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I am often tasked with teaching large enrollment courses, and one thing that becomes obvious quickly is that students attempt to pick up new information and solve problems at much different rates. They also succeed at their tasks with different frequencies. A simple, admittedly naive, classification system is that they exhibit $2 \times 2$ cases:

  1. Students who quickly absorb new information and process it reliably. These students typically have a good background for the course and a strong mathematical aptitude. It is very hard for them not to be successful in the course, though often they will not learn as much because I will not teach the class at a fast enough pace for them.
  2. Students who quickly absorb new information, but often process it unreliably. These students typically rush through problems, often overlooking key reasoning steps along the way.
  3. Students who slowly absorb new information, but are capable of processing it reliably. These students can easily be overwhelmed by a course that moves too quickly, as they cannot understand the reasoning steps when they are presented too quickly for them to process. But when given adequate time to process what they are learning, they are often able to reach mastery.
  4. Students who slowly absorb new information, and are not capable of processing it reliably. Such students often lack adequate background for the course and typically do not have a very strong mathematical aptitude. It is very hard for them to be successful in the course, as there is not adequate time for them to address all the obstacles to success they need to overcome in the short time of the course.

Of course, many students don't fit neatly into one of these categorizations, but to the extent this model gives some approximation to what goes on in the classroom, what are effective methods you have for helping students in cases 2 and 3 to succeed? How do you handle giving lectures to an audience that consists of a mixed group of students that run the spectrum of the four cases. I generally try to encourage students in case 4 to drop the course and, if possible and appropriate, I try to encourage students in case 1 to move to a more challenging course for them.

But I struggle with how to balance the students in the middle two cases. My style preference is to tend toward the slower side of presentation, using the extra time to detail carefully the specific reasoning skills used in each step of the problem and trying to take time to summarize the overall strategy when solving problems that are sufficiently complex. In theory, it works perfectly. It forces the students in case 2 to pay attention to the full depth of the problem, while it gives sufficient coverage so students in case 3 so they can achieve mastery.

In practice, it does not work this way. The students in case 2 often become bored by the slow pace; rather than pay attention to the extra details I discuss, they often disengage from the material. On the other hand, the students in case 3 often become dependent on detailed explanations and typically don't show development in overall problem solving skills. While they do tend to master the specific skills that we spend time in class on, but these skills often don't generalize well beyond narrow application in the context they saw the material. And, of course, a slower pace means that I have to sacrifice some content relative to a faster pace.

So my question is, what factors do you consider when you plan the pacing of your course? Are there ways to use pace in a course to improve performance among some student groups? Is there research suggesting pacing decisions lead to tangible outcome differences?

I typically teach introductory calculus courses, but welcome input from any specific mathematics course at the college or high school level, or general course-independent observations.

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    $\begingroup$ Your question never mentions the fact that the students have a textbook. This is the solution for students of type 3. They can read the book at whatever pace they like. You seem to assume that students must be spoon-fed everything they ever need to know during lecture. Why would you assume this? $\endgroup$ – Ben Crowell Nov 13 '16 at 3:31
  • $\begingroup$ This is a good point. In fact, I am dealing with a problematic textbook at the moment, but that's a temporary problem that can be fixed. My sense is that many students spoon-feed themselves from the textbook -- scanning pages for formulas and worked examples to mimic. Ideally, students would learn to read a math book better and appreciate context, but in reality, the course has a fast-paced syllabus that inhibits such growth. I try to use class time to fill in that context by having students develop the thought process needed to solve problems, but pacing that approach for everyone is hard. $\endgroup$ – Michael Joyce Nov 13 '16 at 18:45
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I don't know how successful it is, but I use a pattern of

  1. short piece of theory and examples
  2. practice questions for the students to complete
  3. talk through the solutions
  4. repeat

I do this by combining slides with writing on the board.

The reason I think this addresses the question is that students can make use of the 'practice question' time in different ways.

Some are still copying down the written material. Some are re-reading the material to process it at a speed they feel happier with. Some are slowly working through the straight-forward questions. Some speed through the easier questions, then ponder the more stretching ones.

If the atmosphere is right, they also use each other as learning resources, and practice actually speaking maths. And some take a break, hopefully regaining some concentration as a result.

By asking for their solutions, you also get a bit of feedback about how they are progressing with the material, helping you to judge whether you need to change pace.

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  • $\begingroup$ So you post several questions of varying difficulty? And you talk through solutions to some or all of the problems? I wonder what the reaction is of the slower paced students who can't make it through all of the problems - are they expected to complete them outside of class and does that typically happen? Appreciate your feedback! $\endgroup$ – Michael Joyce Nov 12 '16 at 16:59
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    $\begingroup$ I doubt the students go back through the questions, except perhaps during revision. I try to go through all answers, but sometimes I need to move on to cover other topics. Students are often happy once they've seen the answer, because verifying a solution is usually easier than coming up with it yourself. They don't learn as much that way, but I'd think they wouldn't do that learning anyway. $\endgroup$ – Jessica B Nov 12 '16 at 19:20

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