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I Rewired My Brain to Become Fluent in Math (2014) (nautil.us)
262 points by paublyrne on Feb 18, 2017 | hide | past | web | favorite | 46 comments



This is related to the Curse of the Gifted. When I was a kid, I could grasp concepts easily and I understood maths very quickly so I didn't really bother doing things like homework or actually studying.

I then went to a decent university (a French Grande Ecole) and for the first year, I was still doing okish but I went from being a student who always had good results to doing average. Then, the second year came, and just understanding the concepts without cultivating fluency by practicing wasn't enough anymore and I failed hard. I hadn't learned how to learn and while I understood the concepts and could easily follow when the teacher solved a problem together with us, I had difficulties doing it on my own. I worked doing exercises and was able to pass but time and time again in my professional life, I've had the same issue where I tend to rely too much on my intuitive understanding until it fails.

I'm not sure how to help kids not to go down this path. As a kid, I would avoid any rote memorization and would probably not do any non graded exercise since I knew that I understood (and also since as a geek wanting to fit in, I tried to be more accepted by my peers by purposefully not doing my homework). In the end, I think the key is play and giving maybe challenging exercises that forces the kid to use his newly found understanding would be a way to get them used to not just rely on their intuition understanding.


This! This happened to me, and good thing that came out of it was that my ego was crushed to pieces. Otherwise I asked myself where did I go wrong, or I lacked motivation to actually sit down, heat the chair and do the heavy lifting. I found it pretty boring and it repelled me in many aspects, so i struggled but eventually made it through.

I always like learning on my own, playing and tweaking with stuff, thinking about problems around me, understanding how things were working on their core level. But with simple "chair and book" studying I suffered a lot. Sadly I don't know any solution and can't give any advice to others...


I had a similar path as yours. Never had to work all that hard through school, did a degree at a university in the sciences which was somewhat the same (had to try a little harder after the reality settled in from freshman year...), but I had always just assumed that was the next step. This was just something I had to do to move on, so a high mark was the end result.

I was thrilled to graduate, took a job out in a semi-related field for about 5-6 years, then slowly began to get this feeling of "being dumbed down". Day by day, doing the same or similar things, not having to learn new stuff; it took a toll. I started listening to podcasts, and learning about all of the advances in science was enough to get me back into school and apply for a graduate degree.

Here is where I think this is something that people have to recognize they want for themselves. Starting my grad classes with other students, mostly fresh from undergrad, I could see a change in the value I took from the classes compared to undergrads. I saw my old mentality in them ("Ugh why do we have to learn this, it won't be useful, just have to get this out of the way to checkmark a box..."). Whereas I spent time in a career, and had to force myself to make a major life-altering decision. I wanted what they had to say, whether it was relevant or not, because these ideas could be weaved together in the future. The properties that dictate the science of one is not fully independent from the other.

This really became infectious. Everything I see, I try to learn beyond just memorization, but to a solid understanding and application level. "I'm running this test, the process isn't working, where is the most likely error? How can I optmize?"

Unfortunately, I don't think you can just teach this. It has to be something that someone actively decides they want to commit to.


This is exactly what happened to me as a kid, but in my case it wasn't curse of the "gifted" but curse of the pushed. In my younger years growing up in Ukraine my parents and granparents pushed us hard to learn math, and so did my school. Endless repetition, hours of homework, Saturday classes, etc. I didn't have any sort of gift for math, this is just how children were taught there.

Moving to the US and going into 5th grade, the first several years were just repetition of things I'd already learned in Ukraine. They were so easy. My grandmother wasn't there anymore and to my parents it looked like I was doing _great_ so they also loosened up and didn't really push me further.

When we finally got to substantially new material I started to struggle. I got too used to everything being so easy and built on concepts I'd already had drilled into me. I failed a math test once - even my teacher was surprised. I ended up falling from perfect score As to Cs, then scraping back up to Bs by the end of high school.

Now I regret not taking math as seriously as I should have after leaving Ukraine, and am now taking time at home to learn on my own. Thankfully I'm used to self-learning in other areas, so I think it's going pretty OK.


I have the same experience. For context, I live in a third world country.

I was lucky in the sense that I was enrolled in a prestigious school from kindergarten to grade 4. The school has a great curriculum that doesn't dumb down math and science while also emphasizing the arts. Arts and Sciences were equally interesting to me. I made no distinction with my enjoyment of solving arithmetic with creating poems or painting.

