A compreensão de crianças sobre adição e subtração
transformações, relações inversas e estimativas
Palavras-chave:
Adição e subtração, Transformações, Relações inversas, Estimativa, Crianças.
Resumo
O presente estudo teve por objetivo examinar como crianças com diferentes perfis de habilidade acerca das operações de adição e subtração lidam com os princípios da complementaridade e da inversão, e como realizam cálculos por estimativa. Os participantes, estudantes do 1º e 2º ano do Ensino Fundamental, realizaram três tarefas: Tarefa A relativa ao princípio da complementaridade, Tarefa B relativa ao princípio da inversão e Tarefa C envolvia cálculos por estimativa a partir de pontos de referência. Observou-se que a compreensão das transformações associadas ao princípio da complementaridade não garantia nem a compreensão das transformações sucessivas e inversas associadas ao princípio da inversão e nem o sucesso com cálculos por estimativa. A conclusão foi que o princípio da inversão e estimar resultados de adições e subtrações são um desafio para as crianças. Os resultados são interpretados a partir das demandas cognitivas relativas a cada tarefa.
Referências
BAROODY, Arthur; LAI, Meng Lung. Pre-schoolers’ understanding of the addition–subtraction inverse principle: a Taiwanese sample, Mathematical Thinking and Learning, v. 9, n. 2, p. 131-171. 2007.
BISANZ, Jeffrey. et al. On “understanding” children's developing use of inversion. Mathematical Thinking and Learning, v. 11, p. 10–24. 2009.
BJÖRKLUND, Camila; MARTON, Ference; KULLBERG, Angelika. What is to be learnt? Critical aspects of elementar arithmetic skills. Educational Studies in Mathematics, v. 107, p. 261–284. 2021.
BRYANT, Peter. Childrens understanding and use of inversion in arithmetic1. Cuadernos, v. 11, p. 231-238. 2013.
BRYANT, Peter; CHRISTIE, Clare; RENDU, Alison. Children's understanding of the relation between addition and subtraction: Inversion, identity, and decomposition. Journal of Experimental Child Psychology, v. 74, n. 3, p. 194–212. 1999.
CANOBI, Katherine. Children’s profiles of addition and subtraction understanding. Journal of Experimental Child Psychology, v. 92, p. 220–246. 2005.
CASE, Robbie; SOWDER, Judith. T. The development of computational estimation: A neoPiagetian analysis. Cognition and Instruction, v. 7, n. 79–104. 1990.
CHING, Boby Ho-Hong; NUNES, Terezinha. Children´s understanding of the commutativity and complement principles: A latent analysis profile. Learning and Instruction, 47(1), 65–79, 2017.
CHING, Boby Ho-Hong; WU, Xiaohan. Concreteness fading fosters children's understanding of the inversion concept in addition and subtraction. Learning and Instruction, v. 61, n. 148-159. 2019.
AUTOR, 1997.
DE CORTE, Erik; VERSCHAFFEL, Lieven. The Effect of Semantic Structure on First Graders' Strategies for Solving Addition and Subtraction Word Problems. Journal for Research in Mathematics Education, v. 18, n. 5, p. 363–381. 1987.
DOWKER, Ann. Young Children's Addition Estimates. Mathematical Cognition, v. 3, n. 2, p. 140–153. 1997.
DOWKER, Ann. Targeted interventions for children with arithmetical difficulties. British Journal of Educational Psychology Monograph Series, v. 2, n. 7, p. 65-81. 2010.
DOWKER, Ann. Young children´s estimates for addition: The zone of partial knowledge and understanding. In: BAROODY, Arthur; DOWKER, Ann (Eds.). The development of arithmetic concepts and skills: constructing and adaptative expertise. Routledge, 2013. p. 243-266.
DOWKER, Ann. Interventions for primary school children with difficulties in mathematics. Advances in Child Development and Behavior, v. 53, p. 255-287. 2017.
DOWKER, Ann. Individual differences in arithmetic: Implications for psychology, neuroscience and education. Routledge, 2019.
GILMORE, Camilla; BRYANT, Peter. Can children construct inverse relations in arithmetic? Evidence for individual differences in the development of conceptual understanding and computational skill. British Journal of Developmental Psychology, v. 26, p. 301-316. 2008.
