Wednesday, July 25, 2007

Understanding Algebra Conceptually?

What do you think would be the results of giving the following Algebra 1 problems to your students before, during, and after the course?
Do you believe that either or both of these could be or have been SAT questions?
Do students normally have exposure to these kinds of problems in their regular assignments?
Do these kinds of questions require a deeper conceptual understanding of algebra?

1. Given: x2 - 9 = 0
Which of the following must be true?

I. x = 3
II. x = -3
III. x2 = 9

(A) I only (B) I, II only (C) I, III only (D) I, II, III

(E) none of the preceding answers is correct

2. Given: (a - b) (a2 - b2) = 0
Which of the following must be true?

I. a = b
II. a = -b
III. a2 = b2

(A) I only (B) I, II only (C) I, III only (D) III only (E) I, II, III


Totally_clueless said...

Is there an implicit assumption that the comma in the choices means 'AND'


Dave Marain said...

sorry 'bout that! I got lazy - shame on me. On a real SAT, it certain would be written "I AND II only"

mathmom said...

The wording of the questions (even if AND were used to replace the commas) is confusing. x=3 AND x=-3 are both solutions to the first equation, but of course neither MUST be true (since the answer could be the other instead) but... well, it just feels like confusing wording to me. I think many kids would get caught up on the wording of the question, even if they understand the concepts.

Dave Marain said...

I agree - it's confusing. However, the wording is as it typically appears on the SAT (you can verify this by looking at released tests). As you pointed out, the key word in this problem is MUST. This is about logical necessity and this is subtle and sophisticated for many students. Students reason:
If x = 3 THEN x^2 - 9 = 0, so I is true! The idea that the logic must go the other way confuses strong high school students as well. That doesn't necessarily make this a poorly worded question. It may indicate that students need more experience with deductive reasoning. I always ask students:
Does x have to equal 3? Yes or No?
Does x have to equal -3? Y or N?
"Have to" is synonymous with 'must!'

I think it takes considerable time and maturity to develop a comfort level with the language of logic.

I suspect this discussion could generate lots of heat about semantics and wording of standardized test questions!

Jackie said...

Hmm, as most "average" students in our state take the ACT, I don't know how my students would do.

I wish I could say that these types of questions were part of our regular curriculum, but... sadly not on a consistent basis.

As to your question of "Do these kinds of questions require a deeper conceptual understanding of algebra?", my response is yes. So many of my students struggle with the difference between and/or statements. The logical thinking is not something with which they are comfortable. When I ask these questions or stress the difference between writing, " x = 3 and x = -3" and " x = 3 or x =-3"they think I'm being overly picky.

mathmom said...

It seems to me that there are 2 things necessary here -- understanding the algebra, and understanding the SAT-style wording. I'd argue that the second is mostly only useful for taking the SAT. It's not only getting the "MUST" part but then also getting the AND/OR combinations of answers. I think if I were posing this in a non-SAT-prep class I'd ask: "Circle all the statements that must be true" or something like that, without getting into the combinations and no correct combination type multiple choice.

The must is not what makes it awkward. It was the different combinations combined with the must that I found awkward. Perhaps because of the sloppiness around the "ands" but overall I do think that adds another degree of confusion to the question that has nothing to do with "understanding algebra conceptually". I know they do that on the SAT, but that's SAT test-taking skills, not algebra. ;-)

Dave Marain said...

Your comments are dead on!
Certainly, there's a a huge mathematical difference between the following choices:
(A) x = 3 and x = -3
(B) x = 3 or x = -3.
This distinction is important but is rarely tested on the SAT or other standardized tests.

It's important for me to point out here that my questions were NOT actual SAT questions (I avoid this for copyright reasons). It was my lame attempt to write questions similar to a question on last October's PSAT that was missed by many. That problem avoided the issues of AND vs. OR! You and others are my excellent 'quality control' team. My question would never have made the cut' over at ETS!!

In my opinion, the underlying concept of this problem is still very important for students to grasp. My careless wording has now become more of the point of all this than my original intent, but maybe the issues of wording and item construction are just as important here! The reality is that these kinds of problems are and will be tested on standardized tests. Semantics and the form of the question are critical, since one does not want the FORM to be more critical than the content. ETS, in my opinion, generally does a good job of checking the wording (certainly better than I did).

You made some excellent suggestions for improving the question. Although it's probably not worth salvaging the problem, I'm wondering if anyone could work this as a Roman Numeral I, II, III type problem or is it not worth the effort! The original problem was multiple-choice by the way! That may be the best way to go. Of course, in my gut, the best way is to avoid this as an objective question. Having students discuss this topic in an open-ended manner is far more worthwhile, but my experience is that they take it more seriously when it appears as a standardized test problem.

Thanks again for highlighting these issues. I am fortunate to have such critical readers!

mathmom said...

Certainly, there's a a huge mathematical difference between the following choices:
(A) x = 3 and x = -3
(B) x = 3 or x = -3.
This distinction is important but is rarely tested on the SAT or other standardized tests.

And... (A) doesn't make sense, IMO. X cannot equal 3 and -3 at the same time. I think that is the crux of what bothered me about the original question. the options that contained both I and II just seemed nonsensical to me, and that felt awkward.

If I were to rework it into a Roman numeral I, II, III problem, I would avoid putting I and II together with an "AND".

Sorry about beating a dead horse....

Anonymous said...

i think it's a very good question
(improved slightly by using "AND").

as to mathmom's concerns:
confusion, and awkwardness,
and even nonsense, are all
part of the process. how else
are we to know clarity
when (if at all) we finally
achieve it?

"x = 3 AND x = -3" might
very well be the result
of calculations done right;
one then concludes (of course!)
that there is no solution for x.

(some) students will always think
we're being "overly picky"
(as jackie observes) when in fact,
we're merely being very picky
(which is often just the right amount).

vlorbik (VME)

jonathan said...

By substituting "are" for "must be" the problem changes. An algebra II course would move a few kids away from choosing I only for each, and I think many, would get the correct answer out of confusion, not knowledge.

I don't remember that combination of "must" with I, II, III questions from back in my day or from prep. I would have been annoyed.

We can make it clearer (although why should ETS?) by asking the almost as difficult "Which of the following does not have to be true..."