RECOGNIZING ALGEBRIZATION LEVELS IN AN INVERSE PROPORTIONALITY TASK BY PROSPECTIVE SECONDARY SCHOOL MATHEMATICS TEACHERS
The aim of this paper is to describe the results of a formative experience with 33 prospective high school mathematics teachers oriented to develop their competence to recognize the objects put into play in solving proportionality problems, and assigning algebrization levels (Godino, Neto, Wilhelmi, Aké, Etchegaray & Lasa, 2015) to the mathematical activity carried out in the resolution. We also tried to develop prospective teachers’ competence to enunciate proportional reasoning problems. Posing and solving problems with different methods, as well as analysing the related mathematical objects is a competence needed in mathematics teaching, since it helps teachers adjust the task complexity to their students’ levels, understand possible learning conflicts and manage institutionalization of knowledge.
Although our research methodology is based on instructional design (Kelly, Lesh, & Baek, 2008), and includes a priori analysis, design, implementation and evaluation, in this report, we focus on the evaluation of the prospective teachers’ competences. The formative experience was developed as a part of the “Máster Universitario en Profesorado de Educación Secundaria”, during the academic year 2016-2017, in Spain and included four two and a half hours long sessions. The first two sessions, where the analysis of objects was introduced, dealt with visualization tasks. The goal of the third session, focused on Algebra, was developing knowledge and skills to recognize elementary algebraic reasoning levels. In the last session, the competence achieved to analyse a proportionality task was evaluated, and finally the educator organized a discussion of the solutions presented. Five additional tasks were proposed as an optional homework, one of which, related to inverse proportionality, is analysed in this paper. The a priori analysis of the tasks and the analysis of participants’ solutions are based on the onto-semiotic approach to mathematical knowledge and instruction (Godino, Batanero & Font, 2007).
The results suggest that the brief formative intervention was not enough for participants to achieve the ability of analyse the problem solutions in order to describe and justify these solutions. Participants neither were able to accurately recognize the different algebrization levels, nor they proposed pertinent variants of the given problem. Consequently, prospective teachers’ knowledge and specialized competences related to inverse proportionality present some weaknesses that may cause some difficulties in the teaching of the subject.
 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., Neto, T., Wilhelmi, M. R., Aké, L., Etchegaray, S. y Lasa, A. (2015). Niveles de algebrización de las prácticas matemáticas escolares. Articulación de las perspectivas ontosemiótica y antropológica. Avances de Investigación en Educación Matemática, 8, 117-142.
 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.