I have turned some of my attention back towards task design. I started my website in September 2018 thinking I’d have a few hundred tasks by now. I hadn’t anticipate leaving the classroom later that year! It’s difficult to find the need to write new tasks when there are no pupils to try them with. However, I’ve maintained an interest in the topic and, coupled with extensive reading have set myself the goal of writing a task-a-day for the whole of April. One of the interesting things about task design is that there are three levels:
The task as intended by the writer
The intention of the teach who selects the task
What the pupil actually does with the task
I often try to write a couple of sentences under my tasks so the teachers downloading them can get a sense of what I had intended.
This evening I wrote a task on asymptotes. You can find it here: Asymptotes PDF
I am basing this upon the assumption that there has been some teaching which has conveyed a few key facts
Vertical asymptotes: Occur when the denominator equal zero
Horizontal asymptotes: We can simply divide the coefficients of the term with the highest power, if the degrees are equal. This gives horizontal asymptote. There is also the common case of y= 0 to consider.
Oblique asymptotes: Long division is ordinarily required, but we can get a rough sense of the asymptote by dividing terms of highest power in numerator and denominator.
I thought it might be interesting to elaborate a little more upon some of the design considerations.
1. At Advanced Higher/ A level, there is so much complexity and depth that it is hard for pupils to get a sense of ideas, when there are SO many ideas to grapple with. Becoming procedurally fluent in all of these is, in itself, a challenge. It might be the case that pupils need to do lots of lengthy practice in order to get a true sense of the principles underlying some topic. With this in mind I decided to write an asymptotes task. I didn’t want pupils to have to do much (if any) written mathematics, until near the end of the task. I wanted them to encounter the key decision process repeatedly in quick succession. This doesn’t happen if we do full questions where we actually establish asymptote equations, find the behaviour as the curve approaches and sketch them from scratch. The intention here was to reduce the technical demand to focus on the key ideas. A matching task seemed about right for this purpose.
2. One of the things I’ve learned from Malcolm Swan’s work on the Standard Units is always to leave a few of the cards blank. This means that as the task nears the end, it cannot be solved by simple process of elimination. Instead, there are two functions which do not have the corresponding graph on a card. This means that pupils will have to generate possible graphs for these ones by themselves. It is not my intention that pupils do this using algebra, but instead, sketch a *rough* sensible graph based upon what they have picked up from the rest of the task.
3. There is an assumption that the majority of pupils engaging with mathematics at this level are interested in the subject and might pursue study of it as part of a mathematics or related degree at university. As such, I like to create opportunities for them to go slightly beyond the bounds of the curriculum and to think deeply and richly about ideas. These opportunities give them a chance to behave mathematically, while at the same time potentially strengthening their understanding of the principles which govern the topic they are working on. John Mason and Anne Watson refer to the example space – the range of examples pupils have encountered related to some idea. By extending this a little, I hope to provide some more insight. There is a real challenge at the end of this task, which I hope serves this purpose. I have included a graph which is of a function with a parabolic asymptote. This isn’t assessed in Scotland at school level, however, the thinking is accessible. If pupils understand that the result of the long division is a quadratic, then all they need to do is produce a polynomial of degree 4 in the numerator and have a quadratic on the denominator.
4. I anticipated that if the questions had thoughtlessly varied vertical asymptotes, then the complexity of it would be greatly reduced. It is easy to see vertical asymptotes and pupils would match using this. Instead, I chose to utilise some variation theory in the design of the task so that there were various rational functions with the same vertical asymptotes. Pupils would then have to focus on some other key features to refine their decisions.
All of this might make sense if you download the task and try it out for yourself. Hopefully I’ve managed to pull something together which is of use. If you can think of any further refinements I’d love to hear about them – please comment below!
5. I create the cards in corresponding order, so that the two grids match up with each other and then I randomise!