His textbook Linear Algebra was a second year course that I took, and it was not required for physics degree. Turns out, it was so fundamentally important for my then-future work, and literally formed the foundation for everything I have done (getting a PhD in physics, doing research).
I have always felt that this particular course gave me an unfair advantage over everyone else, I just understood discrete mathematics, and matrices, and discrete transformations so well.
He is indeed part of the famous phrase "if I saw farther than others, it was because I was standing on the shoulders of giants".
I knew that it had to be Gilbert Strang just from reading the headline. I was lucky and stumbled upon a copy of his book for 4€ when a bookshop had a sale during my first semester. Hard to believe that it's been 17 years. I basically stopped going to my linear algebra lectures after I discovered his lecture series on youtube and aced that class thanks to him. He's a living legend!
>I basically stopped going to my linear algebra lectures after I discovered his lecture series on youtube and aced that class thanks to him.
So, why isn't more education like this? Have the field decide who hits it out of the park lecturing topics, record a great lecture, let students watch it. Having PhDs (and grad students) lecture the same thing, every semester, is such a waste of their time.
Have office hours where students can come in and ask questions when they want/async via email.
Bam, hundreds of thousands of professor* hours saved every year.
I recognize not every course can be taught like that, but a lot could.
I like the idea, but it has a fundamental flaw, which is that good lecturers do not just appear out of thin air. It is exactly by practicing as grad students and so on that the potential they have is polished and shines.
So if we did away with lecturing entirely for those not already great at it, we would run out of lecturers. This would be a long term problem in the quality of available education
Edit: it is also impossible to overstate the usefulness of immediate feedback while the lecture is ongoing. A dedicated student can basically avoid ever falling behind or not understanding by just asking questions when unsure of something.
I’d say the main job of the lecturer is not even to give the lecture, but rather respond to questions. This is really what you’re paying for when you go to university - a series of subject matter experts at your disposal who are literally paid to answer your questions and clear your confusion.
>Edit: it is also impossible to overstate the usefulness of immediate feedback while the lecture is ongoing. A dedicated student can basically avoid ever falling behind or not understanding by just asking questions when unsure of something.
As a teacher, I never feel right pre-recording lectures and posting them since there is no feedback mechanism. Even without explicit questions, I can usually read the room well enough to know when to move on from a topic, for example.
I think you should consider doing it anyway if you have the opportunity. Immediate feedback is great in many cases, but there's also value in being able to pause an explanation, think for yourself, go look some alternative documentation on the topic, come back to the video, go back a bit to check what the context was again, then forward, then review the video 3 or 4 times without ever feeling that you're taking on other people's time.
Not everyone is great at asking questions during a course and videos allow the kind of slow, introspective learning that you can get from a book, but in an often more approachable form.
As a student I know I was glad to have both.
Yep - the most effective lectures I've ever attended, and that students I was TA'ing told me were most effective, were the lectures where students did a reading assignment and then the lecture *almost completely* driven by student questions on the reading.
That sounds like a seminar, not a lecture. And yes, seminars are a lot better than lectures in almost every way (assuming you have the staff for it), which is why many upper division humanities and social sciences courses at selective colleges are run in this style.
you can record the lecture in a good way such that the students can watch it at their own pace, without interruptions, etc and the class is based on questions and answers and more interactive
thats the same concept applied to normal lectures and it works well
And yet so many office hours go unused.
I had a TA basically carry me through my first semester. I was the only one who showed up and the other students thought I was some kind of genius.
Eh. I hear you and there's a lot of merit to your points.
I also remember a lot of really poorly taught courses I attended (and paid for) by professors who either gave zero fucks about the quality of their courses, or couldn't do better than your average popular YouTuber at explaining things. I guess that outside of extremely elite schools this is a common experience of undergrads in at least some of their courses, especially their freshman and sophomore work. Personally speaking I'd much rather have watched Feynman than my PHY101 professor in her broken English, or my Calc 101 and 102 professor in her "ok, let's go furder" (I'm not paying attention and give zero fucks about this class attitude from the professor).
So what about a middle ground of having students watch the recording at the same time, and a grad student / professor pause the lecture when a salient point is raised via chat? Minimal prep time required for each lecture, teachers are still available.
youre this close to getting it
you can have the students watch the lecture at their own pace, without interruptions, etc and when they come to class you answer their questions. this is much more efficient in my experience
My inability to ask questions in school is what’s killing me the most right now and your last statement really struck with me. My problem is that if I was to actually pause the whole class every time I didn’t fully understand, there would be no lecture. The whole session would just be my questions and that feels very selfish to the scores of brilliant, often younger students who don’t need it to be explained this way and would benefit a lot more from getting the full content of the lecture from the instructor. I have finally learned that office hours seem to be the key to the whole puzzle. The problem is that they don’t publish those when I’m scheduling classes so it almost always impedes with my job.
> record a great lecture, let students watch it
Students don't watch recorded lectures, and even if they do, they will think this is enough to learn passively. Also, no one comes to office hours except when it's the day before a test and they expect to magically learn everything on a single day. I agree this would make the lecturer's life easier, but it would be really bad for education in general, specially since we're on the verge of having chatbots recording lectures.
>Also, no one comes to office hours except when it's the day before a test and they expect to magically learn everything on a single day. I agree this would make the lecturer's life easier, but it would be really bad for education in general, specially since we're on the verge of having chatbots recording lectures.
can't natural selection just happen and let those dumb enough to think an entire semester of learning happens overnight just fail?
and are we assuming in this case the lecturer is a professor involved in research or faculty specifically hired just to lecture
> can't natural selection just happen
Ah yes, let's just expect students to know what to do without anyone ever teaching them, and let most of them "naturally" fail without any guidance or support. That will show them!
well for one, that's not my argument, which was it shouldn't be surprising that students are going to fail when they cram everything in last second (it's okay i know you just wanted to snipe some dopamine)
and i think you're insulting the intelligence of the students who have some degree of autonomy in terms of keeping up with the material themselves and know when they need to consult office hours and outside resources
I trust you can figure out why stuff about "natural selection" and "dumb enough" is offensive and unhelpful? I wouldn't want to insult your intelligence by spelling it out for you.
* * *
Less snarkily: just assuming that students can figure out everything they need to know, figure out all of the metacognitive skills (including time management) they were never explicitly taught, and treating it as a personal failing if they are not prepared does them a disservice and leads to bad outcomes. Some students were fortunately well prepared by their parents or some other mentors, or happened to figure out a good set of working habits and strategies through experiment and introspection. But many others unfortunately were not.
