vela said:
I'd give the students zeros who just wrote down the 50 and 50. The point of a test isn't simply for students to write down the right answers. It's for students to demonstrate that they know what they're doing. If they can't or don't explain their reasoning, there's no reason to conclude that they did anything more than make a lucky guess or memorized the answer from the homework.
vela said:
I agree that you have to tell students what your expectations are. In fact, I explain this point about showing work on the syllabus; it's one of the points on the syllabus I go over in class; it's a point I bring up before they turn in their assignments; it's a point I bring up after giving them 0s for not showing work, and it's part of the instructions on every exam. Also, in my experience as a student, you're constantly told to show your work in physics and math courses. A student claiming they didn't know they had to show work is being either incredibly dense or disingenuous.
Then you would certainly have earned my ire if I were your student. In what you write here, you've repeatedly assumed that everyone has the same understanding of what "show work" means. Sometimes a question is so obvious that one cannot envision breaking it into simpler steps. If you were asked "86+27=?" on a calculus exam, would it ever cross your mind that the examiner intended you to write out something like
even if the instructions did say "show work"? On a calculus exam?
We have focused mostly in this thread on the fact that some questions which are obvious to the teacher are sometimes surprisingly un-obvious to the students. But let's not forget that the opposite also occurs: That a question the teacher thought was of the appropriate difficulty (especially taking into account the previous fact) might turn out to be so bleedingly obvious to some students, that it seems silly to "show work".
And as I mentioned, people also have a different understanding of what "show work" means. We can all agree it means "Break the solution of the problem into atomic steps, and write out those steps". But what are the appropriate atomic steps? What counted as atomic in arithmetic shouldn't be necessary to write out in calculus, or even in algebra. But even in the same level, different students will have a different appreciation of what is a "unit of problem solving", and if your grading system relies giving points for seeing specific steps written out, then you are unfairly penalizing students who can think for themselves (and you're also expecting people to read your mind!).
My philosophy, on the 50+50 question, would be this: Is the answer right? If so, and there is no work, do I have reason to believe the student was cheating? If not, then full points. Maybe the student thought the question was more obvious than I did. Maybe the student had an algebra class in high school where the teacher constantly repeated something like "The product of numbers with a fixed sum is highest when those numbers are least different", and has internalized that fact as atomic. (That is not nearly as silly as you might think...a lot of math teaching focuses too much on learning things by rote.)
If you think my above example is too unlikely to appear on a calculus exam, then consider this sort of example, on an algebra exam, that better illustrates what I'm talking about:
Solve for x:
$$36 x + 41 = 113$$
Would you penalize a student who just wrote ##x=2##? If not, then what would you expect? Would it be good enough for them to write "It works when I plug it in"? If that doesn't count, would it be ok to write
$$36 \times 2 + 41 = 72 + 41 = 113$$
? Or would you insist on using a method that doesn't rely on guessing the answer*? Would it be enough to write
$$x = \frac{72}{36} = 2$$
? Or would you need to see
$$x = \frac{113 - 41}{36} = \frac{72}{36} = 2$$
? Or would you insist on
$$\begin{align*} 36 x &= 113 - 41 \\ 36 x &= 72 \\ x &= \frac{72}{36} \\ x &= 2 \end{align*}$$
? Should the student also have to write out long division to find 72/36? Should the student also have to write out long
subtraction for 113-41? What counts as "show work" and what doesn't? Consider also that the exam is
timed, and most students will be looking for ways to be as efficient as possible.
If your answer is "I expect the students to regurgitate the exact sequence of steps that I used on the board when I solved similar problems in class", then you are teaching recipes, not mathematics.
* Oh man, guessing the answer and showing that it works is such a ridiculously valuable skill in mathematics, I really hope you don't kill off students' developing intuitions by penalizing such answers.
