sperglord spergs into blog comments

sigma Says:

Comment #109 June 22nd, 2009 at 1:57 pm
@ Jerry Llevada #99

I just looked at your ‘Algebra I’ DVD set. You say, “Complete Algebra I course based on NCTM** recommendations”.

I don’t know who wrote the “NCTM recommendations”, but it looks like they were going through cases of vodka and smoking kilograms of funny stuff in their meetings.

The list of topics looks just AWFUL: It’s packed with made up, junk-think, busy-work, irrelevant NONSENSE. Some of the nonsense is just contrived; some of it is destructive.

For high school courses in algebra, geometry, trigonometry, and solid geometry there can be some benefit for many people with some passing knowledge. E.g., it can be good to understand perpendicularity and the Pythagorean theorem and the connection with projection and least distance. But mostly the purpose of these courses has to be just to get students ready for calculus. So, if some material is not very important for calculus, then JUNK IT.

E.g., you have

5.1 Systems of Equations: Solve by Graphing (6:23)

and

5.5 Graphing Systems of Inequalities (4:42)

NONSENSE. Made-up nonsense. The idea of ’solving’ by graphing is nonsense. ‘Graphing’ systems of linear inequalities is almost entirely nonsense. You seriously misinterpret the reason for graphs and apply graphs where they are just absurd.

The main purpose of graphs is just to build a little intuition about functions. The goal of graphs is certainly not a method of solution of anything; e.g., we no longer use nomographs. Moreover, graphs are just pictures, and there is nearly no serious question for which graphs provide a solid answer. You are making up work to do, and it’s silly, wasteful work.

For linear inequalities, there is now essentially only one practical reason for that topic, and the reason is not very important in Algebra I or, now, hardly anywhere: The reason is the subject of linear programming, e.g., as in (with TeX markup)

George L.\ Nemhauser and Laurence A.\ Wolsey, {\it Integer and Combinatorial Optimization,\/} ISBN 0-471-35943-2, John Wiley \& Sons, Inc., New York, 1999.\ \

The intuition to be obtained from ‘graphing’ a system of linear inequalities is to see intuitively that the set of all solutions to a linear inequality is a ‘closed half-space’ which is convex and the set of all solutions to a system of linear inequalities is an intersection of closed half-spaces and, thus, closed and convex. Also get to illustrate the extreme points and their role. That’s the point of interest and the point trying to illustrate.

To find a solution to a system of linear inequalities are essentially forced into just linear programming. That is, just finding one solution requires essentially the same effort as solving a linear programming optimization problem. So, there is no royal road here: To find solutions, can’t avoid linear programming. Can’t hope to do anything practical with linear programming with graphs. Instead need at least one good (i.e., effective, not necessarily polynomial!) algorithm, some corresponding software, and a computer.

E.g., the last linear programming problem I solved had 40,013 constraints and 600,000 variables. Actually it was a 0-1 integer linear programming problem, and I wrote some software that found a feasible solution within 0.025% of optimality in 905 seconds on a 90 MHz PC via 500 steps of Lagrangian relaxation and the OSL. Intuition about convexity was crucial; graphs of linear inequalities played no role at all.

Linear programming is part of ‘operations research’, and over several decades the US economy has given its opinion: Operations research is not worth the effort, a ‘late parrot’, a dead subject. Sorry ’bout that.

Thus teaching systems of linear inequalities anywhere, especially in algebra I, is likely a waste of effort.

Saw your topic

10.4 Dividing Polynomials (4:51)

Sure, it can be done. Have any idea why one would want to? Trying to get students ready for rings of polynomials? Galois theory? Poles and zeros in passive electrical circuits? Algebraic coding theory? Chebyshev numerical approximation?

Moreover, you have too much emphasis on polynomials. This topic is partly one that should have been left on the scrap heap of history. Yes, at one time some people were all aflutter with polynomials, roots of polynomials, and finding closed form expressions for the roots in terms of the coefficients. That work is done and over with. The questions were not very important and mostly neither were the answers.

