This study is concerned with how different individuals come to terms with formal axiomatic theory. The whole project was designed to allow the data to speak for itself, in the manner of a Data-Grounded Theory (see Strauss, 1987). It began with a detailed analysis of how students responded to problems involving definitions and deductions, and built into a categorised theory under the headings DEFINITIONS, ARGUMENTS and IMAGERY.
The exploratory study was an analysis of the performance of a group of students training to be teachers, at the end of a course. It intended to refine initial research questions to be followed later in the main study. Issues emerging from the analysis of two sets of data indicate that few students in the research sample appeared to be able to understand and to use the formal aspects of the course content. Instead, students’ responses suggested they were building meaning for the concepts and constructing their arguments from their imagery. It was decided that for the main study, students would be followed during a period of time, focusing on whether they construct concepts and proofs from formal definitions or concept imagery, or a combination of the two.
For the main study, the initial sample was expanded to include students from the Mathematics Department and from the Institute of Education. Eleven mathematics undergraduates and four students training to be school teachers were selected on basis of a questionnaire formulated with the purpose of guaranteeing some very good students to participate in a series of individual interview. Students were interviewed every two weeks, during two terms.
From the analysis of data collected we could identify two different strategies used by the learners in their attempts to build up concepts given by formal definitions, which we named giving meaning (building from earlier experiences) and extracting meaning (building from the formal definitions). As our research progressed, we observed two different styles of learners, which we named natural and formal learners (see also Duffin and Simpson, 1993), distinguished according to whether they regularly used the first or the second strategy to construct knowledge. There is a spectrum of performance in each of these styles of learners; indicating that neither of these routes of learning necessarily lead to success, or failure. Learners in each of these routes revealed different cognitive problems, which are related to their different styles of learning. Both kinds of learners, when unsuccessful in working meaningfully with the formal theory, attempt to rote-learn definitions, images and formal arguments, building idiosyncratic theories with weak conceptual links.
|Advisor:||Tall, David Orme|
|School:||University of Warwick (United Kingdom)|
|Source:||DAI-A 73/04, Dissertation Abstracts International|
|Subjects:||Mathematics education, Educational psychology, Higher education|
|Keywords:||Analysis learning, Cognitive development, Empirical study, Natural and formal learning, Proof, Quantifiers|
Copyright in each Dissertation and Thesis is retained by the author. All Rights Reserved
The supplemental file or files you are about to download were provided to ProQuest by the author as part of a
dissertation or thesis. The supplemental files are provided "AS IS" without warranty. ProQuest is not responsible for the
content, format or impact on the supplemental file(s) on our system. in some cases, the file type may be unknown or
may be a .exe file. We recommend caution as you open such files.
Copyright of the original materials contained in the supplemental file is retained by the author and your access to the
supplemental files is subject to the ProQuest Terms and Conditions of use.
Depending on the size of the file(s) you are downloading, the system may take some time to download them. Please be