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The June 2008 Math B exam is available as a PDF from JMAP.

I already posted three bad problems. (edit: There is ongoing math teacher discussion of individual questions at a Math A/B listserve run by the Association of Mathematics Teachers of New York State). Here are three more problems that bother me:

#25 (2 point, free response)
The accompanying diagram shows the peak of a roof that is in the shape of an isosceles triangle. A base angle of the triangle is 50° and each side of the roof is 20.4 feet. Determine to the nearest tenth of a square foot the area of this triangular region.
Consult the diagram in the link above. An isosceles triangle is drawn, sides are marked 20.4 and 20.4, and one base angle is marked 50 degrees.

Their intent was for kids to find the missing angles (50, 50, 80) and apply the formula (given in the text booklet): Area = (½)(ab)sinC where C=80 and a=b=20.4

But kids dropped altitudes, working slightly longer around, and ended up with a minor rounding error. New York says: 1 point. They did more work, good work, and only get 1 out of 2 points?

The question design is poor. It could have been anticipated that there were two, equally reasonable, but not equally easy, points of attack. It wasn’t. No way a two point question.

(more beneath the fold)

#27 (4 point, free response)
Students have a table with dewpoint data by temperature, and are asked to perform an exponential regression, rounding all to the nearest thousandth.

There’s only one way for kids to do this: memorize procedure. Plug both lists into the calculator, call for exponential regression, round the base and the constant, and plug in 50. Four points for memorizing a word, and a bunch of TI keystrokes?

#28 (4 point, free response)
A right triangle, FEP, is drawn with altitude FM dropped to the hypotenuse. Legs are labeled “Elm” and “Poplar,” altitude is “Fern” and hypotenuse is “Maple”. ( Consult the diagram in the link above.)

Four streets in a town are illustrated in the accompanying diagram. If the distance on Poplar Street from F to P is 12 miles, and the distance on Maple Street from E to M is 10 miles, find the distance on Maple Street, in miles, from M to P.

What makes this one bad is its history. In the 1980s and 1990s this problem, sans streets, was a typical Course II question. One version or other was on every Course II Regents that I can remember. But this is the first time, I think, it was ever on B (in 20 odd exams), and I am not certain it is in the State’s list of topics.

Now, the bizarre rubric allows for 2 points for good work with one “conceptual” error. I would interpret a bad guess that the altitude bisects the angle to be a conceptual error.

Kids whose teachers refuse to let go of this topic had a good chance of knowing a shortcut. Kids whose teachers showed them this, even once, would likely search for similar triangles and proportions, and get it or come close. But kids whose teachers did not teach this were likely to flounder about. Read some actual kid comments here on Yahoo Answers: 123456.

The level of difficulty becomes unpredictable when the test-maker does not adequately communicate the possible range of content to the teachers. And before you start jumping up and down about teachers not covering enough, recall that the “content” of this mile-wide exam goes from circle proofs to trig to functions to regression to complex to … Decisions about what to do and what not to do are high stakes decisions. Adding the occasional “extra” to the curriculum can be risky business. What can we afford to omit?

1. June 22, 2008 pm30 11:02 pm 11:02 pm

OK, my immediate reactions:

Overall: that’s a very respectable-looking paper at first glance, lots of content etc. However your reactions suggest it may normally be *intensely* predictable, which obviously lowers the difficults a very great deal.

First question: “The question design is poor. It could have been anticipated that there were two, equally reasonable, but not equally easy, points of attack. It wasn’t.”

I think we have a fundamental difference in philosophy here. If I had set that question, it would have absolutely been anticipating that there were multiple methods of attack. In what decent geometry question is there not? A major part of the skill to be developed is picking the best one. In any case, the final mark ABSOLUTELY should not be given unless the answer is correct to the required accuracy. If kids don’t know that they need to carry extra precision through a question to round properly at the end they deserve to be dropping marks.

I can’t say whether two marks is reasonable since I don’t know the standards. I think for us at the 16-year-old level that would be three marks (method for finding apex angle OR height, method for second step, third mark correct answer only) but it’d only be two at 18-year-old if it were set at all.

Second one: does seem a little over-rewarded compared to the first, but I wouldn’t have said excessively so – especially in an exam with very predictable questions, as I’m (maybe wrongly) inferring from some of what you’ve said, MOST of these questions are just button-pushing for a well-prepped candidate, after all.

Last one: and this is what I’m basing my inference on. If it’s a non-standard question, great, it’s a thinking test. It’s the sort of question I’d put on a test as a separator for the stronger ones at 14, or as a standardish question at 16+, without them necessarily seeing one like it before. If some teachers will have taught it specifically, presumably they won’t have taught other things in that time that you have, no? Tests SHOULD have unpredictable questions on them, surely!

June 22, 2008 pm30 11:16 pm 11:16 pm

I feel a lot better about writing down nothing for question 28 now. Neither me nor any of my friends I asked had a clue on what to do on that one, and it never struck me to do what was actually necessary to complete the question.

Well, as my sister explained to me going into Math B last year, ‘All you can do is hope to pass the regents, it’ll get better after that.’ I had no idea to the amount of truth that statement would hold after doing a practice regents a couple weeks prior to the examination.

(For the record, I did pass, although whatever it was, I’m sure it made a mockery of whatever I was supposed to learn…which I’m still not entirely sure what I was supposed to)

3. June 23, 2008 am30 12:11 am 12:11 am

@Dr –
I like the questions with multiple approach – however they pointed it as a routine question. At 4 points it would have been fine. The authors clearly (and I have an answer key) did not anticipate multiple approaches. Mistake.

If they wanted a routine question (2 points) they should have nudged kids towards their desired method. I don’t think that would have been a good idea, though.

They should have seen that they had something open-ended, and rewarded it appropriately.

The second question just represents the triumph of Texas Instruments marketing over mathematics. Embarassing.

And the third is not so much a bad question as it is a problem of too many topics and too many disparate curricula. If you are lucky enough to be in a school where they taught this as a routine problem, ho hum, cash in the points. If not, you have a non-routine question. It is simply not fair to broaden the curriculum so far, to make it so diffuse, that we can no longer give a problem and know whether or not it is non-routine.

This is a mess caused by non-explicit content standards. I like non-routine. But I like fairness. And vague standards, broad standards, too many topics – they strip away the fairness.

@Stephen – I think no college out of New York will care too much about the score – doubtful they will no what the exam is, except that it must be higher than “A.” Next year’s math will not be targeted at a NY State exam, so we can hope that it will be a more positive, more intellectually rewarding experience.

4. June 23, 2008 pm30 12:25 pm 12:25 pm

“If you are lucky enough to be in a school where they taught this as a routine problem, ho hum, cash in the points. If not, you have a non-routine question. It is simply not fair to broaden the curriculum so far, to make it so diffuse, that we can no longer give a problem and know whether or not it is non-routine”

Naturally, but surely that’s true of any question? The point I was trying, apparently badly, to make is that ANY problem will be routine to a kid who happens to have seen it before, but an exam with a scattering of unusual questions should have non-routine problems for everyone – just not always the same ones.

5. June 26, 2012 am30 7:30 am 7:30 am

I think Erika Napoletano is especially spectacular at pulling this off. She’ll talk about some awesome/shitty/outrageous that happened to her or someone she knows and leads with what happened and then shows you what you can learn from it.