[HN Gopher] Lessons I wish I had learned before teaching differe...
___________________________________________________________________
Lessons I wish I had learned before teaching differential equations
[pdf] (1997)
Author : itronitron
Score : 283 points
Date : 2023-11-13 09:36 UTC (13 hours ago)
(HTM) web link (web.williams.edu)
(TXT) w3m dump (web.williams.edu)
| itronitron wrote:
| Previous discussion from 2022...
|
| https://news.ycombinator.com/item?id=32530035
| ianai wrote:
| They talk (rightly) about mathematics replacing infinitely
| small but not 0 numbers with limits there. Infinitesimals were
| later reintroduced with rigor through
| https://mathworld.wolfram.com/NonstandardAnalysis.html
|
| I have long thought such would be easier to work with on modern
| computers.
| monista wrote:
| Note though, that nonstandard analysis isn't compatible with
| more "intuitionistic"
| https://en.wikipedia.org/wiki/Axiom_of_determinacy (in place
| of axiom on choice), which free you from Banach-Tarski
| paradox and have some other appealing properties.
| zozbot234 wrote:
| On the contrary, in an intuitionistic setting you can use h
| ttps://en.wikipedia.org/wiki/Synthetic_differential_geometr
| ... and
| https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis
| which aren't nearly as easy to phrase classically. (Sure
| you can phrase things in terms of toposes, but that's just
| a roundabout way of saying that you're operating in a
| constructive setting.)
| yuppiemephisto wrote:
| Can do it in pure ZF:
|
| https://www.youtube.com/watch?v=YE68yptMMeQ
|
| https://u.cs.biu.ac.il/~katzmik/spot.html
| scythe wrote:
| Nonstandard analysis is one of those things that sounds nice
| in the abstract, but even though I first learned about it
| more than ten years ago, I have yet to see a worked didactic-
| level example of how you would, for example, obtain the
| derivative of sine using nonstandard analysis, or derive the
| natural logarithm.
| ducttapecrown wrote:
| I literally saw the sin one yesterday, so I'll rewrite it!!
|
| 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)
| yuppiemephisto wrote:
| https://alok.github.io/2022/09/12/y539043/
| adgjlsfhk1 wrote:
| Infinitesimals are commonly used in scientific computing (see
| forward mode automatic differentiation).
| contrarian1234 wrote:
| "Why is it that no one has undertaken the task of cleaning the
| Augean stables of elementary differential equations? I will
| hazard an answer: for the same reason why we see so little change
| anywhere today, whether in society, in politics, or in science"
|
| 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
| gleenn wrote:
| Came here to quote the same exact part. The author is hilarious
| and savage but it seems appropriately. Amazing prose and sad
| state of affairs
| tux1968 wrote:
| > 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
|
| 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.
| ajkjk wrote:
| The problem isn't writing a better text, it's having
| departments willing to teach it, or to try alternate
| approaches at all.
| tux1968 wrote:
| The OP referenced the Khan Academy as an attempt to
| displace the ossified educational institutions we have. But
| it has its own limitations and challenges, some of which
| will soon be rendered moot by the coming wave of AI
| technologies, perhaps giving it the final push it needs to
| truly take hold.
|
| 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.
| robertlagrant wrote:
| This might work well for knowledge/IC-based attainment.
| There are still other groups, who are possibly the
| majority when combined:
|
| 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.
| tux1968 wrote:
| Agreed. One size does not fit all. It's just the hope
| that options like Khan Academy are made truly viable for
| those who are well served by such options.
| siftrics wrote:
| You have to have a pretty helpless attitude to blame
| someone else (teachers) on your inability to learn from a
| good textbook.
|
| How are you ever going to learn anything truly novel in the
| world if you always delegate teaching to a teacher?
| SaintRomuald wrote:
| I shouldn't need to read a 500page textbook to understand
| a class that a teacher has been 'teaching' for 20 years.
| He's there to give me an understanding of the subject. I
| have at least 5 classes each semester which are very
| technical and difficult, it's impossible for me to read a
| textbook on each while also reading a 20-30 page lab
| instruction each night, all written in a technical
| language which I barely understand. I can try but so many
| times constant consumption of this literature lead to
| burnout
| throwaway98797 wrote:
| no one is there to give you anything
|
| 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
| HelloNurse wrote:
| "Technical and difficult": there are both important
| concepts and broad strokes that your teacher should be
| good at explaining, and important details that should be
| explored in a textbook because they need good pictures,
| data, editing etc. more than good explanations.
|
| 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.
| angra_mainyu wrote:
| The entitlement in your comment is staggering...
|
| Why such a desire to be spoonfed, whining about having to
| put in the work?
| SaintRomuald wrote:
| Am I not supposed to expect a quality lecture from a
| university professor? I have had such a bad experience
| with certain lecturers, I'm certain what they were saying
| wasn't incorrect but the way they were speaking made it
| impossible to follow. It wasn't just me who thought so. I
| admit I've had learning difficulties my whole life but I
| believe it is actually impossible to learn all the
| material from the classes with all the details from
| books. You can pick a field or two which interest you the
| most and delve deep into the details. An extremely
| fundamental topic at my uni is called "introduction to
| circuit analysis". It was taught by a professor who was
| so bad at teaching it there was no point in coming to his
| lectures. You would learn more from watching youtube
| videos. Everyone was aware of the state of that class.
| Even our lab professors would mock his teaching style.
| Now if every class was like this we would be lost. I'll
| give a recent example of another lecturer who I also
| consider bad, but fortunately he is teaching an easy
| subject. First class a kid asks 'what's the difference
| between a router and a switch'. The professor answers 'a
| switch works on layer 2 and a router works on layer 3'.
