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 first understood symbolic derivation after implementing code to apply the rules we'd learned (then I derailed completely and turned the expression evaluator I'd written for it into the core of a compiler).
All through primary school and most of high school, I stayed well ahead of the curve in maths because I could figure most things out from what we'd already learned, and finding algorithms was easier than memorisation.
E.g. I remember being given the task to sum all numbers from 1 to 100, and figuring out the algorithm for summing any series from 1 to n in time to be able to be intolerably smug about how quickly I had the result (I must have been 14-15 at the time so it was nothing out of the ordinary - the teacher assigned the task explicitly to see if anyone would find shortcuts or the formula). I then proceeded to spend the rest of the lesson figuring out how to modify it for all odd numbers, even numbers and various other classes of numbers. Up until then, that was how I spent most of my lessons, and my teachers tended to encourage it.
I lost most interest in maths when the teaching turned to become more about memorisation, and I was not taught the skills needed to figure out the underlying rules to be able to compensate for not memorising. If I'd been more interested, and invested the time in finding suitable resources myself, I'm sure I could've kept up, but it was still "just school" to me, even though I enjoyed math lessons until then.
It went from being my favourite topic to being something I actively avoided because of this.
I work best from examples, and find learning computer languages from documentation very hard.
OTOH, I'm very quick at looking at examples, generalising behaviour, finding edge cases and then looking at the docs to understand those edge cases.
One of the best programers I know is the polar opposite. He will spend days reading every. single. piece. of documentation on something, and then 5 minutes writing a perfect piece of code.
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.
> 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.
I've come to appreciate the importance of rote learning over the past couple of years... but despite its simplicity, I would call it a very advanced technique.
You don't need rote learning to get to understand something. You don't even need rote learning to become very good at something. You only ever need it to be able to think faster about certain advanced topics or if you'd like to become fully fluent in a language.
These are not things a high school student should necessarily spend much time on. Especially considering that rote learning, if not self-motivated, can be incredibly demotivating and have the opposite effect of what was intended.
I believe what is described is repetition, but not rote learning. That, to me, means memorizing facts without connecting the dots. Rote memorization is learning for an exam by memorizing the previous years' problems, or learning history by memorizing dates and events. If I focus on knowing the previous years problems by heart, I may accidentally make the connections to be able to solve more varied problems, but thats rather inefficient. Similarly, knowing historical dates doesn't imply an understanding of why and how things happened.
Learning different conjugations of verbs by using them in sentences isn't rote learning, either. It's rather the opposite, trying to connect the new verb into the existing knowledge of the language.
Rote learning alone is useless for most students. Repetition is certainly important in retaining information, no doubt about it, but as with the math examples, you need to apply basic arithmetic in the context of solving equations, for example, to really get a "feel" for it, a sort of automatism where you not only know how to do 1+1, but also WHEN to use this knowledge.
As far as complex algorithms and such go, I do believe they can be broken down into pieces, but you really do have to know the pieces and their relations and implications innately, and with complex algorithms, it's really hard to know all the pieces of the puzzle, and sometimes even finding out what is missing is difficult. You can drill something ad infinitum without gaining anything if you're missing something.
I believe this to be the secret skill that the article omits. In order to learn effectively, you have to recognize where your knowledge is lacking, find out what it is you should know, and drill those parts. A teacher should help students find these gaps and help fill them. The rest is just a little bit of willpower and discipline.
Spaced repetition algorithms and programs are a godsend. Org-drill and diligence has increased my retained knowledge of vocabulary, grammar, mathematics, and logic.
Other people probably use Mnemosyne which is great but having space repetition in Emacs? Org-mode? Capturing notes to-read or facts from the web that I want to remember and automatically putting it into my drill sequence? Definitely not rote - deliberate, yes.
As a thank-you present to our readers on HN, Nautilus wanted to give you exclusive full access to this Prelude article from our latest print Quarterly online "preview". Subscriptions to our print edition are how we make all our online content free. We thought the HN audience would like this piece so in appreciation for your traffic and your comments, here it is.
I became aware of nautilus via HN and subscribed a few weeks ago.
Please keep up the good work.
While print is exactly the medium I want for something of this sort, I do think it would be wise to look into some sort of digital subscription / supporter option.
