Fractions, show your true colors

Nilza Eigenheer Bertoni; Ana Lúcia Braz Dias

doi:10.46551/emd.v9n18a06

Autores

Nilza Eigenheer Bertoni Universidade de Brasília https://orcid.org/0000-0002-4191-9457
Ana Lúcia Braz Dias Central Michigan University https://orcid.org/0000-0003-0674-0758

DOI:

https://doi.org/10.46551/emd.v9n18a06

Palavras-chave:

Fractions, Representations, Reality, Numbers

Resumo

We present fundamental ideas for a proposal for teaching fractions. The theoretical foundation focuses on the critique of the overemphasized use of geometric models, which are commonly found but difficult to correspond into reality. In contrast, we find theoretical support in Realistic Mathematics Education and Freudenthal's developmental research. As part of the research procedures, we proposed to students realistic problem situations involving real whole objects and special parts of them, allowing students to develop solution strategies through reflection and discussion. We observed the emergence of meaningful concepts and strategies; fractions expressing quantities and being logically placed on the number line, with multiple representations for each; and informal understanding of the density of this set of numbers on the number line.

Downloads

Não há dados estatísticos.

Referências

BEHR, Merlyn; LESH, Richard; POST, Thomas; SILVER, Edward. Rational number concepts. In: LESH, Richard; LANDAU, Marsha. (Ed.). Acquisition of mathematics cconcepts and processes. New York: Academic Press, 1983, p. 91-125.

BERTONI, Nilza Eigenheer. A construção do conhecimento sobre número fracionário. Bolema, 21, n. 31, p. 209-237, 2008.

BERTONI, Nilza Eigenheer. Aprendizagem articulada dos conjuntos numéricos: reflexões, relatos e propostas. In: KALEFF, Ana Maria; PEREIRA, Pedro Carlos (Org.). Educação Matemática: diferentes olhares e práticas. Curitiba: Appris, 2020, p. 13-44.

BERTONI, Nilza Eigenheer. Bloco III, Matemática. Módulo 15. Ensino e aprendizagem dos números fracionários misturados aos números naturais. BNCC de professores para professores. Ipê, s.d.

BERTONI, Nilza Eigenheer. Brincar, pensar, fazer: 3ª e 4ª séries. Apostilas fotocopiadas, 1994.

BERTONI, Nilza Eigenheer. Educação e linguagem matemática IV: frações e números fracionários. Brasília: UnB, 2009.

BERTONI, Nilza Eigenheer. Frações: situações aditivas e multiplicativas. In: MENEZES, Mindé Badauy; RAMOS, Wilsa Maria (Org.). Coleção ProInfantil: Módulo 1, Unidade 7. Brasília: MEC/SEB/SED, 2005, p. 33-58.

BERTONI, Nilza Eigenheer. Um novo paradigma no ensino e aprendizagem das frações. In: Anais do VIII Encontro Nacional de Educação Matemática. Recife, 2004, p. 1-15.

BERTONI, Nilza Eigenheer; FIORENTINI, Leda Maria Rangearo. Somando frações no ábaco dos romanos e auxiliando os babilônios em divisões com resto. In: Actas do 1º História e Educação Matemática. Braga 1996, p. 311-312.

BUNT, Lucas Nicolaas Hendrik; JONES, Phillip S; BEDIENT, Jack D. The historical roots of elementary Mathematics. Massachusetts: Courier Corporation, 1988.

CRAMER, Kathleen A.; POST, Thomas R.; DELMAS, Robert C. Initial fraction learning by fourth- and fifth-grade students: a comparison of the effects of using commercial curricula with the effects of using the rational number project curriculum. Journal for Research in Mathematics Education, v. 33, n. 2, p. 111-144, 2002. https://doi.org/10.2307/749646

DANTE, Luiz Roberto; VIANA, Fernando. Ápis Mais: Matemática: 4º ano. São Paulo: Ática, 2021.

FREUDENTHAL, Hans. Major problems of Mathematics Education. Educational Studies in Mathematics, v. 12, n. 2, p. 133-150, 1981.

FREUDENTHAL, Hans. Revisiting Mathematics Education: China lectures. Dordrecht: Kluwer Academic Publishers, 2002.

