Thanks for the many interesting comments. I'll reply jointly here to a few of the comments, and invite further discussion on some points made in the submitted article that I'm curious about.
As someone who was taught the 'traditional' way of mathematics, can someone give a few pointers of de-programming myself from the traditional way that I was taught? (Although maybe it won't be so hard since I feel like I've forgotten quite a bit)
From a second-level comment, which has already received some helpful replies:
that includes some Frequently Asked Questions articles about learning mathematics for deeper understanding. The FAQ article "Problems versus Exercises"
points to writings by various mathematicians, including the book Numbers and Geometry by John Stillwell, about how to appreciate mathematics as a deep, connected subject.
The submitted article mentioned "Numerous studies over the past thirty years have shown that when people of any age and any ability level are faced with mathematical challenges that arise naturally in a real-world context that has meaning for them, and where the outcome directly matters to them, they rapidly achieve a high level of competence. How high? Typically 98 percent, that's how high. I describe some of those studies in my book The Math Gene (Basic Books, 2000)." The most striking example of this that I remember from a news report was a Wall Street Journal series in the 1990s that followed two young men in an inner city ghetto, one who was a good high school student and the other who was a street criminal. The street criminal usually skipped high school, but happened to show up the day students could take one of the major standardized tests (probably the PSAT, if I remember correctly). The street criminal, who sold illegal drugs among other activities, scored just as well on the test as the more regularly attending student who had learned most of his mathematics from school lessons. That's a rather stark illustration of what's missing in school lessons for children who don't have an outside-of-school environment for learning mathematics.
The article also says that many students say, "You have to be willing to accept that sometimes things don't look like - they don't see that you should do them. Like they have a point. But you have to accept them." I wonder how that relates to the quotation attributed to John von Neumann,
I experience math (and programming) quite differently than learning a language or painting: Once I grasp a concept, I can use it. Before that, it's mostly useless to me.
I ask, because when I studied mathematics in school, I had a drive to understand the general principles first before I launched into working on my homework, while some of my classmates were successful--at least in the context of school--by working on the homework and DEVELOPING some level of understanding as they tried to figure out answers for the homework. (I was in a "tracked" mathematics class, taking algebra in eighth grade in an era when most Americans took algebra in tenth grade, if at all, and most of my classmates had parents who were engineers or medical doctors and could ask their parents for help at home if the school lessons were confusing, as they often were.) I also have a very strongly visual approach to grappling with mathematical problems. So when I first learned algebra, which was presented to me as a bunch of "Do this to the equation, and then do this" with little rationale, I found that very dissatisfying. Later in the school year, we learned about coordinate graphing of systems of equations in the Cartesian plane, and I remember thinking, "Why didn't you tell me this in the first place?" For historical reasons, and perhaps for reasons of what most learners consider most easy, usually purely procedural algebra for solving systems of two equations in two unknowns has been taught in school before graphing systems of equations in the coordinate plane. But for some learners, it would be easier and more accessible to reverse that order. What do you think about the issue of students working first according to instructions, to DEVELOP understanding a la the von Neumann quotation, versus getting the "big picture," perhaps explicitly visually, before working on problems.
I'll comment also that the approach taken to learning mathematics in school in most of the newly industrialized countries of east Asia and southeast Asia is plainly superior to the United States approach for at least two reasons:
1) the school textbooks in those countries explicitly encourage students to THINK about why a procedure will or will not work, and about how many different ways there might be to solve a problem, and
2) the school textbooks show multiple representations of most mathematical concepts, building from "concrete to pictorial to abstract" as in the Singapore Primary Mathematics series
I had a lot of trouble learning to solve 2 equations in 2 variables because I did not see the point. Given any word problem that you were supposed to solve that way, I could solve it in my head. It wasn't until I was shown that I couldn't solve 3 equations in 3 variables that I realized that I needed to learn the boring way.
Secondly the single most useful thing that I did in school was try to generate a table of how likely it was to get various dice rolls when you rolled 4 6-sided dice and took the top 3. I learned a lot from that, and that sparked my interest in math.
Thirdly my biggest complaint about the way we teach stuff is that we present matrices and matrix multiplication with no context. It makes no sense to people. But if you know what a linear function, and realize that a matrix is just a way to write one down, then matrix multiplication turns out to be just function composition.
Just think how surprising the associative law is for matrix multiplication. I remember sitting there thinking, "How on Earth did anyone think it up, and see the associative law?" It becomes something you memorize because it makes no sense.
But the associative law always holds for function composition. Given three functions f, g, h and a thing they act on v, then by definition:
((f o g) o h)(v) = (f o g)(h(v)) = f(g(h(v))) = f((g o h)(v)) = (f o (g o h))(v)
Since matrix multiplication is just a way to write out function composition for a certain class of functions, it likewise must follow the associative law. THAT is how they thought it up!
