mathwonk
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Posts posted by mathwonk
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a quick way to check this is wrong is to plug in s = -2 on both sides. you get zero on the right side where there is afactor of (s+2), but you don't get zero on the left, you get 4, so to make it work you have to subtract off another 4 on the left.
edit: In my opinion, this is probably the most important principle in algebra, called the "root - factor" theorem. I.e. given any polynomial f(x), the linear term (x-a) is a factor of f(x) if and only if x=a is a root, i.e. iff f(a) = 0.
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wow this may be my favorite thread here! i am a professional mathematician but i also get stressed by boring calculations and difficult reasoning, even though i say i love math. i guess i really love the parts that are easy for me and that i am good at, but i have learned to work a bit every so often at the parts that are harder for me. so i could like kevins moms post as well as opposite ones.
my first idea was to ask the kiddo what day of the he was born on, and gradually help him solve that problem by noticing that after 365 days the week day changes only by one, so that if his birthday was on tuesday in 2014 then it was on wednesday in 2015. then of course there is the wrinkle of leap years, where it changes by 2. this is an introduction to the beautiful and useful math idea of casting out nines, or in this case, casting out sevens, and in math - speak: "modular arithmetic"
i also can't help being gobsmacked by how much work you guys put into your teaching, your kids are soooooooo lucky.
finally, i always found harold jacobs' books just intrinsically fun treatments of algebra and geometry. i mean when about 1/5 of the book is taken up by funny cartoons and photos, when i open it it for the first few zillion times i just looked and laughed at the cartoons. and i am, as i said, a professional math geek. i still remember the picture and story about the world's biggest ball of string, that he uses to make the volume of a sphere interesting and fun. in fact all the topics are introduced cleverly like that. and there are plenty of easy as well as a few challenging problems in each chapter.
ok, this is a great thread indeed. when i teach a young person math i just search until i find a topic the kid likes and run with that. e.g. when i had a student who wanted to learn calciulus but really preferred physics, i searched online until i found discussions of how archimedes used a balance beam to find volumes and i learned that and then taught it to him. even though he was brilliant he still got bored just plowing through even an excellent book. one day he said "can we talk about something other than calculus" and brought up a conjecture he had made about fibonacci numbers, so we discussed that for several days, eventually proving it! later when he applied to a prestigious program and they wanted to know "research" what he had done he used that project as his example.
thanks for this discussion and the spirit behind it.
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thank you, i'll check that out when i have a minute.
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This discussion of the usefulness of mental math reminds me of an alzheimers test I heard about recently, where they ask you to start at 100 and subtract down by 7's successively. It dawned on me to thwart it by repeatedly adding 3 and subtracting 10.
I agree conceptual understanding of numbers and grouping is useful, I am just trying to come up with strategies to help a kid who is struggling with it, and all I have thought of is either to preface it by computational practice, or to teach grouping in more concrete examples. For instance, do you think my coca bottle grouping or British money examples could help anyone learn the idea of "grouping" numbers? Can you suggest some more such examples of real life grouping? I guess in the military we have squads and platoons and companies and regiments and armies...
My usual approach when someone does not get something is to back up, make things easier and then gradually come back at the original topic more gently. And I have also changed my viewpoint a lot as time has gone by. I used to always teach everything in full detail, with all theorems completely proven, partly because I myself enjoyed that more and learned from it. But sometimes my students complained they came out a little weak on using the material in practice. My idea now of conceptual understanding is less rigorous proof and more of an informal explanation that still reveals the reasons behind a phenomenon, as I think you referred to in your first sentence. Like that quadratic formula derivation by Diophantus that I like so much. (I also read Riemann's own proof of the Riemann-Roch theorem in his [and his student Roch's] works, because I wanted to know why it was true, and I found he makes it very clear.)
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"Pretty sure that was Fermat, not Galois."
thank you! corrected. I was off also in time by some 200 years! Galois lived in the 1800's, and the first new case of Fermat's theorem (after Fermat's n=4?), i.e. for n=3, was due I believe to Euler in the 1700's.
