One of the reasons many Year 7 and 8 students experience difficulty interpreting and using ratios, rates and percent, is that they have not yet acquired a capacity for proportional reasoning. This is a complex form of reasoning that depends on many interconnected ideas and strategies developed over a long period of time. These features are amply illustrated by Lamon’s (1999) description of proportional reasoning as “the ability to recognise, to explain, to think about, to make conjectures about, to graph, to transform, to compare, to make judgements about, to represent, or to symbolise relationships of two simple types … direct … and inverse proportion” (p.8). At its core, proportional reasoning requires a capacity to identify and describe what is being compared with what. Essentially, there are two types of proportional reasoning problems, both of which require some form of comparison. The first typically involves a comparison of two rates, eg, Which car had a faster average speed, Car A which travelled 217 km in 1¾ hours or Car B which travelled 204 km in 1½ hours? The second type is referred to as missing value problems. These problems typically provide 3 quantities and the fourth is missing, eg, If a supermarket worker can unpack 24 boxes in 1 hour, how many boxes could he unpack in 10 minutes?
Recognising what is being compared with what is not always straightforward. It can be confounded by the types of quantities used, how they are represented, and the number of variables involved. Also, not all problems in which 3 quantities are given and a fourth is missing require proportional reasoning. As Lamon notes, “There are no shortcuts available here! Thought, common sense, and experience must be used to determine whether a situation is proportional or not. You must always bring into play your knowledge about how things work in the real world” (P.225).
While this is undoubtedly true, proportional reasoning also requires a capacity to work flexibly and confidently with the quantities involved (that is, measures, rates and/or ratios expressed in terms of natural numbers, rational numbers, and/or integers), and an ability to recognise multiplicative relationships in a range of problem contexts including the idea of rational numbers as operators (eg, understanding ⅔ × $24 as ⅔ of 24, or 3.5 × 68 as 3 and a half times 68). Neither of which, according to recent research, can be assumed to be in place for all students at this level of schooling.
This could be due to/associated with:
- a lack of access to a broader range of ideas for multiplication, in particular, the Cartesian product or the ‘for each’ idea which underpins rate (see Level 3);
- an over-reliance on the quotition (‘how many groups in’) idea for division, and/or a failure to recognise the role of partition division in relation to fractions and decimals;
- an inability to construct fraction models beyond those in the halving family and/or insufficient experience with making/representing, naming, comparing, and renaming mixed fractions and decimal fractions greater than 1 (see Level 4);
- insufficient exposure to a broader range of continuous (part-whole) and discrete (subset-set) fraction representations where the unit varies (eg, “What is the unit if the brown Cuisenaire rod is 1 third? What is the unit if the brown rod is 2? …);
- a reliance on rule-based procedures to rename fractions (eg, “to change a fraction into a percentage multiply by 100 over 1”) which suggests an inability to reconceptualise a quantity in terms of different sized chunks or ‘units’ (eg, ¾ is 75% because we can think of 3 (25-units) out of 4 (25-units) or 75 out of 100);
- a limited understanding of what it means to multiply a quantity by a rational number, for example, a belief that multiplication always mean bigger (whole number thinking)or that multiplication by a fraction always results in smaller (thinking limited to proper fractions), leading to a reluctance/inability to work with mixed fractions and rational numbers as factors/scalars (eg, recognise that the dimensions of a photo may be enlarged or reduced by rational factors such as 250% or 3/5 respectively);
- a limited understanding of ratio (eg, assumes 1:4 is always the same as ¼) and/or a tendency to think about change in absolute rather than relative terms (eg, 2 is to 3 as 5 is to 6 because “3 was added to both sides”)
By the end of Level 5 students need to be able to formally recognise and work meaningfully with a wider range of numbers, in particular, natural numbers, integers, and rational numbers in whatever form they appear (eg, proper fractions, mixed fractions, finite or recurring decimals, percentages, and rates). They also need to be able to work with a broader range of multiplicative situations including direct proportion and ratio.
A key indicator of the extent to which students have developed a broader range of ideas to support proportional reasoning is the extent to which they use multiplicative strategies such as partitioning to solve problems involving simple proportion (eg, find for 1 or a common composite unit such as 3 in a comparison involving 6 and 9, then multiply as appropriate). Another indicator is the extent to which students can work meaningfully with multiple representations of proportional relationships. For example, given a collection of 7 brown eggs and 5 white eggs, recognise that the following representations can all be used to refer to different relationships in this context depending on what is viewed as the unit:
5/12, 7/12, 5:7, 7:5, 5:12, 7:12.
A major issue here are the meanings students ascribe to the symbolic representation of ratio. Ratio involves the comparison of any two quantities, “it always makes a statement about one measurement in relation to another” (Lamon, 1999, p.13). For example, ratio can be used to compare the number of feral cats to the size of the national park. More specifically, ratios can be used to compare measures of the same type, for example, the number of feral cats compared to the number of feral dogs. In this case, two types of comparison are possible, for instance, one can compare the parts to the parts (eg, cats to dogs) or the parts to the whole (eg, cats to the total number of cats and dogs). Ratios can be also used to compare measures of different types, which is often but not exclusively referred to as rate (eg, the number of feral cats per square kilometre). And, although it is often assumed, ratios are not always rational numbers (eg, the ratio of the circumference of a circle to its diameter)