The entire first year was a lot of math courses, and it did not come with any explanations for how this is important for my profession nor was the math connected to reality (there was relatively little application to real problems). However, at that point I was used to having math presented to me in this way - as a mathematical abstraction in a mathematical world. I like math, I always did, and so as long as it presented enough challenge from the good kind, I did not complain. It wasn't until ODE, and PDE that I started to resent what was asked of me. But there were some elements of resentment in earlier courses as well. Whenever I felt that the emphasis was about mimicking a method and not about understanding it, I got annoyed. However, my feelings are not the point.Did you have the same reaction to evaluating integrals in elementary calculus, or was there enough explanation of how and why they arise in different ways so you understood that you were learning methods for something generally important and which is applicable regardless of the different physical meanings possible in different circumstances?
To give one example: ODE. Ordinary Differential Equations. The entire course was comprised of writing a type of problem, and presenting how to solve it. Then for homework, you had a bunch of problems and you needed to become an expert in applying the right method from your notebook to solve it.
Where did this ODE come from? Why is it needed for an engineer? What does it correspond to in reality? None of these questions were answered. Instead you are required to solve more and more problems until you automatize the different methods. I bet you anything that if I pull an ODE (out of my <let's say notebook, because the word I have in mind is not as nice>) and let you solve it now, you will not remember the method. The memorization of the method lasted just enough for the test. So what's the use of it?
Even if they did explain how ODE is generally important in many fields, it's not enough. They should start with real situations and show how they translate to an equation. Not just give examples, but let students practice But then, after they did that, they should not demand that we memorize the methods to solve all the different problems.
They did. But I barely remember any of it. Why would I care if a solution is unique when I have no idea how such a thing is of any importance to the real world or to my profession? (to describe my perspective from back then)
Did your DE course say anything about how to tell when a solution exists and if so if it is unique?
No, they did not. I picked up a book on the subject and read about real scientific problems (like the heat equation).
Did the course not give examples of how they arise and are a common form of many laws in physics and engineering?
Yes, I've seen it so much the damn equation is spilling from my ears . And I have no idea how second degree circuits or oscillating mechanical systems are of any importance to my profession.
Didn't other courses begin to explain that, or did they all come too late after you already had the DE course? Had you seen, for example, how the same equation for an elementary harmonic oscillator arises in different physical contexts like springs and masses or electric circuits?
Those courses came afterwards. I had one good course (not perfect though, but still pretty good) about transport phenomena (like heat, mass). That course actually had some originality in the problems that needed to be solved, and connection to reality. It didn't teach us how to model reality (no course did), but it did teach us that after reality has been modeled by someone, how to deal with the problem.
Did you have a follow-up course in partial differential equations where you saw formulations of vibrating strings and drum heads, or heat conduction before learning how to solve the common forms of PDEs?
From all the many different DE problems, I think on the large, only one type of problem was used later on.
Didn't the methods you learned for solving DEs come up again in engineering courses where you learned what they were for in different specific contexts, or did the whole subject drop off the cliff as soon as the DE course was over?
There is more that needs to be taken under account when judging if the course was taught well. Mathematics is generalization of many different concretes in reality. So before they start teaching us the math, the first need to demonstrate how it is an abstraction. (This is especially true for kids. Thinking back to my high school education and earlier years, I think the whole approach that they used was wrong, because they taught math as an abstraction living in a bubble. For years I had no idea that 'x' and 'y' have anything to do with reality. I had no idea that they are an abstraction. ) To get back to what else needs to be taken under account - in my DE course they required we memorize and automatize the methods of solving different problems. This was completely useless. An engineer does not need to solve many DEs in a short amount of time. All he needs to know is how to solve an ODE. For that he just needs to look at his notebook/google. Memorization is useless. Second - if an engineer will ever come across a differential equation, it will not be handed down to him as an equation. He needs to be trained in translating the system he is working on to an equation. We got almost zero practice with that.
All these questions relate to how well your courses were integrated into a curriculum and how it and the individual courses observed the proper hierarchy of knowledge in what you were learning.
OK, too spent to continue. But I don't think I can gather enough energy to make this post more coherent, because digging back to my DE course, frankly makes me very agitated . So I'm just going to post what I have, I hope it is coherent enough.