Unfortunately, my family went into hard times and I had to transfer to our public school. In contrast to my previous school, teachers can't personally help each student because there were 50+ students per class. I also found out that the lessons they taught are already covered in my earlier years. I coasted through Grades 5 and 6 using my stock knowledge.

When I got to high school, I was used to leaning back on what I learned. This is where I first tasted difficulty in learning the advanced math subjects. Sure, I got good grades in algebra but when it came to calculus, I was stumped. I could not care less about it because "math should be about calculation, not this symbolic mumbo jumbo". My boredom and inability to deal with the challenges resulted in my delinquency. I often cut classes and plagiarism homework. I still can't remember how I manage to graduate from high school.

Now I deeply regret that I didn't apply myself back then.I am now a software developer and a lot of interesting stuff are closed to me because I can't understand the math behind them. I resolved to return studying high school algebra last year. Armed with experience and a new perspective, I realized how much I have missed and how useful math is.


I'm in the same position as you now, I think - a coder who wants to explore interesting subjects that rely on math knowledge I don't have. How are your studies going now? I have recently (a couple of weeks ago) started going through a textbook on discrete mathematics, spending 1-2 hours a day. So far it feels like I'm going through concepts that I intuitively already know (the first chapter is on logical thinking), but it has been very interesting to learn how to put names and definitions to those concepts, and to prove them. After this I hope to study algorithms and statistics.


This reads like the _exact_ opposite problem to what the parent described, to me.


This part in the original comment resonated with my experience: "When I was a kid, I could grasp concepts easily and I understood maths very quickly so I didn't really bother doing things like homework or actually studying."

This was exactly the case for me after moving from Ukraine. Concepts were easy because I'd already known them, or they were just starting to be built on top of stuff I'd already drilled and knew before. So I did not put any effort into homework or studying. When entirely new concepts came along they crept up on me and my bad habits and I failed hard.


I'm a computer science student and have the exact same problem. I was diagnosed with ~150 points in IQ tests when I was in the 8/9th grade (I was forced to do it to get a seat in a special class - I don't think IQ is relevant for anything that constitutes a wonderful life) and I can imagine that this is one of the reasons I never developed serious perseverance (only exception: software eng) - in school and elsewhere, there was no reason to work hard.

Now in university, it's completely different - if you don't work hard, you're gonna lose. I've learned the hard way that it's not about raw intelligence - which is really nice, because it means that genetic factors can be minimized - but pure diligence. I've met a couple of people who have average intelligence, but hustle like they're betting their lives (which is not so far-fetched, I admit). And I've found it very interesting that they sometimes don't need a real reason to hustle - they don't even know if computer science is what gets them excited, but they do it nevertheless.

Unfortunately I don't have any mechanisms to utilize my potential in academic settings, although I succeed in my narrow software eng scope.

I would like to hear your tips and how you succeeded anyways.


fwiw I had the opposite experience : in school I felt I had to work quite hard (harder than I wanted to for sure) to do well, but in university (EE + CS) life seemed to get easier and easier, I think because the work relied more on intuition and "getting" the material than raw effort, and also because I was interested in the field and so found it more pleasurable to study.


The Japanese learned early on that practice is at the heart of expertise. What you cannot do reliably, you do not understand.

Western culture has almost completely forgotten about practice, unless you're a musician.


Or a magician, a few years ago I watched an interview with a magician who just said his only key to success was daily practice, hard work, forcing himself to get up early every morning and just grind through it for years until he was good enough to invent his own tricks.

This is what I use The Art of Computer Programming books for. I'll watch a lecture in basic computer science I'd forgotten long ago to get the intuition being explained of concepts like generating functions, then look up the subject in TAOCP series to read Knuth's explanation, and do all the exercises possible. Most universities I've found with open calendars or lectures have been locking down their assignments to prevent students from cheating and looking up answers like 15-251 at CMU. They have public lectures but no tests or exercises available so TAOCP complements these lectures for anybody self-learning by giving you hard problems to reason about so you remember the material. https://scs.hosted.panopto.com/Panopto/Pages/Sessions/List.a...


And in reading. We teach reading by consistent, laborious repetition. Most parents are engaged in this almost every day.

We don't do this for math as standard because most parents don't value math as much as reading. We know it works, we just don't care enough as a population.


