Naked math

“Naked math” — computation questions that aren’t phrased as story problems — will make up 20 percent of the new state test in Illinois. Critics say students won’t be asked to demonstrate reasoning abilities. Here are some third grade examples, courtesy of the Chicago Tribune:

What is the missing number in the following pattern?

94, __, 106, 112

A. 96
B. 98
C. 100
D. 102


On Monday, Joe asked his mother if he could go with his friends to the movies on Friday. His mother gave him a list of chores and said he would have to earn the $5 to buy the movie ticket.

Babysitting his sister: 1 quarter an hour
Dusting: 1 nickel
Making his bed: 2 quarters
Washing dishes: 1 quarter
Cleaning his room: 1 quarter
Taking out the trash: 1 quarter
Sweeping the floor: 1 dime
Folding clothes: 1 dime

Make a plan for Joe to earn $5. List the chores he will need to do each day. Explain in words how you found your answer and why you did the plan the way you did.

Scoring open-ended questions costs a lot more money than naked questions, and provides no additional insight into students’ ability, according to state officials. It also penalizes students who can do math but aren’t fluent in English.

Via Education Gadfly.

About Joanne


  1. Curt Wilson says:

    The dichotomy between “naked” and “open-response” math problems is a false one, and the article seems to confuse the issue. At the beginning it talks about the distinction between pure math problems (what is 7% of 350?) and word problems (if there is a 7% sales tax, what is the tax on a $350 item). This type of word problem does have a definite answer, and can even be presented as a multiple-choice question for machine grading.

    IMO, a standardized state test should have both of these types of problems, if only to distinguish what things a student can and cannot do. (I’m tutoring a high-school kid in algebra now who is pretty good at the pure math, but struggling with putting a word problem into equations.)

    But the “open-response” questions don’t belong on such a test. Leaving alone the cost of grading, how would you grade this? Looking at the “chore” example and the instructions to “explain why you did the plan the way you did”, I could easily see a kid responding validly to the question with “I chose mostly babysitting because I like to boss my sister around.” Great insight into the kid’s math skills…

    I get the feeling that these people want the test to be meaningless, so they can then come back and say that it is meaningless.

  2. Joanne and Curt,

    I disagree. I have taught physics and astrophysics classes at Berkeley, as well as high school kids in Upward Bound.
    Even with physics word problems, students are very good
    at memorizing which formulae to use and getting the answer. On tests, we typically put a few questions for which we do not allow calculations, but students must explain a concept in words… you would be amazed at how may students can do the calculation problems, even advanced problems, but not be able to explain in words what they did, why they did it, what was the rational for the method, what assumptions were implicityly made, etc… Students are very good at memorizing patterns to solve problems, but that doesn’t mean they necessarily understand…

    Now, maybe it is far too expensive to grade such exams… fine, I understand that… that is a decent enough reason to only have multiple choice… but open response type questions are incredibly revealing into what a student really understands…

    P.S. This was true for me in grad school as well… I remember advanced quantum theory where I could do the wrote calculations and get all the right answers, but if you forced me to explain what the heck I was doing, or why I was doing that way, I would have been clueless! (Well, not now that I’ve graduated).

  3. Looking at the price list of the open response question, I conclude it must be _really_ hard to get that kid to make his bed.

    But I do think the question is too personal (that is, open response answers have the potential to be tied too tightly to the student’s preferences). It isn’t quite the same thing as explaining math or physics concepts in words.

    Presumably the goal is to get the student to come up with some set of chores, with various repetitions, that will sum to at least $5. An alternative (that tests the same set of math skills, I think) is to offer multple choices of chore sets and let the student analyze them to see which one(s) will add up to a movie ticket.

    Alternatively, if you want to force the student to come up with some scheme on his/her own, you might do better to make the question less personal.

  4. I’m just floored that the kid gets 50 cents for making his bed, but only a quarter for cleaning his room.

    I’m guessing the idea is to also include some concept of “reality” in the question, for example, the boy can only make his bed five times (to earn a total of $2.50) between Monday and Friday. (okay, another question: does “making the bed” mean just straightening the sheets and pulling up the blankets, or does it mean stripping the bed down and replacing the sheets with clean ones? I think in all but the most hygeine-obsessed families, that was done once a week at most. So the question is unclear right there – what does “make the bed” mean?)

    The thing that really bugs me about the question – and this may be just me – is that I was EXPECTED to make my bed (in the sense of pulling up the sheets and blankets each morning), clean my room, and on occasion, babysit my little brother, all for free. To whom do I apply for retroactive pay? Considering that I lived with my parents until I was 18 and 6 months, and I probably made my bed 85% of the time from the age of six on…I’ve got quite a chunk ‘o’ change due me from somewhere…

  5. I agree with jab – over half of the (university calculus) students I’ve taught who have expressed difficulty told me that “[they] could never do word problems”. University calculus is often the first class in which they are confronted with word problems that are not essentially identical to the ones their teacher did on the blackboard. University shouldn’t be the first time students are required to do math that taxes their mathematical reasoning skills.

    However, the open response question about Joe is unnecessarily difficult to grade. It’s possible (and not very difficult) to present word problems that have a single correct answer, and hence can be put in multiple choice tests.

  6. I’m sorry but all I can thing of when I see the phrase “naked math” is boys counting to 21. I always viewed story problems as needless overhead; the algorithm is the same whatever the quantities represent.

  7. It would take me about 3 minutes to write a computer program that would grade the chores problem correctly – but I can think of several teachers, including my 8th grade honors math teacher, that would have misgraded it sometimes.

  8. There are three kinds of mathematical thinking that are all important. (1) One is reasoning with a given model. Naked problems (computation, taking integrals, transforming to a different axis, etc.) are all like this–and you have to know how to do this to solve real-world problems. (2) Another kind of mathematical thinking is figuring out how real-world situations correspond to mathematical models. There is a neat, simple, correct correspondence in most of the story problems kinds see. (3) In lots of real world applications, no single model can capture everything that is going on; there are different possible approaches, each with pluses and minuses. In a meeting about what to do in such a situation (e.g., equating tests with different reliabilites), everyone in the room can do the math, and everyone can establish the correspondence between the situation and any of the alternative models. The real problem then is figuring out, then deciding among, the pluses and minuses of the different options.

    We’d like students to be able to do all of these. You can’t do the third if you can’t do the first and second, at least at the level the problem requires. I think, though, that even little little kids learning addition can do some of the second and even some of the third. Bottom line, if we want kids to learn of this, we need to test it all.

  9. PJ/Maryland says:

    C’mon Ricki, everyone knows that “making the bed” means tucking in the sheets and blankets, and putting the spread on. Putting on new sheets is called “changing the bed”.

    I think my answer would be, “I’ll babysit my sister for 20 hours.” More time to do the other questions on the test. (Tho that 50 cent payoff for making my bed is awfully tempting; no way can it be as boring as watching my sister for 2 hours. And, at a nickel, very little dusting will be done.)

    Oh wait, maybe double-billing is allowed? I could wash the dishes and babysit my sister at the same time!

  10. What about overtime? Does he get time-and-a-half; and double-time for Sundays and Holidays?
    Is it over eight; over forty, or just over forty? What about withholding???