The other day, I was helping my girlfriend with her calculus homework and there was a true/false question about the chain rule:
TRUE/FALSE: To apply the chain rule to sin(x^2) we select f(x)=x^2 and
My initial reaction was “True! Obviously! g(f(x))=sin(x^2) where g(x)=sin(x) and f(x)=x^2”…..but, she quickly pointed out to me that it was false because her teacher had taught her that it was always f(g(x)), which would mean that the question implies sin^2(x) not sin(x^2). And therein lies the problem; we give homeworks with trick questions to check if students have memorized formulas rather than harboring creativity.
We focus so much on making tough subjects like math accessible to every student that we make them actually more difficult to understand. Why do we need to explain the chain rule in terms of d/dx[f(g(x))]= g’(x)(f’(g(x))) when we could much more simply explain it as the derivative of the inside multiplied by the derivative of the outside, e.g. f(x)= sin(x^2); f’(x)= 2xcos(x^2)? A formula list is far more difficult to understand than learning the concepts behind them. We need to make concepts relatable and engaging. A sequence of numbers and letters falls out of our heads as soon as the exam is finished, but remembering that velocity is the derivative of position is something we never forget. Understanding how it works is far more important than just worrying about being able to solve it.
It reminds me of when I was in middle school and was learning algebra. My teacher got mad at me for skipping steps when solving simple problems. She would remind me “the x stands alone!!” even though I knew how to solve the problem and was saving myself some work and writing. I am 100% for teaching people how to solve problems so that they have a template, especially if they are struggling with a concept. I struggled with integration by parts last fall and the quick formula int(vdu)=vu-int(udv) saved me on several quizzes.
However, when those helpful templates begin being taught in place of understanding, I have a problem. We shouldn’t punish students for not following the pre-prescribed template, because this conditions them not to dare to try different methods for problem solving. What this could lead to is a country of calculators rather than innovators. I hope that, in a few decades, when I have children they will be taught to use formulas as a tool for quick problem solving, but not to ruin their ability to think creatively and effectively, because that is what STEM is all about! Thanks for reading!