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Book Review: Mathematics for Computer Science (Suggestion for MIRI Research Guide)
if you’ve read all of a sequence you get a small badge that you can choose to display right next to your username, which helps people navigate how much of the content of the page you are familiar with.
Idea: give sequence-writers the option to include quizzes because this (1) demonstrates a badgeholder actually understands what the badge indicates they understand (or, at least, are more likely to) and (2) leverages the testing effect.
I await the open beta eagerly.
Since this review, Axler has released a third edition. The new edition contains substantial changes (i.e. it’s not the same book being released under “n+1 edition”): though there’s little new material, exercises appear at the end of every section, instead at the end of every chapter, and there’s many more examples given in the body of the text (a longer list of changes can be found on Dr. Axler’s website). I feel these revisions are significant improvements from a pedagogical perspective, as it gives the reader more opportunity to practice prerequisite skills before learning the next thing. The changes also lower the requisite mathematical maturity, which is a good thing (insofar as it makes the book more accessible), although it won’t push the reader to develop mathematical maturity as much. Overall: the third edition came out when I was halfway through the second edition and I felt that the improments merited switching books.
Before we started using SRS I tried to sell my students on it with a heartfelt, over-prepared 20 minute presentation on how it works and the superpowers to be gained from it. It might have been a waste of time. It might have changed someone’s life. Hard to say.
I’m less skeptical. You say that you got a few students to use Anki which, while probably not life-changing, is probably significantly life-impacting. If my tenth grade English teacher had introduced Anki to me… well, right now, I’m reteaching myself introductory biology (5 on the AP exam), introductory chemistry (5 on the AP exam) and introductory psychology (A in the college course) because I forgot the content of each of these courses because I lacked Anki. I obviously don’t know everything you do in your classroom, but it’s entirely plausible that, rather than being a waste of time, introducing your students to Anki might have been (on average) the most impactful 20 minutes of teaching you did all year; you just may not see all the benefit in your classroom.
Writing Collaboratively
I once took a math course where the first homework assignment involved sending the professor an email that included what we wanted to learn in the course (this assignment was mostly for logistical reasons: professor’s email now autocompletes, eliminating a trivial inconvenience of emailing him questions and such, professor has all our emails, etc). I had trouble answering the question, since I was after learning unknown unknowns, thereby making it difficult to express what exactly it was I was looking to learn. Most mathematicians I’ve talked to agree that, more or less, what is taught in secondary school under the heading of “math” is not math, and it certainly bears only a passing resemblance to what mathematicians actually do. You are certainly correct that the thing labelled in secondary schools as “math” is probably better learned differently, but insofar as you’re looking to learn the thing that mathematicians refer to as “math” (and the fact you’re looking at Spivak’s Calculus indicates you, in fact, are), looking at how to better learn the thing secondary schools refer to as “math” isn’t actually helpful. So, let’s try to get a better idea of what mathematicians refer to as math and then see what we can do.
The two best pieces I’ve read that really delve into the gap between secondary school “math” and mathematician’s “math” are Lockhart’s Lament and Terry Tao’s Three Levels of Rigour. The common thread between them is that secondary school “math” involves computation, whereas mathematician’s “math” is about proof. For whatever reason, computation is taught with little motivation, largely analogously to the “intolerably boring” approach to language acquisition; proof, on the other hand, is mostly taught by proving a bunch of things which, unlike computation, typically takes some degree of creativity, meaning it can’t be taught in a rote manner. In general, a student of mathematics learns proofs by coming to accept a small set of highly general proof strategies (to prove a theorem of the form “if P then Q”, assume P and derive Q); they first practice them on the simplest problems available (usually set theory) and then on progressively more complex problems. To continue Lockhart’s analogy to music, this is somewhat like learning how to read the relevant clef for your instrument and then playing progressively more difficult music, starting with scales. [1] There’s some amount of symbol-pushing, but most of the time, there’s insight to be gleaned from it (although, sometimes, you just have to say “this is the correct result because the algebra says so”, but this isn’t overly common).
