[math-fun] hilbertish curve via gray code in radix R
View X as an unsigned integer with <=D digits written in radix R, and also as the D-vector of those digits. The function uint gray(uint X){ return VectorSubtract(X, RightShift(X)) mod R; } converts X to a different such integer. This is a radix-R generalization of the usual binary gray code. This maps the integers which walk from 0 to R^D - 1, to D-dimensional vectors within the hypercube grid {0,1,2,...,R-1}^D which walk in hamilton-tour style via nearest neighbor steps, visiting each grid point exactly once. Extremely trivial implementation of a Hilbertish space filling curve... but it treats the different coordinates very differently, which is annoying.
Did anyone happen to read the article "Wrong Answer: The Case Against Algebra II" by Nicholson Baker, in the Sept. '13 Harper's magazine? I'm curious what other math-fungi think of it. If you'd like a copy just let me know and I'll send you one. --Dan
I read it, partly because I had just read his wonderful novel The Anthologist. I was prepared to disagree with him, but I found it persuasive. His complaint is mainly that modern standards emphasize terminology and memorization over actual ideas... at least that's what I got out of it. Cris On Nov 14, 2013, at 8:41 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Did anyone happen to read the article "Wrong Answer: The Case Against Algebra II" by Nicholson Baker, in the Sept. '13 Harper's magazine?
I'm curious what other math-fungi think of it. If you'd like a copy just let me know and I'll send you one.
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Dan, Send me a copy of "Wrong Answer: The Case Against Algebra II" by Nicholson Baker, in the Sept. '13 Harper's magazine; Thanks, Don
From: dasimov@earthlink.net Date: Thu, 14 Nov 2013 19:41:51 -0800 To: math-fun@mailman.xmission.com Subject: [math-fun] The Case Against Algebra II
Did anyone happen to read the article "Wrong Answer: The Case Against Algebra II" by Nicholson Baker, in the Sept. '13 Harper's magazine?
I'm curious what other math-fungi think of it. If you'd like a copy just let me know and I'll send you one.
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Hi Dan, The article repeats an often-made argument that because the standard high school math curriculum is difficult for some students they should be allowed to avoid it. An important point left out of the article is that to understand dynamic processes in science, engineering, economics, etc., you must become fluent in differential equations by the second year of college, and this in turn necessitates a chain of prerequisite subjects---calculus, trigonometry, functions---down to the dreaded high school algebra. So the cost of allowing students to avoid algebra is that they would be making a decision when 16 years old that affects and limits them for the rest of their lives in ways they don't yet understand. Educators are understandably reluctant to let 16 year olds make such an important decision that sometimes is made from laziness. Some students who might have chosen to skip algebra will find they succeed if they work harder, connect with a motivating teacher, or simply mature for a year. Yet Baker and others are right that many students will never understand algebra and get no value from being forced to sit in the class. I think the resolution is to allow some students to skip algebra, but the education community needs to work out a very good system of selection and counseling, so these students are making a properly informed decision. I would argue that other branches of mathematics should be taught instead to those students (and be available as electives to students in the differential equations track) including fun high-school versions of discrete math, logic, and statistics. George http://georgehart.com/ P.S. My latest sculpture barn-raising: http://www.youtube.com/watch?v=V6kMGDHDPsY On 11/14/2013 10:41 PM, Dan Asimov wrote:
Did anyone happen to read the article "Wrong Answer: The Case Against Algebra II" by Nicholson Baker, in the Sept. '13 Harper's magazine?
I'm curious what other math-fungi think of it. If you'd like a copy just let me know and I'll send you one.
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
New York State is in the process of adopting the "Common Core Curriculum". I first thought this was just about stricter accountability, testing, etc. But then I read some teacher interviews, indicating a significant overhaul of the teaching methods as well. For example, one teacher explained that, in the past, the operation of dividing a/b by c/d was taught as first flipping c/d -> d/c and then multiplying top and bottom, ad/bc. But in the new CCC this is no longer just a fact to be committed to memory, but is explained in terms of more basic principles. I'm going out on a limb here, but it seems to me my grade school teachers (back in the 60's) were already teaching in that mode! The exercise of logic, instead of memory, in performing ever more complex manipulations had the effect that there was no dreaded barrier to learning algebra and beyond. -Veit On Nov 15, 2013, at 9:10 AM, George Hart <george@georgehart.com> wrote:
Hi Dan,
The article repeats an often-made argument that because the standard high school math curriculum is difficult for some students they should be allowed to avoid it. An important point left out of the article is that to understand dynamic processes in science, engineering, economics, etc., you must become fluent in differential equations by the second year of college, and this in turn necessitates a chain of prerequisite subjects---calculus, trigonometry, functions---down to the dreaded high school algebra. So the cost of allowing students to avoid algebra is that they would be making a decision when 16 years old that affects and limits them for the rest of their lives in ways they don't yet understand.
