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Old 2nd March 2012, 06:33 PM   #1 (permalink)
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Words Matter in Math!

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, 30x^5 + 32x^4 + 8x^3 = ?

Well... it's equal to all kinds of friggin' stuff. Like: 30x^5 + 32x^4 + 8x^3 + 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(x^2-9). 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: 4x^2-36).
Factor. (Answer: 4(x+3)(x-3)).
Identify the Degree. (Answer: 2nd).
Determine the Roots. (Answer: +3 and -3).

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Last edited by whitecorp; 2nd March 2012 at 09:55 PM.
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Old 2nd March 2012, 09:37 PM   #2 (permalink)
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Re: Words Matter in Math!

You know,you should have rewrote all the powers to ^X instead of leaving them like this.

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Old 2nd March 2012, 09:56 PM   #3 (permalink)
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Re: Words Matter in Math!

Done. Hope this is better. Peace.

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Old 2nd March 2012, 10:06 PM   #4 (permalink)
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Re: Words Matter in Math!

Lol in summary, the flaw in maths education around the world is that students are taught HOW to do it, but they don't know WHAT they are doing and WHY they are doing it, which results in them not knowing WHEN to do it.

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Old 2nd March 2012, 10:47 PM   #5 (permalink)
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Re: Words Matter in Math!

few words to define what you are looking for is fine but i hate when they start using words to describe math expressions.

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Old 2nd March 2012, 10:54 PM   #6 (permalink)
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Re: Words Matter in Math!

Originally Posted by Betakuwe View Post
Lol in summary, the flaw in maths education around the world is that students are taught HOW to do it, but they don't know WHAT they are doing and WHY they are doing it, which results in them not knowing WHEN to do it.
Clearly you missed the point of the original thread by a mile. It's about how sometimes students cannot be faulted for their inability to solve problems simply because the questions are just too ambiguous.

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Old 2nd March 2012, 11:06 PM   #7 (permalink)
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Re: Words Matter in Math!

Originally Posted by Deadpool View Post
Clearly you missed the point of the original thread by a mile. It's about how sometimes students cannot be faulted for their inability to solve problems simply because the questions are just too ambiguous.
Yeah somewhat, but somewhere in the post says "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." This thing happen when students learn things blindly. So say for example a student is taught how to factorize expressions and solve simple equations, then the student is given a maths question, for example: Solve 2x^3-18=0.
Then the student sees that it can be factorized into 2(x-3)(x+3), but is not sure whether to factorize or not in order to get the solution. So this is an example of knowing how to do it(factorize), but don't know why or when it should be done.

I believe, also, that ambiguity in some poorly set questions can also be due to this mindset in some teachers that, for example, given a certain expression that can be factorized, the expression "should" be factorized, which results in the teacher setting the question poorly.

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