| In my opinion this is a non answer. Yeah he said the
| truth but it didn't mean anything. If that student didn't
| know the difference between a switch and a router then
| surely he doesn't know about the ISO OSI model. Before
| you think he probably didn't have time to answer more
| specifically to make sure he understands. This guy
| notoriously keeps digressing from the subject. As that
| classes subject is one I am familiar with I knew the way
| he taught it was impossible to make sense of because he
| never answered in a concrete way, everything felt so
| drawn out and lacking a focal point, it's hard to
| describe. It took him four hours to go over the ISO OSI
| model and honestly while I wasn't paying attention to
| every single word he said. With all the digressions and
| constant comments about stuff you just couldn't follow
| his train of thought. On the contrary electromagnetism
| class was absolutely amazing. It's still a difficult
| subject, at least for me because I've always been pretty
| bad at physics but the lad who taught it was brilliant.
| He made sure we all understood, he started with very
| basic concepts that everyone knew, just to be certain
| everyone is on the same page. His presentations were
| concise, easy to follow with links to read up/watch stuff
| that you didn't fully get or wanted to learn more about.
| They had plenty of visualizations with gifs of stuff.
| It's a pleasure going to his classes. So I think that I
| should expect the lectures to give me an understanding of
| a subject. If I had to read a book for every class there
| wouldn't be enough time in a day for me to catch up.
| tekla wrote:
| We call this "University"
| ajkjk wrote:
| You are just wrong. There is a lot more that goes into
| teaching a subject better than 'generating a better
| textbook'. There are thousands of hours of work that need
| to happen to figure out _how_ to make the subject make
| more sense. It is not a question of 'writing down known
| result sin a different order'. The subject needs to be
| reworked in its entirely. Advanced results need to be
| made elementary. Simple formulations of complex ideas
| need to be discovered, articulated, and developed into
| exercises. Out of every available model the best models
| need to be picked and refined and tested. Etc. And then
| you have to get everyone on board with the project!
|
| Almost none of the work is 'rewriting the book.'
| gumby wrote:
| Then you could simply read the script -- no need for the
| video at all.
| tux1968 wrote:
| For those who prefer to read the script, of course. For
| those who benefit from a much more multimedia presentation,
| the generated video and associated generated items, will be
| preferable.
| TheOtherHobbes wrote:
| The video won't be generated by AI, because there will no
| video.
|
| 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.
| tux1968 wrote:
| > there will no video...
|
| 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_suffocated wrote:
| Just out of curiosity, what was that classical mechanics
| textbook?
| kdavis wrote:
| Goldstein?
| contrarian1234 wrote:
| Pretty sure it was An Introduction to Mechanics Kleppner,
| Kolenkow
|
| The book was probably okay-ish, the point is that surely
| someone could have made some improvements in the past few
| decades
| selimthegrim wrote:
| Doesn't it use cgs only (I think the new editions finally
| bit the bullet and switched to mks)
| itronitron wrote:
| Khan Academy is excellent for Calculus however their material
| for Differential Equations is rather minimal since they don't
| really go into university level material.
|
| 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.
| Baguette5242 wrote:
| > "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."
|
| 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.
| mnl wrote:
| What's the problem with a good book, age? Baby Rudin's first
| edition is 70 and the latest one is from 1976. It's still
| widely used and will be for a while.
|
| 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.
| contrarian1234 wrote:
| You're conflating things. This issue isn't that Classical
| Mechanics has somehow evolved or changed. It's that people
| continue to find new and better ways to explain and
| illustrate concepts. At lest personally I don't know of any
| field or book where I've felt "Hmm, this is basically perfect
| and I can't imagine a better way to explain these concepts".
|
| 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
| distcs wrote:
| > Have you looked at for instance Khan Academy's Grant
| Sanderson (aka 3Blue1Brown) Math videos?
|
| 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.
| contrarian1234 wrote:
| I didn't mean to say videos are better - just as far as I
| can tell that's where the most creative new teaching
| techniques are on display. I'd definitely prefer they
| were in the written word. Especially if they were an open
| collaborative effort. Books are great for flipping back
| and forth with. You have an "ah-ha" moment and skip back
| several pages and reread something you misunderstood on a
| first read-through. It's somehow clunky and takes you out
| of the flow when you do it in a video.
|
| 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
| threatofrain wrote:
| Note that 3B1B often warns you that his videos bring
| perspective for a book/class that you're doing or are
| already done with. And Khan Academy's focus is for K-12.
|
| If you want anything past Analysis 1 I think you'll find
| that universities guard their content.
| da39a3ee wrote:
| > 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.
| elteto wrote:
| Really? Can you link some for Analysis and beyond?
|
| 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".
| da39a3ee wrote:
| What about these amazing resources from Daniel Murfet at
| University of Melbourne:
|
| 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/
| elteto wrote:
| Fantastic, thanks!
| dboreham wrote:
| > I think you'll find that universities guard their
| content.
|
| 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.
| mindcrime wrote:
| _I 've given up on these online learning media. Back to
| textbooks. The difference is like night and day._
|
| 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.
| distcs wrote:
| > 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.
| mindcrime wrote:
| Fair enough. For me personally, I find great value in
| jumping back and forth between different modalities,
| where the different presentations reinforce each other.
| But what works for me may not work for everyone, and
| vice-versa.
| da39a3ee wrote:
| I think you're just parroting things you've heard other
| people say. 3b1b's videos are universally agreed to be
| excellent, and it's baffling that you think it is a
| choice between watching them and using textbook and doing
| the exercises. Anyone with the intellectual capacity to
| study that sort of material is not going to have a hard
| time comprehending that they are intended to be
| complementary, as Grant Sanderson makes very clear at
| numerous points.
| andyferris wrote:
| Hmm - I do wonder if for very particular things in physics
| their heyday has come and gone? Were there more
| ridiculously talented inviduals deeply steeped in classical
| mechanics and discussing amongst themselves in the past? I
| feel modern physicists move onto high-energy phyics or low-
| energy physics research pretty early in their careers...
| mnl wrote:
| There are excellent textbooks on Classical Mechanics,
| probably because it's a crystal clear subject and you can
| give a detailed account of the essentials in a single
| volume without handwaving. Of course everything can be
| improved, but it also can be muddled. If it works think
| twice before fixing it. Kind of what happens with Rudin and
| introductory Real Analysis.
|
| 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.