This also reminds me of how old the math is we teach high schoolers and younger. The quadratic equation seems to go back at least 1700 years, probably more. Euclidean geometry in a form more sophisticated than we teach goes back over 2000 years.
The common core does suggest using the basic identity above to produce Pythagorean triples, but does not classify all of them, nor use it to deduce the case n=4 of Fermat's last theorem which Fermat himself knew. I haven't checked to see what they say about irrational numbers and their approximation by rationals, again treated geometrically in Euclid. But basically I have thought of nothing yet that we teach that was even discovered in the last thousand years. Maybe complex numbers, are they covered?
Ok, I see the core mentions complex numbers and also vectors and matrices it seems, but I cannot find how complex numbers are defined. It just says students should know they exist, but what it means for them to "exist" is not explained. (They should probably be defined as ordered pairs of real numbers.) It also says they should know a few basic facts about irrational numbers, but I cannot find the idea that even the existence of irrational numbers requires proof, already given by Euclid in the case of sqrt(2). In particular I cannot find any discussion of the meaning of "real" numbers. I also did not find any mention of the key "rational roots" theorem in algebra, an essential tool for finding rational solutions of higher degree p[olynomial equations. In geometry it says students should be able to prove various theorems about triangles and circles, but it does not say what axioms are to be given, without which "proof" is meaningless. It does say one should use geometric transformations such as rigid motions and similarities, but it is not clear to me here either just what precise definitions of these are assumed. In particular a notion of length seems to be assumed, which requires a precise definition of real numbers, that I did not find. The principle of similarity for triangles is to be proved, but it is not stated how this is to be done. An arithmetic proof requires precise understanding of approximation of real ratios by rational ones, and the geometric proof in Euclid uses the theory of area. Perhaps they intend a transformation approach, assuming some properties of similarity transformations. There are some nice topics on volume included like deriving the volume formula for a sphere from Cavalieri's principle, but no mention that this is due not to Cavalieri but much earlier to Archimedes. For some reason the requirement is only to give an "informal argument", whereas the assumption of Cavalieri's principle allows a rigorous proof using Pythagoras. This is very clear in Harold Jacobs' Geometry, where I first learned it. The laws of sines and cosines are mentioned but not that the latter occurs in Euclid in a simple geometric form, as a natural generalization of Pythagoras. So to me there are a lot of good topics in the core standards, but many good ones omitted, and no completely precise description of how the topics are to be covered.
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After consulting the book by Lagrange where I learned Diophantus' solution of the quadratic equation, and the book of Euler, I have edited/corrected the discussion in post #22 above. It turns out Euler does complete the square, but the original solution by Diophantus, in the 3rd century AD, seems to me still the simplest one. Indeed I am reminded that the great modern algebraist Fermat [thanks purpleowl!] used the book of Diophantus, and wrote the famous marginalia "I have found a truly marvellous proof.... but the margin is too small to hold it" in his copy. (Referring to the assertion that X^n + Y^n = Z^n has no positive integer solutions X,Y,Z for any n > 2.)
A quick search reveals that the method of completing the square occurs first(?) in the work of the famous Arab mathematician from Baghdad, Al - Khwarizmi, in the 9th century.
I just want to remind that in the 18th century elementary math books were written by the greatest mathematicians, i.e. Euler and Lagrange, and for some purposes those books are still the best.
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By the way if you are teaching positional notation and grouping, I encourage you to make up your own real life concrete examples. One I came up with, while teaching Professor Beckmann's class, involves coca cola bottles. I.e. they are organized into cartons (sixpacks), then cases of 4 sixpacks, then maybe flats holding 4 cases each, then maybe trucks holding say 48 flats. Ask a kid to compute how many trucks, flats, cases, and cartons would be used to store say 10,000 bottles.
Or if you know about British money, it gives groupings into pennies, or I guess pence, shillings, pounds, and I don't know what else. Ask a kid to say how to rearrange say 1,000 pence into shillings and pounds. Be original!
