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I teach undergraduate and graduate level courses in mechanical engineering at a research oriented university in the United States. I am a teaching track professor and my sole responsibility, on which I am judged for pay raises or promotions, is teaching efficacy.

The courses I teach have a strong mathematical component to them. Some notable examples would be: computational fluid dynamics or finite element methods. For those here familiar with these courses, you would know that calculus background is necessary to understand the nuances.

However, I find that the students I teach this to (mostly 4th year undergraduate students) balk at the prospect of doing some analytical "pen-on-paper" work and would like only to know how to use relevant software skills for these, with no mathematical background.

This leads me to receiving unsatisfactory end of semester evaluations (3.90/5.00) from students with a lot of them complaining that "the course is too math-y"(verbatim). The department requires teaching professors to consistently receive a 4+/5.00 for courses they teach and base promotions, continuing contracts and raises on this.

Unfortunately, I refuse to pander and just teach students "software-button-clicking" alone. Software skills can be learnt from Youtube and they don't really need me to teach them that. I continuously tell them that mathematical/fundamental analytical skills are far more important because the software landscape is quite fickle and ever-changing based on industry whims and fancies. However, fundamental mathematical skills are resolute and robust and may be applied to most engineering problems.

Has this been encountered by other folks on this SE and is there some effective method you have devised to counter this? My university is significantly "engine research" driven and students end up getting jobs at the "big 4" automotive companies. Perhaps I need to ply my trade at a different department like "applied mathematics"?

On my part, I juxtapose software results with analytical results continuously to explain how analytical results are necessary for validating software results. However, students are under the impression that the "software is always right" irrespective of whether or not their problem set-up and boundary conditions are correct or not.

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I have two questions: 1. Does the department want these classes to go over the conceptual, mathematical side of the material? and 2. Do the students have the necessary mathematical maturity/background to understand what you're teaching? Your goal is a noble one, but it's not worth getting killed on the evaluations for it if the material goes over the students' heads anyway. – user37208 Jan 13 at 15:57
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I'm in electronics, and as a colleague of mine is used to saying: to proficiently employ an electronic simulator like Spice, you should know electronics much better than it. First, recall that to the students ;-) – Massimo Ortolano Jan 13 at 16:30
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Have you tried to make more clear either in prereq's or outright in the syllabus the mathematical expectations of the course? You might have to be extremely blunt and state "Paper and Pencil calculation required, will not instruct to specific software" and then possibly offer an office hour or two to those who want industry level software support. – scrappedcola Jan 13 at 17:41
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I'm surprised that the department has a hard requirement of 4+/5.00 on student evaluations. That seems like possibly shooting itself in the foot if that is prioritized over anything else like, say, meeting the course description in content, etc. This makes it effectively impossible to have a challenging, "rigorous" course. One possibility I shudder to bring up is, since evaluations usually occur at the end, make the front part of the class even harder and encourage failing students to withdraw ASAP. – Daniel R. Collins Jan 13 at 18:05
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A quick comment: in the EE classes I teach with similar math-y components, I make it a point to constantly remind students that employers in my sub-field get quite frustrated with junior engineers who rely too heavily on circuit simulations, for example, and remind them that, during an 8-hour long technical interview, they will not be placed in front of a machine and asked to simulate anything: they need to be able to convince interviewers of their skills at the whiteboard. – Mad Jack Jan 13 at 19:14
up vote 13 down vote accepted

The professor that taught me statistics dealt with this basic problem, which is that students would come into the course thinking it was like any other "narrative" style of course (like a history class), but the truth is it was a math class that was not listed as MATH 3XX.

Part of the job of a student is to pick courses, ordering them and balancing them into a semester that allows them to handle all of their classes. If they do not have the information they need to balance their schedule (as some classes are naturally heavy loads and others are light), they cannot do their job. So here's how I learned to help them do their job so that they would feel that you were doing your job.