| contrarian1234 wrote:
| I think you have the perspective of somehow who's
| succeeded in Physics. In your typical introductory
| physics class less than half the students will walk away
| with a very solid understanding of classical mechanics.
|
| 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
| mnl wrote:
| I almost wrote above that, to my knowledge and IMVHO, no
| one has succeeded at writing a book on Thermodynamics
| yet. I self-censored because that would be too flippant,
| wouldn't it? Lmao
| rawgabbit wrote:
| I took thermodynamics over 30 years ago. I remember
| having the feeling of learning a different language using
| a text book whose explanations also needed translation. I
| remember the book explained Entropy using chaos theory or
| randomness and talked about popular philosophy during the
| 19th century. After a bit of mental torture, I realized
| by Entropy they really mean Thermodynamic Stability. It
| is just that heat usually dissipates when materials
| touch, they were using words like chaos or randomness to
| describe the process. But their description was vague and
| poorly conceived.
| AnimalMuppet wrote:
| > In your typical introductory physics class less than
| half the students will walk away with a very solid
| understanding of classical mechanics.
|
| 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.
| BeetleB wrote:
| > 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
|
| 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.
| BobaFloutist wrote:
| Why should pedagogy ever end? That's like saying at some
| point in health care medicine must end and you're
| responsible for your own treatment. What are the
| professors for, doing research and abusing their grad
| students?
| mnl wrote:
| The idea is becoming intellectually independent, arriving
| in the stage of self-pedagogy if you like. Peer learning
| when there's the chance.
|
| 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?
| Folcon wrote:
| You seem to have a very individualistic notion of
| education.
|
| 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?
| mnl wrote:
| I mentioned peer learning, collaboration is that.
|
| 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.
| lordnacho wrote:
| When you get to university, the lesson is very much that
| you have to learn things yourself. I found that the more
| decorated the professor, the worse he was at explaining
| anything, due to some mixture of being unable to go back
| to a state of ignorance and being in a seat where his
| main responsibilities are elsewhere (grant
| applications!). I'm talking about 1or2-to-one tuition
| here, done several times a week to kids who did very well
| in high school.
|
| 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.
| starcraft2wol wrote:
| > people continue to find new and better ways to explain
| and illustrate concepts
|
| 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.
| da39a3ee wrote:
| Then you are objectively wrong. Look at Grant Sanderson's
| explanation of introductory Linear Algebra. Very
| obviously it adds something good to complement the best
| paper textbooks on the subject.
| cyberax wrote:
| > 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.
|
| 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.
| fn-mote wrote:
| 1. I think the research shows that increasing the
| cognitive load increases the retention. So in general,
| when it is harder to learn something, you retain it
| better.
|
| 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).
| WalterBright wrote:
| I've suggested to college students that they leave their
| laptops behind and attend lecture with pen and notebook,
| and take notes. It makes things a lot more sticky in the
| mind.
|
| And do the homework problems. You'll never understand the
| material without doing the problem sets.
| washadjeffmad wrote:
| I had a 1930s edition of a "radio physics" textbook as a
| child, and because there could be no prior assumptions about
| exposure or familiarity, it was filled with very complex
| ideas explained so cohesively and coherently that, well, I
| could understand.
|
| 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.
| narrator wrote:
| Do you have any ore information on that book? I would like
| to dig it up at a local library. Is is this one perhaps? ht
| tps://archive.org/details/dli.ernet.16412/page/n9/mode/2up
| BenjiWiebe wrote:
| I'd like to know which book it was.
| commandlinefan wrote:
| > complex ideas explained so cohesively
|
| 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?"
| Raidion wrote:
| I feel like teaching is mostly a one size fits all
| endeavor, but people learn and think in different ways,
| just like a processor is optimized for some operations
| but not others.
|
| 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.
| marcosdumay wrote:
| The problem is that there is no good textbook for Classical
| Mechanics out there. At least not on an introductory level.
|
| 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...)
| mnl wrote:
| I learnt a lot of physics from Marion's 2nd edition (not SR
| there though). An older and completely forgotten fine
| textbook is W. Hauser's _Introduction to the Principles of
| Mechanics_. Then you jump into Goldstein (the 1980 one,
| again not SR there). It 's a good idea to buy any Schaum
| book from Spiegel about this too, also for vector analysis
| if you can't take a course on that.
| marcosdumay wrote:
| Yeah, in retrospect, I've set myself for failure with
| that universal claim.
|
| But the proof of failure is instructive :)
| elteto wrote:
| Taylor's is fairly "introductory" and it's fantastic. Very
| well written.
| dustingetz wrote:
| No Bullshit guide to Math and Physics
| https://minireference.com/
|
| (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?)
| hoseja wrote:
| The problem is that the pedagogy, as in, the best most
| effective way to teach people things, _should_ improve
| massively in 70 years. Whether it actually does, that is
| another question.
| housecarpenter wrote:
| Should it really though? Maybe pedagogy is just hard and we
| can't expect constant progress.
| peterbell_nyc wrote:
| Pedagogy, like any other field is constantly developing.
| We are continually learning new and better ways to teach
| materials and thus I think the continued evolution of
| teaching materials has the potential to be a good thing
| (and I've created curriculum for professional learning,
| for grad school and for bootcamps).
|
| 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.
| irchans wrote:
| I think that researching pedagogy is very difficult, and,
| like many scientific fields, it is hard to reproduce
| results found in papers. (I am not an expert in this area
| -- I am just a former teacher with around 10 years
| teaching experience.) One of the main things I notice is
| that standardized test scores are not really improving. I
| think that high school students today would score about
| the same as high school students in the 1980's if they
| were given the same multiple choice tests. This implies
| to me that the field has not advanced a lot. I do think
| that LLMs and other computer based teaching could help.
| BobaFloutist wrote:
| The problem is, like many social sciences, people refuse to
| apply the scientific method to pedagogy and would rather
| lean on vibe-based "theories".
| sheepshear wrote:
| Your response does a pretty good job demonstrating the
| internal challenges universities face. The interest in making
| a change was motivated by unmet needs, and it's not going
| away until those needs are met. The only way to deal with
| people who won't meaningfully engage in the problem solving
| process is to go around them.
