How I Rewired My Brain to Become Fluent in Math

My observation from years of experience teaching calculus and mechanics is that every student has a different learning style. In particular, there are "theory people" who need to understand the reasoning behind the concepts first and "practice people" who best understand concepts by looking at worked examples. There may be other subdivisions, but these are the main types.

I had two cases when my students showed no signs of progress after hours of theory-style training, and then became Einsteins after solving some practice problems. I was like... why did we waste so much time with the theory???

I don't necessarily agree with the OP that rote memorization is for everyone. Ultimately both the theoretical/symbolic understanding and the "doing"-experience are necessary, but the order in which students acquire these skills is up to them.

Here's a nice quote:

    不闻不若闻之，      Not having heard is not as good as having heard,
    闻之不若见之，      having heard is not as good as having seen,
    见之不若知之，      having seen is not as good as mentally knowing,
    知之不若行之；      mentally knowing is not as good as putting into action;
    学至于行之而止矣    true learning is complete only when action has been put forth
           -- 荀子                                    -- Xunzi

I come from a very similar background as the writer. As a young man, I showed high potential for languages. So when I went to college, I loaded up on languages, graduating with 6 different languages at at least the 200-level and fluency in 3 of those languages. Unlike the writer, I chose neither military nor government work upon graduation, but instead worked as an ATA-certified translator for 10+ years, primarily translating pharmaceutical documentation and medical studies.

I've since moved into professional software development and am now rusty in the spoken languages I used to be fluent in. I work with a number of different programming languages, primarily doing web development since that's where the money is in remote work (although I also have a strong interest in embedded programming). In between, I also finished graduate school in public health (don't ask!), taking multiple upper-level biostats and epidemiology courses.

Anyway, people were always interested in how I learned so many languages. As it turned out, I used similar techniques as what the writer describes to not only learn languages, but also learn graduate-level quantitative coursework, and now programming languages, algorithms, and data structures (an ongoing task!). People are taking exception to her description of the technique as "rote learning". Perhaps a better emphasis would be on repetition, drilling, and fluency. But for me at least, this did entail things like flash cards -- Quizlet is a godsend -- and repeated, focused practice of short problems.

I have found that the technique doesn't seem to work as well for complex algorithms as it did for math and languages. Maybe because it's difficult to break down some algorithms into small parts, perhaps because I just haven't figured out how to do so yet, or maybe I'm just getting older.

But I am in total agreement with the writer that modern education does a grave disservice to young learners by discarding rote learning and repetition.

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I can recognize some of my experience in what she's saying -- as I've been told several times in math/stats "you don't understand this, you just get used to it."

That's a bit of troubling statement at first, but I think it's sortof a shorthand way of saying that much of the understanding available comes through the process of repeated manipulation and observing outcomes.

I've also noticed that a decade or two after going through a math undergrad, the limited material I've retained best was the stuff I learned my freshman year and had to repeatedly re-apply in other coursework.

Still, I think she may be overstating the case about the limits of conceptual understanding, though (or perhaps bringing an engineer's perspective to the discussion rather than a mathematician's. ;)

Practice is crucial, but when I need to dredge old mostly-forgotten material up out of my brain, the interconnections formed by the mapping of conceptual space we call "proofs" often turn out to be pretty helpful. There's plenty of things I can't remember that I can derive from what I can recall.

I found the article rambling and mostly pointless...

However, if you have a weak background in math and want to get up to speed before going into calculus and beyond, I have 2 suggestions.

1) Lial's Basic College Math[1] is adequate and will get you up to speed. 2) Serge Lang's "Basic Mathematics" is great and will cover all you need to go into a rigorous theory based college math class.

[1] http://www.amazon.com/s?ie=UTF8&field-keywords=lials%20basic... The editions basically the same... pick the cheapest

[2] http://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/038796...

She never really gave a definition of "understanding". Personally, I see "understanding" as exactly the way she described her method of learning. When I learned f=ma I did it the same way she did, but I would've called it "understanding".

