Knowing a Thousand Formulas Is Not The Same as Knowing Mathematics

Abstract

This research is about how students at different levels of mathematical studies view, understand and study mathematics. The research is based on two theories on mathematical understanding, James Hiebert and Patricia Lefevre’s (1986) theory on mathematical knowledge as divided into procedural knowledge and conceptual knowledge and Anna Sfards’ (1991) theory on mathematical understanding as the duality of structural understanding and operational understanding. Rooted in these theories, the mathematical understanding, views of mathematics, and how mathematics is studied by students at the gymnasium, student teachers and students studying advanced mathematics at university in Sweden, are researched. It is analyzed whether views, understandings and ways to study differ between different levels and whether students’ views and understandings impact students' ways of studying. The method used is mixed methods, involving both qualitative and quantitative methods. The students answered a questionnaire and their answers were summarized and analyzed with the help of coding and qualitative content analysis. The results point out that the most common way to view mathematics differs quite a lot between students at the gymnasium and students at the university and so do the ways of studying and the ways of understanding mathematics. The results show a correlation between the way students at different levels view mathematics, how they understand it and how they study it, with the students at the gymnasium having more of a procedural approach to mathematics than students at university.