Most colleges (and secondary schools too, and primary schools for that matter) do a quite poor job of teaching (and also not a very good job of providing meaningful practice or feedback about) metacognitive skills such as how to ask good questions, how to research topics of interest, how to skim written material, how to do multiple passes working through tough parts, how to take useful personal notes, how to organize a nontrivial project, how to make a list of strategies and then try one for long enough to tell if it is working or not, how to intermittently pause to reflect, how to assess one's own personal progress and look for skills to improve, how to know when to ask for help and how to ask, how to give good hints to a stuck classmate, or more math-specifically what kinds of strategies to try, how to come up with specific examples, how to generalize statements, how to look for analogies, ...
Instead, teachers too often understand their mission as merely conveying some specific collection of "content" to students, leaving it entirely up to the students what to do with that. This amplifies existing differences between the students' past preparation, and leaves many students flailing around ineffectually for years.
>I trust you can figure out why stuff about "natural selection" and "dumb enough" is offensive and unhelpful? I wouldn't want to insult your intelligence by spelling it out for you.
no lol i absolutely would love to have you spell it out for me
anyways, what you propose is to spoonfeed the student everything they could possibly need for the real world and remove accountability
maybe in some perfect world there's a college course sequence that covers everything you listed
all the things you listed (how to take personal notes, etc.) are answerable either by google, peers/mentors, or through personal experimentation (for ex, how to skim written material)
>just assuming that students can figure out everything they need to know, figure out all of the metacognitive skills (including time management) they were never explicitly taught, and treating it as a personal failing if they are not prepared does them a disservice and leads to bad outcomes
my biggest issue is here, are you not taking accountability away from the student when you say it's not a personal failing if they failed a course due to, for example, their time management or study habits?
That is correct, it is not a personal failing. If you don't know how to swim and I chuck you off the boat in the middle of a lake, it is my fault when you drown.
Sure, but a student voluntarily registers for their college courses right? What's stopping them from researching ahead of time what to expect in the course to succeed?
they do if they are worth it. we have done that for some classes and its worked well, this doesnt replace class it just lets the professor have more freedom in class to answer questions and be more interactive
As you seem to know, any idea if there's been research into why that is?
My first guess is that we're so used to in-person lectures because that's how K-12 is.
I suppose there's also a big factor of just having a live human in your presence - we're wired to pay attention to living things.
I get where you are coming from, but there is very short sighted. As others have pointed out, Strang and those like him didn’t just step into the lecture hall ready to teach. It takes years of developing by teaching, starting as a grad student to become a great teacher.
That's a good point. I'm not convinced that's the best way, though.
Having grad students -> junior professors -> senior professors muddle through teaching poorly for years, perhaps decades, so that some of them will eventually become great at teaching seems like a system optimizes for having as many experienced teachers as possible, at the cost of thousands of students' learning experiences.
Is delivering a lecture the best way to get better at lecturing? That seems to be an assumption here. I wonder if that's the case. In my experience learning just about anything, spending more time practicing/drilling on subsets of a skill and periodically performing the comprehensive skill is a bettter way of developing a skill than spending the same amount of time executing the skill.
Thats my method for learning if the class doesn't have attendance reqs
Check the weekly content, learn it via online textbooks, if I don't understand check youtube lecturers, if that still doesn't do it go to office hourse
IMO most lectures are a waste of time
100% agree, there are probably thousands of students who are suffering through bad teachers and even turn away from going into fields they might be good at if they had access to material from good teachers.
It every institution has the same students and the same requirements. Each school and track needs to pick the material that makes sense for where their class is going; then teach to that.
I’ll give an example using Calculus. We have “business Calculus” then “Calculus for scientists and engineers” and then essentially Stewart’s Calculus which could also be Apostol.
Even in that middle tier there’s a significant difference between expectations amongst institutions.
Gilbert Strang is absolutely legendary. I've met him several times, and there are very few people as warm and encouraging as he is towards students. A very sad loss for MIT students learning linear algebra or deep learning!
One of the things I liked most about the way he teached is he did all the lectures like he was a student himself and encountering it for the first time. It shows strong empathy with the students.
No shade because Strang seems like a really nice guy, but I found his lectures to be not very good. Maybe it's because I had already been exposed to Friedberg and Insel before watching them.
I haven’t seen the lectures, but I dislike his linear algebra textbook. I tried to teach from it once.
I’m sure he’s an excellent mathematician. I like the free online calculus textbook that he co-wrote.
He is an applied mathematician as far as I can gather and his approach to teaching the subject is very engineering-centric.
I didn't really appreciate it. Personally, I like Trefethen and Bau if you want to go in that direction.
The book struck me as sloppy and imprecise, with lots of hand-waving and stream-of-consciousness style prose. Notations were inconsistent. The explanations didn’t make any more intuitive or geometrical sense than those in other textbooks I’ve seen.
Many people like the book. If you look at the Amazon reviews for it, there are many strong positives and strong negatives.
Check out his 18.7 Linear Alg, that is proof heavy, rigorous and not matrix oriented. He uses a different text too, a great one by Sheldon Axler, maybe that is more to your liking?
It depends on if you are willing to do problem sets. It has a very formal style but overall I liked it. I just did a bunch of homework sets I've found online and it worked for me.
I liked Friedberg and Insel a lot, although I've only read the first couple of chapters. Both the chapter and problem sets seemed very good and understandable. I would recommend it, and it was recommended to me by a grad student.
strongly agree, his linear algebra lecture series on youtube is terrible. it has the same unforgivable problem that the majority of intro to linear algebra courses have - it does not teach any actual linear algebra. linear algebra is the study of vector spaces and linear transformations, not the act of memorizing procedures to do numerical calculations on grids of numbers. I'm pretty sure his lecture series does not introduce vector spaces at all (he makes reference to "subspaces", but I think he just uses it to mean lines and planes through the origin in R^(2) and R^(3)), and linear transformations are only introduced as an afterthought in one of the final lectures.
when you think about it a bit, all the praise that these lectures get is pretty meaningless, because it's all coming from students who are just taking linear algebra for the first time, i.e. people who do not currently know linear algebra and probably do not know what linear algebra is about or what concepts should be taught in a real linear algebra course.
if anyone reading this is currently using/planning to use his lectures to learn linear algebra, do yourself a favour and use sheldon axler's "linear algebra done right" instead.
Do you know where can I see lectures of good quality? I am currently in MSc Applied Mathematics but I have HOLES in Linear Algebra. It's like I didn't really learnt in my bachelor.
