of work developed. Research has showed that what students learn is largely influenced by the tasks given to them (Doyle, 1988; Stein & Smith, 2009; Vale, 2009). Among the different tasks that we use in mathematics classes, problem solving plays an important role in the learners’ competences, involving rich discussions, being considered cognitively challenging and are the primary mechanism for promoting conceptual understanding of mathematics (Stein & Smith, 2009). Problem solving tasks must be revisited through new approaches, encouraging students to be persistent and look for creative ideas in order to increase the flow of mathematical ideas, flexibility of thought and originality in the responses. Mathematics is naturally engaging, useful, and creative, and challenging tasks usually require creative thinking. Creativity can be considered as the ability to produce new ideas, approaches or actions. When we think in students’ mathematical creativity this can be characterized by three components: fluency (ability to generate a great number of ideas and refers to the continuity of those ideas, flow of associations, and use of basic knowledge), flexibility (ability to produce different categories or perceptions whereby there is a variety of different ideas about the same problem or thing) and originality (ability to create fresh, unique, unusual, totally new, or extremely different ideas or products) (e.g. Leikin, 2009; Silver, 1997; Vale et al., 2012). Research findings show that mathematical problem solving and problem posing are closely related to creativity (e.g. English, 1997; Leikin, 2009; Pehkonen, 1997; Silver, 1997). Tasks that can promote fluency, flexibility, and originality must be open-ended and ill structured, assuming the form of problem solving, problem posing, mathematical explorations and investigations. Rather than closed problems with a single solution, students should be provided with open-ended problems with a range of alternative solution methods (Mann, 2006). In this context tasks related to posing and solving problems help develop new approaches and creative ideas, as well as provide multiple (re)solutions, raising the flow of mathematical ideas, flexibility of thought and originality in the responses. Teachers must encourage students to create, share and solve (their own) problems, as this is a very rich learning environment for the development of their ability to solve problems as well as their mathematical knowledge. This way, future teachers should themselves develop these skills and go through the same type of tasks that they will offer their students.
Conway, K. (1999). Assessing open-ended problems. Mathematics Teaching in the Middle School, 4, 8, 510-514. Doyle, W. (1988). Work in mathematics classes: The context of students’ thinking during instruction. Educational Psychologist, 23, 167-80. English, L.D. (1997). The development of 5th grade students problem-posing abilities, Educational Studies in Mathematics, 34, 183-217. Haylock, D. (1997). Recognizing mathematical creativity in schoolchildren. International Reviews on Mathematical Education, Essence of Mathematics, 29(3), 68–74. Retrieved March 10, 2003, from http://www.fizkarlsruhe.de/fix/publications/zdm/adm97 Leikin, R. (2009). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman and B. Koichu (Eds.), Creativity in mathematics and the education of gifted students. (pp. 129-145). Rotterdam, Netherlands: Sense Publishers. Mann, E. (2006). Creativity: The Essence of Mathematics. Journal for the Education of the Gifted, 30(2), p. 236-260. Pehkonen, E. (1997). The State-of-Art in Mathematical Creativity, International Reviews on Mathematical Education, Essence of Mathematics, 29(3), 63–67. Retrieved March 10, 2013, from http://www.fizkarlsruhe.de/fix/publications/zdm/adm97 Pinheiro, S. (2013). A criatividade na resolução e formulação de problemas: Uma experiência didática numa turma de 5º ano de escolaridade (Tese de mestrado não publicada). Viana do Castelo: Escola Superior de Educação do Instituto Politécnico de Viana do Castelo. Silver, E. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM, 3, 75-80. Stein, M., & Smith, M. (2009). Tarefas Matemáticas como quadro para a reflexão. Educação e Matemática, 22-28. Vale, I. (2009). Das tarefas com padrões visuais à generalização. XX SIEM. Em J. Fernandes, H. Martinho & F. Viseu (Org.). Actas do Seminário de Investigação Matemática (pp. 35-63). Viana do Castelo: APM. Vale, I., Pimentel, T., Cabrita, I., Barbosa, A. & Fonseca, L. (2012). Pattern problem solving tasks as a mean to foster creativity in mathematics. Tso, T. Y. (Ed), Opportunities to learn in mathematics education, Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education, vol. 4, pp. 171-178. Taipei, Taiwan: PME.
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