After they get the idea, then the abstract grouping of 1's. 10's. 100's, .... may make more sense. I recommend in teaching these ideas to just use your own good sense. To teach from, and for, real creative understanding, not according to any strict set of marching orders from on high.
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Going out on a limb here, I want to try to give Euler's explanation of the cubic formula, since again the idea is that once you realize you should write the solution as a sum of two other numbers, the formula plops right out. I think you will not find this explanation in any high school math book today anywhere, unless someone somewhere uses Euler's great opus, Elements of Algebra, written for his butler about 250 years ago. (I have taught this method to bright 10 year olds.)
CUBIC FORMULA:
To solve a cubic equation, we start with a simplified one of form X^3 -3bX - c = 0, and again assume we want to find X as a sum X = (p+q). Plugging in gives (p+q)^3 = 3b(p+q) + c, and expanding gives p^3 + 3p^2q + 3pq^2 + q^3 = p^3 + q^3 + 3pq(p+q) =
3b(p+q) + c, and for this to hold means that pq = b, and c = p^3+q^3. Cubing the first of these gives p^3q^3 = b^3, and p^3+q^3 = c. Since we know b and c, we know both the sum and the product of the cubes p^3 and q^3. Can we find p^3 and q^3 from this? If so, then we could take cube roots and find p and q, and finally add them and get our root X = p+q.
Just recall in a quadratic equation of form X^2 - BX + C, that B and C are precisely the sum and product of the desired roots, and we can find those roots from B and C. I.e. we can find any two numbers when we know their sum and product, by solving a quadratic.
Since p^3+q^3 = c and p^3q^3 = b^3, the numbers p^3 and q^3, which can be used to give a solution X = p+q of our cubic, are solutions of the quadratic equation
t^2 -ct + b^3 = 0
e.g. to solve X^3 = 9X + 28, we have b = 3, c = 28, and so we solve t^2 -28t + 27 = 0. Here B^2-4C = 676, whose square root is 26, so we get t = (1/2)( 28 ± 26) = {27, 1}, for p^3 and q^3, so p,q are 1 and 3, and hence X=1+3 = 4 solves the cubic. Of course if we know about complex numbers, there are two more cube roots of 1 and 27, and we get two more complex roots. (Only two more because b = pq, so we must always have q = b/p, i.e. the choice of the cube root q is determined by the choice of p.)
Finally, one can translate the variable in any cubic equation to change it into one with zero quadratic term, so this process works in general. -
Uh oh. I downloaded the common core standards and read them. What a depressing read. It seems like a lot of blah blah and it is not really clear to me what it exactly means. I would have preferred a nice clearly written book actually explaining the math in plain language, over a tract talking abstractly about desirable math skills.
[edited] When I did find an explicit discussion of a topic, like the quadratic formula, it said to be able to derive it in the standard way from completing the square. This is one of my pet peeves since when I was a high school student that particular derivation was hard for me to follow and completely unmotivated. Decades later as a teacher I read algebra books written by some of the greats, like Diophantus, and Lagrange, they don't do it that way at all, but make the idea much clearer as I hope to explain next: the basic idea is you are given the sum and product of two numbers and you want the numbers. From the identity:
(a+b)^2 - 4ab = (a-b)^2, you see that the sum and product of a and b, determines also the difference a-b. Now if you know a+b and also a-b, you can find both a and b. That's all there is to it. Try and dig that insight out of the common core, or any modern algebra book. In more detail:
QUADRATIC FORMULA:
To solve a quadratic equation X^2 -bX + c = 0, first assume the solutions are X=r and X=s, and note that then (by the root-factor theorem), the equation factors as X^2 -bX + c = (X-r)(X-s) = X^2 -(r+s)X + rs, so that we must have b = r+s and c = rs. Thus we know in particular the sum b of the roots. Hence if we only knew the difference, say d = r-s, of the roots, we would be done, since then we would have 2r = b+d and 2s = b-d, so we could get r and s by dividing by 2.
So we assume the roots are expressed as a sum, namely r = b/2 + d/2, and s = b/2 - d/2. Plugging in X = (b+d)/2 and eliminating denominators we are trying to solve (b+d)^2 - 2b(b+d) + 4c, for d, since we already know b and c.
