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>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.
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).
FYI: This comment will get downvotes. Not because people here are senselessly mean. But, because the HN community takes a strict approach to comments that add noise without contributing signal to the conversation. Downvotes on "Nice article!" comments frustrate many new users. So, here's your explanation. Welcome to Hacker News!
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.
This is why the education system will never ever change. Too many people learn too many different ways. Some people can read a formula and instantly memorize it. Other's need a connection to that formula. Some need to figure out how that formula was derived in order to understand and memorize it. And some know the formula before you even show it too them.
Not to rehash what you hashed, but it depends who you are. If you are really good at memorizing, you may not need to make connections. You may just be able to memorize everything. Some people can.
This is a good point. It's important not to overgeneralize. At the same time, for those students who learn best by memorizing, it's best not to completely discard rote learning.
This is very true. A corporate trainer once told me there's specific kinds of learners. They've actually identified each type, and which specific methodologies you need to reach each one. The best teachers find a way to synthesize it all into an effective single presentation. (This may also be why tutors are so helpful -- they can customize their material to the audience...)
A corporate trainer once told me there's specific kinds of learners.
I don't believe there is compelling evidence that this is the case.
Instead, different people have different backgrounds and experiences. Learning requires making connections. Depending on what you have in your head already to which you can connect, you may respond differently about a particular topic than someone else.
Although there is ample evidence that individuals express preferences for how they prefer to receive information, few studies have found any validity in using learning styles in education.[2] Critics say there is no evidence that identifying an individual student's learning style produces better outcomes. There is evidence of empirical and pedagogical problems related to the use of learning tasks to "correspond to differences in a one-to-one fashion".[3] Well-designed studies contradict the widespread "meshing hypothesis", that a student will learn best if taught in a method deemed appropriate for the student's learning style.[2]
Heh. Okay, that's good to know. These comments prompted me to look up "Learning Styles" on Wikipedia, which confirms that scientific studies have not confirmed the validity of the "different learning styles" theory.
That probably says something about corporate trainers. I still remember the HR department at one company where I worked that insisted on giving the Myers-Briggs personality test to every employee. So maybe this also says something about junk science and the way it lingers on in our workplaces...
As a CPGE student I must say that memorization IS key to solving problems (the essence of learning maths); it's not a traditional "dumb" memorization of proofs however.
The whole assertion of "understanding how a proof is done rather than memorizing it is how maths should be done" holds true, BUT only when you are not bound by time or a deadline, which is not the case most of time, whether you are working through a test, an exam or even on a PhD, you cannot afford losing time "reinventing the wheel"; working on proving theorems that have already been proven rather than using them directly.
The factor of time forces you to "memorize" certain concepts/theorems/facts so that you can use them directly as tools.
Personally, my approach consists in working on proving these "tools" at a first stage; understanding why they are true, then, I simply go past that and simply "memorize" them in order to boost my workflow.
In a nutshell, in order to be productive (doing maths or even physics), you must memorize shit, just don't do it blindly.
"In a nutshell, in order to be productive (doing maths or even physics), you must memorize shit, just don't do it blindly."
And voilà, you put it much better than I did in 5 rambling paragraphs. Memorization is key (at least for some learners), but only as one part of a process.
Did you even read the article, or did you read the subheading, which I will admit is a stretch, and immediately cast judgement? You used the strawman of a k-12/college student who learns something and quickly by practicing a particular method then forgets it. What about those who actually do learn this way? There are most certainly visual learners who can see concepts in their head -- a trait I admire personally.
>So many students in other cultures
Do you know every student? Every culture? Again, nothing but over-generalizations here. The article actually referenced the afterschool programs in Japan that had success.
Yes at some point you have to back and understand the why and the how, but solving equations and crafting proofs are actually two different skills in my opinion and I have met mathematicians that were stronger in either or both.
In any case, I think you are missing the point of this author. I think the authors strongest argument which is building that foundation, or "innate ability" and he argues that comes through memorization and rote learning. Quoting, "building well-ingrained chunks of expertise through practice and repetition was absolutely vital to their success" I don't think he just pulled that out of his butt. If you take the time to practice like the author did where it becomes engrained into your mind, you are in fact learning effectively.
Besides all this, what is mathematics anyways? Is it not a language? Does it not have "letters" (variables, operators, symbols), does it not have a grammar? Does it not have "parts of speech?" (an equation must have a left side, and a right side), Is it not used to communicate to others?
In any case, certainly not a terrible read in any way.
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.
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.
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.
The problem is that unless you are also able to approach things like a constructivist, you will end up with a bunch of disjoint facts. I think that most people just don't understand math well enough to understand this key. The more math I know, the more I wish I hadn't memorized anything, but derived more things. There are many cases these days that I will have to go back and derive an equation for the first time because I was never taught the equation.