Polynomials became a quasi-religion: Once computers were widely available, too many people rushed to ‘fit’ polynomials to data. NOT promising since the stupid thing is guaranteed to rush off to positive or negative infinity far too quickly; the fit is unstable; the usual numerical approach (least squares normal equations) is numerically unstable (i.e., leads to the notorious Hilbert matrix); etc. One should need a special license to be permitted to fit a polynomial with degree higher than 3. In particular, all the material in your course on polynomials of degree 3 should be chucked unless you want to mention poor, couldn’t shoot straight Galois for 90 seconds.

Your interest in polynomials is from the characteristic polynomial of a square matrix, Taylor series, analytic functions, that the complex numbers are algebraically closed but the reals are not, the differences between the rational numbers, the algebraic numbers, and the real numbers? Not worth it in algebra I. Besides, often material on polynomials is misleading: E.g., mostly we do NOT use Taylor series to construct numerical approximations of functions. We definitely do NOT find the eigenvalues of a square matrix by constructing the characteristic polynomial and finding its roots.

Noticed your

Chapter 11: Statistics

11.1 Measures of Central Tendencies and Spread

11.2 Introduction to Probability

11.3 The Counting Principle, Permutations, and Combinations

and I’m horrified.

First, 11.2 is definitely NOT a topic under 11.0. Sorry ’bout that. Instead 11.2 is the most important tool in 11.0.

Second, for 11.1, definitely JUNK it. The idea of ‘central tendency’ is some quasi-religious nonsense left over from about 100 years ago. The idea is, in any measurement, should always get the same result except for ‘experimental error’ and by averaging repeated measurements the errors will cancel and will converge to the mean and ‘central tendency’. This is narrow, simplistic nonsense. Instead, the world is just awash in data where we can want to think of a distribution and where ‘central tendency’ and ‘experimental error’ mean nothing at all. E.g., consider system management data, generated by the terabyte in server farms daily. Or consider economic data. Or, what is the ‘central tendency’ of the age of the next person you see on the street? The idea of ‘central tendency’ has misled a lot of people. Your syllabus is drawing from the dregs of nonsense from 100 years ago. JUNK it.

Avoid mentioning unless you also want to discuss the law of large numbers.

Avoid mentioning the Gaussian distribution unless you also want to discuss the central limit theorem.

For ’spread’ or ‘dispersion’, just mention variance and standard deviation. If you want an application, then give the main result of the W. Sharpe capital asset pricing model. The students might enjoy hearing that the SAT scores are supposed to have standard deviation 100.

For 11.3, there is NO easy connection with probability unless you also discuss probabilistic independence, and that is a much more difficult topic than permutations and combinations.

For statistics, if you want some, do some simple hypothesis tests and maybe some confidence intervals. There are some non-parametric ideas that are easy to teach.

Competent teaching of probability and statistics is rare in the US, at every level. Mostly the subject is too difficult for high school and/or high school teachers.

Watched your

1.6 The Number Line (12:36)

Definitely JUNK it. It’s contrived NONSENSE. It’s not just a waste of time but HARMFUL. No student should watch that. You are teaching that we should do numerical addition and subtraction by using our fingers to walk left and right on the number line. Contrived, made-up, busy-work, misleading NONSENSE.

Apparently someone with no significant knowledge of the real numbers was told to tell students about “the number line” and wanted a reason why and a connection with ’solving real world problems’ so cooked up that finger exercise as the ‘reason’ for the number line. Ignorant. Brain-dead. Destructive. While the students will likely watch porn sometime, we have a shot at keeping them from your number line lecture forever.

