Lessons I wish I had learned before teaching differential equations [pdf] (1997)
web.williams.edu
web.williams.edu
But then I happened to do some audio signal analysis, and when I saw a magnitude spectrum of a waveform, it was instantly obvious what's going on (and understanding the phase part after this was no problem). Such practical examples seem to be almost banned from university math, I'm guessing to make sure everything is very abstract and rigorous (e.g. you have to work with finite length and discretized signals where the maths don't strictly apply). And after getting this intuition the formal math started to make sense too.
But when one begins to teach, it becomes quite easy to see why things are like this. All of these things are so obvious to the teacher that it's hard to understand how one thinks before these are obvious and the standard notation/vocabulary is typically a good way to work with these, but only after understanding the stuff.
What I try to do is to probe out something that the student already knows, which may be from a totally different field, and find a simple example in the new topic so I can say that "this is exactly the same thing, but with this different notation/abstraction". This very often causes the things to "click" for them.
This is of course very hard to do with a textbook or a mass lecture. And probably the main thing why we need humans to do teaching instead of just giving out material.
It should be taught in linear algebra, where it does not only make sense, but is a non-trivial example of application that the textbooks have so little of.
I know that, for myself, revisiting Fourier Analysis after going through DFTs in my numerical analysis classes made a lot more sense, and I kinda wish I had started with that angle in the first place.
But for a lot of fields you just need the DFT and the calculus stuff can be mostly a distraction. How I finally figured out FT is something like the infinitesimal limit of DFT.
The calculus course is way too bloated for historical reasons that haven't mattered for more than a century. Pushing everything into that context only serves to make the contents hard to understand and seemingly useless.
I'm not exactly sure where FT should belong in the math syllabus. It's heavily related to trigonometry of course but it is an integral transform, so needs a bit of calculus. Although discrete versions are probably easier to grasp with just multiplications and sums, and there is quite rarely much actual integrating as in find-the-closed-form-antiderivative going on.
Maybe trying to have too "unified" syllabus for engineers/scientists is not the best way in practice. Instead there could be more discipline-specific teaching (something TFA hints to too). With ML (and 3D graphics earlier) something like this seems to be happening to linear algebra. The "mathy" linear algebra (that I studied at least) is mostly about properties and decompositions of (complex) matrices, but in ML/3D engineers very rarely use such things (but do need stuff outside "traditional" linear algebra like tensor products, projective coordinates and rotation groups).
The discrete transformation is quite fitting for finite-dimensional algebra, and honestly I can't understand how anybody every thought teaching the continuous transformation first and the discrete one never was a good idea. The only explanation is that since everything is thrown on the calculus package without any consideration, there's only time for one, so people kept the most general one.
I'd guess math uses continuous forms because it's where the mathematical tools are and many things tend to get simpler in mathematical sense when you let something go infinitesimal or infinite. This could maybe be different if digital computers would have been invented before calculus.
I've learned to appreciate that mathematicians think of maths quite differently to engineers or scientists. To them the interest is in the "mathematical objects" and their (provable) properties, not the applications or relationships to "the real world" (and this is probably a good thing in itself) and for engineers/scientists it's the opposite. Maybe something like how linguists vs novelists approach language.
I'm looking for a book that helps me understand ML algebra, so that when I read research papers I'm not just lost and nodding my head aimlessly. Such a book may not exist, maybe it's a group of books, but if ANYONE would have pointers in this direction, I would be in your debt
My degree was in physics, and I had worked in industry. I told him that I wished the math courses had included some engineering applications of differential equations. He looked at me with a straight face and said: "There are no engineering applications of differential equations."
TA'ing = being a TA
For background, I studied theoretical math. I am also amateur electronics engineer.
I agree, theoretical math students and staff are a bit dismissive about everything else. At least where I studied. Where I studied, we would regularly joke about other students claiming that math is hard saying, "what they are learning isn't even math, it is just a bunch of formulae and rote learning". And this included physics or applied math students. Don't even get started about CS students...
It is unfortunate, because math concepts were frequently invented to solve a real physical problem. Or they were invented and we found them extremely useful for a physical problem. Historically people would not put much distinction and would usually do both physics and math and there would be a lot of cross-pollination.
Just read a bit about how Einstein came up general relativity -- it is fascinating story about how Einstein wasn't good at math but needed a problem solved so he reached out to some friends and those friends basically gave him some private lessons until Einstein clicked and figured it out.
For me, I pretty much understood Fourier analysis before I started doing electronics. But the usefulness and practicality of it only clicked after I moved to higher frequency problems and started using frequency domain when working with my circuits. You don't get much significance of it just studying theoretical math.
I think this does go somewhat both ways. E.g. in the current discussion there are quite a few people complaining about "useless maths". I think a big problem is that different areas of studies don't appreciate that they are, well, different, and it's not trivial to transfer ideas across fields.
> Just read a bit about how Einstein came up general relativity
Also an interesting story about special relativity: Minkowski (Einstein's math professor) proposed very early on to Einstein that special relativity should be formulated geometrically (with what's now called Minkowski spaces), but Einstein deemed it too mathy and unintuitive, only to accept later that it was the right way to go. Without the geometric formulation it's very possible Einstein wouldn't have developed general relativity later on.
> But the usefulness and practicality of it only clicked after I moved to higher frequency problems and started using frequency domain when working with my circuits.
One argument for teaching FT more abstractly is that the "useful parts" are quite different in different fields. E.g. in many applications of sound and acoustics (and probably in EE) the main interest is in the spectrum. But in e.g. statistics the main thing is the convolution theorem and the spectrum has practically no use.
Explains DEs from scratch, physical significance, one or two traditional methods, then goes on to numerical methods.
If you ever want to learn DEs, I HIGHLY recommend this tutorial. This is short, too.
I have never found a resource where the concepts behind DEs are explained better.
This tutorial won't prepare you for a course, or tell you everything there is to know about DEs, of course.
Some areas of mathematics are applicable to physical realities, but they aren’t defined by those realities.
No problem with there being a rigorous math department. For mathematicians. But, quite literally, no one else cares. Mathematicians and everyone else are at crossed purposes, and the mathematicians are wrong to choose their purposes over their students' needs.
Uh... Not that I'm defending the previous poster. Just griping about rigor in general. Physical arguments for mathematical problems are rarely sound, and I'd assume that an undergraduate who thinks they have made a sound one is wrong until reading it to see.
For example the Italian school of algebraic geometry caused the waste of legitimately decades of work by many people because the foundations weren't right: https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge....
Part of the purpose of a maths degree is to teach the students the rigour required to be a mathematician. If you don't want to learn that rigour then thats completely fine, you can use physical or intuitive arguments, but the place to go and do that is in the physics or engineering departments.
I agree mathematicians would be wrong if they were forcing their rigour on physics students or whatever, but I think they're emphatically right to teach it to maths students.
The purpose of a published paper is to explain things to other humans, while that of a computer formalization is to explain things to computers. Just as it is generally not easy for a computer to understand a published proof, it is usually not easy for other mathematicians to understand the code you feed to Lean or whatever.
The stuff you need to focus on to get a computer to accept your proof is usually quite different to the stuff you want to focus on when explaining things to colleagues. Roughly speaking we care about why you're doing things, while the computer cares about the intricacies of what you're doing.
As someone who did an undergrad in mathematics, I have to disagree with you here. People think math is about numbers and computation, but that's like saying literature is about letters and composition. Fundamentally, mathematics is the study of things that are provably true. Without rigor it's not math.
> The idea that math is only about rigor isn't intrinsic; it's a historical accident that people seem to not realize is optional
I mean, it is by definition. To a mathematician, doing calculations is not mathematics, no more than spelling is doing poetry. Which is not to say that doing calculations is without value! I think what we have here is (ironically) an unrigorous definition.
I hope you realize that physics isn't about calculations either? You can put numbers or types in the equations, just like you can put numbers or types in the expressions or results you get in math.
Likewise, when schools focus only on teaching literature by the most artistic definition, at the cost of teaching them basic day-to-day reading and writing skills, the result is a lot of students who can do neither.
Let's treat rigorous math and lofty literature like the specialized skills that they are, and offer them to students who show particular interest in those areas. For the bulk of students, let's teach them skills that will be useful and relevant.
Why would you study such trueness if whether you know it is true or not does not have any added value in the physical world?
