‘By three methods we may learn wisdom: First, by reflection, which is noblest; Second, by imitation, which is easiest; and third by experience, which is the bitterest.’ –Confucius
Since the 1970s, which is as far back as my research took me, maths educators have all been asking the same questions about the same issues on why so many students find ratio and proportion so challenging. Why are the students of each generation making the same mistakes on this challenging topic and how can we fix it?
Well, approximately 50 years into the future and we are (or at least I am) still asking these questions. While the challenge of teaching and learning Ratio and Proportion is not unique to the United Kingdom The National Centre for Excellence in the Teaching of Mathematics (NCETM) has been taking a proactive approach to solving this problem in hopes that the next generation will no longer struggle with these issues. It is by being involved in NCETM Challenging Topics at GCSE Maths Hubs that I have gained confidence in teaching this topic. (See Maths Hubs link to get involved or view their projects: https://www.mathshubs.org.uk/)
Three years ago I would get a bit nervous when it was time to teach ratio & proportion especially to a year 10 or year 11 class. Why? Well, my only key tip was always ‘the key students, is to find the value of 1 unit, then you will be able to find any amount even 1 million’. But even with that tip, some of my best students were not sure how to start the more sophisticated problems. They needed a structure to help them see the problem and
a) I didn’t think of that
b) none of my research then, which was obviously limited, gave me any helpful answers – it was all abstract.
Who got it right?
Singapore mathematics education, the envy of the world, has gotten it right. They have reflected and embed a culture of using Concrete Pictorial Abstract approach to teaching mathematics. Many of us are familar with this CPA approach but don’t use it all the way, especially in secondary schools, we often ditch the C & P and go straight to the A.
However, much to their advantage, the Singaporeans used this approach thoroughly to develop a nation of problem solvers. So how does this relate to ratio and proportion? Two words: Bar Models. The bar models (rectangular boxes) is a pictorial approach that can be used to help students see the structure of a problem and then figure out what missing information is needed. The NCETM have been promoting this model over the years and it is a model that I can now safely and confidently say has helped me to understand ratio and proportion much better and thus, my students have been and will be the recipients of this knowledge.
Do the maths:
Here are three basic, but essential problems that a number of students struggle with, especially number 2 and 3:
1. Will and Olly share £80 in the ratio 3 : 2. Work out how much each of them get.
2. Rajesh and Gudi share some money in the ratio 2 : 5. Rajesh receives £240. Work out the amount of money that Gudi receives.
3. Pritam, Sarah and Emily share some money in the ratios 3 : 6 : 4. Sarah gets $15 more than Emily. Work out the amount of money that Pritam gets.
As you look at these problems, I would like you to think about how you would have solved each problem prior to seeing the bar modelling. Also think about the misconceptions or the mistakes that your students (weak and middle ability) would have likely made at their attempt at each problem and how you would have helped them to understand the problem(s).
Attached is my demonstration of the bar modelling of the problems above:
Finally, the NCETM (https://www.ncetm.org.uk/) is free to subscribe and has more on this and many other useful resources on best practices in maths education.
As always, I am keen to hear from you as I aim to better my best and hopefully inspire reflection and action among other practitioners.