Joana Brocardo, Jean-Marie Kraemer, Fátima Mendes, Catarina Delgado
The project ‘Numerical thinking and flexible calculation: critical issues’ aims to study students’ conceptual knowledge associated with the understanding of the different levels of learning numbers and operations. We follow the idea proposed by several authors that flexibility refers to the ability to manipulate numbers as mathematical objects which can be decomposed and recomposed in multiple ways using different symbolisms for the same objet (Gravemeijer, 2004; Gray &Tall, 1994;). The project plan is based on a qualitative and interpretative methodology (Denzin & Lincoln, 2005) with a design research approach (Gravemeijer & Cobb, 2006). This article focus the preparation of a teaching experience centered in the flexible learning of multiplication. It describes the analysis of a clinical interview where Pedro (9 years) solves the task ‘Prawn skewers’. It illustrates how we identify and describe Pedro’s conceptual knowledge associated with the different levels of understanding of numbers and multiplication/division and analyzes if and how this knowledge facilitates adaptive thinking and flexible calculation.
Multiplicative reasoning, Flexible calculation, Mathematical tasks
Baroody, A. (2003). The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. 1-33) NJ: Lawrence Erlbaum Associates.
Denzin, N. & Lincoln, Y. S. (2005). Introduction: The discipline and practice of qualitative research. In N. Denzin & Y. S. Lincoln (Ed.) The Sage handbook of qualitative research. Thousand Oaks, CA: SAGE.
Freudenthal, H. (1973). Mathematics as an educational task. Springer Science & Business Media.
Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht, the Netherlands: Reidel Publishing Company.
Freudenthal, H. (1991). Revisiting mathematics education. China lectures. Dordrecht, the Netherlands: Kluwer Academic Publishers.
Gravemeijer, K. (2004). Local instruction theories as means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6 (2), 105-128.
Gravemeijer, K., & Cobb, P. (2006). Design research from a learning design perspective. In J. van den Akker, K. Gravemeijer, S. McKenney, & N. Nieveen (Edits.), Educational design research (pp. 45-85). London: Routledge.
Gray, E. M. & Tall, D. O. (1994). Duality, Ambiguity and Flexibility: A Proceptual View of Simple Arithmetic. Journal for Research in Mathematics Education, 26 (2), 115–141.
Hatano, G. (1982). Learning to add and subtract: A Japanese perspective. In T.P. Carpenter, J.M. Moser, & T.A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 211–223). Hillsdale, NJ: Erlbaum.
Hatano, G. (2003). Foreword. In A. J. Barrody & A. Dowker (Eds.),The development of arithmetic concepts and skills: Constructing adaptive expertise, (pp. xi-xiv). London: Lawrence Erlbaum Associates Publishers.
Hatano, G., & Inagaki, K. (1986). Two courses of expertise. In H. Stevenson, H. Azuma, & K. Hakuta (Eds.), Child development and education in Japan (pp. 262-272). New York: Freeman.
Kilpatrick, J., Swafford, J. & Findell, B. (Eds.) (2001). Adding it up: Helping children learning mathematics. Washington DC: National Academy Press.
Lesh, R., Kelly, A. E., & Yoon, C. (2008). Multitiered design experiments in mathematics, science and technology education. In A. E. Kelly, R. Lesh, and J. Baek (Eds.), Handbook of design research methods in education: Innovations in science, technology, engineering, and mathematics learning. (pp. 131-148). New York: Routledge.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1-36.
Thompson, P. W., & Saldanha, L. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, G. Martin & D. Schifter (Eds.), Research companion to the Principles and Standards for School Mathematics (pp. 95-114). Reston, VA: National Council of Teachers of Mathematics.
Vergnaud, G. (1983). Multiplicative structures. In Lesh, R. and Landau, M. (Eds.) Acquisition of mathematics concepts and processes, (pp. 127-174).New York: Academic Press Inc.
Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert & M. Behr (Eds.) Number concepts and operations in the Middle Grades (pp. 141-161). Reston: NCTM.
Verschaffel, L., Greer, B. ,& De Corte ,E. (2007). Whole number concepts and operations. In F. Lester (Eds.), Handbook of research in mathematics teaching and learning, (pp. 557-628). New York: MacMillan.