Tuesday, March 07, 2006

Is Math Necessary?

Well, the 10th Carnival of Homeschooling is up and hosted by Anne at Palm Tree Pundit. As I scrolled through the line-up, I checked out the posts on "Why They Need Algebra" and "Life Without Algebra? I Think Not." I thought of my own child who insists that she "does not need math." When I hear this, I cringe. My father was a mathematician who spent hours (I mean hours) a day teaching me Algebra, Trig and Pre-Calculus. In my first year of college, when I thought I would be an Aerospace Engineer (fat chance of that), my dad still coached me through my first year of advanced Calculus.

This is why I get disheartened when I read journalists like Richard Cohen at the Washington Post giving advice like this to a girl who had failed Algebra six times:

Here's the thing, Gabriela: You will never need to know algebra. I have never once used it and never once even rued that I could not use it. You will never need to know -- never mind want to know -- how many boys it will take to mow a lawn if one of them quits halfway and two more show up later -- or something like that. Most of math can now be done by a computer or a calculator. On the other hand, no computer can write a column or even a thank-you note -- or reason even a little bit. If, say, the school asked you for another year of English or, God forbid, history, so that you actually had to know something about your world, I would be on its side. But algebra? Please.


It's amazing that this journalist is so egocentric that he doesn't see the need for knowing a subject outside of his own field as something important for kids. When reporters at newspapers think that writing is the highest form of reasoning, we have to question where their priorities lie. Knowing something about the world is nice--but knowing how the world works is the ultimate knowledge.

I came to see the beauty, complexity and sheer enjoyment of doing math and have been forever grateful to my dad for instilling in me a sense that math was important. But so many kids hate math and this extends into the teen and adults years. Just go to any check-out counter and ask the clerk to calculate change without a machine of some sort and see the glazed look of puzzlement in his or her eyes. It's a shame because we need more mathematicians and young people in the hard sciences, not fewer. Anyone got any clever ideas on how to interest kids in math?

Update: Joanne Jacobs has more thoughts on life without algebra.

94 Comments:

Anonymous Anonymous said...

When I finally got an A inn Algebra II in high school, I said, "Enough, I'll never use this stuff, and I don't do math. Give me a good book about history or a mystery and I'll show you what's important." I went on to be a political science major struggling through the required stats course, because it had been a solid 3 years since my last math course. I then went on to a Master's in Public Administration and have taken up work in performance measurement doing algebra and some forms of calculus every day. My point? There is absolutely no way for a young person to be certain what he/she wants to be and where he/she will excel. Tell your daughter grade school is not the place to give up on yourself in any way shape or form (I don't need math is really saying I don't like it, I'm not inclined to think that way.)

8:15 AM, March 07, 2006  
Blogger Dave said...

Sorry, any person who claims you don't need algebra is a complete, absolute, abject idiot.

As for calculus, well, obviously, some fields (engineering, architecture, derivatives pricing and trading, risk management) require it. Most fields don't (have you ever measured the integral of your patients??) Of course not.

But to argue that algebra is unnecessary is to ignore the very simple fact that our lives, from credit card payments, interest rates, mortgages, taxes, and all other manner of daily financial matters are priced according to principles of algebra. To assert that one does not need to know algebra is to assert that one should be robbed blind by those more mathematically adept than another.

8:16 AM, March 07, 2006  
Blogger Thom said...

The idea that computers will figure it all out for us is also nuts. Ever tried to set up an Excel spreadsheet of any complexity without knowing at least a little bit about formulas (which is algebra)?

Computers and calculators can do the complex math for you, sure. But you have to be able to tell them what complex math you want them to do. To my knowledge we have not yet written a program that will read your mind.

There are a lot of things you can do without algebra and geometry, but you can sure do them faster with them.

I was a music major, and I didn't like math. But I'm sure glad I at least struggled through Algebra II before I gave up. And the college course on what math can be used for was excellent. Learning about game theory was a lot of fun--and keeps me from being at all tempted to gamble.

8:38 AM, March 07, 2006  
Blogger David said...

Increasingly, people in all professions are responsible for their own retirement investment portfolios. I don't think it's likely that one could be an effective investor without having a good feel for numbers and the relationships among them.

9:06 AM, March 07, 2006  
Blogger Spunky said...

The post Life Without Algebra, I think not was written by my 17 year old homeschooled daughter. She is definitely more of a writer so I was pleasantly surprised when she saw the value of something outside her own world of writing unlike Mr. Cohen. Thanks for linkiing to it.

What Mr. Cohen is actually showing us is how little value there is in the professional punditry. Vive the blogosphere.

9:13 AM, March 07, 2006  
Blogger quadrupole said...

It should also be noted that jobs that require majors that use math start at roughtly twice the salary of jobs that do not. The gap is growing. What most people seem to miss is that not only is math a highly useful skill in and of itself, but the kind of thought it engenders is also highly useful.

As to the journalist, if he can't follow math, I see no reason to believe him about anyting related to economics, planning, policy, etc. He simply isn't equipped to understand most aspects of the world.


In the future, there will be two kinds of jobs, those that require knowing math, and those that involve pampering those who know math.

10:02 AM, March 07, 2006  
Anonymous Edgehopper said...

I find it odd that the same people who would require reading Twain, Hemingway, and other classic authors would criticize algebra as "useless." Because, you know, I use Twain's words in my work every day.

Numeracy, like literacy, is just part of being an informed citizen. A numerate person looks at a "fifteen women are raped every second!" statistic with appropriate skepticism. A numerate person should be able to understand basic concepts of probability, an important part of many decisions in life.

Pharyngula's post has it right: "If I had never heard a poem or listened to a symphony or read a novel or visited Independence Hall, I could probably dumbly write that I don't miss literature, music, or history…never heard of 'em. Don't need 'em. Bugger all you eggheads pushing your useless 'knowledge' on me!"

10:08 AM, March 07, 2006  
Blogger a psychiatrist who learned from veterans said...

Create a degree 'High School Diploma - Algbra' and let the market decide. In college, require accounting or governmant, not just the latter.

10:17 AM, March 07, 2006  
Anonymous Anonymous said...

Anybody remember Lawrence Summers? This journalist writes anti-educational claptrap and gets paid for it. He has done far more damage to people---including women---learning mathematics than Summer ever managed.

11:15 AM, March 07, 2006  
Anonymous Anonymous said...

Welllll, there are simply a lot of students for who the struggle with math does little more than turn them off school in general. I am all for all the math courses the school can handle -- for the benefit of those students who are proficient at it. But, for the rest of the kids, lets spend more time on how to use math in every day life, not just in theory. How to keep a running total in their head at the supermarket, McDonalds, or the blackjack table.

11:16 AM, March 07, 2006  
Blogger dadvocate said...

I work at a marketing research firm. The best paying jobs are for the statisticians, analysts, etc. Virtually everybody here needs to know high school level algegra or better. My company is a "high priced" company. We make our living doing research and analysis others can't or won't, including stastical stuff that is way beyond me.

My oldest son loves cars and working on them. All sorts of math goes into figuring out air intakes, exhaust flow, fuel/air ratios, etc. This stuff used to be guess work but it isn't anymore - math. It will only become more important in the future.

11:23 AM, March 07, 2006  
Blogger Helen said...

Anonymous 11:16:

I agree that students should learn math for everyday life--that is basically arithmetic but apparently, some journalists and others do not even see fit to have students learn at this low level or how to interpret basic stats. They just hear the word math and freeze. How can we trust what a journalist, writer, researcher etc. says if they have so little use for numbers that they cannot even figure out if a percentage is accurate?

11:27 AM, March 07, 2006  
Blogger Jeff said...

"It's amazing that this journalist is so egocentric that he doesn't see the need for knowing a subject outside of his own field..."

This is why the press was so relentless about Bush's malapropisms and general inarticulateness. They measure the world by their yardstick -verbal skill- and literally can't imagine different values or forms of intelligence.

That said, I hate math, but recognise the overwhelming need for it!

11:31 AM, March 07, 2006  
Anonymous dweeb said...

Oh joy, I can't think of a better context for winding Greg Kuperberg's spring.

You have to consider the source. Richard Cohen makes Howard Dean look like Pat Buchanan, and math is anathema to liberals. Math classes can't be politicized. In math, there's only one right answer, and when you're wrong, you're wrong. That's greedy, oppressive, heirarchical, dead-white-male-ism at its worst, and dreadfully harmful to people's all precious self esteem. Math, and the fields that depend upon it, center on real, measurable results, and that's just not PC. It's no surprise Mr. