The Greatest Educational Life Hack: Learning Math Ahead of Time

I was a late bloomer in almost every arena of my life. Developing social skills, having relationships, developing an identity independent of my family, etc. I'm also a late bloomer to mathematics.

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 :)

I went to the most prestigious high school in France. The top 2 students in my maths class shared one thing in common: they would study the curriculum the summer before.

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.

If you are native speaker of any language different from English, the greatest educational life hack is to learn English at the earliest time. It opens one's mind and allows access to content and communication at a global level.

I’m going to push back on the advice to learn higher grade math rather than competition math, as I feel the author is ignoring an important skill that competition math helps develop. They allude it in passing:

> 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.

If you go to any of the wealthy or upper-middle-class suburbs, especially those with large immigrant populations, you'll see half the students secretly doing this, whether it is via Kumon or RSM or something else.

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.

I got good at calculus when I started doing differential equations. I got good at differential equations when I started doing modeling and control theory. In general, you don't get good at a subject when you learn it in class; you get good at a subject when you work on the stuff one level beyond it. So yeah, if you want to be good at the class you're in, start studying for the class after it. This is definitely an effective method.

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".

I work at a private school and will sadly tell you that the author's points are actually pretty severely understated when it comes to the incentives of schools regarding this phenomenon. Differentiation is a word that gets thrown around as some tremendous necessity for schools to implement, yet in the case of math, where one could fairly easily (compared to other subjects) confidently assess the attainment of prerequisites, gauge student progress, comfort, etc., we comically either bound students who have clearly mastered materials OR happily move them along the math curve in which the deficiencies in mastery build on each other to eventually lead to a child who truly has a strong distaste for math.

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.

My wife was a great example of this. She was an undergraduate math major, then went on to get her master's and PhD in engineering. The first year of the master's was largely remedial engineering courses - statics and dynamics, thermodynamics, controls, simple electrical circuits, etc.

I asked if she found them difficult. She quipped, "If you already know the math, it's just nomenclature."

A bit of a sensational title, I would say that Learning to Read as early as possible, then reading well above age level, would be a greater "Educational Life Hack".

My 4th and 5th grade teachers tricked us into learning algebra by calling it "enigmas" and treating it like a fun puzzle instead of a math problem. It definitely worked on me, I was quite shocked when middle school math was just those puzzles under a different name. Made those classes quite easy though.

I am currently learning maths independently. I'm using the book, Maths: A Student's Survival Guide by Jenny Olive. I'm towards the end of the first chapter and feeling confident with basic algebra now! I picked it up after seeing it recommended here.[1]

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!

1. https://news.ycombinator.com/item?id=39050972

This is a hack to create people wha are successful in the education system, I wonder if it is the right approach to create educated people.

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.

The argument made here is there are risks learning math when everyone else does, so learn it earlier. Great, but how? Only the very few have the resources and environment to learn non-trivial math early. What does this displace? Is it more important for a kid to learn calculus, piano or a second language? Are younger people capable of learning math in a no-painful way? Why do they have patient, knowledgeable teachers at this level but not later? Math can be hard because of the required discipline and practice - are younger people better positioned to solve this, or worse?

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?

This is simple but so effective. When I was 5 or 6 years old, my mom would sometimes give me one page of simple math problems. They were all basic arithmetic, things like 12+17 or 99+99 or 8x7, etc. I did them and got on with my life. They probably didn't take more than 15-20 minutes. They didn't feel like much because they really weren't. I think any 5 year-old can do them.

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.

Within a limited range of academic disciplines, it's a great hack. Outside of that, and situations where being a "math genius" is social cred - not so much.

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 did this: I studied pure math in uni because “it could be used for anything.”

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.

Reading between the lines in TFA, it seems that they're implying that university learning is really bad, and pretty much any other way you can use to learn the subject matter before getting to university will serve you better. There's a long discussion to be had there, but for the sake of argument, let's take that as a given.

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?

Perhaps I'm in the minority here, but I've wasted a ton of time in math classes working through way too many academic exercises that have little real world applications. For example, learning a bunch of tricks to solve a differential equation by hand feels like a circus act. Sure it can be done, but only with a limited set of "textbook" equations. When you get into the real world, you'll need to put those equations into a solver like matlab, etc.

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.

> Learning math early guards you against numerous academic risks and opens all kinds of doors to career opportunities.

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!

> Higher Math, Not Competition Math

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.

I agree with this tip. Works great for anyone who can autodidact, and if you're good at finding and vetting resources, autodidacting got easier with the internet, and has only gotten a little harder with the proliferation of nonsense on the internet for topics that aren't hot in business or politically charged

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

Then there's the approach taken by my university's physics department, where they made it a point of pride to always have you using math before you'd learned it from the official math classes...