Proofs themselves are interesting creatures. In most schools, there’s a “transition course” that takes aspiring math majors who have heretofore only done computation and trains them to write proofs; any proofy math book written for any other course just assumes this knowledge but, in my experience (both personally and working with other students), trying to make sense of what’s going on in these books without familiarity with what makes a proof valid or not just doesn’t work; it’s not entirely unlike trying to understand a book on arithmetic that just assumes you understand what the + and * symbols mean. This transition course more or less teaches you to speak and understand a funny language mathematicians use to communicate why mathematical propositions are correct; without taking the time to learn this funny language, you can’t really understand why the proof of a theorem actually does show the theorem is correct, nor will you be able to glean any insight as to why, on an intuitive level, the theorem is true (this is why I doubt you’d have much success trying to read Spivak, absent a transition course). After the transition course, this funny language becomes second nature, it’s clear that the proofs after theorem statements, indeed, prove the theorems they claim to prove, and it’s often possible, with a bit of work [2], to get an intuitive appreciation for why the theorem is true.
To summarize: the math I think you’re looking to learn is proofy, not computational, in nature. This type of math is inherently impossible to learn in a rote manner; instead, you get to spend hours and hours by yourself trying to prove propositions [3] which isn’t dull, but may take some practice to appreciate (as noted below, if you’re at the right level, this activity should be flow-inducing). The first step is to do a transition, which will teach you how to write proofs and discriminate between correct proofs from incorrect; there will probably some set theory.
So, you want to transition; what’s the best way to do it?
Well, super ideally, the best way is to have an experienced teacher explain what’s going on, connecting the intuitive with the rigorous, available to answer questions. For most things mathematical, assuming a good book exists, I think it can be learned entirely from a book, but this is an exception. That said, How to Prove It is highly rated, I had a good experience with it, and other’s I’ve recommended it to have done well. If you do decide to take this approach and have questions, pm me your email address and I’ll do what I can.
This analogy breaks down somewhat when you look at the arc musicians go through. The typical progression for musicians I know is (1) start playing in whatever grade the music program of the school I’m attending starts, (2) focus mainly on ensemble (band, orchestra) playing, (3) after a high (>90%) attrition rate, we’re left with three groups: those who are in it for easy credit (orchestra doesn’t have homework!); those who practice a little, but are too busy or not interested enough to make a consistent effort; and those who are really serious. By the time they reach high school, everyone in this third group has private instructors and, if they’re really serious about getting good, goes back and spends a lot of times practicing scales. Even at the highest level, musicians review scales, often daily, because they’re the most fundamental thing: I once had the opportunity to ask Gloria dePasquale what the best way to improve general ability, and she told me that there’s 12 major scales and 36 minor scales and, IIRC, that she practices all of them every day. Getting back to math, there’s a lot here that’s not analogous to math. Most notably, there’s no analogue to practicing scales, no fundamental-level thing that you can put large amounts of time into practicing and get general returns to mathematical ability: there’s just proofs, and once you can tell a valid proof from an invalid proof, there’s almost no value that comes from studying set theory proofs very closely. There’s certainly an aesthetic sense that can be refined, but studying whatever proofs happen to be at to slightly above your current level is probably the most helpful (like in flow), if it’s too easy, you’re just bored and learn nothing (there’s nothing there to learn), and if it’s too hard, you get frustrated and still learn nothing (since you’re unable to understand what’s going on).)
“With a bit of work”, used in a math text, means that a mathematically literate reader who has understood everything up until the phrase’s invocation should be able to come up with the result themselves, that it will require no real new insight; “with a bit of work, it can be shown that, for every positive integer n, (1 + 1/n)^n < e < (1 + 1/n)^(n+1)”. This does not preclude needing to do several pages of scratch work or spending a few minutes trying various approaches until you figure out one that works; the tendency is for understatement. Related, most math texts will often leave proofs that require no novel insights or weird tricks as exercises for the reader. In Linear Algebra Done Right, for instance, Axler will often state a theorem followed by “as you should verify”, which should require some writing on the reader’s part; he explicitly spells this out in the preface, but this is standard in every math text I’ve read (and I only bother reading the best ones). You cannot read mathematics like a novel; as Axler notes, it can often take over an hour to work through a single page of text.