Educators are understandably reluctant to let 16 year olds make such an important decision that sometimes is made from laziness. Some students who might have chosen to skip algebra will find they succeed if they work harder, connect with a motivating teacher, or simply mature for a year. Yet Baker and others are right that many students will never understand algebra and get no value from being forced to sit in the class.
I think the resolution is to allow some students to skip algebra, but the education community needs to work out a very good system of selection and counseling, so these students are making a properly informed decision. I would argue that other branches of mathematics should be taught instead to those students (and be available as electives to students in the differential equations track) including fun high-school versions of discrete math, logic, and statistics.
George http://georgehart.com/
P.S. My latest sculpture barn-raising: http://www.youtube.com/watch?v=V6kMGDHDPsY
On 11/14/2013 10:41 PM, Dan Asimov wrote:
Did anyone happen to read the article "Wrong Answer: The Case Against Algebra II" by Nicholson Baker, in the Sept. '13 Harper's magazine?
I'm curious what other math-fungi think of it. If you'd like a copy just let me know and I'll send you one.
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
George, I agree with you that people are to quick to make math optional. But what do you think of Baker's critique of the way the curriculum is structured? He gives the example of teaching kids about removable and unremovable singularities in rational functions --- a lot of terminology-heavy stuff that seems more suited to memorization and multiple-choice standardized tests than actual understanding. My suggestion would be for some math to be mandatory, but for the curriculum to be designed around getting a sense of how mathematics and mathematical discovery actually works. For instance, I would give them a taste of combinatorics and abstract algebra, where there are interesting and accessible proofs --- the current focus on algebra and calculus is about calculation but hardly ever about proof. Cris On Nov 15, 2013, at 7:10 AM, George Hart <george@georgehart.com> wrote:
Hi Dan,
The article repeats an often-made argument that because the standard high school math curriculum is difficult for some students they should be allowed to avoid it. An important point left out of the article is that to understand dynamic processes in science, engineering, economics, etc., you must become fluent in differential equations by the second year of college, and this in turn necessitates a chain of prerequisite subjects---calculus, trigonometry, functions---down to the dreaded high school algebra. So the cost of allowing students to avoid algebra is that they would be making a decision when 16 years old that affects and limits them for the rest of their lives in ways they don't yet understand.
Educators are understandably reluctant to let 16 year olds make such an important decision that sometimes is made from laziness. Some students who might have chosen to skip algebra will find they succeed if they work harder, connect with a motivating teacher, or simply mature for a year. Yet Baker and others are right that many students will never understand algebra and get no value from being forced to sit in the class.
I think the resolution is to allow some students to skip algebra, but the education community needs to work out a very good system of selection and counseling, so these students are making a properly informed decision. I would argue that other branches of mathematics should be taught instead to those students (and be available as electives to students in the differential equations track) including fun high-school versions of discrete math, logic, and statistics.
George http://georgehart.com/
P.S. My latest sculpture barn-raising: http://www.youtube.com/watch?v=V6kMGDHDPsY
On 11/14/2013 10:41 PM, Dan Asimov wrote:
Did anyone happen to read the article "Wrong Answer: The Case Against Algebra II" by Nicholson Baker, in the Sept. '13 Harper's magazine?
I'm curious what other math-fungi think of it. If you'd like a copy just let me know and I'll send you one.
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Cristopher Moore Professor, Santa Fe Institute The Nature of Computation Cristopher Moore and Stephan Mertens Available now at all good bookstores, or through Oxford University Press http://www.nature-of-computation.org/
As I recall, Baker spoke (also?) of "points of discontinuity" of rational functions. But I don't think a rational function can have a point of discontinuity. (It's continuous at all points in its domain, no?) And when I took algebra II (then known as intermediate algebra), no one taught about removable singularities. --Dan On 2013-11-15, at 7:24 AM, Cris Moore wrote:
George, I agree with you that people are to quick to make math optional.
