August 2011 - mathfun@mailman.xmission.com

Re: [math-fun] NYTimes: How to Fix Our Math Education
by Henry Baker 29 Aug '11

29 Aug '11

FYI -- Below are some of my favorite quotes on learning. I seem to recall that famed UCLA basketball coach John Wooden said something about banning the word 'teaching' from the language & replacing it with 'learning', but I haven't been able to locate the quote. How is it that little children are so intelligent and men so stupid? It must be education that does it. --Alexandre Dumas Do not train a child to learn by force or harshness; but direct them to it by what amuses their minds, so that you may be better able to discover with accuracy the peculiar bent of the genius of each. --Plato I have never let my schooling interfere with my education. --Mark Twain A child educated only at school is an uneducated child. --George Santayana Tell me and I forget. Show me and I remember. Involve me and I understand. --Confucius Every truth has four corners: as a teacher I give you one corner, and it is for you to find the other three. --Confucius A mind is a fire to be kindled, not a vessel to be filled. --Plutarch Setting an example is not the main means of influencing another, it is the only means. --Albert Einstein Education's purpose is to replace an empty mind with an open one. --Malcom S. Forbes 'Do not go where the path may lead; go instead where there is no path and leave a trail'. --Ralph Waldo Emerson 'Some of us are just less damaged than others.' (by schooling) --Buckminster Fuller 'We can easily forgive a child whom is frightened of the dark; the real tragedy of life is when men are afraid of the light.' --Plato 'Evolution favors the survival of the wisest.' --Jonas Salk 'Remember a dead fish can float downstream but it takes a live one to swim upstream'. --W C Fields 'Children have real understanding only of that which they invent themselves, and each time that we try to teach them too quickly, we keep them from reinventing it themselves'. --Piaget 'I believe babies are born as innovative personalities... But our social processes work to stamp out exploration and questioning.' --Jay Forrestor Prof MIT 'What we want to see the child in pursuit of knowledge, and not knowledge in pursuit of the child.' --G B Shaw 'I am always ready to learn although I do not always like being taught.' --Winston Churchill 'Imagination is more important than knowledge.' --Albert Einstein

4 3

[math-fun] what is this function sqrt(x-2+5/x)?
by Bill Gosper 29 Aug '11

29 Aug '11

Last nite, taking up a chase I abandoned before he was born, Julian derived "the q-cosine of pi/5" (p 16, line 4) according to my version of q-trig<http://gosper.org/symsac.pdf>. (He also got the much messier sine. Unfortunately sin^2+cos^2 /=1 in q-land.) From these I found the missing 4th eqn on p 15. To save you some indigestion, my q-sine and q-cosine are just tweaks of theta functions, and the Pi_q are theta constants, q-analogs of Wallis's product. Both of these new formulae contain the same mysterious sqrt(x-2+5/x) that stymied me when it showed up in the quintuple formula. People still ask whether it's a misprint. Note that by line 8 of p15, the sqrt (in this q-trig context) can be replaced by a rational function if we introduce a Pi_q^10. Does this ring any bells? --rwg

1 0

[math-fun] Bracing for hurricane Irene
by David Wilson 28 Aug '11

28 Aug '11

We're right in it's path. :-(

2 1

Re: [math-fun] NYTimes: How to Fix Our Math Education
by Dan Asimov 27 Aug '11

27 Aug '11

At least in the U.S. we have this problem that most math teachers (at least before 9th grade) are non-specialists in math who often dislike math as much as anyone, and this attitude is conveyed from each generation of teachers to the next. --Dan Fred wrote: << An inspirational, committed, capable teacher. Now why didn't someone think of that before? >> Sometimes the brain has a mind of its own.

