Physics and Mathematics Self-Study Project
diegovera.org
diegovera.org
> One of the things I found useful here was to get immediate practice once I knew just enough. I think this is one of the reasons project-based learning can be so powerful: you have a problem and research your way into it versus researching your way into it and then seeing what to apply.
This. Perhaps it varies from person to person, but this is also how I learn best. I want to start with a problem, struggle to solve it, and then learn how to actually solve it. That way the material immediately clicks and sticks with me longer. Why don't schools teach this way?
Project Euler. ( https://projecteuler.net/ )
Bite sized “specs” calling for an answer to be solved in your new language. Each getting harder and dealing with important concepts to grok and hurdles to overcome.
Hands-on lab courses are something of an exception, as you simply have to do the project yourself in class, but these tend to be long classes (three hour sessions etc.) and again are limited to relatively small groups of students.
Building computer simulations in real-time of physics problems might be a way to do it, but the students would need to have good programming skills beforehand. Basically a physics hackathon using a standardized set of programming tools, that would be fun and educational.
A few notable cases are cited here: https://www.pblfuture.aau.dk/about. As far as I know, some American institutions use the same approach, e.g. Olin College.
IMHO, when done well, i.e. having an equally strong focus on theory, this leads to excellent learning outcomes.
However, at undergrad level, schools are reluctant to teach this way because it requires a lot of supervision.
Schools don't teach project-based learning because schools are not interested in learning; they are interested in child-rearing, not letting them kill each other while the parents are at work, and offering employment for teachers and administrators. In schools, generally, learning is an epiphenomenon, not even an afterthought, simply an accidental side effect.
To truly learn is, in my opinion, to own.
Basically I want to put myself in the shoes of great minds, learn what math tools were available at the time, do their observations, know how their thought processes work.
edit Would appreciate if someone could recommend a textbook that follows such approach.
Readings include Archimedes's "On the Equilibrium of Planes" and "On Floating Bodies" and Nicomachus's "Arithmetic." However, I've heard of some mixed reviews on the effectiveness of learning math [1] [2] via reading historical texts at St. John's, versus more traditional approaches today in good mathematics programs.
A bit separately but as a related idea, I've spoken to a couple people who said they preferred older mathematics books from the 1900s to the 1960s for their studies over modern books. I personally prefer modern books with recent editions that are well-recommended, as they usually have good visualizations of ideas and shouldn't frequently have errors (due to their reputation). If you're studying as a hobby, though, the most effective mathematics book, assuming it's reasonably reputable and heard-of, is the one that interests you enough to stick with it for consistent study in the long-term.
[1] https://old.reddit.com/r/stjohnscollege/comments/ehcem3/how_...
[2] https://old.reddit.com/r/stjohnscollege/comments/ngpu0i/math...
> Mathematics is one of the many subjects studied in the college’s interdisciplinary great books curriculum. There are no majors at St. John’s.
It may be fulfilling from a humanistic/personal development perspective, but you won't really have the tools to do anything useful like cryptography, algorithms, physics, statistical inference, data analysis, etc., which, in my mind are the really cool things you can understand by learning math.
None of this is core to any major in mathematics. These are all tangential/elective applications.
hence it's not incentivized by the current "capital" run markets
The reason it isn't taught this way is because it's generally not seen as an efficient route, and you can understand such theories by studying them directly and in a logical progression.
You might gain that because you compare the two and notice the old books are obviously wrong against new data ... something the current books happily tell you too.
People are constantly talking about chatGPT and reinforcement learning but mathacademy is using AI to teach humans. And from my experience very effectively.
Very impressed with what I have seen of it so far, and the founder's of it have an impressive amount of passion and drive to deliver. You can read more about it here as covered by the washington post https://www.washingtonpost.com/education/2022/10/16/math-aca...
Aggressively positive? Maybe. I have been a paying customer since october and very happy with the results.
In addition to presenting the topics and providing exercises, MA automates some tasks that are usually left to the learner, like spaced-repetition and tracking down prerequisite topics. They have a detailed DAG of math topics connected with dependency relations that enables this. Basically automation of a lot of what a tutor would provide.
For reference, though I now work in tech, I have a physics degree from a top-10 worldwide university, and reviewing linear algebra on math academy has not only been great to re-learn things I've forgotten, I'm confident that it's taught me at least a few things that I've never encountered before.
