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I'm currently teaching a second-semester calculus course in which a significant percentage of my students (over half) either tested into the course just out of high school, took a much easier first semester at a nearby community college, or simply got through the preceding course by the skin of their teeth. For many of them, it has been over a year since their previous math course.

It seems as though there is almost no recollection of limits, differential calculus, or the basics of integration that were included in the first semester material. They seem to be able to keep up with the new topics as long as they are independent of the previous coursework, but whenever I ask them to recall old material, things start to get dramatic. For example, a recent lecture on numerical integration (e.g. Simpson's rule) contained a discussion of error bounds, which involved maximizing a function on an interval. This quickly dissolved into a review of finding extrema while I responded to student questions, and by the time I was done, we had no more time to talk about integration formulas.

I feel like the natural thing to do would be to simply start over, i.e. give a comprehensive review of the entire first semester as quickly as possible before moving on. However, this is a summer course, so it is accelerated, and there is no time for this. Things are already behind schedule.

How can I facilitate a reasonable understanding of the curriculum under these conditions?

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Does the textbook for your course include the material that your institution teaches in Calc I? –  jonsca May 29 at 21:39
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There are many, many free calculus textbooks out there. Some are listed here: theassayer.org/cgi-bin/… Just tell your students they have to read chapter X of free book Y to review topic Z for the next class, and that there will be a 5-minute multiple-choice quiz at the beginning of that class covering Z. –  Ben Crowell May 30 at 1:11
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Classic challenge: Given limited time and manpower, do you teach to the bottom of the class and help those who need most help, at the cost of the better students not getting the full class they signed up for (and are paying for), or do you teach to the middle of the class and let those who realize that they're underprepared either do their own make-up work or drop the course, or do you teach to the top, let the middle struggle, and TELL the bottom to drop? Depends on the school's expectations, I think. Personal instinct is teach to the middle, and perhaps have the TAs run remedial sessions. –  keshlam May 30 at 1:13
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This is tough: if I were one of the students who truly qualified for your course, I'd be pissed as hell if it suddenly turned into a remedial catch-up program. I think you owe it to the upper half of the class to stay on track and encourage the others to either re-teach themselves what they were supposed to have known or to drop out and get into an appropriate-level course. –  Carl Witthoft May 30 at 14:28
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+1, favorited, waiting with baited breath for good answers to one of my life's great teaching challenges. –  Pete L. Clark Jun 1 at 2:32
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3 Answers 3

I don't think there is any way to get students caught up without expending some effort. If you are willing to expend the effort, you can create a simple "you should remember" pre-class activity for days that require it, and preserve your class time for new concepts.

Steps:

  1. Look through your upcoming lectures (even just a few in advance), and determine which ones require a Calc 1 skill.
  2. Track down the Khan Academy video that correspond to the Calc 1 skill for that lecture, and post it on your class website.
  3. Post 2 simple questions for students to work that they need to show you when they walk into class.

There are some pretty awesome videos out there to re-learn things -- frankly it's a good life skill for students to grasp.

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+1 I was in the middle of crafting an answer with the same advice and with the following addendum: if possible assign Webwork problems for the review material since they can be graded automatically and give students instant feedback (so they don't have to wait for your feedback). Especially since this is for a summer course and time is already pretty compressed for you and for them. –  ncr May 29 at 23:07
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+1, but Khan Academy is an extremely inefficient way to review something that you could review just by opening a book. There are boatloads of free calculus books out there on the web. –  Ben Crowell May 30 at 1:29
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There's also been some criticism of Khan Academy - I won't post any specific links, but a search engine will swiftly locate many examples. That said, one should not throw out the baby with the bathwater. –  J W May 30 at 7:13
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@JW I agree with the part about not throwing out the baby with the bathwater. That'd be an open and shut case of infanticide. –  coburne May 30 at 14:44
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The easiest method would be to issue an addendum to the syllabus that highlighted specific chapters of the Calculus I material that would be helpful as prerequisites to the new material. Since you indicated in the comments that you are using a separate textbook, this doesn't seem as feasible, since it's tough to ask students to shell out another $150 on a book that they may not use heavily.

Firstly, I would not worry about the students with advanced placement at all. Given that they've already demonstrated at least some independence in learning the AP material, they should be able to dig up the resources that they need to supplement the current text.

If it were me, I would start the class with a worked-through Calc I "warm up" problem, but quickly finish it so that the lecture doesn't get diverted into a refresher course. For example, a quick optimization problem before the numerical integration lecture would force the students back into the proper mindset and prompt some recall of the older material, I think.

If you did elect to go back through for a firehose-style review of the material, giving a Calc I problems "quiz" as an assessment first would let you gauge where you need to focus your energy. Given the fact that most students take a summer course under various circumstances (early start, remediation, etc.), there may be some students that are just so far behind they need to reconsider their options anyway.

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I have two suggestions... 1) for the students that are having the most difficult time, you might ask them to get a copy of "Hurricane Calculus". This book is immensely helpful in simplifying calculus concepts quickly and succinctly. It is also less intimidating to students because it reads less like a text book and more like a how-to guide. You can get it for a buck, used on Amazon.
2) In my experience, when you have a group of mixed ability students and a short time line, it is best to get the students working together. I do break-out groups and have them report out... usually in something like a game format or non-graded group quiz. This works even better when you can get the groups heterogeneous so that each group has students who get it and who don't get it. The biggest lesson I've learned about education in general is that the more you get students talking to each other about the content and sharing with the class, the quicker they learn, since students generally communicate better with their peers than with their teachers.

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