| mnl wrote:
| A textbook should be provided for reference. Copying its
| contents on a blackboard isn't teaching. You still have to
| design your course. There's goals to meet, you have to
| evaluate where your students come from and your job is
| getting them there.
|
| 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.
| sheepshear wrote:
| I actually completely agree with everything you wrote.
| I'm also very familiar with the ongoing battle between
| education and training.
|
| 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.
| mnl wrote:
| I'm very adaptable nowadays. It's just that I think I
| know which things should work, but I'm not fighting
| society, particularly not on this.
|
| 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.
| sheepshear wrote:
| The old one-size-fits-some approach is as pragmatic as it
| gets. Society's education needs have grown beyond what
| the old model can adequately service. We need new
| solutions that don't cause regressions on the old
| solutions.
|
| 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.
| lanstin wrote:
| I took Lebesgue integration and some pretty high powered
| group theory classes in the 1989s and am retaking them now,
| and I have to say the presentation now is much better. I
| think having Terrence Tao blog about how stuff really works
| makes a difference, at least for lower level grad classes. :)
| and trying to present groups as the widely useful
| abstractions they are instead of just a cute self contained
| theory makes a difference.
|
| 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.
| j7ake wrote:
| This masochism also applies to classical music (eg violin).
| Decades of rote before you're allowed to express your own
| musical thoughts.
| marcosdumay wrote:
| There are plenty of alternative teaching methods for
| instruments. (TBH, I don't even know what you are referring
| to by "classical music", but yeah, some places can be quite
| oppressive.)
| gosub100 wrote:
| I was going to make a variant of this comment.
| $FAMOUS_COMPOSER's $REALLY_HARD suite is right there, you can
| get the sheet music and it clearly says "play this note, then
| play that one, then stop for a specific duration...and so
| on". That _doesn 't_ $EXPLETIVE _mean_ that if you watch Khan
| Academy 's video on how to play notes[1..n] you can play the
| music according to the composer's specification.
|
| 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.
| financltravsty wrote:
| > 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.
|
| 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.
| marcosdumay wrote:
| Yeah, the high-capability outliers are used to coast in any
| subject and may or may not have time to adapt to the
| laborious requirements before failing.
|
| 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.
| pif wrote:
| > Both fields embrace a sort of masochism and active desire to
| keep knowledge impenetrable
|
| 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.
| michaelcampbell wrote:
| > > 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.
|
| > 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.
| pif wrote:
| ???
| jlokier wrote:
| For what it's worth, I was a researcher at CERN with
| colleagues from multiple countries, where I _turned down_
| an offer to do a PhD in particle physics :-)
|
| (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.
| pif wrote:
| What I meant is that I think I know something about
| academics in science in general, and in Physics in
| particular.
|
| 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.
| baz00 wrote:
| I don't think it's that at all. The real cause is just some
| entirely institutionalised academics wrote the textbooks and
| everyone else is copying it rote as if it's the way of doing
| everything because that's all they know.
|
| 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...
| JadeNB wrote:
| > 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.
|
| 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.
| contrarian1234 wrote:
| I don't think the line of thinking is directly "lets make
| this painful on purpose" but there is a sense of - everyone
| has to go through the same gauntlet and some people are just
| not cut out of math/physics/etc. It's just the way of the
| world. There is not direct desire to make the experience
| better/smoother/easier than it was your generation.
|
| 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
| zozbot234 wrote:
| > 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.
|
| 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.
| JadeNB wrote:
| > What you should be "delighted" by is when you find a new
| way to explain something that resonates with most of the
| class
|
| 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.
| quacked wrote:
| I don't know why you're catching downvotes, I think you're
| completely right. In my engineering curriculum in some
| cases the teachers were openly hostile to students that had
| a hard time understanding the material.
|
| 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.
| JadeNB wrote:
| > In my engineering curriculum in some cases the teachers
| were openly hostile to students that had a hard time
| understanding the material.
|
| 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.
| quacked wrote:
| > 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.
| JadeNB wrote:
| > 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.
|
| 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).
| selimthegrim wrote:
| I mean, have you ever taught a class? Do you know how it
| feels when we make sure a topic is covered both in class
| and recitation with plenty of time for questions and the
| students still stuff it?
| quacked wrote:
| No, I've never taught a class, so it's possible that the
| experience of teaching as a professor would irrevocably
| change my opinion. However, I have spent a lot of time
| tutoring math and directing bands and choirs, so I'm
| familiar with the frustration that comes from explaining
| concepts and then assigning necessary work with plenty of
| time to do it and then having people still show up
| unprepared.
|
| 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.
| mhh__ wrote:
| There is nuance to this.
|
| 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.
| yawboakye wrote:
| given the current political environment i think we should
| protect and promote old textbooks as much as possible. they're
| lean, devoid of ceremony, makes no assumptions about
| race/gender/political orientation/socioeconomic status of the
| students as they work their way through the foundations of the
| subject. for a modern example of mumble jumble junk, with so
| much presuppositions about your intelligence, see the rust
| book[0]. it's a tough book to study because it wastes so many
| words saying nothing at all. many people have used it yet rust
| lifetimes and borrow checker semantics is as arcane and elusive
| as the holy grail.
|
| [0]: https://doc.rust-lang.org/stable/book/
| zopa wrote:
| Curricula are hard to change for the same reason widely-used
| apis are hard to change: it's the downstream dependencies.
| There's a shared understanding of what "took undergrad
| differential equations" means, and breaking from that has a
| cost. If you teach differently, your students will be at a
| disadvantage because they won't know the things futre courses,
| tests and employers expect them to know. Maybe that will be
| outweighed by the objectively superior education you'll have
| provided with your bold new approach, and maybe not.
|
| 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.
| dooglius wrote:
| One argument TFA makes is that a lot of the material is not
| ever subsequently used.
| kjkjadksj wrote:
| Otoh biology classes don't suffer from this as much.
