Over the summer I have been reflecting on the 9-1 GCSE papers that were sat back in June. In particular I was remembering hearing about and talking to people back in 2013 and 2014 when we were getting the first details of the 'new' GCSE and one of the key aims being to try and make sure pupils are understanding maths rather than just being taught certain procedures in order to solve certain questions. One of the questions that struck me as evidence of this appeared in the AQA Non-calculator papers:
Those people who have taught GCSE Maths for a while will be familiar with the more typical question about averages from grouped tables from the previous specification, which looks a little more like this:
Both of these questions are worth 4 marks but the way those 4 marks are earned is very different. In the second question from the older spec, the marks are given for:
(1) identifying the midpoints of each class as the average time taken for each person in the group,
(2) multiplying the midpoint of each class by the frequency of each class to work out an estimate for the total time taken for each class of people,
(3) adding these estimates to give an estimate of the overall time taken for all 40 people, then
(4) dividing the estimate of the total time taken by 40 to give an estimate of the mean time taken.
The point here is that many teachers, and I include myself in this during my early career days, would approach the teaching of this concept without any of the explanation I have given above, simplifying the whole thing to a straightforward procedure:
(1) Write down the midpoints of each class
(2) Multiply the midpoint by the frequency
(3) Add your answers together
(4) Divide by the total of the frequencies.
The point is that in the past, maths teachers could get away with this because every question that asked about mean and grouped data was structured in precisely this way, even if the values were different. There was therefore no incentive for teachers (beyond their own intrinsic wish to teach pupils good maths rather than teach them to pass exams) to teach any semblance of understanding for this concept - provided pupils can remember the four steps they can answer the question on this topic.
Contrast that with the first question for the'new' 9-1 GCSE. Provided a pupil is not going to simply guess at the correct answers (which I admit is a possibility), then what should be clear is that the level of understanding of mean and range required to answer the question is significantly greater than the second question. To confidently answer the new question a pupil needs to have quite an understanding of how mean and range link to distribution, what can be inferred about the distribution from the grouped table, and also what mean and range measure about a distribution. If a pupil were to carry out the steps above (getting a correct answer of 34 minutes) they might even come to the mistaken conclusion that the only place the mean could be would be in the 20-40 class. This might be enough to perhaps score 1 mark, but certainly not the 4 marks it would have secured in the past.
This, for me, illustrates the importance of teaching maths for understanding rather than just as a set of procedures. Of course it would be quite right (in my opinion) to say that it was always important to try and teach maths for understanding, and that as teachers we should always be trying to develop understanding in our pupils. What is nice now though is that what many people see as the ultimate 'end-goal' of our teaching, the pupil securing a good GCSE grade, doesn't allow for recourse to procedural only approaches. There have been many critics of the new 9-1 GCSE, and for certain things I have been amongst the most vocal of them, but I will consider it all worth it if it means that teachers have to move away from teaching 'maths' as answering questions by following a sequence of steps and begin to try and teach maths 'these are the concepts, skills and knowledge you need and these are how they relate to each other'.