 # Summer School Insights: Week 7 Ratio Craig Barton

This is the seventh post in our Eedi Summer School series, where I provide insights from the cohort of Year 6 students who are taking part in our seven week programme. My hope is that this may give you an indication of where our Year 7 students might struggle come September..

This week we turn our attention to Ratio.

Our group of 10 and 11 year olds struggled a little with this quiz, scoring an average of 66%, which is the lowest of the series. Was this summer fatigue setting in, or are there misconceptions about ratio lurking that we need to get to the bottom of? Let’s find out!

Here are the proportion of students getting each question correct in reverse order from the most poorly answered to the best answered:

We can see that Question 5 is the one that caused our group the most problems. What is Question 5, I hear you ask? It is this…

Take a moment to consider where you predict students might go wrong with this question.

Here are the results from our Summer School students:

Why did more than half of our students get this question wrong? And what was so alluring about the most common wrong answer of “neither is correct”?

Let’s have a look at some examples of student explanations to see if we can get to the bottom of it:

Tom is not correct because the difference between 4:10 and 6:15 is different. Katie is wrong because 10 and 11 cannot be found in the same time tables.

Neither is correct because none of the numbers are multiples of 4:10

How can 4:10 be equivalent to 6:15 and how can katies be equal?????

As with all these misconceptions, I find it useful to ask myself two questions:

• Why are students going wrong?
• What am I going to do about it?

I love the exasperation in that late explanation.

I find the results surprising here. One of the most significant changes to the Maths GCSE specification in England in the last few years has been the increased emphasis placed upon ratio. I remember the days when so long as students could simplify, find equivalent ratios and do a bit of sharing, they were laughing. Now - rightly so, I believe - students are being assessed on their understanding of ratio at a much deeper level both in terms of the concept itself, and also how it weaves its way into other mathematical areas, including algebra and geometric reasoning.

But to get this particular question correct, you don’t really need any of that deeper thinking. You just need to be able to find equivalent ratios. So, why is it so difficult?

Well, as the explanations hint at, the problem seems to be that there is no obvious way to get from 4:10 to 6:15. Specifically, there is no integer factor. Students who have been introduced to equivalent ratios by going from, say, 4:10 to 8:20 or even 12:30, and then practiced with similar questions are going to struggle with questions like this - questions which either require a knowledge of non-integer factors, or a more flexible approach like simplifying the 4:10 and then working from there.

Once again, for me this all comes down to the examples we choose for our students and the practice questions we give them. They are what shape our students’ understanding of an idea. If we limit the scope of these examples and practice, then we limit our students’ appreciation of the boundaries of the concept.

I have been guilty of this for many years. These days I like to present my students with sequences of examples such as this:

And letting the discussions, arguments, and different representations to support their point of view flow!

By including “the unusual” early on in a child’s experience of an idea, the unusual becomes part of the norm.

You can find similar sequences of questions and examples on my Variation Theory website.

Let me show you the second and third worst answered questions from this quiz, together with the proportion of students opting for each answer:

Again, it might be useful to ask yourselves the questions:

• Why are students going wrong?
• What am I going to do about it?

You can access the whole quiz here, and the Insights page that will give you access to the percentages and student explanations here.

More big news from DQs and Eedi coming shortly, but in the meantime, take care of yourselves, thanks for reading and stay safe

Craig and the Eedi Team Written by
Craig Barton