Most Asian parents focus on math mechanics but forget that math is essentially problem solving. As a child I had the concepts down solid, which helped me when I went back to University to do math classes but doing applied math had been a hard mountain to climb. Learn to build DE models etc. has been v challenging.


I'm a student in university now, and I can relate to your experience.

What helped me a lot as a kid was being motivated to learn things when it didn't seem like "learning". For example, I was able to get the hang of programming by playing a game that incorporated it, and it succeeded because it got me excited about doing cool things. Maybe this kind of "gamification" could be more useful than just adding points and rules to things to make people compete.


> I'm not sure how to help kids not to go down this path

Now that I'm no longer in school I've found way more progress in learning things by learning through curiosity. My best guess for a teacher to successfully facilitate a curiosity-based approach would require using the soctratic method and more importantly encouraging students to use that technique and give them opportunities to come up with their own good questions they seek answers to. None of my teachers or professors ever did this and usually just overloaded everyone with an extreme amount of busy work. It would also require a professor to let go of their own ego in support of a better teaching method. I actually learned about the soctratic method from a good therapist. I've been using it to prepare for coding interviews and it's been working really well so far


Mirrors my university experience quite well and also sometimes in my professional life. I still rely a lot on my intuitive understanding, which works great in mos cases, but leaves me lacking when it fails.


I was never "gifted" but I went down this exact route. I did great in high school and then poorly in college even though I feel like I understood all the material.


Read Mindset by Carol Dweck. She talks about what you experienced and how to overcome it. She also talks about how to shield children from that fixed mentality.


> Gradually, neuroscientists came to realize that experts such as chess grand masters are experts because they have stored thousands of chunks of knowledge about their area of expertise in their long-term memory. Chess masters, for example, can recall tens of thousands of different chess patterns. Whatever the discipline, experts can call up to consciousness one or several of these well-knit-together, chunked neural subroutines to analyze and react to a new learning situation. This level of true understanding, and ability to use that understanding in new situations, comes only with the kind of rigor and familiarity that repetition, memorization, and practice can foster.

This made a lot of sense to me. A pianist plays through these chunks of knowledge - and learns through them as well. These "chunked neural subroutines" can be built, and clearly are.

I think a lot of resistance for me comes from learned helplessness (aka baby elephant syndrome). For me, growing up my inner narrative was one of failure and rejection. But as i've grown and changed this narrative to be more nurturing, I do see more and more, my capacity for a great many things. Am I crazy? A genius? An idiot? Nope. I just figured out that with a lot of curiosity, play and patience, you can become proficient or even a domain expert. That's easy to see, at a certain level.

But I think for a lot of us, we have a concept of who we are, what we are and what we are not. And this, at least for me, this definition of I, has held me back more than anything. Change the definition, challenge assumptions, push boundaries. See who you are.


i agree, and i like your 'see who who are'

i say the same when people cite the taxi cab number(o) story and suggest some kind of intuition or magic

ramanujan simply had worked with cubes enough to recognise the number, still very impressive, but anyone could do it with the kind of interest ramanujan had for number theory

(o) https://en.m.wikipedia.org/wiki/Taxicab_number


a rather interesting aspect of the taxicab number story is that ramanujan had been looking at "near misses" to the fermat equation, and this was the first values of one of the infinite families of near misses he had constructed. The story is often dramatized to make it sound like Ramanujan just knew about all the cubes and their sums and he pulled this out of thin air on the spot, but the truth is probably that he had worked enough with these equations to recognize the number when Hardy mentioned it.

check it out, super neat number thy stuff! https://arxiv.org/abs/1510.00735


I've realized lately that I often fool myself into thinking I'm proficient in some subject by picking up on concepts quickly. I think I "get" it because I can understand something on a conceptual level, but when it comes to practice, my deficiencies become apparent. I guess the tricky part is to push myself past those initial concepts to delve deeper into the subjects with practice. I need to slow down and remind myself that I don't really "get" it just because I can understand the concepts in a very broad sense.


When you're feeling proficient and confident in something, move your goalposts, and force yourself to take the next step:

* Become proficient not just the concept or task, but good enough to where you can teach it to someone else, or write a clear and concise KB article.

* Implement it (perform the task) repeatedly, with consistent outcomes.

* Automate the process.

* Use this as one part of a much more complex system.


Binary trees was that for me. I listened to lectures and understood them very well from a conceptual point of view. But once I started writing them from scratch as an exercise did I learn that understanding the theory alone was woefully insufficient.


