Mathematical modelling processes : implications for teaching and learning
The principal aim of the project has been to investigate formulation-solution processes and the extent to which these processes lead to better guidance and understanding of teaching, learning, and assessment in mathematical modelling. The following main activities have been carried out in support of this aim: the development of case studies of the mathematical modelling approaches that may be used in the solution of practical problems; the design of teaching and learning experiments carried out mainly with undergraduates with some knowledge of physics and teachers on an MSc course in mathematical education; the theoretical development of formulation-solution processes by means of a concept matrix and a relationship level graph; the analysis of a selection of students' modelling attempts; an investigation of assessment methods and the implications of the theoretical development of formulation solution processes for these methods. The case studies were based on possible modelling approaches to practical problems which are connected in some way with every-day reality. These studies were used in seventeen experiments with students working in a genuine educational environment under the usual time constraints. Most of the students involved had little or no modelling experience. Results have shown that students have a common set of difficulties, and a set of learning heuristics has been devised in an attempt to overcome these. The theoretical development of formulation-solution processes has identified the following main characteristics in early model development: distribution of features from global (difficult to quantify) to specific (easily quantified) concepts; basic relationships are often generated as solution proceeds; relationships can occur in either general or specific forms; general progress is gauged by relationship 'level'; most variables and constants are generated with relationships; partitioning a problem into sub-problems may be possible initially, but such break-down into distinct parts is often only possible after having seen a pattern of linkages in a relationship level graph. Finally, the implications for assessment methods are examined, and suggestions for further research investigations are made.