*Débora Tavares, Isabel Cabrita*

#### Abstract

Despite the importance of creativity, its development has not been a priority of Mathematics, especially in what concerns non-‐gifted students. Students often reveal many difficulties in functional reasoning. Some studies have concluded that an adequate exploration of visual patterns may contribute to overcoming these difficulties. Thus, we developed a qualitative case study aiming to find out if the exploitation of tasks focused on visual patterns contributed to the development of creativity and functional reasoning in 8th grade students. It was found that students improved their performance in tasks whose resolution required the mobilization of functional reasoning and their creativity. Some of the beliefs and/or conceptions about creativity have also evolved positively.

#### Keywords

Creativity, Visual patterns, Algebra, Functional reasoning

#### Full Text:

#### References

Adams, D. Hamm, M. (2010). Demistify Math, Science and Technology: Creativity,

Innovation, and Problem Solving. UK: Rowman & Littlefield Education.

Amit, M. Neria, D. (2008). “Rising the challenge”: using generalization in pattern problems to unearth the algebraic skills of talented pre-‐algebra students. ZDM – The International Journal on Mathematics Education, 40(1), 111-‐129.

Barbosa, A. (2010). A resolução de problemas que envolvem a generalização de padrões em contextos visuais: um estudo longitudinal com alunos do 2o ciclo do ensino básico. Tese de doutoramento. Braga: Universidade do Minho.

Blanton, M. Kaput, J. (2004). Elementary grades students’ capacity for functional thinking. In M. Jonsen Hoines & A. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol.2, pp. 135-‐142). Oslo: PME.

Bogdan, R. Biklen, S. (1994). Investigação qualitativa em Educação: uma introdução à teoria e aos métodos. Porto: Porto Editora.

Conway, K. (1999). Assessing open-‐ended problems. Mathematics Teaching in the Middle School, 4, 8, 510-‐514.

Davis, P. Hersh, R. (1995). A experiência matemática. Lisboa: Gradiva.

Devlin, K. (2002). Matemática: a ciência dos padrões. Porto: Porto Editora.

Doyle, K. (2007). The teacher, the tasks: Their role in students’ mathematical literacy. In J. Watson & K. Beswick (Eds.), Proceedings of the 30th Annual Conference of the Mathematics Research Group of Australasia, 1, 246-‐254.

Lee, L. & Freiman, V. (2006). Developing algebraic thinking through pattern exploration. Mathematics Teaching in the Middle School, 11(9), 428-‐433.

Leikin, R. (2009). Multiple proof tasks: Teacher practice and teacher education. In Proceedings of ICMI Study-‐19: Proofs and proving.

Levenson, E. (2011). Mathematical creativity in elementary school: is it individual or collective? Proceedings of CERME 7, Feb. 2011. University of Rzesków, Poland.

Mann, E. (2005). Mathematical creativity and school Mathematics: Indicators of mathematical creativity in middle schools students. Connecticut, USA: University of Connecticut.

Martinez, M. & Brizuela, B. (2006). A third grader’s way of thinking about linear function tables. Journal of Mathematical Behavior, 25, 285-‐298.

Meissner, H. (2011). Challenges to further creativity in mathematics learning. In M. Avotina, D. Bonka, H. Meissner, L. Ramana, L. Sheffield and E. Velikova (Eds.), Proceedings of the 6th International Conference on Creativity in Mathematics Education and the Education of Gifted Students, 1, 143-‐148.

NCTM (2000). Principals and Standards for School Mathematics. Reston: NCTM. [Tradução portuguesa: Princípios e Normas para a Matemática Escolar. Lisboa, APM, 2007.]

Orton, A. & Orton, J. (1999). Pattern and Approach to Algebra. In A. Orton (Ed.), Pattern in the Teaching and Learning of Mathematics (pp. 104-‐124). Londres: Cassel.

