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It’s standardized test time in Connecticut and my child will be busy participating in this painfully drawn-out process over the next couple of weeks. While I should clarify that I think testing *can* be useful for diagnostic purposes, I consider the following question from the eighth grade CMT Mathematical Applications section to be yet another example of why I find my state's manner of assessing students quite useless:
Sample Item 8-5 (Numerical): Buying Tickets
The carnival offers you two different options for buying tickets.OPTION A: $2.00 per person plus $0.75 per ride
OR
OPTION B: $5.00 per person plus $0.25 per ride
If your uncle gave you $10 for the carnival, which option – A or B – would you choose. Show the mathematics you used to determine your answer.
OPTION CHOSEN: _______
Explanation:
This poor excuse for a word problem is just one example of why we started homeschooling. While I’m assuming the objective of the question is for the student to show mathematically that option B is the better choice because you can go on 20 rides as opposed to only 10 rides with option A, the question does not indicate that the goal is to go on as many rides as possible. I can easily imagine any of my children (including the 8-year old) coming up with alternate scenarios that could make either option the better one. Unfortunately, I can just as easily imagine a scenario where the person responsible for grading 200 tests containing these strange open-ended mathematical responses before the end of their shift would mark their mathematically and numerically accurate answer WRONG.
Let's say the student were to choose OPTION A since he really likes to eat junk food at carnivals (just like his favorite uncle who spots him the $10) but hates the rides because they make him dizzy (thereby leaving him with $8 to spend on food instead of $5). Would that be counted as a correct answer? I would argue that either option could be the better choice depending on the objective—which was not made clear. Mathematics is supposed to be clear, precise, and accurate. This question is just silly.
These types of Everyday Math word problems (I'm being generous here by calling it a word problem) used to make me crazy when my child would come home with them in 4th grade. Now here we are in 8thgrade running in circles all over again.
Meanwhile back in Singapore children are answering this:
Hooke's law for an elastic spring states that the distance a spring stretches is proportional to the force applied. If a force of 150 newtons stretches a certain spring 8 cm, how much will a force of 400 newtons stretch the spring? (New Elementary Math 2 Placement Test)*sigh*
13 comments:
Wow, the implied assumptions in that question are staggering. I'm relatively not-old *cough* and even when I was in school we never had such terribly open-ended questions in math. And 8th graders are expected to do this? I suppose they can "think critically" about it, but yeesh.
oh boy - the contrast between the CT question and the New Elementary Math question is horrifying.
Speaking of word problems...I get confused reading the Art of Problem Solving "Venn diagram" problems...
When a question says, "20 students take algebra" and "30 students take French," I want to know whether "20 students take algebra" means "20 students **either** take algebra or algebra and French."
I guess the logic of the phrasing is strict: you should assume that nothing has been said about whether the algebra student(s) also take French.
But from the POV of a writer, it's bad writing.
My AofPS algebra 1 textbook just arrived!
I love these books, but I wish to heck they weren't so UGLY.
The covers are awful.
There are so *many* of these poorly worded questions - at least in the prep materials. (I haven't seen the actual tests.) Then there are the questions with flat out mistakes (ex. The library has 1270 science books and 430 French books. How many more French books than science books does the library have?) Or weirder ones that I've seen too, where the question itself has different nouns than the set-up. (eg. different farmer's name and different plant variety in a question about how many of x plants the farmer should plant.) Really inspires trust in the results...
Though, I thought you were going to talk about choosing to give carnival food to kids in preparation for the tests. I was flabbergasted to hear that my son's class was given doughnuts before their exam yesterday. The reason, he said, was something like, "to calm us down and help us think." Or maybe it was something else equally ridiculous...ugh
and, oddly enough, my word verification for this comment looks like 'vitriol' if I don't look directly at it.
