On Algebra

There have been a whole flock of article recently addressing the question, “Should we teach algebra to all high school students?” It started (I think, anyway) with Andrew Hacker’s Op-Ed post, “Is Algebra Necessary?” in a recent issue of the New York Times. His conclusion: No, algebra isn’t necessary. A careful reader ought to question, “What does Hacker mean by algebra?” and “Do we want an emeritus political science professor to make decisions about the mathematical education of the masses, given that there are so many people whose entire careers are dedicated to mathematics, the research of mathematics teaching and learning, and being mathematics educators?” But today I wasn’t planning on addressing those questions or Hacker’s article. (See Daniel Willingham’s response, “Yes, algebra is necessary” if you’re interested.)

Instead, there are several important issues that I think are worth pondering whenever anyone starts talking about the necessity of algebra.

1. What do they mean by algebra?

Out of curiosity, I asked WolframAlpha to tell me about “algebra”. It gave a lot of responses (but no definition). It did, however, provide a clear distinction that algebra is something more than equation solving. Yet in Schank’s first paragraph, he seems to conflate all of algebra with the quadratic formula:

“Whenever I meet anyone who wants to talk about education, I immediately ask them to tell me the quadratic equation. Almost no one ever can. (Even the former chairman of the College Board doesn’t know it). Yet, we all seem to believe that everyone must learn algebra.”

I’ll skip over a discussion pointing out that he means the quadratic formula but wrote “the quadratic equation” (as if there’s only one).

In any case, very few people have given the word “algebra” a good-enough definition, from my viewpoint. When I put on my research mathematician hat, I like thinking about universal algebra, which is somehow even more broad and amorphous (and beautiful) than the above definition could convey.  (I am even a published, theorem-proving universal algebraist.) I define algebra like this:

Algebra. Noun. The branch of mathematics that deals with the study of structure.

Yes, that includes studying equations. But it also includes wilder animals like  finite fields or nonfinitely axiomatizable equational theories and the varieties they generate!

2. Who is the “they” and who is the “us”?

Schank also asks,

“Are mathematicians the best thinkers you know? I know plenty of them who can’t handle their own lives very well.”

It turns out that, despite lots of evidence to the contrary, mathematicians are people. As in, real people with real lives and real feelings and real kids and real cats and real hobbies. And, sometimes, real problems. I don’t know why this is news. Surely no one would suggest that we ought not listen to music (or teach music in schools) because some musicians have had difficult lives. The people behind the subject is what makes it compelling. If only robots did math, I’d probably be less interested in math. [And more interested in robots.]

How do mathematicians think? Wow, that’s a fabulous question. Look, no mathematician I know claims that we think “the best.” That’s a “they versus us” distinction if I ever heard one. However, many of us do claim that we think differently.

To believe anything, a mathematician requires a proof.

This is different from every other academic discipline. Mathematicians have a very strict code for how we think about problems. Intuition is never enough.

3. Who taught these people mathematics?!

Moving toward his conclusion, Schank writes,

“You can live a productive and happy life without knowing anything about macroeconomics or trigonometry but you can’t function very well at all if you can’t make an accurate prediction or describe situations, or diagnose a problem, or evaluate a situation, person or object.”

(It humors me that the end of the article talks about making “sensible political choices,” but here it turns out we don’t need to understand any macroeconomics. Say what?)

I wonder who taught these people algebra! None of my students will survive my courses unless they are able to demonstrate that they can use the mathematical content knowledge from my class and apply it to real-world problems about situations that involve optimization or diagnostics or evaluations or predictions. Who are these [potentially imagined] math teachers who teach nothing but endless factoring of polynomials without any motivation?

4. What’s a better question to ask?

Schank and I agree that “[t]he ability to reason from evidence really matters in life.” He thinks algebra doesn’t help develop the skills to do this — I disagree hugely. Algebra can help develop this skill. Does every math educator teach it perfectly? No. Could all of us math educators do a better job? Yes.