So yeah, it's legit in my book.
I also think that one of the reasons it gets such rave reviews/etc from folks using it is that it's, frankly, not that well known yet, and we users who like it _really_ like it, so we want to see it stick around and grow further for our own use too!
I was impressed with both the breadth of the courses and the efficiency gains from letting it do the scheduling and making sure I review topics just often enough to hold onto the skills.
I’ve previously written about MA in more length here: https://news.ycombinator.com/item?id=32923684
https://ocw.mit.edu/courses/6-041sc-probabilistic-systems-an...
We're 4 months in and on Lecture 15. These do take a while. I feel like I'm not getting everything and will need another round. I've been binge watching Stat 110 too:
https://projects.iq.harvard.edu/stat110
"All probabilities are conditional"
1.) Verification, standardization, and authentication of work being done and of knowledge being gained.
2.) Networking opportunities and social/professional development and maturation in a a limited-stakes playground surrounded by others in the same social and economic class, and
3.) A sort of IQ test laundering service.
If you want to say that you learned mathematics, you have to do more than this student did. you may have chosen to give him the benefit of the doubt, but I did not. I checked his work, and its mostly wrong. The reason that it is mostly wrong is because the practice of mathematics requires some training to get to the point where you can be relied upon to know if you are lying, and the author of this blog did not reach that point.
Can you show some examples that make you draw this conclusion? Thanks!
The question is to show that R2 is second countable. This means that it contains a countable base for the topology. The usual way to do this is to pick open balls with rational centers and rational radii, and use a little finesse to find one around any point in an arbitrary open ball.
The Author's answer was to take the set of all open balls, and pick only those with natural number radii. This is neither countable nor a base for the (usual) topology on R2. This answer has a check mark on it.
It's about supply and demand. The demand is not there, so nobody is going to invest the significant resources to create the supply.
That is a very workable approach, so long as you consistently study. For motivation, if you'd like to get better at mathematics just as a life goal, that should be valid enough to spend time on the hobby. If you're studying primarily for personal enjoyment and fulfilment (and especially if you find joy in the process of learning), the slower pace doesn't matter as there are no expectations to learn quickly.
If you're studying mathematics or a technical subject for work, I believe it's better to earn a graduate degree from an accredited program for the resume (as a record of passing proctored exams is seen by hiring managers as strong evidence of learning), plus it's nice to be able to learn from your fellow students and ask the professor questions.
For myself, I've found it much easier to find motivation to consistently self-study mathematics outside of accredited coursework by practicing out of personal interest, as it eases up a lot of pressure about learning a certain concept by a certain time. I won't expect to learn as quickly as the person's reported progress in the submitted article, but this also makes the practice more enjoyable and easier to keep up in the long term for myself.
Setting aside 1-2 hours a day should be enough. I think this is not unreasonable for most people. But why? Unless you need it for a resume or degree, I don't think it matters that much. So you can make a blog post about it?
One hour a day should be enough to make steady progress.
Meetup log: https://susam.net/maze/meet/iant/log.html
A few blog posts about it: https://susam.net/maze/tag/iant.html
I have kept an archive of all our meeting notes here: https://offbeat.cc/iant/boards/
I very much agree that attacking problems from different angles is very important in building intuition for the concepts taught in the book. However, that's something that we did not do within the 40 minute meetings. It wouldn't make sense too because I believe solving problems is very much a personal journey where different people need different amount of time to solve problems. I believe solving problems is best done on our own time. Sometimes though we did discuss the solutions of some problems in the meetings just to take a break from the theorem-and-proof style of meetings.
I solved most of the problems in the book in my own time. I know another participant who did too. Of course, that took a lot more than 80 hours. I must have spent an additional 2 to 3 hours everyday for solving the problems.
I've personally enjoyed language learning as a hobby, and found time in my commutes to listen to audio programs (I enjoy this more than anything else I could do on a commute in a crowded train, so it's no loss of time). For more casual gym-goers like myself, oftentimes running on a treadmill doesn't require full attention, so it's possible to watch video lectures or listen to more audio programs. Neither of these are as ideal as concentrated study in a quiet room with a desk, but they're relatively lower-effort ways to get useful practice with a specific subject (so it's easier to practice consistently).
