I could have used this 32(!) years ago when I was struggling in college. (This and 3b1b.)
It amazes me just how many key topics were so inaccessible to the majority of the class at engineering school. I base this on observations from group study sessions and the hyper-aggressive test curves.
I knew lots of people who never got the hang of div/grad/curl, or a Jacobians, or eignenvectors, or Z-transforms... These are key engineering concepts, you'd think colleges would bend over backwards to make sure these concepts are learned as succinctly as possible rather than add a curve to a test that makes a 23 out of 100 an "A" grade.
I'm digressing, and complaining, but the counter argument has always been: you're not supposed to learn everything in college, you're supposed to learn how to learn. Sure, right, but who has time to keep learning advanced calculus after college? (Well, I still study math & physics for fun, but over the course of decades, not years.) Not being able to see the world through these lenses I think means missing key engineering perspectives and relationships.
I feel the same. How can it be that I went to a world famous institution providing 2-to-1 student-teacher ratios, but I still think the best explanations are these modern internet explanations? I guess the best explanations just bubble up in the modern environment.
> you're not supposed to learn everything in college, you're supposed to learn how to learn
But to learn how to learn, you gotta learn some things to a somewhat decent degree. I think at some point you need to have these linalg/divgradcurl things down, if only briefly. You might forget any particular topic, but if you've indexed it you should be able to pick it up again, particularly in the modern learning environment.
Just imagine coding without access to StackOverflow.
Interesting to think a bit about student/teacher ratios (STR).
The up-side is that with a low STR (2:1), the teacher can adapt to the particular strengths and weaknesses of the students, to get the best reinforcement. The /downside/ is that the students will typically also have fewer teachers overall, and are maybe stuck with a bad one. (This is the problem of bad grad school advisors in a nutshell...) In this world, teachers are very expensive, though, so we end up with students competing for access to good teachers, by paying super-high tuitions, dedicating their early childhood to olympic-level basketweaving, etc.
In the medium-STR regime (30:1 or 100:1), we get the worst case: There's no teacher adaptation to individual students, but teachers are still the bottleneck.
The internet has something to say about extremely high STR (1MM:1). In this regime, things flip and any teacher can teach every student: Teachers are no longer scarce, and so have to compete on giving the best instruction. Instruction quality increases as a result. And on the flip side, there's no student competition, which /maybe/ causes student quality to drop.
Not maybe. Absolutely. Even paid-for online courses have a relatively high drop out rate.
But that's okay. It's the price to pay to achieve the volume needed to pay for great instruction. I can take a music theory class from an instructor who would never waste their time teaching someone like me. Even though I may not get much more than entertainment value from it. I'm effectively subsidizing the students who do learn something concrete from the course.
There might be an argument to be made that pandering to an "edutainment" crowd might reduce the quality of instruction, but a good instructor should be able to find the right balance.
The latter regime is less a function of the ratio itself and more a function of the total number of accessible teachers, and the ability to switch teachers at will to find the one most suited to your learning style. If a university had 100 teachers all teaching the same course in the low or medium STR regimes and students were able to easily swap between teachers at will, the effect would be the same.
The ratio is really what matters more than absolute numbers; it tells us where the bottleneck is (supply or demand).
If the STR is 2:1 and you've got 1000 teachers, it means you've got 2000 students. If only 5% of teachers are good (see: sturgeon's law), you've still got competition centered on the student side, as 2k students fight to get into the classes with the 50 good teachers.
Ah, I get your point now. In my hypothetical example, you could still have those 2000 students enrolled with only the 50 good teachers. The overall STR would still be 2:1, but most of the teachers would have no students, so the effective STR would be 40:1.
The main thing distinguishing online education is the ability for students to all flock to the good teachers and completely abandon the crappy ones.
> Just imagine coding without access to StackOverflow.
That was the beginning of my career in the PC industry!
C compilers for PCs were in their infancy, so all of the code I was writing was x86 assembly using MASM 6 on MS-DOS 5.2 (hello TSRs and config.sys).
I forget the company, but some tech house published a giant 500 page PC encyclopedia every year that listed all of the x86 CPU instructions, IO ports, interrupts, DOS interrupts, etc. The last issue I had was white with pink lines on it and weighed about 3 kilos! Then the internet showed up and that all went away.
Well, except the mentors. Mentors will always be a step-function way to learn new material.
Turns out that mathematics pedagogy is poor, in general. Especially for geometry and vector calculus, where it's either obsolete, busted systems or incredibly abstruse ones.
Skipping over div/grad/curl and Gibbs-Heaviside vector calculus by going straight to differential forms and Clifford algebras (geometric calc) would save a bunch of heartache and pointless effort.