Her department is "Industrial & Systems Engineering", she doesn't need to know much more math than how to do arithmetic and a Chai squared. Not that I know more math than she does, it's just not a true part of the mathematical disciplines. She and I probably both just use the tools handed to us and don't understand the beauty or the ugliness. One of the big problems with math today is that nobody understands that arithmetic isn't math. It would be like a baseball player being called a woodworker because she used a wooden bat.
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.
I've been trying to get the .epub working for a long time, but its not easy to convert all the equations and make them look nice. Recently I found some very good new tools[1], so hopefully I'll add .epub/.mobi to the eBook bundle soon.
Do you know of any math books that are available as .epub? I'd like to see how they implement equations... PM me if you would like to be a beta tester.
It's an interesting looking book, and I'm somewhat inclined to buy it. Bookmarked!
I agree that you don't need to read thousands of pages to learn calculus. However, I don't want to stop at calculus. Basically, what I'd really love to have is a "mother of all maths textbook" -- a thick and heavy tome that compacts information from all the other thick and expensive books (which I'm never going to read) and different fields of mathematics. With enough detail that you can actually learn from it -- so it shouldn't be just for review and looking up formulas you couldn't memorize. I'd like to call it a reference book I can forever keep in my bookshelf (under my bed) and always look in it if I'm unsure about something...
I've looked at a bunch of these math compendiums while researching what to include in my book, and this one seemed the best so far: http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do... The writing isn't very hand-holdy, but it covers a lot of important topics, and without too much fluff.
I have both books on the shelf, but not finished reading through all of them so I can't give my full endorsement, but from what I've seen so far, they're good stuff.
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.
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 disagree. We are expecting students to learn more than ever before. Math is a great example. Their ability to handle even a simple concept like multiplication requires fluency in addition. The ability to learn algebra requires fluency in arithmetic. The ability to learn calculus requires fluency in algebra. Tons of kids pass one grade but emerge unprepared for the next. The author hits the nail on the head: they understand the material at the time, but never develop fluency. Then next year when they try to build on that foundation it has evaporated. Rote learning should begin in elementary. I am thankful that I was required to learn my multiplication tables.
Yes, we need to tailor our teaching to the attention span and abilities of our students, but that doesn't mean we shouldn't be using this powerful tool of fluency through rote learning.
It really depends on the goal. From a mathematician's perspective Engineering is still part of the kiddie pool of math. The kind of things some people did in high school mostly though route memorization.
However, for ~98% of the population it's by far the most useful parts of math. Then again you can also say the same thing about just Arithmetic, basic Algebra, Logic and Statistics. So, it's really a question of what your goals are.
There is actually a lot of Math that's been dropped from K-12 education. EX: Understanding logarithms is really fundamental for using a slide rule or understanding floating point arithmetic, but it's not really that useful for most people.
Even though I chafed a bit at the delays it imposed and kept asking the teacher about other more advanced topics, I do not regret a single minute of the time spent chanting 12-times tables in my first couple of school years. Same deal for written exercises and homework - I don't think I'd be so good at mental math, estimating, and 'back of the envelope' calculations done in pencil if it weren't for all that practice as a child.
> You only ever need it to be able to think faster about certain advanced topics
This (and pretty much only this - you don't need rote learning to become fluent in a language, as evidenced by the way people learn their first language). Rote learning is what in tech we call caching things.
I very much prefer calling it like that, because it immediately highlights some benefits and drawbacks of the method:
- you cache things for quicker access, therefore it enables you to think faster
- the biggest benefit comes from caching things that are most frequently accessed; therefore learning multiplication table makes sense, it significantly speeds up every math you do in your head explicitly
- rote learning basics of a given field can help you learn it faster as again, the basic facts you need will be readily available in cache
- this matters because a lot of thinking people do has timeouts (aka "power of will", "energy", "patience", "curiosity") - if figuring out some connections between pieces of knowledge takes too long, you will get frustrated and give up early
- rote learning of history dates usually makes little sense because you end up caching vertices of a graph, while what you really care about are edges and traversing
- there is only so much you can fit in cache without losing its effectiveness, so memorize what matters, not everything
- rote learning is caching things, not developing understanding
- spaced repetition is a hack to force your brain from removing a particular information from cache prematurely; after few iterations it will understand that this piece of information you keep repeating is important and should stay cached
Also note that various memorization techniques are basically very fancy names for data structures optimized for human brains. For instance, "mnemonic link system" (aka. "chain method") is basically a linked list, where you use stories to build cons cells. There are variants of this method that are essentially doubly-linked lists or skip lists, and then you have other methods that implement association lists ("memory palace"), hash tables, etc.
