Using quantitative and qualitative data to construct a developmental learning framework in the context of elementary high school algebra

The study for a PhD thesis aims to identify linguistic characteristics of students' utterances that indicate their levels of understanding of elementary algebra. To do so, both quantitative and qualitative data was collected. Secondary school students were asked to complete a forty-item algebra test, and then a selection of students was interviewed. The responses of the students (n = 222) on the algebra test were analysed using the Rasch model. The results display groupings of types of algebra items that indicate developmental levels. These levels may be described in terms of a qualitative framework such as the SOLO taxonomy, as well as in terms of the mathematical characteristics of the items. Interview data obtained from students (n = 32) is also to be used to explore their levels of understanding, and to support findings from the analysis of the test items. In this paper, issues arising from the use of the Rasch models, and in devising and obtaining valid and reliable interview data are posed and discussed. Some issues are the strengths and limitations of the Rasch model, the structure and conduct of interviews and ways in which interview data may be analysed. Research into human behaviour faces a perennial problem of providing convincingly objective, valid and reliable data. Unlike the physical sciences there is not usually recourse to repeated, and hence replicable experiments. Quantitative data, although objective, and able to be validated and shown to be reliable through repeated applications of the data collection instrument, may provide insight into a limited set of behaviours – those which can be enumerated. Reliance on qualitative data alone leads to the problems of establishing reliability and validity. Consequently both quantitative and qualitative data-collection instruments need to be used in such research in such a way that one complements the other. Quantitative data informs the researcher about 'how much' mainly through counting techniques that are assumed to be free of the influence of the researcher. Qualitative data provides information about 'how well', but is often suspected of being unduly influenced by the will of the researcher. Leaving aside the philosophical and psychological questions these last sentences raise, the use of an analytical tool that takes account of 'how much' and 'how well' provides researchers with a framework of particular value in the field of education. One such analysis tool is the Rasch Model, developed by Georg Rasch and first published in 1980. The model provides a framework of developmental pathways based on objective measures. Like all models, it makes certain assumptions and simplifications that limit the extent of inferences that might be drawn from it. The use of the Rasch model and some of the limitations encountered will be discussed in this paper in the context of research into connections between language use and cognitive development in mathematics. Firstly a brief outline of the research is provided, then a general overview of the Rasch model, as used in the study. Data from the study will then be used to illustrate the power of the model to use quantitative data to provide insight into 'how well' students understand the mathematics and to illustrate some of the consequent questions.