Pehkonen, E. (1997). The State-‐of-‐Art in Mathematical Creativity, ZDM – The International Journal on Mathematics Education, 29, 3, 63-‐67.

Ponte, J. P., Serrazina, L., Guimarães, H., Breda, A., Guimarães, F., Sousa, H., Menezes, L., Martins, M. & Oliveira, P. (2007). Programa de Matemática do Ensino Básico. Lisboa: ME/DGIDC.

Radford, L. (2008). Iconicity and contraction: A semiotic investigation of forms of algebraic generalizations of patterns in different contexts. ZDM – The International Journal on Mathematics Education, 40(1), 83-‐96.

Rivera, F. (2007). Visualizing as a mathematical way of knowing: understanding figural generalization. Mathematics Teacher, 101(1), 69-‐75.

Rivera, F. & Becker, J. (2005). Teacher to teacher: figural and numerical modes of generalizing in algebra. Mathematics Teaching in the Middle School, 11(4), 198-‐203.

Rivera, F. & Becker, J. (2008). Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involving linear and figural patterns. ZDM – The International Journal on Mathematics Education, 40(1), 65-‐82.

Robinson, K. & Aronica, L. (2009). The element: How finding your passion changes everything. New York, NY: Penguin.

Silver, E. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM – The International Journal on Mathematics Education, 3, 75-‐80.

Stacey, K. (1989). Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20(2), 147-‐164.

Stein, M. K. & Smith, M. S. (2009). Tarefas Matemáticas como quadro para reflexão. Educação e Matemática, 105, 22-‐28.

Sternberg, R. & Lubart, T. (1999). The concept of creativity: Prospects and paradigms. In R. J. Sternberg (Org.), Handbook of Creativity (pp. 3-‐15). New York: Cambridge University Press.

Tanisli, D. (2011). Functional thinking ways in relation to linear function tables of elementary school students. Journal of Mathematical Behavior, 30, 206-‐223.

Tavares, D. (2012). Padrões Visuais, raciocínio funcional e criatividade, de Débora Tavares, docente dos ensinos Básico e Secundário. Dissertação de Mestrado. Aveiro: Universidade de Aveiro (http://hdl.handle.net/10773/9929)

Torrance, E. P. (1974). Torrance tests of creative thinking. Bensenville, IL: Scholastic Testing Service.

Vale, I. & Cabrita, I. (2008). Learning through patterns: a powerful approach to algebraic thinking. In K. Kumpulainen& A. Toom (Eds.), ETEN 18 -‐ The Proceedings of the 18th Annual Conference of the European Teacher Education Network (pp.63-‐69). Liverpool, England, april, 2008. (ISBN: 978-‐952-‐10-‐4951-‐4).

Vale, I., Barbosa, A., Barbosa, E., Borralho, A., Cabrita, I., Fonseca, L., & Pimentel, T. (2011). Padrões em Matemática: uma proposta didática no âmbito do Novo Programa para o Ensino Básico. Lisboa: Texto Editores.

Vale, I., Pimentel, T., Cabrita, I., Barbosa, A., & Fonseca, L. (2012). Pattern Problem Solving Tasks as a Mean to Foster Creativity in Mathematics. In T. Y. Tso (Ed.), Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 171-‐178). Taipe, Taiwan, 18-‐22 july,2012. (ISSN: 0771-‐100X).

Warren, E. (2000). Visualisation and development of early understanding in algebra. In T. Nakahara, M. Koyama (Eds.), Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 273-‐280). Hiroshima, Japan.

Warren, E. & Cooper, T. (2008). Patterns That Support Early Algebraic Thinking in the Elementary School. In C. Greenes & R. Rubinstein (Eds.), Algebra and Algebraic Thinking in School Mathematics – Seventieth Yearbook (pp. 113-‐126). Reston: NCTM.

Yin, R. (2010). Estudo de Caso: Planejamento e Métodos. Porto Alegre: Bookman.

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