Cumulative Frequency: Guess the Misconception
This week on Guess the Misconception we turn our attention towards everybody’s favourite way of representing grouped statistical data, the cumulative frequency diagram.
What do you think the most common incorrect answer to the following question is?
And here are the results from a question that has actually been answered over 12,000 times:
So, we have a question that 1 in every 3 students get wrong, with Option B leading the way in erroneous thinking. But why do students get this question wrong? Well, let’s dive into some student responses in a bid to find out:
Answer A
Here we have students who have recalled something very important about cumulative frequency diagrams — they are always plotting on the upper boundary of the group, as this explanation demonstrates:
You have to look at the last number so in this case 20. The first number is 0. So from 0 to 20 it is 20
That may well be true, but of course that does not mean this upper boundary should be placed in the cumulative frequency column.
Answer B
Here we have the most common choice of incorrect answer. Why would a student go for 15? Well, more than likely because they are so used to being presented with a grouped frequency table and going straight into the well-rehearsed algorithm of finding the mid-point, doing mid-point multiplied by frequency, and going on to calculate an estimate for the mean:
I think because its half way between 10 and 20, but not sure
Or, perhaps more worryingly, there were a number of students who correctly spotted that this was a question about cumulative frequency, but still found the midpoint:
Because the cumulative frequency is the number in between the two values
Answer D:
The answer of 10 was equally as fascinating. This seemed to stem from students recalling that they needed to add something together to find the cumulative frequency, but then getting a bit muddled as to exactly what that something should be:
I think this is the answer because the frequency is 10 add 10 which is the cumulative frequency
or:
Because it is the distance between the 10–20, 20–30 etc????
Summary
I love this question. Not only does it do what any good diagnostic question should do and allow me to quickly and accurately understand exactly why my students are going wrong, but it does so by focusing on a very specific, but incredibly important aspect of the multi-step process students must go through to correctly draw a cumulative frequency diagram — namely, correctly filling out the table. And I can assess this in the classroom in about 30 seconds see my blog post about using diagnostic questions in the classroom), without needing to give out graph paper and rulers. And indeed, if students cannot answer this question correctly, then getting out the graph paper and rulers would only be asking for more trouble as we open up the possibility of further misconceptions developing on top of existing ones. Let’s breakdown the complex process, isolate the required skill, assess it, and help students get through it.
So, what do all these wrong answers have in common? They all come from a misremembered or misapplied rule. Whether it is a student vaguely recalling something about the upper bound being important (answer A), or remembering that anytime they are faced with a grouped frequency table it is usually a good idea to find the midpoint (B), such a shallow understanding of cumulative frequency will inevitably lead to problems.
So, I have two suggestions.
The first is to ensure the complex process of drawing a cumulative frequency diagram is first broken-down into several sub-processes, and that each of these are looked at in turn. So, starting with a discussion of what cumulative frequency is, followed by focused practice on completing the table, would be a good start. This then can be followed up by a focus on how to plot specific points, using a question such as this:
Only then do we need to venture into the territory of plotting curves and calculating the median.
And then, when the sub-processes have been mastered, we can then turn our attention to the kind of rich, thoughtful activities complied by Don Steward, such as this one which requires students to work backwards from completed curves to fill out the tables:
Why not try this diagnostic question out on your students, either in class or as part of a homework, and see how they get on? Talk about the correct answer, and also the wrong ones. Better still, you can ensure students receive a regular diet of quality questions like this — together with all the teacher insights you can ever want-by setting up our free schemes of work. We have free maths schemes from Year 1 to GCSE, with all the awarding bodies represented. Just click here to get started.
Subtle advertisement alert: My book, How I wish I’d taught maths, contains an entire chapter dedicated to the practicalities, benefits and considerations when using diagnostic questions in the classroom, is available to buy from Amazon and John Catt Educational Ltd.
Have a great week
Craig
