# How much credit for a single arithmetic mistake?

A question related to this one, on an equivalent problem on a community-college College Algebra exam. In the prior question most readers observed the error as a critical conceptual issue. Now consider this example of student work:

In this case, the distribution has been done correctly, and two of the three combinations of like terms. But in the very last term, a signed-arithmetic error has been made; the student replaced $-3-14$ with an incorrect $-11$.

On a percentage basis, what is the most appropriate amount of credit to award in a case like this? Assume that the problem is worth an appropriately granular number of points (not necessarily 10).

• Again, just a comment: I cannot tell, contextually, whether this is a simple arithmetic error as opposed to another critical conceptual issue. I have seen students mistakenly transfer the notion of "two negatives makes a positive" from multiplication to addition. If cornered, I would give it a 75% by [roughly] following the ideas in the earlier answer here to your previous question. Jun 30, 2017 at 5:45
• As per my comment: matheducators.stackexchange.com/questions/12516/… I would look at the whole exam and see what type of mistake this is. Looking at problems individually and independently of other problems on the exam can lead to a death by a thousand red marks for what could be one single conceptual mistake. If this is the only mistake they've made of this type, I'd nick a minimal amount. If it is chronic, I would deduct a hefty amount from one problem only and circle the error on other problems. Jun 30, 2017 at 21:37

The correct realization of the operations prior to the error indicates that the student does understand how to take sums and differences of negative numbers, how to negate a negative numbers, and how to realize more complicated operations, such as the distribution of the negative sign over a sum, and this suggests that the mistake in the final step is due to carelessness. If one accepts this reasoning, only a minor deduction should be made. Exactly what it should be depends on the context. Were this a final step in the solution of a problem in a university differential equations class, I might not even deduct points (were everything else perfect), but in a class dedicated to teaching precisely this sort of operation, some deduction is in order, on the order of $1/8$ of total value (depending on how one counts there are $6-10$ operations that have to performed, and one of them has been performed incorrectly, and the evidence supports the conclusion that the student understands what to do, simply has made a careless mistake).

• RE: "suggests that the mistake in the final step is due to carelessness" If other items in the same student's paper suggested a conceptual error instead -- e.g., $-5 - 3$ calculated elsewhere as $-5 + 3$ -- would this have an effect on your proposed deduction of points? Jun 30, 2017 at 5:51
• @BenjaminDickman It would. To the extent possible, one wants to contextualize the awarding of points, taking into account as much information as possible. Fine grained scoring is a procedure less robust than one would like (probably in grading stability is more important than precision). If it is clear that a student has repeatedly made errors of a similar type, one can reasonably infer that it is not due to carelessness. The level matters too. In a basic course the error indicated is more serious than it is in a course where manipulations of this sort can reasonably be taken for granted. Jun 30, 2017 at 9:51
• Dan's answer is spot on, and I'd go even further and if possible, give the student the chance to catch and repair the mistake with no penalty: Rather than mark the mistake, just give it back to him/her and say "There's a mistake in the answer and if you can spot it and explain to me, right here and now, what you did, then I'll give you full credit." Then do so; if they can't or don't spot it, take off a tiny amount and then discuss it with them. Jun 30, 2017 at 21:30
• @RobertTalbert: I don't understand when the "give it back" process is meant to occur, or how it would be feasible. Whether during the test or after grading, there will be many, many such errors (class size ~30) and I don't see any way how to process them all simultaneously. Jul 4, 2017 at 1:03