GABRIEL, Florence; VAN HOOF, Jo; GÓMEZ, David M.; VAN DOOREN, Wim. Obstacles in the development of the understanding of fractions. In: ROBINSON, Katherine M.; DUBÉ, Adam K.; KOTSOPOULOS, Donna. (Ed.). Mathematical cognition and understanding: perspectives on mathematical minds in the Elementary and Middle School years. Cham: Springer, 2023, p. 209-225.

GILLINGS, Richard J. Problems 1 to 6 of the rhind mathematical papyrus. The Mathematics Teacher, 55, n. 1, p. 61-69, 1962.

GRAVEMEIJER, Koeno. Educational development and developmental research in Mathematics Education. Journal for Research in Mathematics Education, 25, n. 5, p. 443-471, 1994. https://doi.org/10.2307/749485

GUNDERSON, Elizabeth A; HAMDAN, Noora; HILDEBRAND, Lindsey; BARTEK, Victoria. Number line unidimensionality is a critical feature for promoting fraction magnitude concepts. Journal of Experimental Child Psychology, v. 187, p. 1-29, 2019. https://doi.org/10.1016/j.jecp.2019.06.010

HACKENBERG, Amy J. Units coordination and the construction of improper fractions: a revision of the splitting hypothesis. The Journal of Mathematical Behavior, 26, n. 1, p. 27-47, 2007. https://doi.org/10.1016/j.jmathb.2007.03.002

HAMDAN, Noora; GUNDERSON, Elizabeth A. The number line is a critical spatial-numerical representation: evidence from a fraction intervention. Developmental Psychology, 53, n. 3, p. 587-596, 2017. https://doi.org/10.1037/dev0000252

HANSEN, Nicole; JORDAN, Nancy C.; RODRIGUES, Jessica. Identifying learning difficulties with fractions: a longitudinal study of student growth from third through sixth grade. Contemporary Educational Psychology, v. 50, p. 45-59, 2017. https://doi.org/10.1016/j.cedpsych.2015.11.002

HEMPHILL, John Knox. Educational development. In: HEMPHILL, John Knox; ROSENAU, Fred. S. (Ed.). Educational development: a new discipline for self-renewal. Eugene: Center for the Advanced Study of Educational Administration, 1970, p. 3-15.

HILTON, Peter. Do we still need fractions in the elementary curriculum? In: Proceedings of the Fourth International Congress on Mathematical Education. Berkeley, 1983, p. 37-41.

JACOB, François. Evolution and tinkering. Science, v. 196, n. 4295, p. 1161-1166, 1977. https://doi.org/10.1126/science.860134

JAHN, Ana Paula; SILVA, Maria José Ferreira; SILVA, Maria Célia Leme; CAMPOS, Tânia Maria Mendonça. Lógica das equivalências. In: Anais da 22ª Reunião Anual da Associação Nacional de Pós-Graduação e Pesquisa em Educação. Caxambu, 1999, p. 1-18.

KERSLAKE, Daphne. Fractions: children's strategies and errors. A report of the strategies and errors in Secondary Mathematics project. Windsor, Berkshire. 1986.

LAMON, Susan J. More! Teaching fractions and ratios for understanding: In-depth discussion and reasoning activities. 3 ed. New York: Routledge, 2012.

LAMON, Susan J. Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. 3 ed. Routledge, 2017.

LANGRALL, Cynthia W. Make this your year to review. Journal for Research in Mathematics Education, 46, n. 1, p. 2-3, 2015. http://dx.doi.org/10.5951/jresematheduc.46.1.0002

LAPPAN, Glenda; FEY, James T.; FITZGERALD, William M.; FRIEL, Susan N.; PHILLIPS, Elizabeth Difanis. Bits and pieces I: understand rational numbers (teacher's guide). Hoboken: Prentice Hall, 2002.

LAPPAN, Glenda; FEY, James T.; FITZGERALD, William M.; FRIEL, Susan N.; PHILLIPS, Elizabeth Difanis. Bits and pieces II: using rational numbers (teacher's guide). Hoboken: Prentice Hall, 2004.

LATHER, Patti; MOSS, Pamela A. Introduction: implications of the scientific research in Education report for qualitative inquiry. Teachers College Record, 107, n. 1, p. 1-3, 2005. http://dx.doi.org/10.1111/j.1467-9620.2005.00450.x

MACK, Nancy K. Confounding whole-number and fraction concepts when building on informal knowledge. Journal for Research in Mathematics Education, 26, n. 5, p. 422-441, 1995. https://doi.org/10.2307/749431

MACK, Nancy K. Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education, 21, n. 1, p. 16-32, 1990.