In fact, "rote memorization" is the opposite of how I'd describe her method of learning. She says she learns language by using it in many different situations and learning the situations when it's usable and when it's not to be used. That sounds exactly like a search for intuitive understanding to me.

Rote memorization is more like using flash cards or scanning over a list of words and memorizing their definitions.

Contrast this with her method. She's playing with the subject matter.

I find that playing with a subject is the best way to understand it and to be able to apply it.

A similar comment Joel Spolsky from years ago always stuck with me.

"... Even though I understand all the little bits, I can’t understand them fast enough to get the big picture. And the same thing happens in programming. If the basic concepts aren’t so easy that you don’t even have to think about them, you’re not going to get the big concepts."

http://www.joelonsoftware.com/articles/GuerrillaInterviewing...:

As someone who graduated with a math minor and promptly forgot the vast majority of what he learned, something that really messed me up were gaps in my education.

For example, I took a graduate level course in the foundations of mathematics (proving natural numbers and arithmetic). I grasped the lectures, but one day I was completely stumped by a step in the proof where algebra was performed across an inequality.

I had never in my life seen algebra across an inequality! As an undergrad senior in a 500-level course.

I grew up moving a few times, and even within single schools I found the teachers weren't on the same page in terms of how math would be taught. For this alone, the Common Core sounds like a good idea to me.

I am interested in this kind of topic, but came away disappointed. Not only is it long-winded, but also she never gave clear definitions of 'fluency' and 'understanding', which are the key ideas in the article.

I think the reason she has succeeded is quite simple: she is an avid learner. She has put in huge amount of effort into learning. It's not about some magic methods she discovered.

Carrying the formula f=ma in head all day long, thinking about it, practicing its various forms in different situations, that is not rote learning or simple repetition (as the author claims), it is working one's a$$ off to understand something.

Take a look at Daniel Willingham's material and his book "Why Don't Students Like School"

Here's a review of the book: http://ed-policy.blogspot.com/2009/04/one-of-handfull-one-of...

Here's Dr Willingham's web site with a lot of articles worth reading: http://www.danielwillingham.com/articles.html

Here's his credentials: Earned his B.A. from Duke University in 1983 and his Ph.D. in Cognitive Psychology from Harvard University in 1990. He is currently Professor of Psychology at the University of Virginia, where he has taught since 1992.

It's always the combination of conceptual learning and memorization when it comes to learn math. It's like performing a complex computation job once, and then caching the result for fast retrieval. Memorization is for efficiency: it clears the way for our brains to focus on higher-level thinking. The great Euler "memorized not only the first 100 prime numbers but also all of their squares, their cubes, and their fourth, fifth, and sixth powers. While others are digging through tables or pulling out pencils and paper, Euler could simply recite from memory...". Besides, math is all about making invisible visible, about discovery patterns, or about connecting dots. That means we need to have dots to connect, and need to have patterns to work with. If we don't remember them, what the hell can we use for?

On the other hand, there's no need for rote memory. Just practice by solving interesting problems. There are plenty of opportunities to use math every day. There is also a very effective way of learning: work on slightly harder than usual problems. When learning calculus, I started to work on Demidovich's Problems in Mathematical Analysis, and I thought it was hard. Then, I stumbled upon solution book for college math competitions. Man, that was a huge help. After working through the problems, a lot of concepts became clear to me, and Demidovich's problems became reasonably easy too. It turned out the hard problems were hard because they required me to make non-obvious connections, which nudged me to really understand, from different angles, the concepts that I learned in the classroom.

By the way, when did arithmetic become so hard? It seems kids nowadays are being spoiled by their parents...

He highlights one of the problems I had with mathematics when I reached University. I was in the top 2% of students in the first few months. I considered it a trivial subject. "Here's a class of problems, here's how you solve them." Easy peasy. Then we started integrating and differentiating forms of problems we weren't familiar with. All of a sudden, everything we were taught didn't seem to apply any more.

Our lecturer told us: "I can see that some of you are struggling and are confused. You want to know how you become skilled at solving these problems? Forget your social life. Solve every problem in that 900 page textbook of yours. When you've done that, come to me and I'll give you another 900 page text book and finish that as well."

