Measuring perspective of fraction knowledge: integrating historical and neurocognitive findings

Referencias

ALEKSANDROV, A. D. A general view of mathematics. In: ALEKSANDROV, A. D.; KOLMOGOROV, A. N.; LAVRENT’EV, M. A. (Org.). Mathematics: Its content, methods, and meaning. Cambridge, MA: Massachusetts Institute of Technology, 1963. v.1 p. 1-64. ALQAHTANI, M. M.; POWELL, A. B. Investigating how a measurement perspective changes pre-service teachers’ interpretations of fractions. In: HODGES, T. E.; ROY, G. J.; TYMINSKI, A. M. (Org.). Proceedings of the 40th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Greenville, SC: University of South Carolina & Clemson University., 2018. p. 484-487. BAILEY, D. H. et al. Competence with fractions predicts gains in mathematics achievement. Journal of Experimental Child Psychology, v. 113, n. 3, p. 447-455, 11//. 2012. Disponível em: . BEHR, M. J. et al. Rational number, ratio and proportion. In: GROWS, D. A. (Org.). Handbook on research on mathematics teaching and learning. New York: Macmillan, 1992. p. 296-333. BEHR, M. J. et al. Rational number concepts. In: LESH, R.; LANDAU, M. (Org.). Acquisition of mathematical concepts and processes. New York: Academic, 1983. p. 91- 125. BEHR, M. J. et al. Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, v. 5, n. 5, p. 321-341, 1984. BOBOS, G.; SIERPINSKA, A. Measurement approach to teaching fractions: A design experiment in a pre-service course for elementary teachers. International Journal for Mathematics Teaching and Learning, v. 18, n. 2, p. 203-239, 2017. BOOTH, J. L.; NEWTON, K. J. Fractions: Could they really be the gatekeeper’s doorman? Contemporary Educational Psychology, v. 37, n. 4, p. 247-253, 10//. 2012. Disponível em:. BROUSSEAU, G.; BROUSSEAU, N.; WARFIELD, V. Rationals and decimals as required in the school curriculum: Part 1: Rationals as measurement. The Journal of Mathematical Behavior, v. 23, n. 1, p. 1-20, //. 2004. Disponível em: . CAMPOS, T. M. M.; RODRIGUES, W. R. A idéia de unidade na construção do conceito do número racional. REVEMAT - Revista Eletrônica de Educação Matemática, v. 2, n. 4, p. 68-93, 2007. CARAÇA, B. D. J. Conceitos fundamentais da Matemática. Lisboa: Tipografia Matemática, 1951. CLAWSON, C. C. The mathematical traveler: Exploring the grand history of numbers. Cambrige, MA: Perseus, 1994/2003. COURANT, R.; ROBBINS, H. What is mathematics?: An elementary approach to ideas and methods. 2nd revised by Ian Steward,. ed. Oxford: New York, 1941/1996. CUISENAIRE, G.; GATTEGNO, C. Numbers in colour: A new method of teaching the process of arithmetic to all level of the Primary School. London: Hienemann, 1954. DAVYDOV, V. V.; TSVETKOVICH, Z. H. On the objective origin of the concept of fractions. Focus on Learning Problems in Mathematics, v. 13, n. 1, p. 13-64, 1991. DOUGHERTY, B. J.; VENENCIANO, L. C. H. Measure Up for Understanding. Teaching Children Mathematics, v. 13, n. 9, p. 452-456, 2007. Disponível em: . DUFFY, S.; HUTTENLOCHER, J.; LEVINE, S. It is all relative: How young children encode extent. Journal of Cognition and Development, v. 6, n. 1, p. 51-63, 2005. ESCOLANO VIZCARRA, R.; GAIRÍN SALLÁN, J. M. Modelos de medida para la enseñanza del número racional en Educación Primaria. Unión. Revista Iberoamericana de Educación Matemática, n. 1, p. 17-35, 2005. FLAGG, G. Numbers: Their history and meaning. Mineola, NY: Dover, 1983. GATTEGNO, C. Gattegno mathematics text-book 1: Qualitative arithmetic, The study of numbers from 1 to 20. New York: Educational Solutions, 1970a. ________. Gattegno mathematics text-book 2: Study of numbers up to 1,000; The four operations. New York: Educational Solutions, 1970b. ________. Gattegno mathematics text-book 4: Fractions, decimals, and percentages. New York: Educational Solutions, 1970c. ________. What we owe children: The subordination of teaching to learning. New York: Avon, 1970d. ________. The science of education: Part 1: Theoretical considerations. New York: Educational Solutions, 1987. ________. The science of education: Part 2B: The awareness of mathematization. New York: Educational Solutions, 1988. GATTEGNO, G. C.; HOFFMAN, M. R. Handbook of activities for the teaching of mathematics at the elementary school. New York: Human Education, 1976. GILLINGS, R. J. Mathematics in the time of the pharaohs. Mineola, New York: Dover, 1972/1982. GÓMEZ, D. M. et al. The effect of inhibitory control on general mathematics achievement and fraction comparison in middle school children. ZDM Mathematics Education, v. 47, n. 5, p. 801-811, 2015. Disponível em: . ISCHEBECK, A. K.; SCHOCKE, M.; DELAZER, M. The processing and representation of fractions within the brain. An fMRI investigation. NeuroImage, v. 47, p. 403-413, 2009/07/01/. 