| Textbooks are routinely updated to reflect new
| understandings. Professors seem to have a lot of choice with
| what they cover, often highlighting their own subfield where
| they can speak more to it. Physics classes in comparison felt
| like a factory. You'd also pay twice as high a lab fee as any
| biology or chemistry lab to play with 40 year old brass
| weights and frictionless cars.
| brocha wrote:
| Makes me think of Jackson's electrodynamics - yit always seemed
| to me a hazing ritual. I remember a story of someone asking him
| a question from the book and he basically said "how the hell
| would I know?"
|
| I enjoyed the book and class myself but it did not make me a
| better physicist, just a better mathematician
| BeetleB wrote:
| > 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.
|
| 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.
| trashtester wrote:
| > The system acts an an informal IQ test
|
| 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.
| frognumber wrote:
| I have a lot of expertise in this domain:
|
| > 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.
| Q_is_4_Quantum wrote:
| >I really wish textbooks with open licenses would take over and
| they could be reworked and improved year after year by
| different people
|
| 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...
| EsportToys wrote:
| Speaking of 2nd-order linear ODEs w/ const factor (a.k.a. mass-
| spring-damper systems), I wrote a blog post[0] deriving all
| possible general solutions in a concise matrix that makes it
| easily implementable in code.
|
| 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
|
| [0] https://esporttoys.pages.dev/2022/11/21/damped
| pastage wrote:
| This is a good example. That was math heavy, it took me too
| long to parse for basic understanding and honestly I am not
| sure I got it right. The LUA code was easy to understand even
| though those variable names force you to read all the code, and
| guess what is what.
|
| 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.
| wiredfool wrote:
| My grad school was basically working through F=Ma-Cv-Kx in all
| it's expanding varied glory, working up to full blown finite
| element analysis.
|
| Ultimately it all comes down to choosing the most convenient
| basis functions for the questions you're answering.
| JadeNB wrote:
| > My grad school was basically working through F=Ma-Cv-Kx in
| all it's expanding varied glory, working up to full blown
| finite element analysis.
|
| If I'm understanding correctly, you want another = there for
| a standard damped spring: F = ma = -Cv - Kx.
| denotational wrote:
| F is a forcing function, not the resultant force. It's
| often arranged this way with all the derivatives on one
| side (as opposed to having the resultant force, ma, on one
| side of the equality by itself) so that it matches the
| general form of a non-homogeneous second-order linear
| differential equation.
|
| (At least I assume this is what the original commenter
| meant!).
| thorin wrote:
| If when I started learning about calculus at 14/15 someone had
| explained why we were doing this it would have made it so much
| less confusing. Explaining in terms of
| speed/distance/acceleration seems to make complete sense now (one
| of several potential examples). However discussing a function and
| talking about splitting in into small (infinitesimal) strips of
| delta quantities and a list of equations / proofs made it
| completely dry and uninteresting. I don't think I had any ideas
| one what was going on until it was mentioned in physics several
| years later.
| ajkjk wrote:
| Physics doing the work of actually teaching math is a universal
| embarrassment to math departments. Or, well, it should be.
| f1shy wrote:
| I remember my first calculus exam at the University: I made a
| lot of explanations and justifications based of physics. Of
| course I did not pass. As I asked why? It was all correct,
| right? The prof. said "is perfect reasoning, but this is the
| math department, physics is one floor below"
| DoughnutHole wrote:
| He's right - intuitive physical reasoning is useful (and
| often valid in the applied sciences) but it isn't rigorous
| mathematical proof.
|
| Some areas of mathematics are applicable to physical
| realities, but they aren't defined by those realities.
| ajkjk wrote:
| Yeah it's not rigorous. The tragedy is that the
| organizations responsible for teaching math, like how to
| do it and how to understand things mathematically, _for
| other purposes_ , would rather waste their time on rigor.
|
| 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.
| eigenket wrote:
| I quite strongly disagree with this. Mathematicians
| aren't rigourous just for the hell of it. We have
| examples where a lack of rigour caused a bunch of people
| to waste a bunch of time working of stuff that ultimately
| didn't work at all.
|
| 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.
| zozbot234 wrote:
| On the other hand, if you want full and complete rigor
| that means formalizing things in a computer-based proof
| checker. Now, that can often reveal ways to refactor the
| argument into a cleaner, more elegant, more intuitive
| proof, which is why full formalizations are often
| regarded as publishable work - but the other side of it
| is that dotting all the i's and crossing all the t's does
| take a whole lot of effort since some published proofs
| are not nearly as clear as they're supposed to be.
| eigenket wrote:
| I agree that some published proofs are not as clear as
| they're supposed to be, but I disagree that formalizing
| things in computer-based proof checkers is the way
| forward there. Computer proof checking is cool, but its
| an augment to what we already do and not not a substitute
| for it.
|
| 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.
| ink_13 wrote:
| > No problem with there being a rigorous math department.
| For mathematicians. But, quite literally, no one else
| cares
|
| 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.
| ajkjk wrote:
| That's just false. What physicists (and engineers, etc)
| are doing is also math and it's way more useful and
| insightful.. and (imo) the world would be a better place
| if that's what everyone else was also learning in
| college. The idea that math is _only_ about rigor isn 't
| intrinsic; it's a historical accident that people seem to
| not realize is optional. It is also about understanding
| things and being able to wield concepts to accomplish
| goals or convey understanding to others.
| ink_13 wrote:
| Novelists and lawyers both do writing, but I wouldn't
| call their output the same thing. You might call legal
| writing more useful or novelists more insightful, but
| that's a matter of opinion. One is not intrinsically more
| valuable than the other.
|
| > 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.
| nuancebydefault wrote:
| > Study of things that are provably true.
|
| 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?
| ink_13 wrote:
| I don't even understand the question. Is there in truth
| no beauty? Why do anything that doesn't add value to the
| physical world?
| nuancebydefault wrote:
| It seems you did understand the question and I thank you
| for your answer, it is helpful.
| nuancebydefault wrote:
| If some mathematics are not applicable to physical
| realities, to what are they applicable? Is there any un-
| applicable math and if so, why does it even exist?