I can hear the complaints already. "Oh, but this is too abstract for college students, they can't possibly understand this approach!" Bull. They can, and they do if you have the guts to present it this way. I've done it, with success.
Start with linear transformation F from vector space V to vector space W, each of which has a basis (v1, ..., vn) and (w1, ..., wn). The matrix M representing F in that pair of bases is (F(v1) F(v2) ... F(vn)) where each vector is written as a column.
Given those bases it is easy to demonstrate that every linear function can be uniquely represented that way.
Thanks to the properties of linearity, it is easy to demonstrate that the special case of matrix multiplication of a matrix against a column is the same as applying that linear function to the corresponding vector. Furthermore you can demonstrate that given the basis and the matrix, you have actually defined a linear function. (Therefore completing the demonstration that matrices are a notation for linear functions, and linear functions are what matrices represent.)
With that in mind the matrix representing (F o G) is going to be ((F o G)(v1) ... (F o G)(vn)). And when you unwind that definition you find that function composition turns into matrix multiplication. (As long as all of the bases match up of course, don't forget them!)
At this point you now have a rule for matrix multiplication. Thanks to the correspondence to linear functions, you can derive all of its algebraic properties (including associativity) from the corresponding properties of linear functions.
Incidentally by keeping track of the role of the basis throughout the presentation, you make it much easier to work out change of basis matrices later. Which has a lot of potential to be confusing because they work out to be the inverse of what you'd naively guess them to be. For instance if you rotate your basis 30 degrees clockwise, the change of basis matrix you get is a rotation 30 degrees counter-clockwise. (This happens for the same reason that while you spin clockwise, it looks to you like the world is spinning counter-clockwise.)
So how do you get the change of basis matrix? Well, go back to the definition. Make your function be the identity (everything remains the same, and then you just write out a matrix which has each column being, in the new basis, the coordinates of the basis vectors for the old basis.
Now an exercise to demonstrate to yourself that you really understood this. Let V be the vector space of polynomials of degree at most 2, and W be the vector space of polynomials of degree at most 1. Let F be the linear function called "differentiation". Start with a coordinate system on V which is just (p(0), p(1), p(2)) and a corresponding coordinate system on W which is just (p(0), p(1)). In that pair of coordinate systems, what matrix represents F?
If you can figure that out, you probably understood the whole thing. If not, well...
(Big hint. There is a different pair of coordinate systems in which you can easily write down the answer. Use that fact...)
Thanks for the detailed explanation. I was subconsciously thinking "Matrix multiplication is this weird operation, which happens to be isomorphic to function composition in the space of linear transformations." rather than "Function composition, when functions are represented as matrices, is called matrix multiplication."
“… try to generate a table of how likely it was to get various dice rolls when you rolled 4 6-sided dice and took the top 3.”
I felt the urge to code this. Here is the result: https://gist.github.com/2899137. It doesn’t tell you “likelihood of various dice rolls”; it can either print out the rolls for each trial or tell you how common each of the 6 numbers were in all the rolls.
That’s true. But I thought doing it by hand would require writing a tediously large table because you have 6^3 possible roll results to give the probability of, if you were actually going to write the “likelihood of various dice rolls”. I suppose the appropriate compromise is a symbolic manipulation program like Mathematica, which can work with exact numbers easily while automating the creation of the table. (If anyone can explain the problem, it would be great if they could link to a document demonstrating the solution on somewhere like http://www.mathics.net/ .) Or is there an easier, simpler way to solve this by hand?
Great comment -- chiming in because intuitive math (or the lack thereof) is a hot-button issue for me :).
I really dislike the von Neumann quote "Young man, in mathematics you don't understand things. You just get used to them."
I know what he's saying (there are some concepts you just need to internalize), but taken at face value it implies you stop looking for insights once you've "gotten used to it" (vs really grokking it).
I think learning is a spiral of theory & practice, i.e., present some principles, explain with examples, present more principles, explain with deeper examples, and so on.
Shameless plug, but check out this article on imaginary numbers:
I try to explain imaginaries by starting with negative numbers (something we're familiar with, but was counter intuitive at the time) and building up with examples (3 cows - 4 cows is "absurd", right? sqrt(-1) is "absurd", right?).
I don't think you can just define "i" as sqrt(-1) and give a bunch of problems, or talk about abstract visualizations for pages with any meaty examples: it's an interleaving spiral of both.
