A few months ago I joked, How Long Till the Polynomials?
All kidding aside, I knew there was a pebble in my shoe around the polynomials, I just couldn't pinpoint the issue at the time.
Turns out it was more than a pebble.
Cut to a few weeks ago, and I was attempting to write my own "Solve for Expressions" question. I emailed the first draft to PWNtheSAT so he could take it for a test drive, and I got back the following message:
"I don't see how to get from what you're giving me to what you want......Is there a trick I'm not seeing?"
That was my first inkling that something was very wrong, but I checked my work and sent him my steps:
First this: (a - b)², then that: (a + b)(a - b), etc. etc.
And then he emailed me again:
"Look at your second step!"
And just like that, in the blink of an eye, I had my polynomial epiphany.
(Incidentally, I'm baring my soul here in case there's anyone out there who might benefit from knowing that it's okay not to know everything.)
Mortified, I wrote back, "I'm scaring you now, right? I'm beyond your scope, aren't I?"
Then he told me it's a big distinction, but a common mistake (and I am choosing to believe him about the "common mistake" part, if only to maintain the courage to soldier on, and not die from embarrassment.) And, I'll try not to obsess about what other holes might be lurking.
I called my friend Catherine who attempted to console me. "It's not you," she said, "It's called associative interference. Have you read Wickelgren?"
And then she sent me a post she'd written, from which I will quote, because it did make me feel better: Why is Remembering What You've Learned About Math Hard?
It's the similarity between the facts. That is, the fact 3 + 5 = 8 is not so different from 3 + 6 = 9. They both contain 3's; they both contain +'s, and they both contain single-digit numbers....
Thus, to a child beginning to learn such facts, the facts overlap in the brain, creating a blur that makes it easy to confuse them and difficult to remember any single answer. In cognitive psychology, this "blur" is called associative interference, which occurs when one idea, A, is linked in the mind to two or more other ideas. It's like static on the radio, which often occurs when other stations or electrical impulses interfere with a radio station's music or speech.
Anyhoo, I adapted my "Solve for Expressions" question to incorporate all areas of confusion:
If a² = 4 and b² = 9, which of the following could equal c in the following equation: c(a - b)² = 2(a² - b²)
Hopefully, "the issue" is now resolved. I did have a moment of satisfaction when I ran across a need to know this piece of information yesterday, while taking a full, timed, practice test.
As always, any and all attempts to answer the question above in the comments below, will make my day.
Illustrations by Jennifer Orkin Lewis