My goal as an educator of mathematics is to converge asymptotically on being amazing at my job. Could teaching algebra help students understand logic, reason, and critical thinking? Absolutely. Does it always? Maybe not. But that is not a reason to say, “Don’t teach algebra.” We should say, “We need to teach algebra better.”

And, I promise, I am trying to do just that — along with thousands of my colleagues around the world.

First Day Activity

I really enjoyed today’s “First Day” activity in Precalculus. I found the idea on Becky Lyon’s blog; you can also find her on Twitter: @rhlyon.

I had the students find someone to work with and told them one member of the pair would be the Explainer and the other would be the Grapher. The Explainer was supposed to sit facing the projector screen, while the Grapher was supposed to sit facing the door (i.e., away from the projector where they could not see it at all).

The idea of the activity is this:

  1. Display a picture or graph on the projector screen for about one minute.
  2. The Explainer has to describe the graph only using words — no hand gestures allowed!
  3. The Grapher tries to re-create the picture or graph from the description.

The pictures I used started out easy (a giant smiley face) and got progressively more difficult. To give you an idea, I uploaded the exact graphs I used to my public Dropbox space: It’s http://dl.dropbox.com/u/59433434/111-Day1.pdf. After we were done, we went through the graphs together and talked about what descriptions had been given and what people could have said to make it easier.

This gave us a great opportunity to review vocabulary like “degree” and “vertex” and “parabola” and “quadratic” and “intercept” and “slope” and “local maximum [versus global maximum]”. It also gave me feedback as to what the “groupthink” occurred and at what level my students are starting. (For instance, some of them volunteered the idea that an even-degree root of a polynomial behaves differently on a graph than an odd-degree root!)

They seemed to enjoy the activity. It helped cement for them that I will expect them to do things in class, not just be. And, happily, it gave them the opportunity to practice my Friendship Policy.

My three favorite comments from today include:

  • This will be the most FUN class!” –a student said to her friend, at the end of class
  • An e-mail I received after class said, “What a great first day of class! Super exciting and thank you for your approach!
  • A fantastic Tweet (admittedly from a calculus, not precalculus, student):


Friendship Policy

I have my first course meetings this morning. Right now I’m enjoying a one-hour break between classes in what will become my Office Hours once students figure out what Office Hours are for. I thought I’d take the time to write about an important topic I covered during today’s PreCalculus class.

A Very Important Course Policy:

One of the notable policies I have on my syllabus is called my Friendship Policy: Students in my courses are required to make two friends from class. For those of you who, like me, haven’t been a college student in a number of years, this policy may seem very silly and totally unnecessary! However, the policy has an important function at fixing a “problem” I noticed a few semesters ago.

Before class, I would find students sitting on benches in the hallway for several minutes waiting for the previous class to end. There would be, say, ten or twelve students all from the same course, standing in the same hallway, and it was library silent. No person was talking to any other person! Instead, every single one of them was texting someone on their phone, checking Facebook on their iPad, playing a game on their laptop, etc. Eventually they would all enter the same classroom and continue their technologically dependent anti-social activities.

When I pointed this out to my students, they had never noticed this phenomenon and they didn’t understand why I thought it was weird!

“Back in my day,” says the professor…

There were no cell phones. In order to fill the awkward silence, students in my classes would talk to each other, real-time, face-to-face. Sure, we would talk about course-related things like homework or exam studying, but we would also talk about social activities or sporting events or movies or whatever. This is how we made new friends.

I realize that students in my class have lots of friends. (Otherwise, who would they be constantly texting?) But I still have not figured out how they make new friends. Hence the birth of my Friendship Policy:

Friendship Policy:

You are required to make friends with students in this class. If you are absent from class, your friends will be very happy to lend you their notes to copy! In fact, I think cooperative learning is so important I am going to leave blank space on this syllabus for you to write down the names of two of your class friends and their contact information.

After explaining all this to the students, they usually look at me with confused faces until I say something along the lines of, “Friendship Time: Commence!” and then stare at my wristwatch expectantly. Within seconds, the room explodes in conversation. Occasionally, I have to nudge some of the shy students in the right direction.