| jampekka wrote:
| Maybe depends on what's meant by application, but I'd say
| a lot of, probably most, research-level math has no
| (clear) practical applications and most research is not
| done from the PoV of applications or relationship to
| physical reality. Math is study of formal systems, sort
| of "what can and can't follow from some set of formal
| rules".
|
| 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_Effe
| ctivene...
| nuancebydefault wrote:
| So if the application is not visible today, it might
| appear tomorrow. In fact that makes sense.
| sampo wrote:
| Up until about World War II and use in cryptography,
| number theory was pretty much an un-applicable field of
| mathematics.
| nuancebydefault wrote:
| Pretty amazing that someone discovered and explored
| number theory without knowing if there would be an
| application to it. Hopefully they lived to see its usage.
| guhidalg wrote:
| Fiction writing exists, should it not be enjoyed?
| Sometimes mathematicians invent things before scientists
| discover how to apply it.
| monista wrote:
| And they were right imho. Having a clear notion of a
| subject is good for both physics and mathematics, but
| what's good for physics or engineering, is not enough for
| math, in math, you also need rigorous reasoning, or you'll
| fall into subtle mistakes.
| HelloNurse wrote:
| Fundamentally, without _sufficient_ rigor there can be no
| proofs; "maybe" and "someone says so" aren't good
| theoretical building blocks.
| regularfry wrote:
| It's worse than that. I remember that when I was doing
| physics aged 14/15/16, they weren't allowed to assume that we
| were taking the maths course that included calculus, so while
| you could see these repeating patterns crop up (half-thing-
| squared, you say? How peculiar) they were effectively
| prevented from taking the time to explain it properly. It
| would have made things _so much simpler_.
| Workaccount2 wrote:
| Many math teahcers (and likely mathematicians) could not care
| less about the real world applications of math. My college
| calculus teacher had no interest in physics or science. She
| liked math in the same way someone would like doing a
| crossword or Sudoku. A puzzle game with a well defined rule
| set and the challenge of finding answers.
|
| 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.
| anon291 wrote:
| > 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)
| ajkjk wrote:
| Personally I don't care that it's about _physical_
| things. But I care that it 's about _doing_ things:
| understanding concepts intuitively, solving problems,
| being adept at thinking mathematically. The thing that is
| not useful is tedious definitions and tedious proofs.
|
| 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.
| anon291 wrote:
| > Personally I don't care that it's about physical
| things. But I care that it's about doing things:
| understanding concepts intuitively, solving problems,
| being adept at thinking mathematically. The thing that is
| not useful is tedious definitions and tedious proofs.
|
| 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.
| Koshkin wrote:
| > _nobody needs a proof of how differentiation or
| integration work_
|
| This "nobody" includes a lot of industry workers, e.g.
| those in finance and economics, not only "pure"
| mathematicians.
| Koshkin wrote:
| I would not call this "destructive." Solving problems is
| fun in its own right, and things would start making sense
| in their interconnection, regardless of whether they can be
| applied to sciences or not.
| eszed wrote:
| I think you put your finger on a challenge fundamental to
| teaching itself. Speaking as a (former, at least for now)
| teacher, the concepts that I find most intuitive were
| always the most difficult to teach, whereas skills I
| struggled to master were the ones I was best able to
| instruct. For problems in the second category I could
| diagnose the gaps in my students' understanding or
| application - likely because I'd faced them, or similar,
| myself - and strategize an approach that would help.
| Conversely, the subjects that I "got" without effort, left
| me floundering (at least at first) to find an alternate way
| to present to students.
|
| (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.
| rudolph9 wrote:
| I had a professor freshman year of college who offered the
| question "How many times can you split the distance between me
| and you in half?". Ultimately I arrived at the conclusion
| infinity which forever more made the general idea of calculus
| stick.
| fatnoah wrote:
| > However discussing a function and talking about splitting in
| into small (infinitesimal) strips of delta quantities and a
| list of equations / proofs made it completely dry and
| uninteresting.
|
| 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.
| phkahler wrote:
| >> I don't think I had any ideas one what was going on until it
| was mentioned in physics several years later.
|
| 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.
| BeetleB wrote:
| Although I studied calculus in a pure math perspective, I knew
| enough physics to tie the two concepts together and it may have
| benefited me some. For the most part, though, all I needed to
| know was "integration = area under curve" and "differentiation
| is computing the slope at a point on a curve" and most of the
| first 3 semesters in calculus made sense from that perspective.
|
| 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.
| jrgoff wrote:
| I don't know how or where you learned calculus, but when I
| learned it and when I taught it, the classes were filled with
| physical world examples to motivate and train the material.
| That said, my experience as a student reflected on as an
| instructor, was that a lot of the explanation and example stuff
| didn't have a lot of fertile ground in my mind yet when I was
| first learning the material. I'm sure it helped, but there were
| definitely times when as a grad student I reviewed the
| foundational material and thought "This makes so much sense,
| why didn't they teach me this when I was in high school?" only
| to realize that almost certainly they had taught me that in
| high school and I just didn't have the mathematical maturity
| yet to retain it. I think I was also hampered in my chances of
| learning the conceptual fundamentals because I was able to do
| most of the work through a solid ability at algebraic
| manipulation. While solid skills in algebraic manipulation is
| quite important, I do think it would be a good idea to
| restructure those classes so that a solid conceptual
| understanding is also more necessary to pass the class. Of
| course, easier said than done.
| zzbn00 wrote:
| Nicely written. But isn't a corollary that we should stop exams
| in Maths (and Physics/etc/etc) degrees and just teach people who
| want to learn, how much they want to learn?
|
| Exams are for people designing bridges and doing surgery. Basic
| science just learn and start doing some to see if it is for you.
| zozbot234 wrote:
| Exams are ultimately for the students' collective benefit; they
| ensure that lecturers are teaching effectively, with good
| learning outcomes. If you're someone who "just wants to learn"
| and don't care for the rigor of a full Math/Physics curriculum,
| that's one thing. But if you do care, exams are great.
