My thoughts on College Algebra

I'm 24 and have taught College Algebra 9 times.

I realize that's not some kind of record.  But it is a lot.

When I first started teaching it, I was so excited:  How could anyone possibly find any of this boring?!

I'm finally starting to understand my students' frustrations.  Don't get me wrong, I still love math, and I love teaching College Algebra.  But I'm starting to wonder if it's really accomplishing what it set out to do.

I know this isn't the case everywhere, but in the part of the country where I live (the great Midwest), College Algebra is the peak of most college students' mathematical career.  Any previous or remedial work is a build-up to help students succeed in College Algebra.

But what is College Algebra?

It was designed to help students succeed in calculus.

The problem is that only one in ten students enrolled in a College Algebra class will end up taking a full-length calculus sequence.

So what's the point in preparing students for calculus if very few of them will end up taking it?

I believe College Algebra needs to be revamped (and kudos to the schools who are already working on this).  I'm not one to jump on the "Math education has to be applicable to real-world situations" bandwagon; although I'm not at all opposed to this approach.  What I am opposed to is giving pure math a bad rap.  Pure math can be just as fun as applied math.  The thing is, in a typical College Algebra course, we teach very little of the fun stuff in pure math.  Call me crazy, but I'd love to see a beginning college course that showcases the best of the world of pure mathematics.  I truly believe every college student can do a bit of abstract algebra, a little number theory, a piece of combinatorics, and a snippet of analysis/calculus.

How encouraging would it be for a student to finish her final mathematics course and be able to say, "I can do calculus!"  As opposed to what several of our current students end up saying:  "I barely passed a class that I had already taken in high school."

I currently teach high school seniors who are taking College Algebra for both high school and college credit.  I teach them nothing new.  Nothing.  I've compared the topics I have to cover to the topics taught in their Algebra 2 curriculum and they're identical.  Do we really want to send the message of This is the end-all of math!  Functions and equations.  That's pretty much all math is.  Specifically, rational, exponential, and logarithmic functions and equations.  The end.  In fact, we think this is so important, we're going to teach it to you again!  And again!

That's really not the message I want to be sending.  But what are students to think if that's what every single math class they've taken (both in high school and in college) focuses on?

At the Conference to Improve College Algebra in 2002, Arnold Packer said the following:

Many would skip College Algebra if they did not have to pass it to get the degree they need to enter their chosen career field. Enrollment in CA tends to fall dramatically when colleges make quantitative reasoning or intermediate algebra the requirement. Finally, a few years after finishing the course, getting their degree, and starting their professional life, they cannot recall anything they learned. Or, equivalently, they have never used anything they learned in College Algebra.

All of this is unfortunate and related. Mathematics courses that seem hard, boring, and irrelevant prior to College Algebra establish the expectation that College Algebra will be more of the same. Moreover, the course – as conventionally taught – does nothing but confirm the foreboding.

That's the sad truth.  But I think things will change.  And I hope to be part of that change in some small way.

This fall I will start an adjunct position at my alma mater, where I will be teaching a Survey of Mathematics course.  My hope (albeit lofty) is to showcase some of the best and most interesting topics of pure mathematics.  We'll see how it goes.

In conclusion to this rant, a final plea...

To high school teachers:  Encourage all your students to take their required college math classes ASAP (i.e., discourage them from waiting until their senior year of college).  I've taught too many amazing individuals that were--and probably always will be--one class short of a bachelor's degree:  College Algebra.  Oftentimes this is due to a prolonged break in their mathematics education.  Also, encourage your bright students to test out of College Algebra (CLEP it if allowed) and move on to Calculus or take some other math elective, lest we produce a generation that thinks polynomial functions are the epitome of mathematics.

To College Algebra teachers:  Let's try to make this class as engaging as possible.  It will be more enjoyable for both ourselves and our students if we challenge ourselves in this way.  Also, I try to remember:  my students have seen all of this before.  That doesn't mean they remember it all, but they have seen it.  Thus, according to the law of diminishing marginal utility, we'll have to work harder to sell it the second (or third) time around.  (That's right, my husband's a CPA.  I know my economics talk.  Sort of.)

To college administrators:  Decide who really needs to take this class and who might benefit from a different kind of mathematics course.

Overheard in the Math Lab

In addition to teaching at the college, I also tutor at our Math Lab.  This job is both rewarding and frustrating.  Some days I leave very proud of my work; other days I leave very disappointed by my lack of grace and patience.  But one thing that's almost always a given:  I will hear or see something interesting.

Below are some of the things I've heard or seen in the Math Lab recently.  Let me be clear, my intention is not to make fun of any student.  Think of this as a "Kids Say the Darndest Things:  College Edition."
                   