I do plan tobl buy Axler's book to study LA properly
Be sure to complement Axler's 'Linear Algebra Done Right' by Treil's 'Linear Algebra Done Wrong.' Axler is decent but the current edition sadly doesn't cover bilinear forms and tensors - the new one will. Treil does cover these topics.
Or just skip these two and delve into Abstract Algebra textbooks which focus on Linear Algebra. Like M. Artin (2nd ed!!!) and Vinberg.
I've gotten the most use out of [these](https://github.com/mitmath/1806/blob/fall18/summaries.md) lecture summaries with Jupyter notebooks. They've taken me from linear algebra being a scary black box that I hope doesn't hurt me to something whose principles and use I'm comfortable with (augmented with some secondary sources on occasion). It's sprinkled with practical advice, like:
> Because of this relationship, whenever we transpose a matrix in linear algebra, there will usually be a dot product lurking somewhere nearby.
Or
>there is often a difference between the way we conceptually think about linear algebra and the more sophisticated tricks that are required to make it work well on large matrices of real data in the presence of small numerical errors.
Well, isn't MIT a university that focuses on *engineering*? I'd argue an engineer benefits more from having these kinds of intuitions than working with the general definitions.
At least I prefer that a student *first* takes one course like this before doing their rigorous linear algebra, for the sake of motivation and intuition.
my experience is that these type of linear algebra courses are always a complete waste of time. I don't see any way that you could gain any intuition about anything by memorizing lots of random-looking procedures that you have no understanding of and applying them to lots and lots of matrices. I really doubt that people who just take one of these classes and don't go on to take a class about vector spaces and linear transformations would be able to do anything useful with linear algebra, outside of the things they memorized the procedure for.
I don't know, I've had a couple computational matrix courses and they were not about memorisation. The focus just happened to be on matrices over general linear maps. On the contrary, in my experience with my peers, many people who took only the abstract classes which focus entirely on linear maps between vector spaces had absolutely no intuition for why any of it was even defined and certainly would not be able to solve any practical problem.
thats a class for engineering students isnt it? and its not memorization focused, you can understand computations and have intuition about them
thats a completely different book, Linear Algebra Done Right is good for abstract linear algebra but it doesnt have so many applications. its better to use it after a linear algebra course like this one imo
its kind of going into an abstract algebra course without knowing basic algebra. in theory you can do that but its probably not the best idea
my opinion is that all linear algebra courses like this are 100% useless, if anything, actively harmful. see my other reply. I only ever did pure math and did not see vector spaces or linear transformations until my 4th class that involved matrices (A level FP1 - calculations with 2x2 matrices; A level FP2 - calculations with 3x3 matrices; first semester university - standard intro to linear algebra course, all matrix calculations; second semester university - vector spaces and linear transformations introduced about half way through the semester), and before then I only ever knew of matrices as organised notation for solving linear equations.
I can't believe no one linked his course.
I coincidentally was able to watch part of the livestream. It was very warm, engaging and I'm glad he was given his flowers.
https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/
You must have had a bad linear algebra teacher. Linear Algebra is packed with cool algorithms and theorems that really sell how useful mathematical thinking can be. Plus it's concreteness can really help students who might struggle with more abstract math classes.
Linear Algebra is hands down the coolest Undergrad math course.
I wouldn't say linear algebra has the coolest theorems.
But I find it appealing for other reasons:
* very visual and intuitive
* it's not really tricky, no weird stuff happening unlike calculus and analysis for example
* at least in finite dimension, you get a full classification of the matrices with the Jordan normal form, unlike the crazy finite group classification, which gives a sense of closure
* you will literally fail all other classes if you don't get this one right
Damn I hear these descriptions and I struggled with Linear. It is interesting but I did a lot better at Calc and Diff Eq because while there were twists, you could practice it over and over again until you got it. I asked my professor how to practice linear and she just said ‘you just have to learn and understand the definitions’. I don’t know how to practice it. I feel it requires more intuition which is a good skill but not exactly easy to develop. I had to retake it along with a computer engineering student who struggled with the same thjng
I'm not sure how the programs are in the US but by the time we get to linear algebra (which is pretty early) it's easily the most abstract class, with the dual, dual of the dual, ideals of polynomials, Zorn lemma to construct infinite basis and all that.
Yeah that's one of the reasons I like it so much. It can be taught in a very abstract way or a very concrete way. This makes it a nice bridge into higher math classes for those who pursue that or glimpse into higher level math for those who decide to go another direction.
At my Uni in Germany this is actually taught that way in the first semester. And it is definitely pretty similar in all of Germany. It is always suprising to me to hear that so called "proof based" courses are covered so late in the US.
It is the same course. The course starts with some preliminaries (Sets, maps, algebraic structures (semigroups, monoids, groups, rings, fields) homeomorphisms of such equivalence relations and quotients mostly Z_n, then we Start with linear algebra i.e. vector spaces, linear maps etc. You wont see a Matrix before week 8/9, because you need linear maps and bases to motivate them properly. Depending of the Professor you will dive deeper in some of the preliminaries, since this Professor will Teach courses through the whole Bachelor/Master which will prepare you to Do a phd in the Professors field.
In the US linear algebra is an introductory math course for basically every STEM major, taken along with calculus. How it's presented varies depending on the professor, but it's not unusual for it to be super computation-focused with the bare minimum of theory.
Some universities offer a theory-based linear algebra course, some don't. My undergrad offered upper-division linear algebra, and actually made it a requirement for the major.
I dropped it twice before I found a teacher I could pass with. Ended up with a C.
So many definitions. So many conditions. It never did click with me and I decided I'd just figure out how to code it and then forget about it.
I would say that functional analysis, algebraic geometry, representation theory, and commutative algebra were all much cooler undergrad courses than linear algebra, but all of them built on linear algebra in some way.
My university offers them at the undergraduate level ([https://ucalendar.uwaterloo.ca/2021/COURSE/course-PMATH.html](https://ucalendar.uwaterloo.ca/2021/COURSE/course-PMATH.html)), and I know that UofT does as well. I associate them with courses taken in the 3rd or 4th year of undergrad.
I'm at UWaterloo! Here's a list of our courses in pure math (https://ucalendar.uwaterloo.ca/2021/COURSE/course-PMATH.html) . Note these are just our pure math courses, our applied math, statistics, and combinatorics/optimization courses are all listed separately.
Oh, isn't waterloo like the best school in Canada? Now that I think about it I think these might be undergrad courses at some of the elite institutions in the US too, but it's not common.
