# How to incorporate optional higher level mathematical content in an Engineering Maths course?

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.

• 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.
– J W
Oct 14, 2016 at 16:19

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?

• 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. Oct 11, 2016 at 12:50
• 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. Oct 11, 2016 at 13:58
• 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. Oct 12, 2016 at 23:02
• 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. Oct 20, 2016 at 1:29

Since you asked for different approaches (even anecdotes), so giving mine. (My practical solution is at the end.)

My undergrad school had a similar engine math course. It was semester-long and used Kreyszig as the text. Which is a quite good book, imo, with sound explication and drill, but also wide coverage. Of course the problem is that it would take 4-6 semesters to cover it all! And the students don't have time for that. They have to take statics and thermo and etc. To the extent that it already constrains their humanities electives more than non-engineers (chemists for example). So, the solution is to prioritize. They get a deeper exposure to PDEs (over what the diffyqs course gave) so they've at least dealt with "Jo and Yo" (Bessel functions, J(0) and Y(0) and the vibrating drum problems). And a few weeks on linear algebra (which is basic matrix manipulations). This, because they DON'T have time for a full semester class on LA. (Something math majors are often shocked to learn, but is common.) Complex analysis was NOT covered at all, despite being in the book. Really, only a small amount of the book was covered.

When I went there, the solution for mathies was that math majors NEVER took engine math. They took the same classes through diffyqs (calc 1-3 plus diffyscrews (almost entirely ODEs). But never took engine math. They moved into real analysis (called "theoretical calculus") and abstract algebra and the like. They also had a required full semester in linear algebra, along with the option for additional courses in that topic. PDEs and complex analysis were electives and full semester courses, meaning some of the students never took them, although realistically, the majority did...and if someone prefers taking an operations research course to PDEs, we should realize that "different strokes for different folks". There is not time in the undergrad to take everything, hence why we have majors to start with!

The solution for physics was to take engine math...and then any special topics were done in physics class (e.g. learning calculus of variations in the second mechanics course). This is non-ideal, but life is a constrained by time problem. A few years later, they just felt this was inadequate and installed a two semester math prep course. So essentially physics (and EEs) got a second semester of engine math that covered the complex analysis (of Kreyszig).

So, practical alternatives: diverge the math students out of an engine math course (they never even take it). And for physicists and EEs give them a second semester (and figure out what to cover in it, but it's mostly an engineering-oriented complex analysis course. Maybe a little more on PDEs also. To be practical, I would just do that as an appended course. Not attempting to "enrich" the previous semester, while running or integrate into it. [Note that this is not unreasonable. ODEs is treated again after a short exposure in second semester calculus. And it is normal for physics, chemistry, and biology to be treated at a college level after an earlier high school level exposure. We routinely get questions on this site for "why don't they do it right from the start", but this ignores practical pedagogy and the limits of human comprehension, as well as the option value of those who take a simple course and decide later to go deeper.]