The Greatest Educational Life Hack: Learning Math Ahead of Time
justinmath.com
justinmath.com
I'm in my 30s and getting a bachelor's degree in Math now after a lifetime of math-phobia. Math was my worst subject because it never came easily or naturally to me, and so I assumed I must have been innately incapable of it. I didn't take a single math class during my first bachelor's degree.
I sure wish I could have learned math properly earlier in life, but my point with this comment is that it is never too late to learn math.
Learning mathematics "late" over the last couple of years has enriched my life in so many ways. Learning to write proofs has brought a sense of organization and calm to many other areas of my life. Complex problems and challenges in life feel so much more approachable, because I am much more skilled now in breaking down tasks to manageable components. I can see now how mathematics has influenced programming languages and computer science, and every time I can identify the mathematical underpinning of some program I use or write, I feel like I am peering into the heart of the universe.
Learning math early is a great hack, but so is learning math late :)
Math it's way easier than you think it is, it greatly depends on how you approach it. I really like the style of Robert Ghrist videos on YouTube.
A great tutor/video goes a long way. I wish I could share some resources but am a bit outdated on that.
The overall idea is that some people can explain math concepts in a very clear and straightforward way, while some others will write up a bunch of symbols and let you figure them out. Avoid the latter. As a note, those are usually the lowest performers in academia, lol.
My experience with the comments in this thread, the overwhelming majority of people I know IRL and the widespread sentiment that "Math is hard" does not seem to reflect that.
At age 33, I enrolled in community college and took Calc I-III, Linear Algebra, and Differential Equations. The community college hosts weekly "math jams" and offers free 1:1 tutoring.
I'm currently taking a Discrete Math and Probability class at UC Berkeley for fun this summer (CS70), which would have seemed absurd just a few years ago. The community college system in California is extraordinary; I'm glad I got to experience it first-hand.
Springer Undergraduate Texts in Mathematics and Dover have lots of elegant and concise textbooks that can help you. At the beginning, the key is to move slowly and build some solid foundations.
If you want to learn math, a good place to start would be AoPS curriculum https://artofproblemsolving.com/store/recommendations
Continue with Susan Rigetti's curriculum https://www.susanrigetti.com/math
You can get answers to your questions here https://www.reddit.com/r/learnmath/ and here https://math.stackexchange.com/
I would go as far as to say that most high school “math” and “math” taught in many college courses is borderline irrelevant.
It’s like learning how to paint by memorizing names of colors. Learning to fix a car by reading parts list.
Painters can tell you about colors and mechanics parts but you don’t become like them by making those things your goal.
The only way to learn math is to learn proofs rigorously.
Calculus isn’t math, it’s just calculus. Algebra, linear algebra, they’re not math. Any “math” without rigorous definitions and theorems with proofs for each one isn’t math. (memorizing names of colors isn’t being a painter)
This book seems a good start. This is not advanced math. It’s an introduction to math- if you don’t know this you don’t know math. https://richardhammack.github.io/BookOfProof/Main.pdf#page=8
Stuff like what’s in this book is taught starting in week 1 for Waterloo computer science degree.
It’s life changing knowledge because you can use math to understand almost anything.
That is the midpoint, the core goal of math is getting enough intuition that facts are obvious, the proofs are just a guide to get you there.
This means you shouldn't study proofs, you should study facts, the proofs are just an example of how to support that fact, you can prove things in many different ways and also many things can be constructed in many different ways and still have the same properties. All of that is much easier when you think in terms of facts instead of proofs.
If you struggle with proving something then you don't understand it. If you memorize a proof for it, then you still don't understand it. The right path to take is to build understanding and then the proofs comes on their on.
Well it may be the core but it's not the purpose. As an engineer and later quant I actually use math for practical purposes in everyday life. It wasn't like this in the beginning, I remember primary school was a torment of being fed math olympiad-style problems and hating it. Then somewhere in gymnasium I discovered electronics and everything changed. Math became not just useful but inevitable and from then on learning of math for my own purposes went hand in hand with practical applications in electronics, from simple equations to matrices to differential equations, numeric calculus etc.
Of course there's also always the "standard math" (for passing the SAT/baccalauréat) and entering the good schools, that's inevitable. One can say that "Learning Math Ahead of (the vast majority of) Others" is the way to get ahead :)
That's not to dismiss the importance of arithmetic (and this is what I believe we should call grade school math operations): everyone should know how to add, subtract, multiply, divide, etc. But the core of mathematics is logical thinking and reason, not numbers
Learning math is like learning any natural language. For example, I'm "bad at Russian" because I have devoted all of 6 hours in my life to learning Russian and there are profound gaps in my understanding of Russian writing and grammar.
But I don't believe I am intrinsically incapable of learning Russian. The reality is that I've simply not put the effort into it.
It's truly the same with math. I am personally quite bad at computation by hand. It's exhausting, I often make careless errors, and I find computational problems by hand to be very boring. But that doesn't mean I'm bad at math! I've simply not invested much effort into improving my skill at computation by hand. I'm not terrible at proofs, for example; and the reason for this is that I find them interesting, and have devoted extra time and effort into learning how to write them. The heart of math isn't computation (which I'm not strong at), but proof and abstraction (which I am strong at, only because abstraction is interesting to me).
