Yes, that was a great article. I would like to know what changed in the author's mind. Did he just get good at particular math problems through repeatedly doing them, or did he gain a general math aptitude? Given his performance later in life it seems like he gained the latter.
I became interested in math in a similar manner, and I think learning it this way gave me an edge over those who were just force fed it in school. There is nothing like understanding the practical significance of a subject to make it stick in my mind and give me intuitive links. This aspect of learning seems to be really lacking in public school, at least the ones I've attended.
Thanks for writing! I actually read that article that day, and it was really on time. It's a good motivation, but it doesn't talk about how one should learn Math. I find the commends below very informative and I've concluded that I need to spend some time on finding better ways to learn, and what suits me well.
I have to say, I actually loved this article. Especially “Understanding doesn’t build fluency; instead, fluency builds understanding.”
I love math and majored in it in college. The rest of my family is all scientifically inclined, but I think found/find math itself opaque and somewhat intimidating. I remember my brother asking me at one point how one would ever find, for example, the Pythagorean theorem intuitive. The author’s quote is the response I wish I had. The Pythagorean theorem becomes intuitively true not when you have some deep insight about Euclidean space, but when, on seeing a right triangle, three proofs of it spring instantly to mind. Which happens after a lot of practice.
FWIW I think it’s appropriate that the author talks about herself a lot. She’s trying to explain the subjective, cognitive experience of going from math-phobia to math mastery over her career. She can’t explain that without talking about her background and her perception of the process from inside her head.
Thanks for posting the article. I wish I had known more about this when I was in high school. I have recently found myself becoming much more interested in mathematics when thinking about very simple concepts and asking why it is a certain way vs. memorizing the rule. I think my 10+ year aversion to the subject is rooted in the fact that it felt like nothing more than rote memorization in high school, and thus a dreary subject to learn anything about.
Thanks for a great essay. I'm in exactly the position you describe at the beginning. I have a degree in Math, but always felt like what I was learning was just ahead of what I was understanding. I work in computers, so I get to use the reasoning skills I developed on a daily basis, but I have been wanting to start a rediscovery of Math. Thanks for the great pointers.
Primarily it instilled that love of math, where previously there had been anxiety and ennui. This was sufficient to get me actually complete the work -- and do some extra work for fun -- which was more than enough to get through the math in my university coursework. I didn't get into more advanced maths in university, and for the fields that I've ended up in (and here I need to omit some otherwise too-revealing biographical details), I'm undoubtedly sub-par at mathematics. When it comes to calculation, I'm still definitely slower and more mistake-prone than my colleagues. I attribute this to avoiding mathematics for my first two decades; if math is a language, then I learned it at an age when having a thick accent is unavoidable. I make up for these deficiencies by being much more stubborn than my colleagues -- just hammering away at problems until they're done -- and also having an excellent intuitive grasp of certain domains, which I can then implement via coding rather than formulae. (I learned to code when I was very young, so that's something which I'm much more natively fluent at). After I've solved a problem via intuition and code, I usually try to prove (or at least investigate) it with proper math, to reassure myself that the solution is efficient and correct. That's far slower and more painstaking, but I eventually get there (and nine times out of ten, my intuition was correct). That's probably the wrong way to do things, but it works. I wouldn't be able to have a job where the math necessarily came first, however!
This is insightful and rings true for me as well. A good blend of theory and practical knowledge is important to get the most out of any education. Do you have any recommendations for learning math this way?
I loved reading your comment. Your story is very similar to my own! A little over 8 years ago, I also started at the beginning of Khan Academy. Due to reasons related to my childhood, I had essentially no education growing up. When I was in my early twenties, I had only an elementary education. The highest level of math I knew was basic fraction arithmetic. I had never written an essay and I did not have a scientific understanding.
Having no education, I only did menial work for money. Yet in my early twenties, I was contemplating my lack of scholarship and realized I wanted to fill the holes in my education. I went to Khan Academy and, as you did, started with pre-K and worked my way linearly through up to pre-college math. Thankfully, I was soon laid off from my job, which was an opportunity to start attending community college.
I then transferred to a state school and double majored in applied math and computer science. Now I’m doing theoretical research as a PhD student in computer science.
The 8-year path from pre-K math to graduate-level math classes and now being published has been a journey. And I’m deeply grateful for resources like Khan Academy.
Deciding to commit to a daily study of math transformed my life.
One of the commenters to the original article talks about this phenomenon specifically. Each math course you took taught you some ideas or techniques, but those techniques weren't really learned until you used them in the next course. For example, you didn't learn algebra well until you used it in calc 1; calc 1 skills are really cemented in difeq.
I've read this article several times at this point (it does tend to pop up everywhere) and it resonates with me but I'm not sure what to do about it.
I really want to experience the kind of math the author writes about; can anyone recommend a place to start as someone who has only ever done "fake" high school math? I'm in college now and I'm halfway through a computer science degree; I've tried a few times to break into theoretical math classes but I've found the bar for entry pretty high (especially when I only have room for one or two courses), with most classes and even peers asking for years of experience and "mathematical maturity." Have any of you ever succeeded in learning some math outside formal curricula?