Reset Student Expectations

On week 1 or 2, assign a special test. Include absolutely the most challenging math you expect them to do for the entire semester (not hard for the sake of being hard). Make it absolutely clear that this course does in fact require previous knowledge and ability, and that this test is designed to do just that - test their readiness for this class. You can make it an in-lab assignment and just give credit for doing it - you can give them the answer key and let them grade themselves.

Clearly explain to students that if they cannot complete this test satisfactorily, that is information for them to decide if they are ready for this class. When I ran this lab session I personally explained to students that if they weren't willing to spend at least X hours per assignment per week, they would be unlikely to do well and should reconsider their plans now while they can still re-arrange their schedule. I had a few students drop by the next week, but then everyone who stayed stuck with it all semester.

Don't Make Students Feel Dumb

One of the great dangers of being an expert and a teacher is that you become very distant from that point in time when the material was truly difficult for you, and you lose some of your understanding of what it was like to not know. You are also likely to have an IQ far beyond the average, as well, so you might not ever have experienced this material as difficult - which makes it even harder to avoid this mistake.

I personally remind myself of the time where I had scrawled out about 8 pages of hand-written calculations and equations to solve a factorial analysis of variance by hand, because my professor said it was important. If someone at that moment had looked at what I was doing and even vaguely implied "oh, that's mostly just basic algebra, that's not so hard" I would have stabbed them in the neck with my pencil right then and there. (The pencil was dull and my wrist was tired, so they would have probably escaped serious injury. But still!) Maybe it wouldn't have been hard for them, but it was a challenge to me.

With your advanced understanding of mathematics, you might unintentionally be sending your students a message along the lines of, "oh, this math is pretty basic first year stuff, you shouldn't have any trouble with it". Math is often hard and time consuming and mind-bending and forgotten quickly, even if it is important, and to imply otherwise is insult your students intelligence and character. This will likely result in them not liking you, and worse - they may not learn as much from you as they could have.

Be honest, but take care to honor their struggle with fundamentally non-trivial material. Students appreciate "I know this may seem hard, but you can work through it" more than "this is easy, work harder".

...and make sure the material Is Really Relevant

Yes, math is important and sometimes being able to do it by hand is even important. But I've had teachers include complex material based on the idea that it would be on X industry test that was 3+ years away and that I would not ever be taking, and could otherwise be looked up in < 10 seconds if I really needed it. Sure, now if I get stranded on a desert island I'll be able to calculate proper binary subnet masks so I can build a complex routed network for all my coconuts, but otherwise I'm still a little annoyed I spent hours on that mess when I could have spent it on my research projects.

Motivating the material with a calculation the computer can't solve but they can do it by hand, if such an example exists, can help. It may also be helpful to really drive home the point that if you don't understand the underlying math you'll click stuff that is laughable and makes no damn sense, but the software didn't know any better so it did what you asked anyway.

Still, make sure you really aren't including tough material (according to the students) just because you really think its cool and is technically somehow applicable, but not really necessary or very valuable to the student. Those topics might be a perfect fit in another course - just not this one.


I think you'll find that if students are made to understand what will be expected and necessary upfront, you empathize and honor the challenges your students deal with, and you pair down the material to what truly best serves students, you'll find that not only will students like you more and rate you more highly - but they'll also learn more and the class will be more fun to teach, too.

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I actually am establishing a "2nd week concepts exam" in this course. It tests students in calculus concepts that are relevant to this course. I am also making mathematics relevant by breaking down, say, the heat equation into its individual terms and explaining the physics behind each term and "switching ON or OFF" terms to express the physical effect. I hope these work. Thank you for your answer. – drN Jan 16 at 15:12

Based on your comment that the mathematical ability of the students is poor, to the point that they struggle with 1-D Calculus, I think you need to drastically adjust your expectations. I don't know how deeply you get into the mathematical basis of the Finite Element Method, but doing it "right" requires knowledge of functional analysis at an advanced undergraduate level. For a student who doesn't even understand 1-D calculus well, most any sentence in a functional analysis textbook will be literally gibberish. You couldn't bridge this gap in a one-semester class even if the entire focus of the class were functional analysis, let alone in a course with other priorities. So while I agree that your students should learn the material you're teaching in a perfect world, it's unrealistic to expect them to learn something they don't have the necessary background for. This isn't only a problem for your teaching evaluations--your students are very likely not getting much out of this class (at least the more mathematical parts of it), and in fact they have every right to complain.

