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3.9 My Personal Position on the Use of History 89
mathematics is, naturally, best explained to the students by means of the involved mathematics. Illuminations approaches bear the risk of ‘illuminating’ the externalistic sides of the historical elements in question and leaving untouched the internalistic ones.
And as it has already been argued above, this is not a favorable approach to conducting the history of mathematics, and therefore neither to the use of it in mathematics education. It is my belief, and working hypothesis, that many meta-issue elements of a certain historical case will be better understood by the students if they simultaneously develop some kind of understanding of the related in-issues – the empirical data to be revealed later will be a test of this.
As argued in chapter 2 and in Jankvist (2009a), a purpose of ‘history as a goal’ seems best manifested through some kind of modules approach. The historical cases chosen for such an approach should, however, be exemplary, e.g. in such a way that they embrace as many general topics and issues related to the history and historiography of mathematics as possible. Or differently put, the history of mathematics in general, or the ‘global’
history, should be illustrated through specifically chosen exemplary ‘regional’ cases. As illustrated and discussed in sections 3.6 and 3.8, some of these general topics and issues have no final, correct ‘answers’, e.g. that of invention versus discovery. In some sense, it becomes a matter of personal opinion and therefore also one of personal beliefs. In this respect, the important thing is not always whether the students lean toward one viewpoint or another, but whether they are brought toreflect upon their own personal opinions, views, and beliefs, and then brought to discuss and defend these evidentially among each other. In this way they are not only creating an image of mathematics, they are also ‘living’ it and testing it out. Or in the words of Radford (2008a, p. 165),
“any process towards knowledge (in other words, all processes ofobjectification) is also a process ofsubjectification (or of the constitution of the ‘I’)”. Radford (2008a, p. 165) recapitulates this view in the sentence: “Knowing somethingshould be at the same time being someone.” We may rephrase this sentence into: believing something is the same as being someone. And it is in this way that students’ beliefs about ‘mathematics as a discipline’ become a goal (rather than a tool to enhance in-issue learning), a goal of Allgemeinbildung.
Besides relying on theoretical constructs the research questions, or an answering of them that is, must also rely on some ‘methodological constructs’. The methodology used and developed to answer the three research questions of this dissertation shall be explained in the following chapter.
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4 Methodology
The three research questions posed and analyzed in the previous chapter will be addressed on two levels. First, they will be answered in terms of the implementation of the two specific teaching modules. Second, the answers concerning the specific cases will be used as a basis for trying to address the three research questions on a more general level. In this chapter I shall describe the methods used to address the research questions on both these levels. In order to do so, some more general considerations concerning methodology will also be presented and discussed. However, an answering of the research questions requires, first and foremost, a clarification of the terms on which these are researchable and to what extent. I shall address this matter now and then return to it later in the chapter as well.
4.1 On What Terms are the Research Questions Researchable?
As should be clear from the introductory chapter (1), my approach to researching the use of history in mathematics education is an empirical approach. Due to the setting of this empirical research within the frame of Danish upper secondary school, the focus of the study has been on ‘history as a goal’ rather than ‘history as a tool’ (cf. section 3.5). Thus, with reference to the discussion of the ‘effectiveness’ of using history in mathematics education, as touched upon in chapter 1, this study concerns the effectiveness of using history as a goal.
From the survey of empirical studies on history in mathematics education in section 2.8 we saw that very few empirical studies have focussed specifically on the use – or effectiveness – of history as a goal. This means that the present study must also, at an initial stage, deal with the development of a way for doing that. In this light, it seems reasonable to approach the three research questions (RQ1-3) through a methodology of qualitative research rather than one of quantitative research. Qualitative research may be defined as studies which are concerned with causal relationships on a conceptual level, whereas quantitative research concerns the production of quantitative measurements for practically anything. Mason (2002, pp. 2-3) says, within a social science context, that qualitative research “aims to produce rounded understandings on the basis of rich, contextual, and detailed data”, data gathered by methods which are “flexible and sensitive to the social context in which data are produced (rather than rigidly standardized or structured, or removed from ‘real life’ or ‘natural’ social context, as in some forms of experimental method)”.
Especially research questions 1 and 2 (asking about in what sense, to what extent, on which conditions and levels) seem to call for ‘rounded understandings’ based on rich, contextual, and detailed data rather than data removed from the ‘natural’ social context in which they were generated. Research question 3 is more suitable for quantitative
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methods, since it asks to the possible changes in students’ beliefs. But it also asks how these changes may be brought about, which again calls for more ‘rounded understandings’.
Some authors argue for a combination of quantitative and qualitative methodologies (Brown, 1992, p. 156), e.g. by combining a concentration on large scale data bases with in-depth microgenetic analyses of a few students or a group of students – an approach which to some extent may be applied for research question 3.
In summary, to provide answers for research questions 1 and 2, in-depth studies of single implementations of designed teaching modules is a first step in addressing the
‘effectiveness’ of history as a goal. Furthermore, this together with the actual formulation of the two questions seem to suggest a line of qualitative research rather than one of quantitative research, not least due to the lack of previous empirical, experimental studies on history as a goal and hence of any pre-made methodologies for handling the research questions. For research question 3, on the other hand, it can be argued that a combined strategy of both quantitative and qualitative methods may be suitable (I shall explain the details later). Of course, the quantitative measurements for research question 3 shall not rely on “large scale data bases”. The basis for answering the question will be restricted to the same population used to answer research questions 1 and 2, since question 3 asks after some of the effects of carrying out the research on questions 1 and 2.