I love Nautilus! Gorgeous layouts (print & online) with engaging and opinionated content.
I notice that you only sell print subscriptions. Do you plan on selling any digital-only subscriptions? I would sign up in a heartbeat because your online layout is Really That Good (TM).
>I learned Russian by gaining fluency through practice, repetition, and rote learning—but rote learning that emphasized the ability to think flexibly and quickly. I learned math and science by applying precisely those same ideas. Language, math, and science, as with almost all areas of human expertise, draw on the same reservoir of brain mechanisms.
Author approached learning mathematics in the same manner as learning a new foreign language (Russian), and it served her well.
Student focus on maintaining grades throughout school is sadly a detriment to slow learners who actually do need to spend the time to absorb, savour, play with and drill new concepts. I used to think I was a smart, but after years of successfully guessing the teacher's passwords[1] in class and on exams, I realized I only short-changed myself. Comparing against others with the benefit of age and hindsight, I realize now that I'm one of the slow learners who actually need an inordinately longer duration to thoroughly understand concepts than others.
MOOCs and other forms of self-study are great in this regard since pace-setting is now controllable, but rarely lead to a degree or credential one can leverage later in life. I'm glad that the author was able to apply her effective language learning approach to mathematics and science presumably within course durations (all the way to her PhD).
Thank you for that article. Well written and it hit me straight away. I struggled through getting my B.S. Mechanical Engineering - fighting to learn the heavy math load along the way. Looking back, it would have been nice to have the perspective presented by this article.
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.
Since we are on the topic of math textbooks, I will suggest the No bullshit guide to math and physics which is a math textbook written specifically for adult learners. See http://minireference.com/ for more info.
The article wasn't pointless at all. He's saying that a conceptual understanding is important, but so are things like flashcards. I know growing up my ability to quickly do simple math and recognizing patterns quickly always helped me speed along quickly.
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."
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.
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.
I can't understand how anyone would decide to lose their social life for anything. But then, perhaps I am simply not the right person to become a scholar. Sure, I can put an hour or two a day of personal time keeping up to date, but I wouldn't see myself going "alright, let's waste an entire year of personal time for non paid formations" ... If I really have to do some learning, I'm going to try to do it through projects which I am paid to do.
It was much the same for me with quantum physics. The theory was fine, but the maths was just so much memorisation that I completely lost all enthusiasm. I still get the chills thinking about 4 hour exams with only a single question on them that we pretty much just had to memorise the answer for in advance.
"As a young woman with a yen for learning language and no money or skills to speak of, I couldn’t afford to go to college (college loans weren’t then in the picture). So I launched directly from high school into the Army."
> 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.
I can relate to your long division example. In grade school my mom helped[1] me put in a ridiculous amount of effort in order to get through my school's spelling tests. I have come to believe that the reason that I couldn't spell was because at the time I did not understand the rules behind how the letters form together to get words. Simply memorizing how words are spelled took a lot of painful practice.
A more recent example is when I was first learning lambda calculus I wanted to understand how church encoding worked to allow basic integer operations with just function abstraction and application. Most of the operators are easy except for predecessor, which is absolutely nightmarish to get on your own. I tried memorizing the definition I found on wikipedia for over a year to no avail, but once I understood how it was working ...
pred = λ n . λ f . λ v.
n ( λ a . λ b . b (a f) )
λ k . v
λ i . i
I haven't yet forgotten what it is.
[1] - well, basically dragged me kicking and screaming
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.
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.
And isn't that completely bogus too - you don't want programmers who can regurgitate algos you want ones that can take an overview and know which algos to apply and how to implement them in a timely way using reference material if needs be ... perhaps it's too hard to test for that ability in interview?
It's like upthread someone talked about engineering exams - seemingly you don't need working engineers to run through equation derivations and do arithmetic. You want them to understand the basis for the derivations and to have a mental model that will spot if the calculations are way out, for sure, but really you want them to be able to take an overview: know which factors are important, use the engineering tools they have. A practical engineer is going to be using computers to perform calculations (be that chemical processes or electronics or civil structures or whatever).