I then find more time on the weekends for more dedicated study. If you're dating someone who also shares your interests in studying, they can also spend time studying alongside you (alongside more fun and relaxed dates).
To quantify this, that is about 13 hours a week of non-concentrated study (assuming a 1-hour commute each way every weekday for 10 hours, plus another 3 hours assuming studies during a 1-hour workout three times a week, at the lower end of how often you can consistently exercise). Add in another 1 hour of concentrated study a day (or if you'd like, 3 hours of extended study sessions each weekend day, plus additional scattered study during the week), and you hit 20 hours a week. You may not be as fast as the writer of the submitted article with this, but you can at least get pretty far from consistent practice over long periods of time.
But that is not the point. My point is super-smart people can pick up and master concepts very fast.
I have always wondered what the truth is about this but I have never seen any real answers. On the "linear" side you have median incomes by IQ percentile. On the "not even quantitatively comparable" side you have the fact that your score on an IQ test will not increase that much if you are allowed much more than the standard time to finish it, indicating that there is a wall (defined by your ability) that you can hit. On the "sublinear" side, there's the fact that a lot of moderately smart people have made major scientific discoveries, which you wouldn't think could ever happen when there are 10,000 people with IQ > 160 in the US today - easily more than enough to take all of the places in the history books written about this decade.
I do not think there is any clear-cut answer.
The average IQ among mathematicians is somewhere between 130 and 140.
I'm not debating the claim that smart people can learn faster, but like the above I am calling the specifics into question.
You’re essentially correlating it just to processing speed which is of course important, but you can still have a significant +z score and need more time to absorb material. BTW, g (a statistical construct in psychometrics for general intelligence) according to research is best estimated through the measurement of reasoning ability on a standardized IQ test (WAIS, Standford - Binet) whereas the processing speed part of the test doesn’t significantly contribute to it.
Things like:
* Stellar Nucleosynthesis - S-process, R-process, stellar evolution
* Fluid mechanics - Navier Stokes equations
* Physical Oceanogrpahy - Ocean currents, tides, Pressure and Salinity effects
* Plasma Physics - MHD equations, Alfven waves
* Medical Physics - CAT, PET, MRI scanners. Radiation treatments.
* Planetary Science - Plate tectonics, Gas giant phenomena.
* Geophysics - EM, seismic, anisotropy
* Electron Microscopy - SEM, TEM, AFM etc.
Can you identify gaps - and worse, misconceptions - that you don't know you have?
I think you can "debug" yourself, to some extent, starting with what you do know: you have trouble with that something . Start with the symptom, and diagnose. Narrow down where the problem occurs, by attempting sub-parts of the problem, simpler versions, and prerequisites.
If you can't identify it, it's a sign that the problem is not where you think itis. i.e. you have a misconception about something you think you know.
Then, there are many resources available to try to remedy it: textbooks, courses, subreddits, stackexchanges.
Of course... so much easier to have a perceptive coach who can instantly see where and why you stumble.
In other words, tutoring is spoon-feeding.
This is still great, if you need that skill for something, such as for a specific technical job, or in order to learn something subsequent at which you are talented.
At a meta-level, an intelligent person wants to solve problems, or increase their ability to solve problems. So an intelligent motivated person would appreciate anyone more familiar with a subject strategically removing some of the friction.
Even if it didn't make them innately smarter.
Time, after all, is finite for us all. And nothing is worse for learning, than an unnecessary stall. — Nevermark
Who is this? Is this am unemployed autistic kid with nothing else to do? A brilliantly gifted high school student? A software engineer? A college student at a school without relevant courses? An independently wealthy semi-retired rich kid?
Why? To be macho? As an alternative to college? To level up his skills?
Quick web search answers a few of those, but not all of those.
If OP is the creator, look at the web site from the perspective of a visitor, and give some answers.
As a footnote, there really ought to be a way to give college credit for this sort of thing.
Brilliantly gifted high school student it turns out.
> https://www.scotthyoung.com/blog/2023/02/21/diego-vera-mit-c...
> Interviewer: Tell me a bit about your life situation at the time. Were you working on the project full-time? What did you do for funds?
