I am not a huge fan of rules in maths. Specifically, rules without understanding. There are just too many rules in mathematics to remember, and if the underlying understanding of where these rules came from is not there, then they are likely to be forgotten and/or misapplied.
Working out the mean from a frequency table is a classic example of this. In the past, I have been guilty of essentially saying to students: “just remember: multiply those two numbers together, add up that column, divide it by the total of that column, write down your answer, get your 4 marks, move on and don’t ask any questions”.
But of course, come the exam, with a thousand other dodgy rules swimming around in their heads, before you know it students are adding up and dividing by anything and everything.
This question – and specifically, the discussion that will hopefully follow it in the classroom – is designed to find out whether your students actually understand what they are doing when calculating the mean from a frequency table. It is a lot quicker than asking students to work out the mean themselves. All the calculations have been done for them, and hence we can get straight to the heart of the matter.
Here is what you might learn from each response:
Answer A) This is a classic one. Students who have been taught that to find the mean you divide by “the number of things that there are” may well (understandably) think that the number of things in question here are the rows, and there are 7 of them.
Answer B) is very interesting. In the past, students giving this answer are the ones who are very dependent upon rules. They have remembered that the totals that matter are from the 2nd and last column, but then they have done the division the wrong way around. This may indicate that they don’t really understand what they are doing.
Answer C)is the correct answer
Answer D)suggests the students have remembered the final column is important, but have decided to divide it by the sum of the 1st column, again possibly relating this to adding things up and dividing by how many there are.
When students have given their answers, ask them to explain their reasoning for each one. Give no indication yourself which is right or wrong. Just let the students voting for A give their reasons, then B, then C, then D. Then let the students re-vote. Hopefully they will have resolved their own misconceptions by simply listening to each other.
Of course, much of the confusion emanating from the mean can be eradicated if students look at their answer in the context of the question. A and B give answers that simply do not make sense as a measure of the average hours of sleep, and there is little chance of D being correct if students really think about it. But do students think about it? I know mine often don’t. They get their answer and move on.
So, another interesting way to ask this question might be to present the students with the 4 answers instead of the calculations, and see if they can get the correct answer from that. I bet more would get it right, which sounds counter-intuitive if you think about it, as they have less information.
Right, I am going to create a series of questions that test exactly that!
In the meantime, please try out this question in your classroom and let me know how your students get on. Challenge them to explain where each of the wrong answers come from, to suggest alternative incorrect answers, and to write diagnostic questions of their own.