So really investigate your belief system regarding your capacity for mathematics. It's unlikely you are innately bad at it. Maybe you have knowledge gaps or you, like me, are not innately skilled at computation. But there are strategies you can employ to improve both.
-adventure time
Now I feel vastly more mature than I did before I began my degree! I have that same belief and confidence that no problem I face is unsolvable. I've also discovered a much deeper love of learning itself, and a desire to continue studying long into the future, and that interest includes but is not limited to mathematics! I want to have many different hobbies and learn all about how the world works.
It's like a fat person going to the gym for the first time. But once they start getting into the habit of working out and seeing the improvements, the anxiety goes away.
Anyway, congrats on overcoming your math anxiety.
Unfortunately your experience is atypical. I have seen a few people trying to learn math late(ish) in life, but I haven’t seen a single one succeeding. I am not claiming it is impossible, because I thing everything is possible, it’s just that I haven’t seen it done.
Congrats to you for beating the odds. It is quite a singular achievement.
This learning adventure has been very, very hard. But it is possible.
Because if I can do it, seriously anyone motivated can. I was the epitome of "bad math student".
My precalc teacher in high school actually discouraged me from going on to calculus (I took his advice and took trig, not calculus, in senior year of high school), and I decided during that meeting that I would never take a math class again.
As an adult, I really take umbrage with that lack of faith. I wish someone had told me that math is not any harder than learning a new language (something I was very good at).
It would have given me courage and helped me see math as not some kind of untouchable, elite pursuit, but just a learnable skill set like any other.
Thank you for sharing your experience. I feel the same.
I tried to
Fast forward and I came back to math for Python projects. And by this time youtube and authors like Khan Academy are making learning materials which let you enter in the middle and work your way in whichever direction suits your goals.
But what changed me from "I'm not good at math, so I hate math" to loving it was history of mathematics--the stories of the really interesting problems, and insights into the structure, theories of mathematics. I stopped thinking of math as a black box, and started to see its beauty and simplicity.
I did it one summer, and while I was nowhere near as good as them - something magical happened: even though I hadn't understood all the concepts, my ability to understand the concepts during the class went way up. It was easier to follow what the teacher was saying since no concept was totally new to my mind.
So basically I'm just trying to say it's up to you to make things not boring
Source: I’m a parent of a 3yo who now understands speaks both English and Polish. Me and my wife are Polish and only I speak English. Apart from speaking we also use English audio in all TV content she watches and buy books that contains both English and Polish text.
Edit: as pointed out below I should’ve clarified that this applies when you live in a non-English country where your child does not have any other way to learn English (over here you can’t really learn English in schools - not enough hours, plus it starts way too late anyways).
If you are living in a place where people don't speak your mother tongue but English is spoken everywhere and is the main medium of education, don't do this. The kids will pick-up English anyway because they will be exposed to it for 8 hours daily at school but if you don't speak with them in your mother tongue, they will never pick it up. The older they get, the harder it is. First hand experience.
Bilingual children whose parents don’t speak the language of the community at home may learn languages slightly slower but they quickly catch up once they make friends who only speak X.
By the way, something nobody mentioned, once your kid learns English in an English speaking country, your English gets better (assuming you are non-native). My daughter started correcting my pronunciation and it's getting way better
Source: as a kid I was in that situation, at first my parent spoke only in English with me and I started to forget Portuguese. After my parents realized that they pivoted to speaking Portuguese. I learned English fine at school and never had problems with either languages. Now I'm a parent of a 2yo and 1yo and am speaking Portuguese with both.
If you live in an English speaking country then sure. Over here it’s almost impossible to learn English in school, you only get a few hours per week of English classes.
Looking back at it, after 3 decades, what helped me learn the language was that immersion I mentioned, i.e. I was watching English TV programs (Cartoon Network, Eurosport, MTV Europe) for a big part of the day, without that I wouldn’t have been able to pick it up so easily.
There is absolutely nothing going on in the world to suggest that happen in the foreseeable future though. There is no competition, and it's inherently hard to change like any standard due to chicken-and-egg, so it tends to only happen when the entire world system is completely upheaved, to the point of the old world being a small part of the new world, and to a degree far greater than the possibilities of today. The stuff going on in the world today is mild comparatively.
In particular, politcal falls of the sponsoring empire don't directly lead to much change here, actually. Latin kept being the language used throughout Europe centuries after the Roman state was gone and buried in the west. Scholars were still writing in Latin in the 1600s. Within my parents' lifetime, Catholic mass was still said in Latin.
Let's check back in 30 years, and see if we have to reply in Mandarin, Arabic, Russian, or Malbolge.
The world you are imagining could just as easily be one where Chinese and Arabs dominate, the US and UK are a joke, and yet the dominant coalition paradoxically speaks English at least for elite communication anyway.
