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).
Well said, Delta
ReplyDeleteI hope elementary teachers read this too - it's not just algebra questions that are crap in many places. In fact, some even recommend teaching students to look for "key words" so they can ignore the rest - AAARRRGGGHHH!
Let's start a revolution about teaching math so our students UNDERSTAND it, not so they learn a bunch of tricks to answer questions quickly. :)
^ I fully agree!
ReplyDeleteHere's a related anecdote. First day, first problem in my sophomore statistics class looks like this -- "The following table provides percentages of U.S. adults, by educational level, who believe that evolution is a scientific theory well supported by evidence... Do you think that this study is descriptive or inferential? Explain your answer."
ReplyDeleteAnswer -- "Inferential, because the reported results are for all 'U.S. adults', not restricted to a sample or survey."
So last year this prompts one student to ask: "Do you you always do it like that?" And I'm somewhat at a loss for words, but what I say is, "No, you have to read the problem and do whatever it asks you to do." And the student in question had apparently never confronted that issue.
'...that for any given algebraic expression, there's some implied thing that you always "do" to it'
ReplyDeleteThat's some heavy, hammer-beating-nail-on-head stuff right there, Delta. Many people seem to suffer from this disease and it limits their ability to succeed in advanced math courses and hard-science.
It's also worth noting that math teaches essential problem-solving skills, critical thinking skills, visualization skills, and other abilities useful for life that have nothing to do with solving an equation.
I think part of the problem is that we teach math as a completely isolated subject through algebra II/trig. Most people end their studies at about this point and think "well that was interesting but a complete waste of time."
I find that math is a lot more interesting and digestible in the context of physics lab, real-world examples, and word problems. I didn't really understand algebra until I got to calculus and put all those esoteric techniques to use. I didn't understand series and Fourier analysis until I took engineering courses and was exposed to useful and meaningful applications of math.
Going back to your point, I fully agree that many math problems don't make it clear what you're being asked to do. I have no supporting evidence, but I also believe that poorly-worded problems can lead to an apparent inversion of intelligence in the class where the students who don't really know what they're doing can at least memorize the mechanics and the more capable (but bored) students don't recognize the limited scope of the question and give a wrong correct answer.
Thanks for the comment and appreciation on that point.
DeleteJust today (first day of fall semester for me), I did my usual 1st-class handout of the definitions of terms for my remedial algebra class. One girl comes up at the end and profusely thanks me for it, saying, "I begged my last teacher to tell me what these terms meant, and he said don't worry about that, just do the problems." And my mouth went agape and I started thinking about going to find someone to throttle!
Your realization that this implied thing that you "always do to it" is somewhat useless if taught came to me one day, too. For a bit I used symbolic math software -- both Maple and MathCad, they both use the same engine (or at least used to at the time). The results were usually presented simplified in a certain way, and it seemed to "agree" with how things were expected of us in high school maths.
ReplyDeleteI then started using the open source Maxima, and there the default simplifications were rather minimalistic. Maxima had me explicitly invoke various kinds of simplifications. At first it seemed like a drag, and I thought it was some kind of a drawback. Eventually it came to me that perhaps, just perhaps, the nice form of "usual" or "common" high-school simplification is not very useful except in said high-school math setting. That's because I was using symbolic maths in engineering, and how you'd want the final symbolic result presented was really application specific. Different kinds of post-processing "simplification" applied to different subject areas and types of problems.
There were times when sorting by highest monomial exponent in sums monomial products made the expression unreadable in the context of a problem. Other times taking out common factors would make it blow up to half a page and make it equally unreadable. Various simplifications of rationals/fractions would also backfire. Almost every simplification you'd think of, that we were expected to apply as a matter of fact to produce "clean" final results, would backfire once, and then keep backfiring when one delved deeper into a particular kind of problems. Eventually I realized that pretty much every symbolic math problem that I was solving in grade school was engineered to be amenable to certain mechanistic approach in solving it, including mindless application of simplifications. Real life turned out to be a nice eye opener.
Coincidentally, I agree with Unknown's love for applied math. I couldn't make the heads nor tails of maths until I could apply it.
Wow, what a great observation! I hadn't heard of Maxima before, but awhat a fascinating lesson to draw from it and your work.
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