Maybe you just didn't understand the "theory" as well as you thought you did.

I fell into the "it makes sense when we go over it in class, but not when I do it at home" trap often.


I've often had the experience of thinking I understood something well up until the point I tried to explain it to somebody. Teaching is the best way the learn. I think the "smart" person in a study group often gets the most out of it.


Can highly recommend the book the Coursera course is based on [1]: A mind for numbers [2]. Have read it multiple times back to from and front to back.

[1]- https://www.coursera.org/learn/learning-how-to-learn

[2] https://www.amazon.com/Mind-Numbers-Science-Flunked-Algebra/...


The author is not wrong to emphasize the fundamental necessity of drilling, but in her article it sounds like she, having memorized basic facts (in Russian and math), ultimately achieved fluency through play.


That was my takeaway as well. The author played with the equation f=ma, and understood it inside and out in various scenarios.


This is mentioned in her bio at the end, but worth noting here, I think: Barbara runs a coursera course on learning how to learn (https://www.coursera.org/learn/learning-how-to-learn).

It is interesting in that I find that Coursera courses highlight the flawed learning processes she mentions quite well. I often find myself watching the videos, thinking I get it, buzzing through the usually basic follow-up questions, and moving on. Likely that material won't last in my brain for very long in a quickly usable fashion.


Coursera simply mirrors collegiate pedagogy: a professor lectures you for an hour, you take a quiz here and there, and then there are some larger assessments that prove mastery. Study habits and methods are entirely left up to the student. You could employ Dr. Oakley's methods, whatever works for you, or just breeze by without truly internalizing anything.

Coursera's missing one powerful dynamic of a traditional university, however: incentives to remember beyond a class.

Say you coast your freshman year without internalizing: you'll pay the price the following year or when you take some cumulative assessment like the MCAT.

With Coursera everything still feels very disjointed. Even in the specializations, knowledge doesn't need to compound for success. You can easily succeed in edutainment mode. Why take notes when you can use your hands for popcorn?


Lately I've been picking up the soroban[1], the Japanese abacus, and it's been tons of fun. It feels a little like solving a Rubik's cube with arithmetic, or maybe like working with a finite state machine. There are different algorithms to apply depending on the state of the soroban, and applying the right sequence of these algorithms will get you to the right result.

I find it to be a little addictive, and sometimes find it a bit hard to stop. I always feel like wanting to improve my skills a little more, become a little faster at it, and increase the number of digits I can handle without making a mistake.

The soroban is a great tool for developing concentration, a memory for numbers, a facility for performing a relatively complex series of steps in a certain sequence, and eventually for lightning fast mental arithmetic.

In Japan, soroban use is taught to young kids[2], who after a while develop enough proficiency not to need the physical device any longer and can perform the calculations on an imaginary soroban, and eventually can achieve some really amazing feats of mental arithmetic, such as this example from their national competitions: [3]

[1] - https://en.wikipedia.org/wiki/Soroban

[2] - https://www.youtube.com/watch?v=Px_hvzYS3_Y

[3] - https://www.youtube.com/watch?v=7ktpme4xcoQ


> By interleaving my learning—in other words, practicing so that I knew not only when to use that word, but when not to use it, or to use a different variant of it—I was actually using the same approaches that expert practitioners use to learn in math and science.

This is huge! Even if you learn something so that you can use it without fail today, if you interleave your practice of using it (today) with other things, you'll do much better a week from now.


Interleaving has served me very well during my second year of Engineering. I hadn't attended that much and had a grade of 3/20. There were the final exams where the materials of the whole year was fair game.

I buckled down for a month with a friend at my sister's house which was empty. I eliminated the modules with diminishing returns that I had passed or where I was close (easier to go from 0/20 to 12/20 than it is to go from 8/20 to 20/20 for the same amount of points).

I was left with five modules I hadn't attended: Numerical Analysis (ANAI), Rational Mechanics(MECA), Strength of Materials(RDM), Vibrations-Waves-and-Propagation(VOP), and Atomic and Nuclear Physics(PAN).

I drew a pentagon and organized the modules. Starting at the top, going counter-clockwise: VOP, ANAI, PAN, MECA, RDM.

Each day, I'd do two modules:

Day1: VOP-ANAI

Day2: PAN-MECA

Day3: RDM-VOP

Day4: ANAI-PAN

Day5: MECA-RDM

Day6: Restart cycle.

