Creative Potential of Problem Solving and Problem Posing Tasks

Author(s):

Isabel Vale(presenting / submitting)Ana Barbosa(presenting)

Conference:

ECER 2014

Network:

24. Mathematics Education Research

Format:

Paper

Session Information

24 SES 06 B, Issues of Mathematics Learning

Paper Session

Time:

2014-09-03

15:30-17:00

Room:

B215 Sala de Aulas

Chair:

Ole Kristian Bergem

Contribution

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.

Method

We adopted a qualitative approach to understand in what way an experience, grounded on challenging tasks, is a suitable context for promoting creativity and consequently fostering mathematical knowledge. The aim of this study is to identify potentialities and features in the tasks that develop creative thinking, through the perspectives of elementary pre-service teachers’ performance. Our two main research questions are: (i) Do the proposed tasks have creative potential to use in the mathematics classroom?; (ii) How can we characterize creativity when students solve challenging tasks? The participants in this study were twenty elementary mathematics pre-service teachers during the didactics of mathematics classes. The didactical experience was based on theoretical references on creativity, focused on problem solving and posing, consisting in several tasks which required multiple approaches and/or (re)solutions and were designed/selected with the purposes of identifying one of the dimensions of creativity and being challenging, even enjoyable, for the solvers. This didactical experience was developed during two hours two times a week, along a period of three weeks. Data was collected in a holistic, descriptive and interpretive way and included classroom observations, methodological notes and written productions of the students. This experience assumed an exploratory dynamics in class where: (a) problem solving was the work context; (b) students were asked to analyze and discuss the tasks proposed; (c) communication was privileged, particularly the use of questioning and of different representations; and (d) used a set of challenging tasks which raised multiple (re)solutions within problem solving and posing. To measure creativity we followed the basic ideas of some authors (e.g. Conway, 1999; Pinheiro, 2013; Silver, 1997; Vale et al., 2012). To analyze the potential of tasks related to students' creativity in problem posing the same dimensions of problem solving were used: fluency, flexibility and originality. No score was assigned to the students concerning these dimensions instead we made an overall analysis of the work presented, considering the frequency of the most common and the most original responses.

Expected Outcomes

The tasks used are related to problem solving and posing and were selected with the intention to foster creativity of elementary students, beginning with their future teachers. Concerning structure we can say that they promote flexible approaches, in the search for multiple (re)solutions, and the use of different representations. They also involve diverse elementary mathematical concepts and can be applied in different educational contexts, in the 1st and/or 2nd cycles of basic education. We concluded that not all students had the same performance on tasks. The discussions raised in the classroom resulted in a better understanding of some of the more confusing aspects of the tasks. The productions of future teachers throughout the proposed tasks suggested some characteristics of creativity (e.g. multiple representations, multiple ways of solving, formulate different types of problems), which means that the used tasks have some potential to develop those characteristics in students. The most well succeeded tasks were those related with pattern exploration. The productions related to problem posing tasks revealed less creativity by these students. In this sense, there is still a long way to break “some mental sets of knowledge” that a few students reveal, in overcoming some fixations on some contents and structure (Haylock, 1997). We identified this fact mainly in problem posing tasks, where students presented difficulties in thinking divergently. We mostly found motivated students in looking for many and different resolutions, as they were engaged and willing to overcome the barriers they experienced. In this sense, we can assume that the proposed tasks have the potential to promote creative mathematical activity in the classroom. It’s important to find rich and challenging tasks to develop mathematical creativity, but we also must look for adequate strategies to use in class,in order for teachers to be more confident and efficient in teaching.

References

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.

Author Information

Isabel Vale (presenting / submitting)

Polytechnic Institute of Viana do Castelo, Portugal, Portugal

Ana Barbosa (presenting)