Tuesday, January 27, 2015

Conversations on the Rifle Range 23: The Quadratic Formula Ultimatum, and the Substrate of Understanding

Barry Garelick, who wrote various letters published here under the name Huck Finn, is at work writing what will become "Conversations on the Rifle Range". This will be a documentation of his experiences teaching math as a long-term substitute. OILF proudly presents episode number 23:

It took me about three weeks to learn all the names of my students. Identifiable patterns of behavior took me a little longer. For example, Cindy, who is in one of the two algebra classes, tended to stop me in the midst of explaining a new procedure and say: "Wait, wait, I'm confused, I don't understand."

Over time it got so I could anticipate when this would occur. When showing the factoring of 9x2 –16, for example, I paused after writing down (3x + 4). As expected, I heard: “Wait. I’m so confused. Where did the 3x + 4 come from?” I knew it was Cindy.

“I just don’t understand why it works that way,” she said. I had started the lesson by having students multiply (x-y)(x+y) and other similar problems showing how the middle value drops out. It is not unusual for students to have difficulty extending the pattern of the x2– y2 form to one like 9x2–16. I explained how it worked. Students got it but Cindy persisted. Once she understood something, she got it, but until she did it was painful—particularly when she would get frozen and could not move on until she understood, which was the case here. Students who manage to get it groan when this happens. Someone told Cindy “Because it works out that way; just follow the rule and figure it out later.”

While that exchange tends to bolster my views on how procedures lead to understanding, I can also easily imagine how easy it would be for those in the “students-must-understand-or-they-will-die” camp to make an out-of-context smear campaign against traditional modes of teaching. They would film Cindy saying "Wait, wait, I don't understand!" Freeze frame of Cindy and cue announcer voice-over: "Rote learning is no way to learn algebra. Paid for by Friends of the Common Core.”

Cindy eventually got it as she did most things; she aced all her tests and often came to my classroom before first period to get help. And all-in-all, the algebra classes did fine on the factoring unit, and on both the quiz and the test, the class average was 90. They seemed happy with themselves and with me, and we moved on to quadratic equations. There I was able to have them use what they learned in factoring to solve quadratic equations, all the while continuing with worksheets that I made, drawing from older algebra textbooks. We went through how to “complete the square” and then used that technique to solve quadratic equations that could not be factored.

The day came for presenting the quadratic formula, which, in case you have forgotten, is
x=(-b±√(b2-4ac))/2a. It was fifty years ago when I saw Mr. Dombey present it in my algebra 1 class. I watched dumbstruck as I realized that the solution of a generic equation with tools we had been using yielded a much more powerful tool. For people who think that math education should be about “patterns” rather than “meaningless algebraic symbols that bore students”, I point to the derivation of the quadratic formula as an example of taking pattern-finding ability to the next level. Some problems can be solved from a few small examples, but solving every quadratic by completing the square is too time-consuming. That's where the magic of formalism comes into play. The intuition lets go and math does the work of creating a formula by solving ax2 + bx + c = 0 via completing the square. In my opinion it is a large part of what mathematics is about. I wanted to give at least a few students that same epiphany.

My classes were patient with the explanation and with the predictable interjections from Cindy of "I don't get it" and "I'm so confused". Were I to do it over again, I would start with showing the formula and how it’s used to solve any quadratic equation. But I prefaced my presentation by saying that those who could present the derivation on the next quiz would get 10 percentage points added to their score. This tended to focus concentration.

My presentation went fairly well, though I knew that it was the sort of thing that only a few truly followed, and others would put in the effort afterward, if only to learn it well enough to get the extra credit. Others wouldn’t bother. This is my version of differentiated instruction.

Pamela (one of the students who I suspected complained about me to the counselor) tried to negotiate for more. “Can’t you give the extra credit on the chapter test? You’ll be giving a quiz next and it won’t count as much.”

I gave what I felt was a measured response.

“Well, compared to the ultimatum I gave my daughter regarding the quadratic formula when she was taking algebra, I think this is all quite fair,” I said.

“What was that?” Pamela asked.

I described how in order to entice my daughter into learning the derivation, I told her that once she starts dating, her future dates would have to show me they can derive the quadratic formula. “Now this gives you a choice. You can either date a boy who knows how to derive it, or if he doesn't, you can learn how to derive it so you can show him how, and then he can demonstrate it to my satisfaction when he picks you up."

The classroom became strangely silent and Pamela looked at me in disbelief. “You actually told her that?” she asked.


“What did she say?”

“She made a fist, held it in front of my face and said ‘I will hurt you!’ ” The class was generally appreciative of this and someone said “Good for her!”

Lonnie, a bright boy, asked me how old my daughter is. I saw in his notebook, he had every detail of the derivation copied down. “Too old for you, Lonnie,” I told him. “Go for the extra credit.”