It's more a surprise that some rulesets map to physical world as well as they do [1].
For some there may be very important ones in the future like e.g. number theory got in cryptography. But for example just knowing whether P=?NP or if the Riemann hypothesis is true probably has no "physical reality" direct applications.
[1] https://en.m.wikipedia.org/wiki/The_Unreasonable_Effectivene...
But (integer) numbers and their behavior had huge significance in the ancient worldview, and still do even in our days if you look deep enough. And the Pythagoreans et al applied number theory in e.g. music.
The motivation of most non-applied mathematicians is aesthetic, much like the motivation of someone in the arts. It's done for its own sake ("mathematical beauty"), not because it has any practical application.
Mathematics might turn out to be "useful" more often than the average novel - but that doesn't mean the math was done with any use in mind. For example many developments in differential geometry made in the 19th century were inapplicable to anything physical for decades until Einstein applied them to create general relativity. Plenty of math becomes useful long after its development, and plenty more never finds any practical application.
I cannot overstate how destructive this is, because it basically presents math as this arbitrary logic game rather than as a fundamental language of the universe.
You are making a clear philosophical assumption yet don't realize it. Is math really the 'fundamental language of the universe'? or is it an arbitrary logic game that, in its most common interpretation, describes the universe well? Given that we have no full formulation of the universe in terms of mathematics (no theory of everything), the claim that mathematics as it is today is the 'fundamental language of the universe' seems laughably wrong.
On the other hand, there are thousands of formal mathematical systems that don't describe anything 'real' as far as we can tell. I mean, the entirety of the lambda calculus was developed before there were any computers that could realize any it in any physical system. Yet, its development has proven incredibly important in the entire foundations of mathematics (many theorem provers, etc are based on some variant of lambda calculus).
I'm not denying that some people learn calculus, algebra, etc (i.e., standard high-school/early college maths) with physical systems in mind. However, mathematics is much more than that and limiting the field only to that which we can sense is actually detrimental to mathematical progress as a whole. It seems best to me to encourage purely mathematical thinking itself in the hope that the arbitrary systems humans create may one day be used to describe something. History is littered with 'useless' subfields of math later becoming fundamental to the economy (number theory and elliptic curves to cryptography, lambda calculus to computation, group theory to quantum physics, etc)
A simple example is: nobody needs a proof of how differentiation or integration work. They need to know how to do it, think with it, apply it to problems, and generalize it. The proof is a tool for not being wrong, and that is a perfectly fine thing for mathematicians to do and a waste of everybody else's time (unless they really need convincing).
It is an absolute travesty how many people enjoy math until they get to their first college class which is taught by a disconnected mathematician that ruins the subject for them. At my school everyone stopped at multivariable because that's the level at which mathematicians systematically ruined math -- by teaching and testing the wrong stuff. Personally, I survived by the skin of my teeth, and then later found physics and actually learned multivariable calculus and learned to love math again. Despite the efforts of the people who were paid to teach it.
That's great. Not everyone does. Thinking simply via formal systems is not inherently inferior to 'doing' something with it. Both have their uses. In particular, as I said, quantum mechanics would not be where it's at today if it weren't for many pre-quantum mathematicians pushing around symbols on a page.
This "nobody" includes a lot of industry workers, e.g. those in finance and economics, not only "pure" mathematicians.
I studied physics in undergrad and the general method was, yes, it involved detailed understanding of all the theory, far better than anyone learned it in the relevant math class (differential equations, multivariable calculus, and linear algebra, in particular --- a 1-semester course in QM has to teach everyone linear algebra because the math course on the subject is garbage; a 1-semester course in E&M has to teach multivariable calculus also because the math course was garbage, etc.)
But yeah, absolutely you have to really learn e.g. group theory and algebraic number theory to do cryptography. I actually quit engineering and switched to physics precisely because I was so annoyed that an intro electrical engineering class used the Fourier transform as black magic without bothering with why it worked.
In particular many of the exact theorems that are used in formal math are totally irrelevant and historical accidents. Calculus could have been developed in a thousand different ways; we happened to pick one, the particular names of the theorems or what order the results are constructed in is, IMO, totally irrelevant to anything outside pure mathematics (particular examples: which construction of the integers you use; which construction of the real numbers you use; which definition of the derivative; which definition of the integral, stuff like that. In the case of integrals, yes, different constructions have different theoretical properties in terms of which functions can be integrated --- but, in any application, the answers intrinsically can't be affected based on your choice of definition, so the actual definition can't matter.
If your goal is teach math to the bulk of people, including those who will not go on to be mathematicians, it makes sense to tie math to something that the bulk of people can relate to. And most people DO enjoy thinking about physical objects and physical space (because we ourselves are physical beings who evolved to interact with our physical environment), so this is a really great starting point for introductory math for the average person.
If your goal is to only teach the subset of the population who prefer highly abstract puzzles, and to alienate all others, then our current methods are working fine I guess. But I don't think this is a good goal for general math education.
(I suppose that's where the terrible canard of "those who can't, teach" comes from. There's a kernel of truth to it, of course, but the more generous and relevant point is that success through effort is better preparation for teaching than success through natural ability alone.)
My guess is not that your teacher would disagree that math is "a fundamental language of the universe", just that she spoke it so fluently that she wasn't well able to relate to people who don't.
This was exactly my experience as well. I spent almost an entire first semester with a C average, mostly because I couldn't grasp what I was doing or why I was doing it. I still remember my "Eureka!!!" moment of sitting in the school library and working on a series of problems about water escaping a pool through an every-widening hole in the side. At that point, it all made sense, and I a) finished with mostly A's and b) was the only one in my class to score a 5 on the AP test later that year. I eventually went on to major in EE with a focus on Signals Processing (including grad school), so I always find it ironic that I went from not getting it all, to basically 8 straight years of nothing but Calculus.
This is where kids with an engineer parent have an advantage. Such a parent can offer examples of how some of the math is used in the real world. Once relevance is established it's OK to point out that this stuff is often buried in software but someone had to put that math into code form so we can all benefit from it. "You may never use it, but you use tools that use it under the hood" can be motivating when they're wondering if it's all just weird busy work.
However, my concern with your approach is that those examples help only if you are interested in physics (or whatever field those examples come from). I worked at a math tutoring institute for a few years and often saw that physics examples in textbooks like Stewart's confused a lot of students who were not that interested in physics. Not only do they have to learn the math, they now have to learn physics concepts just to understand the examples! There were plenty of students who could do differentiation/integration, but struggled at the questions involving physics.
Likewise, the department had a separate calculus course for people in finance, economics, etc. That textbook had applications in finance. All the tutors struggled to help those students because we had to learn basic finance concepts to understand the problems. And on the flip side, the students ended up useless at solving calculus problems on their own - but they could do the ones involving finance concepts.
As to your second point - yes, programs solve DE's numerically. I suppose it's still nice to know a bit about how they were solved in the 'olden days'?
DE theory is still useful for designing frameworks within which numerical methods can operate.
At least, that's my understanding.
The following is the complete solution in Lua:
function sprung_response(t,pos,vel,k,c,m)
local decay = c/2/m
local omega = math.sqrt(k/m)
local resid = decay*decay-omega*omega
local scale = math.sqrt(math.abs(resid))
local T1,T0 = t , 1
if resid<0 then
T1,T0 = math.sin( scale*t)/scale , math.cos( scale*t)
elseif resid>0 then
T1,T0 = math.sinh(scale*t)/scale , math.cosh(scale*t)
end
local dissipation = math.exp(-decay*t)
local evolved_pos = dissipation*( pos*(T0+T1*decay) + vel*( T1 ) )
local evolved_vel = dissipation*( pos*(-T1*omega^2) + vel*(T0-T1*decay) )
return evolved_pos , evolved_vel
end
It has been a long time since I did diff eq at work, and I agree with the PDF the more I knew the less I understood. I dont know why I need to have the math tainted by the unclean reality to understand, and if that hinders my understanding of them.
Ultimately it all comes down to choosing the most convenient basis functions for the questions you're answering.
If I'm understanding correctly, you want another = there for a standard damped spring: F = ma = -Cv - Kx.
(At least I assume this is what the original commenter meant!).
The linked article mentions differential forms, which is a perfect example. In engineering courses, these are often introduced without any rigor or formality. They just “appear” as a way of rewriting the equation. What is a “differential”? Who knows. Is there some kind of axiomatic basis for how these symbols can be manipulated in a consistent way? Who knows. But here are the steps to solve the equation; good luck on the test.