Yes, the real numbers are important, one of the most important topics in all of mathematics. Some of the properties are amazing, nearly beyond belief. E.g., see the long, gorgeous dessert buffet in

John C.\ Oxtoby, {\it Measure and Category:\ \ A Survey of the Analogies between Topological and Measure Spaces,\/} ISBN 3-540-05349-2, Springer-Verlag, Berlin, 1971.\ \

One not fully wrong definition of calculus is “some of the elementary properties of the completeness property of the real numbers.”. But at the level of algebra I, it is plenty just to say, “The X axis can be called ‘a number line’.”. DONE. Do NO MORE. Get it OFF THE HEAT. STOP. QUIT. CEASE. DESIST. Saying more is TOO MUCH.

For algebra I, squeeze it down to about six weeks. A good student should get through it in one week, maybe a weekend. The rest is a BIG WASTE.

Use the extra time to move on to plane geometry, algebra II, trigonometry, solid geometry, a little analytic geometry, especially the conic sections, and then calculus. Then linear algebra, abstract algebra, advanced calculus, ordinary differential equations, partial differential equations of mathematical physics, applications in physics and engineering, metric spaces, general topology, measure theory, functional analysis, optimization, probability, stochastic processes, statistics and various applications, filtering, control, exterior algebra, Maxwell’s equations, special and general relativity, quantum mechanics, etc. or a better list.

There is a larger pattern here: Somehow high school math has taken on huge quantities of excess baggage. Somehow there is a ’system’, apparently dominated by graduates of schools of education who have no decent knowledge of math.

In the pursuit of make-work, or some neurotic fear of being criticized for doing too little, these people have dragged in excess baggage, junk-think nonsense. Need to get rid of ALL the high school algebra I up to calculus nonsense and start over. And for AP calculus, JUNK it and just take a respected college calculus text.

S. Eilenberg said, “Elegance in mathematics is directly proportional to what you can see in it and inversely proportional to the effort it takes to see it.” He was correct, and your algebra I syllabus is the opposite of ‘elegance’.

sigma Says:

Comment #110 June 22nd, 2009 at 3:59 pm
@ Patrick Cahalan #97

The problems in ‘teaching’ that you describe are correct and illustrate that the system you describe is nearly hopeless.

Children are TERRIFIC at learning but not in an environment such as you describe.

In one important sense you are trying to do too much, and in another, too little.

Too much: You keep trying to ‘teach’, that is, transfer knowledge from your head to that of the students. So, you talk, write on the board, interact, ‘motivate’, ‘engage’, etc. Mostly that doesn’t work. If a ‘good’ teacher can to it, then good for them; still mostly it doesn’t work. You are trying to make learning too much of a spectator sport. Learning, especially math, is mostly not a spectator sport. In particular, cannot do much in teaching someone math by talking and writing at a board. Instead, the student has to take some basic learning materials and do the learning themselves by whatever magic it is in people that lets them read, think, and learn.

Too little: Most students need some guidance, e.g., help in finding suitable learning materials. Then occasionally a student needs some guidance to say such things as, “You need another explanation of this topic. Try books Y and Z. Rarely does one author give a really good explanation of everything; so, usually need one main source plus two or three others to use when the main source is not so good.” “You are doing too much here; the real goals, needs, and content are simpler than that. Here’s a fast overview of what is going on and what you should get …”. “You seem not to have gotten some of the main ideas in this topic. Here’s some of what you might look at again ….”

You explain some of the problems with the environment well, e.g., disruptive students. One solution is to let a student who is trying to learn go to the library and just STUDY alone. Tell them when the next test is and how they can come to you for help and then just let them get to work.

So, (1) Provide an environment that is not destructive and lets students learn. (2) Provide some good guidance. (3) Be available to let students ask questions. (4) Give and grade some appropriate tests, even standardized tests.

Does it work? It’s about the only way I learned anything about anything. It’s overwhelmingly the foundation of the software part of the US computer industry: People learn from materials on the Internet and books. A huge fraction of most bookstores are just books teaching computer software. LOTS of competing books. The whole system has been a big success for the US, and schools have had next to nothing to do with it. For me, all through school, K-Ph.D., the time in class was nearly always from wasteful down to destructive. We’re talking a LOT of waste here, folks.