So, how can you avoid this problem without pandering to the students' unambitious goals? If I were you, I'd probably do some combination of the following two approaches:

  1. Rename the class something like "Mathematical Methods in Mechanical Engineering" and state clearly in the course syllabus that there is a significant mathematical component to the course. If you can, list a sophomore-level math class or two as prerequisites. This is to attract students who will get something useful out of the more mathematical course you want to teach.

  2. Choose different, less complex, engineering topics to teach, but dig into the mathematical concepts behind them, which hopefully will be closer to earth. Focus on examples where blindly using the software without understanding the math will give the wrong answer. They will be "toy" examples, but if you're lucky, they will instill in your students a lesson that they will remember when they use software on more complex systems.

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^^ I don't want these people designing my skyscrapers and bridges, or cars for that matter. – shoover Jan 13 at 18:23
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@shoover my sentiments exactly.... about my students. However many of them are being hired to do exactly that since they "pass" other courses! Since this FEM course is an elective, it can be left out of the degree schedule. – drN Jan 13 at 18:53
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+1 for "If you can, list a sophomore-level math class or two as prerequisites." – Ghillie Dhu Jan 13 at 21:24
    
I'm always very much motivated by the problems I can solve. There should be examples where, depending on the problem, a different solver/model/approach will yield much better/faster results. I think by showing these cases, and clearly showing how you can re-model things better and solve much more difficult problems, if you understand the math behind it, is a great way to motivate learning the math. – Steve Heim Jan 14 at 2:12

There are several aspects to examine:

  1. Students' expectations of the course
  2. Your expectations of the students' abilities
  3. Course alignment with curriculum

Let's take #3 first. Presumably, you live in a state that licenses Professional Engineers. Since you mention Big 4 auto, I'm going to take Michigan as an example. The Michigan Department of Licensing and Regulatory Affairs declares:

Article 20 of Public Act 299 of 1980, as amended was created, to license and regulate the practice of professional engineering in Michigan. Article 20 defines professional engineering as professional services, such as consultation, investigation, evaluation, planning, design or review of material and completed phases of work in construction, alteration or repair in connection with a public or private utility, structure, building, machine, equipment, process, work or project when the professional service requires the application of engineering principles or data.

Michigan requires PE applicants to pass both the Fundamentals of Engineering Exam and the Principals and Practices Exam of the National Council of Examiners for Engineering and Surveying (NCEES). As you know, these are standardized national exams. Your students will generally take the FE in their senior year of college, which is the year of the students you're asking about.

You should make sure that your courses align with the topics that your students are expected to know for the FE. These include calculus, differential equations, and numerical analysis, which your students already seem to be having trouble with. The bulk of the test for mechanical engineers (pdf) consists of MechE topics: statics, dynamics, mechanics of materials, material properties, fluid mechanics, thermodynamics, heat transfer, measurements and instrumentation, and mechanical design. Your students will need to know these topics cold in order to pass the FE.

As for #2, given that calculus is a series of freshman courses, and all of engineering builds upon it, you are not unreasonable to expect your students to be able to practice it.

As for #1, my suggestion is to clarify in your syllabus for each course that the goal of the course is to have the students understand topics X, Y, and Z that are covered in the FE, with links to the NCEES site. It also might be a good idea, for each new module that you cover within a course, to identify which section of the FE that it addresses.

