Today I attended an MEI CPD event on the “Principles of the New A-Level.” I came away with a lot of ideas and things to think about further as I continue to work towards starting teaching the new syllabus in September. One of the new “Overarching Themes” in the specification for A-level maths is problem solving and it was this that my thoughts have kept returning to this evening. What is problem solving in mathematics, and specifically what is it that makes on question a problem solving task?
Take the picture here (one of the tasks we worked on at the session). What might students be asked to do? As an example we might ask “find the area of the rectangle”. In the previous specification students were likely to be guided to a solution, with parts of a question asking them to find relevant, intermediate, pieces of information such as the equation of the line that joins A and D. The idea of removing this and having to think really appeals to me as a mathematician but scares me as a teacher.
What interested me most about this was my own response to then thinking about how I would approach the question. I initially decided that I should work out the coordinates of the points A, B and D, then work out the lengths of segments AD and AB and multiply them together to find the area of the rectangle. As I started working through the problem it quickly dawned on me that having worked out A, B and D it was much more efficient to work out the area of the triangle made by them, and then double it, rather than using Pythagoras theorem twice. While my initial strategy would have led to a correct solution, my adaptation of it led to an easier process. However, if I had not been able to think of a strategy at all I would not have been able to start the problem.
Another aspect of the new schemes is the use of technology. Some of these problem solving tasks are better explored with dynamic software. We looked at a geometric method for showing that sin (a + b) = sin a cos b + cos a sin b. This looked at drawing a rectangle and dividing the area up in different ways. One area students may struggle to grasp is that this is true for all values of a and b. By creating a dynamic model we are able to illustrate that it holds for different values and we can then generalise.
We are going to have to make sure that we build enough of these sorts of tasks into our lessons so that students have experience of the thought processes required to answer them. This will obviously require time to allow students to develop their skills. Something that I am currently looking at to help free up time in the classroom is flipped learning, but that is a topic for another day.
I am off to try and make a geogebra applet that shows sin (a + b) = sin a cos b + cos a sin b!
[My first attempt at the geogebra is available here: https://www.geogebra.org/worksheet/edit/id/qtNKm2pa Needs some work on the labelling!]