Did you know the original Latin language came from a tribe called the Latins that were vanquished by the Romans?
Reading the history of English after the Norman Conquest might be instructive too. For centuries the common people spoke the language that had been spoken locally, while the elites spoke the language of the old conquerers.
Fundamentally, you underestimate the chicken and egg effects involved, and they happen on a time scale of generations because that's the time scale upon which people learn languages.
Most money is spent in manipulating English media. Only fractions for other languages. It makes a difference.
If you are learning English later in life, you will struggle.
English is probably one of the dead simplest languages of use to learn later in life.
My dad (who learned English as a second language after his native Spanish, and also learned Portuguese and Latin, don't recall if he knew any others) used to claim English was one of the easiest languages. I don't recall his reasoning but I think it might have been because the rules (other than spelling lol) were simple and forgiving.
And I've met people who felt the opposite. Idk.
Still doesn't hold a candle to Mandarin or Japanese, of course. These guys are just on a level of their own. :)
In purely intellectual terms, know thy enemy.
I always say English is what Esperanto wanted to be.
> A student can wrestle with a competition problem for long periods of time, and all the teacher needs to do is give a hint once in a while and check the student’s work once they claim to have solved the problem.
Wrestling with a problem for long periods of time is not just a convenience for the teacher, it is a skill that will serve students well for decades to come. Sitting with a problem that you don’t know how to solve for hours, trying various approaches, failing and failing and trying again, is a life skill that learning calculus two years early won’t teach you.
Many of the tactics used in competition problems are also useful in general quantitative situations: identifying symmetries, invariant quantities, properties that can only increase under perturbations.
Competition math becomes a zero sum game when you are competing with students who have both built strong fundamentals AND then concentrated on technique and problem solving.
You can't run if you can't walk.
> failing and failing and trying again, is a life skill that learning calculus two years early won’t teach you
But learning Calc for 2 years, and getting a 5 on the AP Calc BC exam means you can take 2 additional courses in college or graduate early.
> Many of the tactics used in competition problems are also useful in general quantitative situations
Agreed. But at the end of the day, the kids getting into AIME or USAMCO were already doing high school or even college level math by 9th grade
Anyway, it seems like a shame that there’s a problem solving strategy beyond fundamentals for competitive math. What makes the puzzles in the game different from the sort of typical math somebody in STEM might do?
And just by the way: competition math is definitely "higher math" in a lot of cases. To be competitive at a decent level you have to know stuff like "real" algebra (groups, fields, etc., stuff like Burnside's lemma is pretty much table stakes), vectors, barycentric coordinates and so on for geometry problems, how to handle recursion for combinatorics, generating functions etc. It's by no means only silly tricks.
Burnside's lemma is a silly trick, in the sense that it's easy to memorize without knowing why it's true.
No, those are table stakes. To reach that level you have to know the theory and be good at the game.
> Burnside's lemma is a silly trick, in the sense that it's easy to memorize without knowing why it's true.
By that standard pretty much 100% of mathematical education is silly tricks unless you are literally getting a PhD in pure math.
Calculus without real analysis? Silly trick! Statistics without measure theory? Silly trick! Learning how to code without an in-depth understanding of computability theory? Silly trick!
Applying Burnside's lemma in a competition setting requires at a minimum a fairly sophisticated understanding of what is a group action and an orbit, which is definitely "real math". The broader point is that competition math isn't some kind of parallel universe where you learn "fake math" that doesn't exist in the real world. You learn the real thing -- perhaps less rigorously than in a university course, but the real thing nonetheless.
And one of the best ways of developing that skill is... learning higher-level math. This can also include 'competition math' topics of course, but they should be approached as self-contained subjects of their own, not just as a bundle of disconnected "tricks" to be applied solely in a competition- or puzzle-solving context.
I took courses in topology and number theory in undergrad that were set up this way—the professor did almost no lecturing; we were given a series of results to prove and expected to wrestle with them ourselves (mostly alone as homework). Once you thought you had a proof you presented it to the class. But this is very atypical. Your typical calculus or differential equations or linear algebra course does not develop this skill.
my child is very good at math, able to grasp advanced concepts quickly, years ahead of his school curriculum, etc.
there is __0%__ chance I could get him to do the above for __hours__
And of course, you (usually) don't have to literally sit! I spent many lunch breaks in school pacing around thinking about problems.
One challenge is the amount of distractions today (internet, devices, etc.) seem to have greatly shortened children's capacity to 1) handle "boredom" in creative ways (like trying to solve a problem, as per your experience) and 2) have the sufficient attention span to see things through. (I used to spend hours as a teen doing math problems on my own because I enjoyed it; but this was pre-internet and my parents didn't even have a TV (out of principle). There was only books, math, and my bike. I feel like kids today are at a huge disadvantage.
Getting into the "higher math" world was really painful for me actually. Seeing how some "modern" techniques (which has already existed for ~400 yrs, not literally modern at all) have solved the problem I struggled with in such an ultimate way made me feel overwhelmed. I felt like a track athlete, gloating over how fast I was running, realizing how modern transportation has revoluted - people no longer move long distances with human power. The quality a good driver needs is attention - on the road and the car simultaneously - not a pair of sporty legs.
