Multiplicative reasoning

Multiplicative reasoning is a key competence for many areas of employment and everyday life, and for further mathematical study. It is however a complex conceptual field. The ICCAMS project, with multiplicative reasoning as one of its two focus themes, has in Phase 1 conducted a broadly representative survey of attainment which suggests that standards in this area have not risen since the 1970s and that relatively few students are achieving competence in the relevant areas of the national strategy Key Stage 3 framework. Student difficulties are illustrated by evidence from group interviews in Phase 2 of the project.

The ESRC project ‘Improving Competence and Confidence in Algebra and Multiplicative Structures’ (ICCAMS) selected these two areas of the curriculum because of their pivotal role in the Key Stage 3 (age 11-14) mathematics curriculum, and especially for the further study of mathematics and for its functional application. While algebra centrally underpins the whole of the further study of mathematics, it is the basis for only the more sophisticated applications. In contrast  multiplicative reasoning is the foundation of most mathematical applications and is relevant to all pupils. This paper will examine aspects of performance in multiplicative reasoning that students find difficult and which form a major barrier to developing competence and confidence in functional applications. The delineation of the conceptual field of multiplicative reasoning is complex (Harel and Confrey 1994, Confrey et al. 2009). While there are other aspects of multiplicative reasoning e.g. those concerned with combinatorics or calculation of areas and volumes, applications of the ratio/rate model are by far the most common and only these will be discussed in this article.

Broadly speaking, the main contexts of ratio/rate application are those where two or more values are being compared, and/or where one value is being scaled up or down to give another. Sometimes these comparisons or operations are appropriately expressed additively and involve the ideas of difference (‘a is d more/less than b’); more often they are multiplicative (‘a is r times bigger/smaller than b’). The values being compared may be of essentially the same quantity and the comparison may be related to two or more different ‘things’, or to one ‘thing’ at two or more different times (numbers of boys and girls; changing numbers of boys). Especially where the quantity is a discrete variable, or where the multiplying factor is a whole number or familiar fraction, this relationship is often expressed as a ratio a:b. Where the comparison is between two variables which refer to different quantities measured in different units, the relation is usually expressed as a rate a/b which takes the form of a single number to which is attached a composite unit like miles per hour or £ per capita derived from the units in which the two variables are measured. These comparisons are equivalent in reverse to scaling up a value to get another value, respectively by using a ratio or scale factor, or by using a rate.

Whereas a specific value of a rate is a single number with a composite unit, a specific ratio a:b can be regarded as a set of pairs of numbers which is associated with two ‘dimensionless’ rates a/b and b/a. Such a dimensionless rate is often described as a proportion, especially when it is expressed as a fraction or percentage and refers to a  comparison between one contributing part and a whole collection (‘what proportion of the class are boys?’), or to similar geometrical figures (‘have the same proportions’) . ‘Direct proportion’ is also used more generally to describe a multiplicative relation between two variables.

Rates often but not always involve the variable of time and are very commonly used in finance and economics (e.g. GNP per capita, rates of interest, and exchange), in the physical sciences (e.g. speed, density, power, pressure) and in health (e.g. rates of growth, medicine doses). Other applications use dimensionless rates (proportions), for example probability and risk, and enlargement through scaling in spatially focused professions such as architecture, design, and engineering. Thus all the ratio/rate applications have in common the two processes which constitute multiplicative thinking: the derivation of a rate (or ratio or proportion) from two corresponding values (a/b) of two or one variables, and the use of a rate (or ratio or proportion) to calculate an unknown value of one variable given a corresponding value. These require respectively the operations of division and multiplication, with division if anything taking the predominant role.

In our primary curriculum the aspect of multiplication which is still most emphasized is that of repeated addition (‘add three five times’) rather than that leading to multiplicative reasoning and ratio (‘five for every one of three’, ‘five times larger than 3’). This is even when the ratio meanings have been demonstrated to be easier (Nunes and Bryant 2009).

Multiplicative reasoning starts in the primary school with whole number quantities and whole number scale factors. Later in primary schools and especially in Key Stage 3, it becomes tied in strongly with rational number reasoning. Dickson et al (1984, 287) note there are two meanings of rational numbers, those related to measurement (1.7 metres) and those to operators – essentially rates or scale (multiplying) factors (1.7 times as large); Confrey et al. (2009) separate off from the latter a further meaning, that of ratio, but there seems little justification for doing so since ratios are so closely related to the operators of rates and scale factors. Again, the non-ratio ‘measurement’ meanings of rational numbers as a measure of a fractional part of a spatial whole, or as a decimal number on a number line, are the ones that tend to be emphasized in the primary curriculum, although there is some mention of meanings concerned with ratio (‘3 for every 5’, ‘3/5 of’). Perhaps not surprisingly, given the complexity of the conceptual field on which it depends, and current emphases in primary schools, in the early years of secondary school students experience problems with multiplicative reasoning. 

Multiplicative reasoning is clearly an area which is key to both a large number of mathematical applications and to further study in mathematics. Yet the evidence presented here suggests that students have a very weak grasp of important aspects of the conceptual field and that understanding in this area has not improved since the 1970s. This is not because these things are not taught as most of the students in the sample will have experienced teaching of the relevant ideas. We have to conclude that the teaching has not been very successful, and it is possible to speculate on many reasons why this should be so. For example it may not have been appropriately related to students’ prior understandings, or may not have been sustained enough and broad enough to have had a permanent effect. In Phase 2 of the project we will be developing an intervention based on formative assessment and more sustained periods of teaching to see if this improves the learning of these important ideas.