In one fell swoop Mathematics had lost all its appeal. I switched to Applied Math and aced it.

On the other hand, my brother actually followed this advice. For an entire year he spent a few hours every night doing Calculus till he recognized every different form they could conceivably throw at him. I asked him how he knew how to solve some problems. He said he didn't. It was all just memorization and recognition.

> memorized the equation so I could carry it around with me in my head and play with it. If m and a were big numbers, what did that do to f when I pushed it through the equation? If f was big and a was small, what did that do to m? How did the units match on each side? Playing with the equation was like conjugating a verb. I was beginning to intuit that the sparse outlines of the equation were like a metaphorical poem, with all sorts of beautiful symbolic representations embedded within it.

I cannot emphasize how useful this approach is. What's more, I am very surprised by how many students lack the basic ability to sit and "play around" with math concepts.

I used to tutor algebra, trig, and pre-calc. I was surprised by how few students had the ability to look at something and break it into pieces. If you gave them piece A which they knew, piece B which they knew, and put them together, the result was something new the student couldn't understand and they'd sit and wait for an explanation.

This was sad because this is largely what math learning is. Given pieces you do understand, you put together bigger pieces and then build an understanding of those bigger pieces.

I have always advocated something similar to what what the author said here:

> What I had done in learning Russian was to emphasize not just understanding of the language, but fluency. Fluency of something whole like a language requires a kind of familiarity that only repeated and varied interaction with the parts can develop.

I love this point. I am borderline convinced it it is near impossible to learn math concepts more than one or two steps beyond the point that you are fluent in. Beyond that, it's just memorization, guessing, and poor heuristics that just get you to skate past the test and do little or nothing for understanding or retention.

The bigger picture here is that the author has learned how to learn. It amazes me how significant a divide there is between people who know how to learn and people who don't. Forget IQ or test scores, I think knowing how to learn is the biggest indicator of how far someone will go in life. So perhaps most of all I love the fact that they began this journey relatively late (most students going through a similar math path would have done in their late teens or very early 20s what the author was doing in their late 20s).

Case studies in education as applied to non-atypical brains are essentially worthless. You'd think that someone with a newfound scientific disposition might realize that you probably shouldn't make sweeping prescriptions based on personal anecdotes (though honestly, based on the quality of the educational literature I've seen, it's not uncommon even for professional education researchers).

I think those of us who aren't a fan of "memorize/repeat" is that it is extremely inefficient. And you lose as many people as you gain.

The classic example is long division. In elementary school I did a thousand long division problems, but I never understood it. I just knew the pattern. I didn't really know the math. It wasn't until much later did I learn how and why it worked. Having done those thousand problems didn't really help my understanding at all. And honestly, it's not a skill I use today.

Rote learning is useful when what you memorize is useful in itself. Learning what 2+2 is useful, but because adding 2 to another small number, including itself, is something you do a lot.

Addition/times tables are great because the scope of numbers is something you will run across a lot of day to day. A thousand long division (or multiplication) problems is just rote work for little benefit. It's better to teach the concepts, and then build on them.

Just saw that the author, Barbara Oakley, has a (free) Coursera course titled "Learning How to Learn" starting Oct 3. Sign-up here - https://www.coursera.org/course/learning

For those interested in this learning approach, I recommend the book "The Talent Code" by Daniel Coyle [1] which covers 3 areas for mastering new skills.

Here the author only mentions about the importance of repetition (which she refers to as fluency). In the Talent code book Repetition is the first step, but we can learn that the brain is wired to master new skills by taking advantage of 2 others areas as well: A/ Ignition (or passion and motivation), and B/ Discipline and long term commitment.

[1] http://www.amazon.com/The-Talent-Code-Greatness-Grown/dp/055...

Understanding is "facile" [superficial], but rote learning is the real way....?

Seems reversed. An explanation is she's in Electrical Engineering - my ugrad experience of EE was a emphasis on using tools (formula etc) as opposed to understanding them (that's for Science). Engineers, after all, are paid to get stuff done, not just sit there and grok it. "Fluency" with tools works well for EE.

