Horizontal and vertical mathematisation in inquiry based pedagogy
Oduor Olande2 and Jöran Petersson1
1Malmö University, Sweden, Faculty of Education and Society; joran.petersson@mau.se 2Linneus University, Sweden, Department of mathematics; oduor.olande@lnu.se
Keywords: Active learning, didactical engineering, mathematisation, statistics education
IBP (Inquiry Based Pedagogy) is a growing trend and is “…defined loosely as a way of teaching in which students are invited to work in ways similar to how mathematicians and scientists work.” (Artigue and Blomhøj, 2013, p.797), which means that the learner engages in a process of self-reflection and discovery. IBP open for a plethora of possible research issues and we illustrate this with our work with formulating and using the following task in IBP and analysing the outcome.
The village association has decided to build a common well somewhere along the street in the village (see figure), where all households can go and fetch water. Your task is to suggest a rule for where to dig the well. It should be of mathematical nature and the villagers should find it fair.
When formulating a task for IBP, we first needed to consult cognitive research on concept formation in statistics education. We also considered the students’ prerequisites needed for IBP to work (Kirschner, Sweller, & Clark, 2006). In next phase, we used didactical engineering for refining the problem formulation and the lesson arrangement. While exploring the collected classroom data during one of the first iterations of refinement, we identified several aspects of research interest in students’ group work. We felt that besides describing this process in terms of Pólya’s cognitive phases of problem solving in Schoenfeld’s timeline protocol and the social co-operation following socio-mathematical norms, another unexplored option is to describe the students’ work process in terms Treffer’s horizontal and vertical mathematisation. Namely, that one student group seemed to work only mechanically while others also in a structural vein created what for them was new mathematics and regarded the realistic aspects of the problem. In particular, we saw that the working process in the latter groups shifted between realistic, structural and mechanical phases (and an intermittent idle phase) in a way that we think deserves a closer analysis on how these phases are distributed in time throughout the students’ group work.
References
Artigue, M., & Blomhøj, M. (2013). Conceptualizing inquiry-based education in mathematics. ZDM Mathematics Education, 45(6), 797–810. doi:https://doi.org/10.1007/s11858-013-0506-6
Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41(2), 75–86.