In a way I think we're on the cusp of a change in how we need to approach job focused learning. We're getting to the point where we've stood on the shoulders of so many giants that if you want to operate at the top you can't see the floor any more. By which I mean that maybe the human mind can't cope with the full vertical stack in some fields, that we need to have people either focus at the bottom - understanding the basis on which particular knowledge is built. Or understand the top - be able to work with the tools for implementation.
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.
[+] [-] ivansavz|11 years ago|reply
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:
[+] [-] Create|11 years ago|reply
[+] [-] vidarh|11 years ago|reply
All through primary school and most of high school, I stayed well ahead of the curve in maths because I could figure most things out from what we'd already learned, and finding algorithms was easier than memorisation.
E.g. I remember being given the task to sum all numbers from 1 to 100, and figuring out the algorithm for summing any series from 1 to n in time to be able to be intolerably smug about how quickly I had the result (I must have been 14-15 at the time so it was nothing out of the ordinary - the teacher assigned the task explicitly to see if anyone would find shortcuts or the formula). I then proceeded to spend the rest of the lesson figuring out how to modify it for all odd numbers, even numbers and various other classes of numbers. Up until then, that was how I spent most of my lessons, and my teachers tended to encourage it.
I lost most interest in maths when the teaching turned to become more about memorisation, and I was not taught the skills needed to figure out the underlying rules to be able to compensate for not memorising. If I'd been more interested, and invested the time in finding suitable resources myself, I'm sure I could've kept up, but it was still "just school" to me, even though I enjoyed math lessons until then.
It went from being my favourite topic to being something I actively avoided because of this.
[+] [-] nl|11 years ago|reply
I work best from examples, and find learning computer languages from documentation very hard.
OTOH, I'm very quick at looking at examples, generalising behaviour, finding edge cases and then looking at the docs to understand those edge cases.
One of the best programers I know is the polar opposite. He will spend days reading every. single. piece. of documentation on something, and then 5 minutes writing a perfect piece of code.
[+] [-] Expez|11 years ago|reply
[+] [-] pbhjpbhj|11 years ago|reply
The "put forth" in the above is not entirely clear to me but I imagine that fits in there as a sub-division.
[+] [-] xsace|11 years ago|reply
闻: hear
若: good
见: see
之: have
知: know
looks easy :)
[+] [-] eric_bullington|11 years ago|reply
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.
[+] [-] stdbrouw|11 years ago|reply
I've come to appreciate the importance of rote learning over the past couple of years... but despite its simplicity, I would call it a very advanced technique.
You don't need rote learning to get to understand something. You don't even need rote learning to become very good at something. You only ever need it to be able to think faster about certain advanced topics or if you'd like to become fully fluent in a language.
These are not things a high school student should necessarily spend much time on. Especially considering that rote learning, if not self-motivated, can be incredibly demotivating and have the opposite effect of what was intended.
[+] [-] dognotdog|11 years ago|reply
Learning different conjugations of verbs by using them in sentences isn't rote learning, either. It's rather the opposite, trying to connect the new verb into the existing knowledge of the language.
Rote learning alone is useless for most students. Repetition is certainly important in retaining information, no doubt about it, but as with the math examples, you need to apply basic arithmetic in the context of solving equations, for example, to really get a "feel" for it, a sort of automatism where you not only know how to do 1+1, but also WHEN to use this knowledge.
As far as complex algorithms and such go, I do believe they can be broken down into pieces, but you really do have to know the pieces and their relations and implications innately, and with complex algorithms, it's really hard to know all the pieces of the puzzle, and sometimes even finding out what is missing is difficult. You can drill something ad infinitum without gaining anything if you're missing something.
I believe this to be the secret skill that the article omits. In order to learn effectively, you have to recognize where your knowledge is lacking, find out what it is you should know, and drill those parts. A teacher should help students find these gaps and help fill them. The rest is just a little bit of willpower and discipline.
[+] [-] Ixiaus|11 years ago|reply
Other people probably use Mnemosyne which is great but having space repetition in Emacs? Org-mode? Capturing notes to-read or facts from the web that I want to remember and automatically putting it into my drill sequence? Definitely not rote - deliberate, yes.
[+] [-] nautilus|11 years ago|reply
[+] [-] incision|11 years ago|reply
Please keep up the good work.
While print is exactly the medium I want for something of this sort, I do think it would be wise to look into some sort of digital subscription / supporter option.