Another example is the quantum chemistry class I took (my first introduction to quantum mechanics). The professor mentioned something about how you can take a measurement and collapse the wave function of an electron; the probability of measuring its position at any location in space is non-zero. I followed up with some question about an experiment to confine the wave function collapse to certain regions of space which could be used to transfer information faster than the speed of light, which I thought wasn’t possible. My question was mostly met with a blank stare and something along the lines of “this course doesn’t cover that” before the professor moved on.
The most egregious example was a grad-level course on statistical mechanics. The professor mentioned something about the overall wave function of the system being a Slater determinant of individual wave functions. A student (not me) asked why we have wave functions for individual particles and a separate one for the overall system, and which was more “correct” fundamentally? The professor replied that the wave functions for the individual particles were more fundamental, and the overall wave function for the system was just an approximation. I replied, sure, the Slater determinant is a way of enforcing certain constraints that must be met for the whole system, but one of the defining features of quantum mechanics is that the function describing the state of the entire system is generally inseparable; without this we couldn’t have entanglement. The professor dismissed my response as incorrect and said students shouldn’t challenge professors on topics they don’t know anything about. This was a guy whose research career was built on dozens of computational chemistry papers that heavily utilized DFT by the way (and by “research”, I mean plug a file with atom coordinates and atom types into the DFT software, run the software, and publish the results).
> Is there some kind of axiomatic basis for how these symbols can be manipulated in a consistent way?
Yes.
> This was a guy whose research career was built on dozens of computational chemistry papers that heavily utilized DFT by the way (and by “research”, I mean plug a file with atom coordinates and atom types into the DFT software, run the software, and publish the results).
Again, large portions of academia simply stick to their small subfield, publish there, and don't think much about the broader implications, if they think about them at all. This is why so many discoveries are made by new entrants to a field with a slightly different background.
Lessons I wish I had learned before I started teaching differential equations [pdf] (1997) - https://news.ycombinator.com/item?id=32530035 - Aug 2022 (177 comments)
10 lessons I wish I had learned before I started teaching differential equations - https://news.ycombinator.com/item?id=19005798 - Jan 2019 (2 comments)
Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf] - https://news.ycombinator.com/item?id=15163979 - Sept 2017 (108 comments)
Ten lessons I wish I had learned before teaching differential equations (1997) [pdf] - https://news.ycombinator.com/item?id=11207183 - March 2016 (118 comments)
Yes! Jesus. People so often forget you can literally take variables and just fill in easy constants to get a sense of how they work. I can't believe this isn't, like, the first thing they teach in DE.
A few years ago I experienced a perfect example of this phenomenon in a glassblowing museum in Sandwich, Massachusetts. The museum takes visitors through the history of glassblowing in New England, mostly by static exhibits but also with live glassblowing demonstrations. At one point in the tour I read a placard that said something like "in 17XX, John Newenglishman invented a stamped glass method that allowed for much higher production rates of glass. The already-employed glassblowers resisted this development as it meant infringement on their industry." Then I looked up at the wall and read a quote from John Newenglishman that said something like "after demonstrating my stamped glass method, I hid in my room for weeks... gangs of blower-men hunted for me, no doubt to inform me by force their displeasure with my invention." I am paraphrasing in the extreme, but I was struck by how sanitized the placard was compared to the actual words of J.N. It reminded me of the history textbooks I used to read, which removed all of the flair and human-ness from the teaching of history.
If I could change one thing about modern education, I would somehow disabuse students of the notion that the development and archival of information has ever been "regimented" or "unbiased", and take great pains to expose them fully to the tremendous wit and wisdom of the past authors whose work they pore over. Anyway, great essay, saved.
(As a somewhat related point, I have also consistently found mathematicians and engineers to be much funnier than artists and authors, with the exception of artists and authors devoted to comedy. Somehow the tragic hilarity of writing math textbooks or working in a late stage engineering company never made its way into the material I was reading about math and engineering until I was out of my aerospace engineering degree by about three years.)
As a former teacher, I think the reason for this is the educational system as we know exists to (attempt to) have reproduceable learning outcomes, at scale. We try to transmit the hard-won knowledge of centuries of geniuses into the minds of adolescents. It's kind of a weird thing to do. It's actually surprisingly successful when I think of it that way, even though I think that it is not great at transmitting the actual insights behind the knowledge.
I suspect a better way to educate people is the more student-led, discovery driven approach. But that is both seemingly harder to scale and less deterministic. Instead, we play out the same cycle of boring, mildly effective education over and over.
Have you ever read Ivan Ilich? I find his style almost too iconoclastic for even my contrarian tastes, and I can't stand his sentence structure. But he makes some very coherent points about exactly what you're talking about. John Gatto is a former teacher who writes in a somewhat less brazen style but keeps a lot of the same opinions. It was actually on HackerNews where I first read "the seven-lesson schoolteacher".
I have long thought such would be easier to work with on modern computers.
Let e be an infinitesimal.
We write st(x) for the function dropping the infinitesimal part of a number.
Then f'(x) = st(1/e (f(x+e)-f(x)))
Now use the angle sum identity and cos(e) = 1 - e^2, sin(e) = e. I don't know how to justify these values other than the power series identities for sin and cos...
So st(1/e (sin(x+e)-sin(x))) = st(1/e (sin(x)cos(e)+cos(x)sin(e)-sin(x)) = st(1/e (sin(x)(1-e^2)+cos(x)e - sin(x))) = st(1/e (sin(x) - sin(x) - e^2 sin(x) + cos(x)e)) = st(e sin(x) + cos(x)) = cos(x)
Isn't that from the definition of cos and sin, even geometrically?
1 - x^2 < cos(x)^2 < 1 / (1 + x^2)
from ordinary trigonometry, although you can't just do this with nilsquare infinitesimals; you need a more sophisticated setup.
No wonder there is super-high barrier of entry. And that is why us mortals are just using numerical methods instead.
At the end of the day, most practitioners will work with just a bag of tricks, so if that is what the class teaches most of the time, I really don't mind at all. Yeah there is deeper theory to changes of variables and all that, but for an intro class that is aimed at a wide audience (physicists and any sort of applied math), then yeah some light theory and a bag of tricks is perfect. The students that need more will learn that theory later anyways in a second or even third class.
You don't have to sacrifice that much rigour either.
Not every class needs a narrative either. I agree that concrete examples to go with the abstract is more needed in mathematics, but differential equations isn't a victim of that.
I see other comments say that differential equations are victim to :
> why we are we learning this?
I don't think this really applies to differential equations at all to be honest. And people say the same complaints about learning scales in music or wtv. Just learn shit and figure out meaning later works too.
Reasonable people can probably disagree, and this might be biased by their personality and learning style.
I personally forgot every trick from my bags mere moments after walking out of my graduate exams. Later, when I got a job that required me to actually /do/ things, I (re-)learned the theory and tricks I needed for the task at hand. The courses that have had a real impact on me were not the "bag of tricks" ones, they were the ones that planted a small grain of understanding and/or appreciation in my brain. I think these are like seed crystals: They make later learning orders of magnitude more efficient and are far less tangible but far more important than "bags of tricks" learning.
That's exactly how I was taught differential equations 20 years ago. And as expected I have totally forgotten all of them. I do wish I'd gotten some deeper lessons from it. Even a concept or two that stuck with me would be better than remembering I learned a bunch of math tricks a long time ago that I can no longer do.
* Keep deriving circular functions until they cancel out. * Completing the square * Use the quadratic formula to solve degree-2 polynomials ... No formula for degree-5 or higher. * Use the Laplace transform here... for reasons...
Let me give you an example. Both Dart and Flutter borrow from Google's own lessons learned and innovations they did with the native Android OS and widget stack. In fact the testing stack of widgets, goldens, and re-using storyboarding as BDD come directly from the same practices on native Android.
In programming we have a legacy of borrowing from the language and frameworks that came before the current stack that is helpful to know about as everyone uses human knowledge and human language shortcuts to refer to past programming languages and frameworks.
The problem with this area of math is that you need a good basic grasp of geometry as that is how it was built and discovered, see this history chart to see why I state that from wikipedia:
https://en.wikipedia.org/wiki/Geometric_calculus#/media/File...
