Noteworthy:

- The kids have a
*ton*of confidence, even in the stuff that they haven’t formally studied in class yet. (For this survey, Questions 1-3 had been covered formally, and Questions 4-5 had not.) To my mind, this continues to reaffirm that the most annoying mistakes aren’t the distortion of instruction; they’re the*failure*of instruction to override preconceptions. - Kids like to say that , and teachers like to say that this is due to overuse of the Distributive Property. That might be true, but those teachers also have to recognize that kids said that with almost the same verve and frequency. It’s hard to blame exponents or notation for
*that*mistake, right? So where does this intuition come from? - A couple of kids included a term in Q4 and a term in Q5. I find this interesting, but I’m not exactly sure what its significance is. Is the temptation to add and when the binomials are in the same visual position that they are for addition problems?

The idea that kids walk into our classes with these intuitions is, I think, counter to the way that most math teachers talk and think about these mistakes. I think that realizing that these mistakes are the result of deep intuitions about how math *should be *is important. I also think thinking about where these intuitions come from is important, because maybe we can avoid setting them in earlier years.

I hope that some of you will give this survey to your students who haven’t yet received instruction on how to multiply polynomials. The original survey can be found here.

You’ll disagree with me in the comments, right? I’m counting on you all…

Yeah, we absolutely have to break ourselves free of the idea that kids have no intuition about how math works before we attempt to teach it to them (through whatever method we feel works). I’m not totally sure we can avoid the intuitions in the earlier years so easily though. What helps a student understand division may hurt them when understanding algebra (although I have no proof of this claim, I am using it as an example).

http://www.learner.org/resources/series26.html describes the attempt to teach photosynthesis and the idea that the solid in the stick in that tree is built with carbon atoms from the C02 in the air… and fails miserably, despite concept-rich teaching and very smart student. They suggest that the fundamental misconception wasn’t addressed — that air doesn’t have mass (so can’t be turned into something that does).

I remember recognizing the awesomeness that “holy cow, the C02 is turning into FURNITURE *and* storing up energy!” … in a senior level undergrad course in Plant PHysiology… and I also remember having been imbued from childhood in the benefits of scraping out common misconceptions for that ego gratification of GETTING THINGS RIGHT. Not sure how that can translate into instruction, though. One thing I thought of to try for computer exercises was to have setups like this, where the misconception was tagged so when you hover over it, it reminds you… perhaps you’d get a big smiley face the first time you just click on the right answer (this would come after the visual buildign up of the concept of course).

http://www.resourceroom.net/2012math/misconceptionjun8.swf

(oh… the drawing is “Miss Conception,” who works nights and keeps getting dragged out to the wrong place at the wrong time. She wishes people would just get to understand her instead of trying to avoid her…)

I’m spitballing here. But I wonder what would happen if we changed our approach in Algebra from “prove this” to “disprove this”. What would it be like if we had a problem like:

Find a counterexample to (x + 7)^2 = x^2 + 49.

and then talking about what (x + 7)^2 is equal to. I find that approach works in my stats class (i.e. find the flaw in this study). That is, I’m trying to play up my kids’ level of cynicism for good.

I give a lot of “compare and contrast” practice. Then they can tell me things like “they are the same only if x = 0″ and “the difference is 2x” and “one of them is always nonnegative, the other is always greater than or equal to 49″ and whatever else they can come up with.

I have always tried to introduce a different representation that makes it visible that x^2 + 49 = (x+7)(x+7). The missing linear terms are quite obvious with algebra tiles or a drawn equivalent.

Yeah, an area model of some sort for multiplication seems to be helpful to a lot of people. It would be nice if we chose algorithms for integer multiplication that would carry over nicely to polynomials.

I’ve certainly seen kids who do things analogous to these mistakes with integer multiplication too, like multiplying the hundreds place, tens place, and ones place, as in 312 * 231 = 632.

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