Results and Analysis

After several classes over several semesters, this policy seems to make a big difference. First, no one sits before class in techno-quiet. They talk to each other, get to know each other, and occasionally I have caught them teaching each other how to do math problems. Second, I no longer get e-mails asking, “What did you cover in class yesterday?” Third, I learn a lot from my students by participating in before class conversations. For example, in this morning’s class, one student is here on a golf scholarship from Sweden! (How awesome is that!)

I still have two more classes this morning. We’ll see how those groups take to forced friendship-making time.

Project Based Learning

Our classes for the Fall 2012 semester start today. Thankfully, my teaching schedule doesn’t include Tuesdays, so I don’t start until tomorrow! I’m hoping to use Tuesdays this semester to work on several other projects, including adding more blog postings. Wish me luck.

This semester I’ll be teaching two sections (Section 05 and 17) of our Pre-Calculus class (Math 111) and one section (Section 05) of our Calculus I class (Math 120). Each class meets for 50-minutes per day on Mondays, Wednesdays, and Fridays, and an additional 75-minutes on Thursdays. The longer meetings on Thursdays will be useful in my current quest to incorporate Project Based Learning (“PBL”) into my classes.

I’ve begun the task of designing “Lab assignments” for students to work on, in small groups, during our Thursday meetings. Ideally they would be assignments that require no pre-lecture and ask the students to draw from their course content knowledge to form connections between ideas. By working together in a group, the students could collaborate (hopefully allowing for some peer instruction), ask questions, have a discussion, and digest what we’ve talked about during our other class meetings. According to my calendar, the students will have ten lab assignments over the course of the semester.

Yesterday I began working on the third lab assignment for Calculus. The topics covered earlier that week will be limits at infinity; asymptotic behavior; and continuity. I found an activity called “Carousel Game” from the NCTM‘s Illuminations series and modified it for my class. Here’s a brief overview of this lab:

  • Topic: Graphing rational functions
  • Goal: To correctly determine the equation that corresponds to the problem situation or graph
  • Technology Required: None allowed!
  • Warm-up: Vocabulary assessment, including: asymptote,  rational function, exponential function, end behavior, domain, range
  • Activity: Students will use a description or a graph to find the equation for twelve functions
  • Assessment: After finding the functions, students will find domain, range, vertical asymptotes, horizontal asymptotes, and all intercepts. This will be turned in and graded.

I also uploaded a copy of the lab instructions to my public Dropbox. If you are interested in seeing the entire lab, check it out here: http://dl.dropbox.com/u/59433434/120-Lab2.pdf. (Notice that it’s 120-Lab2, even though I mentioned before it is really our third lab — I start numbering things with zero.)

I’m hoping to reuse this activity in Pre-Calculus later in the semester, once we cover material about rational functions.


Earlier today, Derek Bruff (@derekbruff) tweeted a link to a Wordle done by graduate student Jessica Riviere. Jessica blogged about her Wordle, so check out this link for what she had to say. Her Wordle contained data from her teaching evaluations and what her students had commented. This was clever and fun and it inspired me to make one as well.

I used my course evaluations done by College of Charleston students during the last academic year (Fall 2011 through Summer 2012). Altogether I have data from eight courses (covering several sections of Elementary Statistics, Pre-Calculus, and Linear Algebra) for a total of 114 evaluations. To make the data collection easier, I restricted my focus just to the “Comments on Instructor” and “Comments on Teaching” prompts. This meant ignoring data from sections called comments on “Organization,” “Assignments,” “Grading,” “Learning,” and “Course.”

The most frequently used words were: and, the, to, I, is, she, a, class, was, of, her, Owens, with, Dr. Several of these were removed by Wordle since I had chosen to “Remove common English words.”  I also removed my first name and corrected some misspellings (ex: “explaiend” to “explained”). I enjoyed the following word counts: awesome, 6; funny, 5; humor, 5; and enthusiastic, 9.

Wordle: Eval Cloud