Operating Axioms, the Death March to the Quadratic Formula and an Unimpressed Student Teacher

Editor’s note: After a long hiatus because he teaches during the school year, Mr. Garelick returns once again in presenting the fourteenth chapter in a series called “Out on Good Behavior: Teaching Math While Looking Over Your Shoulder”.  Garelick is a second-career math teacher in California.  He has written articles on math education that have appeared in The AtlanticEducation NextEducation News and AMS Notices.  He is also the author of three books on math education.  Says Mr. Garelick: “At its completion, this series will be published in book form by John Catt Educational, Ltd. Also, to my devoted readers, I decided to name the first school I taught at as Cypress School rather than “my previous school” to reduce confusion and irritation with the author.” The previous chapters can be found here: Chapter 1 , Chapter 2 , Chapter 3 Chapter 4 , Chapter 5, Chapter 6, Chapter 7Chapter 8Chapter 9Chapter 10,  Chapter 11Chapter 12 and Chapter 13.

Ch 14 Operating Axioms, the Death March to the Quadratic Formula and an Unimpressed Student Teacher

In math, assumptions held to be self-evident are accepted without proof and are called axioms. In teaching, as in many situations in life, one also makes many assumptions. I accept them without proof not because they are self-evident, but because 1) they seem like they could be true, and 2) I lack absolute proof. My axioms change from day to day depending on circumstance and observation but eventually they coalesce into a consistent set. Associated theorems then follow with the proviso that I could be dead wrong.

During my first year at Cypress I was forming many axioms particularly for my algebra class. It was a small class—only 11 students—and included two sets of twin girls. One set determined the tone of the class—they were somewhat dour and projected a “don’t mess with me” vibe. The other twins were very bright, and students clamored to be seated next to or near them whenever I changed the seating. 

The class was unusually quiet and for the first few months I was always in doubt as to where I stood with them. It wasn’t until about March when we hit the chapter on quadratic equations that I felt I was hitting my stride with them.

At the start of the chapter I announced that we would now be continuing on our death march to the quadratic formula. “We’ve learned to solve some types of quadratic equations by factoring, but now were going to look at more complicated cases when we can’t factor,” I said. “By Friday of this week we will learn the quadratic formula.”  I then wrote it on the board: 

I tend to stay away from things like posting “Today’s Objective” since most students ignore them as do I. I find it far more effective to let them know what they’ll be doing in a week’s time, or even a month’s. Plus the looks of horror and disbelief on their faces tell me I have their attention. “By Friday this will not look as ominous to you as it does now,” I said.

And sure enough, by the time Friday came, and after working with solving quadratic equations by completing the square, they were ready for the much easier way of solving equations by the formula. After they felt comfortable with the formula I told them that next week I would show them how the formula is derived.

“What does deriving the formula mean?” one of the dour twins asked.

“It means solving the equation ax2 + bx + c = 0 using the steps of completing the square.”


“And the derivation of the quadratic formula will be an extra credit problem worth 10 points.”  

“It must be hard,” the other dour twin said.

“I don’t think it’s hard,” I said.  “If you can complete the square, you can derive the formula.”

“I love completing the square,” one of the bright twins said.

Earlier that week the third grade teacher, Sandra, had asked me if the student teacher she was mentoring in her class could observe one of my lessons.

“She’s interested in teaching middle school math and wants to see a class.”

“She doesn’t want to teach elementary school?” I asked.

“She’s exploring options.”

“Does she know what middle school is like?”

“I think that’s why she wants to observe a class,” Sandra said. Sandra had in fact taught algebra at Cypress a few years before, team teaching with James, the union representative. An opening for a third grade teacher came up and Sandra went for it, apparently preferring it to middle school.

“Fine,” I said. “I’m deriving the quadratic formula next Monday in my algebra class.  Have her come by.”

The student teacher was in her twenties and projected an aura of confidence that comes from a belief that the (forgive me) crap ideas she had been fed in ed school were actually worth following. (I’m assuming this as an axiom and feel fairly confident in doing so.)

I started my lesson that day by pointing to a poster I had made which bore a quote from Rene Descartes: “Each problem that I solved became a rule which served afterwards to solve other problems.” 

“Nowhere is this more evident in Algebra 1 than in the derivation of the quadratic formula,” I said, and proceeded to show the steps. The students knew how to complete the square, having done it as part of last week’s death march. As I worked through the derivation I asked them for the next steps. For the most part, they knew them, though it often took some prodding on my part. I note that this is how I normally teach but I was particularly aware of keeping up a dialogue lest I be judged guilty of too much “teacher talk” as traditional teaching has come to be characterized.