This is important because you can apply knowledge about those data structures to evaluating those memorization techniques. For example, the reason it's difficult to insert another item in the middle or at the end of the list memorized with "chain method" technique is that you have to first traverse linked link to the place of insertion and then re-cons it there (develop two additional stories). It suggests that a/ maybe try to add new elements to the beginning of the list, and b/ you shouldn't use the "chain method" for remembering things you'll need to have random access to.
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.
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 got tired of not having an intuitive understanding of particular mathematical concepts. The rote memorization was very limiting for be because I wasn't as easily able to grasp when it might apply to certain concepts.
Pretty sure Kalid is a commenter here, but I have to give him a huge thanks and plug for BetterExplained.com--really helped me grasp some of these higher-level concepts in a more intuitive manner, even if I didn't walk away with a full understanding of the intricacies.
Wow, thanks for the plug and the kind words! Glad to hear the site is helping -- most of the time I miss the intricacies too, and end up adding things years later [after a re-read] or in response to a comment from the article.
As a more general reply to the article, I was fortunate enough to work with Prof. Oakley on her Coursera Class (https://class.coursera.org/learning-001/lecture, I'm the last guest interview at the bottom) and I really, really like her learning strategy.
Her article didn't use the exact phrase "deliberate practice" but I think that captures the essence of what she means by repetition. You need enough conceptual understanding to make sure you're following the path correctly, but then you want to practice -- at the edge of your comfort zone, with feedback, etc. -- to make sure it's really clicking.
It's really easy to fool yourself into thinking "I've got this" when it's untested. In my own case, I realized I didn't "get" imaginary numbers and exponents when I couldn't estimate a^b [a raised to the bth power] in my head with equal fluency for all numbers a and b. I had never really tested every type of number in every type of exponent position (base and power).
For example,
3^4 => this should be a positive real number greater than 1
3^(-4) => this should be a positive real number, very close to 0.
3^i => Hrm.
i^i => Uh oh.
I knew that unless I had fluency with all of these scenarios, I didn't truly "get" exponents or complex numbers. Sure, maybe I had a baby version where I could use them in well-defined ways, but I had a subconscious fear of i appearing as a base and/or exponent. I had to challenge myself and practice thinking through the various permutations before I recognized the gaps. Then I had to deepen my conceptual understanding, and practice again.
(For the previous questions, 3^i should be a complex number on the unit circle, maybe around 50 or 60 degrees, but less than 90, and i^i should be a real number, greater than 0 but less than 1. I can estimate these without calculating them, see http://betterexplained.com/articles/intuitive-understanding-... for more details.)
Very happy to give the plug--the quality of your content and your generosity for putting so much out there for free is something I wish more people did.
I hope the business side of it is going well for you.
Side note--have you considered trying to work with Khan Academy? I feel like Salman's style flows well, but sometimes substitutes learning tricks vs. internalizing the underlying logic. Khan Academy could benefit from a touch of BetterExplained.
Thanks! The business side of things is coming along, I've just finished up a hectic few months (wedding, honeymoon) and am looking to crank on the site again. Long term I want the site to be an ever-growing part of my life.
I'm very interested in collaborations, and helping people sprinkle an intuitive approach into whatever teaching they are doing. (I like Bret Victor's approach, where he provides guidance on what programming tools could look like, which might work its way into new programming languages like Swift, etc.). I'm in touch with an internet buddy at Khan Academy and have been meaning to work together, I'll be reaching out soon I think :).
The entries on complex number, exponential functions & e, and many other subjects at betterexplained.com are simply amazing! It helped to clear my decade-long confusions or fears when dealing with them, Thank you for the great work.
In my opinion the three essential characteristics of a great teacher are expert level in the subject at hand, great communication skill and enthusiasm for teaching. Possessing two of them already makes a good teacher. You have all three.
Yep, I think so. My philosophy, at least, is that any idea can eventually be intuitive, no matter how difficult at first, if we find the right analogies.
I see any exponent like a^b as starting at 1.0, intending to apply a rate of change of (a), but modifying that rate by (b).
For example, 3^2 is an initial rate of change of 3x, which is then applied for 2 units of time [leading to 9]. So, 1.0 would turn into 9.0.
Well, our rate of change is "i", which means we plan on starting at 1.0 and having our rate of change be a rotation: 1.0 * i = 90 degrees on the unit circle.
If we are using i^i, that means we are rotating our rate of change. That is, we intended to rotate around the unit circle at 90 degrees, but now I'm going to turn the "rocket boost" which is applying that rotation by another 90 degrees (the i as the exponent). This means the rocket is facing 180 degrees (backwards) and our growth is going to be an exponential decay. We'll be on the real number line, but shrinking.
