Posted on 03/15/2008 1:58:58 AM PDT by neverdem
American students’ math achievement is “at a mediocre level” compared with that of their peers worldwide, according to a new report by a federal panel, which recommended that schools focus on key skills that prepare students to learn algebra.
“The sharp falloff in mathematics achievement in the U.S. begins as students reach late middle school, where, for more and more students, algebra course work begins,” said the report of the National Mathematics Advisory Panel, appointed two years ago by President Bush. “Students who complete Algebra II are more than twice as likely to graduate from college compared to students with less mathematical preparation.”
The report, adopted unanimously by the panel on Thursday and presented to Education Secretary Margaret Spellings, said that prekindergarten-to-eighth-grade math curriculums should be streamlined and put focused attention on skills like the handling of whole numbers and fractions and certain aspects of geometry and measurement.
It offers specific goals for students in different grades. For example, it said that by the end of the third grade, students should be proficient in adding and subtracting whole numbers. Two years later, they should be proficient in multiplying and dividing them. By the end of the sixth grade, the report said, students should have mastered the multiplication and division of fractions and decimals.
The report tries to put to rest the long, heated debate over math teaching methods. Parents and teachers have fought passionately in school districts around the country over the relative merits of traditional, or teacher-directed, instruction, in which students are told how to do problems and then drilled on them, versus reform or child-centered instruction, emphasizing student exploration and conceptual understanding. It said both methods had a role.
“There is no basis in research for favoring teacher-based or student-centered instruction,” Dr..
(Excerpt) Read more at nytimes.com ...
I have no idea what that gibberish meant in your last post. Maybe you forgot to put 7 dots and only put 6, but that is still ridiculous. Maybe, you can explain the nonsense with the dots.If you don't understand what it means, how do you know it's nonsense? It is simply seven lines of six dots each…
To do that for simple multiplication or division - as per what you advocate - is just insane.I am not advocating that. I want people to be able to figure out what 6*7 is, in case they have forgotten the answer. If somebody is not able to figure out what 6*7 is without using a calculator, then something went seriously wrong in his math education. That's what I'm saying.
For two discrete time functions, the z transform of their convolution is equivalent to polynomial multiplication of their z-transforms. Convolution of x(1) = 1, x(2) = 1, zero otherwise with h(1) = 1, h(2) = 1, h(3) =1, zero otherwise gives y(n) = x*h = [1 2 2 1], for n = 1, 2, 3, 4.
X(z) = z^-1 + z^-2,
H(z) = z^-1 + z^-2 + z^-3,
Y(z) = z-1+ 2*z^-2+ 2*z^-3+ z^-4 = X(z)*H(z).
Make any sense at all?
A decimal number is itself a polynomial. For example 121 is really ( 1 times x0 ) plus ( 2 times x1 ) plus (1 times x2), where x0 is 10^0, x1 is 10^1 and x2 is 10^2.
And the factors of a polynomial, when the polynomial is power series, or follows some other positional rule -- those factors are just a sequence of numbers. A series of numbers in a set order.
What you call a convolution can also be called an "inner product".
Yes it is. If you need a calculator then you don't have the basics. My daughters math teacher allowed calculators and they didn't need to write out the whole calculation on homework. How can you correct what you can't see? How can you tell them why their answer is wrong if you can't see what they did? Shortcuts lead to errors. She was in a MATH class not a class on how to successfully use a calculator.
I don't think I'd go that far. Lots of folks do well with "rote" memorization". Some never get any farther. (I worry about my great niece that way). Others, like me, have to have lots of practice, we learn by doing, not by looking. Not everyone is the same. But the basic facts must be learned, if one is to ever grasp the principles behind them.
I have my own theory about that, one which which my wife, the math teacher and teacher of math teachers, agrees only to a small degree.