The transfer from high school math olympics towards "higher math" requires not only practically a major upgrade in knowledge and toolsets, but also some shift of thinking paradigm - the task is not looking for an elegant and ingenious shortcut to a particular problem, but a highway that is general and inspiring to a fully new field (like how equations opened the door to linear algebra and everything subsequent). I pushed myself to embrace the transition, but it didn't seem to work. I managed to pass the exams and obtained a higher degree, with some expertise in a particular field of application. Still, I always have the feeling that my mathematic understanding is like house built on sand and lack a solid foundation. There is some sort of chasm that I failed to break through...
I agree with the part about mental skills, partially. Experience with math competition improved my concentration and persistence. I found piano practice more contributing in this means.
In many ways it skews the ratings of the schools because they can be lazy and not teach as well...but still show great school average scores, since so many kids are already enriching externally. Before you know, the school is just a motion and the real learning is at home. I suppose it is idealistic to think teachers "should" teach well, of course, since in reality not all do.
Amongst the participants, it isnt secret -- you see all the other participants at the center weekly, or more. I think a lot of it is a class thing that runs side by side.
For the outsiders, it is a secret. I was part of a group in K12 that didnt even always have consistent nutrition. The Kumon kids were a strange breed -- folks who had money to "splurge" on "private" education.
This is how I was raised and how my family went from sugar cane peasant farmers to school teachers in one generation and from school teachers to software engineers and doctors in just one more. From 6th grade dropouts to high school degrees in one generation, and from there to masters degree in one more.
Culture matters. Values matter.
There was a time in my young life when my parents slept on the couch so the kids could have a bedroom, so we could live in a tiny place so we could pay less rent and afford to live in a really really good public school district where many kids' parents were ivy league college professors.
It paid off. It usually does. Culture. Matters.
I'd be a little careful with venue effects on a discussion like this: this is a group of people that have, as a cohort, a particular fixation on academic and especially mathematics status signals.
My son then attended RSM from first grade. RSM instruction started from problems like "there are 3 birds on a branch. 1 bird leaves. how many birds are now on the branch" and progressed onward. By grade 7 he is learning logarithms and, at a very basic introductory level, abstract concepts such as function
(by function I mean the real definition of a function, not the easy f(x) = 2 x + 1 - https://en.wikipedia.org/wiki/Function_(mathematics))
I stuck with it for 6 months during self study and now I see it everywhere in the world.
Working class too if you're Asian American.
Asian American kids in SF public schools and the closest suburbs (eg. Daly City, SSF) skewed working class but the parents would also push their kids to attend Kumon or cram schools.
Same story in working class Asian neighborhoods of SoCal and Boston like SGV or Quincy+Malden respectively.
EDIT. Below is an example (not from his instructor but the same material). Remember this is for "business calculus". It just seems like silly math tricks to me. https://people.tamu.edu/~jdkim/math142fall2019/math142week11...
It's true you won't do a symbolic integration ever again, but practice learning stuff like that will serve them well when they're learning intricate tax code rules and principles of managerial finance, cash flow, rates of return etc.
I mean, now that I say all that, they'd probably be best served by just jumping into the complex business and finance math and skipping symbolic integration, once they totally understand integration conceptually. But if they don't, they're still getting good practice out of it, especially if they find it challenging. That means they are learning problem solving techniques they didn't know, and need to know.
The only way you're right and it's a waste of time is if they find it easy.
If you are growing up as an average working class or lower middle class American, the education-hating culture is hard to escape except in a small number of places.
If you are growing up upper middle class, your family can just avoid those problems by living in a nice suburb where everyone values education.
Believe me, I'm knee deep in this dilemma right now. You can inculcate certain values as parents, but your kids WILL absorb the culture from the kids around them too to some degree.
And that is why we're in no hurry to buy into that kind of neighborhood and don't see it as any better than decent inner city neighborhood -- in fact, based on experience the education is better in some Philly public schools than some Nice White Rich Suburb schools (although the food and the A/C sure ain't, but that's not what school is for), and the values is why.
That said I spent part of my childhood in Princeton, NJ and that is one sort of "white suburb" that breaks that pattern and actually seriously emphasizes education. Unfortunately the truly nice places like that are truly expensive. Although, to your point... Princeton had a huge Asian and Indian population, and plenty of the white kids had parents who were Princeton professors and such -- not a very common or average job, even among other people of the same race, class, and income.
Perhaps I just need to find an affordable very Asian suburb, somewhere near an H-Mart or something?
The emphasis on sports in Texas is a big turn off for me. We opted to live inner city “bad school” area and just send our kid to a private school. They have sports but it’s basically for fun, exercise and team building life skills. We don’t care about how good or bad we are for the most part (of course we celebrate the wins… but hope you see what I’m saying).
Cost wise, it’s expensive. It wouldn’t be feasible for an average income family but we can swing it with one kid.
