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I noticed that in my undergraduate class a few students understand things quite fast and some times see the proof before I even explain things.

But some of them also have trouble understanding quite basic ideas.

I did a survey among students about the speed of lecture. Half of students thinks it's a bit slow, the other half think it's good.

So do you have any suggestion about how to balance the need of students of very different levels?

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  • $\begingroup$ One strategy is called Differentiated Instruction. The NCTM document "Why and How to Differentiate Math Instruction" mostly pertains to secondary education, but some of the examples might help you think about ways to implement this method at a college level too (assuming your class size allows for it). $\endgroup$
    – Aeryk
    Commented Jan 24, 2019 at 21:45

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For students who quickly solve the basic problems, what about having a list of "next level" problems for them to solve? Maybe if you want students to solve (or follow your argument for) five problems, try to have problems 6 and 7 be challenging enough for them, but accessible enough that they are possible without knowing other, special information.

I've tried this, and my stronger students (almost!) always go for it and try to solve the challenging problems. My goal, of course, is to teach what is in the course description, which means that everyone (regardless of ability) has been exposed to the basics. If a student has strong skills, I think they usually appreciate the opportunity to test that strength.

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  • $\begingroup$ The question is about lectures. What "problems for them to solve" you are talking about? $\endgroup$
    – Rusty Core
    Commented Jan 25, 2019 at 17:39
  • $\begingroup$ I was positing a situation where an instructor gives on paper (or somehow lists in advance) the problems they are going to demonstrate in class. Also, because I (almost) never simply demonstrate solutions to problems without class participation, I may use this list of problems to: (a) demonstrate from, (b) pause for students to try one on their own before returning to demonstration, (c) engage the class in some Socratic dialogue, (d) give part of class time for an extended student problem-solving session (i.e. they do multiple problems), (e) begin building a homework set, (f) etc. $\endgroup$
    – Nick C
    Commented Jan 25, 2019 at 19:37
  • $\begingroup$ If you do not have a plan of problems to look at, but rather create it on the fly or let class discussion lead you in this regard, then my answer above would require you to prepare for class differently. $\endgroup$
    – Nick C
    Commented Jan 25, 2019 at 19:38

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