As a young teacher, I was awesome at teaching factoring. My mentor teacher had a system that was brilliant — she’d analyzed every factoring problem and broke them down into types — greatest common factor, difference of squares, easy trinomial, hard trinomial, grouping, sum/difference of cubes, two-step, etc. So she explicitly taught the kids how to identify what type of problem they were looking at, and then she taught them sweet tricks to factor each type efficiently.
So that’s how I taught. And boy, did I think I was good at teaching factoring. Because when I was going through school, I was taught using the tried and true “figure it out” method. This was where the teacher put up a factoring problem — like 3×2 – 5x + 2 — put a couple parentheses below it ( )( ) — then said “Figure it out!” This process was then scaffolded by smacking the blackboard and saying “Figure it out!”, except louder, and then eventually jumping up on the desk, waving the hands wildly and yelling “FIGURE! IT! OUT!”…. because learning.
I’d share this story with my own students and it had the desired effect — it made them super-glad that they had me, extra-helpful teacher, with my rules for identifying types of problems and my efficient tricks for solving them. And all was good until I moved to a small school district, had kids for more than one consecutive year, and realized that none of my former Algebra 1 students could factor.
Sure, they all could factor during the factoring UNIT, and many could factor on the Algebra 1 exam, but in Geometry? In Algebra 2?
Yeah, after review some of it would come back, there was always trouble with those sneaky two-step GCF problems (“stands for Gotta Check First,” I’d say, for the billionth time, debating the utility of jumping on my desk for emphasis), but what got them, every time, no matter how much we reviewed and no matter how many reminders and hints and prompts and blahs I gave were those damn trinomials with a leading coefficient greater than 1.
Because no matter how slick I could make the trick, the cognitive demand was too high for many of my students. They could remember the easy tricks, but when you started talking about rainbows, or cows jumping around, or any of the crazy “best ways” I heard from other teachers how to teach these problems, it was just a procedure and it was a procedure that did not make sense.
So I burned it all down. I threw out my beloved sequence: 1. GCF (Gotta Check First), 2. DOS (get it? “dos” means “two”, there are two terms, it’s the second type), 3. ET (easy trinomials), 4. HT (hard trinomials, or rainbows, or whatev) 5. Grouping (let’s put this last because it has more than three terms so it looks really hard), and 6. SODOC (eh, leave this for the Algebra 2 teacher, poor sucker, oh wait that’s gonna be me in two years).
And I threw out the “Field Guide to Factoring” we’d make every year, I threw out all the tricks that I used as a student and that I diligently had taught my kids, and I taught every factoring problem the same way. EVERY. SINGLE. ONE.
With that one switch, I all of a sudden noticed something. Some of the “easier” problems under my old system became harder. But all of the “harder” problems became easier. Much easier. And I found that even when kids didn’t know how to complete a problem, they knew how to start one — without hints, without prompting, without any help but my standard stonewall: “Well, what do you think you should do here?”
And I felt dumb. Because what I was teaching was dumb. It was basic, basic, basic. No special teacher skills required, no flashy tricks, no special techniques. No secret knowledge. But, by gosh, it worked.