This hits close to home for me. I was one of those kids who was 'good at math' throughout pretty much all of school. I tested out of Calc I thanks to AP classes and began university in Calc II.
It didn't go well. Somehow this thing that had seemed so natural and intuitive now just seemed totally incomprehensible, mainly because I really just didn't understand the level the abstraction was at or something. I did poorly and it really shook my confidence. I changed majors and wound up a designer instead.
It's not a surprise to me that I've worked my way back into a field with deep math roots, though I think I would have found it much sooner if it hadn't been for that initial roadblock.
It gave me a new appreciation for what many of my classmates were struggling with in the early math that I breezed through in grade school. It's very difficult to see what the concepts you're learning are building towards if you don't have a sense of the bigger picture.
I thought I was absolutely the worst person in the world at math. Turns out, crappy teachers and a teaching methodology that is diametrically opposed to one's optimal learning style count for a lot in school.
However what was amazing to me was how I went from zero confidence in my math skills to actually being excited about math when I took geometry. I never studied once in that class--just absorbed and immediately internalized what the teacher said. I could look at a proof and it just sort of visually made sense to me and clicked and I could step through it because of the pattern recognition. To this date I've never experienced anything intuitive in that fundamentally primal manner.
What has been great lately is Khan Academy and a growing interest in teaching myself software development has rekindled my interest in math. I also owe a huge debt of gratitude to Kalid @ BetterExplained.com (he frequents HN) for getting me past my fear that higher-level maths were beyond my capabilities. They weren't--I just needed to find a way of applying them in an intuitive manner that I could easily internalize vs. staring at equations and their definitions.
I wouldn't be surprised if I would have ended up as an engineer if I hadn't had such a poor experience with math when I was younger. I am pretty resentful of it. Fortunately I can take steps to change that, and I am.
Recently I read comments about "the art of learning" by child chess prodigy and expert in martial arts Josh Waitzkin about coaching and how to conquest excellence. My take on that is that if your family or society inspire on you the value of hard work and incremental improvement, you will obtain a strong motivation to get better in whatever you do and that will give you an enormous opportunity to succeed.
Speaking from my own experience, in my high school our math teacher had a voice problem and on top of that he used to speak very quickly, hence it was really difficult to understand what he was saying, but that was not a problem for me, I learned all the math in a book and became the right hand of that teacher, my duty was to try to teach to other people in class. Another nice anecdote was how I discovered the multiplication table, I was a dumb body in my childhood, I repeated several years until I was nine or ten years. I didn't know what was 2 x 2, then someone told me that the multiplication table was written in our pencil and that to answer that you have to memorize that table, that was my start on math. It seems that all the ways to a solution are written somewhere.
Anyway I wouldn't change any bit my wild education, I have really loving memories from my chilhood and school. I would never change that for a formal education, let nature teach you, something in the spirit of Mark Twain novels.
> For most people—who won't solve complex math problems daily at work—the takeaway from learning math is not their mechanical ability at solving math problems. The takeaway is their understanding of math concepts and ideas, which will shape their thinking skills in general.
Agree, but for some others, there are real world consequences, e.g. whether they get accepted into a university or whether they can read and properly understand an academic paper.
I enjoyed reading the article; overall good advice and a powerful change of perspective regarding one’s own learning... But I have always wondered how one can find a powerful why to more abstract subjects e.g. topics in mathematics.
I’m having a hard time finding intrinsic motivation to study at this (Master’s) level, even if I absolutely know how good it feels when you have internalized a couple of famous results...
This exact theory made me utterly sick of math at a very early age. The response of the school was to assign increasingly more time to math bookwork, which I had completely stalled on. The more I was forced to do it, the more it made me sick, and the slower I progressed. For me at least, having a real application for knowledge is key.
My experience in school seems similar except I didn't get nice math courses until very late on and I was miserably bad at the memorization-based Calculus work (and I still am, but I can do abstract math very well). By then I was pretty disillusioned by school in general.
His luck probably helped quite a lot. Nobody really knows what they are good at or what they will enjoy until they do it. Even if it's 'harder' or 'easier'. Having a good mentor is difficult as well; they have to know their stuff, want to teach and mesh well with you.
> 1) It helps to build a sort of mental "muscle memory", and 2) it helps with developing intuition, since your mind eventually gets bored with the mechanics and starts to notice patterns.
muscle memory is super important in math.
the last thing you want is to be struggling with foundation when you're trying to progress through concepts. everything needs to be available to you at the snap of your fingers.
trying to progress math knowledge without the basics is doable, but it's a ton of mental overhead and frustration that you can cut down if you just get the practice out of the way.
plus it's super cool if you're one of those guys who have the ability to quickly apply 10-20 different models on a real world problem. that's what we think of when we think "good at math".
I became interested in math in a similar manner, and I think learning it this way gave me an edge over those who were just force fed it in school. There is nothing like understanding the practical significance of a subject to make it stick in my mind and give me intuitive links. This aspect of learning seems to be really lacking in public school, at least the ones I've attended.
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