Linear algebra is essential since the whole point of differentiation is to construct linear approximations of functions...among other things.
It's not that there aren't massive lectures, tutorials are in addition to those.
Not bad value actually, despite what I said earlier. You do get to ask about whatever your mental block is, and the tutor is gonna know. But studying is time consuming, an hour is not as much time as it seems. You probably learn the most on your own, which nowadays ought to mean on the internet.
The answer is statistics: what is more likely, that the best explainer of a certain topic is within a group of people you have access to, or that person is somewhere else in the world?
This is very analogous to the problem industrial research groups face trying to answer a certain problem, e.g. ‘how do we ensure that our team is the most likely to solve a particular problem first?’
This is why start up acquisitions are so common even among the best funded tech companies.
Might also have something to do with the explanation size: this Poisson thing is one little thing. With the internet, it's perfectly acceptable to just do a blog post on one little thing.
Previously, if you were to write a textbook or teach a tutorial, you needed to teach a bunch of things.
So in the internet age there's a bunch of fine grained "best explanations" coming from a variety of authors that beats the best that one guy can do across a range of topics.
You are assuming that the best explanation can be given without interaction. I don’t have to be a great explainer but only have enough social skills to do an iterative search thru explanations, using my learning friends face and questions as metric, and find a very good explanation for that person at that time. If but only if they missed some but of linear algebra, i can sketch it out. And while
Many people learn well by listening and note taking, many others also
learn well by doing. We have documents and so on at work but when I see a good PR then I know the message is understood.
Start ups are better than large companies not because the people are so much smarter, but because the structure enables for so much more rapid learning and search of the solution space.
Agreed! It's one reason I'm bullish about our children learning and applying math better than us.
Btw, 3 Blue 1 Brown's videos on linear algebra [1] are similarly awesome and of course his video on Fourier is magnificent [2]. Another awesome math explanation is on game theory by Nicky case [3].
30 yrs ago the internet was still in its infancy and good engineering teachers were available to only a chosen few and closed out to the rest of the world.
I stayed curious and learning. Better and better teaching methods became accessible thanks to the maturing internet. I started to understand ideas in engineering much better.
Alas, there is no way I can express this progress in a resume but learning and understanding satisfies my curiosity.
>I could have used this 32(!) years ago when I was struggling in college.
I was struggling with this material more like 15 years ago, but same. I wish someone had explained the LaPlacian like this when I was in Multivariable calculus:
>Find me a function f where every value everywhere is the average of the values around it.
It's so simple and easy to grasp, yet provides so much insight into what's going on when you're doing the actual calculation behind the operation, but is so easy to lose sight of when you're overwhelmed with figuring out the 'mechanics' of it and everything else that was covered in the day's lesson plan.
Obviously depends on the school, and how much grade inflation (or deflation) they've gone through. When I studied Electrical Engineering, class average was 2.8 or so. The few students that actually had anything close to a 4.0 were terrific students.
I remember in our real analysis class, the professor started with a comment (to the class) in the lines of:
"This is a demanding class. Top performing students usually spend 25-35 hours a week on the problem sets alone, and top grades are rarely awarded - some years there are zero A's. Please take the weekend to consider if you really need or want to take this class."
FWIW, this was no top University - but then again, grade inflation is not that bad in STEM, from my experience.
Hmmm. I took a real analysis class at UNC Charlotte and one at Oxford and the latter one transformed my understand of maths. I felt if I ever taught a “maths for humanities” class Lebesgue integration and measure theory would be key components. I think if they are taught well by a person that really understands it backwards and forwards, well enough to explain it like a physicist, it is accessible.
That's not true at all. When someone drops 200 resumes on your desk and says, "Find 10 candidates to interview by tomorrow", you need some initial sort criteria.
The truth, my criteria were: #1: university, #2: GPA, #3: keywords. Sure, I was bitten a few times (I hired an MIT master's student who was utterly helpless), but over the course of years doing this, some patterns emerge, and high-GPA absolutely correlates with good candidates.
Sure, there might be a 2.0/4.0 who is a whiz, but sorry charlie, I'm not gonna picky your resume, so apply yourself or start your own company, because a low GPA means you don't give a shit or have some other problem.
I agree that GPA is useful as a binary indicator—if it’s abominably low, like a 2.0 in your example, it’s a red flag. But a 4.0 is no more predictive of being a good hire than, say, a 3.3. It’s also very school-dependent. Some places inflate grades a lot more than others.
In my own anecdotal experience, I omitted my GPA entirely from my CV when looking for jobs straight out of college and got interviews at every single place I applied. It’s not nearly as important as a lot of people think.