How about (i^i)^i? That's another 90-degree twist on the rocket, so it's pointing 270 (downward) and we'll rotate around the circle clockwise. It's probably a negative imaginary number, and (i^i)^i actually equals -i.
This is a quick brain dump, and not very clear without diagrams, check out the articles above if you'd like!
Your intuition is nice for figuring out the directions, but not so nice for explaining the numbers. By which I mean, I can understand why it's real, but I can't explain why i^i = e^(-pi/2)... I would expect it to be a nicer number instead (like 1/2, or 1/e, or something like that). I can't really explain why both pi and e end up int he formula.
Oh yeah, after getting the direction, figuring out the numbers is the next step :).
Having i as a base means "we plan on rotating 90 degrees" which actually means pi/2 radians.
e^rt models growth rate of r, for time of t. so e^(i · pi/2) creates a 90 degree turn (we intend on rotating, i, and do this enough to get a full 90-degree turn, pi/2).
This is all a fancy way of saying:
90 degree turn = i = e^(i · pi/2)
Now, with i^i, we're planning on modifying that growth rate (e^(i · pi/2)) that we just figured out! We're going to twist the "rate" from i (90 degrees) to i · i (180 degrees):
1. I said earlier that I already know how to find the numbers mathematically. What I don't have any intuition for is why the numbers are correct (such as why both e and pi should be in the answer).
2. You can't just distribute the exponents without justifying that it's valid. (I don't know the appropriate conditions for doing this myself either, but I know they exist.) For example (e^(2 i pi))^(1/2) is not e^(i pi), it's e^0. In other words, you need to first reduce it mod 2pi. Now if you're dealing with complex numbers I have no idea what the conditions should be.
(Replying to myself since we hit the nesting depth.)
1. Here's a deeper intuition: any circle is just the unit circle, scaled up or down. Any number is just 1.0, scaled (and rotated, if complex) by the exponential function that was run for some rate and for some amount of time. e^rt is a rocket ship of constant change, we just decide how long to stay on for, and we can get to any number.
In other words, for any number a: a = 1.0 * e^ln(a)
This formulation is useful if we know we're going to be taking exponents on our number a, i.e. we really want a^b. (If we know we'll be rotating our number, maybe we write it in polar coordinates, etc.)
So, the intuition is: "I know I'm going to be taking my number to various powers, so let's get the base settings for e^rt dialed in. pi/2 is the setting for how long we'd ride e^i for in order to get to 90 degrees. I should expect pi/2 somewhere in the answer as I take it to various powers."
2) I haven't taken complex analysis, so my understanding isn't nuanced enough here either. Technically, i^i can be multi-valued, for this graphical analogy let's settle on the principal root (https://www.math.hmc.edu/funfacts/ffiles/20013.3.shtml).
> (e^(2 i pi))^(1/2) is not e^(i pi), it's e^0.
(e^(2 i pi))^(1/2) is asking for the square root of 1, which is both 1 [e^0] and -1 [e^(i pi)]. Again there may be a subtlety here, but I'm not sure how the above statement is incorrect (barring a technicality like "we always mean the positive root"). For the purposes of an intuition it makes sense, I think.
It's interesting that that famous quote about math is a von Neumann quote. I always use the same quote with language and I think that there is a strong connection between language learning, especially ultra logical and systematized ones like Russian. Still, for the sake of saying the full quote:
Young man, in mathematics you don't understand things. You just get used to them.
If understanding is about "connecting the dots" then you need to learn about all the pertinent dots before that can happen. You can't be expected to have a deep understanding when first introduced to something, you have to trust that it will probably all make sense later.
My German teacher at school did an accelerated course from scratch at university (no previous knowledge) - a full language degree. It involved rote memorisation of German texts. It worked! She said that the advantage was that you later recognised phrases and grammar and were able to analyse what was previously just rote memorisation.
I spent a couple of years in Italy in my early 20s and learnt fluent Italian in that time. I took a couple of classes a week in the evenings, but spent the day programming with a team of English colleagues and perforce spoke English. I made an effort to socialise with Italians outside of work.
I went out with groups of Italians many a time, and experienced many a boring evening where if people didn't speak to me specifically I couldn't understand a thing.
I realised that after 9 months I could hold a good 1-to-1 conversation but that it was another 9 months before I was comfortable in a group situation - in your native tongue you can keep tabs on the various threads of a group conversation with ease, whereas if you are having to explicitly process it, it's so easy to fall behind and then you're lost.
From experience with exchange students, staying with families and going to regular local secondary school (ages 15-18, look at www.afs.org for more information) -- that sounds typical for an adult in a non-native work setting, making an effort.
