Knowing and teaching fractions: conceptions and practices of elementary school teachers
DOI:
https://doi.org/10.1590/S1678-4634202349261007Keywords:
Teaching fractions, Teachers’ conceptions, Didactic knowledge, Elementary schoolAbstract
A teacher’s knowledge determines the quality of his or her practices. However, a teacher is not always able to put her or his ideas into practice. Concerning rational numbers, it is recognized that this is an important topic, a difficult one to teach, though. Often, we see a misalignment between teachers’ ideas about fractions and their ideas about how to teach this topic. This paper analyses elementary teachers’ conceptions and ideas about practices regarding fractions, whose goal is to answer the following questions: 1) What ideas do teachers have about fractions? 2) How do they understand their teaching should be? For this purpose, thirty-one (31) Portuguese elementary teachers from public schools, with several years of teaching experience, were interviewed. An individual semi-structured interview was conducted, seeking to explore concepts and properties of fractions as well as problem-solving involving fractions. The results obtained revealed teachers’ weaknesses in mathematical knowledge associated with the concept of fraction, namely, in the domain of the different meanings of fraction, as well as in the interpretation of the different ways of representing fractions. Regarding problem-solving related to fractions, teachers’ weaknesses were also identified, particularly in situations involving discrete quantities and in marking fractional numbers on the number line. These results seem to appeal to an ideal of teacher training capable of continuously supporting and updating teaching practices.
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Ball, Deborah; Hill, Heather; Bass, Hyman. Knowing mathematics for teaching. Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, Washington, D.C., v. 29, n. 3, p. 14-46, 2005.
Ball, Deborah; Lubienski, Sarah; Mewborn, Denise. Research on teaching mathematics: the unsolved problem of teachers’ mathematical knowledge. In: Richardson, Virginia (ed.). Handbook of research on teaching. New York: Macmillan, 2001. p. 433-456.
Ball, Deborah; Thames, Mark; Phelps, Geoffrey. Content knowledge for teaching: what makes it special? Journal of Teacher Education. Thousando Oaks, v. 59, n. 5, p. 389-407, 2008.
Behr, Merlyn; Harel, Guershon; Post, Thomas; Lesh, Richard. Rational number, ratio, and proportion. In: Grouws, Douglas (ed.). Handbook of research on mathematics teaching and learning. New York: MacMillan, 1992. p. 296-333.
Behr, Merlyn et al. Rational-number concepts. In: Lesh, Richard; Landau, Marsha (ed.). Acquisition of mathematics concepts and processes. New York: Academic Press, 1983. p. 92-127.
Bogdan, Robert; Biklen, Sari. Investigação qualitativa em educação: uma introdução à teoria e aos métodos. Porto: Porto Editora, 2010.
BORN, Bárbara; PRADO, Ana; FELIPPE, Janaína. Profissionalismo docente e estratégias para o seu fortalecimento: entrevista com Lee Shulman. Educação e Pesquisa, São Paulo, v. 45, 2019. https://doi.org/10.1590/S1678-4634201945002003
» https://doi.org/10.1590/S1678-4634201945002003
CARDOSO, Paula. Ensinar frações: um olhar sobre o conhecimento de professores do 1º. ciclo do ensino básico. 2017. Tese (Doutoramento em Ciências da Educação - Educação Matemática) – Instituto de Educação, Universidade do Minho, Braga, 2017.
CARDOSO, Paula; MAMEDE, Ema. As interpretações de fracção na construção do conceito: um estudo com alunos do 6º. ano do ensino básico. In: Seminário de Investigação em Educação Matemática, 20., 2009, Viana do Castelo. Actas […]. Viana do Castelo: CIEd – Universidade do Minho, 2009. p. 452-461.
CARDOSO, Paula; MAMEDE, Ema. Ensinar frações nos primeiros anos de escolaridade. In: MAMEDE, Ema; PINTO, Hélia; MONTEIRO, Cecília (ed.). Contributos para o desenvolvimento do sentido de número racional. Lisboa: Associação de Professores de Matemática, 2021. p. 247-268.
COPUR-GENCTURK, Yasemin. Teachers’ conceptual understanding of fraction operations: results from a national sample of elementary school teachers. Educational Studies in Mathematics, Cham, v. 107, p. 525-545, 2021.
DGE. Direção Geral de Educação. Programa de matemática do ensino básico. Lisboa: Ministério da Educação e Ciência, 2013. Disponível em: http://dge.mec.pt/metascurriculares/index.php?s=directorio&pid=17 Acesso em: 25 fev. 2015.
» http://dge.mec.pt/metascurriculares/index.php?s=directorio&pid=17
DGE. Direção Geral de Educação. Aprendizagens essenciais. Lisboa: Ministério da Educação, 2018. Disponível em: https://www.dge.mec.pt/sites/default/files/Curriculo/Aprendizagens_Essenciais/1_ciclo/aemat_1a_2021_.pdf Acesso em: 22 set. 2019.
ERICKSON, Frederick. Qualitative methods in research on teaching. In: WITTROCK, Merlin (ed.), Handbook of research on teaching. 3. ed. New York: Macmillan, 1986. p. 119-161.
GUBA, Egon; LINCOLN, Yvonna. Competing paradigms in qualitative research. In: DENZIN, Norman; LINCOLN, Yvonna (ed.). Handbook of qualitative research. London: Sage, 1994. p. 105-117.