Student:  If physics makes sense to you, then chemistry won't. Just like if you like geometry, you can't do algebra.

Apparently liking geometry and doing algebra are now mutually exclusive.
                    
Student coming up to the tutor table:  So, are you good at math?
                    
A problem a student worked:

I just love the third line.

                    
Student to a man with his PhD in mathematics:  Are you a math tutor?
Professor:  I try to be.
                    
Me:  What's zero minus zero?
Student:  2.
(Me:  frantically trying to figure out how to salvage that answer)
                    

And how does one plot a y-intercept at (-infinity, infinity)?

                    
Me:  What does x^2 divided by x simplify to?
Calculus student:  One-half!
Me:  Maybe let's try that again.
                    
Some of my favorite repeats:

"So, can you show me an easier way to do that?"

"Can you just write it and I watch?"

"I've only missed like one week of classes..."

We're getting closer with recursive definitions

Last semester was the first time I used a College Algebra curriculum that taught sequences and series.  The first section of this chapter was a nice little intro to sequences and series, just to get students used to notation.  I thought I did such a good job explaining sequences that are defined recursively.  Until I saw the test.

Not.  So.  Hot.

Then, as I helped students in our Math Lab, I realized something--recursive definitions are not that obvious to students.

I think what happened here was a classic case of it's-so-obvious-to-the-teacher-she-automatically-thinks-it's-obvious-to-everyone-else.  We've all had teachers like this.  My absolute favorite prof from grad school loved the phrase, "Oh, this is kindergarten stuff!"  Which usually had one of two effects on me:  (1) Ahhhh!!  This is NOT kindergarten stuff!  I just spent the majority of my weekend trying to figure this out!  (2)  Where in the world did you go to kindergarten?  Remind me to send my kids there.

But I digress.

What hit me was that when I see something like:

an=an−1+an−2

I automatically think, "If I want to find a certain term, I need to sum up the two previous terms."  Furthermore, I know that

an+1=an+an−1

means the same thing as the previous equation.


On the other hand, when my students saw a recursive definition, I'm pretty sure they thought, "WTF.  Skip it."

So, this semester I paid much more attention to these types of sequences.  The very first thing I did regarding recursive definitions was show a slide with this at the top:

I asked students to fill in the blanks and then asked them three questions:

1. What do the dot, dot, dots mean?
2. What do we call the term before a_n?
3. What about the term after a_n?

Maybe this is an obvious starting point, but it was such a revelation to me.

We then did some examples with

an=an−1+an−2

I had them tell me what they thought it meant (with a lot of guidance from questions like, "a_(n-1) is related to a_n how?").  Then we wrote a few equations in symbols and in words.  For example for,

a4=a3+a2

an=an−1+an−2

I made them write "The fourth term is equal to the third term plus the second term."  And so on.

Then came

an+1=an+an−1

which everyone was convinced was a totally new problem (darn you, indices!).  But, once we did the same examples (finding a_4, etc.), I think/hope all minds were changed.


After working some specific examples, where initial values were given, I gave them an exit ticket of something like:

List the first five terms of the sequence defined by:
a_1 is the number of boys in the room; a_2 is the number of girls;

an=an−1+2an−2

Finj 

I think about 80% of students got it with zero help from me.  Not perfect, but I'll take it this time around!

Conic Sections: Parabolas

Much like with circles, I need some major help in the area of teaching parabolas (through the lens of conic sections).  Maybe I'm just not cut out to teach geometry.  It's quite possible.

In any case, here are a couple of things that I found/made that did work nicely:

1. This graphing paper from MathEdPage.org is awesome.  It gives a very nice low-tech option for discovering the geometric definition of a parabola.  I split the class into groups of two or three and gave each group a sheet of this graphing paper with one of the lines darkened (which is to be the directrix).  I told the students to plot seven or so points that are equidistant from the point in the middle and the darkened line.  We found a couple together, and then they were good to go for the rest.  When I gave them the following definition, they were able to fill in the  blanks no problem:
   
   
   
2. I made a little graph with sliders that shows what happens to a parabola in the form x^2=4py when you change p.  It wasn't a ton of work, but I'm still pretty proud.  Plus, I continue to absolutely adore Demos graphing calculator at abettercalculator.com.  I also love their new "Projector Mode" under Settings.

I got to borrow one of these from my college. 
I really want one.  Unfortunately they're a little pricey.

So, those are two things that worked.  The 5-10 minute intro.  But once we got to working examples, I wasn't too pleased.  I feel like I jump all over the place when I work these problems.  "What's the vertex?!  How do we find the focus from there?  And the directrix?"  I think students get it during class, but that's with me asking all the right questions at all the right times.  Ideas for making them do more of the work?