Posts tagged ‘multiple choice’
How Would You Answer These Questions?
The following is one of my alltime favorite assessment items:
Which of the following is the best approximation for the volume of an ordinary chicken egg?

The reason it’s one of my favorites is simple: it made me think. Upon first look, I didn’t immediately know the answer, nor did I even know what problemsolving strategy I should use to attack it.
I used estimation, first assuming that the egg was spherical — and, no, that is not the start of a math joke — and then by attempting to inscribe the egg in a rectangular prism. Both of those methods gave different answers, though, and not just numerically; each led to a different letter choice from above.
Not satisfied, I then borrowed a method from Thomas Edison — I filled a measuring cup with 200 mL of water, retrieved an ordinary egg from my refrigerator, and dropped it into the cup. The water level rose by 36 mL. This proved unsatisfying, however, because although choice B is numerically closer to this estimate than choice C — only 29 mL less, compared to 34 mL more — it was five times as much as B but only half as much as C. For determining which is closer, should I use the difference or the ratio?
It was at this point that I decided the answer doesn’t matter. I had been doing some really fun math and employing lots of grey matter. I was thinking outside the box, except when I attempted to inscribed the egg in a rectangular prism and was literally thinking inside the box. And, I was having fun. What more could a boy ask for?
On a different note, here’s one of the worst assessment questions I’ve ever seen:
What is the value of x?
3 : 27 :: 4 : x
I can’t remember if it was a selectedresponse (nee, multiplechoice) item, or if was a constructedresponse question. Either way, it has issues, because there are multiple possible values of x that could be justified.
On the other hand, it’s a great question for the classroom, because students can select a variety of correct responses, as long as they can justify their answer.
The intended answer, I’m fairly certain, is x = 36. The analogy is meant as a proportion, and 3/27 = 4/36. (Wolfram Alpha agrees with this solution.)
But given the format, it could be read as “3 is to 27 as 4 is to x,” which leaves room for interpretation. Because 27 = 3^{3}, then perhaps the correct answer is 64 = 4^{3}.
Or perhaps the answer is x = 28, because 3 + 24 = 27, and 4 + 24 = 28.
Don’t like those alternate answers? Consider the following from Math Analogies, Level 1, a software package from The Critical Thinking Company that was reviewed at One Mama’s Journey.
If this analogy represents a proportion, then the correct answer is $10.50, but that’s not one of the choices. Instead, the analogy represents the rule “add $1,” and the intended answer choice is $10.00.
What amazing assessment items have you seen, of either the good or bad variety?
Wait, Wait… I’ve Got a Math Question
“Not My Job” is a segment on the NPR game show Wait, Wait… Don’t Tell Me! During the segment, host Peter Sagal asks a celebrity three questions, on a topic about which they likely have no clue. For instance,
 Cindy Crawford was asked questions about scale models, not supermodels;
 Rob Lowe was asked questions about bratwurst, not the Brat Pack;
 Stephen King was asked questions about the Teletubbies; and,
 Leonard Nimoy was asked questions about the other Dr. Spock (you know, the celebrity pediatrician).
My favorite of these segments, however, was with singersongwriter Will Oldham, better known by his stage name Bonnie “Prince” Billy. Sagal explained, “You sing mostly sad or mournful songs, interspersed with the occasional tragic one. And we were thinking, who’s the singer least like you? […] So, we’re going to ask you three questions about Ms. Doris Day, the sweetfaced, sweetvoiced singer most famous in the 50s and 60s.”
Oldham answered two of the three questions correctly, so he won.
As Sagal congratulated him, Oldham pretended that Doris Day was in the room with him. “Hey, Doris, you were right!” he said. “All the questions were about you!”
Now, that’s funny!
(As an aside, let me share with you a slide that I sometimes show during presentations about classroom technology:
Yep, that’s Doris Day in the 1958 movie Teacher’s Pet. Hopefully that image looks a little odd to you in 2015. Honestly, if you’re teaching math to adolescents and still using chalk and a wooden pointer, I’ll kindly ask you to consider a different career. There are other options for you. For instance, you could become a dentist and specialize in square root canals. But, I digress.)
So, back to the point.
The celebrity quiz contains three multiplechoice questions, each with three choices. If the guest answers at least two questions correctly, he or she wins. Which got me to thinking…
What is the probability that a celebrity guest will get at least two questions correct if she guesses randomly?
Brute force is definitely an option here. Write down all possible answer choice combinations, randomly decide which configuration will be correct, and then figure out how many of the possible combinations would yield a winner. Not pretty, but it works.
Speaking of combinations, here’s a joke that just has to be shared:
Courtney Gibbons’s comics used to appear at Brown Sharpie, but then she got a job.
Prime Number Problem and Gender Bias
This morning, my friend AJ called to ask for help in solving a problem from his tenyearold daughter’s homework. When he explained his dilemma, the first thing I did, of course, was laugh. “Wow,” I said. “You really aren’t as smart as a fifthgrader, are you?”
AJ and his daughter are both intelligent, and his daughter loves math. The problem they were trying to solve was this:
What is the units digit of the product of the first 21 prime numbers?
You can use this list of prime numbers if you need some help. As a hint, the 21st prime number is 73 (which, incidentally, is the Chuck Norris of numbers).
Once you solve the problem, of course, you realize that the problem would have the same answer if asked as follows:
What is the units digit of the product of the first n prime numbers, for n > 3?
This made me think that this could be a good problem for the classroom. Have all students randomly generate a positive integer, and then have them solve the problem above using their random number to replace n. It would be impactful for students to see that everyone gets the same answer; and those who multiplied things out might be compelled to look for a pattern and figure out why everyone got the same answer.
But then I realized: this problem is gender biased. Well, maybe. The problem asks for the units digit of the product of the first 21 prime numbers. The choice of 21 was very deliberate, I’m sure. It’s small enough that an industrious student might actually try to calculate the product. In my experience, female students are more industrious than males and therefore more likely to do the computation. But the number is large enough that male students, who are lazy like I am, will think, “That’s too much work. There’s got to be a trick!”
I mentioned to AJ that if a larger number were chosen — for instance, if it involved the product of the first 1,000 prime numbers — then it might be more obvious that students ought to look for a pattern. “You haven’t met my daughter,” he said. “She’d still try to compute it.”
You may think my assertion is crazy. There is nothing in the problem that appears inherently biased against females.
A few years ago, the AAUW published a report about gender bias in math questions. One of the selected questions was something like, “What is the value of n if n + 2 = 7?” Despite the neutrality of the content, girls scored significantly lower than boys on this question, so it was deemed to be biased. (Sorry, I wasn’t able to find a reference to the report. If anyone knows the report to which I’m referring, please share in the comments.)
Further, FairTest claims that the gender gap all but disappears on all types of questions except multiple choice when other question types were examined on Advanced Placement tests. What is it about multiple choice questions that makes them implicitly unfair to females? I have no idea.