> But a 4.0 is no more predictive of being a good hire than, say, a 3.3
Ironically I graduated with a 3.4, and I did feel guilty for not passing on resumes with GPAs as low as mine. But as I said, when I had many other tasks to do for work, and then had to stop them all to sort resumes (we all took turns), it was hard to justify excursions when there were so many 4.0s. It is a sad truth that new college grads almost all look the same on paper...
For me at least, at a large public university with a highly-rated engineering department, it mostly came down to math courses being taught by TAs/new faculty who were not primarily interested in teaching.
When I had actual engineering dept courses on these topics (frequency domain comes to mind), I grok'ed it pretty quickly. But the math lectures were basically spent doing practice problems from the book, without any sort of insight or practical application.
The generation growing up today is the first that can broadly learn mathematics from mathematicians rather than from teachers, and we should encourage them to do so.
When I was 12-16, I was interested in physics enough that I would study it in my free time just for fun, solving problems from the Physics Olympiad. I could solve lots of the problems with just intuition, until I stumbled upon the sliding chain problem. I spent about a week pondering over it with no results. I approached my math teacher. He admitted defeat and referred me to the physics teacher. If I remember correctly, the physics teacher avoided admitting he didn't know how to solve it and didn't give me any pointers. That was the last Physics Olympiad problem I tried to solve, after dreaming of attending the competition for years (I should mention that at that time my primary interest already were computers; physics was just a hobby). I wonder whether this could still be a problem today when anything can be found on YouTube. There's a lot of noise, too. How long does it take the average curious kid to find the signal that is 3b1b, MIT OCW, etc?
It is certainly much easier. But if that kind of self study culture doesn't already exist in one's own school, either encouraged by teachers or classmates, it may take a much bigger leap to discover that material.
> you'd think colleges would bend over backwards to make sure these concepts are learned as succinctly as possible rather than add a curve to a test that makes a 23 out of 100 an "A" grade
School is not about learning. It's about money. Seating as many students as possible for as long as possible to suck in as much student loan money as possible. For this you need to be a prestigious school. Of course they want to boost their grades.
These are the people who have the gall to claim a moral high ground when they find and punish "cheaters".
What's a "test curve"? How does it make 23/100 be an A grade, is it some kind of CDF-based transformation that makes certain % of students pass the test?
Unlike in high school where the grade percentages are pre determined. ie a 90% is the cutoff for an A and it is expected that students will score that or higher, College professors see that as making the test too easy with the potential of having too many students get A's. Instead they will make the test longer or harder such that the best grade is 60/100 and the average grade is around 40. The will then scale things such that the top 10% get A, 20% B etc.
Sometime the prof screws up and the test is too hard where the aveage grade is 30. Sometimes some over-achieving git 'busts the curve' and scores a 95 and the next highest grade is a 60 - making it hard for the prof to justify giving A's to the top 10%
That sounds like what it is from my experience as well but it's really up to professor.
One exam was really hard such that your new grade was sqrt(#correct/#total). Another was to make new total the value of highest correct score, e.g. everybody in class got 0-20 out of 100, new total is 20.
Your 30 years of experience is why the article makes so much sense. If you saw this in college you wouldn't have also learned everything else about div and grad.
> these concepts are learned as succinctly
?
Succinctness is *why" you didn't learn multiple interpretations of everything.
> ? Succinctness is *why" you didn't learn multiple interpretations of everything.
Good catch. I think that was the wrong word because I was learning s-transforms in three classes sophomore year: differential calculus, linear systems, and thermodynamics. It was the opposite of succinct because the same concept was being thrown at me in three different classes from three different perspectives.
to throw one other, maybe-less-obvious, spin on it:
When you take a class, you see one wrong way to do things. When you teach a class, you see fifty wrong ways to do things.
Puzzling through all the different wrongs ways to think about a problem really helps cement the core ideas... And first-year calc students are masters of creating interesting-but-wrong interpretations.
It amazes me just how many key topics were so inaccessible to the majority of the class at engineering school. I base this on observations from group study sessions and the hyper-aggressive test curves.
I knew lots of people who never got the hang of div/grad/curl, or a Jacobians, or eignenvectors, or Z-transforms... These are key engineering concepts, you'd think colleges would bend over backwards to make sure these concepts are learned as succinctly as possible rather than add a curve to a test that makes a 23 out of 100 an "A" grade.
I'm digressing, and complaining, but the counter argument has always been: you're not supposed to learn everything in college, you're supposed to learn how to learn. Sure, right, but who has time to keep learning advanced calculus after college? (Well, I still study math & physics for fun, but over the course of decades, not years.) Not being able to see the world through these lenses I think means missing key engineering perspectives and relationships.
Anyway, very well written article.