OTOH, all discplines have "tools" - even pure maths has algebraic manipulation. If you're not fluent, it will slow you down (fortunately, you'll have years of algebraic practice from school).

OAH, most professional mathematicians don't think symbolically (a survey found, IIRC, about 70% visual, 25% kinesthetic, 5% linguistic) - notation not so much a tool of thought as a serial representation (serialization/data format), for recording/communicating. So practice in thinking is what's helpful.

I think fluency with "standard" modes of thought is a double-edged sword. Yes, you become expert with those tools, fast and able. But that very expertise biases your perception and reasoning in terms of them. Thus, it's hard for you to see another way; you'll tend to build on top of them instead. Fortunately, since our tools are pretty good, this approach works well. It's just that you're less likely to see fundamentally new approaches (though to be fair, that's pretty damn unlikely anyway).

>>student may have thought he’d understood it at the time, and perhaps he did, but he’d never practiced using the concept to truly internalize it.

I'm changing jobs and I've interviewed at quite a few places off late. Having faced with so many Algorithm and Data Structure questions, I took it upon myself to read upon a good deal of material before I started interviewing. So I did the usual approach one takes while studying math, understand all the basic concepts and look for places where you could apply them, rather than studying individual problems.

Recently I interviewed at one place, where I guess a guy who was just out of college asked a few questions. My general approach is to look for how the problem would fit in the concepts I've learned. I gradually derived the solution out of the concepts I knew. And to my surprise, the solution was simply unacceptable to the interviewer, he just clung to his position that I was wrong. He then proceeded to write the answer, he even wrote the test cases.

Now I go back home and check it out of the internet and it was hardly surprising for me to see what I had suspected. The guy didn't want the answer to the question. He wanted exactly the same answer he had memorized. The guy had even memorized the variable names, even the exact input values.

Most people interviewing you on your Algorithm and Data structure skills. They are simply testing your algorithm and data structure rote memorization skills. The same goes in exams.

I have a crappy memory. I tried med school for a year. In all my pre-med and medical courses, they constantly warned us to understand, not memorize. I gladly took their advice, tried to understand rather than rote-memorize, and I ended up flunking 5 exams, which finished my medical career.

The fact is, in many fields of human endeavor, you need to memorize before you can think. Would we have better physicians if they weren't required to memorize and spit out so much rote information? Not sure about that. I think a good doc not only has great personal and intuitive skills, he or she needs good analytical skills and a "think outside the box" problem solving mentality similar to that of a really good computer programmer (which I thought I had, hence crossing over into medicine from computers).

But, a doc also needs to remember stuff. There's so much data and you need to remember signs and symptoms, case histories, and see the similarities and differences with previous cases you've had.

The OP was about math, and I'm definitely in the memorize-the-formula camp; I've never really grasped the theoretical niceties of mathematics and am more of a connect-the-dots kind of guy. But definitely there are similarities between learning math and learning the physical and biological sciences. Some of it is intuitive, but some of it is just wacky stuff out of left field and you have to just take it on faith.

I think we need to emphasize critical thinking and analysis more, but we also need to teach the ancient Greek and Roman methods of memorizing, e.g. the memory palace method -- see Moonwalking with Einstein for more details.

It reminds me a lot of the Pimsleur method.

http://en.wikipedia.org/wiki/Pimsleur_method

It's rote memorization and repetition, but it also incorporates interaction and active thinking. Additionally, the repetition is non linear. As you learn new terms they throw old terms at you once in a while to make sure you remember them, and the spacing is set out for maximum retention.

I believe that math is something a lot of people can learn and become effective at using, way more than Americans currently believe about themselves.

I think it all comes down to making sure people understand the actual abstractions and concepts rather than formulas and rote learning. The advantage of math is that you can figure stuff out using what you already know, without having to learn every little detail - while that's impossible in other subjects, such as history.

The missing piece is, in fact, granularity and feedback. And real world studies support my position:

http://opinionator.blogs.nytimes.com/2011/04/18/a-better-way...

http://opinionator.blogs.nytimes.com/2011/04/21/teaching-mat...

