More Anti-Factoring Trees

Follow-up: Consider the article by McNeil, et. al. in COGNITION AND INSTRUCTION, 24(3), 367–385 (2006), on "Middle-School Students’ Understanding of the Equal Sign: The Books They Read Can’t Help" (link). In the conclusion they write:
In this study, the operations on both sides context was most effective in eliciting a relational understanding of the equal sign... Although the jury is still out, we argue that middle-school students would benefit from seeing more equal signs in an operations on both sides context.
Consider: Factoring seems like a golden opportunity to practice writing and reading operations on the right-hand side of the equals sign (i.e., use for anything other than simplifying/evaluating). And for that reason, using factoring trees instead of standard equation-writing is even more of a huge lost opportunity than I first thought.


Words Matter!

Here's a thing that irritates me more and more over time: When a math problem doesn't have any words in yet. Most specifically: when it lacks an action verb on what you're supposed to do with it. For example, here's a problem that comes from a textbook I use (and similar stuff even pops up from time to time on our department-wide final exams):

For all x, 30x5 + 32x4 + 8x3 = ?

Well... it's equal to all kinds of friggin' stuff. Like: 30x5 + 32x4 + 8x3 + 1 - 1 and an infinite number of other things. Now, in this particular case, if it's a multiple-choice problem, then you can look at the proposed answers and infer that what's being requested is for the expression to be factored. Although you can still get in trouble if one of the options is only partly factored, but it's still technically equal to the original expression. Stuff like that. But this sample problem is definitely not a fair question, because you could not tell what action to take if it were posed completely alone, outside the context of a multiple-choice test (plus: many of our students' abilities to look at multiple-choice responses and back-infer intent will be shaky at best).

I think that this is a major symptom of a scurrilous disease that lets students get away with the false impression that for any given algebraic expression, there's some implied thing that you always "do" to it -- when that's absolutely, totally not the case. Different use-cases will require different actions to be taken (e.g.: sometimes to factor, and sometimes to simplify, which are opposites).

So once again: It really all comes down to a matter of reading. If students think they can "do" math through rote mechanical processes without reading the words -- at least a requested action to take, a single verb at minimum -- then they are tremendously, grievously in error. The #1 skill that I tell my algebra students they're expected to master is learning new vocabulary, so that we can have an intelligent discussion about math, and so they can follow the instructions on a test from me or anyone else (and more generally: make use of that learn-new-vocabulary skill elsewhere in their lives). Failing to phrase our math questions with clear, well-defined action requests in words is simply an atrocious example to set.

One last example: Take the expression 4(x2-9). There's all kinds of things we might have to do with this at different times, including but not limited to any the following (so: get in the habit of reading & writing the words carefully for any of these):
  • Simplify. (Answer: 4x2-36).
  • Factor. (Answer: 4(x+3)(x-3)).
  • Identify the Degree. (Answer: 2nd).
  • Determine the Roots. (Answer: +3 and -3).


Dead Grandmother/Exam Syndrome

There was a discussion the other day where I mentioned a student that a colleague and I had many years ago. The colleague once said, "I just feel terrible for his mother; every time I give a test she has a heart attack."

So apparently this problem is more widespread than I first thought; someone on Slashdot kindly linked to an article by Mike Adams of the Eastern Connecticut State University Biology Department, on the subject of "The Dead Grandmother/Exam Syndrome and the Potential Downfall of American Society" (Connecticut Review, 1990). A truly stellar piece of work; highly recommended.