Hats off, I think it takes some real dedication to stick with it.
> 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?
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.
Anecdata, however: project-based learning is powerful because getting something to work brings pleasure and failing to make it work brings pain. Being able to manage cycles of pain-pleasure from various stages of building and refactoring a project (hardware can also be refactored) is the basis of learning.
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.
I don't think you can learn your way to the cutting edge of science in a lifetime with project-based learning. In my experience it just takes too much time.
I don’t think you are at the cutting edge if you are not comfortably getting your hands dirty with it, to the point that you can clearly see what is missing and how you can further improve it.
The issue is that the "low hanging fruits" such as general relativity have been picked, the high hanging fruits require at least some kind of ladder, nuts and bolts. Perhaps a theoretician should not wire themselves the electromagnets or write the ladder logic of the PLCs running the experiments, but going up and down the stack certainly helps one's understanding and appreciation of the larger scope.
You could be hard-pressed to find a more project-based learning project than the Large Hadron Collider/James Webb Space Telescope. No amount of Gedankenexperimenten could ever prepare you for the cutting-edge science&engineering at LHC/JWST/ITER/etc.
well, you probably learn more in the end if you aren't killed as a child by your peers or something happen to you while wandering around randomly all day while all adults at work
Some universities teach this way. In Europe, there are a few institutions that shifted to problem and project-based learning during the reforms of the 1960s.
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.
It's indeed the best way to learn, but schools are charged with issuing grades to students and there are just too many problems with take-home projects (i.e. paying someone else to do it, etc.), and with group collaboration on such projects (one person doing 80% of the work, etc.), that it generally doesn't ever happen except in a few small-sized high-level undergrad or perhaps graduate courses, where the students are more motivated about learning the material then they are about gettting a good grade, or just passing the class.
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.
It's impossible to learn the full breadth of material in a project-based way. You'd have huge gaps. Projects work for a sampling of the material and an motivation for the field, not everyone little piece of it.
I'm thinking about a historical approach. For example what exactly is Ptolomy system? What's the math involved? Why did the phases of Venus rebutes the model? How did Tycho make his observations? Can we repeat it for say a few months? How did Newton come up his ideas? What are Einstein's thought experiments.
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.
Though I lack personal experience with a historical approach to studying mathematics, a starting point could be the reading list by St. John's College as part of the materials the faculty uses to teach mathematics to their undergraduates: https://www.sjc.edu/academic-programs/undergraduate/subjects...
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.
If you go through for the "great books" approach, just be aware that it's not the same curriculum as a major in mathematics:
> 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.
An interesting book (2 volumes) would be: 2019, June Barrow-Green et. al., The History of Mathematics. A Source-Based Approach, American Mathematical Society.
People who want to study physics want to understand such theories, so would be heavily incentivised by the market.
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.
What's important about the old books and theories is that they teach you that science is invention and discovery and being wrong, not memorizing rules and solving already -solved puzzles. If you only learn from modern textbooks, you get the wrong impression that these biblical tomes are what science is. Science is about finding patterns and creating order out of chaos, not memorizing the order someome else creaymtes.
They don't teach you that any more than current books do.
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.
If people want to replicate the same feats as this kid you should check out http://mathacademy.com
People are constantly talking about chatGPT and reinforcement learning but mathacademy is using AI to teach humans. And from my experience very effectively.
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!
Math Academy seems like it could be a game-changer for learning, it's a paid service and in beta-testing. They only cover math, though, at least right now.
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.
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...
I used Math Academy to relearn some math I’d forgotten since initially studying it decades ago. It covers pretty much an entire undergraduate curriculum.
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.
The testimonials on the homepage are aggressively positive, what has your existence been like? I’m interested in revisiting the various University topics, particularly DiffEQ/stats/LinAlg, and have been looking for an excuse/resource to study.
My experience has been that I never fail to make progress whenever I encounter something I didn’t know or was weak in. The system offers you plenty of practice problems, at just the right level, to make sure you master the material before moving on to material that builds on the previous lesson.
Aggressively positive? Maybe. I have been a paying customer since october and very happy with the results.
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:
Maybe this isn't the exact right place to bring it up, but is related. But there is a lot of self-paced learning things coming up between schools and alternative to schools. But an issue I have is that this self paced learning is really limited in scope. Why can I take classes so easily for an MS in computer science or IT, or get an MBA or take data analysis courses, but I can't easily get a degree / certificate for say a bachelors in Mathematics? Or literature? There seems to be a big split where so much is focused on job training but very little is being done on education.
The answer is that mathematics is a thing that you can actually assert that someone is doing incorrectly, but for which it will not be obvious to non-mathematicians. If a programmer writes a bad program, the user can tell. If a mathematician writes a bad proof, basically it takes a mathematician to tell. It doesn't take an incredible mathematician, but it does take one.
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.
Sure, but I will limit myself to one, which is fairly exemplary.
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.
The oft cited refrain is that colleges/universities serve several purposes aside from education:
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
I would love to learn a much deeper level of mathematics but where do people find the time for that? If it's not related to work, or future work I couldn't find the large amount of time required. And self study doesn't get recognized on resumes very well. Perhaps spread out over the course of many many years on your own personal time.