2009. Disponível em: . JACOB, S. N.; NIEDER, A. Notation-Independent Representation of Fractions in the Human Parietal Cortex. Journal of Neuroscience, v. 29, n. 14, p. 4652-4657, 2009a. Disponível em: . ________ . Tuning to non-symbolic proportions in the human frontoparietal cortex. European Journal of Neuroscience, v. 30, n. 7, p. 1432-1442, 2009b. Disponível em: . KERSLAKE, D. Fractions: Children’s strategies and errors. Windsor, England: NEER- NELSON, 1986. KIEREN, T. E. On the mathematical, cognitive and instructional foundations of rational number. In: LESH, R. A. (Org.). Number and measurement. Columbus, Ohio: ERIC Clearinghouse for Science, Mathematics, and Environmental Education, 1976. p. 101-144. ________. Personal knowledge of rational numbers: Its intuitive and formal development. In: HEIBERT, J.; BEHR , M. J. (Org.). Number concepts and operations in the middle grades. Reston, VA: Lawrence Erlbaum, 1988. p. 49-84. LAMON, S. J. Presenting and representing: From fractions to rational numbers. In: CUOCO, A.; CURCIO, F. (Org.). The Roles of Representations in School Mathematics - 2001 Yearbook. Reston: National Council of Teachers of Mathematics, 2001. p. 146- 168. ________. Rational numbers and proportional reasoning: Toward a theoretical framework for research. In: LESTER, J. F. K. (Org.). Second handbook of research on mathematics teaching and learning: A Project of the National Council of Teachers of Mathematics. Charlotte, NC: Information Age, 2007. p. 629-667. ________. Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. 3rd. ed. New York: Routledge, 2012. LEWIS, M. R.; MATTHEWS, P. M.; HUBBARD, E. M. Neurocognitive architectures and the nonsymbolic foundations of fractions understanding. In: BERCH, D. B.; GEARY, D. C.; KOEPKE, K. M. (Org.). Development of mathematical cognition: Neural substrates and genetic influences. San Diego, CA: Academic Press., 2015. p. 141–160. LIN, C.-Y. et al. Preservice Teachers’ Conceptual and Procedural Knowledge of Fraction Operations: A Comparative Study of the United States and Taiwan. School Science and Mathematics, v. 113, n. 1, p. 41-51, 01/01/. 2013. MAHER, C. A.; YANKELEWITZ, D., Eds. Children’s reasoning while building fraction ideas. Boston: Senseed. 2017. MATTHEWS, P. G.; CHESNEY, D. L. Fractions as percepts? Exploring cross-format distance effects for fractional magnitudes. Cognitive Psychology, v. 78, p. 28-56, 2015. MATTHEWS, P. G.; ELLIS, A. B. Natural alternatives to natural number: The case of ratio. Journal of Numerical Cognition, v. 4, n. 1, p. 19-58, 2018. MATTHEWS, P. G.; ZIOLS, R. What’s perception got to do with it? Re-framing foundations for rational number concepts. In: NORTON, A.; ALIBALI, M. W. (Org.). Constructing Number: Merging Perspectives from Psychology and Mathematics Education. Cham: Springer International Publishing, 2019. p. 213-235. MCCRINK, K.; WYNN, K. Ratio abstraction by 6-month-old infants. Psychological Science, v. 18, n. 8, p. 740-745, 2007. MOCK, J. et al. Magnitude processing of symbolic and non-symbolic proportions: An fMRI study. Behavioral Brain Functions, v. 14, n. 9, p. 1-19, 2018. MORRIS, A. K. A teaching experiment: Introducing fourth graders to fractions from the viewpoint of measuring quantities using Davydov’s mathematics curriculum. Focus on Learning Problems in Mathematics, v. 22, n. 2, p. 32-83, 2000. NATIONAL MATHEMATICS ADVISORY PANEL. Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: US Department of Education, 2008. NI, Y.; ZHOU, Y.-D. Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, v. 40, n. 1, p. 27-52, 2005. OBERSTEINER, A. et al. Understanding fractions: Integrating results from mathematics education, cognitive psychology, and neuroscience. In: NORTON, A.; ALIBALI, M. W. (Org.). Constructing number: Merging perspectives from psychology and mathematics Education. Cham: Springer, 2019. p. 135-162. OECD. A profile of student performance in mathematics. In: PISA 2012 results: What students know and can do—Student performance in mathematics, reading, and science. Paris: OECD Publishing, 2014. v.1, Revised edition, February 2014 p. 31-144. POWELL, A. B. Melhorando a epistemologia de números fracionários: Uma ontologia baseada na história e neurociência. Revista de Matemática, Ensino e Cultura, v. 13, n. 29, p. 78-93, 2018a. ________. Reaching back to advance: Towards a 21st-century approach to fraction knowledge with the 4A-Instructional Model. Revista Perspectiva, v. 36, n. 2, p. 399- 420, 2018b. POWELL, A. B.; ALI, K. V. Design research in mathematics education: Investigating a measuring approach to fraction sense. In: CUSTÓDIO, J. F. et al. (Org.). Programa de Pós-Graduação em Educação Científica e Tecnológica (PPGECT): Contribuições para pesquisa e ensino. São Paulo: Livraria da Física, 2018. p. 221-242. RESNIKOFF, H. L.; WELLS JR., R. O. Mathematics in civilization. New York: Dover, 1973/1984. RITCHIE, S. J.; BATES, T. C. Enduring links from childhood mathematics and reading achievement to adult socioeconomic status. Psychological Science, v. 24, n. 7, p. 1301- 1308, 2013. ROQUE, T. História da matemática: Uma visão crítica, desfazendo mitos e lendas. Rio de Janeiro: Zahar, 2012. SCHEFFER, N. F.; POWELL, A. B. Frações nos livros do programa nacional do livro didático brasileiro (PNLD). Revemop, in press. SCHMITTAU, J.; MORRIS, A. The development of algebra in the elementary mathematics curriculum of V.V. Davydov. The Mathematics Educator, v. 8, n. 1, p. 60- 87, 2004. SIEGLER, R. S. Magnitude knowledge: The common core of numerical development. Developmental Science, v. 19, n. 3, p. 341-361, 2016. SIEGLER, R. S. et al. Early Predictors of High School Mathematics Achievement. Psychological Science, v. 23, n. 7, p. 691-697, 2012. SIEGLER, R. S. et al. Fractions: the new frontier for theories of numerical development. Trends in Cognitive Sciences, v. 17, n. 1, p. 13-19, 1//. 2013. Disponível em: . SIEGLER, R. S.; LORTIE-FORGUES, H. An integrative theory of numerical development. Child Development Perspective, v. 8, n. 3, p. 144-150, 2014. SON, J.-W.; SENK, S. L. How reform curricula in the USA and Korea present multiplication and division of fractions. Educational Studies in Mathematics, v. 74, n. 2, p. 117-142, 2010. Disponível em: . SOPHIAN, C. Perceptions of proportionality in young children: Matching spatial ratios. Cognition, v. 75, n. 2, p. 145-170, 2000. STRUIK, D. J. A concise history of mathematics. 3rd Revised. ed. New York: Dover, 1948/1967. SWELLER, J. Cognitive load theory, learning difficulty, and instructional design. Learning and instruction, v. 4, n. 4, p. 295-312, 1994. SWELLER, J.; VAN MERRIENBOER, J. J.; PAAS, F. G. Cognitive architecture and instructional design. Educational psychology review, v. 10, n. 3, p. 251-296, 1998. TORBEYNS, J. et al. Bridging the gap: Fraction understanding is central to mathematics achievement in students from three different continents. Learning and Instruction, v. 37, p. 5-13, 6//. 2015. Disponível em: . TUCKER, A. Fractions and units in everyday life. In: MADISON, B. L.; STEEN, L. A. (Org.). Calculation vs. context: Quantitative literacy and its implications for teacher education. Washington, DC: Mathematical Association of America, 2008. p. 75-86. TZUR, R. An integrated study of children's construction of improper fractions and the teacher's role in promoting that learning. Journal for Research in Mathematics Education, v. 30, n. 4, p. 390-416, 1999. VAMVAKOUSSI, X.; VAN DOOREN, W.; VERSCHAFFEL, L. Naturally biased? In search for reaction time evidence for a natural number bias in adults. The Journal of Mathematical Behavior, v. 31, n. 3, p. 344-355, 2012/09/01/. 2012. Disponível em: . WU, H.-H. How to prepare students for algebra. American Educator, v. 25, n. 2, p. 1-7, 2001. ________. What’s sophisticated about elementary mathematics? Plenty—That’s why elementary schools need math teachers. American Educator, v. 33, n. 3, p. 4-14, 2009.