Expanding gives b^2 + 2bd + d^2 -2b^2 -2bd + 4c = d^2 - b^2 + 4c = 0, or
d^2 = b^2 - 4c. Thus 2r = b + sqrt(b^2-4c) and 2s = b - sqrt(b^2-4c), the usual formula, if you notice we had a minus sign in the linear term of our original equation.
Notice how the reason for the goofy square root expression (b^2-4c), becomes completely clear here as opposed to the completing the square method. Note also that the basic idea here is already given in the common core standards, in a more complicated form, as a stand alone trick for producing Pythagorean triples, i.e. it is based in the identity (a+b)^2 - 4ab = (a-b)^2, (which they give with a = x^2 and b = y^2). So the common core contains the germ of this method but does not make the connection between it and the quadratic formula, as the great ancient mathematicians did.
[edit: I just looked back in the book by Lagrange on Elementary Mathematics, and found his account of the derivation by Diophantus, much simpler than mine above. Namely he says indeed that in the equation X^2 - bX + c = 0, that b is the sum of the two roots, so we only need to find their difference d, since then the roots themselves would be (b+d)/2 and (b-d)/2. However then he observes that since their product is c, we have [(b+d)/2][(b-d)/2] = c, i.e. [b^2-d^2]/4 = c, or d^2 = b^2-4c. hence d = sqrt(b^2-4c). Rather shorter and simpler then my version, and makes more clear the use of both coefficients b and c, and their meanings, as the sum and product of the roots sought. It always helps to consult the great authors.]
[edit: Let me admit however that it is a useful skill to know how to derive the formula by completing the square, as in the common core. Indeed, after checking I found this method in Euler, contrary to what I said yesterday. I just advocate also giving the more transparent method here for the benefit of those to whom it may be found preferable. The point is not just to make one certain approved presentation and quit, but to try different strategies until the child "gets" it.]
So someone who knows more and tries to make these connections runs the risk of being out of compliance with the core. This is why those of us who think we understand something about the subject, or at least think of it as an open topic for experimentation, have trouble when we are told it must be done a certain way. Now I know some of the authors of those standards personally and they are well trained research mathematicians, so I don't know why the core standards read as they do, without any visible benefit of some of these historical insights. Sometimes there is a power struggle among the various authors as to what should be there. It is also possoble I would agree completely with how one of them actually implements the standards if I saw them at work. I have noticed people argue about teaching in the abstract, mostly out of poor ability to communicatre what they mean. In practice they often actually do the same things in class. But if I were a teacher and were handed this document, I would not be a happy camper. Apologies to my friends who worked so hard to help produce it.
Actually an example of this puzzling dichotomy, is that my friend who participated in writing the common core standards which I find disheartening, wrote her own book on teaching math which I recommend to everyone as absolutely wonderful:
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I won't go on forever about this, but I want to give an example where computation is universally agreed to be more important than understanding why. In my field of algebraic geometry the most famous and important theorem is the "Riemann-Roch" theorem. Every algebraic geometry student and researcher knows the statement of this theorem, but few know the proof. Here is a quote from the standard graduate text on the subject by Robin Hartshorne, then a Professor at Berkeley:
"If a reader is willing to accept the statement of the Riemann-Roch theorem, he can read this chapter at a much earlier stage of his study of algebraic geometry. That may not be a bad idea, pedagogically, because in that way he will see some applications of the general theory, and in particular will gain some respect for the significance of the ...theorem, In contrast, the proof... is not very enlightening."
As a matter of fact, I audited the course from this professor at Harvard in the 1960's that preceded his writing the book, and he taught it in exactly that way, stating and using the theorem, but never proving it at all. As another famous example, Archimedes used ideas of integral calculus to make correct and useful calculations that were not theoretically justifed clearly for centuries, as did Newton.
Forgive me, the trouble with philosophy of teaching is it is undecidable, and the discussions can become lengthy and sometimes argumentative. I certainly do not insist on any of my own opinions, and I greatly appreciate all the help I have received here, thank you!