At the end of the class, the student teacher left without a thank you—or anything. Her head was held obnoxiously high. I assume but cannot prove that she thought that all I was doing was promoting memorization and imitation of procedures, but not “deeper understanding”.

I never heard from Sandra on what the student teacher might have thought. And in fact, I noticed that she was no longer as friendly to me as she once had been. I assume (but cannot prove) that I was somehow discredited in her eyes.

 I did ask Sandra how her student teacher was doing, hoping to get some feedback.  “Oh, she’s doing some innovative things in the class room,” she said. What these innovative things were she didn’t say, nor did I ask. I assume with some degree of confidence that it involved group work, collaboration, student-centered, inquiry-based projects and not answering students’ questions.

As it stands, four out of my eleven students got the derivation correct on the test. Two or three more got partial credit for getting halfway through. I hold out belief that at least one person was as fascinated as I was years ago in seeing how a method for solving problems could be turned into a formula.

I have no proof of this of course.

4 thoughts on “Operating Axioms, the Death March to the Quadratic Formula and an Unimpressed Student Teacher

  1. Barry,
    As a retired math teacher myself, I applaud your work and effort. I now have four grandsons with a love of math, and this book would be a a blessing to them. I attempted to access the previous chapters, and it told me the site was unavailable. Can you help me?

    Alan Haberecht


  2. I would say that individually doing the actual steps of the derivation correctly shows more mathematical understanding than knowing about completing the square. I use those skills on every problem all of the time. I know I can use the quadratic formula (which we had to memorize) because I was shown the derivation and understood where it came from. Having students derive the formula is good because often times the problems with actual numbers turn out to simplify along the way. Using the letters a, b, and c force you to better “understand” what you’re doing algebraically.

    In our state, that lower grade student teacher would not be able to teach 7th or 8th grade math without receiving a math certification. Schools are required to provide math classes that can lead into a proper geometry class as a freshman in high school, where all IB/AP track math classes are taught in a traditional way. Parents might not have caught on to (or they deal with) the full inclusion fuzzy-ED ideas and problems of K-6, but they and their kids catch on fast when faced with the high school sort into traditionally taught honors and AP classes – the only path to a STEM career. Meanwhile, we still see a complete disconnect of reality with the magic fairy dust world of K-6.

    So, a more interesting question is what Sandra thinks of this episode? She must have a sense of what’s going on with the change from low and fuzzy expectations of K-6 to the real world influenced high school traditionally taught classes. I remember for my son that the 7th and 8th grade teachers talked about toughening them up. They knew that K-6 was anything but rigorous. Middle schools that go from grades 5-8 must be strange worlds of conflicts and rationalizations.


  3. Beyond proof-based Euclidean geometry, derivation of the quadratic formula was part of what set me on a career of teaching mathematics. That is what I thought I wanted to do after high school geometry (it “fit” so well and helping other kids “get it” when I considered them to be very bright) but confirmation was with calculus in college (there was no high school calculus in schools such as mine). I was certainly a so-so calculus student, not doing much recommended homework and just brushing up a little before exams, but I did okay. It was in the middle of an exam where I knew that the problem, that had come down to a quadratic equation, was done provided I had remembered the quadratic formula but I hadn’t used it – hadn’t even thought of it – since high school Algebra II a couple years earlier. I had time at the end of the exam so derived it from scratch. Voilà! Mathematics really works!


  4. Barry, this was set me on my math path!
    I was in 9th grade, algebra 1, and our teacher had just told us to memorize the quadratic formula because we would have a quiz the next day. She did examples of course, but because of my various activities, I never studied and showed up to class the next day not knowing the formula.
    So I am looking at a quiz with like 10 problems and I don’t know the formula. But I remembered her saying that if you know how to complete the square you can figure it out. (I am a totally auditory learner.) So I turned the paper over, and completed the square for ax^2 + bx + c = 0. I had a formula, but I wasn’t sure I did it right. But what did I have to lose? I was looking at a zero for the quiz. I had a formula now. Maybe it wasn’t right but maybe it was so what the heck?
    I got 10/10 on the quiz. That was when I realized that math wasn’t about memorizing, but about figuring stuff out. To use social justice speak, that was one of the most empowering moments of my life!


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