Of course I always found it easier to think and solve things in geometrical terms and methods. It drove one of my Calculus professors crazy, that and sleeping in class and yet still passing class with an A, getting As on all the tests.
It’s not the abstraction that gets me usually, it’s the lack of an explicit relationship between math and physics. I’m sorry but personally I am not satisfied by what I learned in my math and math/science classes on this. It started for me day 1 of high school physics and I’m now graduating in 1 month in undergrad applied mathematics.
I felt reinforced when I read that Plato clearly delineated the two “realms”. I was hopeful a single math teacher would discuss this idea for one lesson, but it never happened. Plato was a champion of math who recognized the division.
I’m not one of those disbelievers in complex numbers or anything like that, but if some model formulates precise physical scenarios, we should be able to have ample cross-over language. Even the explanation for why honeycombs have six sides: there is always some kind of fumble from going from the mathematical reason to the physical reason. It’s the paradigmatic example of mathematical explanations for physical phenomena, and between two heavily studied fields, and yet we fumble it continually. What is it about the mathematical hexagon which determines the physical honeycomb? These questions are at the height of philosophy of mathematics, e.g. Mark Colyvan, yet are still disputed and we act like it’s all so obvious. I get it works, but don’t tell me it’s not mysterious, because 6 years in now I’m not at all satisfied and have little confidence it will be mentioned in future math/science classes.
Anyhow, Aristotle had another take on the problem of universals, and taking his view that every concept abstract or otherwise can only exist as part of our physical world (I'm not a philosopher, and there are variations on both themes that different people subscribe to), then your supposition would be nonsense in itself. Stated as a question: What is your reasoning for preferring a Plato-like model of the world vs. a Aristotle-like one?
(If it needs to be said: This is not a gotcha or anything - your opinion intrigued me, and I'm curious as to your reasoning)
Another related notion is that there is no way to 'realize' an irrational 'number' -- sqrt(2) can be defined but we can never draw a length of sqrt(2).
> Another related notion is that there is no way to 'realize' an irrational 'number' -- sqrt(2) can be defined but we can never draw a length of sqrt(2).
Again, it's unclear what anyone means by things like 'sqrt(2)'. Why is drawing a thing of length sqrt(2) any different than drawing a thing of length 1? If I draw a thing of length 1 and say it's a line of length 1, then why is that different than my drawing the same line, claiming its length is sort(2) and then pointing out that it's actually now impossible to mark where 1 would appear along its length?
One is a map, and the other is the territory. Both 'real' in some sense but a map without the territory feels less 'grounded' (pun?).
> Again, it's unclear what anyone means by things like 'sqrt(2)'. Why is drawing a thing of length sqrt(2) any different than drawing a thing of length 1? If I draw a thing of length 1 and say it's a line of length 1, then why is that different than my drawing the same line, claiming its length is sort(2) and then pointing out that it's actually now impossible to mark where 1 would appear along its length?
Perhaps a more precise way to describe the situation is that one can define one or the other as a base 'unit' but you can never get one from the other (they are 'incommensurate', as the greeks would say). Irrational numbers can be defined but not 'realized' (or 'constructed') in the same way that the rationals can.
Once again, I'm not 100% sure what you're saying. If you have something that's of length 1, then you can easily construct the line with a ratio sqrt(2):1. Draw another line of length 1 (use compass and straightedge) at a 90 degree angle. Repeat 4 times until you have a square. Now draw the diagonal. You have successfully constructed the square root of 2 in your own setup.
There are numbers that cannot be constructed geometrically using only a compass and straightedge (e for example). However, again these are no less 'real'. Drawing lines is not the only measure of real. There are other methods. In computing, we often say a number is computable if you can define a function that, given any rational number can tell you if the number it represents is greater than or equal to the rational number. The square root of two and e and pi, etc are easily representable this way. There are some numbers that are not, and perhaps these can truly be said to not exist.
However, the field of computables is closed anyway, so it really doesn't matter if you don't want to believe the reals exist.
> There are some numbers that are not, and perhaps these can truly be said to not exist.
So then we have a real issue because the vast majority of the real line is composed of these uncomputable numbers which you've suggested don't exist.
As for the 'problem'... I personally don't view it that way. In my opinion (and it's just that, since there's no mathematical 'truth' here), I don't believe non-computable reals exist in any meaningful way. I believe this is similar to how we talk about a 'program that can check if another one halts'. Anyone can make that statement and claim that such a thing exists, but it's not at all clear that such a thing exists. But that's a lot different than saying there are X particles in the universe, thus the number X + 1 does not exist. Because x + 1 does exist and you can write a turing machine that can compute it to any precision (or a lambda calculus function that'll give you the next church encoded representation of it, etc).
My point is two fold. Firstly that there are certainly numbers that are greater than the total number of 'stuff' in the universe. Secondly, that there are some numbers that cannot be described in any meaningful way. These can be said to not exist (my belief), but others disagree.
10^241: I can't find the paper from years ago I thought I would be able to re-find (if it even exists). But, Seth Lloyd has written about there being a hard limit of 10^120 operations on 10^120 quantum bits as a max computational capacity of the entire universe up to today. Whether one just multiplies the two to get 10^240 or there was another paper I can no longer find where this is explicitly done, I'm not sure. The number stuck though. Since it uses plank areas and times, it is a maximum "quantity" for a human to say physically exists according to their best scientific theories. No other physical quantity in this modest metaphysics is larger than this. So we can and must get a little more creative in how we think math relates to the world since I can talk of 10^241. It represents the universe at max capacity since inception to today, what could be greater. (If you want to get really pedantic, we could say well a person thinking of 10^241 is physical too, and science is probably somewhat wrong, ergo 10^241 is physical as is any mathematical object we can think of. What about infinity? Does it take infinity of something to think of infinity? It couldn't as there aren't infinite physical anything at our disposal up to now according to science. So if it doesn't take infinite resources to think of infinity, where is infinity in the physical. We can't just say in our thoughts. This seems hard to argue).
Regarding Aristotle, is 10^241 in the physical world? How? I personally don't want to have to argue that side. Plato's division between physical and abstract (math) say, allows me to better understand math as its own thing, and leave the mystery to how it relates to the physical world as less of a scientific question a la Aristotle.
For something like Aristotle's potential infinities existing, I'm not very knowledgeable on how he envisioned they actually do exist. Maybe it can explain how to find vast quantities in the physical. Maybe. But then is 10^241 only potential?
Funnily enough I am not a platonist about math, nor however do I believe in psychologism which I would attach to Aristotle. I'm more of a fictionalist by default but always trying to challenge this. Plato I see as setting math free from physical conceptions, which is why I gravitate toward him vs. Aristotle.
What do you mean, reduce to the minimum first order equations? I went to the best technical uni in my country (a country known for quite some orthodoxy in teaching), most people would have had some grasp of differential equations before they left high school. Having the most basic equations revisited with actual rigor was eye-opening to whoever I've spoken to about it.
What do you mean, concepts not tricks? Is looking for a solution in a suitable Fourier representation a trick? Certainly is if you ask me. A mother of all tricks hinting at the spectral fabric of the universe.
Last but not least, he sounds like he has some sort of metric on how well his students apply his enlightened teaching later in life vs orthodox inefficient professors'. I have some doubts on this.
This has been something that always seems a bit magical to me, when someone comes up with an obscure change of variables that completely simplifies the problem. I'd love to learn this in a more structured/rigorous way. Does anyone have any recommendations for picking this up?
There are two important concepts that can be generalized: - Substitution, the method of renaming and replacing variables - Abstraction, replacing a variable with another term (function) that can also serve as output
This should give you a more natural feel to concepts like parametrization or what it means that the solution to one function may be another function. Perhaps you might even look a bit into the simply typed lambda calculus which has given me yet more perspectives on the fundamentals of mathematics.