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Thank you! I went to that link and found explicit descriptions of methods to be used to teach addition and subtraction at each grade level. No wonder there is controversy over this stuff. I had no idea. It is very hard for adults who learned one way, and who have used that knowledge profitably for decades, in my case even at the international mathematics research level, to accept someone's theory that we should teach it another way. Of course I could always be quite wrong, but I do have a lifetime of experience that to me recommends allowing a good deal more flexibility in approaches.
I did find there the statement that conceptual understanding and computational facility are both important, which I agree with. I would suggest however after a very long career of practice, with thousands of students of almost every level of ability, that for many students, indeed most, computational facility should usually come first. I would go out on a limb and suggest that in real life, even research level math, being able to compute accurately, even without complete understanding, is more useful than understanding a principle that one cannot carry out computationally. I wonder how many practicing engineers, who use calculus and differential equations every day, remember the theory of differentiation and existence and uniqueness of solutions of d.e's? I guess I could ask my brother the EE, but I already know he has always been very dismissive of overly theoretical math.
I confess too that having been a lazy student, who contented myself with abstract formulations of many mathematical ideas, without taking the time and effort to compute examples, I was always hampered in my research and highly dependent on other stronger mathematicians for help.
Anyway I think I see the difficulty, namely telling people not only what result is desired but exactly how to go about it, without flexible accomodation of different learning styles, is asking for problems. Another challenge is that it takes more teacher training to produce teachers who can teach in a variety of ways with different children, and who are not dependent on the specific method they are given. Good luck to us all!
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This example will perhaps seem goofy, but it is drawn from an actual experience in my life of training in math. My university teachers insisted on the conceptual approach to every topic, and at one point I just lost contact with reality. Here is the example, still burned into my brain after more than 50 years, of a subject I could not deal with until I finally saw the computational approach years later. It is called "exterior algebra". The first approach is the one I was taught. Afterwards I could not do the exercise. In the second approach I easily did it correctly. See which one you prefer. i encourage tackling approach II first. i believe everyone here can do it with a little effort. approach I , i doubt any of us can do. if you can, contact me for referral to a grad math program with free ride support.
I. Abstract approach:
Let V be a finite dimensional vector space spanned by the linear forms x1,...,xn, and consider the symmetric algebra S(V) they generate. In that algebra consider the ideal J of finite sums of all products involving a repeated factor. Define an equivalence relation on the algebra such that two elements are equivalent if and only if their difference belongs to that ideal and denote the set of equivalence classes by S/J = E(V), called the “exterior algebra†on the space V. Then this algebra is naturally graded from degree 0 to n, according to how many linear factors occur in a representative for a given equivalence class of products.
Ex: compute the dimension of the graded piece of degree n.
II. Concrete approach ("wedge" multiplication):
Or, just consider all sums of form adx + bdy + cdz, for all numbers a,b,c, and multiply them with the usual rules except require that dx^dy = - dy^dx, dx^dz = -dz^dx, dy^dz = -dz^dy, and dx^dx = dy^dy = dz^dz = 0. Thus also dx^dx^dz = 0, etc...E.g. (2dx + 5dz)^(4dy - 6dz) = 8dx^dy -12dx^dz +20dz^dy -30dz^dz = 8dx^dy -12dx^dz -20dy^dz.
Ex: show that any triple product, e.g. (3dx-2dy+4dz)^(dz-2dy+7dx)^(3dy+6dx-2dz), can be rewritten as a numerical multiple of dx^dy^dz. -
Thanks for all the feedback. It sounds as if most of your kids have had lots more preparation for this mental math than the child in question, who is still counting on her fingers. I guess I myself, even after a roughly 50 year career of research and teaching pure math at all levels from second grade through graduate school, and having learned the traditional way myself, tend to go with the "whatever works for the particular child" approach. I.e. no matter what theory I brought to the classroom in all those years, there was always some child for whom it did not work. I guess the only general principle I have is "start where they are". I just assumed that "common core" referred to the body of material that should be known, not the method of teaching it. Thanks again!
