> A course taught as a bag of tricks is devoid of educational value. One year later, the students will forget the tricks, most of which are useless anyway. The bag of tricks mentality is, in my opinion, a defeatist mentality...In an elementary course in differential equations, students should learn a few basic concepts that they will remember for the rest of their lives...
I hated the DE cleass I took in college and it was largely because I felt like it was nothing but a bag of tricks. I very distinctly remember one problem that seemed unsolvable until the teacher showed that you had to substitute a "2" with "1/2 + 3/2". And then, to make matters worse, he put the exact same problem on the test. So we were being rewarded, not for really understanding the core basic concepts, but for memorizing the tricks needed to solve specific problems.
I was working as a programmer at the NIH for a senior scientist who was an MD & math PhD. In the course of the work he was doing, we came across a differential equation that needed solving. My undergraduate math degree was not enough to crack it in a day. I tried Mathematica, which also choked on it (probably should have tried Mathematica first.) My boss spent a couple of hours and came up with no explicit solution. I was preparing to move forward with numerical approach, but my boss made a phone call to a friend who came into his office and copied the problem off the whiteboard.
Two days later he had an explicit solution based on two really non-obvious (bizarre) substitutions. I came away very, very impressed. The guy earned a 'we are indebted to ...' footnote in the paper.
I guess the point is someone has to come up with the tricks.
Nothing so quickly emphasizes that a bag of tricks can turn into actual math more than a linear algebra course. In a math degree you can literally take the same course twice. At a first year level you will learn to perform all these tricks with matrices and get a hint of inner product spaces. In 2nd or 3rd year you'll do the reverse and justify why matrix algebra works in the first place.
But the harder a subject, the longer it feels like learning a bag of tricks. The first partial differential equations course feels like you are working on only 3 problems for 4 months.
edit: I had a prof whose first DE course was at the graduate level. At the oral exam he was asked to give an example of a differential equation and all he could do was point to phi on the blackboard.
I'm thankful that my calculus courses were extremely heavy in showing how the tricks actually worked. It wasn't on the level of grad school analysis, but the professor pulled no punches and we went extremely in depth.
Years later when I was learning multi-variable calculus, I found most of it easy because, even having forgotten most of the tricks within those years, the method behind the madness was still there.
I don't know how correct this is http://www.math.harvard.edu/~knill/pedagogy/pechakucha/ but it makes the whole pre-college math education looks like a bag of trick. Here principles are both concrete and abstracted yet it all blends as one.
I'm not sure what that problem would have been, but an integral part (no pun intended) of DEs is expansions. It's not part of a bag of tricks, it's a very common technique used to solve a problem.
I think the problem is that not enough time is spent explaining these very common techniques - what the rationale behind them is, hot to intuit that it could be applied to a specific problem, and how to get from that intuition to the actual answer / application.
Often, there's a very large skill gap between the student and the teacher (especially at college level, where many of the career mathematicians live and breathe maths), and these things are hand-waved away as obvious. Even worse is when the teacher doesn't actually know, and is just presenting the material straight from a guide. The way most course material seems to be set up is to skip over the middle part, and as a result the best short term strategy is to learn it as a bag of tricks.
I love maths, but I think its universal applicability and beauty get lost due to the way it's generally taught.
You know that from study and experience. If that underlying concept wasn't the lesson, like the parent indicated, it would I deep end up a trick in the bag.
The toughest part of teaching, I think, must be really knowing that your students are internalizing the core rules/principles/concepts behind the examples you teach with.
I remember a similar problem in Calc 2. I forget the specifics now, but I think it was an integral of some combination of sin/cos that ended up being circular. You had to recognize an opportunity to swap one of the steps for an equivalent, which would lead you to the final solution.
Probably the second example here [1] for those curious (I think the integral of sin(x)*e^x dx is the only place I've seen this used, would love to know if there are other examples).
