Really, isn’t a lot of what we teach in a typical high school math class just a lot of “shortcuts” to solving problems? Take graphing for example. In Algebra 1 kids learn about slope-intercept form and point-slope form and all the other little things that can make graphing a line quicker. They spend so much time on these shortcuts that by the end of it they don’t even remember what they are doing or how a graph connects to an equation (and maybe they never knew that to begin with). An even simpler example is absolute value. I’m not sure when students are first introduced to absolute value, but I’m going to assume that they are initially taught that the absolute value of a number is its distance from zero on the number line. Before too long they just remember the shortcut that “absolute value just means make the number positive.” I was shocked last week when I asked one of my pre-calculus classes what absolute value means and no one could tell me. All they knew was that you make the number positive. So we talked about the meaning of absolute value and then extended that meaning to distances between any two numbers. Once they understood absolute value in terms of distance, they had no trouble figuring out how to graph inequalities such as . (Yay, I just typed my first latex equation!)
One thing that I have been thinking about a lot lately is the fact that we so often teach kids the shortcuts or the quick way to figure out the answers. I don’t think there is anything wrong with doing this, but I am realizing that I need to make sure that they understand the concepts before they start learning the quick ways of doing it. For example, kids will have all sorts of problems solving absolute value equations and inequalities if they just see absolute value as “make the number positive.” On the other hand, once they understand what absolute value means, solve equations with absolute value seems obvious to them. So my challenge to myself for this year is to slow down and not be so quick to show them the “shortcuts.” Instead, give them time to understand the big ideas and concepts first.