Sandra reflects:

“It’s really not an ERROR in the ANSWER, just an ERROR in thinking that they have to do any work at all to get the answer. Perhaps an over-achiever?”

What say you all, Calculus people?

Thanks to Sandra for the submission!

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## 5 replies on “Derivative of a Constant”

I personally prefer that a student recognize the derivative of a constant function as a special case of the power rule, rather than solely a separate rule to be memorized. This work makes it believable that the student recognizes that a constant is being viewed as a function, while many students won’t. I agree that there are times when being too verbose can get in the way of understanding the main idea of an argument. But here, when the main argument is, “you can just know that the derivative of a constant is 0”, nothing is lost by giving some detail, either that given here or a geometric description.

I find @barrysmith’s rationale persuasive. Good stuff. Also, I like such questions because they test understanding (compared to plug-‘n-chug questions); but one possible worry though: they may be seen by students as ‘trick’ questions, and a bit ‘smart alecky’?? Additionally, it seems to me that such questions also test understanding of notation: to the uninitiated, the symbols for pi and e look far more like variables (x say) than they do numbers (1,2, …). The formalisms are barriers to be hurdled, so I wonder how teachers are coping with this little issue.

In any case, I can’t see how it’s an error in any way at all. Clearly the answer is correct, so no error there. And also definitely not an error of the second sort i.e. an error in having to do any work, because the need to do work on the question is implicit in the presentation i.e. it is a test question of some sort and the name of the game is to show an answer, with some sort of justification: they cannot simply just write ‘0’ (points out of 3 suggests this). (What justifications are considered adequate here I wonder).

In fact, to obtain any type of answer, work has to be done on the question no matter what! (No work would only mean no attempt). So also not necessarily an over-achiever either; more likely to be someone who is conscientious and careful. Full marks surely?

I have to disagree with cam scott here. By the time a student reaches the Calculus level, I want them to be clever and efficient as well as knowledgable about technique and properties. I would be unhappy with a student who did so much work here. As far as rationale or support work, I would want a simple statement that pi and e are constants and that the derivative of a constant is 0

“By the time a student reaches the Calculus level, I want them to be clever and efficient as well as knowledgable about technique and properties.”

What you want doesn’t necessarily jibe with reality. A significant fraction of high school students take Calculus, and probably most of them are not “clever and efficient”, but instead are reasonably smart and careful. Most of them are probably risk-averse, and would rather do extra work to guarantee full credit than take shortcuts that they may get marked down for, or which may be incorrect. (I think most of the math teachers here would probably prefer their students to be careful and err on the side of extra work, rather than follow their intuition, as that leads to a lot of “careless” mistakes.)

It may also have been the case that the student realized it was a constant halfway through the second line, but it was easier to just continue the train of thought than erase everything and write that the function is constant.

I think “efficient” is a complex idea that’s highly individualized. The student’s work here is 38 characters, but the statement “pi and e are constants and the derivative of a constant is 0.” is nearly 50 characters. Switching gears from writing math to writing words has a cognitive load, and the work shown benefits in efficiency from having a repetitive structure that is likely shared in other problems in the assignment.

I think this discussion hinges entirely on the expectations established by this teacher. If the teacher subtracts points for not showing work, it’s easy to imagine the student felt that answering the question without showing work would have been an unnecessary risk. (And I think that if one were to take the “efficient” discussion to any serious depth, risk would become a major factor. i.e.: It’s faster if I don’t check my answers, but then I’m risking errors…)