How do we explain the difference between ratio and proportion to our pupils? In what sense can the fraction ¾ be interpreted as a ratio, and if so is the corresponding ratio 3:4 or 3:1 – or both? Is it sometimes useful to think of a decimal as a ratio, and does it depend on the context: should a “trig ratio” such as sin 30˚ = 0.5 be written as a decimal or a fraction or a ratio or a scale factor? Is the gradient of a line a number or a ratio or a measure of the rate of change – and if the latter, what’s the “thing” that’s changing?
“Ratio, proportion and rates of change” has its own dedicated section amongst six in the new GCSE Mathematics Scope of Study (Department for Education, 2013). The related concepts, however, are present throughout mathematics, in number, algebra, geometry, statistics and probability. This article will look at the detail of one piece of research, and provide some pointers to various resources for exploring ratio and proportional reasoning.
As Watson et al point out in “Key Ideas in Teaching Mathematics” (2013, p. 42), ratio and proportion are extremely difficult to pin down as definitions, which gives an indication as to why pupils have such difficulty with the concepts. Commonly, ratio is described as the quotient of two numbers while proportion is the comparison of ratios. However, these definitions fail to convey the richness of mathematical experience. “Students learn ratio and proportional reasoning through repeated and varied experiences, over time, so that multiple uses of the words and the associated ideas and methods are met, used, and connected.”
The Rational Number Project, based at the University of Minnesota, “advocates teaching fractions using a model that emphasizes multiple representations and connections among different representations” (see Figure 1).
Figure 1. From Cramer (2003)
The project has been running since 1979 and the associated website collates not only a large number of research papers but also provides sets of lessons, based on the research findings, with activities, comments and actions for teacher and student worksheets. One of the papers (Cramer and Post, 1993) explains how three types of task were developed to examine proportional reasoning:
(1) Missing value. This is based the work of Karplus et al (1974). Mr Tall measures 6 buttons high; Mr Small is 4 buttons in height, which is the same as 6 paperclips in height. How high is Mr Tall in paperclips?
(2) Numerical comparison. In these problems the pupil is to compare two given rates. A typical task might involve the comparison of orange drinks made by combining different ratios of concentrate and water and deciding which is the stronger mixture (Noelting, 1980).
(3) Qualitative prediction and comparison. “These types of problems require comparisons not dependent on specific numerical values … Qualitative prediction and comparison problems require students to understand the meaning of proportions” (Cramer and Post, 1993). For example, “What will happen to the fraction 7/8 if the top number gets smaller and the bottom number gets bigger?” (Heller et al, 1990).
Each problem type was posed in four different contexts: speed, scaling, mixture, and density. For example, in Figure 2, Problem 1 is a combination of missing value and speed, Problem 2 combines numerical comparison and scaling. Problems 3 and 4 both use qualitative type tasks – one uses prediction with mixture, the other comparison with density. Potentially this gives sixteen styles of question (given the two types of qualitative task).
Figure 2. From Cramer, K. & Post, T. (1983)
The researchers gave a number of problems to students in US grades seven and eight (equivalent in age to years 8 and 9 in England) and found four distinct solution strategies were used. These can be described as (Watson et al, p. 54)
- identifying a unit rate (“how many for one?”)
- identifying a scale factor (e.g. “3 times as many”)
- matching equivalent fractions (treating rates as fractions and using equivalent fractions to find the missing number)
The researchers found two particular factors stood out: the context of scaling was more problematic while, as might be expected, the complexity of the numerical relationships was critical. Whole number multiples suited the more intuitive unit rate and scale factor strategies and when faced with non-integral numerical relationships students often reverted to inappropriate additive methods. For example, in Problem 1 in Figure 2, an extra 2 miles might be interpreted as adding an extra 2 minutes onto the journey time, rather than multiplying by a factor of 3/2. The cross multiplying strategy, while effective with non-integral ratios, had a tendency to be applied incorrectly.
The researchers recognise the need to start with the more intuitive strategies and familiar contexts, but emphasise the need to ensure that all strategies are experienced and that unfamiliar contexts and non-integral numerical relationships are encountered. They advocate the use of the qualitative problems on the basis that these require the understanding of concepts and do not allow for the blind application of a procedure. Furthermore, this type of qualitative reasoning is a necessary step in numeric problems before the setting up of the actual calculations.
The Nuffield website Key Ideas in Teaching Mathematics, associated with the book of the same name (Watson et al, 2013), also has a section devoted to these concepts, Ratio and proportional reasoning (RPR for short). The authors treat the topic through eight themes, providing research-based guidance on issues to look out for and approaches to use. They also include a number of student activities, from SMILE, NRICH and Bowland, all available on the web, to address various teaching points. Their conclusion (p. 66) from the research is “Nothing suggests that one way of teaching is better than another” but “that learning RPR is a medium term project and not something that can be ‘sorted’ in a few lessons”. “Students learn ratio and proportional reasoning through repeated and varied experiences, over time, so that multiple uses of the words and the associated ideas and methods are met, used, and connected”.
To finish, an anecdote from the above book (pp. 60 – 1). “One of us recently observed a teacher ask a class of 12-year-olds what they knew about ratio. One student said ‘it is something to do with division but I cannot explain it very well’ and the teacher replied ‘well neither can I’, which seems a reasonable response given the complexities above.”
How do you explain ratio and proportion? What activities have your pupils found especially helpful? Which have prompted them to reason deeply and successfully? Let us know, by email to firstname.lastname@example.org or Twitter @NCETMsecondary.
Cramer, K. (2003) Using a translation model for curriculum development and classroom instruction. In Lesh, R., Doerr, H. (Eds.) Beyond Constructivism. Models and Modeling Perspectives on Mathematics Problem Solving, Learning, and Teaching. Lawrence Erlbaum Associates, Mahwah, New Jersey.
Cramer, K. and Post, T. (1993). Connecting Research To Teaching Proportional Reasoning. Mathematics Teacher, 86(5), 404-407.
Department for Education (2013) Subject content and assessment objectives for GCSE in mathematics for teaching from 2015.
Heller, P., Post, T., Behr, M. and Lesh, R. (1990) Qualitative and Numerical Reasoning about Fractions and Rates by Seventh- and Eighth-Grade Students. Journal for Research in Mathematics Education 21, 388-402.
Karplus, E., Karplus, R. and Wollman, W. (1974) Intellectual Development Beyond Elementary School IV: Ratio, the Influence of Cognitive Style. School Science and Mathematics 74, 476-82.
Noelting, G. (1980) The Development of Proportional Reasoning and the Ratio Concept: Part 1-the Differentiation of Stages. Educational Studies in Mathematics 11, 217-53.
Nuffield Foundation Key Ideas in Teaching Mathematics: Research-based guidance and classroom activities for teachers of mathematics.
The Rational Number Project.
Watson, A., Jones, K. and Pratt, D. (2013) Key Ideas in Teaching Mathematics: Research-based guidance for ages 9 – 19. OUP.