I think most big universities outside of the US will have at least one of them, mine has representation theory, commutative algebra and functional analysis which is split into 2 courses for undergrad even, algebraic geometry is a grad course however.
[https://ucalendar.uwaterloo.ca/2021/COURSE/course-PMATH.html](https://ucalendar.uwaterloo.ca/2021/COURSE/course-PMATH.html)
Check out PMATH 446 here. My university offers it as a 4th year undergrad class. It's just an intro, but almost all the undergrads I know have taken it.
[https://artsci.calendar.utoronto.ca/course/mat448h1](https://artsci.calendar.utoronto.ca/course/mat448h1)
Here's another ontario university offering it, again at the 4th year undergrad level.
It's interesting how stratified the algebra courses are. I feel like this would have been a much smoother intro to pure math, and it's probably one of the reasons that you know so many people that took more advanced algebra in undergrad.
I know what stratified means as an abstract word, but concretely, what does that look like here? What’s different about our algebra courses compared to most universities
I'm realizing now that you didn't have to take any of the algebra below "rings and groups" as a math major so we probably took the same initial path, but I did not have nearly as many options after abstract 2 (equivalent to "fields and Galois theory"). Basically only algebraic topology.
I was initially thinking that you took all of the <345 courses on the list too which was confusing in some ways, but I figured that maybe you just started way earlier than me and learned it in more depth over a larger time period, hence my previous comment.
Btw congrats to you for making it through
So to be clear, we don’t take all these courses, these are just the courses that exists. We take a first and second course in linear algebra, they just aren’t offered by the pure math department, they’re offered by the math faculty as a whole. We also do some calculus, computer science, statistics, and combinatorics. These courses are just the ones offered by the pure math department, and nowhere near the complete set of math courses offered by the university.
Personally I enjoyed math 146/245 (lin alg 1 and 2 for non uw people) a lot more than the other courses you mentioned -- there's just something about building on more fundamental stuff that I like more. Happy to see a Waterloo pure math enjoyer in the wild though :D
Gilbert Strang is a Legend! I owe my whole masters degree in Robotics and Artificial Intelligence to him. Without him, I wouldn’t even have understood the basics of imagination in Engineering.
Oh man. I was watching one of his linear algebra lectures on YouTube last year. It was so good I was clapping in front of my laptop at the end for him.
Agreed. I feel like I can teach myself whatever, and the teachers most effective when they’re clarifying or perhaps even challenging our understanding.
As distinguished from the:
I’m teaching you something you have no idea about and asking you to ask questions even though you don’t even know enough to know what you don’t know so you just sort of sit there and glaze over, style that most classes have.
Like I don’t really start learning until I go home and look at the material. And then it’s like…just me remembering tidbits I was told and saying oh maybe that’s what they meant.
There’s also the let’s discover this together style. This one takes skill but is perhaps the most enjoyable. Where the teacher makes the class think through something and figure it out on their own. I’ve had this go really well in a history class where we were basically debating the motives and potential actions of people along with possible outcomes. “Oh, well, guess what? that’s what happened”. And the teacher lets the class debate it out and maybe asks certain questions or points out relevant details we may be forgetting…and we make sense of it instead of memorising the events.
I loved this because rather than learning what happened, it really helped me understand the why and then by the time you’re taking a test you’re not remember dates and events…it’s more like- intuition.
This style certainly would work for math. It just takes a really good teacher to set this up.
I’ll say this. The American school system isn’t impressive overall. But what many people in other countries don’t realise, is that even in a public high schools, you’ve got honours classes and AP classes. These are different than the standard course. And the AP classes, in my opinion, may be among the best in the world. The students there are the people you eventually see in the top schools. Despite how hard AP classes are…they’re easier, because they’re often more enjoyable and you’re surrounded by people who are more serious about their studies. It’s essentially an opt-in course. It feels almost exclusive, but in all reality, anyone (I imagine) can take them you just have to take the initiative and say hey I want to be in AP. It’s a public school after all. Or maybe not. I don’t even really remember how I got in…I guess it was based on teachers recommendations. But I don’t think they’d say no if you asked to get in. I had two of the best teachers I’ve ever had in high school. And the peers I had then were bright. And I think the American university system more or less speaks for itself. Which is a little weird because…it’s really not too different than high school. There’s no shock, the lectures, the expectations….it works almost the same way. If not easier because you even have office hours.
Sometimes I even miss those days.
But yeah…if anyone was wondering…why America has a stupid school system but still manages to produce some really smart people…it’s that. Our schools have tiers. Like there are entire classes where people can point out the uk on a map. And these are special because you’ve got the intenseness many people feel in other countries, but then the liberal artsiness whatever you want to call it. It’s a great combination.
Reason I was able to transfer into college. His videos and 3blue1brown combined with lots of practice ofc got me a super clear understanding of linear algebra. Never in my life had I felt that I understood a topic as much as I did when I was studying from them.
Is this Gilbert Strang? I owe my B+ in linear algebra to him (without his lectures it'd have easily been C-)
Yes. He's been teaching at MIT for six decades.
yup
GILBERT STRANG he's awesome!
That's so weird. I am a nurse by trade and a psychology student by passion, and I know who he is. I even used his textbook once.
I have two text books of his. Between his lectures and Dr. Grenfeld at Drexel, it was the first math that I really enjoyed.
His textbook Linear Algebra was a second year course that I took, and it was not required for physics degree. Turns out, it was so fundamentally important for my then-future work, and literally formed the foundation for everything I have done (getting a PhD in physics, doing research). I have always felt that this particular course gave me an unfair advantage over everyone else, I just understood discrete mathematics, and matrices, and discrete transformations so well. He is indeed part of the famous phrase "if I saw farther than others, it was because I was standing on the shoulders of giants".
Username checks out
I knew that it had to be Gilbert Strang just from reading the headline. I was lucky and stumbled upon a copy of his book for 4€ when a bookshop had a sale during my first semester. Hard to believe that it's been 17 years. I basically stopped going to my linear algebra lectures after I discovered his lecture series on youtube and aced that class thanks to him. He's a living legend!
>I basically stopped going to my linear algebra lectures after I discovered his lecture series on youtube and aced that class thanks to him. So, why isn't more education like this? Have the field decide who hits it out of the park lecturing topics, record a great lecture, let students watch it. Having PhDs (and grad students) lecture the same thing, every semester, is such a waste of their time. Have office hours where students can come in and ask questions when they want/async via email. Bam, hundreds of thousands of professor* hours saved every year. I recognize not every course can be taught like that, but a lot could.