Some students asked what if they simply memorize the derivation? I suppose I could have told them they had to supply reasons for each step, but I decided not to. Anyone willing to put in the time to reproduce the derivation was going to pick up something. Even things one learns by rote represent the substrate, the raw material, of understanding. Not the popular view, I realize. In my case, Mr. Dombey didn’t require us to derive it. But his presentation of it fascinated me enough that I tried to reproduce the derivation on my own. It also played into my decision to major in math.


SteveH said...

Lots of interesting points.

Since you did a number of cases of (x+y)(x-y), then they should have "discovered" that it produced a "pattern" of X^2 - y^2. However, this leads to my problem with student-driven class discovery - that just a few might discover anything and then directly teach it (badly) to people like Cindy. They will say things like: “Because it works out that way; just follow the rule and figure it out later.”

"In my opinion it is a large part of what mathematics is about. I wanted to give at least a few students that same epiphany."

"Give?" I can hear the pedagogy alarms going off. I agree with you, however, and would say that discovery is too important to leave up to the students. A few might discover not quite the right thing and then directly teach it to other kids badly. Fixing misunderstandings is much more difficult than getting it right the first time.

Wayne Bishop said...

Completing the square is worth spending some time becoming comfortable with in anticipation of putting quadratic equations in 2 variables in standard form (comics sections) but I had a better use (not as good as getting a date however). I was in calculus class in college and was done with a problem provided I had remembered the quadratic formula correctly but I hadn’t used it for a couple years and wasn’t sure. I had time to spare and was able to derive it on the spot. I wasn’t sure that I was remembering how to complete the square correctly either but, when I got the same answer, I was sure of both and forever convinced that mathematics really does “work”.

Anonymous said...

BG, Very well written with kudos to your ability to share your experiences with sensitivity and humor.
Your passion for your subject material is exhibited clearly in your choice of words and your delivery.

I interpret that your passion invigorates you and supports your strong desire for positive outcomes for your students.

Exceptional teachers certainly have, among their numbers, those who themselves maintain a presentation-grade understanding of several alternate methods of instructing how to achieve a goal of communicating a sometimes difficult-to-understand concept during math class(or any class).

Those instructors who, in addition, are also blessed as thoughtful, clear communicators as well as possessed of a robust and quick-witted sense of humor are truly a gift from Olympus, and those instructors will be recalled with some passion by their students throughout their subsequent brushes with "real-life", even long after their student days are past and they find themselves in the role of parent/instructor.

This type of exceptional instructor, accomodating but still on-point(and on-schedule) could be(may I say should be) bottled, like fine wine, to be decanted(among age-appropriate-participants) unto future generations of math instructors who need only to enhance their teaching repertoire a bit beyond the "standard-allocation" of teacher understanding plus an added measure of humorous patience, like a spoonful of sugar, to make the "medicine" of sometimes intimidating math concepts become absorbed.

To those advanced-degree students now graduating (Ed.D , Ed.M. and, of course, the less lofty math-teaching-credential), fully intending to become members of the math-teaching fraternity, Please take note.

I have never regretted having both matches, flint-and-steel as well as magnifying-map-reader-sheet as options I may choose among when it comes time to build my campfire on a hiking trip.

To build a bit upon the hiking analogy, we are here, student and teacher, starting out together at semester beginning in this math course described in the course catalog. Certainly we comprehend that there are multiple ways to arrive at our selected destination(skill-level)as anticipated and hopefully adequately described in the course-catalog. I recall vividly that the environment often helps(sometimes forces) us to select one method over another, one path over another, and will surely continue, over time, to affect our choices along the remainder of our successful hike and certainly also be instrumental in achieving success in accomplishing class-teaching goals.

It's clear that this description is the description(brief though it is) of life itself.

On a map, you must know where you are (you are HERE!) as well as your destination(goal) to enable you to make choices and allocate resources to be successful.

As on any map, there are pre-printed routes that may be selected to move between starting location and destination and typically you may select from among several alternate travel routes. Of course, you may also determine that moving between several pre-printed routes may hold some advantage, with consideration given to changing environment, resources, skills, goals, and always, unanticipated events.
Flight Captain Sullenburger springs to mind. Maintaining some flexibility in your thought processes and execution can, indeed,be helpful, and intuitive, deeply-embedded skills are lifesavers, in this example.

Thoughtful selection of alternate paths can be rewarding, conserve resources and save time. Practice improves performance. Rote or right? Labels at some point, aren't helpful, but may be polarizing and resource-sapping.

Please, let's use all the gifts we are possessed of to enable upcoming generations of math students to grasp, to their greatest extent, the full palette of math concepts.
Chris L.