GILMORE, Camilla; SPELKE, Elizabeth. Children’s understanding of the relationship between addition and subtraction. Cognition, v. 107, n. 3, p. 932–945. 2008.
KULLBERG, Angelika. et al. Effects of learning addition and subtraction in preschool by making the first ten numbers and their relations visible with finger patterns. Educational Studies in Mathematics, v. 103, n. 2, p. 157–172. 2020.
LAI, Meng Lung; BAROODY, Arthur; JOHNSON, Amanda. Fostering Taiwanese preschoolers’ understanding of the addition subtraction inverse principle. Cognitive Development, v. 23, n. 1. 2008.
LEFEVRE, Jo-Anne. et al. Home numeracy experiences and children's math performance in the early school years. Canadian Journal of Behavioural Science, v. 41, n. 2, p. 55–66. 2009.
LEMAIRE, Patrick.; LECACHEUR, Mireille. Children’s strategies in computational estimation. Journal of Experimental Child Psychology, v. 82, p. 281–304. 2002.
NOGUES, Camila; DORNELES, Beatriz Vargas. Estimativa numérica, memória de trabalho e raciocínio quantitativo: relações no desempenho matemático. Zetetiké, v. 28, p. 1-17. 2020.
NUNES, Terezinha et al. Teaching children how to include the inversion principle in their reasoning about quantitative relations. Educational Studies in Mathematics, v. 79, n. 3, p. 371–388. 2012.
PALIWAL, Veena; BAROODY, Arthur. Fostering the learning of subtraction concepts and the subtraction-as-addition reasoning strategy. Early Childhood Research Quarterly, v. 51, p. 403–415, 2020.
PIAGET, Jean; MOREAU, Albert. The inversion of arithmetic operations. In: CAMPELL, Robert (Ed.). Studies in Reflecting Abstraction. Psychology Press, 2001. p. 69-86.
ROBINSON, Katherine; DUBÉ, Adam. Children's additive concepts: promoting understanding and the role of inhibition. Learning and Individual Differences, v. 23, p. 101–107. 2013.
SEKERIS, Elke.; VERSCHAFFEL, Lieven; LUWEL, Koen. Exact arithmetic, computational estimation and approximate arithmetic are different skills: evidence from a study with 5‐year‐olds. Infant and Child Development, v. 30, n. 5, e2248. 2021.
AUTOR, 2006.
AUTOR, 2018.
AUTOR; AUTOR; AUTOR, 2021.
AUTOR; AUTOR, 2018.
TORBEYNS, Joke et al. Children’s understanding of the addition/subtraction complement principle. The British Journal of Educational Psychology, v. 86, n. 3, p. 382–396. 2016.
TORBEYNS, Joke et al. Subtraction by addition strategy use in children of varying mathematical achievement level: a choice/no-choice study. Journal of Numerical Cognition, v. 4, n. 1, p. 215–234. 2018.
TORBEYNS, Joke; VERSCHAFFEL, Lieven; GHESQUIÈRE, Pol. Strategic aspects of simple addition and subtraction: the influence of mathematical ability. Learning and Instruction, v. 14, n. 2, p. 177-195. 2004.
VAN DER AUWERA, Stijn et al. The remarkably frequent, efficient, and adaptive use of the subtraction by addition strategy: a choice/no-choice study in fourth- to sixth-graders with varying mathematical achievement levels. Learning and Individual Differences, 93, 2022. https://doi.org/10.1016/j.lindif.2021.102107
VAN DER AUWERA, Stijn et al. Children’s subtraction by addition strategy use and their subtraction-related conceptual knowledge. Educational Studies in Mathematics, 2023. https://doi.org/10.1007/s10649-023-10276-3
VERGNAUD, Gérard. The acquisition of arithmetical concepts. Educational Studies in Mathematics, v. 10, p. 263– 274. 1979.
VERGNAUD, Gérard. A classification of cognitive tasks and operations of thought involved in addition and subtraction problems. In: CARPENTER, Thomas; MOSER, James; ROMBERG, Thomas (Eds.). Addition and subtraction: a cognitive perspective. Hillsdale: Lawrence Erlbaum, 1982. p. 60-67.