If you want to be able to reproduce every clever mathematical trick of the last 200 years you probably can (I believe in you!). But everyone short of Ramanujan learns them by studying many problems, repeated exposure, and either systematically or organically memorizing them. Afterwards, using them in a problem is more about pattern recognition. This is my experience as a physics phd-master-out ymmv
Secondly, a related problem is that a lot of "classical" mathematics was born as an answer to the big questions of the day, and the way it's taught is often just the answer, without any context about the question, and less still about why anyone would care about the question. For example, there's basic logic courses that show 0th-order logic is complete and sound w.r.t the usual rules, but don't give any intuition that this links back to the Entscheidungsproblem and Goedel's theorem and fundamental questions about the nature of the world and philosophy - the mathematical community cared about this at the time! It would be like teaching Turing's halting problem just by giving a definition, a proof and perhaps an example without any idea what question Turing (and Church) were trying to answer in the first place, or the implications for proof-checkers and code-verifiers these days (Coq, TLA+ and so on all exist so Turing's negative result doesn't imply that all is lost).
For differential equations in particular, Cauchy, Sturm-Liouville and others mentioned were all answering (or trying to answer) particular questions that the mathematical community at the time cared about, and without this context the subject can indeed seem dry. I agree with G.-C. Rota (the author of the PDF) that these are not necessarily questions that today's engineers care about, and so yes we can mention that uniqueness theorems exist and move on to linear systems with constant coefficients. If you can't find space to teach and motivate the question, then sure, you don't need to teach the answer either.
As for "functions" that have integrals but no graphs, I also don't like avoiding them by mumbling something about different types of integrals and measures, as if these objects were some kind of organs for sexual reproduction that we can't name because Queen Victoria would blush. No, an ordinary function is something with signature A -> B, and a density on A is a quite ordinary function f with signature S(A) -> B where S is a sigma-algebra that is a subset of the powerset of A and f has certain properties. Usually, but not always, evaluating a density at a set with a single element just gives you 0 and is not very helpful. Measure theory is useful because we can work out certain properties of these functions without always falling back to the powerset definition - that is we can pretend they are a kind of function A -> B in those cases where it does something useful and doesn't break anything else, as long as we understand what's really going on. Case in point: a distribution on the reals that is 1/2 the uniform distribution on [0, 1] plus 1/2 probability mass on the point 2. The probability of any measurable set X is 1/2 the intersection of X and [0, 1], plus 1/2 if 2 is an element of X. There is no mysterious delta function whose value at 2 is exactly "one half infinity" or anything like that, it's just a function from sets to sets where one particular set with one element happens to have a nonzero value. Once you can do this, you are allowed to write 1/2 * U_[0,1] + 1/2 * \delta_2 if you want to.
I suppose he could argue that they could wait for a second course, perhaps.
And yeah, physics students need to be able to solve QM and EM equations, can't get away from (some) special functions.
I see a lot of parallels with the Physics department and I think the reason is much more depressing. Both fields embrace a sort of masochism and active desire to keep knowledge impenetrable b/c it acts as a mechanism to feed out the dummies. The system acts an an informal IQ test - that maintains the prestige of the departments. If you're pigheaded and clever enough to get through the masochistic torture is that their undergrad textbooks then you're probably pretty clever and so the prestige of the degrees is maintained.
There is also a certain veneration of the establishment and traditions. I remember on my first day of Classical Mechanics the 50+ year old teacher beamed with pride when he told us he used the same exact textbook when he was in school. As if it was written by the Shakespeare of textbook writing and nobody has managed to surpass it in decades. It' frankly should be an embarrassment
As he observed, other departments will step in an do much better. The best linear algebra class I had was a graduate course in the electrical engineering department
I had a lot of hope for things like Khan Academy, but the issue is video is not text and it's hard to iterate and improve on. I really wish textbooks with open licenses would take over and they could be reworked and improved year after year by different people
When the entire video is generated by an AI, that dynamic changes. It's no issue to improve "the script", and have it seamlessly regenerated. We can't be that far away from this capability, and it could have a profound impact on available learning resources.
Can imagine a world where it doesn't matter where or how you get your education, all that matters is that you can have your knowledge verified by an accredited _testing_ institution. This would open education up to a world of creative competition for students, and allow individuals to find a learning paradigm that best works for themselves.
Some people are there for "the experience", no matter how much it costs, nor how little benefit it results in. They will still be paying $80k for an experience that qualifies them for nothing, even though they had that qualification already before they started.
Some other people are there for the network, and/or will learn leadership skills in that environment. They can't build/learn those things in an online environment.
How are you ever going to learn anything truly novel in the world if you always delegate teaching to a teacher?
they are there to give you a chance
maybe it is impossible for you to do this, but it’s clearly not impossible for others
that challenge is valuable information
maybe a different area is a better pursuit for you
or maybe a better way to view learning
good luck
Taking a longer time than normal to learn a topic and prepare university exams might be a good alternative to burnout; or maybe your lifestyle and/or intellect are incompatible with studying.
Why such a desire to be spoonfed, whining about having to put in the work?
Almost none of the work is 'rewriting the book.'
The mid-game is personalised, conversational, interactive tutoring.
The end-game will be deciding how much humans need to know about any of this, because AI will be far better at 99% of it.
If this seems unlikely, consider that Cauchy, Gauss, Laplace, and so on were all super-elite 0.001%-ers in terms of ability.
No one knows whether AI will eventually have the creative ability to work at their level of insight.
Personalised training could select for prodigies, and - whatever we decide to do with them - we'd have more super-talents than we do now.
Currently a lot of talent is wasted because of political and economic inefficiencies.
There will likely be video. Visual presentations can be an important part of learning. Perhaps it will be generated JIT, but likely AOT compilation of video will represent a useful savings in processing power. Students can always "put their hand up" to interrupt a presentation for a little one-on-one time with professor AI.
The book was probably okay-ish, the point is that surely someone could have made some improvements in the past few decades
Perusing online resources for Differential Equations shows that there doesn't seem to be much agreement on how to define the core subject, and there is no grounding on 'how we got here' to provide a path forward to people just starting out. My experience so far, at least.
So true, in many areas. I know a couple of young people that got destroyed by this kind of meat grinding machinery. Typically applies to every medical studies in France, where many kids (with parents that have the money to pay) are going to Spain, Romania or Belgium to study in a less oppressing environment and better teaching methods.
Honestly your problem was that you didn't know any Classical Mechanics yet and you were assuming that the volume of recent developments made old books obsolete. Maybe in Biology, in Physics getting to recent developments would mean that you're familiar with Goldstein, Landau's Vol. I... Abraham-Marsden? Arnold? Those are old.
Often newer editions actually worsen textbooks and then only a few contemporary books become references in the long run. It's always been like this, there's tons of great books from the 70s that aren't used today and could definitely do. At least they're not ~1,000 pp. of waffle, which is what you usually get for your first textbook on anything nowadays.
Have you looked at for instance Khan Academy's Grant Sanderson (aka 3Blue1Brown) Math videos? it's really apparent there is a LOT of room for improvement in pedagogy.
As the linked PDF illustrates, most people are teaching along a set formula and sequence of concepts. Good teachers will try to tweak and iterate on these formulas and evolve a better curriculum that sinks in better for students.
Naturally as time goes on, if each author has to start from scratch, then it becomes harder and harder to beat "the best book on BLAH" from the last 100 years. (Though I refuse to believe it's a monumental task to write a better textbook than Rudin)
If you have open copy-left books, then in theory people could start with a Rudin, fork it, tweak it and improve it. 70 years of improvement could yield some amazing forks!
"Often newer editions actually worsen textbooks"
That's typically because they select a random new author to in-effect update their copyright date.. and the new author is rarely of the same caliber as the first
I have. I went through Khan Academy, Brilliant and 3Blue1Brown. After spending more than 100s of hours I started getting the feeling that these are all good for elementary level math.
But for any serious math (think real analysis, complex analysis, group theory and beyond), all these platforms did was leave me with a warm fuzzy feeling of having learned something cool but in reality that warm fuzzy feeling was not good enough for solving actual exercises that come in textbooks or really deeply understand the material.
I've given up on these online learning media. Back to textbooks. The difference is like night and day.
Critically, you can read/listen to something and come away with the false impression you understand it. Sitting down and doing problem is .. not always fun.. but can be critical for the concepts to sink in. I think this is the main point of what you're saying
I could see in the future it being something like watching a video and then doing a programming exercise
If you want anything past Analysis 1 I think you'll find that universities guard their content.
Not so; there's an absolutely vast amount of freely available undergraduate mathematics resources available at all levels. Honestly, so much that it makes it confusing to choose and not get distracted by the options -- perhaps AI-mediated distillation could be helpful in the future.