Your students may protest that they are only getting the degree for its own sake, or they will immediately go to business school for their MBAs, or whatever. Emphasize that many engineering jobs will be closed to them if they don't follow the PE path. Also, if they are going for an MBA, a PE+MBA combination would facilitate their going into business for themselves.

Disclaimer: I have an engineering degree but am not in academia and am not a P.E. Also, I am from Texas and not Michigan, but the tests and accreditation are national.

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I certainly agree that you should not pander to a teach me the 'software-button-clicking' mindset in an engineering course. When I took Finite Element Analysis during my undergraduate engineering degree, commercial FEA packages were not covered. Everything was done by hand with some work done in Matlab. I am personally a strong supporter of this fundamental approach.

As others have pointed out, it would be worth checking that the mathematical rigor you are using in your course is appropriate to the students at your university. Some schools/countries are much more mathematically focused. Perhaps go through your lecture notes with a lecturer who is popular with the students, or with a graduate student who did their undergraduate degree at the same school. From the universities I have been at the level of mathematics in a text such as Hutton - Fundamentals of Finite Element Analysis has been appropriate for upper year undergraduates.

I don't know what material you are covering in your course and how you are approaching it, but perhaps a few adjustments could boost your ratings. Some things that I have found useful in the past:

  1. Include a project in the course where the students use a commercial software package to solve some problem. Don't teach them the software in class, let them learn it themselves (or give them some instructions to follow). Have them solve the same problem by hand and compare the results.

  2. Focus more on applications in assignments and exams, but cover derivations in lectures.

  3. We wrote a direct stiffness method FEA program for analysing trusses in Matlab as part of a course. I thought this was useful, but some students struggled with the programming aspect.

  4. Nonlinear stuff is likely beyond undergraduate capabilities.

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Nonlinear stuff is also where, imho, software fails (or simply comes up with sub-optimal solutions) a lot whereas some pen-and-paper analysis can give a lot of insight. It might be a good idea to show a few of these examples (without focusing on it, but giving the students a taste of it). – Steve Heim Jan 14 at 2:06

Well if you insist on delivering content that your students dont want, and not the content that they do want, then you should expect a low score.

Most CFD users see it as a problem solving tool, and you are not really preparing your students to solve problems if they have no software tool experience.

You could spend more time showing them how to actually solve real world problems with it, even without getting too hung up on what software package you use.

The counter point to "software can be taught from youtube" is that the math can be taught from any of a dozen reference textbooks. Software is inherently more useful to the students. As for the "you always have to check the software" approach, you are right that the software cant be trusted alone, but it can be validated through other means, such as prototyping.

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Well in most first year math courses the content our students want is the emptyset. Our job is to teach the course curriculum, and not to skip things just because the students find hard..... Mathematics is about understanding the material, not blindly using tools without the faintest idea why. – Nick S Jan 13 at 16:19
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Students seldom know what they want. I doubt education is a democratic system where students get to vote on what they want and don't want. I also disagree that math can be taught from any dozen reference books. It takes expertise to explain to students for example that "the Biharmonic equation can be used for BOTH evaporating liquid films and vibrations. Isn't it cool that you can solve two different engineering problems with the same equation?". As an instructor, it is my duty to distill "complex" math information to provide perspective. – drN Jan 13 at 16:45
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Your comment is completely nonsense. Of course that the students have absolutely nothing to say about what is to be taught. – Joka Jan 13 at 17:17
    
@Jonas, it's not necessarily that students should have control over course content, but that their evaluations will reflect their opinions, however misguided or not those opinions may be. An instructor whose fate is significantly determined by such stuff is in a tough spot. – paul garrett Jan 13 at 19:56
    
@paul garrett Quite right. Often, we think twice before doing certain things. Unfortunately I have been there often. Basically, universities put the pressure on the wrong place, the right place would be the students. – Joka Jan 13 at 20:04

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