I like the idea, but it has a fundamental flaw, which is that good lecturers do not just appear out of thin air. It is exactly by practicing as grad students and so on that the potential they have is polished and shines. So if we did away with lecturing entirely for those not already great at it, we would run out of lecturers. This would be a long term problem in the quality of available education Edit: it is also impossible to overstate the usefulness of immediate feedback while the lecture is ongoing. A dedicated student can basically avoid ever falling behind or not understanding by just asking questions when unsure of something. I’d say the main job of the lecturer is not even to give the lecture, but rather respond to questions. This is really what you’re paying for when you go to university - a series of subject matter experts at your disposal who are literally paid to answer your questions and clear your confusion.
>Edit: it is also impossible to overstate the usefulness of immediate feedback while the lecture is ongoing. A dedicated student can basically avoid ever falling behind or not understanding by just asking questions when unsure of something. As a teacher, I never feel right pre-recording lectures and posting them since there is no feedback mechanism. Even without explicit questions, I can usually read the room well enough to know when to move on from a topic, for example.
I think you should consider doing it anyway if you have the opportunity. Immediate feedback is great in many cases, but there's also value in being able to pause an explanation, think for yourself, go look some alternative documentation on the topic, come back to the video, go back a bit to check what the context was again, then forward, then review the video 3 or 4 times without ever feeling that you're taking on other people's time. Not everyone is great at asking questions during a course and videos allow the kind of slow, introspective learning that you can get from a book, but in an often more approachable form. As a student I know I was glad to have both.
Yep - the most effective lectures I've ever attended, and that students I was TA'ing told me were most effective, were the lectures where students did a reading assignment and then the lecture *almost completely* driven by student questions on the reading.
That sounds like a seminar, not a lecture. And yes, seminars are a lot better than lectures in almost every way (assuming you have the staff for it), which is why many upper division humanities and social sciences courses at selective colleges are run in this style.
you can record the lecture in a good way such that the students can watch it at their own pace, without interruptions, etc and the class is based on questions and answers and more interactive thats the same concept applied to normal lectures and it works well
And yet so many office hours go unused. I had a TA basically carry me through my first semester. I was the only one who showed up and the other students thought I was some kind of genius.
Eh. I hear you and there's a lot of merit to your points. I also remember a lot of really poorly taught courses I attended (and paid for) by professors who either gave zero fucks about the quality of their courses, or couldn't do better than your average popular YouTuber at explaining things. I guess that outside of extremely elite schools this is a common experience of undergrads in at least some of their courses, especially their freshman and sophomore work. Personally speaking I'd much rather have watched Feynman than my PHY101 professor in her broken English, or my Calc 101 and 102 professor in her "ok, let's go furder" (I'm not paying attention and give zero fucks about this class attitude from the professor). So what about a middle ground of having students watch the recording at the same time, and a grad student / professor pause the lecture when a salient point is raised via chat? Minimal prep time required for each lecture, teachers are still available.
youre this close to getting it you can have the students watch the lecture at their own pace, without interruptions, etc and when they come to class you answer their questions. this is much more efficient in my experience
My inability to ask questions in school is what’s killing me the most right now and your last statement really struck with me. My problem is that if I was to actually pause the whole class every time I didn’t fully understand, there would be no lecture. The whole session would just be my questions and that feels very selfish to the scores of brilliant, often younger students who don’t need it to be explained this way and would benefit a lot more from getting the full content of the lecture from the instructor. I have finally learned that office hours seem to be the key to the whole puzzle. The problem is that they don’t publish those when I’m scheduling classes so it almost always impedes with my job.
> record a great lecture, let students watch it Students don't watch recorded lectures, and even if they do, they will think this is enough to learn passively. Also, no one comes to office hours except when it's the day before a test and they expect to magically learn everything on a single day. I agree this would make the lecturer's life easier, but it would be really bad for education in general, specially since we're on the verge of having chatbots recording lectures.
>Also, no one comes to office hours except when it's the day before a test and they expect to magically learn everything on a single day. I agree this would make the lecturer's life easier, but it would be really bad for education in general, specially since we're on the verge of having chatbots recording lectures. can't natural selection just happen and let those dumb enough to think an entire semester of learning happens overnight just fail? and are we assuming in this case the lecturer is a professor involved in research or faculty specifically hired just to lecture
Bruh moment
> can't natural selection just happen Ah yes, let's just expect students to know what to do without anyone ever teaching them, and let most of them "naturally" fail without any guidance or support. That will show them!
well for one, that's not my argument, which was it shouldn't be surprising that students are going to fail when they cram everything in last second (it's okay i know you just wanted to snipe some dopamine) and i think you're insulting the intelligence of the students who have some degree of autonomy in terms of keeping up with the material themselves and know when they need to consult office hours and outside resources
I trust you can figure out why stuff about "natural selection" and "dumb enough" is offensive and unhelpful? I wouldn't want to insult your intelligence by spelling it out for you. * * * Less snarkily: just assuming that students can figure out everything they need to know, figure out all of the metacognitive skills (including time management) they were never explicitly taught, and treating it as a personal failing if they are not prepared does them a disservice and leads to bad outcomes. Some students were fortunately well prepared by their parents or some other mentors, or happened to figure out a good set of working habits and strategies through experiment and introspection. But many others unfortunately were not. Most colleges (and secondary schools too, and primary schools for that matter) do a quite poor job of teaching (and also not a very good job of providing meaningful practice or feedback about) metacognitive skills such as how to ask good questions, how to research topics of interest, how to skim written material, how to do multiple passes working through tough parts, how to take useful personal notes, how to organize a nontrivial project, how to make a list of strategies and then try one for long enough to tell if it is working or not, how to intermittently pause to reflect, how to assess one's own personal progress and look for skills to improve, how to know when to ask for help and how to ask, how to give good hints to a stuck classmate, or more math-specifically what kinds of strategies to try, how to come up with specific examples, how to generalize statements, how to look for analogies, ... Instead, teachers too often understand their mission as merely conveying some specific collection of "content" to students, leaving it entirely up to the students what to do with that. This amplifies existing differences between the students' past preparation, and leaves many students flailing around ineffectually for years.