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Gosh, I cannot think of any reason to prevent kids from accessing the benefits of traditional positional notation, developed over centuries. I mean in second grade, you learn it the easy way, then later maybe you branch out. But what do i know? I just live here.
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A mother told my wife, who is a math tutor, that her 2nd grade daughter is struggling to add 2 and 3 digit numbers like 23+48 = ? without arranging them in columns over and under each other in the usual way, but that she is expected to do so at school because "it's common core". I never heard of this and I taught under the supervision of a well known expert in elementary math ed who wrote some of the common core. Has anyone else encountered this from schools?
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Comment: I have a friend who co - wrote a standard book for an advanced undergraduate class. I assume he does use his own book in the class when he teaches it, because I think he believes it to be the best book for implementing his vision of how the subject should be taught. He also told me he has never received over $1500 in royalties in any one year from this book, and that next time he thinks it would make more sense just to give the unpolished book away rather than work hard enough on it to have it published. In particular, he specifically discouraged me from bothering to write up any of my own sets of notes as a book. Since I just checked on amazon that his book sells new for over $175, I would deduce that most of the profits go to the publisher.
Another friend who wrote a short entry level paperback book that sells for under $20 new said I believe that he receives about a nickel per sold copy, or it may have been another author friend whose book sold at the time for $5, so 1%. The one acquaintance who seems to have made money off his book, wrote a freshman level book that was quite well written, well motivated, packed with interesting examples and applications, and full of challenging and instructive exercises. He spent years writing it and then lots more time revising it to accomodate the publisher and the students who commented on it. Basically the revisions were devoted to making it easier to read (possibly simplifying the language), and added new and easier problems, more computer oriented ones, and even removed some of the deeper scientific applications. Hence although I myself preferred the first edition, it became and stayed popular for many years and sold widely. Needless to say the prices of the newer editions are far higher than for the old ones, say over $100 versus under $5. Had he not revised it regularly in accord with the publisher's wishes, the book would long ago have gone out of print and his income from it stopped. He himself was not responsible for deciding whether his book would be used even in his own classroom, that duty belonging to a committee he did not serve on, and of course by far most of the sales were at other schools.
My advice to a student in a class using a book like this and wanting to save money, would be to use an old edition for learning and copy the problem sets from someone else or a library copy of the new edition, unless of course the student cannot read the more sophisticated prose in the earlier edition. In such cases however there are usually so many different editions that a cheap version of a recently rewritten one should be available.
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in regard to recommended book lists, this link to the preface of a book celebrating excellent math writing includes a list of books on math and math education that i had never heard of, but that seem to be excellent. see especially page xxiii.
heres maybe a better link for that introduction:
http://press.princeton.edu/chapters/i9821.pdf
i looked up one of them, math from three to seven, by zvonkin, and it did seem interesting, the account of a professional research mathematician who ran an experimental math school in his apartment for preschoolers.
in general the russian tradition in teaching math through interesting problems seems wonderful.
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boy things have changed. regentrude, that is an amazing story.
in my day, they did not reissue the same book every year or two with trivial changes just to jack up the price. Or maybe some did, but my professors did not use those books. If you compare popular calc books llike the one by George B Thomas, it has gone through lots of editions and the new ones are expensive and the old ones are not. But the books they used in honors classes at Harvard, like the books by Apostol, and Courant, have not been rewritten essentially at all. They are now very expensive however, as outlandish academic book pricing has become normal practice by greedy publishers. I bought my used copy of Courant for about $6 in 1960 and it can be found now for well over $100, and I saw an ad today for Apostol for over $200. Nonetheless when I tried to sell my own copies of my books to a used bookstore charging these prices they offered me $4 or less for my books, even in new condition for some books I had not yet read, and most of the time nothing at all.
interestingly here are copies of courant for a low price, maybe not quite kosher:
here is a reasonably priced collection of copies of spivak's calculus:
maybe it's the publisher of apostol that is out of line, who is that? hmmm... wiley.
but this exercise shows one does have options, i.e. use courant or spivak instead of apostol. (to boost my own university's math major, i donated copies of all these books last year to the undergrad math tea room library.)