Generally students starting at zero will use 3/5/7 months to approach fluency, approaching native level at the end of a stay. The difference being the "jump distance" -- so German to Norwegian or Spanish to Italian is 3 -- Arabic to Norwegian might be 5 -- and Japanese to Norwegian might be closer to 7.
Coming to Norway, sometimes students with poor English skills will do better - not being tempted to fall back to English, breaking the immersion - and not having such a hard time "forcing" class mates to speak Norwegian.
Having spent a year in Japan myself, it's fascinating how one can go from seemingly "nothing" to fluent after those months of no apparent progress. But language learning tends to be like a staircase -- you can feel stuck at one level for a long time (and being in a foreign setting it can be incredibly frustrating to be constrained to a preschool level of speech) -- only to seem to jump up a level. And then you'll be stuck there... etc
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:
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
> I couldn't spell was because at the time I did not understand the rules behind how the letters form together to get words.
One of the main goals of this kind of learning (at least for me) is to recognize patterns and rules on your own. When I repeat something often enough, patterns and rules began to jump out at me.
"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.
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Depending on what you want to learn, sure. Googling for "[someLanguage] practice problems" appears to turn up good results for most languages.
For data structures, write some unit tests and then make them pass.
I personally wouldn't recommend you bother learning the implementation details of algorithms: just learn the names and use-cases. Make some Quizlet flash cards and you're good. If you're having trouble finding some to learn, try Googling "sorting algorithms" or "sorting algorithm for [someDataStructure]". Go from there. I'd recommend a similar approach for design patterns.
If you want to get better at building applications, practice by reimplementing apps you use every day in your favorite language (or just come up with your own). "Building applications" encompasses a wide variety of things not mentioned above that you'll only be exposed to by building stuff, so do that.
which is a collection of exercises that slowly introduce you to a programming language.
I can read stuff out of a book, but I won't have that 'muscle memory' until I've spent that time in Vim and the compiler/REPL. So that when something new gets thrown my way, I'm not having to think back to the basics.
When I was learning C, I would write my program, then run it step-by-step in the debugger (I was using Visual Studio at the time) until it was pointless to do so because I knew what would happen next. If I made a change and did not have a mental model of what would happen, I would run the debugger.
Learning how to solve problems is a different process, but this gave me good understanding of basic code structures.
Drop in to the REPL, this is why languages that have Read, Eval, Print, Loop are super popular. BASIC, Python, LISP, SQL, Ruby, etc. You can get proficient in such languages faster than C, Java
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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 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.
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.
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.
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...
151 comments
[ 4.2 ms ] story [ 171 ms ] threadI 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).
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.
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/
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!
Not to rehash what you hashed, but it depends who you are. If you are really good at memorizing, you may not need to make connections. You may just be able to memorize everything. Some people can.
I don't believe there is compelling evidence that this is the case.
Instead, different people have different backgrounds and experiences. Learning requires making connections. Depending on what you have in your head already to which you can connect, you may respond differently about a particular topic than someone else.
http://en.wikipedia.org/wiki/Learning_styles#Criticism
That probably says something about corporate trainers. I still remember the HR department at one company where I worked that insisted on giving the Myers-Briggs personality test to every employee. So maybe this also says something about junk science and the way it lingers on in our workplaces...
The whole assertion of "understanding how a proof is done rather than memorizing it is how maths should be done" holds true, BUT only when you are not bound by time or a deadline, which is not the case most of time, whether you are working through a test, an exam or even on a PhD, you cannot afford losing time "reinventing the wheel"; working on proving theorems that have already been proven rather than using them directly.
The factor of time forces you to "memorize" certain concepts/theorems/facts so that you can use them directly as tools.
Personally, my approach consists in working on proving these "tools" at a first stage; understanding why they are true, then, I simply go past that and simply "memorize" them in order to boost my workflow.
In a nutshell, in order to be productive (doing maths or even physics), you must memorize shit, just don't do it blindly.
And voilà, you put it much better than I did in 5 rambling paragraphs. Memorization is key (at least for some learners), but only as one part of a process.
>So many students in other cultures
Do you know every student? Every culture? Again, nothing but over-generalizations here. The article actually referenced the afterschool programs in Japan that had success.
Yes at some point you have to back and understand the why and the how, but solving equations and crafting proofs are actually two different skills in my opinion and I have met mathematicians that were stronger in either or both.
In any case, I think you are missing the point of this author. I think the authors strongest argument which is building that foundation, or "innate ability" and he argues that comes through memorization and rote learning. Quoting, "building well-ingrained chunks of expertise through practice and repetition was absolutely vital to their success" I don't think he just pulled that out of his butt. If you take the time to practice like the author did where it becomes engrained into your mind, you are in fact learning effectively.
