Misconceptions. (Or, teaching the shortcuts without teaching the concepts)

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 $|x-2|\le{3}$.  (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.

4 thoughts on “Misconceptions. (Or, teaching the shortcuts without teaching the concepts)”

1. Congrats on the LaTeX!

I am soooo in total agreement with this post! Often times I don’t “ever” show them a shortcut. Sometimes kids will figure one out and share it/explain it to the class, and that fine with me. What really frustrates me is when someone outside of the class (parent, older sibling/friend, student in another class) just tells someone a shortcut without ANY support or reasoning. Really LEARNING math is all about making sense of it yourself and not just following a bunch of rules/steps 😦

In terms of your absolute value situation, I am not sure many Middle School teachers will realize the benefit of really retaining the “distance from zero” definition of absolute value and how it will pay off in a deeper understanding of concepts (limits, anyone?) in the future!

2. Nice Post!
I am part of the blogger initiative, and I just posted on misconceptions also. I also try to go for conceptual, but, at times, there is just not the resources or time, so I have to go back to shortcuts (I try to do this judiciously). However, the slope, y-intercept shortcut seems to create so much confusion for students. I tried to use LaTeX too but somehow erased my post in the process…oh well…always learning 🙂
Lisa

3. Nice post! I really think that you hit the nail on the head when you said:

“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.”

The longer I teach, the more I realize that students need to understand the “why” behind what they are doing for the shortcuts to make more sense. Otherwise, they just memorize the shortcuts.

Thanks for sharing!
–Lisa

4. Loved your thoughts and totally agree! The first time I “confronted” some teachers about the shortcuts, I thought I was going to start an argument. I know we sometimes get rushed or we’re trying to prepare for THE TEST, but understanding the concepts is so, so crucial. Thanks for your post!

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