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Re: [Phys-L] Evaluation tests



Obviously should have been 1.a, 1.b and 1.c in the third para from the bottom ... sorry!

On 12/17/2013 12:52 PM, Ze'ev Wurman wrote:
It's interesting that there seem to be a misunderstanding what a success on each of the following indicates.

1) Which is bigger:

(a) 5/7 or 7/9 ; (b) 25/27 or 15/17 ; (c) 12345/12347 or 23457/23459?

and

2) Add 198 plus 215

Problem (2) checks if the student (i) understands addition, (ii) can perform addition, and (iii) has a repertoire of mental shortcuts, often (but not necessarily) stemming from deeper understanding of addition (if we can measure the response time on the test somehow).

I doubt that (i) or (ii) are predictive on future success in college. Even a first grader *understands* integer addition and most can perform the addition of 3 digit numbers by the time they are in college. Some perhaps slower -- or less reliably -- than others, but integer addition not a stumbling block for most. So, at most, (iii) has some potential as a predictor. Even if we assume that we can somehow measure the response time to identify those who used a mental shortcut, what does it really tell us? At best, that the student understand ADDITION to a slightly deeper level. As likely it might be that he has been trained in this particular mental trick, as it is quite common in today's early grades. So it may not even mean much about "deeper understanding addition" but simply having mental dexterity to add longer (and a limited subset of) integers. Bottom line, I would consider (2) useless as a predictor for anything in college.

Variants of problem (1) are somewhat different. They try to measure *understanding of fractions.* Now, I doubt anyone will argue that first graders -- or most sixth graders -- *understand* fractions. Further, understanding fractions is key to forming, simplifying, and solving functional relationships and polynomial expressions, and hence relevant to high school and college math and science. So it is reasonable to think that a good understanding of fractions might be predictive of some college success.

The problem of (2.a) is too easy to solve numerically rather than abstractly, so it will not have good discriminating power. (2.c) is too hard to solve numerically which gives a strong hint not to solve it that way. Its large numbers are also bound to scare-off students, so some may simply guess or skip the question. I realize that John thought that giving a hint that it is a logical problem is a "good thing." I challenge that. I think that if a students needs such hint then much of the predictive power of this item may be gone. So I find the original (2.b) of 25/27 vs 15/17 in the "just right" category: not too scary numbers; complex enough for manual division that if you are foolish to do it that way you may make arithmetic errors, or waste enough time, so the overall results will be discriminating; small enough to visualize the fractions if someone uses visualization.

As an aside, it may help BC and others to treat denominators as the "names" of fractions. This helps young students to realize that you can't add <something>/6 with <something>/7, the same way you can't add 2 bananas and 3 apples to get five of anything. It also helps to make clear that <something>/6 is in bigger units than <something>/7 and hence two units of the former are bigger than two units of the latter.

And remember what started this thread. It is NOT ABOUT SOLVING THE PROBLEM, IT IS ABOUT THE PREDICTIVE POWER OF THE PROBLEM.

Ze'ev

On 12/17/2013 9:14 AM, John Denker wrote:
On 12/17/2013 09:44 AM, Robert Cohen wrote:
For a simpler question, how about 5/7 vs. 7/9?
OTOH one could argue that it would simplify the task to
ask which is larger, 12345/12347 versus 23457/23459.

That makes it more obvious that it is a logic problem not
a simple division problem, and increases the value of the
devious solution relative to the brute-force solution.

It really doesn't matter as long as you talk to the student and ask
how he/she approached the problem.
There are lots of questions in that category, questions
that work fine one-on-one but are less reliable and more
difficult to interpret in a mass-production written-test
environment.
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