General Calculus Advice (and even some Linear Algebra)
Have the student you’re tutoring show you what their class has been working on and some of the problems and concepts that have been giving them the most trouble. Make sure to have them show you the textbook and their notes, so that the language and notation you use with them is consistent with what their professor is using. (And also to help remind yourself about any details you may have forgotten!) Working through problems from a problem set is good, but whenever possible, start by working through a different problem that wasn’t assigned. Then you can have them try the actual homework problem themselves, with you there to guide them and make suggestions and catch errors.
Many students struggling in a math course are embarrassed not to be understanding, and sometimes they’ll fake understanding (nodding along, responding “yes” automatically if you ask whether they understand, etc.) to avoid feeling stupid. So gently press them, saying things like “OK, good, but just to be sure, let’s have you practice that on this next problem,” or “Really? Cuz I thought that was pretty hard” when they say they understand; sometimes that prompts them to ask for clarification. (Or sometimes they’ll explain the concept back to you, confirming that yes, they really do get it!)
Also remember that many struggling students (sometimes the same ones who fake understanding you) have gaps from earlier math courses, often including from high school algebra. So be sensitive to that; a lot of calculus tutoring ends up being algebra tutoring.
Math 111 and 105/106
Students in Introductory Calculus often have difficulty with word problems. One common mistake is that students try to memorize problems, and they often write down equations and formulas before they actually understand what the problem is asking for. Ask the student where each equation comes from and why they wrote it down. If a student is seriously confused about the word problems, start by going over a problem (say one from the student's notes) very carefully. Then do a problem or two from the assignment, giving small hints where appropriate, with the student doing the bulk of the work. Finally, have the student do a problem completely independently of you. One thing to beware of is that in doing minimum‑maximum problems, it is important to prove that you have found the desired minimum or maximum. You may have been taught a different method for doing this, so that you may need to talk to the course instructor about what method to use (note also that the methods used in class may differ slightly from those in the book!). Remember: the student’s goal when doing these problems is not just to get an answer, but rather to understand each problem in the context of the ideas and methods taught in class.
Students struggling in these courses often have *both* math-phobia *and* shaky algebra skills. So be gentle and friendly, and be ready to go back and review some high school algebra frequently. Some such students also often want to skip steps and omit words, equals signs, limit signs, etc. Don’t let them get away with it; be relaxed and friendly about it, but emphasize to them that they’ve got to take the time to write out all the steps and all the symbols. And repeatedly (but gently) remind them to show all their work, define their variables explicitly, and USE WORDS, especially on word problems and max/min problems.
Math 121
There are a lot of little topics here that students struggle with, but the biggest ones are integration methods and series. For both kinds of problems, many struggling students simply have no idea where to start, and on top of that, they are shaky on the methods themselves. So you need to get their fluency up by having them do a LOT of problems. (For example, demonstrate yourself with an unassigned problem, then have them “help” you do one or other unassigned problems, and then have them do an assigned problem under your watchful eye.) Also, get them in the habit of making multiple attempts on a given problem, WRITING STUFF DOWN on each attempt; demonstrate this yourself! You want to get them to feel comfortable with trying several different strategies before solving a problem, AND you want to help them out of the trap of staring at a blank page, trying to devise a strategy in their head.
Math 211
The 3D diagrams in this course are hard for most students. Encourage them to take care in drawing their graphs, but with emphasis on the important features (e.g., get the equation right for the surface on one side of the solid, but don’t obsess about exactly how wide or narrow the parabola is). Another tough spot are the multivariable algebra computations that show up in max/min and Lagrange Multiplier problems. As always, do lots of examples with your tutee, including some assigned and some unassigned problems.
From Your Professor
Check in with your tutees to make sure they’re actually going to every class meeting, on time. They should also be taking extra time to go over their class notes every day.
Finally, remember that seeking help from a tutor is often uncomfortable for your tutee — it feels like an admission of defeat or even stupidity to some students, even though it is ABSOLUTELY NOT that — and so they might not be as motivated as they should be to keep up a regular meeting with you. So set a regular time to meet, and check in on them every time beforehand to confirm/remind. Otherwise there’s a risk that they’ll start letting their meetings with you lapse.
- Professor Rob Benedetto