Reconstructing logarithms using number lines as mediated tools

The aim of the research reported in this licentiate thesis was to create an environment that could support students’ learning about logarithms. To develop such a learning environment, Davydov’s ‘learning activity’ was used as a theoretical framework for the design. A new tool was created, that was used by the students to unfold and single out some of the unique properties of logarithms when solving different learning tasks. The construction of the model was inspired by Napiers original idea from 1614, i.e. exactly 400 years ago, by using two number lines; one arithmetic (i.e. based on addition) and one geometric (i.e. based on multiplication). The research approach used was learning study where teachers and researcher worked collaboratively in an iterative process to refine the research lesson. The study was conducted in six groups with six teachers in upper secondary school in a major city in Sweden. The sample comprised about 150 students and data were collected by filming lessons and by interviews with some of the students. The data were analysed using an analytic framework derived from ‘learning activity’ and the results show what supports, but also what does not support, the creation of an environment for supporting students’ learning of logarithms. The results from the study are related to former research regarding instrumental/procedural vis-à-vis relational/conceptual understanding and also about research about students’ ‘errors and misconceptions’. It is argued that the formal definition of logarithms, y = 10x <-> x = lgy (y > 0), should not be used to introduce the concept, instead a new way is proposed. One conclusion is that it is possible to reconstruct logarithms without using the definition as a tool. The results from the analysed lessons show how students looked for ways to solve learning tasks using the new tool. The definition and the identities regarding logarithms appear as bi-products of the students learning activity. When analysing students actions, they rarely over-generalised mathematical rules, e.g. used the distributive law, or separated log-expressions, e.g. adding log expressions part by part, that seemed to be an issue according to former research