I wanted to find good analysis video lectures from a real university complete with problem sets, homeworks and their solutions. I couldn’t. I think MIT OCW now has one analysis course like that, but it’s relatively “recent”.
http://therisingsea.org/post/mast30026/
They have videos as well as everything else. I'd love to study them with someone/some group of people one day.
But, you don't need videos if there are carefully-written course notes PDFs.
Try Oxford: https://courses.maths.ox.ac.uk/course/index.php
E.g. two random Analysis-related courses (second more advanced than first)
https://courses.maths.ox.ac.uk/course/view.php?id=65
https://courses.maths.ox.ac.uk/course/view.php?id=4988
And there are tons of others, but with videos is a bit harder.
Berkeley exam papers with solutions: https://tbp.berkeley.edu/courses/math/113/
Hmm. All the way back to when I was in college there was advanced content available from the Open University. You had to be awake at 2am and it was in black and white, but it was there.
Are there people who think this is an "either/or" choice, as opposed to a "use both" thing?? I ask, because it's pretty well established that learning is enhanced by use of multiple media types and it seems self-evident to me that books and videos are complementary.
Can't speak for others but for me it is more about efficient utilization of time rather than complementing multiple learning methods.
I've found that time spent in learning math from videos have poor return of investment. That time is better spent re-reading a chapter or that thing that I couldn't fully understand the first time and doing more exercises.
On the other hand, there's for instance Optics where you basically have to condense an encyclopaedia and there's always prettier pictures. Or Thermodynamics, Fluid Mechanics etc that can be taught in different ways depending on the curriculum.
There definitely should be pedagogical considerations in higher education, that's lacking because it's usually an afterthought. And it also should be very clear to people getting into higher education that at some later point pedagogy must end and you have to be capable of working your way through the material.
To my mind, if the textbook was actually excellent then that would be 80%+. We're nowhere near there. I think there is LOT of room for improvement
But sure.. Thermodynamics.. things could be worse :)
Sometimes things are just hard because they're complicated and you need to buckle down and learn your multiplication tables. But at least in my own life experience, the vast majority of the time things are a problem because their poorly explained - often by people that poorly understand it themselves.
Once you truly understand something inside and out - and look back on it - it all generally looks relatively simple. But it takes a special talent to be able to go back and reexplain it from the naiive perspective
That's probably true.
> To my mind, if the textbook was actually excellent then that would be 80%+. We're nowhere near there. I think there is LOT of room for improvement
In my view, that's probably false. I don't think the problem is masochism, gatekeeping, and people holding on to old textbooks. I think the problem is that classical mechanics is actually hard, at least for most people. If you come in to beginning classical mechanics wanting to have learned it, rather than wanting to learn it, no textbook can save you. And I think that many people come in that way. They want it out of the way as a prerequisite for something else, rather than really wanting to know it for itself.
I think you overestimate the capabilities of students entering university (even 20 years ago), and underestimate how poor high schools can be in preparing said students.
I went to a mediocre university. A 50-80% drop out rate was there for both physics and EE - I don't know how it compares to the other engineering. And I did not even consider it challenging. Almost all the classes were a breeze for someone like me who was well prepared going in. At least in that EE department, the teachers were very dedicated to teaching. They would allocate 3-9 hours a week for office hours, and the pace they taught as was slow (probably only covered 70-80% of the material that is covered in a top university).
Students were given lots of chances.
The reasons they drop out are:
- Poor preparation at the high school level
- Poor discipline. A lot of students didn't transition well to independence, and didn't have an authority figure (e.g. parent) controlling their schedule.
- Realizing too late what it means when courses are built on top of other courses. Thus you'd have people getting an A in Calculus I, but almost failing Calculus III because they didn't realize they needed Calculus I beyond the course.
- In high school you can get far with a cursory understanding of the material. At university, you could get a B, or even an A, with that approach for introductory courses, but that approach will start trending towards an F in junior/senior level courses.
Sure, I agree with you that pedagogy can be improved, but I expect that 80% would at best become 60% if all you focus on is pedagogy.
You can't realistically expect that there will always be someone up the ladder to explain things to you. I mean, who explains stuff to the professors if it worked like that?
Even at the research level we are not independent islands of learning and discovery. People collaborate, some pickup certain concepts better than others and vice versa. So we teach and aid each other.
It seems you're firmly against this notion? Or if not please clarify your position?
I think everyone should be capable of working alone as well, and that has been the general assumption around as far as I've noticed. Of course collaboration is usually way more productive and also unavoidable.
But we were talking about education. Theses are individual for a reason.
Yes, you have to shed the expectation that others will teach you, I agree with that. In the end, people slogged through by doing a bunch of reading from various sources. It is maybe the main lesson of university for everyone: you're not in high school anymore, you won't just learn whatever the guy says while talking to you. It's quite the shock if you had actually good teachers at school.
The thing is though, you can still demand good teaching materials. Textbooks have to explain things in the clearest way possible. They shouldn't be confusing, especially considering they end up being the main source for just about everything. In this modern world where there are online lectures and textbooks, there's no reason we can't all have the very best explanations of every relevant concept. Yes, of course as a student you still have to put in the time, but the materials ought to be the very best.
I strongly disagree. All the best math and CS books I have are old.
New books about old topics tend to be less informed about the context and core ideas that led to their development.
There is a study showing that you actually understand material better, if you use the most primitive methods: chalkboard and a lecture. Because you are forced to visualize the material yourself, instead of being presented with a ready-made animation.
It probably makes sense to use visual aids for students that just can't grasp the concept, but I believe this will only help in elementary math.
2. When I struggle with books it is because they do not present the motivation behind what they are doing. Videos and "more popular" articles can both provide the big-picture motivation and overview. Sometimes, you have to construct a motivation for yourself, based on what you read. That's hard. Maybe you even invent something new in order to understand a concept better. This approach is slow, though. It's easier if someone explains to you why a certain concept is "hard" or a point of view from which the concept is "easy".
3. I think students who build on a partial understanding are not going to have a better time with videos. They are in greater need of learning how to learn something than they are of facts, but school does not teach that skill (afaik).
And do the homework problems. You'll never understand the material without doing the problem sets.
The author knew that this book was for people who might be as involved in the business of radio as its science or engineering, so they wrote as much about the application across every industry, breaking down the systems to the component level and manufacturers, deployment, and ordering, as they did the design and theory.
I learned how to evaluate a textbook from its structure and style. It certainly wasn't designed for discrete lessons, and the professor would have needed a diverse and practical understanding to teach it effectively.
I wonder if the same thing isn't happening with computer science. When I started studying the topic in the late 80's, I was part of the earliest generation that actually did, and everything seemed to be explicitly written with the goal of making sense. Some things (like recursion and pointers) were fundamentally complicated, but they were made as simple as they reasonably could be.
My son is studying computer science in college right now and I look at the way they present the material and it often seems designed to confuse - I'll read it over and then explain it to him the way _I_ was taught it and he'll say, "oh my gosh, why don't they explain it that way?"
Take simple arithmetic like 12x17. Some people do the long form multiplication (carry the one..), some people say it's 12x10+12x7. Some remember 12x12 from times tables and go 12x12+12x5. Some people make it 24x8+12 => 48x4+12 => 50x4-8+12 etc. Some do it on the abacus in their heads.
All valid, though some are slightly more optimal than others. Good teachers empower alternative solutions and try to help people connect what they already know to what they already understand.
Oh, and 12×17 = 10×17 + 2×17 = 10×17 + 2×10 + 2×7 = 170 + 20 + 14 = 204.
It's funny that for the 19th century and the first few decades of the 20th one physicists were so eager to simplify and generalize their knowledge. With the side-effect of learning quite a few surprising things from the work. And yet for almost a century the goal is explicitly the opposite. (It's almost like if Academia is in crisis...)
But the proof of failure is instructive :)
(serious reply, if it's insufficient for a freshman course i propose following up with Feynman https://www.amazon.com/Feynman-Lectures-Physics-boxed-set/dp..., any objections?)
[1] https://archive.org/details/physics-for-mathematicians-mecha...
It is true that just because a book in newer it is not necessarily pedagogically better
It is also true that a poor selection of content or understanding by the author could doom a book even with better pedagogy.
All that said, I love the idea of OSS books/exercises for teaching - I don't know if a sufficiently engaged and competent (domain + pedagogy) would evolve around and/all of them, but it'd be a fine experiment to try!