In elementary school, I got contempt and criticism from teachers. So I tried harder. I still got contempt and criticism, even when it seemed to me I did well. So I concluded that the opinions of the teachers were not accurate or fair and that, sometimes, the teacher was really just my enemy. As I look back I can see that I was not fully wrong on these points. In K-8 the main problem was simple: I was a boy and nothing like a girl. All the teachers were women, and about half of the girls were teachers pets.

Here’s a complete course in high school math teaching by example:

In plane geometry the teacher was possibly the most offensive person I have ever known. Her favorite statement was a nasty, “Get that mess OFF the board.”. But the book was good enough — that is, it was actually about proving theorems. So, each day in class I had my head down with my eyes closed mostly just resting. Then that evening, alone, I read the section for the day and then attacked the exercises. I started with the easiest exercises and skipped until I found one that took more than 10 seconds to see. Then I found solutions to all the rest. If an exercise took more than a minute, then I’d write out a solution, in the margin of the book or on scrap paper. Then I attacked the more difficult supplementary exercises in the back of the book and did all of those. If an exercise took me more than an hour, then I’d write out a solution on a clean sheet of paper. The teacher’s idea of homework was for students to write out, in the usual laborious format, never seen earlier or later in math, three not very difficult exercises. I only used that format for the most difficult exercises, and in class I never showed any homework. The teacher viewed me with scorn and contempt.

One of the supplementary exercises took me the weekend, all Friday evening, most of Saturday, most of Sunday, and into Sunday evening. I got it. Early in the book there was an easy exercise with the same figure. So, in class on Monday, after the class did the easy exercise, I mentioned, the first time I spoke in class, that there was an exercise with the same figure in the back of the book. About 30 minutes later, with the teacher at the board shouting, “Think, class. THINK.” with no real progress, I didn’t want to be accused of ruining the whole class period so raised my hand and gave a hint. The teacher was livid, and shouted, “You knew it all the time.”. Of course I knew it all the time. I never said I didn’t. If I didn’t know it, then I’d certainly would not have mentioned it in class. I got every non-trivial exercise in the book, never let myself miss even one. Period. Given the hint, she didn’t bother to complete the solution. Payback time, sweetheart!

She was a case: She started the course saying that she was “Miss” but still hoping. She had all the feminine prettiness and figure of a city bus. Try the lottery, sweetheart, where you have a MUCH better chance. Also, that class was half full of some of the most gorgeous women on the planet. E.g., Cybil Sheppard went to that school and was NOT one of the prettiest girls. The irony, that offensive city bus standing before drop dead gorgeous Hollywood casting material and hoping to get married.

On the state test, I did second best in the class. The last problem on the state test was to inscribe a square in a semi-circle. I didn’t get it until after the test. So, I asked to show my solution to the teacher after school. She was angry and confused at my interest and asked why I still cared about the problem but gave in. I started by constructing a square, and she said “You can’t do that”. I continued: Bisect one side of the square, take that point as a center of a circle with radius the distance to a non-adjacent corner of the square. Now have a figure similar to the desired one. For the crucial length, say, half the length of one side of the square, in the original figure, construct a fourth proportional. Done. Still she said, “You can’t do that.” Her solution was to construct a square with one side the diameter of the given circle and then draw a line from the center of the circle to essentially the same square corner I used. So, she did the same but her similar figure was positioned so that she didn’t have to use the usual construction of a fourth proportional. Okay. My solution used a general approach and hers was more specific. By usual mathematical values, my solution was better. The test was multiple choice and asked for the first step in “the” solution; my first step was likely not theirs which means that the test question was badly written.

In college freshman chemistry, a senior English major struggling with math was taking a strange course in geometry and gave me a problem: “Given triangle ABC, construct D on AB and E on BC so that lengths AD = DE = EB.” So, I started with angles CAB and CBA, went to the side, constructed a figure similar to the desired one, and found the crucial length AD in the original figure by constructing a fourth proportional, again. Next day in chemistry, I showed her my solution, and she said: “How’d you know to do that? You just reinvented ’simitude’ that we are studying now.” I explained that I had invented the technique when I was in high school but the teacher said, “You can’t do that”.