Fascinating. I read the problem a totally different way: that the writer was an expert hoping to get the student to write "it depends", and then talk about how for a small number of rides, one answer is good, but for a large number, the other is good. I assumed they wanted the discussion and thought that handling the ambiguity would. demonstrate "critical thinking".
An expert could make the mistaking in thinking something like this was manageable by an 8th grader.
The question is totally inappropriate. A related question like "a) graph two lines to indicate the two pricing policies. b) what happens at location (x_important, y_important)? c) which policy is better for x values below x_important? d) which policy is better for x values above x_important?" should be appropriate for kids having taken algebra 1.
Catherine -- If I read that "20 students take algebra and 30 students take French", I would interpret that to mean that the enrollment in algebra is 20 students and the enrollment in French is 30 students, especially since Venn diagram problems usually have you to figure out how many students are doing both, and stuff like that.
I am using the Art of Problem Solving beginning algebra with my 12-year old son now and I am very impressed with it. Sometimes he has been frustrated with being given problems to solve that he hasn't been shown how to solve, so sometimes I ease it up on him by short lectures on the relevant material, and often we do some review of past previous things. On the whole, though, he is learning that he needs to pay attention to problems to see what he can learn from them, not just do them and go on. Often a problem is given to make a specific point.
I compare this to some algebra textbooks that I looked at from major publishers. They seemed so dreary, with what seemed to be chapter after chapter of "evaluating expressions". My son does not like drudgery, and AoPS is short on that. He does like the more sophisticated problems, such as one that have an equation or 2x2 system of equations plus a parameter, and you have to determine for what value of the parameter does the equation have no solutions, things like that.
This problem is a good example of why you need to have at least one parent who is a math professor--so they can explain to you while you are doing homework, what the problem was that the author _intended_ to ask.
... and then there are the scatter plots, stem and leaf plots, and looming blank spaces where you are to explain and demonstrate (the longer and more convoluted the better). Because showing your work with an efficient equation and solving it accurately is just never enough.
I'm very impressed with the Art of Problem Solving books so far.
The Counting & Probability book has been fantastically helpful.
I don't mind drudgery -- I like it, in fact.
But the Art of Problem Solving books are fantastically helpful for --- wait for it! --- problem solving!
I continue to think they're the best preparation for SAT math I've seen. Have no way to prove that, but it sure seems that way to me.
I'll start looking to see whether AofPS problems are similar to SAT problems...
Grade 8 CSAP constructed response item for your enjoyment:
Elaine is shopping for new office supplies. She has made a list of the items she will purchase.
Computer paper $29.25
Hole punch $10.99
Calculator $89.99
Folders $14.49
Estimate how much Elaine will spend. In the space below, explain or show your work and write
your answer on the line.
And the scoring rubric included:
Exemplary Response:
• Estimate $144
AND
• If the cost of an item ended in 50–99 cents, I rounded up to the nearest whole dollar amount. If the
cost of an item ended in 0–49 cents, I rounded down to the nearest whole dollar amount.
Then I added $29 + $11 + $90 + $14 = $144
OR
• Other valid response
Score Points:
Apply 2-point holistic rubric.
Want to torture yourself even more? Here's the page where CSAP released items are listed. Colorado Department of Education They haven't released a math item since 2005.
"... which option – A or B – would you choose." ... to maximize the number of rides you could take.
Either that, or all you have to do is justify your answer with no mathematics. You know how they luv no one right answer questions.
I choose Option A because there is no way in hell I will wait 45 minutes for a ride that lasts 2 minutes. Let's see, for 6 hours at the park and 45 minutes wait time per ride ... why don't we define an entire "fun" merit function for the entire day. There are ways to change "no one right answer" into one right answer for you - mathematically.
I think that SteveH is getting at the general problem with such problems. They are supposed to be "real world" but they are not, since they allow various "real world" considerations (reserve as much money as possible and hope you can leave the stupid fair as soon as your little brother has ridden the Dinky Donkey two more times) that the grading authorities would probably not really accept.
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