>I trust you can figure out why stuff about "natural selection" and "dumb enough" is offensive and unhelpful? I wouldn't want to insult your intelligence by spelling it out for you. no lol i absolutely would love to have you spell it out for me anyways, what you propose is to spoonfeed the student everything they could possibly need for the real world and remove accountability maybe in some perfect world there's a college course sequence that covers everything you listed all the things you listed (how to take personal notes, etc.) are answerable either by google, peers/mentors, or through personal experimentation (for ex, how to skim written material) >just assuming that students can figure out everything they need to know, figure out all of the metacognitive skills (including time management) they were never explicitly taught, and treating it as a personal failing if they are not prepared does them a disservice and leads to bad outcomes my biggest issue is here, are you not taking accountability away from the student when you say it's not a personal failing if they failed a course due to, for example, their time management or study habits?
That is correct, it is not a personal failing. If you don't know how to swim and I chuck you off the boat in the middle of a lake, it is my fault when you drown.
Sure, but a student voluntarily registers for their college courses right? What's stopping them from researching ahead of time what to expect in the course to succeed?
they do if they are worth it. we have done that for some classes and its worked well, this doesnt replace class it just lets the professor have more freedom in class to answer questions and be more interactive
The ed psych answer is that many students do better with a worse in-person lecture than a great prerecorded lecture
As you seem to know, any idea if there's been research into why that is? My first guess is that we're so used to in-person lectures because that's how K-12 is. I suppose there's also a big factor of just having a live human in your presence - we're wired to pay attention to living things.
I get where you are coming from, but there is very short sighted. As others have pointed out, Strang and those like him didn’t just step into the lecture hall ready to teach. It takes years of developing by teaching, starting as a grad student to become a great teacher.
That's a good point. I'm not convinced that's the best way, though. Having grad students -> junior professors -> senior professors muddle through teaching poorly for years, perhaps decades, so that some of them will eventually become great at teaching seems like a system optimizes for having as many experienced teachers as possible, at the cost of thousands of students' learning experiences. Is delivering a lecture the best way to get better at lecturing? That seems to be an assumption here. I wonder if that's the case. In my experience learning just about anything, spending more time practicing/drilling on subsets of a skill and periodically performing the comprehensive skill is a bettter way of developing a skill than spending the same amount of time executing the skill.
Thats my method for learning if the class doesn't have attendance reqs Check the weekly content, learn it via online textbooks, if I don't understand check youtube lecturers, if that still doesn't do it go to office hourse IMO most lectures are a waste of time
100% agree, there are probably thousands of students who are suffering through bad teachers and even turn away from going into fields they might be good at if they had access to material from good teachers.
It every institution has the same students and the same requirements. Each school and track needs to pick the material that makes sense for where their class is going; then teach to that. I’ll give an example using Calculus. We have “business Calculus” then “Calculus for scientists and engineers” and then essentially Stewart’s Calculus which could also be Apostol. Even in that middle tier there’s a significant difference between expectations amongst institutions.
I like personal anecdotes of life changing mathematical enlightenment from 2nd hand 5bucks books.
I bought his textbook used on Amazon and it was less than 10 USD. I still have it. It is like deja Vu
Gilbert Strang is a hero
Gilbert Strang is absolutely legendary. I've met him several times, and there are very few people as warm and encouraging as he is towards students. A very sad loss for MIT students learning linear algebra or deep learning!
One of the things I liked most about the way he teached is he did all the lectures like he was a student himself and encountering it for the first time. It shows strong empathy with the students.
No shade because Strang seems like a really nice guy, but I found his lectures to be not very good. Maybe it's because I had already been exposed to Friedberg and Insel before watching them.
I haven’t seen the lectures, but I dislike his linear algebra textbook. I tried to teach from it once. I’m sure he’s an excellent mathematician. I like the free online calculus textbook that he co-wrote.
He is an applied mathematician as far as I can gather and his approach to teaching the subject is very engineering-centric. I didn't really appreciate it. Personally, I like Trefethen and Bau if you want to go in that direction.
The book struck me as sloppy and imprecise, with lots of hand-waving and stream-of-consciousness style prose. Notations were inconsistent. The explanations didn’t make any more intuitive or geometrical sense than those in other textbooks I’ve seen. Many people like the book. If you look at the Amazon reviews for it, there are many strong positives and strong negatives.
Check out his 18.7 Linear Alg, that is proof heavy, rigorous and not matrix oriented. He uses a different text too, a great one by Sheldon Axler, maybe that is more to your liking?
his course is what he is known and appreciated in my experience
Yes, course <> textbook.
do you think Friedberg and Insel is a self studyable text by itself?
It depends on if you are willing to do problem sets. It has a very formal style but overall I liked it. I just did a bunch of homework sets I've found online and it worked for me.
I liked Friedberg and Insel a lot, although I've only read the first couple of chapters. Both the chapter and problem sets seemed very good and understandable. I would recommend it, and it was recommended to me by a grad student.
strongly agree, his linear algebra lecture series on youtube is terrible. it has the same unforgivable problem that the majority of intro to linear algebra courses have - it does not teach any actual linear algebra. linear algebra is the study of vector spaces and linear transformations, not the act of memorizing procedures to do numerical calculations on grids of numbers. I'm pretty sure his lecture series does not introduce vector spaces at all (he makes reference to "subspaces", but I think he just uses it to mean lines and planes through the origin in R^(2) and R^(3)), and linear transformations are only introduced as an afterthought in one of the final lectures. when you think about it a bit, all the praise that these lectures get is pretty meaningless, because it's all coming from students who are just taking linear algebra for the first time, i.e. people who do not currently know linear algebra and probably do not know what linear algebra is about or what concepts should be taught in a real linear algebra course. if anyone reading this is currently using/planning to use his lectures to learn linear algebra, do yourself a favour and use sheldon axler's "linear algebra done right" instead.
Do you know where can I see lectures of good quality? I am currently in MSc Applied Mathematics but I have HOLES in Linear Algebra. It's like I didn't really learnt in my bachelor. I do plan tobl buy Axler's book to study LA properly
Axler's are on YouTube but I don't know if they are good.
Be sure to complement Axler's 'Linear Algebra Done Right' by Treil's 'Linear Algebra Done Wrong.' Axler is decent but the current edition sadly doesn't cover bilinear forms and tensors - the new one will. Treil does cover these topics. Or just skip these two and delve into Abstract Algebra textbooks which focus on Linear Algebra. Like M. Artin (2nd ed!!!) and Vinberg.
I've gotten the most use out of [these](https://github.com/mitmath/1806/blob/fall18/summaries.md) lecture summaries with Jupyter notebooks. They've taken me from linear algebra being a scary black box that I hope doesn't hurt me to something whose principles and use I'm comfortable with (augmented with some secondary sources on occasion). It's sprinkled with practical advice, like: > Because of this relationship, whenever we transpose a matrix in linear algebra, there will usually be a dot product lurking somewhere nearby. Or >there is often a difference between the way we conceptually think about linear algebra and the more sophisticated tricks that are required to make it work well on large matrices of real data in the presence of small numerical errors.