It would also be great training material for LLMs to help them to tutor using more thoughtful metaphors and examples.
Besides pedagogy, in college you have to respect your students as studying adults and give them a proper bibliography, emphasizing references for independent study if they don't like your lecture notes, nor your approach, nor whatever.
I understand what a university is, I also understand and am qualified in secondary education and it would be incredibly depressing turning colleges into extended high schools because of business models. That would be exploiting students, I never agree with that.
What I'm talking about is expanding options to meet additional education needs. Since universities are a shared resource, any solution must be carefully designed to preserve the ability to continue providing existing services. That's difficult to achieve, so I understand the obstructionist response.
All I'm saying is that if you position yourself as an obstructionist, don't be surprised when you're treated like one.
The thing with giving the public what they want and being too much of a pragmatist is that we've seen it before.
Consider Western universities in the 17th century, they were still there churning out degrees, but modern science, mathematics and technology developed elsewhere.
You're right to protect your existing solution against regressions, and there's value in revisiting old topics in the new discussions, but you're not going to constructively contribute much if you're unwilling to engage with why so many people feel the need for something different in the first place.
Interesting to note we haven’t got a text book for these classes just lecture notes and a number of text books recommended if we want additional presentations.
I think people who are frustrated they can't grok diff eq in a n-week course are likewise deluded. There is such a deeper meaning those symbols on the page, and I suspect there is simply no way to spoon feed it.
In a twist of irony anyone could have predicted: this selects for the middle of the bell curve, and not the right-hand tail.
I feel as though there are a set of personality qualities that must not be present in order to achieve anything worthwhile. Wasting time, frivolous activities, and fostering an outsized ego are just a few.
Also, anybody with "street-smarts" can see the cost-benefit ratio and perceive the benefit is kept much smaller than some other, easier subjects, while the cost is artificially inflated.
What it does select is people that really love the subject. And is at least a little bit smarter than average.
I have a PhD in particle physics, and I've been a researcher at CERN with colleagues from multiple countries.
And I cannot understand what you are saying there.
> I have a PhD in particle physics, and I've been a researcher at CERN with colleagues from multiple countries. And I cannot understand what you are saying there.
Hrm.
(A foolish career decision in retrospect.)
I think they are saying that if you were clever enough to do physics at that level, you wouldn't understand the problem being discussed. So when you said you don't understand, you gave a live demonstration.
When you emphasised your physics credentials as if to say the above didn't make sense even to a physics-smart and well-credentialed person, that pattern-matched even more strongly with the idea that those who go far through higher education institutes don't relate to the problem so don't tend to work on improvements.
Getting my PhD was not easy for me, as it wasn't for the most people I met, because none of us is a so-called genius. Rather, all of us had to put a lot of effort and discipline in order to move on.
And never did we met any artificial obstacles or deliberate difficulties. Physics is hard; it is as hard as many other academic fields. You need focus, you need discipline, and yes, you need passion to be able to keep focus and discipline.
But you will be welcome if you try, and you will receive a lot of help. Maybe you will find out it was not the right choice for you and you will change your target, but you will not be screened out and rejected just because.
You need people from outside the echo chamber. I found The Open University to be a considerably better education provider on that basis than the red brick I attended quite frankly. The material and tuition is far far far better.
Incidentally your point in Khan Academy is spot on. That's basically OU but the material is miles better and it actually leads to a qualification.
Example free course: https://www.open.edu/openlearn/science-maths-technology/intr...
I have been a on the mathematics faculty, including at some decent places, for almost 20 years. I have never met a single university mathematics teacher who thought this way. To the contrary, we are delighted when someone shows even a spark of interest or aptitude (hopefully both). Granted there are high bars for reaching professional competence, but that's intrinsic to the subject—and there is a welcoming place, in math probably more than in any physical science, for amateurs as well as professionals.
> As he observed, other departments will step in an do much better. The best linear algebra class I had was a graduate course in the electrical engineering department
It's worth noting that this isn't necessarily because EEs are better teachers—although of course in any particular case they might be—but because they can give you a course more focused on your interests. Math departments wind up teaching many courses populated largely, if not entirely, by non-math majors, and we cannot be discipline experts in every field of application in which students might be interested—nor, even if we were, could we simultaneously teach one course in a way that appealed simultaneously and particularly to the diverse applications needed by every student.
I don't know how it is in the Math department, but in Physics there is almost a sort of hazing that goes on, where some subjects are just known for how grueling they are and how you just have to go through it
"To the contrary, we are delighted when someone shows even a spark of interest or aptitude" Not to read too much into it, but this kinda hints at the problem. You're delighted at the students that have passed your IQ test. There is generally very little care given to the 70%+ of students that aren't making the cut. The true horror is how many students are in the class and not getting it. And the teachers are not freaking out
From my university experience (which was a while ago) it was abundantly clear that most students hate their math/physics classes, were bored out of their skulls and the teachers are only interested in the engaged students that are getting it.
What you should be "delighted" by is when you find a new way to explain something that resonates with most of the class
That's because happy students are all alike, but every unhappy student is unhappy in their own way. It's just not feasible to care in depth about why any one of the 70%+ is not learning effectively. They're probably missing some prior topic that's effectively a prereq for the class, but that's something that should be addressed by the student themselves.
You are right, and I am! I don't think that's incompatible with also being delighted when someone shows interest or aptitude. In fact, the synergy is the best part, when someone shows interest or aptitude because they are willing to put in the work to follow the pathway that I have tried to open up for them.
Professors of undergraduates don't seem to think from the undergraduate perspective. Most undergrads have only ever known school, and are just following directions while fumbling their way to their first interview. From that perspective it doesn't make any sense why some classes have to be immensely difficult and high-stakes, and others can be a little easier. From the graduated perspective, you can see "the big well-intentioned lie"--in truth, no field can be condensed into 16 weeks, even if you're studying it and nothing else 14 hours per day. The more difficult a class is, the closer it is to the truth that the class itself is a carefully structured playground, and the real field is more dizzyingly wide and complex than any student can imagine. However, I don't see why this lesson has to be so painful to students.
While there's no excuse for it, I think faculty see so much apathy on such a regular basis that sometimes it's easy to mistake sincere struggle for a lack of desire to engage. For precisely that reason, I try very hard to recognize and reward my students who are willing to take the time to work with me, whatever their existing comfort or proficiency level with the subject is, but I know that there are times (probably many more than I'd like to imagine) when I don't rise to that ideal.
> Professors of undergraduates don't seem to think from the undergraduate perspective. … However, I don't see why this … has to be so painful to students.
I think you might underestimate how painful it is to faculty, too! Most of us are in this for the love of our subject, and many are even in it for the love of teaching, and it's painful to have something that's so beautiful and beloved for us be the cause of suffering in others.
Anyway—while I completely agree that it would be better if the learning experience could be the joy it should be, and free of the pain that it often carries, I do wonder if some of this might be intrinsic. There are certain topics that are just difficult, and on their first encounter with which most learners will find themselves lost and confused. But those topics have to be, or are believed to have to be, understood to be successful in the field, so that loss and confusion will have to be felt some time. Isn't it better if that's in the classroom rather than on the job?
> The more difficult a class is, the closer it is to the truth that the class itself is a carefully structured playground, and the real field is more dizzyingly wide and complex than any student can imagine.
This is a beautiful description. I know as a teacher that I try to communicate this to my students, but also that I surely fail much more often than I succeed.
Thanks. There's an element of hopelessness in trying to explain the immensity of human knowledge to someone who's lived their entire life captured by mandatory schooling. It's just not an interesting thought to most of them. Their worldview has been so artificially limited that attempts to explain the limitations of their circumstances just appear to be more limitations. "Guys, there are more than six hundred thousand mathematicians working right now in the US, and they're working with concepts first thought of as long ago as 3000 BC and maybe earlier!" "Okay, will that be on the test?"
> Isn't it better if that's in the classroom rather than on the job?
If only we were having this conversation in a bar instead of a text forum. That's a huge question and it's clear you're actually interested in talking about your thoughts on it. It's also a subject that interests me.
When you say:
> But those topics have to be, or are believed to have to be, understood to be successful in the field,
I think that's where the big disconnect comes from. What's the point of school? Is it to know enough to be useful at a job, or to plumb the depths of knowledge? Is it some third thing? Ask any recruiter and they'll tell you the new hires aren't prepared to actually do anything useful, and ask any advisor and they'll tell you the new graduate students aren't prepared to actually do anything useful, so we can at least conclude that there's some kind of disconnect going on. Students spend thousands of hours and hundreds of thousands of dollars doing something that does not actually adequately prepare them for what people want them for.