If you want to read how I'd reform education and improve the economy, these are my thoughts:

http://magarshak.com/blog/?p=158

Assuming the author's convictions are true, how would one apply a 'rote learning' technique to programming?

I believe in repetition, but, personally haven't been able to find a useful way of implementing it to become a top-notch programmer.

Why? Well, take calculus. Buy a book, pound problems. Do 1,000 derivatives. Next chapter. 1,000 integrals.

Is there a similar "exercise" for programming?

"Sorry, education reformers, it’s still memorization and repetition we need." This is very true. For example, doing the advanced engineering calculus can often boil down to manipulating polynomial, which are still arithmetic. A person with "understanding" but no fluency in basic arithmetic is setup to fail.

I thought thus would be about the guy on HN who used a psychedelic trip to gett over math anxiety. Still a good read.

I think that it is strange that she is saying that there is too much focus on understanding at school. From what I could see, most people learned maths by just memorising the process. Except for a few who invested an exceptional amount of effort, they scored poorly. In contrast, everyone who I knew who focused on gaining a deep understanding was able to pick up maths quickly.

There is one area, though, where there is probably too much focus on understanding at the cost of memorisation and that is in the Olympiads. I remember that many of us held the rather uncharitable attitude that the people who tried to succeed in maths by memorising it were 'stupid'. But clearly, understanding combined with targeted memorisation of the key building blocks will lead to the most success.

How is "rewiring your brain to become fluent in maths" different from "learning maths"? Can someone explain this? I read the article, and I don't understand it. To me it sounds like she learned maths, in much the same way one learns anything else.

It's not so hard:

(1) Pick a topic at about the right level.

(2) Get one main, maybe plus a few extra, good books on the topic.

(3) In a good chair in a quiet room, in the good book, read the material on the topic and try to understand it well enough to make sense. Then work the exercises.

If something doesn't make sense, look in one or more of the other books.

(4) At times, review the last several topics and try to get more understanding, e.g., what else is true, what is not true, what makes the true statements true and the false statements false.

Maybe take a class that covers the main book and check work and understanding with the professor.

Rinse, repeat with other topics.

Done.

Worked for me.

I liked her point of repetition being the main problem. It makes sense as we strengthen the new synopsis that are made. Josh Foer author of moonwalking with Einstein proposed an interesting theory that, essentially fucked up imagery of any kind helps increase and strength synaptic connections.

Interesting watch (first 5mins gives good context) http://books.google.com/books/about/Moonwalking_with_Einstei...

I don't understand why there seems to be a languages/math dichotomy. I remember in school teachers used to bucket people in one or the other.

But from what I can tell, and also from the article, they have a lot of common ground. Learning either requires understanding of structure, but also a fair bit of memorization. Learning conjugations by heart is not all the different to learning trig relations. You also need both on the fly when you're using your language/calculus.

Memorization is not learning. It is a part of learning. Being able to memorize things and understand their relationships in context is learning.

I wonder if part of the effect is simply studying a lot. It sounds as if she devoted a lot of time to studying Russian and then maths. Perhaps the actual way she spent the time is not that relevant? Not saying it is irrelevant, but for example maybe simply doing problems in maths for the same amount of time would have been just as effective?

Reading this reminds me of how much I really want to be better at math, and especially the quick fire recall and understanding. I'm taking courses with a lot of math that I've learned before, and encountering similar struggles this time around as last time. Maybe I'll try a new approach, one such as this, to insure my success.

I get this sense that mathematicians tend to focus on abstracting relationships, whereas programmers tend to focus on abstracting objects/entities. Hard to explain though. Like programmers are constantly building models of the world, whereas mathematicians are constantly building new functions and general transformative techniques.

