# Effective Final Review

I'm currently teaching a small section of calculus (~35 students). As the semester comes to a close, I'm looking for an effective way to use one or two classes to help my students review for the final exam. In the past, I've used the time as a question/answer period, but this doesn't seem to accomplish anything that couldn't be done in office hours.

Do you have an effective way to use class time to review for a final exam? Answers specific to calculus are most welcome, but I imagine there are good generic answers as well.

In my calculus class, I give a practice test that covers (almost) the same material as the test. I give some class time to do the practice test but they largely do it in their own time. In the class before the test I discuss the practice test. The students know this is the last time for them to ask any questions.

This system isn't perfect, but what is? It has worked pretty well for me.

• great suggestion! another thing I will do sometimes is give the practice test the class prior to the review so they have time to do it on their own and then can come to the review session with any questions – celeriko Dec 1 '14 at 18:26
• I do something very similar, and it seemed to produce better questions for discussion during the review. – JPBurke Dec 1 '14 at 19:33

I spend one entire class period with the students working in groups on a massive stack of problems. More problems than is really reasonable to complete in the time allotted. In this "review mode," certain kinds of problems are more effective than usual.

One type: where the problem is easy once you identify the correct strategy, but the correct strategy is elusive. Often students struggle on exams because they can't distinguish two fundamentally distinct questions, and this is a good time to delve into the issue. For example

1a. Find $\int_{-5}^5 25 - x^2 \, \mathrm{dx}$

1b. Find $\int_{-5}^5 \sqrt{25 - x^2} \, \mathrm{dx}$

1c. Find $\int_{-5}^5 x\sqrt{25 - x^2} \, \mathrm{dx}$

1d. Find $\int_{-5}^5 (\sin x)(25 - x^2) \, \mathrm{dx}$

Hint: (1d) is only reasonable to do in one way, and it doesn't require any computation. (1c) is actually doable by the same method, although there are other options.

Second type: can also use up your elegant "trick" questions on these review packets since they are legitimately a learning exercise, so none of the negative feelings of the trick question happen but you get all of the educational benefit. For example, in an algebra class, I can give this on a review day (but maybe not on an exam, although the ideas under the trick question will appear on the exam):

2a. Solve for $x$: $x^2 = 81$.

2b. Solve for $x$: $\sqrt{x} = -3$.

Hint: neither answer is simply "$x = 9$".

I will often give two review sessions.

In the first session, I will reorganise the content of a particular section of the course into an order more useful for review. I tell the students that often we learn things in a certain order because it's the way our brain has to learn them, or because that's how the topic is built up axiomatically. However it's often more useful to remember it in a different order. So I will talk through the ideas in the topic not necessarily in chronological order, and highlight the connections between them.

In the second session, I will actually do a question and answer session. I take requests about anything in the course, including questions from past exams or assignments. [If there aren't past exams available I would recommend (like @celeriko) giving them a practice exam to do in their own time.] I encourage students to email me questions they have beforehand, so I can prepare and also so I can give them at least an indication of at least one thing that will be covered.