But what do you think of Baker's critique of the way the curriculum is structured? He gives the example of teaching kids about removable and unremovable singularities in rational functions --- a lot of terminology-heavy stuff that seems more suited to memorization and multiple-choice standardized tests than actual understanding.
My suggestion would be for some math to be mandatory, but for the curriculum to be designed around getting a sense of how mathematics and mathematical discovery actually works. For instance, I would give them a taste of combinatorics and abstract algebra, where there are interesting and accessible proofs --- the current focus on algebra and calculus is about calculation but hardly ever about proof.
My son is taking Algebra II now and just finished rational functions. Which means I just finished rational functions, too. I think the textbook called a function like f(x) = (x^2-1)/(x+1) discontinuous. Or maybe it simply said it had a "hole". Will this be on the test? WILL I EVER USE THIS???? On Fri, Nov 15, 2013 at 8:01 AM, Dan Asimov <dasimov@earthlink.net> wrote:
As I recall, Baker spoke (also?) of "points of discontinuity" of rational functions.
But I don't think a rational function can have a point of discontinuity. (It's continuous at all points in its domain, no?)
And when I took algebra II (then known as intermediate algebra), no one taught about removable singularities.
--Dan
On 2013-11-15, at 7:24 AM, Cris Moore wrote:
George, I agree with you that people are to quick to make math optional.
But what do you think of Baker's critique of the way the curriculum is structured? He gives the example of teaching kids about removable and unremovable singularities in rational functions --- a lot of terminology-heavy stuff that seems more suited to memorization and multiple-choice standardized tests than actual understanding.
My suggestion would be for some math to be mandatory, but for the curriculum to be designed around getting a sense of how mathematics and mathematical discovery actually works. For instance, I would give them a taste of combinatorics and abstract algebra, where there are interesting and accessible proofs --- the current focus on algebra and calculus is about calculation but hardly ever about proof.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
Right. It's infinitely better to ask questions like let f(x) = (x^2-1)/(x+1). . is this function ever zero? where? . does it ever blow up? where? . does it ever switch from being positive to negative or back again? where? . what does this function look like when x is large and positive? what about when x is very negative? On Nov 15, 2013, at 9:34 AM, Thane Plambeck <tplambeck@gmail.com> wrote:
My son is taking Algebra II now and just finished rational functions. Which means I just finished rational functions, too.
I think the textbook called a function like f(x) = (x^2-1)/(x+1) discontinuous. Or maybe it simply said it had a "hole".
Will this be on the test? WILL I EVER USE THIS????
On Fri, Nov 15, 2013 at 8:01 AM, Dan Asimov <dasimov@earthlink.net> wrote:
As I recall, Baker spoke (also?) of "points of discontinuity" of rational functions.
But I don't think a rational function can have a point of discontinuity. (It's continuous at all points in its domain, no?)
And when I took algebra II (then known as intermediate algebra), no one taught about removable singularities.
--Dan
On 2013-11-15, at 7:24 AM, Cris Moore wrote:
George, I agree with you that people are to quick to make math optional.
But what do you think of Baker's critique of the way the curriculum is structured? He gives the example of teaching kids about removable and unremovable singularities in rational functions --- a lot of terminology-heavy stuff that seems more suited to memorization and multiple-choice standardized tests than actual understanding.
My suggestion would be for some math to be mandatory, but for the curriculum to be designed around getting a sense of how mathematics and mathematical discovery actually works. For instance, I would give them a taste of combinatorics and abstract algebra, where there are interesting and accessible proofs --- the current focus on algebra and calculus is about calculation but hardly ever about proof.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/ _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Cristopher Moore Professor, Santa Fe Institute The Nature of Computation Cristopher Moore and Stephan Mertens Available now at all good bookstores, or through Oxford University Press http://www.nature-of-computation.org/
Of course, we've seen attempts to put discrete math into the curriculum before. When I was in grade school, it resulted in stupid busywork like computing the Cartesian product of {1,2,3} and {a,b,c}. But hopefully it could be done better. Cris On Nov 15, 2013, at 8:24 AM, Cris Moore <moore@santafe.edu> wrote:
My suggestion would be for some math to be mandatory, but for the curriculum to be designed around getting a sense of how mathematics and mathematical discovery actually works. For instance, I would give them a taste of combinatorics and abstract algebra, where there are interesting and accessible proofs --- the current focus on algebra and calculus is about calculation but hardly ever about proof.
