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I will give a 2-hour long revision lecture on a Mathematics area. The aim of this lecture is to help the students prepare for the exam test. The course was rather long, covering 5 chapters and a number of topics. I have been thinking of giving the students a refresh of key definitions and results, or to present instead some particular exercises.

In general, what are efficient structures for a revision lecture? Are there any guidelines about how should the material be presented in order to make the learning more efficient?

Of course, students would like to see the same exercises that will appear in the exam. I am trying to avoid serving them the exam test in a silver tray but at the same time helping them learn the material in the course and prepare for the exam.

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    $\begingroup$ I believe the format of these types of lectures would change depending on the level you are teaching and also on the course's objectives. For instance, if you are teaching calculus to people who will need to use calculus in their future careers (but not necessarily become mathematicians), the test could contain 'authentic' problems where calculus needs to be used (in alignment with the course's objective). Then, the revision lecture is about how one can use calculus to their advantage to solve problems, not definitions. In an analysis course, things might be completely different... $\endgroup$ – orion2112 Mar 26 at 15:12
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    $\begingroup$ Just for information, I think in some English-speaking countries this might be called a "review" session. $\endgroup$ – kcrisman Mar 26 at 16:07
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...what are efficient structures for a revision lecture?

Things I've actually tried, sorted by increasing student engagement:

  1. Merely provide list of topics for students to study on their own time.

This was simple and fast, and it made it clear that they should expect questions from the entire course's material. It also put the review entirely on them to perform -- no "silver tray" of problems to memorize. However, calling this review was really false advertising, and it seemed quite ineffective. Students were not engaged and basically had to figure out what problems to study on their own.

  1. Provide list of topics (see (1) above) and spend the session answering student questions.

Without fail, the questions students asked were "Can you work a problem from topic X"? Then I would have to make one up or find one in the text and solve it (or we would solve it as a class). This was a better use of time, but it worked slowly and wasn't tuned to the needs of everyone in class. For example, a few students dominated the questions and the session could have become a de facto tutoring session for them only. Mostly, students sat as observers -- not at all what their upcoming exam would be like.

  1. Provide list of topics (see (1) above) and prepare a list of problems to demonstrate/solve in lecture. Pause between each problem to answer questions.

This was basically a problem-based lecture, and it had the benefit of exposing students to problems I felt were important for them to review. By varying the problem type and depth (all the way back to first-day or even prerequisite material), I could emphasize how cumulative the exam would be, and we could revisit the older or more basic material.

  1. Provide list of topics (see (1) above) and break class session into two parts. In the first 30 minutes, break your class into groups of 3 or 4 and randomly$^*$ assign to each group two or three topics. Have them create a problem for each topic and write them out on paper. Collect all problem sheets. During break time (or have a helper) make copies of all problems for the class. After break, display the problems with a document camera and quickly read over them with the class. Identify which problems are easy, medium or difficult. Use the second half to have students work on those problems in groups.

(*) Knowing your students' abilities pretty well by the end of the term, you may "stack the deck" by giving certain topics to students who you know have a good grasp of the material -- enough to create a problem that isn't impossible, for instance.

This was much more engaging for students, and they actually seemed to enjoy the creative aspect. Some struggled to write clear questions, so I was checking in on them as they wrote, and we (as a class) had to "clean up" some of the language in their problems after copies were made. I now had a good idea of what students thought of when presented with a list of topics, though they did not know what I thought about those topics.

  1. Provide list of topics (see (1) above) and a large list of problems, sorted by topic. Problems should involve some "easy" things (definitions, basic calculation, etc.) but should be more difficult than the exam, overall. Break class into groups of 3 or 4, have them look over the problems, and then give the entire period for them to work the problems in an order of their choosing. Walk around and answer questions as they come up. If a question is particularly insightful or generally helpful, address the whole class in response.

This has been the most effective method for me. By this point in the term, students have already seen me solve dozens or hundreds of problems, and they struggle with different things, student-by-student. The flexibility of being able to work with others and still focus their time on problems they struggle with, all while having me to answer questions, means there's no "down time". Students often thank me for these class sessions. Of course, this means you must create a bunch of problems for this review, but over time your list will grow. [One of my former professors turned his problem sets into books.]

  1. Do (5) on one day, and then (4) on the next day.

This has only worked out twice, as I don't usually end the term with an extra day to spare. On the first day (item 5), students finished the review problems for homework and we were able to discuss a few of them at the start of the last day. Then I was able to have them create problems (item 4) with the understanding that they could not merely duplicate something from my review problem set -- it could resemble an old problem, but it had to be something new. Many students thanked me, saying it "put everything together" for them, and other things you want to believe as your class goes into final exams.

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    $\begingroup$ Note: The topic and level of your course will definitely have something to say about which methods can work for you. I have used my options 4 and 5 from arithmetic through linear algebra (full complement of community college curriculum). $\endgroup$ – Nick C Mar 26 at 15:58
  • $\begingroup$ Correction: I've done (5) in all classes and (4) in a few classes. $\endgroup$ – Nick C Mar 26 at 17:26

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