True story: it’s possible to fly through your own secondary math education – honor roll bumper sticker on your mom’s minivan and all – but miss some of the Very Big Ideas of secondary math.
For one example: in our last post on simplifying rational expressions, the process of turning a lengthy rational expression into a simpler one, Bill F writes:
Another benefit of evaluating both expressions for a set of values is to emphasize the equivalence of both expressions. Students lose the thread that simplifying creates equivalent expressions. All too often the process is seen as a bunch-of-math-steps-that-the-teacher-tells-us-to-do. When asked, “what did those steps accomplish?” blank stares are often seen.
Past a certain point, those operations are trivial. But it’s only past a point much farther in the distance that the understanding – these two rational expressions are equivalent – becomes trivial.