Besides all this, what is mathematics anyways? Is it not a language? Does it not have "letters" (variables, operators, symbols), does it not have a grammar? Does it not have "parts of speech?" (an equation must have a left side, and a right side), Is it not used to communicate to others?
In any case, certainly not a terrible read in any way.
This is the essence of the article. It is chunking.
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.
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...
Her department is "Industrial & Systems Engineering", she doesn't need to know much more math than how to do arithmetic and a Chai squared. Not that I know more math than she does, it's just not a true part of the mathematical disciplines. She and I probably both just use the tools handed to us and don't understand the beauty or the ugliness. One of the big problems with math today is that nobody understands that arithmetic isn't math. It would be like a baseball player being called a woodworker because she used a wooden bat.
<discl>I'm the author</discl>
Do you plan to release an ePub or mobi-format book? I'd prefer to read it on my ereader, and PDFs don't reflow on smaller screens.
Do you know of any math books that are available as .epub? I'd like to see how they implement equations... PM me if you would like to be a beta tester.
[1] https://github.com/softcover/softcover
I agree that you don't need to read thousands of pages to learn calculus. However, I don't want to stop at calculus. Basically, what I'd really love to have is a "mother of all maths textbook" -- a thick and heavy tome that compacts information from all the other thick and expensive books (which I'm never going to read) and different fields of mathematics. With enough detail that you can actually learn from it -- so it shouldn't be just for review and looking up formulas you couldn't memorize. I'd like to call it a reference book I can forever keep in my bookshelf (under my bed) and always look in it if I'm unsure about something...
If someone has book suggestions, I'm all eyes.
http://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_math%C3%A...
For a more "math for general culture" I'd recommend this one: http://www.amazon.ca/Mathematics-1001-Absolutely-Everything-... which covers a lot of fundamental topics in an intuitive manner.
I have both books on the shelf, but not finished reading through all of them so I can't give my full endorsement, but from what I've seen so far, they're good stuff.
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.
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.
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.
Yes, we need to tailor our teaching to the attention span and abilities of our students, but that doesn't mean we shouldn't be using this powerful tool of fluency through rote learning.
However, for ~98% of the population it's by far the most useful parts of math. Then again you can also say the same thing about just Arithmetic, basic Algebra, Logic and Statistics. So, it's really a question of what your goals are.
There is actually a lot of Math that's been dropped from K-12 education. EX: Understanding logarithms is really fundamental for using a slide rule or understanding floating point arithmetic, but it's not really that useful for most people.
This (and pretty much only this - you don't need rote learning to become fluent in a language, as evidenced by the way people learn their first language). Rote learning is what in tech we call caching things.
I very much prefer calling it like that, because it immediately highlights some benefits and drawbacks of the method:
- you cache things for quicker access, therefore it enables you to think faster
- the biggest benefit comes from caching things that are most frequently accessed; therefore learning multiplication table makes sense, it significantly speeds up every math you do in your head explicitly
- rote learning basics of a given field can help you learn it faster as again, the basic facts you need will be readily available in cache
- this matters because a lot of thinking people do has timeouts (aka "power of will", "energy", "patience", "curiosity") - if figuring out some connections between pieces of knowledge takes too long, you will get frustrated and give up early
- rote learning of history dates usually makes little sense because you end up caching vertices of a graph, while what you really care about are edges and traversing
- there is only so much you can fit in cache without losing its effectiveness, so memorize what matters, not everything
- rote learning is caching things, not developing understanding
- spaced repetition is a hack to force your brain from removing a particular information from cache prematurely; after few iterations it will understand that this piece of information you keep repeating is important and should stay cached
Also note that various memorization techniques are basically very fancy names for data structures optimized for human brains. For instance, "mnemonic link system" (aka. "chain method") is basically a linked list, where you use stories to build cons cells. There are variants of this method that are essentially doubly-linked lists or skip lists, and then you have other methods that implement association lists ("memory palace"), hash tables, etc.
This is important because you can apply knowledge about those data structures to evaluating those memorization techniques. For example, the reason it's difficult to insert another item in the middle or at the end of the list memorized with "chain method" technique is that you have to first traverse linked link to the place of insertion and then re-cons it there (develop two additional stories). It suggests that a/ maybe try to add new elements to the beginning of the list, and b/ you shouldn't use the "chain method" for remembering things you'll need to have random access to.
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.
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.
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.
Pretty sure Kalid is a commenter here, but I have to give him a huge thanks and plug for BetterExplained.com--really helped me grasp some of these higher-level concepts in a more intuitive manner, even if I didn't walk away with a full understanding of the intricacies.