Boy, you’d think some of those college educated people would have, I don’t know, modeled some of this and analyzed it before they sold their parking for a billion bucks that they promptly and entirely spend on extra dorms they now can’t fill.
So no, you are using outdated data. The fact is demand around the country is plummeting for schools. This is straight from the department of higher ed in Ohio.
As for global demand, it’s not the eductions it’s the chance to get citizenship. Someone with analysis skills like you must have a degree.
My kids use Kumon and RSM too, only because what their school covers are pathetic. The content may be okay, but the teachers certainly didn't give good enough homework to help the kids deeply understand the math concepts and to get valuable problem-solving skills.
That we rely on Kumon and RSM says a lot the abysmal state of the education quality in the US. Case in point, I would not need Kumon or RSM at all when I grew up, as my school covered way more and way deeper math. Note the US is still the best country for the top students and those who struggled with academics. It sucks only for the majority -- the students in the middle like me. They could've got trained hard, yet the school squandered the opportunity.
If you didn't already know, SES is the rating (with relevant cultural differences that override wealth). Everything about the school is statistical noise by comparison.
Culture matters, and not all cultures produce the same outcomes. That's not a judgement on any culture. Each culture is extremely effective at acquiring that which they value [2].
[1] My wife who grew up in a mostly white area is realistically less bigoted than me; and in my view, this is because she had less exposure to people from different cultures and so just assumes everyone's like her. I honestly think mainly being in a homogeneous society makes you less bigoted.
[2] This is not to say we should be close-minded. There are always exceptions, and the exceptions can be exemplary in their own right. But the exception does not make the rule
[3] There are many subgroups of whites. The German-descended ones have very different views than your 'standard' American white
But then again, that's really difficult to actually do. For anyone who grew up surrounded by resources, that might sound like a really easy and obvious suggestion. "Just listen to the tutors your parents bought for you." But for the students who can't afford books for this year's classes, you might as well be telling them to "just grow wings and fly, it's not hard".
Me personally, I knew plenty of people who did this, learned a year ahead so they looked extra good in class. Most of them had parents who had PhDs, paid their rent for them, and explained what problems they were going to face far ahead of time. For the students who leave class and go to work to pay their own rent and then go back to campus to study and do research at night, this is not very helpful advice. Like so many educational "one simple tricks", the unspoken prerequisite is "just be born rich".
More even than pre-teaching, I would encourage any parent to very actively be involved to ensuring that their child maintains a reasonable comfort with math throughout their study, and to the extent possible, pitch in to help those gaps beyond "passing" or doing "ok" in class, but to earnestly try to see if their child is comfortable. The reality is schools will very frequently PASS your child and given them fine enough grades, but I would argue that it is oftentimes almost orthogonal to how comfortable your child genuinely feels with what they've learned.
I asked if she found them difficult. She quipped, "If you already know the math, it's just nomenclature."
That test was just multivariate calculus I'd already aced, with funny names. I got one of the top scores in the class. So I decided to study an extra hour next time, just to be responsible. Oops! I flunked a test that was differential equations with funny names.
I didn't really learn ODE's till Columbia assigned me to teach them as an assistant professor.
Success in early learning is heavily correlated to how invested your parents are in their kid's education.
It's not a money thing (as plenty of us 1.5 gen Asian American kids can attest to)
I learned to read early because my immigrant mom read to me in her non-native language every single night, and that's because she came from a culture that lauds education.
I wish every child was lucky enough to have a parent like this, but so many kids only get their first exposure to education in public school.
As opposed to Math - which keeps going and going well beyond college....
Kids hate math because teachers and textbook writers hate math, who put no fun into it.
Anyone know an alternative?
The book explains a topic concisely and then gives exercises. Importantly, the exercises don't assume previous knowledge and you can solve them by applying previous explanations. Highly recommended!
I work in science and often work with highly skilled people from China and India. Theses people are much better in applied math than I ever was, but somehow my erratic highly derivative style of problem solving is at least as good at getting the job done and I am much better in thinking out of the box than most of them.
It seems insincere to frame this as math is important, and earlier > later without focusing on what this means, or the opportunity costs. Could we just do a global search & replace on 'math' with 'literature' and end up in the same place?
I believe that whatever little "edge" that gave me in learning math in school compounded exponentially over the years. I always felt "ahead" of the standard school curriculum, and that created a virtuous feedback cycle of success, which bred confidence, which bred success, and so forth.
Just a little nudge here or there at home can make a big difference.
The article's pretty good on why institutionalized education doesn't like students who are seriously ahead in learning math. (Or any other subject.)
But it's pretty much silent on the self-discipline and self-study skills (or parent-paid tutors) required, to seriously learn math years ahead. And the former are probably far better indicators of long-term success than the early math skills are.
I hugely regret this.
1. I didn’t find it that interesting, and so I don’t feel like I got much out of it. 2. I found later that I learn math much better when I can “hang” the ideas off practical examples. For example, I learned math for the sake of understanding deep learning far better than I ever understood math before.