> Diego: The year COVID hit was the most transformative year of my life. I was 15 at the time. A combination of both personal circumstances along with isolation gave me so much clarity—I transformed 180 degrees. During this time, I really got into self-improvement and started working out, meditating, reading, taking cold showers etc.
Personally, I did some of what he did when I was his age, but not to the same extent and I mostly decided to chill out and enjoy college.
At that age it's really risky to do. Really hard to go to college and sit in classes for 4 years if you already know everything. I guess you could go straight to grad school or industry, but you miss out on a lot of the social maturity and friendships you develop in college. Learning all this is almost a curse; he will always, in some sense, be alone in his newfound abilities.
(Sorry for the edits. Done editing before any child comments.)
I think it's extremely cool though and really a great idea but I am cautious about saying he has the equivalent of those undergraduate degrees.
This is the stupid, broken part of school, that only exists as a cost-cutting measure. Real learning and creativity doesn't have this nonsense. It's like saying that living in a nice house is bad because you don't get to smell your poop while you eat. No one needs that. Diego is getting a better education because it isn't being arbitrarily cut short and of track before the going gets good.
Various colleges (ex: Reed, Brown) in the U.S. don't have grades. Their graduates do just fine, afaik.
In defense of grades, they are a good extrinsic motivator for learning boring subjects. Grades are a good consequence for phoning in it. I would probably have skipped reading most of the books I was assigned to read in school if there were no consequences, and would have ended up an (even) less educated person if not for grades.
I think it's better to pre-study, as if you go to the right university, you can take honours-level courses or enter more rigorous, challenging programs, which should still be challenging enough to engage you. Alternatively, if you don't like the challenge (though if you're the type of person to achieve that amount of self-study, you probably would enjoy it), you can take a more normal program and focus on more deeply learning the material over a longer period of time. With the higher grades from deeper learning, you can stand out and earn scholarships and grants to get practical experience by working in a professor's lab.
So, overall, I think it's better to pre-study if you can, as you can keep the benefits while minimizing the risk of boredom by finding ways to challenge yourself. Though in reality, the main issue for someone around 18 is that they might not even know about the risks of boredom, or how to challenge yourself in this way. Hopefully such people who are succeed in self-studying in advance, are around good people who can guide them to find ways to challenge themselves in a healthy way.
So, while grades don't really matter for the vast majority of great employers after graduation, good grades do make it easier to find opportunities earlier on. If it's not overly stressful for a student to achieve higher grades, it's worthwhile to score them for better early career opportunities (e.g. access to top labs and competitive internships).
But if it's too much of a burden for the student, you can be extremely successful career-wise regardless of grades, especially if you work hard and creatively to find ways to gain experience and demonstrate your abilities. For example, one successful former classmate found great internships via networking through their engineering design team, where the companies overlooked their GPA in favour of their demonstrated experience with engineering with the team.
He could go to any big college where he can take very difficult classes right away, and if he’s somehow still bored he can do research.
The author basically did all of undergrad math and physics, and now apparently they're planning to self-study all the grad math. The thing with self-studying is you're taking yourself out of the system, and at some point you have to inject yourself back in. I hope the author is able to do that and doesn't miss out on undergrad college too much, because it's really enjoyable if you find the right friends and you'll look back on it fondly.
Plus, he clearly is self-motivated and able to study by himself. University will give access to a ton of material and researchers. He should thrive.
Given the opportunity, learning is rarely a wrong choice.
Or you can do what many of us did, and take more advanced classes.
> I guess you could go straight to grad school or industry, but you miss out on a lot of the social maturity and friendships you develop in college.
There is a standard pathway. Diverging from it doesn't cripple people, at least anyone I know. The pathway was different 100 years ago, or 400 years ago. It's all good.
Personally, though, if I were him, I'd do something different, like a field of engineering. The math and physics will give a huge edge, while he's learning new stuff.
Before I gave up on teaching as a career, it was kids like this that kept me going.
But more to the point you're addressing, physics tends to be easier to grade than math because there is a big focus on calculation, and there you can have answers in the back with no subtle questions about things like whether a student's proof skipped a step because they thought it was obvious or because they didn't know it was necessary. That is not to say physics doesn't have those questions, just that a lot of times the answer is clearer-cut. If the whole thing had been about topology it'd be different.