> "Perhaps spread out over the course of many many years on your own personal time."
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.
Indeed, “work” prevents learning. One of the reasons I like freelancing is that I can intentionally set aside 15 hours a week to catch up on areas I’m very interested in. Yes I don’t get paid for this “learning” time but I see this as an investment.
I would love to learn a much deeper level of mathematics but where do people find the time for that?
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?
People love self-study materials, but do you really have the time to absorb it? Yeah you could do this stuff every day after work, but then you may not go to the gym, go on that date, etc... Not enough time in the world to go through even a single learning resource people promote.
In 2021, a bunch of us, software engineers, did a small book club to learn analytic number theory from a book written by Apostol. We met for 40 minutes a day and read every line of the book and every page of the book in great detail. We did it for the joy of learning. It took us 79 hours spread across 120 days to complete reading the book and gain a good understanding of analytic number theory along with the analytic proof of the prime number theorem.
Sorry, but you don't properly learn any math topic in 80 hours. I've spent more than 10 hours on some problem sets in difficult courses. Solving hard problems takes a lot of time and is absolutely essential to learn math. When you read a book or a script you feel as if you have grasped the content, but most of the time you really haven't. It's only when you attack problems from different angles that you really build intuition for the concepts taught in that book.
The time recorded in the meeting log includes only the time spent together in reading the book and working through the proofs. There are some more details about our reading style here: https://susam.net/maze/journey-to-prime-number-theorem.html
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 skimmed your blog posts and logs, so apologies if I missed it. How many problems or exercises did you work through in the book? Did people put in any additional time individually between sessions?
I can't speak for all participants of these meetings and I haven't kept an exact count of the problems I solved but I believe I must have solved somewhere between 160 to 200 problems from the exercises in the book. See also https://news.ycombinator.com/item?id=34908191 for more details on this topic.
It's not so hard to find time. Many people with full-time jobs practice martial arts an hour a day for three to five times a week, while others practice a musical instrument for half an hour to an hour a day. Other people also complete part-time Master's degrees in highly technical subjects while working full-time (many programs like these are designed for full-time workers).
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.
it sounds like you have a good attitude for making smart choices about how you use your discretionary time. I feel I should just point out however that generalisations about how much discretionary time people have don't always apply as widely as you may believe. There's probably a certain bias to the "average" person you meet at, for example, marital arts clubs, as they're the people with that time. In other circumstances or phases of life people count their discretionary time in minutes per day, and that is probably right at the end of the day. Not criticising, just hoping to widen your perspective a touch.
If you're smart enough you can retain and master material very fast. People like Ed Witten and Terrance Tao probably mastered year's worth of material in months . IQ does not scale linearly, meaning going from 130 to 166 is not a 20% gains more more like multiple orders of magnitudes. It's hard to appreciate or comprehend what truly smart people are capable of doing. It's like magic, to put it bluntly.
>going from 130 to 166 is not a 20% gains more more like multiple orders of magnitudes.
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.
a discovery is not the same as raw intellectual horsepower. If the goal is to learn as much as possible in the shortest period of time and understand it well, that requires horsepower. But being smarter increases the odds of finding a major discovery for more abstract concepts like math or physics, as Terrance Tao did many times. An IQ of 130 is probably not enough for pure maths, but may be enough for something like medicine.
Enough to make an important discovery in math, which gets citations and such. Having read math papers, there is a wide range of ability. Compare the stuff featured in Quanta magazine to your typical pdf in google. 130 is probably the min. to maybe get a master's degree.
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.
consider the possibly that the other components of intelligence aid in learning ability too. Spatial reasoning probably helps when trying to understand geometry.
Can you tell a field that either Ed Witten or Terrance Tao knows a significant amount about outside their field of specialization? And don't say math for Ed Witten because his field is mathematical physics.
A lot of my favorite physics classes were Applied Physics. These were great as you used the building blocks from 1st/2nd year level to show complex phenomena in the universe. It is a shame this list has none of these.
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.
One thing about the 2 sigma effect of tutoring is that it doesn't generalize; it doesn't make you smarter, just better at what you were taught. A different mechanism, but similar effect is learning the "trick" of solving a puzzle/problem - you just look smarter (and, well, know that specific trick).
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.
Yet, tutoring seldom makes someone dumber. And a good tutor doesn't spoon feed.
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
This is inspiring. I'm a student in the Arizona State University online Physics BA degree (soon to transition to BSc). And by student I mean adult learner with a full plate of other commitments doing this for fun and I dont know when I'll finish.
Maybe a long shot but is there a course on “Geometric Algebra” offered by ASU by any chance? The founder of GA, David Hestenes is a former professor there (emeritus).
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.
> 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 did something like this and ended up just substituting graduate classes for my math and physics requirements while taking regular undergraduate classes for everything else. I made a lot of great friends and was not the only undergrad in those graduate classes. I don't see it as a curse at all. I think the author will be fine if he stays motivated and finds the right peers and the right path.