VERGNAUD, Gérard. La théorie des champs conceptuels. Recherches en Didactique des Mathématiques, v. 10, n. 2-3, p. 133 -170. 1990.
VERGNAUD, Gérard. A criança, a matemática e a realidade: problemas do ensino da matemática na escola elementar. Curitiba: Editora Universitária da UFPR. 2009.
VILETTE, Bruno. Do young children grasp the inverse relationship between addition and subtraction? Evidence against early arithmetic. Cognitive Development, v. 17, p. 1365–1383. 2002.
BISANZ, Jeffrey. et al. On “understanding” children's developing use of inversion. Mathematical Thinking and Learning, v. 11, p. 10–24. 2009.
BJÖRKLUND, Camila; MARTON, Ference; KULLBERG, Angelika. What is to be learnt? Critical aspects of elementar arithmetic skills. Educational Studies in Mathematics, v. 107, p. 261–284. 2021.
BRYANT, Peter. Childrens understanding and use of inversion in arithmetic1. Cuadernos, v. 11, p. 231-238. 2013.
BRYANT, Peter; CHRISTIE, Clare; RENDU, Alison. Children's understanding of the relation between addition and subtraction: Inversion, identity, and decomposition. Journal of Experimental Child Psychology, v. 74, n. 3, p. 194–212. 1999.
CANOBI, Katherine. Children’s profiles of addition and subtraction understanding. Journal of Experimental Child Psychology, v. 92, p. 220–246. 2005.
CASE, Robbie; SOWDER, Judith. T. The development of computational estimation: A neoPiagetian analysis. Cognition and Instruction, v. 7, n. 79–104. 1990.
CHING, Boby Ho-Hong; NUNES, Terezinha. Children´s understanding of the commutativity and complement principles: A latent analysis profile. Learning and Instruction, 47(1), 65–79, 2017.
CHING, Boby Ho-Hong; WU, Xiaohan. Concreteness fading fosters children's understanding of the inversion concept in addition and subtraction. Learning and Instruction, v. 61, n. 148-159. 2019.
AUTOR, 1997.
DE CORTE, Erik; VERSCHAFFEL, Lieven. The Effect of Semantic Structure on First Graders' Strategies for Solving Addition and Subtraction Word Problems. Journal for Research in Mathematics Education, v. 18, n. 5, p. 363–381. 1987.
DOWKER, Ann. Young Children's Addition Estimates. Mathematical Cognition, v. 3, n. 2, p. 140–153. 1997.
DOWKER, Ann. Targeted interventions for children with arithmetical difficulties. British Journal of Educational Psychology Monograph Series, v. 2, n. 7, p. 65-81. 2010.
DOWKER, Ann. Young children´s estimates for addition: The zone of partial knowledge and understanding. In: BAROODY, Arthur; DOWKER, Ann (Eds.). The development of arithmetic concepts and skills: constructing and adaptative expertise. Routledge, 2013. p. 243-266.
DOWKER, Ann. Interventions for primary school children with difficulties in mathematics. Advances in Child Development and Behavior, v. 53, p. 255-287. 2017.
DOWKER, Ann. Individual differences in arithmetic: Implications for psychology, neuroscience and education. Routledge, 2019.
GILMORE, Camilla; BRYANT, Peter. Can children construct inverse relations in arithmetic? Evidence for individual differences in the development of conceptual understanding and computational skill. British Journal of Developmental Psychology, v. 26, p. 301-316. 2008.
GILMORE, Camilla; SPELKE, Elizabeth. Children’s understanding of the relationship between addition and subtraction. Cognition, v. 107, n. 3, p. 932–945. 2008.
KULLBERG, Angelika. et al. Effects of learning addition and subtraction in preschool by making the first ten numbers and their relations visible with finger patterns. Educational Studies in Mathematics, v. 103, n. 2, p. 157–172. 2020.
LAI, Meng Lung; BAROODY, Arthur; JOHNSON, Amanda. Fostering Taiwanese preschoolers’ understanding of the addition subtraction inverse principle. Cognitive Development, v. 23, n. 1. 2008.
LEFEVRE, Jo-Anne. et al. Home numeracy experiences and children's math performance in the early school years. Canadian Journal of Behavioural Science, v. 41, n. 2, p. 55–66. 2009.