2 1

[math-fun] NYTimes: How to Fix Our Math Education
by Henry Baker 27 Aug '11

27 Aug '11

FYI -- There may be something to this. Abstraction too early leaves many (most?) students in the dust. Humans are pretty good at generalizing from the specific to the abstract on their own, but they aren't very good about learning in the abstract without any concrete examples. http://www.nytimes.com/2011/08/25/opinion/how-to-fix-our-math-education.html August 24, 2011 How to Fix Our Math Education By SOL GARFUNKEL and DAVID MUMFORD THERE is widespread alarm in the United States about the state of our math education. The anxiety can be traced to the poor performance of American students on various international tests, and it is now embodied in George W. BushÂs No Child Left Behind law, which requires public school students to pass standardized math tests by the year 2014 and punishes their schools or their teachers if they do not. All this worry, however, is based on the assumption that there is a single established body of mathematical skills that everyone needs to know to be prepared for 21st-century careers. This assumption is wrong. The truth is that different sets of math skills are useful for different careers, and our math education should be changed to reflect this fact. Today, American high schools offer a sequence of algebra, geometry, more algebra, pre-calculus and calculus (or a ÂreformÂ version in which these topics are interwoven). This has been codified by the Common Core State Standards, recently adopted by more than 40 states. This highly abstract curriculum is simply not the best way to prepare a vast majority of high school students for life. For instance, how often do most adults encounter a situation in which they need to solve a quadratic equation? Do they need to know what constitutes a Âgroup of transformationsÂ or a Âcomplex numberÂ? Of course professional mathematicians, physicists and engineers need to know all this, but most citizens would be better served by studying how mortgages are priced, how computers are programmed and how the statistical results of a medical trial are to be understood. A math curriculum that focused on real-life problems would still expose students to the abstract tools of mathematics, especially the manipulation of unknown quantities. But there is a world of difference between teaching ÂpureÂ math, with no context, and teaching relevant problems that will lead students to appreciate how a mathematical formula models and clarifies real-world situations. The former is how algebra courses currently proceed Â introducing the mysterious variable x, which many students struggle to understand. By contrast, a contextual approach, in the style of all working scientists, would introduce formulas using abbreviations for simple quantities Â for instance, EinsteinÂs famous equation E=mc2, where E stands for energy, m for mass and c for the speed of light. Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering. In the finance course, students would learn the exponential function, use formulas in spreadsheets and study the budgets of people, companies and governments. In the data course, students would gather their own data sets and learn how, in fields as diverse as sports and medicine, larger samples give better estimates of averages. In the basic engineering course, students would learn the workings of engines, sound waves, TV signals and computers. Science and math were originally discovered together, and they are best learned together now. Traditionalists will object that the standard curriculum teaches valuable abstract reasoning, even if the specific skills acquired are not immediately useful in later life. A generation ago, traditionalists were also arguing that studying Latin, though it had no practical application, helped students develop unique linguistic skills. We believe that studying applied math, like learning living languages, provides both useable knowledge and abstract skills. In math, what we need is Âquantitative literacy,Â the ability to make quantitative connections whenever life requires (as when we are confronted with conflicting medical test results but need to decide whether to undergo a further procedure) and Âmathematical modeling,Â the ability to move practically between everyday problems and mathematical formulations (as when we decide whether it is better to buy or lease a new car). Parents, state education boards and colleges have a real choice. The traditional high school math sequence is not the only road to mathematical competence. It is true that our studentsÂ proficiency, measured by traditional standards, has fallen behind that of other countriesÂ students, but we believe that the best way for the United States to compete globally is to strive for universal quantitative literacy: teaching topics that make sense to all students and can be used by them throughout their lives. It is through real-life applications that mathematics emerged in the past, has flourished for centuries and connects to our culture now. Sol Garfunkel is the executive director of the Consortium for Mathematics and Its Applications. David Mumford is an emeritus professor of mathematics at Brown.

14 18

Re: [math-fun] NYTimes: How to Fix Our Math Education
by Bill Gosper 26 Aug '11

26 Aug '11

Alex Bellos>Perhaps this is a good argument for introducing study of the abacus into the maths curriculum. Calculation with the abacus uses the "visiospatial" hemisphere of the brain, unlike calculation with pen and paper, which uses the "linguistic" hemisphere. Given the links between memory and calculation, if spatial memory is an untapped resource maybe spatial calculation is too. When I was researching this for my book, I failed to find anyone doing serious research into the educational benefits of the abacus. Recently, Michael Frank at Stanford has been looking into it however: http://news.stanford.edu/news/2011/august/abacus-mental-math-080311.html ------- Naah, if there were anything to this, the Chinese would have discovered Pascal's triangle centuries before Pascal, found the value of 2F1[-n,b;c|1] centuries before Vandermonde, and found pi ~ 355/113 a millennium before *Adriaan Anthonisz.* --rwg

2 1

[math-fun] quintic solver safety recall, sin 1 (almost)
by Bill Gosper 26 Aug '11