Ultimately, I think it’s far more important to study something that interests you, and to learn the tools you need as you go.
Assuming that is true, but that there is still a significant benefit to attending a good university - in terms of connections, social experiences, status etc. - should we maybe strive to decouple the university experience from course enrolment - e.g. make it easier for people who have pre-learned the content, to prove their competency and essentially jump directly into a free-form experience similar to grad school?
It can also feel incredibly demoralizing to be toiling in those trenches. Feeling like you are qualified for the job but you just need to get these damn experiments to finally work so you can actually leave and no longer be impoverished.
It would be nice IMHO to see a more hybrid approach at Universities to teach math and application at the same time. It's strange to send students through YEARS of math classes without strong application. It's like learning music theory without playing an instrument.
Our academic system in general is still modeled after old-school institutions, based on textbook-style learning that all pretty much follow the same recipe. Is it not crazy that we have classrooms in this day and age with 300 students sitting in desks listening to a single professor? It's insane.
We are ripe for an educational system that is truly disruptive - especially with the rise of AI systems.
I agree that we are ripe for an educational system that is truly disruptive. Our current educators are so disconnected from the real world and have no idea how to apply what they teach.
Learning math, just so you can learn it again is quite pointless!
Much better hack is to skip academia completely, and go self educated. No debt, no pointless extra classes, no risk of being misaccused, no politics! You can even move to cheaper country, with nice weather, to have better environment for studying!
Academia does pander to the masses, and it provides a path to take a person off the street and turn them into a somewhat of a knowledge expert in a range of disciplines.
You also hope that your nurse practitioner, physician or surgeon didn’t take a self-taught path.
However, I would love to have a doctor who was so passionate about it they taught themselves before going to school.
Self learning a topic is largely an ability of those who have been taught advanced knowledge in some area.
But as a grown-up, it's more efficient to get help learning hard things. And some things are harder than others. I think you can learn calculus on your own, and certainly computability theory, and point set topology, but learning finite-group theory, which has a lot of numeric details, or measure theory at a really solid level, would be getting harder. Still doable if you have the inner drive, but lot more efficient to take grad level classes where you turn in homework. Also doing a lot of homework does give you a sort of muscle memory "a function is continuous iff the inverse image of open sets are open".
I wouldn't tell everyone to become a professor, but I'd certainly recommend US grad level classes as an extremely efficient way to learn a lot.
Sure you can get a head start on some preclinical subjects or may study them as part of an undergrad, but that isn’t the “hard” part. You simply can’t teach yourself to be a doctor, since the job is so intimately tied to a complex setting you must participate in, and there’s no Linux kernel or GitHub equivalent.
> provides a path to take a person off the street and turn them into
That was true maybe 40 years ago. Today students are asking for debt forgiveness! Academia ruins people financially for decades!
> somewhat of a knowledge expert in a range of disciplines.
University graduates are pretty much useless in practical disciplines. They need years of additional training to become employable.
> You also hope that your nurse practitioner, physician or surgeon didn’t take a self-taught path
Medical professionals have several years of extra training in hospitals. They have to "self study"!
And I'd hire a math major with limited software experience over a boot camp or self-taught person that only knows code any day. In fact, I'd take a math major over most people with MS in CompSci. They know how to learn very difficult stuff, and didn't do it in an environment that is mostly people wanting to be highly paid, but mostly people that have a love of complicated but beautiful abstract structures (hence less weird resume lying and so on; also, tends to be a bit of a salary arbitrage opportunity). (Hiring for experienced people is of course a different problem.)
Of course, trying for a professor job in the US is very likely to a difficult career path; I'm taking some math classes just for fun and the professors are usually grading our papers at insane hours, 3 am and then office hours at 9 am). I could not have done that much work and been a good parent.
But academia is great training. One of the best project managers I've worked with had a PhD in Anglo-Saxon english; her dissertation was on masculinity in the court of the Anglo-Saxon king (or something, I've not worked with her in a long time); surprisingly relevant to trying to get the mostly male dev teams to coordinate to finish projects when she didin't have the feudal power of the technical managers, just the soft power of the travelling minstral.
Nah, you wasted 5 years of your life.
This is very true, especially now. So many families, at least in the competitive places like the Bay Area, push their kids to spend enormous amount of time on AMC, AIME, and etc. Other than viewing competition math as a way for their kids to get into elite universities, they often think that doing competition math as a way to be really good at math and they can cite many examples kids who are good at competition math also would have a bright career. Unfortunately, they got it backwards: kids who are naturally good at maths will like do well in competition math (think about Schulz or Terence Tao), but really not the other way around. For people like me, who have limited talent on maths, focusing on learning higher math and the associated essential problem-solving techniques will have a much higher return on investment.
Also, this really shows how the incentives in "education" are deeply misaligned with the way we talk about it. At least in the US, the point of education seems to be mostly gating outcomes and sorting people. Learning is incidental and game theory suggests it's better to never take a class that's truly new material for you, because getting a bad grade can harm you, but learning something new isn't captured at all
Ok, I get the principle but learning multiple years worth of university math is starting to sound unrealistic? I understand learning something in advance to have an easier time, but this is almost the same as finishing a degree before starting it.