MUNIZ, Cristiano Alberto, COSTA, Edilene Simões, SILVA, Erondina Barbosa, CARVALHO, Rosália Policarpo Fagundes; BACCARIN, Sandra Aparecida de Oliveira. Professora Nilza Eigenheer Bertoni: sua contribuição para o desenvolvimento da Educação Matemática no Distrito Federal e no Brasil. In: Anais da 32ª Reunião Anual da Associação Nacional de Pós-Graduação e Pesquisa em Educação. Caxambu, 2009, p. 1-15.

NAMKUNG, Jessica; FUCHS, Lynn. Remediating difficulty with fractions for students with Mathematics learning difficulties. Learning Disabilities, v. 24, n. 2, p. 36-48, 2019. https://doi.org/10.18666/LDMJ-2019-V24-I2-9902

OLIVE, John; STEFFE, Leslie P. The construction of an iterative fractional scheme: the case of joe. The Journal of Mathematical Behavior, 20, n. 4, p. 413-437, 2001. https://doi.org/10.1016/S0732-3123(02)00086-X

PIRES, Enam Lima. Meus registros para frações e decimais: entre o que eu penso e o que eu escrevo; entre o que eu escrevo e o que você lê. 2004. 150f. Dissertação (Mestrado em Educação). Universidade de Brasília. Brasília.

RIPOLL, Cydara Cavedon; SIMAS, Fabio Luiz Borges; BORTOLOSSI, Humberto; RANGEL, Letícia; GIRALDO, Victor Augusto; REZENDE, Wanderley; QUINTANEIRO, Wellerson. Frações no Ensino Fundamental. v. 1. Rio de Janeiro: Instituto de Matemática Pura e Aplicada, 2016.

SANTOS, Yanna Karla de Oliveira. Matemática: por que uns gostam e outros não? 2006. Trabalho de Conclusão de Curso (Pedagogia). Universidade de Brasília. Brasília.

SCHUTZ, Richard Edward. The nature of educational development. Journal of Research and Development in Education, v. 3, n. 2, p. 39-64, 1970.

SIDNEY, Pooja G.; THOMPSON, Clarissa A.; RIVERA, Ferdinand D. Number lines, but not area models, support children’s accuracy and conceptual models of fraction division. Contemporary Educational Psychology, 58, p. 288-298, 2019. https://doi.org/10.1016/j.cedpsych.2019.03.011

SILVEIRA, Ênio. Desafio Matemática, 4º ano, Manual do Professor. São Paulo: Moderna, 2021.

SINGHA, Parmjit; HOONA, Teoh Sian; NASIRA, Nurul Akmal Md; HANA, Cheong Tau; CHEONG, Tau Han. Obstacles faced by students in making sense of fractions. The European Journal of Social and Behavioural Sciences, 30, n. 1, p. 34-51, 2021. http://dx.doi.org/10.15405/ejsbs.287

STEFFE, Leslie P. A new hypothesis concerning children’s fractional knowledge. The Journal of Mathematical Behavior, 20, n. 3, p. 267-307, 2001. https://doi.org/10.1016/S0732-3123(02)00075-5

STREEFLAND, Leen. Fractions in Realistic Mathematics Education: a paradigm of developmental research. Dordrecht: Kluwer Academic Publishers, 1991.

TIAN, Jing; BARTEK, Victoria; RAHMAN, Maya Z; GUNDERSON, Elizabeth A. Learning improper fractions with the number line and the area model. Journal of Cognition and Development, 22, n. 2, p. 305-327, 2021. http://dx.doi.org/10.1080/15248372.2021.1890603

VAN DEN HEUVEL-PANHUIZEN, Marja. Reform under attack: forty years of working on better mathematics education thrown on the scrapheap? No way! In: Proceedings of the 33rd Annual Conference of the Mathematics Education Research Group of Australasia. Fremantle, 2010, p. 1-25.

VERGNAUD, Gérard. A comprehensive theory of representation for Mathematics Education. The Journal of Mathematical Behavior, 17, n. 2, p. 167-181, 1998. https://doi.org/10.1016/S0364-0213(99)80057-3