Uma trajetória na aprendizagem dos números racionais através da percentagem
A trajectory in the learning of rational numbers enhanced by percentage

Helena Gil Guerreiro, Lurdes Serrazina, João Pedro da Ponte

Resumo


Este artigo tem como objetivo indicar os contributos que uma trajetória com um foco inicial na percentagem, que faz emergir de seguida o numeral decimal e posteriormente a fração, traz para a compreensão da natureza relacional dos números racionais. Trata-se de uma investigação baseada em design na modalidade de experiência de ensino na sala de aula. A recolha de dados resultou da observação participante, apoiada num diário de bordo, de gravações áudio e vídeo e da recolha das produções escritas dos alunos. Os resultados revelam que esta abordagem, partindo da percentagem, permite integrar os conhecimentos numéricos prévios intuitivos dos alunos na compreensão dos números racionais e apoia a construção de uma aprendizagem das diferentes representações, de forma interrelacionada, numa perspetiva de desenvolvimento de sentido de número.


This article aims to figure out the contributions that a trajectory with an initial focus on percentage which leads to the emergence of decimals and later of fractions brings to the understanding of the relational nature of rational numbers. A classroom teaching experiment was developed as a design based research. Data were collected through participant observation, supported by a logbook, audio- and video-recorded lessons and collecting students’ written productions. The results show that this approach, based on percentage, allows integrating students' previous intuitive numerical knowledge into the understanding of rational numbers and supports the construction of learning of the different representations, in an interrelated way, in a perspective of the development of number sense. 


Palavras-chave


percentagem, aprendizagem, sentido de número, grandeza numérica, números racionais

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Referências


ABRANTES, P.; SERRAZINA, L.; OLIVEIRA, I. A matemática na educação básica. 1.ª ed. Lisboa: ME–DEB, 1999. 113 p.

BERCH, D.B. Making sense of number sense: Implications for children with mathematical disabilities. Journal of Learning Disabilities, California, v. 38, n. 4, p. 333-339, jul. 2005.

BROWN, B. The relational nature of rational numbers. Pythagoras, Centurion, v. 36, n. 1, p. 1-8, jan. 2015.

COBB, P. et al. Design experiments in educational research. Educational Researcher, Washington, v. 32, n. 1, p. 9-13. Jan.2003.

COBB, P.; JACKSON, K.; DUNLAP, C. Design research: an analysis and critique. In: ENGLISH, L. D.; KIRSHNER, D. (Eds.). Handbook of international research in mathematics education. New York: Routledge, 2016. p. 481-503.

COMMON CORE STATE STANDARDS INITIATIVE. Common core state standards for mathematics. Washington, 2010. 93 p. Disponível em: http://www.corestandards.org/Math/. Acesso a: 15 nov. 2017.

CONFREY, J.; LACHANCE, A. Transformative teaching experiments through conjecture-driven research design. In: KELLY, A.; LESH, R. (Eds.). Handbook of research design in mathematics and science education. Mahwah: Lawrence Erlbaum Associates, 2000. p. 231-266.

CONFREY, J.; MALONEY, A. The construction, refinement, and early validation of the equipartitioning learning trajectory. In: INTERNATIONAL CONFERENCE OF THE LEARNING SCIENCES, 9, 2010, Chicago. Proceedings… Chicago: International Society of the Learning Sciences, 2010. p. 968-975.

FREUDENTHAL, H. Revisiting mathematics education: China lectures. Dordrecht: Kluwer, 1991. 199 p.

GALEN, F. et al. Fractions, percentages, decimals and proportions: a learning-teaching trajectory for grade 4, 5 and 6. 1.ª. ed. Rotterdam: Sense, 2008. 163 p.

GOETZ, J. P.; LECOMPTE, M. D. Ethnography and qualitative design in educational research. 2.ª ed. New York: Academic, 1984, 292 p.

GRAVEMEIJER, K.; VAN EERDE, D. Design research as a means for building a knowledge base for teachers and teaching in mathematics education. The Elementary School Journal, Chicago, v. 109, n. 5, p. 510-524. may 2009.

HOWDEN, H. Teaching number sense. Arithmetic Teacher, Reston, v. 36, n. 6, p. 6-11, feb. 1989.

INSTITUTO DE EDUCAÇÃO DA UNIVERSIDADE DE LISBOA. Carta ética para a investigação em educação e formação. Lisboa, 2016. 2 p. Disponível em: Acesso a: 15 nov. 2017.