Learn Latin and you can fake your way through so many situations.
Kind of a dick statement
Each paperback book costs less than $20 on Amazon.
Well, yeah, of course.
But this is basically the "draw the rest of the horse" meme.
What about any discussion of how to learn the material in advance, why self-guided learning is better than course-driven learning, or just how to prioritize advanced learning with everything else going on in your life.
Why is this on the front page today?
It tries to substantiate the ahead-of-time learning with how it will benefit you on a larger scale than a course or even a degree.
This is why hobbiests and apprentices are higher skilled professionals than people with mere educational certifications.
Need is the key here in my opinion. Kids usually don't like math unless there is a need for math for something they do like.
https://www.nationalreview.com/2021/09/the-folly-of-woke-mat...
https://www.edweek.org/teaching-learning/seattle-schools-lea...
Damn Chinese, Arabic, Indian and Mesopotamian people, they ruined everything with their Geometry and Algebra.
Oh, wait...
Dear Gutierrez, Science and Math doesn't give a crap about race/ethnics and even less to crybabies as you. And as I say this being a Spaniard, an odd blend between an Iberian, an Atlantic/Mid-European (Goth) and Mediterranean (Who knows, point a huge chunk between Tartesos and Rome) people.
In the Hispanic world (the actual one, not the joke invented in the US) no one gives a shit about the race. It's all about nurture against nature. Since the old times. (Uno no es de donde nace, sino de donde pace) -Lit. one does not belong from where he was born, but where he is lying - - -> Home is where the heart is.
BTW. Latinx -> US creation, not Hispanic. We usually do Science subjects in Spanish AND in English once we reach University/College, thanks. No one it's hurt. Skills on technical English are a must, period.
Black and Latino students here (inmigrants from overseas) do it perfectly fine in Spain. First they study in Spanish, and later in English which is much harder to achieve at the age of 18-20. Stop the ethnic bullshit, please.
Our country invented Algebra, please. European Spaniards learnt it fine from the Moors in ARABIC more than five centuries ago. Later they translated it into Latin and into Castillian Spanish. Are the American children challenged, or what?
You look like the sickos who put "White Only/Colored" labels on everything.
The actual struggle for these children is not the race. It's money and parents being underpaid.
This definitely created a lot of tension along the years. He just couldn't understand why people don't like learning Math, and I just couldn't understand why I couldn't watch TV every night. LOL.
My father actually wanted to teach me programming too. But similar to teaching Math, he wanted me to go competitive programming, which I absolutely hated and still hate. If he tried teaching me game programming then it was going to be a completely different story. I eventually taught myself programming decades later. My first language was C++ and my first project was a 2d game engine.
IMO, all those teaching he tried to feed me, not only did not increase my motivation or learning techniques, but decreased them. Throughout my childhood (starting from maybe 9), I absolutely hated summer and winter vacations. While my friends were enjoying, I had to go through TONs of extra-curriculums. I used to practice piano 4+ hours a day (as long as I don't have school), and some other hours for extra homeworks. I absolutely hated that, to the point that I hated playing piano and completely dropped it after actually achieving a lot. My father simply doesn't understand why would a normal human-being hate piano, music and Math, when he couldn't even get them when he was young. I didn't bother to explain.
You were probably not that tough to your children though, so I guess they fared much better.
So, here's my hot take (which probably isn't terribly original): Compulsory school math should end before algebra, and the rest of the curriculum should be taught the same way (or better) to how we teach art or music.
If you need advanced math for your career, teach advanced algebra or calculus as needed. At the very least this will force post-secondary schools to be honest about how prepared students are leaving secondary school. Right now, it "those people's fault" for how poorly prepared for advanced math most kids are.
Basic math literacy is incredibly important. But being able to solve quadratics or discover geometric proofs is colossally unimportant to 98% of humanity and it's importance can usually be determined based on personal interest in a career. Let's be honest with ourselves that most people well and truly will never need advanced math. Exposed kids to it as a fun game or art form, not a tool that they will never use.
Should learning to use a belt sander be an educational requirement to move from 9th to 10th grade? No, no it should not.
The only class I'd legitimately believe we should teach is labor organizing/union participation, since every career involves labor.
Math, despite what some say, is not that fundamental, but reading and writing well is(and then helps to get math and others).
In foreign language and elective courses (such as history) doing the reading before the lecture meant I could focus on what the lecturer thought was important rather than absorbing new information.
Now I've gone back to grad school (30 years later) and I also have kids (older but not completely ignorable :) and a job and a wife I am determined to keep happy, so I have to optimize for time, so I'm mostly going into lectures blind except for whatever foreshadowing "motivation" they've done, so it's a constant stream of completely new stuff, but a lot of "wow, that's cool" moments.
> The Greatest Educational Life Hack: Learning Math Ahead of Time (justinmath.com)
worked for me, can work for a lot of people, and is a good idea.