LEMAIRE, Patrick.; LECACHEUR, Mireille. Children’s strategies in computational estimation. Journal of Experimental Child Psychology, v. 82, p. 281–304. 2002.
NOGUES, Camila; DORNELES, Beatriz Vargas. Estimativa numérica, memória de trabalho e raciocínio quantitativo: relações no desempenho matemático. Zetetiké, v. 28, p. 1-17. 2020.
NUNES, Terezinha et al. Teaching children how to include the inversion principle in their reasoning about quantitative relations. Educational Studies in Mathematics, v. 79, n. 3, p. 371–388. 2012.
PALIWAL, Veena; BAROODY, Arthur. Fostering the learning of subtraction concepts and the subtraction-as-addition reasoning strategy. Early Childhood Research Quarterly, v. 51, p. 403–415, 2020.
PIAGET, Jean; MOREAU, Albert. The inversion of arithmetic operations. In: CAMPELL, Robert (Ed.). Studies in Reflecting Abstraction. Psychology Press, 2001. p. 69-86.
ROBINSON, Katherine; DUBÉ, Adam. Children's additive concepts: promoting understanding and the role of inhibition. Learning and Individual Differences, v. 23, p. 101–107. 2013.
SEKERIS, Elke.; VERSCHAFFEL, Lieven; LUWEL, Koen. Exact arithmetic, computational estimation and approximate arithmetic are different skills: evidence from a study with 5‐year‐olds. Infant and Child Development, v. 30, n. 5, e2248. 2021.
AUTOR, 2006.
AUTOR, 2018.
AUTOR; AUTOR; AUTOR, 2021.
AUTOR; AUTOR, 2018.
TORBEYNS, Joke et al. Children’s understanding of the addition/subtraction complement principle. The British Journal of Educational Psychology, v. 86, n. 3, p. 382–396. 2016.
TORBEYNS, Joke et al. Subtraction by addition strategy use in children of varying mathematical achievement level: a choice/no-choice study. Journal of Numerical Cognition, v. 4, n. 1, p. 215–234. 2018.
TORBEYNS, Joke; VERSCHAFFEL, Lieven; GHESQUIÈRE, Pol. Strategic aspects of simple addition and subtraction: the influence of mathematical ability. Learning and Instruction, v. 14, n. 2, p. 177-195. 2004.
VAN DER AUWERA, Stijn et al. The remarkably frequent, efficient, and adaptive use of the subtraction by addition strategy: a choice/no-choice study in fourth- to sixth-graders with varying mathematical achievement levels. Learning and Individual Differences, 93, 2022. https://doi.org/10.1016/j.lindif.2021.102107
VAN DER AUWERA, Stijn et al. Children’s subtraction by addition strategy use and their subtraction-related conceptual knowledge. Educational Studies in Mathematics, 2023. https://doi.org/10.1007/s10649-023-10276-3
VERGNAUD, Gérard. The acquisition of arithmetical concepts. Educational Studies in Mathematics, v. 10, p. 263– 274. 1979.
VERGNAUD, Gérard. A classification of cognitive tasks and operations of thought involved in addition and subtraction problems. In: CARPENTER, Thomas; MOSER, James; ROMBERG, Thomas (Eds.). Addition and subtraction: a cognitive perspective. Hillsdale: Lawrence Erlbaum, 1982. p. 60-67.
VERGNAUD, Gérard. La théorie des champs conceptuels. Recherches en Didactique des Mathématiques, v. 10, n. 2-3, p. 133 -170. 1990.
VERGNAUD, Gérard. A criança, a matemática e a realidade: problemas do ensino da matemática na escola elementar. Curitiba: Editora Universitária da UFPR. 2009.
VILETTE, Bruno. Do young children grasp the inverse relationship between addition and subtraction? Evidence against early arithmetic. Cognitive Development, v. 17, p. 1365–1383. 2002.
Publicado
2025-03-10
Como Citar
CRUZ, M. S. S.; SPINILLO, A. G.; CORREA, J. A compreensão de crianças sobre adição e subtração. Perspectivas da Educação Matemática, v. 18, n. 49, p. 1-21, 10 mar. 2025.
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