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Our department teaches two very large first-year "Mathematical Methods" courses (600-ish students) to Engineering students. The syllabus is dictated by their (future) needs and covers a huge array of topics, but none to any great depth.

For resourcing reasons, we cannot put on a separate course for the 10-15% of students who might go on to study Maths or Physics, but we feel that a breakneck tour through a huge number of mathematical methods is not an ideal start for these students.

We are exploring methods of adding (optional) depth to this course to better serve this small subset of the students. So far we've tried having optional "explore by yourself" sections in the weekly tutorial sheets, and we've tried having out-of-class optional "enrichment lectures", but neither of these has worked really well.

So my question is: given the constraints outlined above, what are possible methods of adding optional mathematical depth to an Engineering maths course to better serve future Maths students?

I know that this is rather vague and wishy-washy, but I'm really after anecdotal experiences from people who have tried this sort of thing.

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  • $\begingroup$ I have offered optional enrichment lectures in the past to well-prepared and interested Engineering students, but stopped after a couple of years because I became too busy. It was enjoyable though and time permitting I would consider resurrecting the practice. $\endgroup$ – J W Oct 14 '16 at 16:19
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In our department, large introductory math courses, such as calculus, linear algebra, and discrete mathematics, come together with little satellite courses called "advanced investigations in *", where * is the main course.

Students who want to explore the subject in depth register for the main course and the satellite course. Example: in main Calculus I, students learn the definition of a continuous function, in Advanced Investigations in Calculus I they also learn the proof of Intermediate Value Theorem.

The main course is worth 3-4 credits while the satellite course gives them only 1 credit. Example: if a student gets B+ (4.0) in Calculus I (4 credits) and A- (4.5) in Advanced Investigations in Calculus I, the average grade will be $$\frac{4}{5}\cdot 4+\frac{1}{5}\cdot 4.5=4.1$$

I know that this arrangement is not perfect because it requires to create an extra course, but it is a little course and it takes little effort to teach it.

Will it work for you?

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    $\begingroup$ That sounds like a great solution, but sadly not one that we could implement, as new courses are unlikely to be approved in our current dire financial situation. $\endgroup$ – Gordon Royle Oct 11 '16 at 12:50
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    $\begingroup$ Oh it's a pity. Then I can only speculate that the reason why your approaches didn't work is because optional tutorial sections and enrichment lectures were not graded. Then, if you want to introduce grades for such optional tutorial sections, you'll need a system that encourages strong students to do optional sections and discourages weak students from trying it. I would suggest that optional tutorial questions are hard, but students who try them get immediate feedback from the instructor and unlimited attempts to resubmit. $\endgroup$ – Fedor Duzhin Oct 11 '16 at 13:58
  • $\begingroup$ Good idea. We would not get away with assigning credit for optional work not to be done by all students, but we could certainly give feedback perhaps via an additional tutorial session for students who attempt some of the "extension" questions. $\endgroup$ – Gordon Royle Oct 12 '16 at 23:02
  • $\begingroup$ Ok here is another idea how you can assign credit for optional work. Let's say that each homework assignment has 10 easy questions on computation and 5 advanced questions on definitions and proofs. Each question is worth, say, 10 points. The score for the entire assignment is the minimum of the sum of scores for questions and 100. Besides, advanced questions can be submitted many times via sharelatex.com, each time the tutor gives feedback to the student and the student is allowed to update the solutions and resubmit. $\endgroup$ – Fedor Duzhin Oct 20 '16 at 1:29

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