# How to award points for sense-making at the end of a problem?

Here is a statistics question I asked on a recent exam:

How high should the doorway be to allow 97% of men to fit through it?

I got a very large number of answers like "3 inches" or "0.0001 feet". This makes me sad. I'd like to know if these students notice that their answers are nonsense, and reward them if they have this level of sense-making.

How can I set up my grading scheme to reward people who notice their answer is nonsensical? Is there any way to do this besides the side benefit that nonsense answers are a clue that the work is probably wrong?

• There is an article in which Henry Pollak similarly wonders about an algebraic error that leads to an answer in the thousands for a question about integrating inside the unit cube. (No spectacular remedy is proposed!) I think R. Daulton's technique is quite reasonable; all one needs is to incentivize the checking stage. That said, what counts as "sensical" is sure to vary by person and problem. A favorite of mine that springs to mind: A truckload of watermelons weighs 1000 kg, which are 99% water. If, after evaporation, they are 98% water, then what is the new weight? – Benjamin Dickman Nov 10 '14 at 2:21
• Everybody knows that 100 - 97% = 3 inches. The height of the door is trivial! (sorry, couldn't resist). Make them show their work, then it won't be possible to misinterpret how they got the answer, how they nearly got the answer, or how they missed the point entirely. I wouldn't sweat the "nonsensical" side of things, as common sense is often not common, nor "sensical". It's entirely cultural, and asking students to align to it is asking about experiences rather than problem solving. "Common sense" says it's a 80" door, cause that's what hardware stores sells. – Edwin Buck Nov 11 '14 at 6:30

Introducing the slogan/motto of "sanity check" may be useful, especially if/when mathematics is being taught as something that is entirely compatible with the students' prior experience, general sense, and so on. Situations (all-too-often upper-division undergrad...) in which one's physical intuition is utterly disallowed and students are "pranked" on syntactic or semantic technicalities have many severe faults, not-the-least of which is conditioning to disregard "sanity checks" ... because, apparently, such things are irrelevant or disjoint from the enterprise.

So, in fact, to exhort people that their physical intuition is basically reasonable, and to look at mathematics as an extension of that, making "sanity check" sensible, is a good thing.

(A similar issue is prioritization of concern for details. I.e., not all details are of equal importance, despite the "logical" principle that if any single thing is "wrong", then the whole is wrong, etc. In real life, it's not that boolean...)

• I think this is a really good point, particularly in Calculus II we are fond of using sequences and series to show things that don't pass sanity checks, even though most situations would pass sanity checks. – Chris Cunningham Nov 10 '14 at 16:28

In my grading scheme, any answer that is wrong, checkable, and not checked gets $1/4$ off the partial credit. (I give partial credit for work shown, depending on how much knowledge is shown, the pettiness of the mistake, and so on.)

If the student gets a wrong answer, checks it and sees that it is wrong, and notes this in his answer or work, I do not remove the $1/4$.

Perhaps I should add that I accept a "sanity check" as a check. It is only partial, as it does not catch a "close but no cigar" answer, but it is still a valid check.

• I have been doing something similar for a number of years. Volume/area/length calculations by integrals are a particularly rich source of problems, where a suitable sanity/reality check often is available. I lead up to it by systematically doing it with lecture examples and HW problems. And I announce that by catching that something went wrong in this way they will salvage 1/4 partial credit from an otherwise totally botched attempt (within reason). – Jyrki Lahtonen Nov 10 '14 at 20:07

Students tend to only pay attention to feedback that affects their grade. In your example, where the student calculates the height of the door to be 0.0001 inches, showing no sign of realizing that the answer is impossible, I would give a zero on the problem. If the student writes, "Hmm...obviously this is wrong, but I've run out of time to track down the mistake," I would give appropriate partial credit.

Might you be able to offer an initial "guesstimate" as a sanity check for their final answer? In this example, they are forced to consider the question absent any math. Their initial guess should be about 6ft, in my opinion. Once they solve the problem if they come up with even 5ft, they should realize something is wrong.

The 3 inch answer is another issue, they are not applying any sanity check at all to their work.

Similarly, I've seen a result of 900 for the number of seconds for a rock to fall to the ground off a building. When the student is forced to ponder whether this makes any sense, they agree, 15 minutes is too long. A calculator error was made, 9 seconds was correct. In the end, problems that are real-life need a forced sanity check somewhere along the way.

In accounting and finance work (my field) "order of magnitude" (thousands, millions, billions), is nearly as important as the actual number itself.

Basically, I would give partial credit for answers in the correct order of magnitude, and zero for answers outside it. In this example, people should know that the correct order of magnitude is 70-80 inches, or even "a large multiple of 10 inches, but less than 100," even if they can't do the calculations for means and standard deviations, etc. Or a somewhat acceptable answer might be "mean, plus about two standard deviations," either in those terms, or using numerical values if given, or calculable.