Square root is a fundamentally geometric idea: you need to have a concept of area before you can find the ratio of an area to the side of a square. Much better than talking about what is or isn’t absurd in terms of pure symbols (like √-1 or whatever) is to give some geometric motivation of the definition of area in terms of orthogonality, which brings in the link to rotation, and finally results in complex numbers. By embedding complex numbers in so-called “geometric algebra”, you put them in their proper coherent and comprehensible context. http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
Can you recommend an introduction to geometric algebra & calculus? (preferably online) I've quickly read some but they were vague and in general quite unlike other mathematical texts. For example the document you linked to doesn't actually define the operations it uses. You can sort of conjure up a definition by piecing various statements together (by using the laws plus the operations' behavior on basis elements), but surely there must be a clearer introduction available? Other introductions were just plain wrong, for example one claimed that the curl of the curl is always 0. All that I've seen were strongly trying to persuade the reader of the awesomeness of GA rather than teaching it.
I don't think von Neumann was saying "some concepts you just need to internalize" at all; his idea was more along the lines of
"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" (Jerry Bona)
In other words, intuitive notions of "understanding" aren't necessarily useful when dealing with "pure" mathematical ideas that lack real-world antecedents. (N.B.: the axiom of choice, the well-ordering principle, and Zorn's lemma are equivalent)
Thank you, I really love this kind of thing (former physics major here who sadly, I suppose, went through college never really peering deep into the meaning of tools I used on a daily basis).
An anecdote from my father who helped 'slow' kids learn.
One time he dealt with a kid who struggled with maths. But he happened to know that this same child went to the horse races with his Dad and could calculate the pay-off from fairly complicated trifecta bets in his head.
Over and over my father found it was only motivation that was missing, not brain-power.
Fairly common story, buddies that were quitting school around 15 couldn't compute percentages, yet they did all kind of crazy calculation on the fly with their own non-regular currency unit system to sell drugs. I couldn't follow any of them.
More than motivation (indeed money is a strong one) they had a purpose and a concrete/tangible object to reason about. You can and will test your ideas otherwise you will experience failure. Nobody make mistakes twice when they sell.
Seriously, it's one of the most insulting Internet-names I know-of and I automatically vote down everything I see by you.
If you do good stuff, I suggest you don't begin every single post sneering at us. If you really need reminding, your name is a clever way of saying "I'm smart, you're dumb".
Maybe I'm really just a crank but in a perhaps illusory fashion I think I'm a moderately serious hn commentator.
And oddly enough, an adult.
And sure, I'm happy to be voted down to oblivion for this but hey, just keep this in mind.
Please, tweak your presentation. It would make totally anonymous and irrelevant me happy. I'm sure that doesn't mean anything.
Would you change yours if I changed mine? My screen name here comes from another site, where it was mandatory to use screen names (I'm actually more accustomed to online interaction with my real name and other people who use their real names), and then I carried it over to another site I heard about on the first site, and then over to here. I chose the screen name on the original site (the Art of Problem Solving site) because literally there a great many participants are young people below the age of majority, and it was useful there to distinguish who was an adult. At the second site where I used this screen name, I was again among users of mixed ages, with many minors participating, and it seemed helpful to keep this same screen name. On both of those online forums, most participants have correctly picked up the implication that "token" is part of my screen name to remove offense from the "adult" part of my screen name, by self-deprecation.
I kept the screen name here (although I could just as well have used my real name) simply by Occam's Razor, not desiring to choose a new screen name in case anyone recognized me from AoPS over here.
I automatically vote down everything I see by you.
I had no idea I have such a persistent fan club. I upvote based on the content of comments rather than on the screen name of the person commenting. I'm happy to hear from other users here any time they think I have given offense. I beg your pardon for any offense I have inadvertently caused by my choice of screen name here back 1300 days ago (just shortly before you came on board, it appears). I cherish the opportunity to learn from you and from other participants, and look forward to seeing your further comments.
P.S. Previous HN polls about participant ages suggest the age range here is almost entirely above the age of majority but almost entirely below my age. I have commented on the lack of sure-fire assurance that anyone here knows the full details of the age distribution here.
P.P.S. I'm sure there are other examples, but my family is the only example I know of personally of two generations both participating on HN. My adult son is rather busy with his work this summer, but he has posted here from time to time and often exceeds me in average comment karma score, I think.
I hope you don't feel the need to change your user name, I can't see anything remotely offensive, to be honest. Your comments seem very well thought out and respectful also.
Thanks, Joe, I didn't see a way to reply to you off-forum in your user profile, so I'll follow up here. I sent pg the link to the subthread opened by your grandparent comment to your comment here, and asked him if the HN software allows changing a username with preservation of records of user submissions and comments. I told him I'd be happy to change my user name (and suggested a new one) if the HN software makes that possible. The software is not configured that way, pg told me, so I guess after 1301 days here I'm at risk to my reputation with this same user name I started with. On his part, pg also told me he didn't think my user name is a problem for the HN forum, but I appreciate your concern, and I thank you for drawing it to my attention. See you in the threads. (P.S. I looked up some of your previous comments in the same threads where I've commented before, by site-restricted Google search, and I see we share some interests and you have a thoughtful perspective on several issues I care about.)