Some business people discovered that we academics do value highly good books and just raised the prices enormously in recent decades. Research level monographs of small distribution from Springer used to sell for $4 that now go for way over $100, for thin paperbacks.
A thin paperback I used in an intro to proofs course , worth maybe $15, now goes for $140. The ownership of Springer publishing has apparently passed from a man dedicated to being of service to academics, to the hands of someone bent only on gouging money from them.
But as we have said before, there are usually good used copies of excellent books out there that can be viable alternatives to the exorbitant ones. You do have to communicate to the prof who got his copy free what is going on, and in my experience most profs will make an effort to accomodate you. Of course some of us are tempted to get on this bandwagon and supplement our salaries by publishing our own books and selling them.
There is an argument that a really excellent and unique book deserves a higher than average price, and in the case of the excellent book Calculus, by (my friend) Michael Spivak, who is not in academia as a salaried professor, the sales of his book pay for his living. So we might make a distinction between outstanding books that command a high price by virtue of their quality and demand, versus books that are intellectually worthless but are popular because they are considered easy to read, or are cynically required for a course. Even the second category may receive some argument as to merit of a pedagogical variety. The ones which are reissued in new editions frequently just to make the others go technically out of date are inexcusable. I might mention however that my friends who write such books say that these new editions are demanded by the publisher as a condition not to let their books go out of print.
University libraries are major victims of this price gouging since they try to keep collections of all the needed books. Researchers do the research while on university salary, and submit it to journals for free, or are even charged "page charges" for the privilege. Then the journals publish the research and sell it back to the universities at a high price. Some journals, especially springer and Elsevier I believe, charge so much that there is an organized movement by researchers to boycott them and not submit articles to them. it has over 15,000 signatories including some Fields medalists.
http://thecostofknowledge.com/
Individual researchers are also victimized and charged for access to articles. After years of computers going down and moving from place to place, I do not even have copies of all my own articles that I myself wrote. When I go online to see them at the journal's website I am frequently asked to pay $20 or more for a copy of my own article, from a journal to which I have granted the copyright. Sometimes I am blocked even from posting a copy of my own work on my own website.
here is a link to a springer website asking me to pay about $30 for a copy of my own work:
http://link.springer.com/chapter/10.1007%2FBFb0075005
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This is a sad comment but has a ring of truth. As I probably have recalled before, after I learned to try to accomodate my classes by using a book for homework, I had a student upbraid me for correcting a problem they had presented erroneously, on the grounds that the student had taken the answer straight from an online answer book, so it must be correct. I consulted this answer book, written by a grad student at Princeton, located his error and explained it to the student. I was gobsmacked that a student would not only take work from a cheat sheet, but use that fact as an argument against having it found incorrect.
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This is why it took me a while to understand student comments about my "not following the book" were meant apparently as criticism, when I was trying my best to construct the course in the most useful way possible for that individual class. I had never taken a single class in college where the prof followed a book. Indeed if the prof needed a book to follow, he/she would probably not be teaching the class. Ok, I can think of one class where the math prof followed a book. That was probably the worst math class I ever took in college, but the book was excellent, and has been considered the classic text for many years.
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well its raining here so i'm trolling old threads with comments. this one interests me both as a former professor and student. i have already posted my professor comment so here is my ex student one. i feel like a dodo in the sense of having extinct reactions, but i have a remark about objections to books that are not "used" by the professor. in my day the idea was for the student to use the book by reading it, not for the professor to "use it" by assigning homework from it.
i.e. the textbook list in a course, at least at harvard in the 1960's, was a valuable resource where one could learn what were the absolute best books on a given subject. You bought the book and were supposed to read it and benefit from it even though the professor often never mentioned it at all and never referred to it, and did everything in a completely independent way in class. Having two different presentations of the same material was considered a plus. I still own and treasure the great calculus book by Richard Courant that my professor (John Tate) recommended for us in 1960, even though he never assigned a single problem or reading from it, but created the whole course from scratch for us on the board from his vast expertise.