Partly:
(1) One way to win a 100 yard dash is to start running half way to the finish line and have no one object. The US educational system will usually overlook something like that starting half way to the finish line.
(2) Reasons: (a) The system assumes that their teaching is crucial and that no student really can learn on their own, i.e., the student didn't actually start half way to the finish line. (b) The system so wants more good students that they will overlook the evidence that the student was ahead at the start of the class. But, research in math mostly requires working alone directly from original papers, and working from a highly polished text is usually much easier -- so profs learn on their own, and students can too.
(3) Generally in math, independent study can work well. Basically for each lesson, (a) study the text, (b) work most of the exercises, especially the more challenging ones, and check the answers in the back of the text (need a suitable text or just get the book for teachers), and (c) in a quiet room, lean back, relax, and think a little about what the value, purpose, content of the lesson was, say, be able to explain it to someone who never studied math.
(3) So, take calculus in high school. And visit, call, whatever, and see what the popular college calculus texts are, get one or two of those (used can be a lot cheaper), and before college have worked hard on both the high school course and the college text(s). Then in college, right, take calculus, likely from a text have already done well in. So, will likely be one of the best students in the class. Then will get a good reputation that can be valuable.
(4) Will be ahead, so continue this way and stay ahead.
(5) Next math course, say, modern, abstract algebra, i.e., set theory, groups, fields, Galois theory, elementary number theory, maybe a start on linear algebra.
Next, linear algebra, maybe the most important and useful course. Work through a popular text that is relatively easy. Then work carefully through one or two of the classics, e.g., Halmos, Finite Dimensional Vector Spaces.
Likely next, "Baby Rudin", W. Rudin, Principles of Mathematical Analysis, calculus and somewhat more done with depth and precision. See the roles of open and closed sets, closed and bounded sets, i.e., compact sets, continuous functions, the powerful results of continuous functions on compact sets, Fourier series.
Advanced calculus, i.e., partial derivatives and Stokes formula.
Analysis, e.g., the real part of W. Rudin, Real and Complex Analysis, Lebesgue's alternate and nicer way to define integration (in short, partition on the Y axis instead of the X axis), the Fourier integral, Banach space, Hilbert space, the Radon-Nikodym theorem (can be used for grand approaches to information, Bayes Rule, the Neyman-Pearson result in best statistical hypothesis testing, and with von Neumann's proof based just on polynomials is charming, ...).
More, e.g., differential equations, probability, statistics, stochastic processes, optimization, complex analysis, number theory, whatever.
One consequence: Will learn how to write math. Too often people who don't know advertise that they don't know much math.
At some point in business, some of that math might be valuable. E.g., current AI uses steepest descent via calculus and optimization, linear algebra, and hypothesis testing.
This isn’t a life hack. This is the sign of a failing system. Jesus fucking Christ.
The OP is talking about taking pre-emptive maths course and accelerated learning not particularly taking a full course degree in math.
Personally, I think engineering students will benefitted more if learning engineering maths in their pre-universities courses namely matriculation, A-Level or Baccalaureate, rather than spending few years learning the subjects during their main engineering degree. It helps the student focus and minimize the pre-requisites.
"Outliers: The Story of Success" ( Malcolm Gladwell, 2011 )
i.e. the curriculum lesson plans naturally evolve to exclude individuals that don't need introductory lessons, because they are on average 3 years ahead of their peers by the time they enter undergraduate programs.
The kids that need to "catch-up" in introductory Math/English material are no longer failed/held-back a year in some municipalities, but rather given a remedial curriculum over the summer. If those kids parents can afford to put them through an early tutorial program, than excluding the "poor kids" from a seat at the more lucrative faculties is rather guaranteed.
https://www.youtube.com/watch?v=qEJ4hkpQW8E
Mind you explaining to privileged kids why they _get_to_ attend additional instruction can be difficult. As social media normalizes lack of impulse control, and rewards group-think biases. Our little ingrates think they can con/hack their way through life, as some fool on the web is telling them to take the easy path.
Some university kids that rely on student visa programs to access the US immigration process, will get desperate and try to outright cheat their way through a Bachelor of science degree. The real scandal is some folks get 50% of the final problems from $18.74 USD gray market course manuals out of HK, as many institutions must structure their exams this way for credit-transfer compatibility. The myth of natural talent deteriorates further with some fraternities also gaming the system to out-compete the rest of the student body when possible. Indeed, some people do hack/cheat their way to a better life using underhanded tactics, and are rarely held accountable. Some places are even removing the barrier where one needs to be fluent in English.
You are probably still thinking this can't be right, and seats for becoming a physician/pharmacist/lawyer are open to anyone. Yet I can assure you that while the faculties will take your money, the probability of getting into a Masters/Doctorate level program quickly drops while you worked hard to catch up... Note your GPA took the hits along the way.
People need to recognize there is a subtle yet important difference between intelligence and academic performance. No one ever claimed life was fair, but the hypocrisy of many meritocrats can be intolerable at times.
Stealing Einsteins chalk does not make one Einstein... but does silence talent.
Have a great day, =3