Hart, Kathleen. Fractions. In: Hart, Kathleen (ed.). Children’s understanding of mathematics: 11-16. London: John Murray, 1981. p. 66-81.
Kieren, Thomas. Fractional Numbers: from quotient fields to recursive understanding. In: Carpenter, Thomas; Fennema, Elizabeth; Romberg, Thomas (ed.). Rational numbers: An integration of research. Hillsdale: Erlbaum, 1993. p. 49-84.
Kieren, Thomas. On the mathematical, cognitive and instructional foundations of rational numbers. In: Lesh, Richard (ed.). Number and measurement: paper from a research workshop. Columbus: ERIC/Smeac, 1976. p. 101-144.
LAMON, Susan. Rational numbers and proportional reasoning toward a theoretical framework for research. In: LESTER, Frank (ed.). Second handbook of research and learning. Charlotte: NCTM, 2007. p. 629-667.
LAMON, Susan. Teaching fractions and ratios for understanding: essential content knowledge and instructional strategies for teachers. 2. ed. Mahwah: Lawrence Erlbaum, 2006.
Li, Yeping; Zhang, Huirong; Song, Naiqing. In-service elementary teachers’ knowledge in mathematics and pedagogy for teaching: the case of fraction division. In: Conference of the International Group for the Psychology of Mathematics Education, 44., 2021, Khon Kaen, Thailand. Proceedings […]. Khon Kaen: PME, 2021. p. 226-235.
Mack, Nancy. Building on informal knowledge through instruction in a complex content domain: partitioning, units, and understanding multiplication of fractions. Journal for Research in Mathematics Education, Reston, v. 32, p. 267-295, 2001.
MAMEDE, Ema. The effects of situations on children’s understanding of fractions. 2007. Tese (PhD in Psychology) – School of Social Sciences and Law, Oxford Brookes University, Oxford, 2007.
MAMEDE, Ema; PINTO, Hélia. Pre-service elementary school teachers’ ideas about fractions. Quaderni de Ricerca In Didattica - Mathematics, Palermo, v. 27, n. 2, p. 257-260, 2017.
MAMEDE, Ema; RIBEIRO, Carlos; PINTO, Hélia. Conhecimento de futuros professores do ensino básico sobre frações. In: MAMEDE, Ema; PINTO, Hélia; MONTEIRO, Cecília (ed.). Contributos para o desenvolvimento do sentido de número racional. Lisboa: Associação de Professores de Matemática, 2021. p. 205-219.
MERRIAM, Sharan. Qualitative research: A guide to design and implementation. 2. ed. San Francisco: Jossey-Bass, 2009.
NCTM. National Council of Teachers of Mathematics. Princípios para a ação: assegurar a todos o sucesso em matemática. Lisboa: APM, 2017.
NUNES, Terezinha; BRYANT, Peter. Rational numbers and intensive quantities: challenges and insights to pupils’ implicit knowledge. Anales de Psicología, Murcia, v. 24, n. 2, p. 262-270, 2008.
Nunes, Terezinha et al. Vergnaud’s definition of concepts as a framework for research and teaching. Paris: [S .n.], 2004. Paper presented at the Annual Meeting for the Association pour la Recherche sur le Développement des Compétences, Paris, jan. 2004.
OLANOFF, Dana; LO, Jane-Jane; TOBIAS, Jennifer. Mathematical content knowledge for teaching elementary mathematics: A focus on fractions. The Mathematics Enthusiast, Missoula, v.11, n. 2, p. 267-310, 2014.
Pinto, Hélia; Ribeiro, Carlos. Conhecimento e formação de futuros professores dos primeiros anos: o sentido de número racional. Da Investigação às Práticas, Lisboa, v. 3, n. 1, p. 80-98, 2013.
Post, Thomas et al. Intermediate teachers’ knowledge of rational number concepts. In: Fennema, Elizabeth; Carpenter, Thomas; Lamon, Susan (ed.). Integrating research on teaching and learning mathematics. New York: State University of New York Press, 1991. p. 177-198.
RIBEIRO, Carlos. Abordagem aos números decimais e suas operações: a importância de uma eficaz navegação entre representações. Educação e Pesquisa, São Paulo, v. 37, n. 2, p. 407-422, 2011. https://doi.org/10.1590/S1517-97022011000200013
» https://doi.org/10.1590/S1517-97022011000200013
SIEGLER, Robert; LORTIE-FORGUES, Hugues. Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology, Washington, D.C., v. 107, n. 3, p. 909-918, 2015.
Shulman, Lee. Those who understand: knowledge growth in teaching. Educational Researcher, Washington, D.C., v. 15, n. 2, p. 4-14, 1986.
Streefland, Leen. Fractions in realistic mathematics education: A paradigm of developmental research. Norwell,: Kluwer, 1991.
Tirosh, Dina et al. Prospective elementary teachers’ conceptions of rational numbers. Jerusalem: [S. n.], 1998. Disponível em: http://jwilson.coe.uga.edu/Texts.Folder/tirosh/Pros.El.Tchrs.html Acesso em: 15 nov. 2020.
» http://jwilson.coe.uga.edu/Texts.Folder/tirosh/Pros.El.Tchrs.html
VERGNAUD, Gérard. The nature of mathematical concepts. In: NUNES, Terezinha; BRYANT, Peter (ed.). Learning and teaching mathematics: An international perspective. East Sussex: Psychology Press, 1997. p. 5-28.
YIN, Robert K. Case study research: design and methods. Los Angeles: Sage, 2014.
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