Agree with everything you said. I also did the same thing basically, undergrad CS and grad math classes.
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.
I disagree, I think it will provide him with a great foundation for college. As great as online learning is, I don’t think it’s the same as taking a class that you go to multiple times a week and get in person feedback on. It also takes multiple passes at a given subject to properly understand it. I’m sure he’d learn many new things he didn’t quite get from his self study if he enrolled in an actual analysis course or w/e.
this makes sense to me. I skimmed what he took and it says he learned real analysis, a pretty challenging course. It's one thing to listen to the lectures and solve some problems and it's another to be given a difficult homework on a very tight deadline and perform well, same for exams.
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.
Yeah, I have unpleasant real analysis memories of frantically taking notes while the lecturer stood facing the blackboard writing down proofs. But, I made it in the end and I have it on good authority now that 1 is indeed bigger than 0!
> it's another to be given a difficult homework on a very tight deadline and perform well, same for exams.
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.
I agree with you about it being broken. We’re so focused on grades because they are a believed to be an important part of the credentialing process for getting jobs. An ideal learning environment would likely be something closer to Plato’s academy, imo. Probably not possible under capitalism except for the very rich with a lot of leisure time.
At least in the fields mathematics, physics, and to a lesser extent CS (which has a huge number of students and is becoming like a new business degree), professors view undergraduate degrees as a way to find good students who can go on to graduate school. So there is a tendency to be adversarial in classes. The GPA is one way to measure student aptitude but it's not perfect. Typically more than one metric is taken. If you just love these topics that's a great thing to learn. But if you want to contribute to the fields, it will be hard unless you go through the credentialing process.
> We’re so focused on grades because they are a believed to be an important part of the credentialing process for getting jobs.
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.
> 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.
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.
nah, i have plenty of similarly smart friends who are doing just fine. he'll fit right in in some college environments, and he can take the time he's not wasting doing class and pursue real things that matter, like research.
I can see both sides of the riskiness of studying the material before taking the course. On the one hand, a person who I see as highly accomplished person in mathematics told me that he did very well in his courses by pre-studying much of their material in the summer before they started, because he was both interested in the subjects and wanted to do well. But on the other hand, I also know of some people who took an introductory language course after previously studying the language, and received worse grades than people new to the language because they didn't put as much effort into the assignments as they didn't see the value (though they did well with minimal preparation for writing assignments).
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.
Haha. The importance of grades depends on your goals: on the one hand, I know two engineering grads who scored multiple great internships with a sub-3.0 GPA, with one landing a very high-paying job thanks to these internships. But on the other hand, a mathematics professor persuasively said to my class that your life can be a lot easier if you do get higher grades. They can help with graduate school admissions, scholarships, and getting certain internships.
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.
If he can write a good application and gets great scores at his tests (which he should considered how above the required level he is now), he should get a good scholarship and knowing the material should get him noticed pretty quickly which will open opportunities to publish quickly at least in mathematics.
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 he could lead group study sessions. I graduated last year and for many of my classes there were a couple of people like this, and everyone loved them.
As a former teacher of math and comp sci at a pretty low socio-economic school in (US) Colorado's front range, I can attest that kids like this are out there.
Before I gave up on teaching as a career, it was kids like this that kept me going.
Mr. G! I was just looking back at Robby the Robot stuff from my time at skyline! You should email me so we can catch up! Pick nearly anything at my lastname dot com to reach out!
I looked through his topology problems and notes, and he could really use the guidance of a professor, or at least someone to give him feedback. It's not practical to teach yourself mathematics to start before you can really check your own work. There are quite a few mistakes and misquoted/misunderstood definitions that have no way of ever getting corrected.
If he was learning physics and that stuff about topology came up, he'd notice his definitions weren't adding up and fix it. If he never saw it again then the maybe didn't need to learn it to accomplish his goal.
This is very optimistic. I watch people fail to learn things all the time without noticing, especially if they do not have the tools to be honest with themselves, which is a thing that is almost exclusively learned through formal training. Sure, not everyone arrives at that state, but they are much more likely to with guidance.
I'm inclined to be very optimistic, because the guy is fifteen years old and the opportunity cost is League of Legends. :-)
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.
I disagree. Moreover, it is rather unfortunately the case that the comments section here has decided to throw a party for the death of formal higher education. I'm here to piss in the punch at that party.
I wonder if you could build a service that ingests self taught students’ notes, uses ML to investigate their displayed level of mastery, and upon decision associates with a university that will give them credit for a course, or even a full on degree.
Action creates motivation, rather than the reverse. Just take the first few steps. Goals? Nice to have, but you don't need them either (from a fellow Nihingo student).
I started on numbers since I need that when buying stuff and when the locals says the numbers it still hard to understand. Seems like I need hundred of hours exposure to it before it becomes easy.
It's not really a time issue. It's more of a learning issue. I tutored math one-on-one and even of the smartest kids in my cohort found it hard to get calculus; there was a major learning curve for sure. No amount of time will help if the concepts do not click. Feynman-level of physics means you need to understand stuff like variationals, etc which is well above your typical calc course. Shows how truly exceptional this person is.
> 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?