I think that with the opening of math, science and engineering careers to women, the women with good math skills/aptitude moved to those careers, which pay better. A partial solution might be in paying math and science teachers more than English or History teachers. But the Unions won't hear of that. Plus it doesn't address another aspect of the problem, elementary teachers who are math phobic, because their teachers, even their college teachers, were as well. Eliminating the latter is my wife's crusade. She does pretty well at it by all accounts, but it certainly irks her to have to teach the math as well as how to teach it to her students. But she makes those wanting to be high school math and science teachers to also take the "teaching elementary math" course. Most of them thank her for it later.
My introduction to convolution was the classical continuous time version. These days everything is discrete time, so I just naturally think in those terms...
Honestly, when I first saw the problem, “What is the square root of 12345678987654321?”, I just about immediately recognized it as (de)convolution problem.
We tend to cast problems in familar terms.
I do. I developed/derived the algorithm, (it was mostly high school algebra) which flies the F-16 and F-15E down at below 200 feet at around Mach o.9. I also developed parts of a simulator used to train AWACS back end crew. That involved lots of spherical trig for the inter visibility algorithm, and a background in stochastic processes for the tracking filter. Then I developed a modification to allow the simulator to be fed ground based radar/beacon data from multiple radar sites, more spherical trig and the mathematics of an ellipsoid as well.
I agree with cartan. Being able to figure something out more or less from "first principals" means you really understand it, and if you do it enough, eventually you remember the "formula", that you might have memorized, but not really understood, earlier.
The sine of Pi/2 is. (It's "1" BTW)
But they wouldn't *DO* it *every* time. Just when they hadn't used a fact for awhile, or the first time they needed to use it. Otherwise they'd remember.
Besides math professors hardly ever need to do arithmetic.
> Memorization has absolutely no place at all in mathematics.
This Ph.D. disagrees with you. Memorization puts tools in your toolbox, so that you can spend creative thought solving conceptually advanced problems, rather than 8x6.
In most states they have to take something equivalent before they, or non-government teachers to be as well, before they can get their "license", or in some cases their degree. In Texas even elementary teachers must major in something else, not "Elementary Education". At my wife's college, they make the early teaching courses "electives" in other programs, and then the final education courses, as well as student teaching, come in graduate school. Her department is the only one on the campus with a grad program. Math and science teachers must have an undergraduate degree in their subject matter. Same for other secondary teachers. The graduates of her program are in great demand. They are even prized as student teachers.
I tutor middle school math. I find that many of the kids are not really fluent in their multiplication tables. I don’t bet they are required to memorize them (’too boring’)(’not conceptual enough’). But kids are concrete. Give them memorization, they will do it and master it and then they will have success.
(don’t get me started on math textbooks)...
I love Saxon too. I was reading that Mr. Saxon was so fed up with the way math was being taught/ written up in textbooks, that he wrote his own.
I did have one principal tell me the kids had complained about Saxon (the 9th graders!!) because ‘it wasn’t entertaining enough.’ NINTH GRADERS. I said, “I didn’t think math was supposed to be entertaining.” The only way I would have taught that class was with Saxon.
Our local public school district uses Saxon Math.
Higher-level, conceptual understanding cannot take place until the brain is developed sufficiently, starting about age 12. Before that, memorization builds the foundation. Without a foundation, the kids with math are like balloons without tethers — they have these grand concepts but they cannot multiply. Makes no sense.
Not entirely arbitrary. Few people have 12 or 16 fingers. Counting on toes would make base 20 possible, but it's a such a bother.
We use Math-U-See. The emphasis is heavy on place value notation in the elementary years, and skip counting to lead naturally to multiplication. My fifth grader is very strong, never having had to memorize a single field from a multiplication table. Yee haw!
Memorization of tables is not concrete, it's the ultimate abstract. Now if you use something like base10 blocks or other manipulative to illustrate what addition, multiplication, and even area and volume really mean or are, that's something concrete. Not something that will work with all kids, but it helps to some extent with all, and it does "work" with many or most.
Then once they understand what they are memorizing, the memorization goes faster, and they can use what they've memorized.
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