Learning the whole course ahead of time sounds easier said than done. But I definitely recommend pre-learning the next chapter in the course instead of relying on the teacher's explanation. Personally, I could never understand a relatively complicated math concept just by listening to the teacher. I usually need to think about it, draw things, read several different explanations, etc, to really get it. But when I was already familiar with the topic, then I could benefit from another repetition and ask questions if there were some complicated aspects.

The greatest failure of our time is that there isn't a viral, ad-free website or app for children and teens to just go and learn math on their own. Everything worthwhile requires a credit card, user account, and monthly subscription. Children don't have credit cards, email addresses, and access to the latest iOS device. They do have time and at minimum sporadic Internet access. If we managed to create Wikipedia, we can manage to create a similar site for enjoying and learning math.

I confirm this! My son is 10, finishing 4th class. We're constantly 6-9 months ahead of his class. I think he once in those 4 years got note 2 (one below highest), every other one was the highest. Vast majority of his math classes look like "oh I know that" or "oh I remember that, just need a 5 min refresher". Thanks to it, he has more time for other subjects. His stress level at school is close to zero.

Learning ahead definitely helps me a lot. For some reason I am not capable of learning things from scratch in one swoop. I always need to learn things a little, let them somehow settle in my brain for a while, and then go further. I always had trouble in school when things moved linearly.

> why stop learning one year ahead?

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.

The Even Superiorly Greatest and Lovely Educational Life Hack: Learning Latin Ahead of Time

This presumes an educational career that benefits from engineering math. It's interesting to me that even a lifetime in computer science doesn't necessarily reward this strategy (it might, it might not, depending on focus areas).

One of the best, most cost effective ways to do this is by enrolling at your local community college. Faculty there are primarily focused on teaching, and WANT you to “get it”. In addition to math, I recommend you take ALL the STEM courses you can that you’ll touch in university. I took separate classes in Unix and C at community college before my university quickly introduced them in systems programming. Boy, that was time and money well spent.

Beyond the general idea that the more time you have to think about a problem the more likely it is you will do better at solving it. How does this translate into an ability to solve more emergent problems? Isn’t this “hack” somewhat similar to the idea of people who have never had to step up and learn to work harder. And in fact the hack gives a false sense of confidence in the ability to solve more typical real world problems when it matters.

"When a middle or high school teacher has a bright math student, and the teacher directs them towards competition math, it’s usually not because that’s the best option for the student. Rather, it’s the best option for the teacher. It gives the student something to do while creating minimal additional work for the teacher."

Kind of a dick statement

As a PSA for anybody who wants a very readable introduction to "real" math, check out Jay Cummings' two books: Proofs and Real Analysis at https://longformmath.com/

Each paperback book costs less than $20 on Amazon.

Spending all of your time studying isn't a "hack". Not saying it's a bad idea, but it's a ton of work

I'm just confused by this article. It's basically "Learn a course before you take the course so the course is easy."

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?

Tried teaching my young nephew about math. He just bashed me in the head with the abacus. Then started crying.

I had to lean match for writing programs at 12 and after just a few weeks of trying to make a game that had some higher math, I was leagues ahead of my classmates.

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.

I remember having trouble in a electricity & magnetism course because I needed to learn some math concepts (divergence, gradient, curl) at the same time as the physics. It would have helped to have studied multivariate calculus before the E&M class.

And there are places that have or are trying to ban algebra in Jr. High School (e.g. SFUSD)

My father (a Mathematician) used to teach Math to me early. But somehow I was not motivated to learn Math myself so every year I got a very good mid-term grade but terrible final grade. He also taught competitive Math to me (the Olympics) but to be frank I was totally uninterested.

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.

EDIT: Nevermind, this whole thing is just an add for a tutoring service :(

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.

It definitely makes the first couple of years in university that much easier, although limited to the science and engineering disciplines.

Greatest Educational Life Hack is getting your children to love going to school.

Is there anything specific to mathematics about this?

Slightly galling that people write this kind of drivel without examining any of the shaky premises it's logic relies on. Yes, in a perfect world, we can all learn our course material in advance and skate through our in-class education. More practical advice would be to build strong study habits and networking skills. Being able to get your work done with more time for editing/revisions and having access to other perspectives on the course work would have definitely improved the quality of my education. Building those habits and community take time and energy. I guess no simple hack there.

Yes,

> 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.

Fuck this fucking article. You know one reason I know my math degree is bullshit? Because I figured out quickly my profs wouldn’t explain jack shit and I would have to “learn the math early” which meant just teach my self with other kids the god damned material.

This isn’t a life hack. This is the sign of a failing system. Jesus fucking Christ.

This is basically an ad

The bimodal distribution of student entrance performance correlating to stratified fiscal castes has been observed for sometime:

"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