| zzbn00 wrote:
| "The course is turned into a game of skill, where
| manipulative ability outweighs understanding"
| asplake wrote:
| (1997)
| Xcelerate wrote:
| I went to grad school for chemical engineering a decade ago and
| part of my frustration with the courses is that the math was
| never rigorous enough; when I asked for clarification, the
| inconsistencies were glossed over and hand-waved away.
|
| 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).
| sesm wrote:
| How big a system can be while maintaining a quantum
| entanglement (also called 'quantum coherence') - is still an
| open question. A very important one, because it affects quantum
| computers. Experimentally researchers see, that a big enough
| system can't maintain quantum coherence for long. If you want
| to know about how this process is described mathematically,
| search for 'quantum decoherence'. If you want to know about
| possible physical interpretations, search for 'objective
| collapse theory'.
| BeetleB wrote:
| For those not in the know: DFT = Density Functional Theory
| anon291 wrote:
| My experience as a math major is that most non-mathematicians,
| including engineering and chem/phys professors, have a working
| understand of math and don't think too deep about it.
|
| > 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.
| selimthegrim wrote:
| DFT uses the Kohn-Sham approximation which is single particle
| and is a ground state theory. There's definitely been research
| along the lines you bring up in quantum chemistry but not in
| DFT. You want to look at the Levy-Lieb density and maybe some
| of Mazziotti's papers
| dustingetz wrote:
| they shipped the org chart
| quacked wrote:
| Nearly every educator I've ever learned from has unconsciously
| insisted that learning material remain sterile, and never become
| as funny and opinionated as this essay. For most students,
| "learning" is outrageously boring until you get out of college
| (or into graduate school) and suddenly learn through essays and
| memoirs that what was presented to you for 12-16 years in a bone-
| dry format as established gospel truth was actually the life's
| work of sometimes dozens if not thousands of people, each of whom
| fought through their own careers, formed their own views on
| society, joked, married, divorced, died, squabbled, sniped, and
| felt immense passion for the subjects they ended up distilling
| into textbooks for us. Much of the information we learn as
| students was at the time of its discovery controversial in the
| extreme.
|
| 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.)
| acjohnson55 wrote:
| > Nearly every educator I've ever learned from has
| unconsciously insisted that learning material remain sterile,
| and never become as funny and opinionated as this essay.
|
| 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.
| quacked wrote:
| I couldn't agree more with you. I personally will attempt to
| circumvent "boring, mildly effective education" by
| homeschooling my children (who are at present still too young
| to even speak in words), but homeschooling is an ugly
| compromise to a stupid problem. A true comprehensive
| education would take place everywhere in society.
|
| 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".
| sixo wrote:
| The kind of people who fix fundamentals are not bothering to
| become math professors.
| samvher wrote:
| > 4. Teach changes of variables
|
| 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?
| tihal wrote:
| People might disagree strongly, but my recommendation for you
| is to look into lambda calculus for this.
|
| 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.
| xaellison wrote:
| I think there is less of unifying theory than an assortment of
| artful tricks - and it's a pretty small assortment, too.
|
| 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
| jampekka wrote:
| There seems to be something similar in other branches of
| mathematics too (and lots of other fields). A clear example I
| recall was studying Fourier transforms in math and I couldn't
| make any sense of it. To me it was just some by-rote algebra with
| integrals of exponentials of complex numbers.
|
| 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.
| marcosdumay wrote:
| I'd say that the Fourier Transformation doesn't belong in the
| calculus program at all.
|
| 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.
| starcraft2wol wrote:
| You need analysis to make sense of infinite series,
| indefinite integrals, exponential functions, etc.
| Chabsff wrote:
| At the end of the day, analytical Fourier analysis doesn't
| get nearly as much practical use as numerical approaches,
| and you don't really need THAT much background to make
| sense of DFTs.
|
| 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.
| jampekka wrote:
| Analytical fourier analysis is needed for many other
| fields of math though. For example in statistics it's
| very central for dealing with density functions. I'd
| guess in many fields of physics it's also very crucial.
|
| 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.
| yuppiemephisto wrote:
| Related to the DFT view: https://www.sciencedirect.com/sc
| ience/article/abs/pii/S00492...
| marcosdumay wrote:
| If there's one place for Fourier analysis to be is on
| linear algebra.
|
| 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.
| jampekka wrote:
| Why should it be in linear algebra? You mean as in doing
| linear algebra with functions in infinite-dimensional spaces?
| What I recall mostly (not) learning from linear algebra were
| various classifications of matrices and their properties
| (which made very little sense to me at the time too).
|
| 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).
| marcosdumay wrote:
| In fact, there is a lack of linear algebra in infinite-
| dimensional spaces on the general curriculum, and it is an
| important subject for physics and a few engineering areas
| besides the relevance for mathematicians. It wasn't clear
| to me how the dependency between calculus and algebra was
| supposed to be, but your comment makes it crystal clear.
|
| 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.
| jampekka wrote:
| Infinite-dimensional spaces do indeed come up in e.g.
| Gaussian processes and the kernel trick. I have to admit
| I've never really understood them the rigorous form. For
| ML for example it would be likely more useful than e.g.
| most matrix decompositions.
|
| 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.
| sudosysgen wrote:
| It also comes up a lot in the foundations of RL - the
| basis of how it is justified (or in some cases proven) to
| work is contraction mappings and functional operators.
| red_admiral wrote:
| There's a lot going on here. First, one of the problems with
| engineering - and CS has it even worse - is that new ideas and
| methods get added all the time, but the length of an
| undergraduate degree stays constant. Compounding the problem,
| since schools choose or are made to teach more and more things
| all the time as well (including computing, and topics downstream
| from computing like "digital media"), one way to deal with more
| and more topics is to sacrifice depth/rigour on some of the old-
| fashioned ones like math, so universities have to teach more and
| more things that they used to rely on students knowing from
| school as well. The discussion last year about California math
| reforms and the open letter signed by Scott Aaronson et al. which
| I'm sure was discussed on HN as well, may stand as an example of
| this (yes there's a "social justice" angle to it, but the roots
| of the problem are far deeper than that).