From a top-level comment:
http://news.ycombinator.com/item?id=4084559
As someone who was taught the 'traditional' way of mathematics, can someone give a few pointers of de-programming myself from the traditional way that I was taught? (Although maybe it won't be so hard since I feel like I've forgotten quite a bit)
From a second-level comment, which has already received some helpful replies:
http://news.ycombinator.com/item?id=4084426
After Vector Calc, I wanted to go back to the fundamentals, to understand instead of remembering.
There is a FAQ page on the Epsilon Camp site
http://www.epsiloncamp.org/FAQ.php
that includes some Frequently Asked Questions articles about learning mathematics for deeper understanding. The FAQ article "Problems versus Exercises"
http://www.epsiloncamp.org/faq/faq_1.php
relates to what kind of work to set for yourself to build deeper understanding, and the FAQ article "Learning Mathematics"
http://www.epsiloncamp.org/faq/faq_3.php
points to writings by various mathematicians, including the book Numbers and Geometry by John Stillwell, about how to appreciate mathematics as a deep, connected subject.
The submitted article mentioned "Numerous studies over the past thirty years have shown that when people of any age and any ability level are faced with mathematical challenges that arise naturally in a real-world context that has meaning for them, and where the outcome directly matters to them, they rapidly achieve a high level of competence. How high? Typically 98 percent, that's how high. I describe some of those studies in my book The Math Gene (Basic Books, 2000)." The most striking example of this that I remember from a news report was a Wall Street Journal series in the 1990s that followed two young men in an inner city ghetto, one who was a good high school student and the other who was a street criminal. The street criminal usually skipped high school, but happened to show up the day students could take one of the major standardized tests (probably the PSAT, if I remember correctly). The street criminal, who sold illegal drugs among other activities, scored just as well on the test as the more regularly attending student who had learned most of his mathematics from school lessons. That's a rather stark illustration of what's missing in school lessons for children who don't have an outside-of-school environment for learning mathematics.
http://www.ams.org/notices/200502/fea-kenschaft.pdf
The article also says that many students say, "You have to be willing to accept that sometimes things don't look like - they don't see that you should do them. Like they have a point. But you have to accept them." I wonder how that relates to the quotation attributed to John von Neumann,
http://en.wikiquote.org/wiki/John_von_Neumann
"Young man, in mathematics you don't understand things. You just get used to them."
And from a third-level comment:
http://news.ycombinator.com/item?id=4084865
I experience math (and programming) quite differently than learning a language or painting: Once I grasp a concept, I can use it. Before that, it's mostly useless to me.
I ask, because when I studied mathematics in school, I had a drive to understand the general principles first before I launched into working on my homework, while some of my classmates were successful--at least in the context of school--by working on the homework and DEVELOPING some level of understanding as they tried to figure out answers for the homework. (I was in a "tracked" mathematics class, taking algebra in eighth grade in an era when most Americans took algebra in tenth grade, if at all, and most of my classmates had parents who were engineers or medical doctors and could ask their parents for help at home if the school lessons were confusing, as they often were.) I also have a very strongly visual approach to grappling with mathematical problems. So when I first learned algebra, which was presented to me as a bunch of "Do this to the equation, and then do this" with little rationale, I found that very dissatisfying. Later in the school year, we learned about coordinate graphing of systems of equations in the Cartesian plane, and I remember thinking, "Why didn't you tell me this in the first place?" For historical reasons, and perhaps for reasons of what most learners consider most easy, usually purely procedural algebra for solving systems of two equations in two unknowns has been taught in school before graphing systems of equations in the coordinate plane. But for some learners, it would be easier and more accessible to reverse that order. What do you think about the issue of students working first according to instructions, to DEVELOP understanding a la the von Neumann quotation, versus getting the "big picture," perhaps explicitly visually, before working on problems.
I'll comment also that the approach taken to learning mathematics in school in most of the newly industrialized countries of east Asia and southeast Asia is plainly superior to the United States approach for at least two reasons:
1) the school textbooks in those countries explicitly encourage students to THINK about why a procedure will or will not work, and about how many different ways there might be to solve a problem, and
2) the school textbooks show multiple representations of most mathematical concepts, building from "concrete to pictorial to abstract" as in the Singapore Primary Mathematics series
http://www.singaporemath.com/Primary_Mathematics_US_Ed_s/39....
and the follow-up New Elementary Mathematics series
http://www.singaporemath.com/New_Elementary_Math_s/47.htm
which interleave arithmetic, number theory, geometry, and algebra in increasing depth and interconnection throughout all grade levels.