It would be great if that were how it was taught, but when I took the class it was taught as a trick. No theory behind it, just the prof on the board saying, "But look! :swap: And now you can integrate it."
It becomes a technique once you realise it is a specific case of change of bases.
Even just getting to basic theorem of algebra does not provide this insight.
The common techniques ARE a bag of tricks. Feynman was famous for being really good at integrals, because he had memorized the huge bag of tricks. Today, we have Mathematica for that, you don't need to be Feynman.
I don't remember it being presented that way at all. If it really is a core technique, I would expect it to have been emphasized strongly and the test to have a problem with a different expansion instead of the exact same 2 -> 1/2 + 3/2 problem that he had done in class.
>I felt like it was nothing but a bag of tricks... had to substitute a "2" with "1/2 + 3/2".
Many techniques in math are "tricks" like this. Think of solving a quadratic by completing the square, or integrating by substitution or integrating by partial fractions, etc. You could arrive at these techniques on your own, but that is a lot of trial-and-error, deep understanding of theory, and applying it, which all takes a huge amount of time. Meanwhile, previous mathematicians figured this out and we get to benefit from their work. ;)
Maybe your instructor didn't present it well - plopping out the answer without a good enough explanation of the technique, why it works, etc.
The implication seems to be that students can't do the trial-and-error, understand then apply the theory. I agree with parent comment that this is awfully defeatist.
But even if it is futile to teach true understanding, why are we knowingly teaching computation in its place? What if we left the computation to the computers from the get-go? Could we then have enough time to teach true understanding?
I can't tell you how much time I spent trying to memorize my multiplication table – a 12x12 grid of numbers that for most students became arbitrary 3-number sequences. "3, 3, 9" is different from "3 sets of 3 is equal to 9." Most students learn the former in place of the latter.
>The implication seems to be that students can't do the trial-and-error, understand then apply the theory. I agree with parent comment that this is awfully defeatist.
All teaching is a balance between various factors, and perhaps the instructor didn't get right in this specific instance.
Ideally the instructor would teach the theory, allow some trial-and-error as students grapple with new information, and then step in with prodding towards how to do it - but fundamentally, there is a limit on how the instructor can let the class wander without needing to move on to cover the rest of the material (this wasn't a special topics class covering how to solve this one specific problem).
Hence, go through the information but provide the technique involved after suitable time passes. That's just the nature of teaching/learning when the students don't have infinite time to essentially re-invent the material they are trying to learn.
But Google is a better teacher of that kind of stuff than any professor can claim to be. If this is the kind of knowledge you pass on during education, it's basically worthless.
I had an almost identical experience with my ODE class. This is exactly when I started hating math, and moved out of computer science (which was part of the math department at the university I was at).
It took me a long time to return to learning higher maths, this time on my own according to the needs of my job, which is far less an ideal environment than University.
If ODE contains "a few basic concepts that they will remember for the rest of their lives" I never learned it, and I wonder what those concepts would be.
Indeed, Rota's approach to teaching seems to mirror the application of mathematics in general...
He acknowledges that concepts have prerequisites, and that certain topics are misleading or dead ends. You might even call the act of building a pedagogical scope and sequence for a topic graph traversal!
I wish I could have taken one of his classes. I wonder if there are other mathemetician/philosophers out there who have taken the problem of teaching as seriously as the math itself.
> A course taught as a bag of tricks is devoid of educational value. One year later, the students will forget the tricks, most of which are useless anyway. The bag of tricks mentality is, in my opinion, a defeatist mentality...In an elementary course in differential equations, students should learn a few basic concepts that they will remember for the rest of their lives...
I hated the DE cleass I took in college and it was largely because I felt like it was nothing but a bag of tricks. I very distinctly remember one problem that seemed unsolvable until the teacher showed that you had to substitute a "2" with "1/2 + 3/2". And then, to make matters worse, he put the exact same problem on the test. So we were being rewarded, not for really understanding the core basic concepts, but for memorizing the tricks needed to solve specific problems.