# Multiple extra credit questions in a homework set

Extra credit questions sometimes motivate students to study more. Occasionally I find several valuable questions which are all good as extra credit question choices but I have to limit the number of points that a student can possible receive from doing extra credit work. Say the full mark of a homework set is $$10$$ and there are $$n$$ questions which are suitable as extra credit questions.

Here my goal is:

1. Encourage students to try as many of them as possible.
2. Do not give a student more than $$1$$ point for all the extra credit work.

Here I am asking if the following is a good strategy:

If the student attempted $$k$$ out of the $$n$$ extra credit questions, the student will receive up to $$1/2$$ points for his best finished question, up to $$1/4$$ points for his second best finished question, etc. In general, I will give the student up to $$1/2^m$$ points for the $$m$$-th best answered question. So a motivated student can try most extra credit questions out of the set of $$n$$ questions and no student can receive more than one point for his extra credit work.

• Experience has shown me that students react very badly to complicated grading schemes. My guess is that this will leave students confused, and will not serve as an incentive to attempt more problems. – Xander Henderson Apr 16 at 2:22
• IMO it's no longer really appropriate at all to count homework as a significant portion of students' grades. Everything is on Chegg. – Ben Crowell Apr 21 at 16:44
• I have had good experience with telling my students they only needed to complete x total problems, and giving them x+n problems, where n problems were challenge problems. The catch is that some "problems" were multiple parts consisting of boring (for an advanced student) repetitive exercises. So the students who really needed the repetition got the repetition, and the students who hated repetition did the interesting problems. – Opal E Apr 21 at 20:52

I agree with Xander’s comment; there’s no need to introduce complicated grading schemes, it just makes everyone’s life difficult.

I would even go further and say that extra credit is generally only ever attempted by students who were already going to get an excellent grade anyway, so I don’t even make extra credit worth a significant amount of points. My rule of thumb is that I don’t offer more than 1 extra credit for 20 normal questions, and I make them worth the same as all other problems.

Sometimes I offer “challenge questions” which aren’t graded at all, but only when I think the statement of the question seems actually interesting to students.

1. As I said in a comment, experience has shown me that students react badly to any grading scheme which they see as "nonstandard." You can fight these battles with students, and you might win, but it comes at a cost: every single nonstandard thing you do in a class is going to be a battle, and if you try to fight more than one or two of those battles in a given term, your students are going to start checking out pretty quickly. So, before you implement such a grading scheme, ask yourself, "Is this the hill I want to die on?"

Your answer may be "yes". Maybe you are teaching a bog-standard curriculum out of a bog-standard text using bog-standard lectures. If so, then a "weird" grading scheme may be worth it. Otherwise, you might want to reconsider.

2. What is it that you hope to gain from such an extra credit scheme? It might benefit the high achieving students (if the problems are enriching, they'll learn more or better), but it won't effect their grades, and they would probably have done the problems, anyway. It probably won't benefit the low-achieving students, because they won't bother doing the problems. As such, I am of the opinion that offering extra credit for extra problems is a waste of time and effort.

3. Students don't like fractions of points. If you are going to implement some kind of exponential decay, I would suggest inflating the value of everything, and assigning only whole number point-values. For example, give 10 points per problem (for a total of 100 points). Then make the first extra credit problem worth 5 points, the next worth 2, and every one after that worth 1 point up to a total of 10. This is roughly the same scheme you proposed (with questions 2 and 3 getting a little less weight, and questions 4 and 5 getting a little more weight—everything after that is less than epsilon in either scheme).

4. An alternative is to "lie" with big numbers. I have long offered extra credit to students who type up their homework assignments. Every assignment is worth 3 points (1 point for attending the discussion workshop where they first encounter the homework problems in small groups, 1 point for turning in a completed assignment, and 1 point for giving (mostly) correct solutions). A student can earn another point by typing their work (this is a WHOPPING 25% bonus).[1]

The trick is that homework only constitutes 5–10% of their final grade, which means that if a student types every single assignment, then they get an extra 1.25–2.5% towards their final grade. On a standard grading scale ($$x \ge 90\% = A$$, $$80\%\le x < 90\% = B$$, and so on), this extra credit might be the difference between a B+ and an A-. It looks like a huge bonus, because it is large per assignment, but the ultimate value of that extra credit is low, because it is in a category which doesn't count very highly.

[1] I said above that I don't offer extra credit for extra problems. However, I do often offer extra credit for performing tasks which are tangential to the main content of the class. Learning to type mathematics is a useful skill, but not everyone needs to learn it. However, for the students who are willing to learn, I feel they should be rewarded.