26 Aug '11

That terse solver sailed right through the examples in Quintic function - Wikipedia, the free encyclopedia<http://en.wikipedia.org/wiki/Quintic_function> until x^5 + x^4 - 4*x^3 - 3*x^2 + 3*x + 1, whose sol'ns are 2*Cos[2*k*π/11], 0<k<11. Watson-Williams's correct solution gives a quartic with four roots all of magnitude (5/Sqrt[11])^5, whereas the terse one finds four *different* roots of that magnitude, and then oviposits when it can't figure out the phase corrections. Fudged manually, the first solution gives Out[859]= Sin[7 Pi/22] == -1/10 + ( (-1)^( 2/5) (11 (-89 - 25 Sqrt[5] - 5 I Sqrt[2 (205 - 89 Sqrt[5])]))^(1/5) - (-1)^( 3/5) (11 (-89 - 25 Sqrt[5] + 5 I Sqrt[2 (205 - 89 Sqrt[5])]))^(1/5) + (-1)^(2/5) (11 (-89 + 25 Sqrt[5] - 5 I Sqrt[2 (205 + 89 Sqrt[5])]))^(1/5) - (-1)^(3/5) (11 (-89 + 25 Sqrt[5] + 5 I Sqrt[2 (205 + 89 Sqrt[5])]))^(1/5))/ (10 2^(2/5)) In[860]:= N[{%[[1]]], %[[2]], Sin[1]}, 22] Out[860]= {0.8412535328311811688618, 0.8412535328311811688618 + 0.*10^-23 I, 0.8414709848078965066525} Titus Piezas, a name I wish I'd learned much earlier, has an apparently much neater rigorous solution http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.2.3197&rep=rep1&ty… which he found by walking through a 22nd degree wall with computer algebra. It will be interesting to try on 2 cos(2k pi/11). His paper cites http://www.math.carleton.ca/~williams/papers/pdf/244.pdf but apparently slightly predates http://www.math.carleton.ca/~williams/papers/pdf/276.pdf He seems to have only one exceptional case, and not even that if you always take a limit where you might divide by 0. --rwg PS, I believe the image WFL is trying to attach is http://gosper.org/flowsnakefill_4.gif . What I'm looking for is a recursive construction of (one third of) http://gosper.org/hexflo.png (to arbitrary order) by clustering solid color tiles. PPS, there's a much higher boundary dimension variant of the France fractal, grouped three around three around one vs six around one. I'll try to unearth one.

3 8

Re: [math-fun] NYTimes: How to Fix Our Math Education
by Dan Asimov 25 Aug '11

25 Aug '11

I do think the most important kinds of high school math are logical thinking, and also statistics (so as to be able to be wary of false claims in advertising and in news reports). Applications to real life, like mortgages, taxes, and running a business are also things worth teaching to all students. I'm less convinced that knowing how to deal with an unknown quantity x is valuable for the students whose future will clearly never include a math-based career. It used to be thought that traditional Euclidean geometry with axioms and proofs is the best way to teach logical thinking. I'm not convinced that this trickle-down based approach is best; perhaps addressing logical thinking directly through real-world problems is better. I do agree with Gene (below) that we do need to try many different approaches, since amidst all the debate, nobody really knows the best methods to use the resources available (which vary widely among school districts, anyway) to get the best results -- without actual experience in trying said methods. I don't know if free market competition is the way to find the best ways. It might be; certainly free market competition has led to a great deal of valuable innovation in certain cases. But on the other hand, privatization (e.g., of prisons) has also often led to the contract being awarded to the lowest bidder, resulting in very low quality of the services provided. SO ... I would conclude that among the many different approaches I mention in my first paragraph, privatization should be one of them, but care should be taken to avoid the low bidders who are likely to provide poor services. --Dan Gene wrote: << The dismal state of math and science public education shows that the current process is a failure, and its deterioration over several decadesindicates that the system is incapable of self-repair. The tragedy is that students who might have become mathematicians and scientists are frustrated and discouraged by a system that is geared towards the dumbest of the dumb. There is no "one size fits all" solution. Only by breaking the public education monopoly, and opening up education to competition will we discover the best processes. . . . >> Sometimes the brain has a mind of its own.

1 0

[math-fun] Does Joe Blow ever need Algebra I?
by Bill Gosper 25 Aug '11

25 Aug '11

>From childhood, I recall my great aunt's agonized Odyssey of successive approximations to compensate her husband's forgetting to charge someone NY State sales tax. Everybody's different. If there's one thing everybody needs, it's a roadmap of what kinds of problems are solved by what kinds of mathematics. --rwg

1 0

Re: [math-fun] computer algebra
by Bill Gosper 25 Aug '11

25 Aug '11

Victor Miller> Veit, I gave this to SAGE (which actually uses the Groeber Basis stuff > in SINGULAR) and it fairly quickly calculated a Groebner Basis, and > showed that the dimension of the ideal is 3. I.e., triply underdetermined? That scotches the plan to PSLQ the exact algebraics from 1000 digit approximations. --rwg > Right now I'm waiting > for it to produce a primary decomposition which should shed some light > on the matter. > > Victor

4 12