I got sucked into the online buzz generated by this July 28th New York Times Op-Ed by Andrew Hacker, titled “Is Algebra Necessary?” It struck a nerve. Many nerves. Hacker’s main premises were: 1) Algebra is too hard, and is responsible for many a talented student’s academic failure – backed up by statistics, and 2) Algebra is not necessary for most careers, backed up by flimsier statistics.
Of course Algebra is necessary, we cried! Algebra teaches you to make real-life situations abstract, to simplify them to their core. It teaches you to play with the variables involved and to quantify cause-and-effect. It teaches you to argue logically and reasonably. It teaches you the life skills of discipline and hard work. It teaches you organized thinking and linearity. It’s good exercise for your brain.
These are noble goals. Everyone should learn these skills. These are not taught in high school mathematics. Some students learn them. But look at a high school Algebra 1 or Algebra 2 exit exam and tell me where the teacher is assessing the student’s ability to abstract real-life situations and argue logically. You’ll find a couple of word problems in the problem set on systems of equations, and I can almost guarantee that will be the extent of the assessment on abstracting real-life situations. There’s an assessment here, and also here, and here’s a standardized practice exam for Algebra II here. The students are required to do very little other than notice the structure of a problem and follow the right algorithm. If you’re required to show work, you can show evidence of organized, linear thinking… or, you might just show evidence of being very very good at memorizing a sequence of steps. Students who are already good at pattern-finding and generalizing may excel at this, as they’ll recognize the structure of a problem and know right away it’s, say, a trinomial with integer coefficients they can easily factor. Will this teach a student to become good at pattern-finding? Students for whom this does not come easily will sweat over the same trinomial, trying to recall the steps needed to get the problem into a form the teacher will like. They’ll probably mix up one or two of the steps and arrive at an answer that would make no sense in context, but there is no context around the problem, so one answer looks just as meaningful as another. Students learn a set of rules to apply for a certain structure of problem. Then another set of rules to apply for another structure of a problem. The noble, higher purposes of Algebra – the abstract reasoning, the beauty of a logical argument, the modeling of a complex situation into a symbolic form that can be manipulated to apply to other situations or prove other truths – are not actually ever taught.
There’s more. All of this algorithm-crunching and polymonial-factoring and imaginary-number-figuring by hand would make a lot of sense to keep in the high school curriculum if this were how anybody actually used mathematics. But here is the dirty little secret: nobody does math this way in the real world. Who solves a quadratic or a double-integral by hand? Who shades in those cartesian graphs of systems of inequalities? Who converts a cartesian graph to a polar graph with a pencil, or finds the inverse of a radical expression on paper? NOBODY. It’s time-consuming and error-prone. Computers are better at it, and we have better things to do with our time. You know how long we’ve been teaching algebra by hand, with paper and pencil? The earliest evidence I could find was from 1923. The application of math in the real world has changed a little since the 1920’s.
Inevitably, you’ll be teaching the simplification of expressions using complex numbers, and a student will raise her hand and ask when we’re ever going to use this. You’ll be grumpy about this, because they might use it someday, if the student becomes a theoretical physicist or a mathematician, and anyway, that isn’t the point. The point is that when a solution can’t be found in the set of real numbers, look, there is a whole other infinite set of numbers that are imaginary that have limitless possibilities. It’s beautiful and expands the horizons of what your brain can do.
It’s too bad students don’t actually ever discover that. They learn a set of rules for simplifying expressions if they contain the letter i.
So then you’ll be having coffee with some parents, and a parent will say Johnny has been having trouble with complex numbers, and will ask you honestly when their child might ever be expected to use this. And you’ll say the kid might, if they become an engineer, and that math is important if a kid wants to get into a STEM career. Algebra II success is a great predictor of college success. Then you’ll change the subject.
You know you’ve done this. I have!
I have a story about the way you use math in engineering. I would like to share it so you understand why we so badly do need to completely restructure mathematics education.
I worked as an engineer in an electronic test and measurement company, during the late 1990’s and early 2000’s. I was a software engineer and I had to study a lot of math to get there, because engineers use a lot of math. Once, I was on a project where my job was to write the drivers that took measurements on the power output of a fiber-optic amplifier. I remember there being a function that described the power output in terms of another variable, and although I didn’t need to solve it because it was already done, I solved it anyway to prove to myself that the equation worked. I remember the equation contained a log function which I struggled to remember how to “undo” (is it a base 10 if there’s no base specified, or is that a natural log??), but once I refreshed my memory using the information superhighway, I was good to go. I balanced the equation carefully and tested it with a couple of pairs of variables, and indeed the equation was derived correctly just as I received it. I was satisfied. I spent about an hour on this task, if you also include the time I spent searching online for cool information and simulations of logarithmic and exponential functions. And watching a funny cat video.
So I spent an hour using all that precious knowledge I had gained in the many, many years of math I took in high school and college. Next I had to write the program that retrieved the output from a spectrum analyzer and calculated the power from it. The spectrum analyzer produces a curved trace, which you receive as an array of numbers. I actually had to calculate the area under the trace. I didn’t integrate it by hand… I used a computer program to calculate the area of a lot of tiny rectangles under the trace and add the areas together. With a few lines of code, this took a very short time and really just used seventh-grade math. I was going back to the very basics of what calculus is all about, and it was humbling to see I spent so much time studying integrals and here I was, a professional engineer, adding rectangles together. But in the digital world, this is what you do. Almost nothing is really curved in the computer age.
I finished the software and then had some decisions to make. The more little rectangles I measured, the longer the program took. Did the customer need the result in milliseconds, or did they want a more precise answer that took several seconds to get? How precise was the spectrum analyzer and could I make it more precise? Was one measurement enough? Should I take three traces and average all my answers? Should I take ten traces, drop any outliers, and average the rest? How would I know if a trace were an outlier? What result should I be expecting? How could I independently verify the results I got? Under what extreme conditions did I need to test this? In what format did the customer want the data saved? What was a pass versus a fail? These kinds of questions filled the rest of the development cycle.
The project lasted several months. I spent one hour of that time doing algebra. The rest of the time I spent puzzling over what information I needed, what my results meant, and how to make them presentable and understandable. Since that project, it has never been lost on me that I spent years and years of my life manipulating equations by hand, and I used that knowledge in my career once. And I was an engineer in a bona fide STEM career.
Think about that, and think about how we teach algebra. A STEM career involves making lots of decisions about messy data with unknown sources of error. Understanding how to abstract and idealize a real-life situation is certainly an important part of math. This is what algebra is supposed to teach us to do, and it doesn’t even do that well. We teach symbolic manipulation via a bunch of algorithms instead. We rarely get to having students truly abstract a real-life situation, and we completely neglect teaching communcation, challenging the reasonableness of answers, sources of error, estimation, precision, tradeoffs, and general problem-solving.
I have some half-baked ideas, of course, of what a 21st century math education actually looks like. But I’d like to hear from you: If you were king, how would you structure secondary math?
For some great inspiration, enjoy this TED talk by Conrad Wolfram. You’ll start to get the idea on where I stand.
For more inspiration, read a piece that has challenged me from my early days of becoming a teacher: