"There is no one way"

Thursday, May 12, 2016

In Defense of Algebra 2

Novelist Nicholson Baker wrote a cover story on "The Case Against Algebra II" for Harper's in 2013. More recently, political science professor Andrew Hacker wrote a book (The Math Myth) challenging Algebra 2 and its sequels. Some day, I may respond to their arguments, but I am addressing this post to the math teachers who don't like Algebra 2.

I already had a conversation on this blog about a related topic in 2013. Rereading those posts, I find I still agree with what I wrote. Today, I want to get more specific and zero in on Algebra 2. I may end up repeating some of the points I made back then, so I ask for your forgiveness if you read those posts and still remember them.

One objection to Algebra 2 is that the course is a hodgepodge of unrelated topics. As I stated in a recent post, I don't consider this to be a matter of principle. There's a place for hodgepodge courses, and a place for themed courses. I only know one proposal to give a theme to Algebra 2, which is to center it on the concept of function. I agree that functions should be a significant concept in the course, but as you will see below, I believe there are other valid topics to include. In other words, I'm OK with a hodgepodge Algebra 2.

Eliminating the Algebra 2 requirement will not affect the children of privilege: no matter what is said in magazines, on social media, or on blogs, they will not be deprived of this course. In practice, the elimination of Algebra 2 would just mean that the less well-off will be kept out of careers in science and technology. But this argument is not what I want to write about. I want to argue that Algebra 2 can and should be a meaningful part of the curriculum. It can help prepare our students to be well-rounded, well-educated adults and citizens. I do not claim most will need it in their careers or for their daily life. Rather, I see math as one of the humanities, a way of thinking that helps us make sense of the world around us, and that is the point of view I take when I think of Algebra 2.

Let me start by agreeing that there are many valid reasons to object to Algebra 2. It is often taught poorly. It can be boring. It can consist of arcane techniques that seem completely divorced from meaning. Some of the topics are obsolete. Some are better postponed to later courses. And so on. I do not intend to defend poorly taught, boring, meaningless memorization of highly technical topics.

In fact, it took my department a long time to sort out how to teach that course. We tried a couple of different textbooks, but no matter what we did, this was the course that students and parents complained about. It was too hard for some students, too easy for others, and boring for all. It was, frankly, the sort of course I described in the previous paragraph.

Things started to improve when we designed our own version of the course:
Bits of curriculum I created to accomplish this can be found on my Web site. Other parts we lifted from books which had not worked for us as textbooks, but nevertheless had some good content. And I will share some specifics below, with a selection of topics which I consider worth teaching in Algebra 2. Some of those help cement previous learning. (If we don't think it's important to do that, why did we teach those topics in the first place?) Some help prepare those students who will pursue math, science, or engineering in their future education. But all are intrinsically interesting, and can be taught well to 10th and 11th graders.

Linear Programming. This is a topic for which it is not too difficult to find worthwhile problems in standard textbooks. It can make a great unit to apply and review many basic algebraic skills and understandings, especially linear inequalities and systems of equations. Better do this in an interesting new context than by rehashing the way these topics were taught in Algebra 1. I wrote an introductory activity for linear programming, and created an interactive GeoGebra applet to accompany it: Letters and Postcards

Exponential Functions and Logarithms. Understanding exponential growth and decay is fundamental to topics that range widely: population growth,  radioactive decay, compound interest, depreciation, and so on. It makes it possible to grasp some issues about the environment and the economy. I live in California. Shouldn't my students understand that one more point on the Richter scale represents an earthquake with ten times the amplitude? Here's an effective intro to logs: Super-Scientific Notation.

Introduction to Dynamical Systems. Iterating linear functions is a great topic to explore in Algebra 2. It yields delightfully unexpected results, introduces limits in a very accessible context, and provides a great environment to introduce subscript notation. Students enjoy some of the applications, such as figuring out the amount of caffeine in one's system if one drinks one cup of coffee every six hours. At the end of the unit, arithmetic and geometric sequences can be introduced as a particular case.

Complex Numbers. We approach the topic visually, as in Chakerian, Stein, and Crabill's Trigonometry: A Guided Inquiry textbook (now out of print.) This rests on an understanding of polar coordinates, and offers a chance to review basic trig. For some students, Algebra 2 is the last math class they'll ever take. All the more reason to end with a bang, completing the journey that started when they learned to count. As they moved up the grades, they became familiar with broader and broader sets of numbers, expanding their understanding of operations and the sorts of equations that had solutions. Complex numbers is a great final stop. (And check out the fun complex numbers games!)

I just picked a few of my favorite units to share here in order to perhaps convince you this can be a great course. You can find more links to Algebra 2 ideas here, or you can sign up for one of my summer workshops: one day on Algebra 2 in Oakland, or two days on Algebra 2 and Precalculus in Saint Louis.


PS: lest you think this course strays far from Algebra 2, here is a list of the topics:  Exponentials, Logarithms, Linear Programming, Parabolas and Quadratics, Unit Circle, Laws of Sine and Cosine, Polar Coordinates, Vectors, Direct & Inverse Variation, Sequences and Series, Functions, Complex Numbers. The one topic I'm sure is non-standard is Iterating Linear Functions, but it's worth it as it helps introduce Sequences and Series.

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