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As I grade my final exams, I sometimes find calculus 1 students who will pass with a C but lack critical skills like the chain rule. This can be for a number of reasons, for instance:

  • They understand the concepts but cannot do the computations.
  • They are able to memorize processes well, but they missed this process.
  • They understand geometric intuition but their algebra skills are lacking.

When I pass these students with a C, I am doing them a disservice -- without the ability to do the chain rule, or the product rule, or whatever other basic skill they need, this student will definitely be doomed in the next calculus course. However, I don't want to increase the weight on these "easy" basic concepts, since they are critical but they are also just a small part of the material we cover.

The question:

How can we structure the grading system in our courses so that students who pass the course definitely have the basic skills required for the next course, without the bad plan of massively increasing the grading weight on these concepts?

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    $\begingroup$ This question can be generalized back a few levels. I see students who haven't mastered the prior level, but passed. It's tough to teach any level student who didn't quite understand the lower level, but passed and moved on. You've hit upon a key issue with math education in general. $\endgroup$ – JTP - Apologise to Monica Aug 3 '14 at 20:00
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    $\begingroup$ Isn't the straightforward answer the following? Designate certain tasks (such as a subset of the exam problems) as "basic" tasks. For each grade, set a minimum score for the basic tasks and a minimum score for the non-basic tasks. $\endgroup$ – Ben Crowell Aug 3 '14 at 21:55
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    $\begingroup$ @BenCrowell If you are attempting to answer the question, why not post an answer to the question instead of a comment? I think it is fair to say that your answer is not straightforward enough to be widely implemented. $\endgroup$ – Chris Cunningham♦ Aug 3 '14 at 23:48
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    $\begingroup$ There is an assumption here that being ready for the next class is not their responsibility. Considering that it is possible for students to learn outside of your classroom, it would seem like your responsibility is giving them a grade that includes a useful assessment of what they need to be stronger in to continue on. I disagree that it is a disservice to give them a C in your class. Students should have the option to seek help outside of your class before the next one. If you take that option away, that is the disservice. (Guesses about how likely they are to use the option are a cop out.) $\endgroup$ – JPBurke Aug 4 '14 at 0:01
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    $\begingroup$ I didn't add this as an answer because your question has the assumption that the grading system is the solution to ensuring that all skills are in place for the next course. It seems like the problem could be with prerequisite knowledge, not grading systems. I allow I may be wrong, but the possibility certainly exists. $\endgroup$ – JPBurke Aug 4 '14 at 0:04
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The notion of "gateway" (exam or other thresh-hold) is not universal, and generates some conflicts, but addresses the issue head-on. That is, throughout a course (perhaps at the beginning, but this is in some ways the most harsh and least hopeful/optimistic) separate "exams" are given that test exactly and only the absolutely indispensible skills. Not only is everyone expected to get nearly everything right, but, if they don't, they are required to keep doing it until they do succeed. In the harshest version, the very registration for the course is not allowed without passing gateway exams.

One can make successful completion of all the gateway exams some sort of prerequisite for taking the final, or for getting a passing grade, or... hence the name "gateway".

This makes considerable sense to me, but I would anticipate considerable backlash from students who're too accustomed to Xtreme Mediocrity (really, semi-incompetence) being ajudged good-enough to be allowed to pass on...

If you don't have quite a bit of authority over the grading system, or are somehow vulnerable to skepticism about your teaching ideas, this could cause you a lot of trouble...

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  • $\begingroup$ We usually had special rules regarding homework in our courses. Like "fail to deliver your homework 3 times and you get an F" So you could also employ these as assignments and if they fail 1 completely they fail the course... $\endgroup$ – Falco Aug 4 '14 at 7:55
  • $\begingroup$ I've decided to accept this; explicitly implementing a few gateway exams during College Algebra is a good idea, assuming they can repeat these exams as many times as they like. I don't want to pass anyone who can't factor -- but I don't want to fail people for having a bad exam. Thanks! $\endgroup$ – Chris Cunningham♦ Oct 18 '14 at 15:28
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In my view, the method to make sure they leave calculus I with all the required skills is to simply weave them into all the questions. I warn them, if you don't know the basic computations cold then it will hurt your grade. By the time of the final, it ought not be a struggle to do chain rules and such. If it still is a burden to them mentally then it's still time for them to study calculus I. This is my draconian view. Actually, since they don't tend to study before the next course (especially if they need to), it is the kind thing to keep them back before they find themselves trapped in the much more harsh environment of calculus II.

At each stage of the course I try to test them on everything they've learned up to that point. By the time we reach the final, they should see it coming. Of course, you have to be sensitive to the difficulty of the topic, look for the right places to layer concepts.

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I have the same concern as you have, and even more so I guess: in the French system, not only various subjects of a given course can compensate one another, but student can even pass into the next year by compensating a bad grade in their majors by good enough grades at their minors (which are numerous in first years).

I agree that a passing grade should mean that the student is ready for the next stage, and it makes little sense to average over different skills and pieces of knowledge.

One suggestion I read somewhere in this site is the following: make an exam divided in three parts, C B and A. Part C is made of basic questions (10 say) that are expected to turn out almost all alright. Part B is made of slightly more advanced questions (6 say), corresponding to what you expect a reasonably good student to achieve, and part A is made of advanced questions (4 say) that needs more independent thinking. Then the grading goes like this:

  • you grade part C; if less than say 9 (maybe 8) answers out of 10 are correct, then the student fails and you stop grading. Otherwise, the student passes, and then:

  • you grade part B; if less than say 4 (maybe 3) answers out of 6 are correct, then the student has a C and you stop grading. Otherwise:

  • you grade part A; the student has an A if one (maybe 2) answer out of 4 is correct, otherwise he or she has a B.

One of the main difficulty would be to have student truly understand and accept this grading system. I have not used such a system myself (but we grade from 0 to 20, which is less prone to this method).

A lot of adjustments can be made of course; for example above in part A, the important part is to be able to solve one difficult problem among several; this is most suited for graduate courses, as in research you succeed by finding a problem you can solve rather than by being able to solve whatever problem you encounter.

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