I Rewired My Brain to Become Fluent in Math (2014)
nautil.us
nautil.us
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.
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 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.
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...
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.
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.
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 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.
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
Western culture has almost completely forgotten about practice, unless you're a musician.
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.
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...
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 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
check it out, super neat number thy stuff! https://arxiv.org/abs/1510.00735
I fell into the "it makes sense when we go over it in class, but not when I do it at home" trap often.
* 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.
[1]- https://www.coursera.org/learn/learning-how-to-learn
[2] https://www.amazon.com/Mind-Numbers-Science-Flunked-Algebra/...
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'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?
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
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.
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.
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.
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.
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!
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 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.