# College Math Teaching

## September 19, 2013

### What we mean about poor algebra skills…

Filed under: basic algebra, calculus, student learning — collegemathteaching @ 4:47 pm

Yes, mathematics professors have been complaining about their students lack of algebra skills as long as there have been calculus courses.

No, we aren’t talking about a student who, in a moment of panic, decided to write $\int \sqrt{x^2+1} dx = \int \sqrt{x^2} + \sqrt{1} dx$ because they were stuck on an exam. And yes, I once saw a professor walk into an analysis class, write $\sqrt{x^2+1} dx \ne \sqrt{x^2} + \sqrt{1}$ on the board (while grinding the chalk into the board) while saying “the next person who makes this mistake will get an F for this class, ON THE SPOT! 🙂

But the weakness is more of the following: in class today, I wrote
$\int (sec^2(x) - 1)tan(x) dx = \int (sec^2(x)tan(x) -tan(x)) dx$ $= \frac{1}{2}tan^2(x) - ln(|sec(x)|+C$

The student actually understood the integration, but didn’t understand where the first equality came from! I said “it is just algebra” and he STILL didn’t get it.

I have a hard time believing that this student doesn’t understand the distributive axiom of algebra; what I think is going on is that they don’t have the concept as a regular working part of their math/science/engineering mind.

1. I think this happens because many students attempt to memorize math. They think it works for them. And finally, often at the level of Calc II, it stops working.

In all the math classes I teach, I talk about math not being about memorizing, and I emphasize why things work the way they do. Also, I often tell the students “Algebra is the hardest part of calculus.”

I wonder if the student would see it better if the tangent were in front of the parentheses. (Scary that that might matter.) I never say “It’s just algebra.” Instead, I might say something like “Uh oh, another dreaded algebra bog.” And then I’d give a quick mini-lesson on why the distributive property makes sense. I have not seen students having trouble with this step, though. Most often, it’s when we need to go in the opposite direction, of pulling out a common factor. They want to leave out that one.

Comment by suevanhattum — September 20, 2013 @ 4:32 pm

• I did mention the distributive property and tried to emphasize that the distributive property applies even when doing algebra involving trig functions and the like. I now understand my saying “it is just algebra” might not have been helpful to THIS student; I was hoping he would hear “ok, I can figure this out with a moment’s thought” but he may have well heard “YOU ARE STUPID”, which wasn’t my intent.

Comment by blueollie — September 20, 2013 @ 9:39 pm

• Yeah, I hear you. (And I hope you didn’t hear my comment as critical. I guess I was thinking out loud about it. Teaching is hard, and even the things I know I shouldn’t do, I sometimes slip on. I’m trying not to say students are wrong, but to ask the class what they think. Seems I can’t help myself sometimes.)

Comment by suevanhattum — September 21, 2013 @ 12:14 am