Chris, Yes, personally experiencing mathematical discovery is the key to learning what math really is, so a place must be made for it in schools. Paul Lockhart's book, _A Mathematician's Lament_, is essential reading on what mathematics education should be. Unfortunately, (as Paul is the first to admit) relatively few teachers below the college level actually are trained to think like a mathematician, so they are not capable of leading a class in an exploratory approach to mathematics. Those who can usually find other careers. It is a difficult problem for society to solve, but still it is part of our jobs as mathematicians to help work towards a public understanding of what mathematics is. George http://georgehart.com/ P.S. Someone who helped me and many of my generation think like a mathematician is Martin Gardner. So here is sculpture construction at Princeton in his honor: http://www.youtube.com/watch?v=PzYzCEgTitA On 11/15/2013 10:24 AM, Cris Moore wrote:
George, I agree with you that people are to quick to make math optional.
But what do you think of Baker's critique of the way the curriculum is structured? He gives the example of teaching kids about removable and unremovable singularities in rational functions --- a lot of terminology-heavy stuff that seems more suited to memorization and multiple-choice standardized tests than actual understanding.
My suggestion would be for some math to be mandatory, but for the curriculum to be designed around getting a sense of how mathematics and mathematical discovery actually works. For instance, I would give them a taste of combinatorics and abstract algebra, where there are interesting and accessible proofs --- the current focus on algebra and calculus is about calculation but hardly ever about proof. ...
I also grew up on Martin Gardner. We would be far more impoverished if it weren't for him. I also like Lakatos' "Proofs and Refutations" --- a classroom dialogue where students and a teacher explore Euler's law F-E+V = 2, figure out counterexamples, and slowly move towards a careful formulation and a proof. Cris On Nov 15, 2013, at 12:19 PM, George Hart <george@georgehart.com> wrote:
Chris,
Yes, personally experiencing mathematical discovery is the key to learning what math really is, so a place must be made for it in schools. Paul Lockhart's book, _A Mathematician's Lament_, is essential reading on what mathematics education should be. Unfortunately, (as Paul is the first to admit) relatively few teachers below the college level actually are trained to think like a mathematician, so they are not capable of leading a class in an exploratory approach to mathematics. Those who can usually find other careers. It is a difficult problem for society to solve, but still it is part of our jobs as mathematicians to help work towards a public understanding of what mathematics is.
George http://georgehart.com/
P.S. Someone who helped me and many of my generation think like a mathematician is Martin Gardner. So here is sculpture construction at Princeton in his honor: http://www.youtube.com/watch?v=PzYzCEgTitA
On 11/15/2013 10:24 AM, Cris Moore wrote:
George, I agree with you that people are to quick to make math optional.
But what do you think of Baker's critique of the way the curriculum is structured? He gives the example of teaching kids about removable and unremovable singularities in rational functions --- a lot of terminology-heavy stuff that seems more suited to memorization and multiple-choice standardized tests than actual understanding.
My suggestion would be for some math to be mandatory, but for the curriculum to be designed around getting a sense of how mathematics and mathematical discovery actually works. For instance, I would give them a taste of combinatorics and abstract algebra, where there are interesting and accessible proofs --- the current focus on algebra and calculus is about calculation but hardly ever about proof. ...
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I think they all need to be taught probability and statistics because it's so important in political debate. But even that requires algebra and maybe even a little calculus. Brent On 11/15/2013 6:10 AM, George Hart wrote:
Hi Dan,
The article repeats an often-made argument that because the standard high school math curriculum is difficult for some students they should be allowed to avoid it. An important point left out of the article is that to understand dynamic processes in science, engineering, economics, etc., you must become fluent in differential equations by the second year of college, and this in turn necessitates a chain of prerequisite subjects---calculus, trigonometry, functions---down to the dreaded high school algebra. So the cost of allowing students to avoid algebra is that they would be making a decision when 16 years old that affects and limits them for the rest of their lives in ways they don't yet understand.
