The Inverse Relation Between the Size and the Number of Parts

Ema Mamede, Isabel Vasconcelos


This study analyzes children’s understanding of the inverse relationship between size and number of parts when fractions and division situations are involved. A survey by questionnaire was conducted with 42 Portuguese fourth-graders trying to address two questions: 1) How do children understand the inverse relation between size and number of parts in partitive and quotitive division situations? And 2) How do children understand the inverse relation when fractions are involved in part-whole and quotient interpretations? Results suggest that these distinct situations have different impacts on children’s understanding of the inverse relations between the size and the number of parts.


inverse relation, fractions, division

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Behr, M., Wachsmuth, I., Post, T. & Lesh, R. (1984). Order and Equivalence of Rational Numbers: A Clinical Teaching Experiment. Journal for Research in Mathematics Education, 15 (5), 323-341.

Correa, J., Nunes, T., & Bryant, P. (1998). Young children’s understanding of division: The relationship between division terms in a noncomputational task. Journal of Educational Psychology, 90, 321-329.

Direcção Geral de Inovação e Desenvolvimento Curricular (2007). Programa de Matemática do ensino básico. Lisboa: Ministério da Educação.

Fonseca, H. I. (2000). Os processos matemáticos e o discurso em actividades de investigação. (Dissertação de Mestrado, Universidade de Lisboa). Lisboa: APM.

Kornilaki, E., & Nunes, T. (2005). Generalising Principles in spite of Procedural Differences: Children’s Understanding of Division. Cognitive Development, 20, 388-406.

Mamede, E. & Cardoso, P. (2010). Insights on students (mis) understanding of fractions. In: M. M. Pinto & T. F. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 257-264). Belo Horizonte, Brasil: PME.

Mamede, E., Nunes T. & Bryant, P. (2005). The equivalence and ordering of fractions in part-whole and quotient situations. In: H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 281–288). Melbourne, Australia: PME.

Mamede, E. & Silva, A. (2012). Exploring partitive division with young children. Journal of the European Teacher Education Network, (8), 35-43.

National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Virginia: NCTM.

Nunes, T., Bryant, P., Pretzlik, U., Evans, D., Wade. J. & Bell, D. (2004). Vergnaud’s definition of concepts as a framework for research and teaching. Annual Meeting for the Association pour la Recherche sur le Développement des Compétences, 28-31. Paris.

Ponte, J. P., Boavida, A., Graça, M. & Abrantes, P. (1997). Didáctica da Matemática. Lisboa: Ministério da Educação – DES.

Spinillo, A.G. & Lautert, S. L. (2011). Representar operações de divisão e representar problemas de divisão: há diferenças? International Journal for Studies in Mathematics Education, 4 (1), 115 –134.

Categories: 2016, Articles - JETEN, Mathematics Education

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