Indeed it was not unusual for students not enrolled in a class to show up on day one just to get a copy of the reading list, since that was considered one of the most valuable benefits of the class, just to find out what that famous scholar thought you should read to learn the subject. ahhh,... those were the days, but perhaps this still rings a bell somewhere with someone. ... of course books probably did not cost quite as much then.
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Basic advice used to be to read math books actively, i.e. with pencil and paper in hand, working out examples, trying different hypotheses, basically doing rather than just listening. basically one must read them far more slowly than other books. a single sentence in a math book, such as the statement of a significant theorem, can contain challenging and important content worthy of an entire session of study. As a concrete guide, the best teacher i had in grad school in math said one should write 3-5 pages for every page read. In fact he went further and said essentially no one ever learns anything just by reading a math book passively. He said the only exception to this rule he had known in his life was the famous Fields medalist Paul Cohen, and my teacher was himself a brilliant mathematician and famous researcher (Maurice Auslander).
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there are several salient (i love using words I barely understand, but i think that means relevant) features to this discussion. the key one is regentrude's point that zillions of excellent books are available for a pittance at sites like abebooks.
here are a few excellent calculus books from an MIT professor, some for under $10:
but many points are up for grabs: does your student want to learn the material, get a good grade from a particular professor, or something else? and (presumably not so big an issue on this site), does he/she know how to read?
I.e. although the subject matter has not changed in decades, the ability of the average student to read a text has, and mostly for the worse. Also the standards of textbook writing in terms of user friendliness have gone up accordingly, so sometimes there is an excuse for using a newer book that contains the same content, but explains it more clearly. but i digress for those of you dedicated to teaching discerning reading.
i am a big advocate of using good well written scholarly books from past years that cost a fraction of what new books cost. I will go on abebooks and find cheap books and tell my classes to get them. then i will write up problem sets for them. i also write notes for my classes which are a substitute for at least mediocre books, and i give them away. (few students have ever expressed appreciation for this effort. i guess the parents are paying their tab.)
still this is a challenge. i have had students complain that i am not savvy enough to use the most currently beautiful fonts in my notes and complain that makes them hard to read. come on...
bottom line.. there is no reason to spend more than a small am ount to obtain outstanding books to learn any science and math topics. but you need to wise up your somewhat clueless teacher as to what you are being charged for the book he/she is requiring. it's like a doctor who gets some drug for free and suggests it to you and then it costs your insurance $1000 a tube. just talk to them.
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Grade Inflation, Higher and Higher
in The College Board
Posted · Edited by mathwonk
this reminds me of an article in the harvard alumni mag a few years back. they noted that in the 1960's the average undergrad grade at harvard had been maybe a C+, whereas at the time of the article it was maybe an A-. The opinion of current students was that they were just smarter now than we were in the 60's, but average SAT scores had gone down over the same period.
It also depends on which department you survey. In recent times at UGA Athens, I believe the average gpa in the math dept was some kind of C, maybe C-, but in several education departments it was an A or A-, at least for majors.
I have learned today as well that there is a current grade scandal at UNC, which just played in the national title basketball game, concerning an 18 year period in which students, including a much greater proportion of athletes than others, received high grades for classes that did not even meet. So protecting the money associated with athletics is also a motive. Nothing happened yet to Roy Williams at UNC, and the NCAA charges were mainly directed at minor figures, but at UGA a similar scandal involving head coach Jim Herrick some years back resulted in his firing. Sadly, it also resulted in decimating the prowess of the basketball program which is now apparently run honestly but cannot any longer compete with professional programs like that at UNC.
Indeed there was a plagiarism scandal recently even at Harvard involving a large number of athletes including the stars of the mens basketball team. The coach, Tommy Amaker, of the Harvard team has had great success recently for the first tme in Harvard history. Nothing happened to Mr Amaker, and the guilty basketball team stars were allowed to leave school for one year and re - enroll and go back to winning for the basketball team. This seems to contrast with my memory from the 1960's that plagiarism was then an offense resulting in expulsion.