Something is working

This fall I'm teaching Calculus I for the third time here at Coastal.  Perhaps because I just taught it at the end of the summer, I am finding that it is working particularly well.  I'm not entirely sure why.  Student interaction is good, and I'm sure that is helping.  But beyond that, the lectures seem to be flowing in a way they have not previously.

One possibility is the way I just happen to be presenting the material.  I thought I'd take a moment and record a couple of examples here. 

1. To introduce limits, I started by asking how we might graph $f(x) = \frac{sin(x)}{x}$.  We had previously talked about the graph of $f(x) = \frac{1}{x}$, and what happens at $x = 0$.  We said that while you cannot plug in 0, you can ask what happens near 0.  So we tried that again.  What happens to $f(x)$ as $x$ gets closer and closer to 0.  We made a table, everyone agreed that the $y$-values were getting closer to 1.  Only then did I start using the language "limit."  I gave them the notation as a way to quickly write down what we just did.  This seemed much more natural than giving them a definition of a limit (out of thin air) and then showing them a bunch of examples.
2. For left/right-hand limits, I only introduced them after we discovered the problem in finding limits if the $y$-values approach different values from each side.  I gave them a graph and asked them to find limits in a variety of cases.  We all agreed that at the jump discontinuity, the limit did not exist.  But of course, we can be more descriptive than this.  Coming from one side, the limit does exist.  Then I introduced that notation.
3. For continuity, I drew two graphs, one continuous, and one with a jump discontinuity.  I asked the class what was different about the graphs.  We agreed that the continuous one was continuous and the other not.  I asked what other ways a graph might not be continuous.  We came up with a graph with a removable discontinuity and an infinite discontinuity.  Now, how might we say something about these in terms of limits?  
4. Continuing with continuity, to introduce the difference between continuous at a point vs interval, I covered up the discontinuity and asked if the rest of the graph was continuous.  We agreed it was, so decided that we needed to express continuity at specific points.  This led us to the limit definition of continuity.  This was so much more natural than giving the definition, and then figuring out what it means.

That's all we have done so far.   I'm not sure if this luck will continue.  The common theme seems to be that instead of giving the definition and trying to apply it (which of course has it's place and value), I'm asking them to solve problems, then coming up with the math-way of saying what we are doing. 

I don't know if this is helping the students (we have not had any exams yet) but it definitely feels better to me.   I plan to make an effort to continue using this "technique" for the rest of the class and see how it goes.

The problem = equal signs

New research out of Texas A&M suggests that 70% of middle school students don't understand what the "=" sign means.  The article can be found here.  The key point, as pointed out in the article:

The problem is students memorize procedures without fully understanding the mathematics.

It is interesting that the problem runs this deep.  My calculus students have trouble solving velocity problems because they try to memorize the procedure instead of understanding the problem.  Of course, when they forget one step, they are completely lost.  This study suggests that even from the first time students are exposed to math, they are only taught the steps for solving math problems. 

I am not suggesting that we move to a "new math" style of instruction, where answers don't matter, and all the emphasis is on concepts.  However, if students really have this little understanding of even the simplest of math concepts, we are really missing the point of teaching math entirely.

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...