|
| 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.
| maweaver wrote:
| > What can we expect students to get out of an elementary course
| in differential equations? I reject the "bag of tricks" answer to
| this question. A course taught as a bag of tricks is devoid of
| educational value. One year later, the students will forget the
| tricks, most of which are useless anyway.
|
| 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.
| BeetleB wrote:
| How often have you used them. It's been decades for me as well,
| but I still remember how to solve basic linear differential
| equations as I needed to use them often (in other courses).
| mrkeen wrote:
| The further I got into maths (not very far mind you, but I
| think I at least did a few differential equations,) the more it
| felt that maths really was just a bag of tricks, moreso at the
| advanced end.
|
| * 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...
| ducttapecrown wrote:
| Many famous mathematicians have quotes along the lines of
| "it's not the result that mattered here, it's the new
| method/trick that is cool!" and essentially it's because
| solving math problems requires a bag of tricks and a new one
| can open new doors in every field it hasn't been applied to
| yet.
| hiAndrewQuinn wrote:
| >LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS ARE THE
| BOTTOM LINE
|
| 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.
| selimthegrim wrote:
| OK, then wait until you run into the students who somehow
| passed algebra two but don't understand the difference between
| a constant coefficient and a variable coefficient
| aj7 wrote:
| I had him for second term calculus, which was the first math
| course I took in college. So I get to listen to him speak it in
| my minds eye --- his voice is unforgettable. He was a great
| teacher. 2. I have an interesting take on this subject. I became
| an engineer at 50. Out of personal choice. What I found was that
| engineering had changed and what mattered was your savvy in
| manipulating expensive programs on the computer. In those
| programs, differential equations were solved numerically. No one
| even thinks about solving them any other way. There isn't any
| time.
| GeorgeTirebiter wrote:
| I, too, had Rota for his "exploring higher mathematics"
| seminar. What a remarkable human being! I retain excitement for
| hierarchies of infinity to this day.
|
| 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'?
| meroes wrote:
| How math relates to the physical world is not at all clear. Like
| yes 2 apples clearly makes sense, but non physical quantities
| like 10^241 of anything start to break down the relationship.
|
| 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.
| kriops wrote:
| Can you elaborate on why you describe 10^241 as non-physical?
| Just because it's too large to describe anything physical?
|
| 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)
| gbjw wrote:
| Well, based on a quick Google, there are approximately 10^186
| Planck-length volumes in the known universe. So 10^241 is not
| physical in the sense that there is no way to 'realize' a set
| of 10^241 things.
|
| 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).
| anon291 wrote:
| It really depends what you consider 'real'. It is actually
| pretty straightforward to realize a representation of
| 10^241 (you just did). Why does that make it any less real
| than the digit 2, which represents two things, but is
| actually just itself one thing.
|
| > 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?
| gbjw wrote:
| > It really depends what you consider 'real'. It is
| actually pretty straightforward to realize a
| representation of 10^241 (you just did). Why does that
| make it any less real than the digit 2, which represents
| two things, but is actually just itself one thing.
|
| 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.
| anon291 wrote:
| > 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.
| gbjw wrote:
| Yes, apologies -- using the term 'construct' muddied the
| point as sqrt(2) is a 'constructible' number as you point
| out. The term 'real' is what's at issue here and I am
| arguing for a distinction between a 'map-like' real and a
| 'territory-like' real, the latter of which has some sort
| of spatiotemporal grounding.
|
| > 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.
| whatever1 wrote:
| Just like solving analytically integrals, (O,P)DEs require
| tremendous skill of pattern matching. You need to know all of the
| available tricks, and pick the one that seems to fit. You test it
| and it just works.
|
| No wonder there is super-high barrier of entry. And that is why
| us mortals are just using numerical methods instead.
| SpaceManNabs wrote:
| This is why I found this article kinda pointless.
|
| 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.
| zozbot234 wrote:
| Only a small subset of O/PDE's can be solved analytically in
| the first place, and few of those are anything but theoretical
| curiosities. Linear w/ constant-coefficients are the one class
| that's of big practical interest, which is why OP argues that a
| well designed course should be focusing on these.
| ducttapecrown wrote:
| When I was a kid, one of my parents told me diff eq was a course
| they had not even gone to class for and passed, which set me up
| for disappointment in diff eq pedagogy for sure. It didn't help
| when I read this article, and then took sophomore diff eq and it
| was exactly as described. Later in grad school I'd actually teach
| the course to my complete chagrin. I think I mentioned this essay
| to my students, because how could I not? Of course I did not try
| to redo the diff eq pedagogy, but I did put a lot of emphasis on
| linearity.
| __rito__ wrote:
| Here's the most intuitive introduction to Differential Equation
| that I have ever seen:
| https://www.complexityexplorer.org/courses/31-introduction-t...
|
| 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.
| JadeNB wrote:
| For a course-level approach by some of the early pioneers of
| the pedagogical approach of using numerical methods to
| understand differential equations, I strongly recommend
| Blanchard, Devaney, and Hall's book (http://math.bu.edu/odes).
| not2b wrote:
| Lots of good criticisms of the traditional way of teaching in
| here. The only issue I would raise is his claim that Bessel
| functions should be dropped. In physics and engineering they come
| up all the time when the geometry is cylindrical.
|
| I suppose he could argue that they could wait for a second
| course, perhaps.
| eternauta3k wrote:
| I agree Bessel functions are not excessive, they are just
| enough to scare you straight.
|
| And yeah, physics students need to be able to solve QM and EM
| equations, can't get away from (some) special functions.
| fredgrott wrote:
| Interesting, I wonder if the same things exist in teaching
| programming? Well, yes!
|
| 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.
| dang wrote:
| Related. Others?
|
| _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)
| jojobas wrote:
| This sounds misguided. What do you mean, obsolete? It's largely
| 17th century maths, 19th century understanding was already pretty
| good. Have there been a pedagogical breakthrough allowing for
| igniting a spark in a brain much easier? I don't think so.
|
| 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.
___________________________________________________________________
(page generated 2023-11-13 23:00 UTC)