Proportional reasoning as a threshold to numeracy at university: A framework for analysis

Abstract

There is a generally acknowledged need for students to be quantitatively literate in an increasingly quantitative world. This includes the ability to reason critically about data in context. We have noted that students experience difficulty with the application of certain mathematical and statistical concepts, which in turn impedes progress in the development of students’ critical reasoning ability. One such concept, which has the characteristics of a threshold concept, is that of proportional reasoning. The main focus of this article is a description of the development of a framework using an adapted phenomenographic approach that can be used to describe students’ experiences in the acquisition of the concept of comparing quantities in relative terms. The framework has also helped to make explicit the elements that constitute a full understanding of the requirements for the proportional comparison of quantities. Preliminary results from using the framework to analyse students’ responses to assessment questions showed that many students were challenged by proportional reasoning. When considering the notion of the liminal space that is occupied en route to a full understanding of a threshold concept, about half of the students in the study were at the preliminal stage of understanding the concept and very few were at the threshold

Proportional reasoning ability of school leavers aspiring to higher education in South Africa

“…In their literature review, Tourniaire and Pulos (1985, p. 185) noted that this strategy of simply comparing the numerators (and ignoring the denominators) is 'a developmentally primitive strategy', but that 'it is still used with some frequency in adolescence'. We have also seen that this strategy is used by about half of the students in a university quantitative literacy course for law students (Lloyd & Frith, 2013). The results for Question 7 reveal that even in the top quarter, under two-thirds of the candidates could successfully compare fractions by reasoning alone (admittedly in a fairly complex, reading-intensive question).…”

“…From our experience of teaching quantitative literacy to university students, we have observed that proportional reasoning is a troublesome concept; it is difficult to learn and takes a long time to learn. Consequently, we regard proportional reasoning as a 'threshold concept' (Meyer & Land, 2003) for all academic quantitative literacy (Frith & Lloyd, 2014;Lloyd & Frith, 2013). This threshold concept, once mastered, opens a gateway to thinking differently about quantities.…”

Investigating Proportional Reasoning in a University Quantitative Literacy Course

“…The threshold is reached when proportional reasoning is used automatically and the logical process of this reasoning can be expressed coherently and concisely, making use of appropriate language. We were thus able to conclude that most of our students, in reasoning with absolute numbers rather than relative numbers, had not reached the threshold of reasoning qualitatively about quantities involving fractions -in fact, many were still at the pre-liminal stage of understanding proportional comparison (Lloyd and Frith 2013).…”

“…The proportion of students who were reasoning with absolute quantities was disappointingly high. For a detailed description of this initial process, see Lloyd and Frith (2013).…”

“…In question [8] students had to quantify the relationship between the population sizes (d) given values for murder rates (r) and number of murders (n) thus: 1 ≈ 2 and 1 ≈ 4 2 → 1 ≈ 4 2 . Student responses were coded using a refinement and synthesis of the frameworks used previously Frith 2013, Frith andLloyd 2014), which is shown in Table 2. Unlike those used earlier in our study, this more generic framework can be used for analysing responses to questions where the fractions describe percentage change or rates, such as birth rates or mortality rates.…”