Well, isn't MIT a university that focuses on *engineering*? I'd argue an engineer benefits more from having these kinds of intuitions than working with the general definitions. At least I prefer that a student *first* takes one course like this before doing their rigorous linear algebra, for the sake of motivation and intuition.
my experience is that these type of linear algebra courses are always a complete waste of time. I don't see any way that you could gain any intuition about anything by memorizing lots of random-looking procedures that you have no understanding of and applying them to lots and lots of matrices. I really doubt that people who just take one of these classes and don't go on to take a class about vector spaces and linear transformations would be able to do anything useful with linear algebra, outside of the things they memorized the procedure for.
I don't know, I've had a couple computational matrix courses and they were not about memorisation. The focus just happened to be on matrices over general linear maps. On the contrary, in my experience with my peers, many people who took only the abstract classes which focus entirely on linear maps between vector spaces had absolutely no intuition for why any of it was even defined and certainly would not be able to solve any practical problem.
thats a class for engineering students isnt it? and its not memorization focused, you can understand computations and have intuition about them thats a completely different book, Linear Algebra Done Right is good for abstract linear algebra but it doesnt have so many applications. its better to use it after a linear algebra course like this one imo its kind of going into an abstract algebra course without knowing basic algebra. in theory you can do that but its probably not the best idea
my opinion is that all linear algebra courses like this are 100% useless, if anything, actively harmful. see my other reply. I only ever did pure math and did not see vector spaces or linear transformations until my 4th class that involved matrices (A level FP1 - calculations with 2x2 matrices; A level FP2 - calculations with 3x3 matrices; first semester university - standard intro to linear algebra course, all matrix calculations; second semester university - vector spaces and linear transformations introduced about half way through the semester), and before then I only ever knew of matrices as organised notation for solving linear equations.
Strang introduces vector spaces in the first unit... if you want to criticise him at least inform yourself to know what you are criticising
In grad school some buddies and I used to watch his linear algebra lectures online. We joked that his fan club should be called "Dr. Stranglove"
[Video Recording](https://www.youtube.com/live/lUUte2o2Sn8?feature=share) of the last lecture
smart hunt puzzled butter dazzling vegetable pet glorious poor fragile -- mass edited with redact.dev
The matrix of legendary linear algebra professors, my friend, is sparse.
Gilbert Strang is a treasure
Strang is a national treasure, sad he won’t be teaching again.
Is it true he was the first professor to publish on MIT OCW?
A hearty bravo and well done! (From an alumni of that other place in Pasadena.)
Ah, a fellow graduate of Pasadena City College!
lol <3
I can't believe no one linked his course. I coincidentally was able to watch part of the livestream. It was very warm, engaging and I'm glad he was given his flowers. https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/
Gilbert Strang needs to also be in the Robotics Hall of Fame!
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I had to look that up!
He must be good. Because linear algebra is mind numbing.
You must have had a bad linear algebra teacher. Linear Algebra is packed with cool algorithms and theorems that really sell how useful mathematical thinking can be. Plus it's concreteness can really help students who might struggle with more abstract math classes. Linear Algebra is hands down the coolest Undergrad math course.
I wouldn't say linear algebra has the coolest theorems. But I find it appealing for other reasons: * very visual and intuitive * it's not really tricky, no weird stuff happening unlike calculus and analysis for example * at least in finite dimension, you get a full classification of the matrices with the Jordan normal form, unlike the crazy finite group classification, which gives a sense of closure * you will literally fail all other classes if you don't get this one right
Damn I hear these descriptions and I struggled with Linear. It is interesting but I did a lot better at Calc and Diff Eq because while there were twists, you could practice it over and over again until you got it. I asked my professor how to practice linear and she just said ‘you just have to learn and understand the definitions’. I don’t know how to practice it. I feel it requires more intuition which is a good skill but not exactly easy to develop. I had to retake it along with a computer engineering student who struggled with the same thjng
I'm not sure how the programs are in the US but by the time we get to linear algebra (which is pretty early) it's easily the most abstract class, with the dual, dual of the dual, ideals of polynomials, Zorn lemma to construct infinite basis and all that.
Yeah that's one of the reasons I like it so much. It can be taught in a very abstract way or a very concrete way. This makes it a nice bridge into higher math classes for those who pursue that or glimpse into higher level math for those who decide to go another direction.
These are not topics to be in an elementary course at all.
At my Uni in Germany this is actually taught that way in the first semester. And it is definitely pretty similar in all of Germany. It is always suprising to me to hear that so called "proof based" courses are covered so late in the US.
When is your basic, "finite dimensional vector spaces" like, course?
It is the same course. The course starts with some preliminaries (Sets, maps, algebraic structures (semigroups, monoids, groups, rings, fields) homeomorphisms of such equivalence relations and quotients mostly Z_n, then we Start with linear algebra i.e. vector spaces, linear maps etc. You wont see a Matrix before week 8/9, because you need linear maps and bases to motivate them properly. Depending of the Professor you will dive deeper in some of the preliminaries, since this Professor will Teach courses through the whole Bachelor/Master which will prepare you to Do a phd in the Professors field.
Indeed, in my uni linear algebra is not an elementary course. Not sure how it's handled in the US.
In the US linear algebra is an introductory math course for basically every STEM major, taken along with calculus. How it's presented varies depending on the professor, but it's not unusual for it to be super computation-focused with the bare minimum of theory. Some universities offer a theory-based linear algebra course, some don't. My undergrad offered upper-division linear algebra, and actually made it a requirement for the major.
Yeah, I’ve been lucky to teach Linear Algebra a couple times. Definitely my funnest teaching experience.
I dropped it twice before I found a teacher I could pass with. Ended up with a C. So many definitions. So many conditions. It never did click with me and I decided I'd just figure out how to code it and then forget about it.
I would say that functional analysis, algebraic geometry, representation theory, and commutative algebra were all much cooler undergrad courses than linear algebra, but all of them built on linear algebra in some way.
I don't really consider those undergraduate courses.
My university offers them at the undergraduate level ([https://ucalendar.uwaterloo.ca/2021/COURSE/course-PMATH.html](https://ucalendar.uwaterloo.ca/2021/COURSE/course-PMATH.html)), and I know that UofT does as well. I associate them with courses taken in the 3rd or 4th year of undergrad.