Educators are understandably reluctant to let 16 year olds make such an important decision that sometimes is made from laziness. Some students who might have chosen to skip algebra will find they succeed if they work harder, connect with a motivating teacher, or simply mature for a year. Yet Baker and others are right that many students will never understand algebra and get no value from being forced to sit in the class.
I think the resolution is to allow some students to skip algebra, but the education community needs to work out a very good system of selection and counseling, so these students are making a properly informed decision. I would argue that other branches of mathematics should be taught instead to those students (and be available as electives to students in the differential equations track) including fun high-school versions of discrete math, logic, and statistics.
George http://georgehart.com/
P.S. My latest sculpture barn-raising: http://www.youtube.com/watch?v=V6kMGDHDPsY
On 11/14/2013 10:41 PM, Dan Asimov wrote:
Did anyone happen to read the article "Wrong Answer: The Case Against Algebra II" by Nicholson Baker, in the Sept. '13 Harper's magazine?
I'm curious what other math-fungi think of it. If you'd like a copy just let me know and I'll send you one.
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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There are several problems with teaching Algebra II today. The first is the complete lack of motivation of virtually everything that is taught. The second is the lack of an easy-to-use algebraic manipulation tool -- e.g., Maxima/Maple/Mathematica -- which can automate tedious algebraic manipulations and graphing. Solving the motivation problem is going to be extremely difficult, because very few people have ever had to work out what the proper motivation for each element should be. The usual motivation -- following the historical order of development -- can work, but these tend to get side-tracked with stories about Cardano instead of stories about what motivated Cardano to take the steps he took. What is missing is someone of the caliber of Richard Feynman motivating the development of each step. One powerful modern motivation for analytic geometry is computer games & computer graphics. One possible motivation for Algebra II in high school might be the development of the computer graphics for a computer game. For example, so-called "homogeneous coordinates" never made any sense to me until I saw how elegantly they handled many problems in computer graphics. Historically, the "core" of algebra goes from solving linear equations to solving quadratic equations, to solving cubic & quartic equations & non-solving quintics. The nice thing about this sequence is that we first build N, then Z, then Q, then quadratic extensions of Q (including the complex numbers), then cubic extensions of Q (including cubic roots of unity), and trigonometric solutions. If we have enough time, we can do quartic equations and then get into the unsolvability of the quintic -- e.g., Klein's icosahedron. In one "smooth" progression, we can quickly bring in an incredible amount of mathematics, including rational functions, complex numbers, circular & hyperbolic trigonometry, group theory. So long as we can automate much of the drudgery with computer tools, and use the computer to do pretty animated pictures & diagrams, I think it should be possible to teach many people who hate math all of this knowledge. It isn't necessary that most of them be able to reproduce it, but if they can follow the pretty pictures, they can see that there is beauty there, and can come back later to do it more rigorously. Probably the worst thing that ever happened to high school math in the 20th century was the "rigor mortis" that was introduced in the middle of the century. Rigor is fine for professional mathematicians, but most mathematicians I know don't start with a rigorous argument. They try to "see" a path to a solution, and then fix up the loose ends later. This is how math should also be taught: basic insight, followed by details & fixups. --- If you haven't tried the Khan Academy math courses, you really should. The Khan Academy algebra & geometry courses are really quite good, and one heck of a lot better than 99% of the textbooks & coursework available to high school students today. https://www.khanacademy.org/ --- I've been trying to play a very small part in this process with some web pages. Here is one: Solution of Cubic Equation with Three Real Roots http://home.pipeline.com/~hbaker1/cubic3realroots.htm I'm also working on another one for solving the general cubic with complex coefficients: What Affine Day to Solve a Cubic (I posted the main portion of this page to math-fun a number of months ago.) I hope to have this page up in a month or so. At 07:41 PM 11/14/2013, Dan Asimov wrote:
Did anyone happen to read the article "Wrong Answer: The Case Against Algebra II" by Nicholson Baker, in the Sept. '13 Harper's magazine?
I'm curious what other math-fungi think of it. If you'd like a copy just let me know and I'll send you one.
--Dan
participants (9)
-
Cris Moore -
Dan Asimov -
don skow -
George Hart -
Henry Baker -
meekerdb -
Thane Plambeck -
Veit Elser -
Warren D Smith