Another thing: assume that the people designing the curriculum might possibly know a little more than you do about what subjects are useful later on. I.e., don't say stuff like this:
"But why do I need to learn about matrices? I'm a meteorology major! We don't use matrices at all, it's all just partial derivatives!"
(This is an actual complaint I received when TAing ODE's, which was required by the meteorology major. I didn't even bother explaining how you discretize a linear PDE into a system of ODE's, or why that matters for weather prediction, I just referred her to her department head. )
They are right to be frustrated about not seeing the abstract material connected to concrete applications if your course's title is "Differential Equations for Electrical Engineers."
I recommend to college freshmen that they select courses that are relevant to their chosen field, but also relevant to other fields. Concentrate on fundamental knowledge, not detail knowledge.
For example, don't take a class that teaches how to use AutoCAD. Take a class that teaches analytical techniques.
My engineering physics curriculum required three semesters of Calculus, a differential equations course, and a numerical methods course. Notably lacking? A course in Linear Algebra, which was not required (and most in that program did not have time to take it, as the program was more like a double major with additional technical electives).
Same with my EE undergrad. Then I got to grad school and decided to take a course in Linear Systems, which is when I realized my ODEs course taught me nothing.
In Germany you get to your advanced courses for the last two years of High school. I picked Math and Physics.
We did a bunch of linear regression(with maple?) i believe. And some manual differential equations. I was one of the best in that class, but when I asked what we need this stuff for the other people in the class looked at me and said the following:
"if you ask this question you're in the wrong class"
I went on to study engineering, and i guarantee you the other two still don't know what that stuff is good for. They learned the formulas by heart and then went on with their lives.
I mean it was fun for me, it was basically a coding exercise. And I was happy because I was faster than everyone else, but I didn't get why we were doing it.
The same actually bothered my about the studies. Just one example: folding is a fairly easy concept. But for some reason they first want to drill it into your head, you learn a bunch of techniques and then if you're lucky and you stick with it, eventually it clicks and you'll realize why in a completely unrelated class.
Why can't we just provide a simple real life example first and then go on explaining the details?
Aren't things easier to grasp when you can have a real understanding/connection to it? Isn't that precisely why people that learn coding at home tend to be better than those that just studied it, because they were told it's a solid profession to study?
I think with a good example and visual representation you can probably teach most of the stuff that's taught in Uni to young kids. But then you would be forced to admit that you wasted a lot of time during your own life, who'd want to admit such a thing.
In EECS every undergrad does learn a bit of control/systems theory in their sophomore and junior requirements. I loved it but it was mathematically challenging and unfortunately the better background a student already had, the better and the more they got out of the courses. I wish there were a better way, because the result was students eventually got weeded out by an (inhumane, alienating) competitive system, rather than have each student actually learn something well on their own level.
I disagree. One of the most enjoyable classes I took was Weather and Climatology. Completely irrelevant to my current work and I paid a significant sum of money to take that required science credit.
I went to a top school in the US and I got a major in CS without ever taking linear algebra, which in hindsight seems completely crazy. Every major in science or eng. should have this as part of the mandatory curriculum.
1. Just because I didn't believe they could do differential equations doesn't mean that I didn't believe they could learn college-leve. algebra with some coaching. I had more faith in them than they did of themselves, in many cases.
2. There have been many studies that have tried what you say. The problem is that the class must go slower to accommodate the slower students. Engineering degrees already have so much packed into them that they often can't take classes of interest -- are we going to make it even longer?
TBH I was struggling to come up with a class that was relevant enough that I recalled it, but not relevant enough that I never used it ;). It's been a while since I graduated, so I probably suppressed a lot of memories about courses like Advanced Partial Differential Equations.
My best advice is to take the fun math, whatever that may be. If a class is fun, you learn it properly, so you will be much more able to adapt the concepts and strategies to other problem domains.
So apart from the mandatory calculus, statistics and linear algebra, as an engineer I took functional analysis, operator algebras, group theory, and intro to hilbert spaces.
Hmm, maybe. How would that impact the larger curriculum? Are you thinking a new class, or just change how differential equations is taught?
I think there is a little bit of an annoying situation where at least Electrical Engineering students are going to want Differential Equations pretty early on as they are pretty important to circuits (IIRC, I don't touch analog stuff anymore). Like maybe as a first semester 200 level class. This doesn't afford space to put a Linear Algebra class in beforehand (needed for numerical analysis).
Maybe the symbolic differential equations stuff could be stuck at the end of integral calculus, but
1) curriculum near the end of the semester is risky (students are feeling done, and it can suffer from schedule shifts).
2) Transfer students or students who satisfied their calc requirements in highschool (pretty common for engineering students) wouldn't be aware of your curriculum changes.
Or, a numerical-focused PDE class could be added to elsewhere. I bet most math departments have one nowadays, but as an elective.
Be careful when saying things of the form "$DEGREE_PROGRAM should also include at least one class in $SUBJECT." You can make persuasive arguments for the relevance of a dizzying range of subjects, and at some point you have to prioritize.
Personally, I think this is bad advice, because without an undergrad+ background, the projects above will either be impossibly frustrating or you will make up some crackpot bullshit. Plus, the undergraduate curriculum is its own reward.
It makes me curious about one point. It sounds as if the class you took covered what my engineering program called "Communication Theory". We were expected to have two terms of differential equations and one each of linear algebra and statistics under our belts as prerequisites. (I think. This was a long time ago...)
Were you missing, at least, having at strong basis in differential equations? If you didn't, that would have made the class double-tough.
I run screaming from courses that cover mathematical background I don't have
I had the opposite attitude when it came prerequisites in college. For example, I had no problem taking a course that required linear algebra, or signal analysis, or data structures (none of which I had taken; eg, I took Data Structures II before I). You pick up the missing pieces as you go. I actually had more fun that way - "Signal Analysis" was much more interesting when I already knew the applications!
BTW, I didn't do this on purpose. I was trying to get these classes in to fit my schedule.
Show me an American university that doesn't spend the first two years of a STEM major's life on the three-semester calculus sequence, linear algebra, and ordinary differential equations, and I will then come up with a secondary school math curriculum targeting admission to such a program.
The problem with what you're asking is that it's one more course you're trying to fit into a curriculum. My engineering department already gutted their first year calculus course by making most of it (about 3/4) pre-calculus after finding out that a majority of students weren't prepared.
I was asked to tutor a first year engineering student in programming as she was failing. I ended up spending most of the time helping her conceptualize the problem and breaking it down into steps. Once she learnt that, she could manage the syntax aspects of C++. She ended up with a grade in the 60s which isn't bad considering she had a grade in the low 40s after the midterm.
When I was a TA for thermodynamics, I noticed that we would get a bimodal distribution on the midterm exam. Thermodynamics isn't hard once you understand it. We'd get the people who didn't understand with grades in the 40s, but the people who did had grades in the 70s and 80s. The prof said that was pretty common. The objective was to get as many people from the lower curve to the upper curve by the end of the term. You'd typically have around 3 to 5 of a class of 80 fail by the end. Those were the people who just didn't get it. I'm sure that thermo is one of many courses like this.
"But why do I need to learn about matrices? I'm a meteorology major! We don't use matrices at all, it's all just partial derivatives!"
(This is an actual complaint I received when TAing ODE's, which was required by the meteorology major. I didn't even bother explaining how you discretize a linear PDE into a system of ODE's, or why that matters for weather prediction, I just referred her to her department head. )
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