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Question

Dear Mitch,

A couple of years ago, my older sister read an article about very bad questions that are on some of the most important tests that kids have to pass to be allowed to go to the next grade, and she said that you were one of the people interviewed in the article for the part about the bad math questions. She doesn't remember the exact questions you talked about, but she remembers you said some tests have math questions that are so stupid you thought they could only be answered 'correctly' by a kid who forgot to turn on the light in his head! But my sister says that even though you explained what made the questions stupid so everyone would understand, she still sees the exact questions you talked about, because they're still on really big tests! Now she thinks they are so dumb that they make her laugh! Well, I sure would like to know what she thinks is so funny. Would you happen to remember the questions you talked about?

Answer

Oh yes, I remember the questions I talked about. I remember every one of them, and not because I have such a great memory (I don't) but because I get a fresh reminder several times a year when I look over the new tests as they come out (or, in some cases, before then, to proofread final edits). Anyway, I could describe several dozen questions that I consider too weak to be included in any serious test of math skills. But I think just a couple of the questions will give you an idea of how little respect is given to the fact that people think in different ways -- even sometimes in the world of mathematics, which prides itself on how universal and consistent it is in its "right/wrong" determinations.

Ridiculous math questions can be found in any area -- from arithmetic to trigonometry -- and from the most basic level written for young children to the most sophisticated problems designed to challenge experienced post-graduate students. Keep in mind that determining whether a question is truly stupid or not is subjective, of course. Therefore, I'll tell you the criteria that I use when tagging a question "stupid".

First, though, please note: as an educator, I have always held to the belief that there is no such thing as a 'stupid question'. However, this belief refers to questions asked by any person trying to learn anything. It does not refer to questions 'asked' by a team professional test-writers at big companies who very often do not appear interested in learning new information that could help them improve, as they already have decided on the "correct answer" and are only interested to see if the student will interpret the words the way the authors of the test had meant for them to be interpreted to arrive at the conclusion they liked. Many of these questions can indeed be called stupid, because they actually have some power to make people try to think in only the narrow way the test-writer happened to think, regardless of a more insightful way that may bubble up.

Okay, now that I've blah-blah-blahed, Here are 2 of my favorites, both from the world of probability.

Example One

An assortment of colored marbles (or candies) is depicted, usually in some form of jar, glass or transparent bag, and the question usually goes like this: "If you were to reach into the bag and pick out one marble, which color would it most likely be?"

First of all, the container is CLEAR.

CLEAR!

Like this:

Clear Jar

OY!

To me, the word 'pick' has an active flavor to it, and implies some form of volition (voluntary behavior with a decision to do it).

QUESTION: If each item in the jar was red, except one item which was purple, which would you pick?

ANSWER: If you're like most people, you'd probably pick your favorite color, even if there's only one of that flavor in the jar along with 20 of another color. In fact, especially as one often discovers with time and experience, it is often the rarer flavor that has the greater value. Why not pick the only purple one in the collection? Then, next time, you'll be sure to get one of the reds, because that's all that will be remaining!

I've seen some students bypass that thought however, and intelligently decide they'd pick the marble or candy nearest to the top. After all, in some circles it's considered an unrefined move demonstrating ungracious manners to reach around or across to take the one you want, especially when other people are nearby waiting for their turn... ("You're lucky to get any," some might think, "so don't be so choosy.").

Finally, the great twist of logic that many of the question's variations include arises from a setup like this: 1 red, 2 black, 3 green, 4 white.

And the question usually reads:

If Sally selects one without looking, which color would she be most likely to choose? The "correct answer", of course, is white, because there are more white marbles than any other particular color. But I find it a bit unsettling that a student can just as easily and correctly argue that with one "blind" selection he is less likely to pick a white marble than he is to pick a marble that is not white. In fact, the chance of a white is 4 out of 10, or 2 out of 5, depicted as 2/5.

I believe that this thinking is as insightful as the "correct reasoning", Wouldn't you?

(It should be noted that in recent years many of these questions have been tweaked to narrow down the meaning of both question and answer. Now, for example, one often sees the phrase "without looking" or blind-folded. That helps. Still, often the question does not include such modification.

Example Two

Spinner

(Another probability question-writing blunder, this one involving an item commonly known as a "spinner". We've all seen them in board games, but one of the most commonly seen spinners is the one that comes with the large format game known as "TWISTER". The spinner is the round or square piece of cardboard with a circle printed on it and a movable arrow affixed to the center like a clock with only one hand. With a flick of the finger the arrow is sent spinning until it stops or "lands" on a particular section of the circle, and that section determines the spinner's next move or choice of moves. Well, as most people know, someone many years ago began including a spinner in math lessons for probability, and made it more interesting than coin flipping by giving difference sections of the circle different fractions of the circle to increase or decrease the likelihood of its being the region in which the arrow's journey ends. I think it's a great idea. Spinners are familiar and fun, and they certainly are perfect for demonstrating differences in likelihoods of ranges of outcomes. BUT there's a problem. It's a problem that really is a big problem, and a really stupid problem. It's this:

Perhaps because people have seen dart boards and other games of mostly skill (but at least some chance) hanging vertically on walls and perhaps so many of the most common mathematical tools are most often seen vertically affixed to classroom walls (multiplication charts, graphs, metric to standard conversion tables, number lines, etc.,) but most probably due to the fact that they are often stored on a hook or with Velcro, etc., to a wall, a classroom's math corner, or perhaps they call to mind windmills, electric fans and roulette wheels one might see in a movie. Also recall how closely spinners tend to resemble pie charts, which are very often vertically shown on screens in presentations, many students logically picture spinners with a vertical orientation. So what? Take a look at a spinner hanging on a hook in a classroom and see if you notice anything you never consider...

GRAVITY!

G-R-A-V-I-T-Y

With the exception of compasses, which rely on the force of magnetism to rebel against gravity, all arrows point down. And, unfortunately, the spinners on almost every example I've encountered appear far more vertical than horizontal. Why? I'm not sure, but a circle drawn to appear horizontally as it would be positioned on a tabletop would be distorted by foreshortening. (Put a round plate on a table and step back to look at it: it's an egg.) The opposing argument is that the test-taker is directly above the spinner, like a bird flapping his wings as steadily as he can in his effort to learn math.

Could be.

But to me, regardless of how the circle is divided up, the arrow is most likely to land hanging down. I respect gravity. Fortunately, though, I -- like most of us -- find that I am capable of dimming the lights in my mind just long enough to figure out which answer is "correct" enough to receive credit despite how dumb such an exercise is.

Hope this helps, amuses, or makes you think a bit differently,

Mitch