I have a hundred different ideas about how to address this. One thought I've been mulling over recently is that a redefinition of grades is in order. From essay that we're commenting on:
"My colleague’s error consisted of believing that the more testable the material, the more teachable it is. A wider spread of performance in the problem sets and in the quizzes makes the assignment of grades “more objective.” The course is turned into a game of skill, where manipulative ability outweighs understanding...
In an elementary course in differential equations, students should learn a few basic concepts that they will remember for the rest of their lives, such as the universal occurrence of the exponential function, stability, the relationship between trajectories and integrals of systems, phase plane analysis, the manipulation of the Laplace transform, perhaps even the fascinating relationship between partial fraction decompositions and convolutions via Laplace transforms. Who cares whether the students become skilled at working out tricky problems? What matters is their getting a feeling for the importance of the subject, their coming out of the course with the conviction of the inevitability of differential equations, and with enhanced faith in the power of mathematics. These objectives are better achieved by stretching the students’ minds to the utmost limits of cultural breadth of which they are capable, and by pitching the material at a level that is just a little higher than they can reach."
At the undergraduate/introductory level, the only important question is "what awareness do you have of the breadth of this field, and what mastery do you have over the concepts that most agree are its "most fundamental"? You and I both agree that class is a "playground", and any "grade" is in fact meaningless. Rather than assigning As, Bs, Cs etc. as a percentile of subjective completion of arbitrary problem sets, As-Fs should be assigned at the discretion of the instructor as a holistic assessment of oral and written examination, completion of problems and problem sets, participation in the class, and general wisdom.
Students would of course resist this. They want to be graded on impartial, meaningless criteria. That's how they're taught from third grade, and it's the method that allows for the least possible interaction with the material. The only reason they want these grading criteria is so they can plan to spend as little time as possible on the class. This method of approaching learning simply has to be broken at every level of education. You shouldn't even have a GPA until college.
I'm not completely sure I agree that this is the solution—if we're re-inventing grades anyway, then I'd like to do something more radical than using the same old A–F and just interpreting them differently (although, even if given free rein to do whatever I liked, I don't know what I would do!)—but I definitely agree that grades, and the standard approach to them, are the most pernicious part of "education" (in the sense of the current schooling system). If there were any way to get away with it, then I would be happy to—indeed, I would prefer to—have all evaluative exercises be diagnostic and informative, only for the students' benefit, and to assign no grade at all, or an A for everyone; but this seems incompatible with a modern university structure (and anyway is essentially forbidden by university administration).
I believe that there is a negative feedback loop in the current model of schooling: unprepared students produce jaded instructors produce unprepared students. The big problem with the current lecture method is that any interruption of the momentum of the course for the students' own personal benefit comes with a social cost, so it's better to just shut up and pretend you know what's going on. Furthermore, many lectures build on themselves, so if you misunderstand a concept at minute 3, by minute 20 you're checked out and by minute 50 you're clock-watching. That's why so many math lectures are silent with the exception of the occasional interjection from a star student.
Combine this with the fact that most students are insufficiently prepared for course material in the first place and you end up with modern STEM college: kids who don't get the material slogging through piles of completion-based assignments and exams and putting in the minimum amount of work to get the degree, after which most get the exact same job they'd have gotten if they'd worked harder anyway. They're almost incentivized against deep examination of any one topic, because all time spent working on one assignment incurs a cost against other assignments, or against leisure.
There are so many issues with the way education works at scale in the first world that going down any pathway would take a thousand words, so I'll sum up by saying that I believe that you have a point, and I also believe that the majority of undergraduates are underprepared, entitled, have underdeveloped work ethics, and lack both the discipline and drive necessary to really get something out of their education. However, I still think that the extreme burn-out classes cause more detriments to higher education than they bring as a whole.
It's not so much a habit of being deliberately arcane as, in my opinion at least, badly attempting to keep the sense of the sublime that engineering departments often lose touch with.
I went to a pretty shit physics department, but there were glimpses of beauty. The engineering department was more professional, better run, definitely more fun if you like the thrill of actually making something, but as engineers theory is just a means to an end, so I would've been bored on some fundamental level.
Aside:
As a (I suspect) dyslexia diagnosis in-waiting I am a sucker for really good, crisp typesetting. Old books often struggle with that (e.g. particularly curly non-latin characters are almost unreadable for me), but reasoning is timeless.
Landau & Lifshitz is old, ugly, a bit terrifying, and yet timelessly brilliant. Difficult, but in a physically challenging way rather than the more modern, Grecian-thinking, rigour-by-nomenclature style found in modernity.
Not that we should give up and resign ourselves to outmoded received pedagogy, but these are hard breaking changes and we should set our expectations accordingly.
I enjoyed the book and class myself but it did not make me a better physicist, just a better mathematician
Did you go to a top university? And did you study physics in grad school? I find this statement amusing given how easy the undergrad curriculum was compared to grad school physics.
In my undergrad, physics was challenging only in that you needed a good command of the mathematics. If you had that, the actual instruction (and textbooks) were of average difficulty.
But again, experience may vary from university to university. Certainly I can see professors who could have made it much tougher if they wanted to.
As for the rest of your comments in this thread: Sorry, but to me this is another HN thread where people insist it can be taught better, and teachers are being irresponsible in not finding such approaches, but with very little actual proof that it can be as good as imagined. It's not like you have concrete examples of better pedagogy to pointed out.
Undergrad textbooks in Physics may be somewhat challenging. But the graduate texts (at least in theoretical Physics) tend to be MUCH harder, especially if your undergrad degree didn't include several courses of abstract algebra and topology.
Ideally, physicists should have Geometric Algebra (Clifford Algebra) as part of their undergraduate classes. But at least when I went to Uni, there was no space in the undergrad tracks for this level of math.
> I had a lot of hope for things like Khan Academy, but the issue is video is not text and it's hard to iterate and improve on.
Not the issue.
> I really wish textbooks with open licenses would take over and they could be reworked and improved year after year by different people
The issue.
There actually isn't even a good open content license, analogous to the GPL-style licenses. Improving on video means having access to the source files. Ditto for interactive activities. Khan Academy is designed to look as open as possible, while withholding just enough and being just mean enough with license to make any sort of reuse a hopeless endeavor.
With the proper piece in place, video is very possible to iterate upon.
Could anyone explain to me how they think this might work in practice?
I am presently producing an undergrad textbook in quantum theory. I have two motivations: 1. IMO the "qubits first" (ie teach finite-dimensional QM before wave mechanics) approach to introducing the theory is superior (basically only Feynman did it of all the "classic" books) and 2. I'm involved in third world education and I want the book to be freely downloadable.
Now its a lot of work despite having taught the course multiple times and produced comprehensive lecture notes etc. Once its done I am sure I will not have the time to keep updating it, expanding on the problem sets and so on. A former student on the course is helping with the conversion and he will be a co-author, but like me he sees it as a service not at all about producing a product. So I think we're both very open to the idea of such "open license".
Given all that here are the kinds of questions that immediately arise:
- Mechanically how should one make the book available for such re-working? Put the source files on github? (Not something I've ever used, but I know roughly how it works).
- Via what mechanism does someone get to be credited for work they might do on better versions?
- Who decides what is the current "definitive" or "best" version? I will have a separate website for the book so I guess new versions can be announced there. But one way or the other I won't be involved forever.
- QM is fraught with crackpots, people who have whacky ideas on how to explain things and so on. Can they be prevented from "taking over", rewriting large chunks into (what I would view as) nonsense and so on? Note that presumably my name would still be associated with the new versions, so the issue is primarily not lending credence to stuff I fundamentally disagree with, not that they shouldn't be allowed to go do their thing.
- We will make a POD service available for purchasing hardcopies, the (expected to be small) royalties from which would be donated to third world physics/math education. Is there some license that can ensure any subsequent use of the material is also similarly non-profit?
I can see some (though not perfect) analogies with open-source software, so perhaps someone here has useful ideas about this kind of thing already...
Exams are for people designing bridges and doing surgery. Basic science just learn and start doing some to see if it is for you.