Computer software is increasingly used in mathematical work by researchers and other professionals, and (in part because of that) appear also more or less extensively in secondary and tertiary mathematics teaching. However, in teaching, mathematical software is not primarily used for its efficiency to carry out complex mathematical calculations, although learning about efficient tools can also be a goal; the main purpose is to facilitate students’ development of mathematical knowledge (not necessarily involving the use of technology). Therefore, how teachers use computer software becomes very important and requires didactical knowledge about the software – that is, about methods to use it to further other people’s learning of mathematics. This kind of relation to the software is not automatically developed by attending high school or university mathematics classes, even if these involve technology use – just as you do not learn to play the piano by attending a concert.
My PhD project aims to investigate the ‘discontinuity’ or gap between the mathematical knowledge acquired by university students and the knowledge about didactical uses of computer software which is relevant in upper secondary school, where many of them become teachers; and also to develop end experiment ways to fill this gap. The project will be based on anthropological theory of the didactic (ATD) introduced by Chevellard (1992), both to analyse the didactical and mathematical techniques related to software use, and to develop new didactical designs. More specifically, I will focus on how CAS (Computer algebra systems) is or could be used (in high school) to visualize properties functions of one or more variables, and the teacher knowledge required to do realise the associate materialization potential (cf. Winsløw, 2003).
I plan to investigate the following research questions:
RQ1. What instrumented techniques, relevant to visualizing properties of functions, do undergraduate mathematics students master (after completing their Calculus courses)? To what extent can they deploy these techniques autonomously, in order to investigate more theoretical problems? (cf. Winsløw and Gronbæk, 2014)
RQ2. What didactical knowledge about instrumented visualization techniques is relevant to make use of CAS-tools in inquiry oriented teaching of Calculus in high school? How does it relate to the answers found to RQ1?
RQ3. What didactical interventions can be efficient to help future or current mathematics teachers develop the didactical knowledge identified in RQ2?
The first two questions will be investigated in Danish high schools and universities. The last question will be also mainly investigated in the Danish context, with the possibility of extending the study to other countries at a later point in time – in particular, to the context of China where mathematical software is currently not typically used in upper secondary or university teaching.
C. Winsløw, N. Gronbæk (2013). Klein’s Double Discontinuity Revisited: Contemporary Challenges for universities preparing teachers to teach calculus. Recherches en Didactique des Mathématiques, 34(1), 59-86.
John Monaghan(2007). Computer algebra, instrumentation and the anthropological approach. International Journal for Technology in Mathematics Education, 14(2), 63-72.