Thinking vs Doing

Word problems are hard for students.  I've never really understood why before.  You read the problem, figure out what it is asking, figure out how to answer the question, and do it.  Usually the mathematics part is not that hard.  So why to students have so much trouble?

One of the reasons might be that many students have never been shown how to think about the problem.  So much of mathematics is about doing. Here are the steps you need to follow to solve an equation.  Here is the process to find the equation of the tangent line.  When you want to find the absolute maximum of the function, first... then... and finally... etc.  Many problems in mathematics are complex and require multiple steps.  When we teach students how to solve these problems, it is very tempting to teach them the algorithm for arriving at the solution.  No doubt there is value in learning how to follow a recipe, but as a method for solving a problem, it can be perilous.  If we simply follow the steps, it is all to easy to miss one without realizing it, resulting in a wrong, or even meaningless answer.

As professional mathematicians, this is rarely a problem, since while applying the problem solving algorithm, we realize why we are performing each step.  If we miss a step, we will notice immediately (most of the time).  But for a student who is first encountering a type of problem, this is near impossible.  They are worried about following the steps, not why the steps matter.  If they would just stop to think about what they are doing, they would realize that something has gone wrong.

Consider this example: in calculus, we ask our students some simple velocity problems.  We give them a formula for the height of an object at time t, ask them to find a formula for velocity, then ask them,

1. What is the maximum height of the object?
2. How fast is the object going when it hits the ground?

This semester, I asked how fast the object was traveling when it was 35 feet above the ground.  A student promptly plugged in 35 for t in the velocity formula.  Why would she do such a thing?  This particular student is usually very bright, and definitely hard working.  I suspect that her thought process went something like this. "Okay, one of these problems.  I need to plug something into an equation.  Here is a number.  Here is equation.  Here we go..."

I pointed this out to my class when talking about the problem the next day, and almost all of them agreed that at least one of their math teachers in the past taught them how to do word problems in this way.  To solve the word problem you need to find the equation, and plug in the number, and evaluate.  Quick, don't think about it, just follow the directions.  Now move along.  We cannot blame our students for this poor mathematics upbringing.  I show my students too many algorithms -- it is a fast and easy way to get them to find the right answer, before they forget how the algorithm goes.  In fact, last time I taught this topic, I think I pointed out to my students that questions 1 and 2 above are similar in that you set one of the equations equal to zero, solve for t, then plug that value into the other equation.  Now reflecting back on it, I feel embarrassed to have fallen into the trap.

I am not entirely sure how to fix this problem.  Certainly it is important to, whenever possible, challenge our students to think.  Give them time to struggle with a problem.  Let them figure out the solution without giving them a recipe.  Through practice, the students will come up with their own shortcuts.  This takes a lot of time, and can be frustrating for all parties involved.  But giving them yet another algorithm will only make the problem worse. 

I am reminded of a certain proverb about fish and fishing...