The guy who beat me on the state test in geometry also beat me by a little on the Math SAT. We were 1-2 in the school.

When the teacher with the SAT scores opened my Math SAT envelope, she said, “Uh, there must be some mistake.” That’s right, sweetheart, “some mistake”, and not mine, for 12 long, agonizing, painful, wasteful, destructive years for me.

An important lesson here is, the way I learned plane geometry was really the best way. For all the ‘teaching’ the teacher tried to do, I just put my head down, closed my eyes, and waited until the nonsense stopped and I was permitted to get back to learning plane geometry. It’s NOT a spectator sport. Instead, just have to study it, especially to do good exercises. This work is best done alone in a quiet room.

That learning technique is the main way I learned all the math I’ve learned and, of course, similar to how I’ve done all applications and peer-reviewed research I’ve done. Net, there’s just NO WAY to be any good as a professional mathematician, in academic research or in business, or really go very far in math, without doing nearly all the learning just the way I did in plane geometry. In the course I was fully correct: I did the right stuff. The teacher thought I was a terrible student, but I was likely one of the best she ever had.

Why did I like math so much? I part because when I got correct answers, which I nearly always did, the teachers had one heck of a time saying I was wrong. They had to swallow all their nonsense conclusions from their nonsense

irrelevant evidence, whatever that was, and admit I was correct. However painful that admission was for them, it was not nearly painful enough. I liked standardized tests because the grading was objective; my ‘reputation’ as circulated in the ‘teacher’s lounge’ was irrelevant.

Those 12 years were a disaster: No student should go through it. No taxpayers should pay for it. In K-12 I was a motivated, talented, hard working math student, eager, even desperate, to acquire competence in the fields said to be important — math, physical science, engineering — but as I look back I can see that not one of the teachers I had was any good at math.

For our society, here are some ways to do better:

The students, starting at age 10 or so, can do MUCH, MUCH better than they are, EASILY. The bottleneck is the K-12 system where the teachers just do not know anywhere near enough math.

So, there is a big issue about quality.

For college math, the situation is fairly good: Math quality, teaching quality, and math variety are all high. There is plenty of connection with applications in physical science and engineering. There does need to be better connection with applications in social science and business and maybe computing.

In graduate math in US research universities, the quality is very high although in the pure departments the lack of connection with applications is crippling.

So, we know where the quality is and where it isn’t. For the quality in K-12 after basic arithmetic and its applications, there’s no practical fix — no way to turn all those babysitters into mathematicians — and we have to circumvent.

So, we need some external materials. Past arithmetic and it’s applications, the goal is just to rush to get to college calculus. For the rest, JUNK it.

Is Bill Gates listening? Robert Compton? Barack Obama? So, we need some funding to develop several, really MANY, sets of good materials and some associated tests.

For each of the courses, we cannot permit just a single list of topics and need several competing lists and, for each, several competing sets of materials. E.g., we can’t have just a single ‘algebra I’ course: If someone wants to drag themselves through some algebra I full of excess baggage, so be it. Similarly for someone with good sense who just wants to get on to calculus and knock off algebra I in a week or weekend. Once a student has done well in college calculus, NO ONE will care that they junked the excess baggage in algebra I. There should be a lot of Internet and PDF access. No doubt there will be a lot of fora, blogs, and clips on YouTube. TERRIFIC! For the students’ discovering what’s available, ah, the Internet! The K-12 teachers can maybe administer the standardized tests and, maybe, for some really weak students tutor them a little or arrange some tutoring. Teachers: Do no harm.

Then in K-12 we can cut out 6-8 years of highly destructive waste and nonsense and get our country going in high school math.

this thread = tl;dr

sperglord spergs into world4ch
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