Most math books present definitions, state theorems, and give proofs. In general, you definitely want to spend a bit of time pondering definitions; notice why they’re correct/how the match your intuition, and seeing why other definitions weren’t used. When you come to a theorem, you should always take a few minutes to try to prove it before reading the book’s proof. If you succeed, you’ll probably learn something about how to write proofs better by comparing what you have to what the book has, and if you fail, you’ll be better acquainted with the problem and thus have more of an idea as to why the book’s doing what it’s doing; it’s just an empirical result (which I read ages ago and cannot find) that you’ll understand a theorem better by trying to prove it yourself, successful or not. It’s also good practice. There’s some room for Anki (I make cards for definitions—word on front, definition on back—and theorems—for which reviews consist of outlining enough of a proof that I’m confident I could write it out fully if I so desired to) but I spend the vast majority of my time trying to prove things.
It helps to explicitly visualize people who I perceive as being skilled in X failing at it over and over again
Some of the greatest value I’ve gotten out of attending math lectures comes from seeing math Ph.Ds (particularly good ones) make mistakes or even forget exactly how a proof works and have to dismiss class early. It never happened often, but just often enough to keep me from getting discouraged.
Paul Graham writes that studying fields with hard, solved problems (eg mathematics) is useful, because it gives you practice solving hard problems and the approaches and habits of mind that you develop solving those problems are useful when you set out to tackle new (technical) problems. This claim seems at least plausible to me and seems to line up with me personal experience, but you seem like a person who might know why I shouldn’t believe this, so I ask, is there any reason I should doubt that the problem-solving approaches and habits of mind I develop studying mathematics won’t help me as I run into novel technical problems?
It seems conventional wisdom that tests are generally gameable in the sense that an (most?) effective way to produce the best scores involves teaching password guessing rather than actually learning material deeply, i.e. such that the student can use it in novel and useful ways. Indeed, I think this is the case for many (most, even) tests, but also think it possible to write tests that are most easily passed by learning the material deeply. In particular, I don’t see how to game questions like “state, prove, and provide an intuitive justification for Pascal’s combinatorial identity” or “Under what conditions does f(x) = ax^3 + bx^2 + cx + d have only one critical point?″, but that’s more a statement about my mind than the gameability of tests. I would greatly appreciate learning how a test consisting of such questions could be gamed, thereby unlearning an untrue thing; and if no one here can (or, at least, is willing to take the time to) explain how such a thing could be done, well, that’s useful to know, too.
It has been requested that I post my own take on efficient learning. As I spend half a page describing, this is not yet ready for publishing, but I’m putting out there because there may be (great) benefit to be had. After all, there is low-hanging fruit if you’re willing to abandon traditional methods: simply doing practice problems in a different order may improve your test score by 40 points.
- 7 May 2015 23:28 UTC; 6 points) 's comment on Stupid Questions May 2015 by (
Turns out you’re not the only one who wants to know this. Seems your best bet is to use C-S-v to paste raw text and then format it in the article editor.
There’s two problems here. First, we have duplication of labor in that we have something like 1% of the population doing essentially the same task, even though it’s fairly straightforward to reproduce and distribute en masse after it’s been done once. This encompasses things like lesson plans, lectures, and producing supplementary materials (e.g. a sheet of practice problems).
This leads into the second problem, which is a resulting quality issue: if you have a large population of diverse talent doing the same task, you expect it to form some sort of a bell curve. As noted above, we can take any lecture, tape it, and broadcast in en masse fairly easily. When we choose a system where each student is subjected to their instructor’s particular lecture, a relatively small portion of them get an excellent lecture, a very large portion get an average lecture (rather than an excellent lecture), and a relatively small portion get an execrable lecture (rather than an excellent lecture). If you’re really ambitious, you could even get the top, say, ten lecturers together and have them collaborate to make a super-lecture, and then get feedback on that particular unit, so they can improve the superlecture into a super-duperlecture.