His Cheatsheet is a great starting point if you are interested in a particular topic: http://betterexplained.com/cheatsheet/
As a more general reply to the article, I was fortunate enough to work with Prof. Oakley on her Coursera Class (https://class.coursera.org/learning-001/lecture, I'm the last guest interview at the bottom) and I really, really like her learning strategy.
Her article didn't use the exact phrase "deliberate practice" but I think that captures the essence of what she means by repetition. You need enough conceptual understanding to make sure you're following the path correctly, but then you want to practice -- at the edge of your comfort zone, with feedback, etc. -- to make sure it's really clicking.
It's really easy to fool yourself into thinking "I've got this" when it's untested. In my own case, I realized I didn't "get" imaginary numbers and exponents when I couldn't estimate a^b [a raised to the bth power] in my head with equal fluency for all numbers a and b. I had never really tested every type of number in every type of exponent position (base and power).
For example,
3^4 => this should be a positive real number greater than 1
3^(-4) => this should be a positive real number, very close to 0.
3^i => Hrm.
i^i => Uh oh.
I knew that unless I had fluency with all of these scenarios, I didn't truly "get" exponents or complex numbers. Sure, maybe I had a baby version where I could use them in well-defined ways, but I had a subconscious fear of i appearing as a base and/or exponent. I had to challenge myself and practice thinking through the various permutations before I recognized the gaps. Then I had to deepen my conceptual understanding, and practice again.
(For the previous questions, 3^i should be a complex number on the unit circle, maybe around 50 or 60 degrees, but less than 90, and i^i should be a real number, greater than 0 but less than 1. I can estimate these without calculating them, see http://betterexplained.com/articles/intuitive-understanding-... for more details.)
Again, thanks for the mention!
I hope the business side of it is going well for you.
Side note--have you considered trying to work with Khan Academy? I feel like Salman's style flows well, but sometimes substitutes learning tricks vs. internalizing the underlying logic. Khan Academy could benefit from a touch of BetterExplained.
I'm very interested in collaborations, and helping people sprinkle an intuitive approach into whatever teaching they are doing. (I like Bret Victor's approach, where he provides guidance on what programming tools could look like, which might work its way into new programming languages like Swift, etc.). I'm in touch with an internet buddy at Khan Academy and have been meaning to work together, I'll be reaching out soon I think :).
In my opinion the three essential characteristics of a great teacher are expert level in the subject at hand, great communication skill and enthusiasm for teaching. Possessing two of them already makes a good teacher. You have all three.
I see any exponent like a^b as starting at 1.0, intending to apply a rate of change of (a), but modifying that rate by (b).
For example, 3^2 is an initial rate of change of 3x, which is then applied for 2 units of time [leading to 9]. So, 1.0 would turn into 9.0.
More here: http://betterexplained.com/articles/understanding-exponents-...
So what's i^i?
Well, our rate of change is "i", which means we plan on starting at 1.0 and having our rate of change be a rotation: 1.0 * i = 90 degrees on the unit circle.
If we are using i^i, that means we are rotating our rate of change. That is, we intended to rotate around the unit circle at 90 degrees, but now I'm going to turn the "rocket boost" which is applying that rotation by another 90 degrees (the i as the exponent). This means the rocket is facing 180 degrees (backwards) and our growth is going to be an exponential decay. We'll be on the real number line, but shrinking.
How about (i^i)^i? That's another 90-degree twist on the rocket, so it's pointing 270 (downward) and we'll rotate around the circle clockwise. It's probably a negative imaginary number, and (i^i)^i actually equals -i.
This is a quick brain dump, and not very clear without diagrams, check out the articles above if you'd like!
Having i as a base means "we plan on rotating 90 degrees" which actually means pi/2 radians.
e^rt models growth rate of r, for time of t. so e^(i · pi/2) creates a 90 degree turn (we intend on rotating, i, and do this enough to get a full 90-degree turn, pi/2).
This is all a fancy way of saying:
90 degree turn = i = e^(i · pi/2)
Now, with i^i, we're planning on modifying that growth rate (e^(i · pi/2)) that we just figured out! We're going to twist the "rate" from i (90 degrees) to i · i (180 degrees):
i^i = e^(i · i · pi/2) = e^(-pi/2)
which is the real number less than 1 (about 0.2).
I'm summarizing on the fly, but the full explanation for i^i and i^i^i are here: http://betterexplained.com/articles/intuitive-understanding-...
1. I said earlier that I already know how to find the numbers mathematically. What I don't have any intuition for is why the numbers are correct (such as why both e and pi should be in the answer).