Many benefits:

- You only do a module for half a day. Intensely. Then switch to another module and you sort of hustle your brain for a fresh start. It's not tired because you're doing something else now. "It's not like you've been studying all day" is the impression.

- Mixing modules gives new insights. Especially in second year, there's a bootstrapping phenomenon: to understand a module of Physics, you had to understand a module in Maths.

- You study a module hard. You don't see it the next day, but the day after. Not too soon to be sick of it and burn out, but not too far in the future not to remember any of it.

The problem with the methods most other students followed was that it violated their brains and common sense: they'd do one module exclusively for a week (all chapters, all exercises). Then go on to the next one and do the same. By the time the exam comes: they're sick of the modules, and they remember nothing for the most part because it's been 3 weeks since they've last done the first module they started with.

I, on the other hand, have seen any given module at most 3 days before.

This allowed me to study 13 hours per day during a month without burnout (reading the course material for the first time, going over the exercises and exams, etc). The key was keeping a schedule.

Up at 0500.

0500 - 0700: Study.

0700 - 0800: Breakfast.

0800 - 1200: Study.

1200 - 1300: Lunch and nap.

1300 - 1700: Study.

1700 - 1800: Afternoon snack and chill.

1800 - 2100: Study.

2100 - 2200: Dinner.

2200 lights out, going to sleep.


``At some point, self-consciously “understanding” why you do what you do just slows you down and interrupts flow, resulting in worse decisions.``

I experienced this phenomena while learning the dvorak keyboard layout. The GNU Typist dvorak lesson can be completed in just one day. A mental map of where all the keys are in dvorak was developed very quickly. The problem I was faced with was the relatively slow thought process of envisioning the key layout, moving my typing finger to where it needed to be, and then continuing this thought process as I went on to type full words. Knowing I could type much faster in regular Qwerty layout, this was frustrating. After a month, I was thankful many words no longer required much thinking to write. Months later, I now very much prefer to turn on Dvorak layout on whatever computer I'm logged into.

Great article. I'm also keen on brushing up on my math. Kahn Academy is an amazing learning resource and math is their biggest offering. Not taking advantage of it seems like a sin.


I read her book, watched her videos and i like her.

But while i do already know a shit ton of how to learn, my level of knowledge is not caped by learning issues or by my iq.

It is caped because my stamina is where it is. I have enough stamina to learn and understand a shit ton of stuff but not to sit down day/every second day after day to learn. To Exercise. To do it on a regular base.


Well that's an entirely different problem to attack.

Exercise: I used to do all kinds of programs incl "45 mins a day, 6 days a week" but found "sufficiently enjoyable results" while keeping the process enjoyable and in full balance with the rest of my life with just 3 reps a day, no break days, cycling through 8 exercises. Good enough for both health and looks if no Mr. Olympia goals.

For learning, the right balance is a lot harder to strike IME. Going too hard on oneself or too soft is a real danger. The motivating goals for learning something or other will have to be consistently and sustainedly present to settle into the right balance over time. Whether (perceived) low stamina is merely due to "the undertrained stamina muscle", mismatch of expectations and results, or some deeper real physiological/psychological factor is also the the likeliest found out by yourself. Not reason not to go meta on this roadblock!



Barbara Oakley's MOOC is great. I taught me to take Pomodoro seriously. Also taught me that taking breaks is important to help the brain integrate material.


> your mind constructed the patterns of meaning. Continually focusing on understanding itself actually gets in the way

For an engineer, who only needs to use the math, maybe. But what if you need an understanding too? Once you have "intuition" it's easy to stick with that, rather than challenge your understanding.


I've realized exactly the same thing. A starting intuition is necessary, but not sufficient, for understanding.

I can watch as many lectures as I want, but nothing beats sitting down with a pen and a piece of paper, playing around with equations and developing a true, intimate understanding.


This is why a lot of the best mathematics books won't just state theorems, but will actually heavily encourage the reader to solve problems as well, as that is considered an indispensable part of the learning process.


I seem to recall that this post was posted here previously, I don't remember when. Well googling 15 seconds,https://news.ycombinator.com/item?id=12508776


Every time I see Barbara Oakley I feel the need to post her "Learning How to Learn Talk".

https://www.youtube.com/watch?v=O96fE1E-rf8


Check out the book Make It Stick. It's the recent research, summarized.




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