“If I am given a formula, and I am ignorant of its meaning, it cannot teach me anything, but if I already know it what does the formula teach me?” ~St. Augustine aka Augustine of Hippo (354-430), De Magistro ch X, 23.
I had the opportunity to teach accelerated algebra during summer school this year – a class full of twenty-one 10-13 year olds who wanted to move a year ahead in math, and so they chose to spend five hours a day for six weeks of the summer in a classroom. It was a great experience in many ways but one of the things I noticed was how often the students would say things like “just tell me the formula!”. The students were, generally, very good at math – but not always so good at answering my favorite question, “Does your answer make sense?”
This reliance on formulas is something that drives me a little bit batty, and something I hope to work on this year is finding new ways to teach that don’t rely on me writing down formulas and students copying them. I want students to figure the formulas out – to be able to get to the answer the long way if they forget a formula on an assessment, to understand why the formula works the way it does and how to use it correctly. Coming from a physics background, something that I find particularly useful is the idea of dimensional analysis and I hope to have a chance to introduce it to my students this year, too, as a further aid to understanding.
I think St. Augustine was on to something all those many years ago…