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I've seen that some math teachers design tests which punish errors with negative points.

Why do they assign negative points? What are some pedagogical reasons why teachers might do this?

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    Some standardized tests use negative points to prevent guessing on multiple choice problems. Other times stuff it "required" but not given any points (pre-lab work for example). Ultimately it's up to the teacher, and they can assign any grade they want - but it's generally better to be consistent (i.e. a wrong negative sign is always -.5 pts, etc.), as that makes it easier for students to understand and creates less back and forth discussions over the grades. – user2813274 Nov 5 '14 at 21:37
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    I'm afraid I have to give +1 for the user name in confluence with the question. – Pete L. Clark Nov 6 '14 at 2:38
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    At my university a student got a negative grade due to wrongly answering too many multiple-choice questions. Then a "root factor" was applied to everybody's grade, i.e. newGrade = sqrt(grade) + grade. This student therefore had the dubious honour of being the first student in university history to receive a complex number as his final grade. – dotancohen Nov 6 '14 at 14:51
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    @DavidRicherby: Germany doesn't have case law - I forgot what its type of law is called: one ruling doesn't set necessarily enforceable law for other cases. Also, being German, as a kid I remember our biology high school teacher (also a monk) once giving an F+F for an exam where "one fail wouldn't have done justice to this level of incompetence" (which I'd call the extreme level of negative points; albeit at high school level). And he was everyone's favorite teacher. – gnometorule Nov 6 '14 at 15:57
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    @SteveJessop IIRC, that exact approach is used for some American high school math contests. – cpast Nov 6 '14 at 17:42
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Grade that goes below zero doesn't make sense.

What makes you say that? In some real-world scenarios, thinking you know the answer and being wrong is worse than realizing you don't know. For example, I'd rather have my doctor or lawyer recognize when something goes beyond their expertise, so I can consult a specialist rather than following mistaken advice. I imagine the same is true for most professionals, such as engineers.

If not answering at all yields a grade of zero, then it's reasonable to award negative points for a truly bad answer. (Of course an insightful but flawed answer may still deserve a positive score, just not as high as the correct answer.)

In practice the most common case I've seen negative scores used is multiple choice exams, for the reason given by GeneMachine, but one can make a philosophical case for applying them much more broadly.

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    +1 The answer makes a lot of sense, but a sidetrack question is - how many docs would admit before their patients that they have no clue what ailment they are suffering from? They'll diagnose something and recommend pills even if they are not too sure! – 299792458 Nov 6 '14 at 7:03
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    This is an opinion-based generalisation, and is inappropriate for this forum. – GeneMachine Nov 6 '14 at 8:28
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    @New_new_newbie It's not a doctor admitting they have no clue. It's a doctor saying, "My training and expertise leads me to believe that you have a problem with your [insert body part here], so I'm booking you an appointment with a bodypartologist." That happens all the time. Likewise, any time you meet a lawyer socially and ask them a question about the law, the first thing they'll say is, "I don't do that part of law. You'd need to speak to an expert on that to find the real answer." – David Richerby Nov 6 '14 at 12:24
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    I remember as a student seeing hard questions posed as "Can you {do something difficult}?" As students we knew this was just the professor's polite way of saying "{do something difficult}", but I was always tempted to simply answer "Yes" or "No" and move on to the next question. Perhaps the question should have been broken up into "Can you {do something difficult}." and "{do something difficult}" and students could get points for an honest "No". – emory Nov 6 '14 at 14:22
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    @New_new_newbie any decent doctor knows and admits when he's not the one to help the patient, and will send you to a colleague who can help. Only a quack will kill a patient in order to sell more pills and consults. – jwenting Nov 7 '14 at 7:11
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It's to discourage guessing, and to avoid mark inflation.

There are many discussions of negative marking available; here is one:

http://teach.southwales.ac.uk/assessment/negativemarking/

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To give an (imperfect) analogy, on Stack Exchange posts can get negative points, thus making the poster lose reputation. The reason is to prevent users from posting low-quality posts in the hopes of getting a few upvotes. Downvotes force the user to only post if he's confident it's a good idea.

Similarly, giving points for correct answers on a test, while ignoring incorrect answers, encourages random guessing. Taking away points for wrong answers forces the students to be sure they really know the answer.

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    +1, though I was tempted to give -1 just for the irony. – reirab Nov 8 '14 at 8:59
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There are many positives and drawbacks to negative grading, even for free response tests. It encourages academic honesty and self assessment which are important for learning and discourages "BS" answers where the student knows the answer is wrong but is trying to confuse the grader into awarding some points. Some view "BS" answers as cheating. Unfortunately, negative points for wrong answers can punish students who are under-confident, and choose not to write an answer when they actually can demonstrate some understanding. It can also reward students for not even showing up to the exam if the test is hard enough.

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I see two parts in this question,

Why negative points? Grade that goes below zero doesn't make sense.

Negative Points can make sense, as a punishment. Grades below zero not so much IMO.

I have a lecturer, who gives us assignments before the exam, we hand them in and get some points for it. You then start your exam with that score, but if you fail a question you get negative points. This is for the reason that the exam only tests for topics we dealt with in the assignments. But you don't drop below zero.

This method prevents people from coping assignments from other students beforehand.

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