Saturday, July 27, 2013

When Algebra Was Fun

I have heard the claim that most mathematicians either liked algebra or liked geometry in their early training. Part of my personal mathematical mythology is that I didn't much care for mathematics until I got to geometry.  (Partly, or perhaps largely, because of a really awesome geometry teacher.)  To this day, I have to admit that I mostly find (high school) algebra somewhat tedious, although obviously useful and necessary.  (What is sometimes called abstract algebra, starting with groups, rings, and fields, is obviously a totally different creature.)  I can appreciate some of the patterns and such in high school algebra, although of course at this point finding this type of algebra interesting is a little like finding the alphabet interesting: I'm much more interested in using it as a tool to do other things with.  But when I look back on my algebra classes in junior high and high school, I found those classes sort of boring, although not as bad as the arithmetic classes which preceded them.  I guess I just didn't find algebra that interesting.

Or did I?

I was recently talking about an "enrichment" program I participated in for one year in fifth grade, in which a small group of students from across the school district were gathered and bussed off to a special class one day a week.  On program days, we got to do all kinds of great things, like reading and discussing cool books, engaging in research projects, doing experiments, and working on a computer. It was actually pretty awesome. There were two real problems with the program, 'though.  The biggest problem was that the fantastic educational opportunities we got in this alternative class were really what everybody probably should have been doing all the time, instead of a special one-day-a-week pull-out activity for whosoever was judged to be the "best and brightest."  The second problem was that the program was a set of additional pull-out activities, because the students in the program had to make up all the work we missed in our regular classes.  (This, by the way, is why I only participated in fifth grade: I didn't do so well with keeping up with the other stuff, which was frankly mind-numbingly boring.)  So in the enrichment program, we'd research and report on ancient Egyptian burial practices, then come back and have to read a passage out of the social studies text book to fill in the blanks on a mimeographed worksheet.  Or we'd collect cell samples from our mouths and examine them under microscopes to learn about cells, and then come back to have to copy a diagram of the human nervous system out of the textbook.  (Interestingly enough, I remember that picture because I remember coming to the conclusion that we must be less sensitive in our forearms than our upper arms, because the diagram clearly showed more nerves in the upper arms.  This was not a misconception that I ever got to discuss in class.)  Or we'd go learn how to solve problems using algebra, only to come back to "Do the following 25 fraction addition problems."

Wait, what was the last one?  I'd forgotten about that! We actually learned some algebra in the program.  I don't remember all the details, but I think we had a worksheet, and I remember the idea of introducing a variable for an unknown quantity, setting up an equation to represent a problem, and how you could go about finding out what the x (or whatever) represented.  The problems were puzzles, and they were wonderful.  Some were quite difficult; I'm not sure we solved all of the problems.  I remember being fascinated by the very idea of working in some sense "backwards" to figure out an unknown quantity.  It was an exciting adventure for us to figure out, a marvelous mystery.  We were figuring stuff out, guided (loosely) by the teacher, who introduced just enough hints for us to make it through each new challenge. Each new idea and discovery was shared and traded with great relish.

I remember wanting to learn more about algebra and thinking it was wonderful.  Until of course I had some problems with finishing up the necessary arithmetic by hand, which led to various adults tut-tutting to me about how I obviously should have been doing more arithmetic drills. That was pretty much the end of my interest in algebra  since it was clear to me that expressing interest in algebra would lead to being punished with more arithmetic drills first.  So instead of picking up some of the arithmetic incidentally as I studied more interesting stuff, I just ground my way through the required math classes as best I could, hoping they would be over with soon, and forgot about algebra.

I finally took a regular algebra class in the eighth grade, but I'm not quite sure if I remembered how much I had liked it once.  But my eighth grade algebra class was a bit of a nightmare, taught by a man who was best known for yelling at the students and picking his nose.  (I suspect the latter would have been more tolerated and ignored were it not for the former.)  It was definitely not an adventure, and the problems were definitely not puzzles.  There were just a bunch of rules, and an algorithm of some sort for solving every sort of problem.  Every day was a new type of problem, and mostly an expectation to memorize an algorithm for solving it.  There was no "figuring" anything out, and the techniques were no longer mysteries to be discovered, but miseries to be endured.  Hell, I barely passed that class.

But as I think back on it, I realize that my interest was not completely crushed, even if I didn't realize it at the time.  I remember at one point during a summer vacation suddenly thinking about graphs, and wondering what feature in an equation made a graph "straight" versus "wavy."  I actually developed a hypothesis (by experimenting) that equations in x and y which didn't have any powers except for "1" were the only straight lines, and other powers gave bent curves.  (I have no idea whether I had already been told this before or not, but if so it hadn't stuck until I noticed it myself.)  And it's also clear that I must have had some interest left in math, because seriously, what high school student spends part of his summer vacation plotting multiple graphs by hand to test out a hypothesis about which graphs will be shaped which way?

So my personal mythology is wrong.  I did once love algebra  almost if not as much as I later loved geometry.  And I wonder:  What if my early interest in algebra had been allowed and encouraged, even if I was yet unsteady at arithmetic?  What if my first formal algebra teacher had been the same teacher who later taught my geometry class in high school, who encouraged my exploration and experimentation?  In retrospect, what I relished so much about the geometry class was that the problems were once again puzzles: No algorithms, no sequence of steps to memorize, just a statement starting "Prove that...," and it was up to us to figure out some way of getting from Point A to Point B.

In fact this spirit of investigation, of figuring things out, is at the heart of my favorite movement in mathematics education, known as Inquiry Based Learning, or IBL.  In IBL, students are set problems of some sort to solve, something to figure out.  The steps are small enough for the students to figure out on their own, and they are led along a path of discovery.  That's what happened back in the fifth-grade enrichment program:  We were introduced to the idea of using a variable, or of "doing the same thing to both sides of an equation", and asked to figure out how to solve the next problem using what we knew.  We figured the stuff out "on our own" (in actuality with plenty of guidance), and we were excited to be doing it.  When I got to the eighth grade class, what I got instead was "Day 23: How to Solve a Digit Problem.  Step 1: Let t be the tens digit in the unknown number...."

Now I have a bit of a dilemma:  I now remember what joy in algebra felt like, but can I bring that to my students?  In particular, I've recently been teaching a remedial algebra class.  It's required for many students who have poor math placement scores on entering the university, and it covers a great deal of material in fairly short order to make sure the students have all the needed algebraic skills for their next mathematics class.  Because of this, it is very algorithmic, using a very step-by-step, one-topic-at-a-time approach--the very approach I was bored to tears with. Can I bring any of the joy of algebra to my students?  I can imagine running an algebra class in the spirit of that first encounter I had, following an IBL approach, but I also think it would require more time than the one semester I would generally have.  (Now in high school, algebra is usually spread over two years, which I think would be ample time for a careful, and ultimately quite rigorous and thorough IBL algebra course.)

I'm sure that if our remedial students had a more inquiry oriented algebra class, they would be more likely to find some enjoyment in the mathematics (as I once did), and they would probably grasp some of the basics more fully.  I wonder what the longer term effects of such a remedial program would be.  Would the students with a stronger basic foundation in algebra and an interest in the material do fine in a later class without learning all the needed techniques, more or less filling-in material as they went?  Or would they end up struggling and failing to keep up because they did not know the assumed prerequisite?  Maybe I need to think about this question.