For a long time I’ve held negative views towards numerical methods, particularly “trial and improvement” and the trapezium rule, but I’ve been reconsidering those views. This has been quite a long process that began when Tom Bennison (@DrBennison) questioned negativity towards them, probably around a year ago. We had a brief discussion around them and some of the thoughts have been stewing since.
Tom reminded me that numerical methods are important as in the real world there are many things that cannot be done another way (yet!). The discussion left me thinking that rather than numerical methods themselves being bad, it’s could be more to do with the way they are framed.
I remember when I was studying towards my own A levels I was taught the trapezium rule for numerical integration. My teacher said it was what was used before calculus was invented and that it had no real use now but was still taught, it wasn’t until I got to university I discovered that actually there are many intergrands that cannot be integrated and that the trapezium rule is an excellent method for approximation. This was a fact I’d forgotten between university and entering the teaching profession, but a fact Tom reminded me of.
This seems to me to be a very good reason to keep the trapezium rule in the syllabus. I was teaching it last week and I was thinking about this, and I realised that the way we assess the trapezium rule at A level is silly. We always ask students to approximate an integral than integrate it using calculus, oven via substitution or parts. This can only add to the feeling among students that the trapezium rule is pointless, as they can instantly see a way to find a much more accurate value. I now make a concerted effort to examine it’s importance and to state why I feel it gets a bad run, this had a positive effect on my class this year and they were much more engaged with it than previous classes.
This is not the only numerical method that gets a poor deal on our exams, another that jumps to mind is trial and improvement, a simple iteration method that can be used to find a reasonably accurate solution to an equation, however at GCSE the equation is often a quadratic, which students can find an actual solution to relatively easily via the formula or by completing the square. Why not use an equation they can’t solve otherwise?!
What are your views on numerical methods? Have you had similar thoughts? Is there anything you used to dislike teaching but have changed your mind on? If so, I’d love to hear in the comments.