(IMO, this is still a suboptimal way to do things. Try that process on textbooks (which are much easier to write collaboratively), and instead of getting feedback on hour-long chunks, get feedback on section-sized chunks (which, depending on the subject, can something like one-tenth the size). A good textbook is also cheaper to write, cheaper to distribute, more updateable, and better didactic material to begin with.)
It’s worth noting that there’s still a few wrinkles. Most importantly, there’s really no such thing as a “best” lecture, lesson plan, problem set, or textbook; the “goodness” quality depends, not just on the lecture’s content, but the intended audience. Think of this as a callibration issue. For instance:
Last I checked, MIT uses Sadava as their introductory biology textbook. If you dig around the reviews, you will find endorsements of another introductory biology book by Campbell that claim it’s “SO much easier to understand. It’s better organized, more clearly written”. When I found myself needing to relearn introductory biology (this time with Anki so I actually retain the knowledge), I tried Campbell, since that’s what my high school used, but gave up not halfway through the first chapter, frustrated by the difficulty I had understanding, the poor organization, and unclear writing; I find Sadava, however, to be much easier to understand, better organized, and more clearly written. Is the quoted reviewer lying, perhaps paid off by Big Textbooks? Perhaps, but a much better explanation is that Sadava is more technical; it’s much closer to the “definition-theorem-proof” feel of a math text. This makes it a fantastic text if you’re most students at MIT (or a typical LWer), but much less so if you’re in the other 99% of the population. This also solves the callibration problem: write two (or more) supertextbooks.
(This also neatly explains why MIT sometimse seems like the only school that uses good textbooks and why SICP only has 3.5 stars on Amazon.)
A second wrinkle is individual attention, which I tend to be dismissive of (if the textbook is good enough, you shouldn’t need any individual attention! And it’s not like the current education system, with its one-way lectures, is very good at giving very much individual attention), but if we’re optimizing education, there probably is more individual attention given to every student. However, because of reasons, I suspect that most of it should come from students in the same class, not staff. Also, it belongs after the reading.
A third wrinkle is a narrowing of perspectives. In any particular domain, there’s usually several approaches to solving problems, often coming from different ways of looking at it. In the current system, if you wind up on a team and come across a seemingly intractable problem, there’s a good chance that someone else has happened across a nonstandard approach that makes the problem very easy. If we standardize everything, we lose this. This is somewhat mitigated by the solution to the callibration problem, wherein people are going to be reading different texts with the different approaches because they’re different people, but we still kind of expect most mathematicians to learn their analysis from super!Rudin, meaning that they all lack some trick that Pugh mentions. The best solution I have is to have students learn in the highly standardized manner first, and once they have a firm grasp on that, expose them to nonstandard methods (according to my Memory text, this is an effective manner for increasing tranfer-of-learning).
- 3 Jan 2015 1:18 UTC; 0 points) 's comment on Stupid Questions January 2015 by (
Yes. Lots of them. Right now, my memory deck has about 200 cards, and I’m only about 2⁄3 done with the course. I’ll point again to Baddeley Eysenck Anderson. You seem primarily interested in long-term memory (although that may be an artifact of not knowing a lot about memory; a large benefit of having a textbook on memory is to point out “unknown unkowns”), so here are some big ones off the top of my head.
Implicit and explicit memory (also known as declarative and nondeclarative, respectively).
Episodic and semantic memory (are subsets of explicit/declarative memory)
Also procedural memory (a subset of implicit/nondeclarative memory).
You should also be aware of the testing effect and distributed practice, which, along with forgetting curves, form the basis of Spaced Repetition Software. Since many things don’t lend themselve to Anki, like riding a bike, it’s enormously beneficial to know about these independently.
