proportional reasoning, quantitative literacy, numeracy, threshold concepts, verbal reasoning


The ability to reason with proportions is known to take a long time to develop and to be difficult to learn. We regard proportional reasoning (the ability to reason about quantities in relative terms) as a threshold concept for academic quantitative literacy. Our study of the teaching and learning of proportional reasoning in a university quantitative literacy course for law students consisted of iterative action research, in which we introduced various teaching interventions and analysed students’ written responses to assessment questions requiring students to explain their reasoning in situations that call for proportional reasoning. For this analysis we used a modified phenomenographic method to develop and refine a framework to code the responses. This enabled us to broadly describe the responses in terms of the concept of the liminal space that a student must traverse in coming to a full understanding of a threshold concept, and to further define the liminal space to facilitate finer description of students’ responses. Our latest analysis confirmed that many university students cannot reason with proportions, that this kind of thinking is difficult to learn, and that it takes more time than is available in a one-semester course. The context and structure of the questions have a marked effect on students’ ability to apply proportional reasoning successfully. The fraction of students who were classified as ‘at or over the threshold’ (i.e., fairly competent at proportional reasoning) after instruction ranged between 8% for the most difficult question and 48% for the easiest.



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