[+] [-] otoburb|11 years ago|reply
I notice that you only sell print subscriptions. Do you plan on selling any digital-only subscriptions? I would sign up in a heartbeat because your online layout is Really That Good (TM).
[+] [-] otoburb|11 years ago|reply
Author approached learning mathematics in the same manner as learning a new foreign language (Russian), and it served her well.
Student focus on maintaining grades throughout school is sadly a detriment to slow learners who actually do need to spend the time to absorb, savour, play with and drill new concepts. I used to think I was a smart, but after years of successfully guessing the teacher's passwords[1] in class and on exams, I realized I only short-changed myself. Comparing against others with the benefit of age and hindsight, I realize now that I'm one of the slow learners who actually need an inordinately longer duration to thoroughly understand concepts than others.
MOOCs and other forms of self-study are great in this regard since pace-setting is now controllable, but rarely lead to a degree or credential one can leverage later in life. I'm glad that the author was able to apply her effective language learning approach to mathematics and science presumably within course durations (all the way to her PhD).
[1] http://lesswrong.com/lw/iq/guessing_the_teachers_password/
[+] [-] robodale|11 years ago|reply
[+] [-] wwweston|11 years ago|reply
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.
[+] [-] oldbuzzard|11 years ago|reply
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...
[+] [-] ivansavz|11 years ago|reply
<discl>I'm the author</discl>
[+] [-] blazespin|11 years ago|reply
[+] [-] Lambdanaut|11 years ago|reply
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.
[+] [-] simoneau|11 years ago|reply
"... 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...:
[+] [-] was_hellbanned|11 years ago|reply
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.
[+] [-] Chinjut|11 years ago|reply
[+] [-] kev6168|11 years ago|reply
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.
[+] [-] mfrankel|11 years ago|reply
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.
[+] [-] g9yuayon|11 years ago|reply
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...
[+] [-] ilitirit|11 years ago|reply
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.
[+] [-] Raphmedia|11 years ago|reply
[+] [-] adwf|11 years ago|reply
[+] [-] pessimizer|11 years ago|reply
[+] [-] chestervonwinch|11 years ago|reply
What sort of integration and differentiation were you doing originally that is not included in an applied math curriculum?
[+] [-] B-Con|11 years ago|reply
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).
[+] [-] x1798DE|11 years ago|reply
[+] [-] kenjackson|11 years ago|reply
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.
[+] [-] Verdex|11 years ago|reply
A more recent example is when I was first learning lambda calculus I wanted to understand how church encoding worked to allow basic integer operations with just function abstraction and application. Most of the operators are easy except for predecessor, which is absolutely nightmarish to get on your own. I tried memorizing the definition I found on wikipedia for over a year to no avail, but once I understood how it was working ...
I haven't yet forgotten what it is.[1] - well, basically dragged me kicking and screaming
EDIT: formatting
[+] [-] wz1000|11 years ago|reply
[0]- https://www.maa.org/external_archive/devlin/LockhartsLament....
[+] [-] ahmadss|11 years ago|reply
[+] [-] mw67|11 years ago|reply
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...
[+] [-] hyp0|11 years ago|reply
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).
[+] [-] kamaal|11 years ago|reply
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.
[+] [-] pbhjpbhj|11 years ago|reply
It's like upthread someone talked about engineering exams - seemingly you don't need working engineers to run through equation derivations and do arithmetic. You want them to understand the basis for the derivations and to have a mental model that will spot if the calculations are way out, for sure, but really you want them to be able to take an overview: know which factors are important, use the engineering tools they have. A practical engineer is going to be using computers to perform calculations (be that chemical processes or electronics or civil structures or whatever).
In a way I think we're on the cusp of a change in how we need to approach job focused learning. We're getting to the point where we've stood on the shoulders of so many giants that if you want to operate at the top you can't see the floor any more. By which I mean that maybe the human mind can't cope with the full vertical stack in some fields, that we need to have people either focus at the bottom - understanding the basis on which particular knowledge is built. Or understand the top - be able to work with the tools for implementation.
Maybe.
[+] [-] vidarh|11 years ago|reply
[+] [-] blisterpeanuts|11 years ago|reply
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.