In 2006, then-Education secretary Alan Johnson (had a lot in the mean time) announced that he was going to scrap the coursework element of the GCSE Mathematics in favour of 100% examination-based assessment. This move then came to fruition the following year.
This was a move welcomed by the examination boards, who acknowledged the level of dishonesty that was developing in the system, thanks largely to the proliferation of internet access to students; this provided less scrupulous students with easy access to solutions and pre-written assignments, and arguably devalued the GCSE Mathematics qualification.
However, the tasks have never completely disappeared, and can easily be found within the recommended resources from each of the exam boards. This is due to the fact that, regardless of their value in terms of external assessment, they still provide us with an opportunity to get students exploring, applying their understanding and organising their thinking.
The two of us have considered whether we could potentially use some of the tasks in our own classrooms and across the department as a means of assessing students towards the end of Years 7-9, as well as providing rich tasks to challenge our Year 10s.
Coincidently, I was recently at a training course on “Proof and Reasoning” hosted by FMSP and was discussing GCSE coursework in the classroom. Once we had got over the fact I was the last year to actually complete the coursework as a student, we discussed the benefits of using the coursework investigation to develop not only reasoning skills but also to allow students to progress through the levels described by Waring (see table below) and furthermore the investigations allow students to see the need for a proof, rather than proofs for proofs sake.
So in order to gain firsthand experience of using GCSE coursework, the OCR “Opposite Corners” task was trialled with a top set year 10 class in the final week of term.
All students in the class had the mathematical understanding necessary to engage with the investigation at a high level, which was important in selecting a task to run with them. As with almost all of the coursework investigations available, I believe their real strength lies in their ambiguity – in this case, “Investigate opposite corners on grids,” or often, after an initial set up, “Investigate further.”
We decided to give the task some structure in order to give students a route in; we first of all provided a 10 x 10 grid, and demonstrated to students how the task worked by getting them to pick a 2 x 2 box within the grid, multiply the opposite corners, and find the difference.
What was striking was the prompting required for students to ‘generalise’ the problem algebraically, although this prompting certainly differed in its extent; some simply needed to be asked the question “How could I make this grid more general?” while others needed some further support to see how this could be done. Once prompted, the majority of students were able to demonstrate a proof as to why their difference between the products was always the same. They also seemed to value the proof far more than if they were ask to prove it from the outset.
When students had reached this stage, they were provided with a ‘lucky dip’ of variables to adjust in order to investigate further, rather than simply throwing it completely open, since this was a trial – it was felt that a guide would keep students focused on the problem, although we were happy for students to then decide where they wanted to take their investigation next if they had an idea in mind. For example, they could see what happened with a 3 x 3 set of numbers on their grid, or maybe even a 2 x 3. They might also change the size of the grid to a 12 x 12 or a 15 x 15. We emphasised the importance of only changing one variable at a time.
In my experience, one the aspects of problem solving that students struggle with the most is in their understanding of how useful it can be to generalise the variables within a situation, and when this is an appropriate approach to take. From this experience, and also in some of the lack of resilience we often come across in students, it became obvious that this is something that really needs to be developed in our students, not simply because it will allow them to breeze the proof questions on the GCSE paper. I also believe that it is a skill that needs to be taught, with practice, and perhaps earlier than we tend to approach problems involving such algebraic manipulation.
The GCSE coursework tasks certainly provide students with an opportunity to get to grips with the ideas of mathematical investigation. I like the fact that the opportunity is given to really investigate mathematically, without the “pseudo-contexts” that we sometimes fall into the trap of using – I’m planning a post involving my thoughts on Greg Ashman’s recent post on real-life mathematics to follow soon. It is for this reason that we are also considering how some of these investigations, or similar, could be used with our Year 7 to 9 groups; just to expose them to this kind of reasoning and investigation, to assess their ability to do so, or even both. Our initial thinking is that we would have to select the tasks carefully for each year, as it is important that students possess the necessary mathematical skills in order to really engage deeply with each tasks.
We would be really interested to hear your thoughts on this use of the GCSE Mathematics tasks. Do you see them as being a potentially viable route for assessment (provided they remain entirely classroom-based)? Have you tried something similar in your own departments that has worked fantastically well, or even fallen flat? Let us know, we would love to hear from you.