LAMON, S.J. Rational numbers and proportional reasoning: Toward a theoretical framework for research. In: Lester, F. (Ed.) Second handbook of research on mathematics teaching and learning. Reston: NCTM, 2007, v. 1, p. 629-667.

LAMON, S.J. Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. 3.ª ed. New York: Routledge, 2012, 254 p.

MARKOVITS, Z.; SOWDER, J. Developing number sense: An intervention study in grade 7. Journal for Research in Mathematics Education, Reston, v. 25, n. 1, p. 4-29, jan. 1994.

MCINTOSH, A.; REYS, J.; REYS, E. A proposed framework for examining basic number sense. For the Learning of Mathematics, White Rock, v. 12, n. 3, p. 2-8 e 44, nov. 1992.

MIDDLETON, J. A.; VAN DEN HEUVEL-PANHUIZEN, M.; SHEW, J. A. Using bar representations as a model for connecting concepts of rational number. Mathematics Teaching in the Middle School, Reston, v. 3, n. 4, p. 302-312, jan. 1998.

MOSS, J. Percents and Proportion at the Center: Altering the Teaching Sequence for Rational Number. In: LITWILLER, B. (Ed.). Making sense of fractions, ratios, and proportions 2002 Yearbook. Reston: NCTM, 2002. p. 109-120.

MOSS, J.; CASE, R. Developing children’s understanding of the rational numbers: a new model and an experimental curriculum. Journal for Research in Mathematics Education, Reston, v. 30, n. 2, p. 122-147, mar.1999.

NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS. Principles to actions: ensuring mathematical success for all. 1.ª ed. Reston: NCTM, 2014. 140 p.

PONTE, J. P. et al. Investigação baseada em design para compreender e melhorar as práticas educativas. Quadrante, Lisboa, v. XXV, n. 2, p. 77-98, 2.º semestre. 2016.

PONTE, J. P.; SERRAZINA, M. L. Didáctica da Matemática do 1.º ciclo. 1.ª ed. Lisboa: Universidade Aberta, 2000. 257 p.

PREDIGER, S. Focussing structural relations in the bar board – a design research study for fostering all students’ conceptual understanding of fractions. In: EIGHT EUROPEAN CONFERENCE OF RESEARCH IN MATHEMATICS EDUCATION, 8, 2013, Ankara. Proceedings… Ankara: ERME, 2013. p. 343-352.

PREDIGER, S.; GRAVEMEIJER, K.; CONFREY, J. Design research with a focus on learning processes: an overview on achievements and challenges. ZDM, Berlim, v. 47, n. 6, p. 877-891. oct. 2015.

SIEGLER, R. S. et al. Fractions: the new frontier for theories of numerical development. Trends in Cognitive Sciences, Cambridge, v. 17, n. 1, jan. 2013.

SIEGLER, R.S.; THOMPSON, C.A.; SCHNEIDER, M. An integrated theory of whole number and fractions development. Cognitive Psychology, Amsterdam, v. 62, n. 4, p. 273-296, jun. 2011.

SIMON, M. Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, Reston, v. 26, n. 2, p. 114-145, mar. 1995.

SOWDER, J.; SCHAPPELLE, B. Number sense-making. Arithmetic Teacher, Reston, v. 41, n.6, p. 342-346. Feb. 1994.

STEPHAN, M.; CLEMENTS, D.H. Linear and area measurement in prekindergarten to grade 2. In D. H. CLEMENTS (Ed.), Learning and teaching measurement: 65th Yearbook, Reston: NCTM, 2003. p. 3-16.

TIAN, J.; SIEGLER, R.S. Which Type of Rational Numbers Should Students Learn First?. Educational Psychology Review, USA, p. 1-22, jul. 2017.

VAMVAKOUSSI, X.; VOSNIADOU, S. Understanding the structure of the set of rational numbers: A conceptual change approach. Learning and Instruction, Cambridge, 2004, v. 14, n. 5, p. 453-467, oct. 2004.

VAN DEN HEUVEL-PANHUIZEN, M.; DRIJVERS, P. (2014). Realistic mathematics education. In: S. Lerman (Ed.), Encyclopedia of mathematics education. Netherlands, 2014. p. 521-525.




DOI: http://dx.doi.org/10.23925/1983-3156.2018v20i1p359-384

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