Author makes generalizations based on her own individual experience - the process which she doesn't understand. Rot-learning in music (mentioned as example of good practice) produces millions of classically-trained musicians that don't understand anything about music, and can't play even Christmas song by ear.

so many aspects of this article echo my own experiences, though i was likely a far worse example of a student than the author of this article. i excelled at literature (and took this talent for granted) and i had a vicious disdain for math. my time in high school was spent anticipating a “mathless” career in the NHL that never quite came to fruition… sometimes you just don’t grow any taller after your freshmen year :)

here’s how a gift can actually be a curse: i’ve always had an extremely strong visual memory and it enabled me to get by (and sometimes even excel) at courses like algebra and pre-calc. i’d simply memorize the exact structure of a solution, and “plug and play” from memory when teacher asked identical questions on exams with different numbers. it wasn’t until i was an undergrad and i decided to do my own “rewiring” experiment that i realized i hadn’t ever learned a thing about math my entire academic career. it made for quite the rude awakening, being that i was a college sophomore who’d just switched from pre-med biology to electrical engineering!

i struggled with basic problems that required even a little bit of mathematical intuition because i’d never developed an intuitive understanding of anything–to paraphrase bart simpson, it was like i was cheating on tests by using my own mind, since i could remember my notes with such vivid detail.

how’d i fix it? i became a voracious reader. i started reading “popular” science books about math that actually interested me, and i realized that textbooks were only part of the puzzle. once i was able to grasp the theoretical concepts (aided by the “pop” books) i could return to the textbooks without being bored to tears with the dry, abstract concepts–i could actually relate the concepts to reality! i also took a keen interest in physics because it connected calculus to the real world. when i realized that a parabola could actually be the path of a cannon ball, it changed my entire outlook on math. things started to click.

so how’d it work out? getting through b.s. degrees in both electrical and biomedical engineering (i was a masochist and picked up a second degree) was nonetheless a struggle, due to the fact i was essentially several years behind my peers in terms of my mathematical ability. also, i constantly had to fight the urge to rely on my memory. old habits die hard or simply just go into remission, but i made it!

where am i now? as an MBA student (please bear with me here) i was fortunate enough to be at a university that recently embraced data science. i fell in love with the field because data can tell a story, and my (now) strong mathematical background enabled me to grasp the underlying math & algorithms that yield the wonderful insights you can uncover if you look in the right places.

was it worth it? absolutely. i feel as though i am a different person that’s capable of using both sides of their brain! although i must admit i’m still a natural “right brainer” and i need to use a calculator more frequently than i’d like to admit. i’m an engineering manager at a 9-5, i work as a data science lead for a startup, and i dabble with inventing products. i’m convinced that had i not put myself through the wonderful torture that was rewiring my brain, i’d likely not have achieved a single thing in my career.

if you’re really curious to see how someone like me thinks i’ve done some writing here: www.takeiteasythursday.com/liox

What an incredibly short-sighted sentence: "Sorry, education reformers, it’s still memorization and repetition we need.".

I could not disagree more with this article. Memorization helps, but MAKING CONNECTIONS between concepts is _HOW YOU MEMORIZE EFFECTIVELY_.

Memorizing is great for learning different languages. NOT MATH.

Sure, maybe students are bullshitting in understanding concepts, and forgot quickly since they didn't learn anything. However, just memorizing shit doesn't do anything. So many students in other cultures are being forced to memorize everything, and since they don't understand the connections, they cannot come up with the answer on their own. There was a piece on HN about a math teacher that was amazed by how little students knew: They memorized how to solve a particular problem instead of actually knowing why something was true, and therefore couldn't work through basic proofs and stupidly basic, foundational stuff.

A terrible article!

I remember struggling with Calculus in high school. I sat in the library next to the smartest guy in the class and explained to him that I couldn't figure out why a certain formula was to be used to solve a problem. It went something like:

Steve: "You have to use that formula to solve this."

Me: "I understand that but why do we use that formula?"

Steve: "Cause that's the only formula that works."

Me: "OK but how did they come up with that one formula?"

Steve: "Cause it just works!!"

He didn't know. That's when I first realized a lot of his knowledge was just memorization while I needed to know the why. It wasn't till my first year in college that I finally found a math instructor who always shows us the reasons why that I was able to take off and grasp math by connecting the dots.

So, no, I do not think memorization is the key but, like most zen things, understanding is.

Theorems are muscles of the mind.

Nice!

I could hardly stomach the self-congratulatory tone of the introduction. Jeez.