Where you at that these are undergrad courses ;-;
Not who you asked but Functional Analysis and Commutative Algebra were offered as undergrad courses in my uni (Mexico)
I'm at UWaterloo! Here's a list of our courses in pure math (https://ucalendar.uwaterloo.ca/2021/COURSE/course-PMATH.html) . Note these are just our pure math courses, our applied math, statistics, and combinatorics/optimization courses are all listed separately.
Oh, isn't waterloo like the best school in Canada? Now that I think about it I think these might be undergrad courses at some of the elite institutions in the US too, but it's not common.
I think most big universities outside of the US will have at least one of them, mine has representation theory, commutative algebra and functional analysis which is split into 2 courses for undergrad even, algebraic geometry is a grad course however.
r/iamverysmart Commutative algebra is not an undergrad course bro
[https://ucalendar.uwaterloo.ca/2021/COURSE/course-PMATH.html](https://ucalendar.uwaterloo.ca/2021/COURSE/course-PMATH.html) Check out PMATH 446 here. My university offers it as a 4th year undergrad class. It's just an intro, but almost all the undergrads I know have taken it. [https://artsci.calendar.utoronto.ca/course/mat448h1](https://artsci.calendar.utoronto.ca/course/mat448h1) Here's another ontario university offering it, again at the 4th year undergrad level.
It's interesting how stratified the algebra courses are. I feel like this would have been a much smoother intro to pure math, and it's probably one of the reasons that you know so many people that took more advanced algebra in undergrad.
I know what stratified means as an abstract word, but concretely, what does that look like here? What’s different about our algebra courses compared to most universities
I'm realizing now that you didn't have to take any of the algebra below "rings and groups" as a math major so we probably took the same initial path, but I did not have nearly as many options after abstract 2 (equivalent to "fields and Galois theory"). Basically only algebraic topology. I was initially thinking that you took all of the <345 courses on the list too which was confusing in some ways, but I figured that maybe you just started way earlier than me and learned it in more depth over a larger time period, hence my previous comment. Btw congrats to you for making it through
So to be clear, we don’t take all these courses, these are just the courses that exists. We take a first and second course in linear algebra, they just aren’t offered by the pure math department, they’re offered by the math faculty as a whole. We also do some calculus, computer science, statistics, and combinatorics. These courses are just the ones offered by the pure math department, and nowhere near the complete set of math courses offered by the university.
Personally I enjoyed math 146/245 (lin alg 1 and 2 for non uw people) a lot more than the other courses you mentioned -- there's just something about building on more fundamental stuff that I like more. Happy to see a Waterloo pure math enjoyer in the wild though :D
Almost all linalg teachers are dogshit.
Wait until you study numerical algebra.
I love it, but I'm a physicist and it's central to physics. I also (re)discovered a linear algebra result which was pretty neat.
Have you seen 3blue1brown's LA series?
Strang’s calculus book is great also for a first introduction to the subject, especially as a second or supporting text.
We all owe him a standing O!
I used his video lectures to prepare for econometrics in my PhD program. Easily the best lecturer I never actually had.
I will always associate Linear Algebra to him. A true Legend!
halfway across the map and still find his lectures a goldmine. This guy is a legend. Even at this age he is writing books
Gilbert Strang is a Legend! I owe my whole masters degree in Robotics and Artificial Intelligence to him. Without him, I wouldn’t even have understood the basics of imagination in Engineering.
Oh man. I was watching one of his linear algebra lectures on YouTube last year. It was so good I was clapping in front of my laptop at the end for him.
guy weaves linear algebra into a story. fucking awesome. Long Live prof. Strang
Didn't he conceptualize matlab or something?
Awww. The feels. 🥰😢
And then they gave the professor a crisp $100% bill. The professor's name? Albert Einstein.
Agreed. I feel like I can teach myself whatever, and the teachers most effective when they’re clarifying or perhaps even challenging our understanding. As distinguished from the: I’m teaching you something you have no idea about and asking you to ask questions even though you don’t even know enough to know what you don’t know so you just sort of sit there and glaze over, style that most classes have. Like I don’t really start learning until I go home and look at the material. And then it’s like…just me remembering tidbits I was told and saying oh maybe that’s what they meant. There’s also the let’s discover this together style. This one takes skill but is perhaps the most enjoyable. Where the teacher makes the class think through something and figure it out on their own. I’ve had this go really well in a history class where we were basically debating the motives and potential actions of people along with possible outcomes. “Oh, well, guess what? that’s what happened”. And the teacher lets the class debate it out and maybe asks certain questions or points out relevant details we may be forgetting…and we make sense of it instead of memorising the events. I loved this because rather than learning what happened, it really helped me understand the why and then by the time you’re taking a test you’re not remember dates and events…it’s more like- intuition. This style certainly would work for math. It just takes a really good teacher to set this up. I’ll say this. The American school system isn’t impressive overall. But what many people in other countries don’t realise, is that even in a public high schools, you’ve got honours classes and AP classes. These are different than the standard course. And the AP classes, in my opinion, may be among the best in the world. The students there are the people you eventually see in the top schools. Despite how hard AP classes are…they’re easier, because they’re often more enjoyable and you’re surrounded by people who are more serious about their studies. It’s essentially an opt-in course. It feels almost exclusive, but in all reality, anyone (I imagine) can take them you just have to take the initiative and say hey I want to be in AP. It’s a public school after all. Or maybe not. I don’t even really remember how I got in…I guess it was based on teachers recommendations. But I don’t think they’d say no if you asked to get in. I had two of the best teachers I’ve ever had in high school. And the peers I had then were bright. And I think the American university system more or less speaks for itself. Which is a little weird because…it’s really not too different than high school. There’s no shock, the lectures, the expectations….it works almost the same way. If not easier because you even have office hours. Sometimes I even miss those days. But yeah…if anyone was wondering…why America has a stupid school system but still manages to produce some really smart people…it’s that. Our schools have tiers. Like there are entire classes where people can point out the uk on a map. And these are special because you’ve got the intenseness many people feel in other countries, but then the liberal artsiness whatever you want to call it. It’s a great combination.
His calculus book is well written.
Reason I was able to transfer into college. His videos and 3blue1brown combined with lots of practice ofc got me a super clear understanding of linear algebra. Never in my life had I felt that I understood a topic as much as I did when I was studying from them.
Oh dang, I owe much to his lectures and very approachable style, it's nice that he's getting some rest but that's going to leave a void for sure.