Also Source monitoring, which leads to my favorite term, cryptomnesia.
The best textbook on memory I’m aware of is Baddeley Eysenck Anderson. It is quite good, but some of the definitions are vague, so you’ll need to reference Wikipedia,.
Memory palaces, more formally known as Method of Loci, are well-supported by the academic literature. Brienne’s presentation is a fantastic introduction, in line with all the academic literature I’ve read.
I use Anki. It gets the job done quite well, and although other software may be just as good or better, I’m left with no desire to try anything else. See janki method for implementation suggestions.
I’m in the middle of a course on memory; according to my notes, making outlines is a good way of studying for a test and thinking about things in terms of future plans is “perhaps the best way of remembering stuff” (so, if I wanted to remember regular expressions, I might imagine doing this with them).
According to Scott, bacopa is “a memory-enhancing drug that performs very well in studies”—assuming you take it consistently for 3 months. According to my soylent spreadsheet, this is the most cost-effective source. According to Reddit, this is source with the lowest amounts of heavy metals (which are well within limits set by FDA). Reddit also has dosing recommendations. Apparently is also an axiolytic, so yay. Note that bacopa tastes nasty, so many people pay a bit extra for pills, although I find the taste trivial to deal with if I have a glass of water to wash the powder down with.
And yet, humans currently have the edge in Brood War. Humans are probably doomed once StarCraft AIs get AlphaGo-level decision-making, but flawless micro—even on top of flawless* macro—won’t help you if you only have zealots when your opponent does a muta switch. (Zealots can only attack ground and mutalisks fly, so zealots can’t attack mutalisks; mutalisks are also faster than zealots.)
*By flawless, I mean macro doesn’t falter because of micro elsewhere; often, even at the highest levels, players won’t build new units because they’re too busy controlling a big engagement or heavily multitasking (dropping at one point, defending a poke elsewhere, etc). If you look at it broadly, making the correct units is part of macro, but that’s not what I’m talking about when I say flawless macro.
Introductory discrete math textbook (pdf) courtesy of MIT. I prefer it to Rosen, which is currently recommended in the MIRI research guide, although I think there exist students who would do better with Rosen’s book.
(How to tell which book you should choose? Well, since this one is Creative Commons, and therefore free, I’d try this one. If you find it’s not saying enough words per theorem, try Rosen. If you think it’s saying too many words per theorem, try these lecture notes. A recommendation to LW’s list of best textbooks is forthcoming, which will contain a complete discussion.)
An earlier version of the book corresponds to these videos lectures, which I find to be excellent, as far as lectures go.
If I’m understanding you correctly, you don’t think people who can’t individually affect the equilibrium are evil? Scientists who would be outcompeted, and therefore unable to do science, if they failed to pursue a maximally impressive career seem an example of this. If they’re good (in terms of both ability and alignment), there’s some wiggle room in there for altruism when it’s cheap, but if err too far from having an impressive career, someone else, probably someone not making small career sacrifices for social benefit, gets the attention and funds instead, thereby reducing the total amount of good being done.
I’d be interested in some concrete examples of appealing to people’s conscience getting us less-bad equilibria so I can better understand what you’re getting at. Is the scientists who all resigned from the board of an Elsevier-owned journal and started their own an example?
Also interested in your thoughts where we’re in a suboptimal equilibrium that we’re trying to get out of and there’s 2+ optimal ones (in the sense that there’s no other equilibrium which is a Pareto improvement), but each competes with the other. For instance, suppose it’s several years ago and we agree that it’s better that gay couples have the same right to legal marriage as straight couples, but the two better equilibria are (1) elimination of marriage as a legal institution and (2) extension of marriage to gay couples, which is good for gay couples who want the legal benefits of marriage and less good for those who don’t want marriage but suffer from e.g. less favorable tax treatment. (I’d really like a better example for this, especially after the examples EY gave, but lack a large, responsive group of fb followers I can get to brainstorm examples for me.)
Extremely interested, would move anywhere rationalists would set one of these up.
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