SOLVING COLLABORATIVELY INTRODUCTORY PROBLEMS TO DEVELOP PROPORTIONAL THINKING IN PRIMARY SCHOOL
This research intends to provide some knowledge related to the didactic interactions that should be implemented to optimize mathematical learning in instructional processes. In the case of proportional reasoning, Lamon poses the following outstanding questions, “Instruction must build upon children’s intuitive knowledge but push them beyond it as well. What are the limits of children’s intuitive knowledge and qualitative reasoning? How far can that intuitive knowledge be pushed, and what critical mathematical content and ways of thinking will not occur without instruction?” (Lamon, 2007, p. 662).
In this paper, we describe a mixed-type instructional model that suggests a first phase of collaborative work between teacher and students, in which they jointly solve a problem situation, followed by other tasks in which students work more autonomously. This model has been experimented with 5th-grade primary school students, to create a first encounter with the direct proportionality problems.
The experience is based on the instructional design research methodology (Kelly, Lesh & Baek, 2008) and on the Onto-semiotic Approach to mathematical knowledge and instruction (Godino, Batanero & Font, 2007; Godino, Rivas, Arteaga, Lasa & Wilhelmi, 2014).
The teacher-researcher started by presenting the basic notions of proportionality, linked to similarity and scales, which were familiar to the students. The introductory situations tasks were intended to provide the students with a first encounter with proportionality. The design required that the students paid attention to the information provided by the teacher and, also they listen to their classmates contributions, while judging their answers validity and contrasting with their own solutions. We believe, in light of the results obtained, that this cooperation model between the teacher and the students (and among the students themselves), with regard to the situation-problem to be solved and the content that must be put into play, involves high levels of suitability in its interactional, cognitive and affective facets.
 Godino, J. D. Batanero, C., & Font, V. (2007). The onto-semiotic approach to research in mathematics education. ZDM. The International Journal on Mathematics Education 39(1-2), 127-135.
 Godino, J. D., Rivas, H., Arteaga, P., Lasa, A. & Wilhelmi, M. R. (2014). Ingeniería didáctica basada en el enfoque ontológico-semiótico del conocimiento y la instrucción matemáticos. Recherches en Didactique des Mathématiques, 34(2/3), 167-200.
 Kelly, A. E., Lesh, R. A. & Baek, J. Y. (Eds.) (2008). Handbook of design research in methods in education. Innovations in science, technology, engineering, and mathematics learning and teaching. New York, NY: Routledge.
 Lamon, S. (2007). Rational number and proportional reasoning. Toward a theoretical framework for research. En F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 1, pp. 629-667). New York, NY: Information Age Pub Inc.