2. You can't just distribute the exponents without justifying that it's valid. (I don't know the appropriate conditions for doing this myself either, but I know they exist.) For example (e^(2 i pi))^(1/2) is not e^(i pi), it's e^0. In other words, you need to first reduce it mod 2pi. Now if you're dealing with complex numbers I have no idea what the conditions should be.
1. Here's a deeper intuition: any circle is just the unit circle, scaled up or down. Any number is just 1.0, scaled (and rotated, if complex) by the exponential function that was run for some rate and for some amount of time. e^rt is a rocket ship of constant change, we just decide how long to stay on for, and we can get to any number.
In other words, for any number a: a = 1.0 * e^ln(a)
This formulation is useful if we know we're going to be taking exponents on our number a, i.e. we really want a^b. (If we know we'll be rotating our number, maybe we write it in polar coordinates, etc.)
So, the intuition is: "I know I'm going to be taking my number to various powers, so let's get the base settings for e^rt dialed in. pi/2 is the setting for how long we'd ride e^i for in order to get to 90 degrees. I should expect pi/2 somewhere in the answer as I take it to various powers."
2) I haven't taken complex analysis, so my understanding isn't nuanced enough here either. Technically, i^i can be multi-valued, for this graphical analogy let's settle on the principal root (https://www.math.hmc.edu/funfacts/ffiles/20013.3.shtml).
> (e^(2 i pi))^(1/2) is not e^(i pi), it's e^0.
(e^(2 i pi))^(1/2) is asking for the square root of 1, which is both 1 [e^0] and -1 [e^(i pi)]. Again there may be a subtlety here, but I'm not sure how the above statement is incorrect (barring a technicality like "we always mean the positive root"). For the purposes of an intuition it makes sense, I think.
I spent a couple of years in Italy in my early 20s and learnt fluent Italian in that time. I took a couple of classes a week in the evenings, but spent the day programming with a team of English colleagues and perforce spoke English. I made an effort to socialise with Italians outside of work.
I went out with groups of Italians many a time, and experienced many a boring evening where if people didn't speak to me specifically I couldn't understand a thing.
I realised that after 9 months I could hold a good 1-to-1 conversation but that it was another 9 months before I was comfortable in a group situation - in your native tongue you can keep tabs on the various threads of a group conversation with ease, whereas if you are having to explicitly process it, it's so easy to fall behind and then you're lost.
Generally students starting at zero will use 3/5/7 months to approach fluency, approaching native level at the end of a stay. The difference being the "jump distance" -- so German to Norwegian or Spanish to Italian is 3 -- Arabic to Norwegian might be 5 -- and Japanese to Norwegian might be closer to 7.
Coming to Norway, sometimes students with poor English skills will do better - not being tempted to fall back to English, breaking the immersion - and not having such a hard time "forcing" class mates to speak Norwegian.
Having spent a year in Japan myself, it's fascinating how one can go from seemingly "nothing" to fluent after those months of no apparent progress. But language learning tends to be like a staircase -- you can feel stuck at one level for a long time (and being in a foreign setting it can be incredibly frustrating to be constrained to a preschool level of speech) -- only to seem to jump up a level. And then you'll be stuck there... etc
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
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.
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
One of the main goals of this kind of learning (at least for me) is to recognize patterns and rules on your own. When I repeat something often enough, patterns and rules began to jump out at me.
[0]- https://www.maa.org/external_archive/devlin/LockhartsLament....
Done, and done better and faster than you could possibly do it.
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?
http://blog.jenniferdewalt.com/post/56319597560/im-learning-...
For data structures, write some unit tests and then make them pass.
I personally wouldn't recommend you bother learning the implementation details of algorithms: just learn the names and use-cases. Make some Quizlet flash cards and you're good. If you're having trouble finding some to learn, try Googling "sorting algorithms" or "sorting algorithm for [someDataStructure]". Go from there. I'd recommend a similar approach for design patterns.
If you want to get better at building applications, practice by reimplementing apps you use every day in your favorite language (or just come up with your own). "Building applications" encompasses a wide variety of things not mentioned above that you'll only be exposed to by building stuff, so do that.
such as 'Études for Elixir' http://chimera.labs.oreilly.com/books/1234000001642
which is a collection of exercises that slowly introduce you to a programming language.
I can read stuff out of a book, but I won't have that 'muscle memory' until I've spent that time in Vim and the compiler/REPL. So that when something new gets thrown my way, I'm not having to think back to the basics.
Project Euler?
Learning how to solve problems is a different process, but this gave me good understanding of basic code structures.
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.
Interesting watch (first 5mins gives good context) http://books.google.com/books/about/Moonwalking_with_Einstei...
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...
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.
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...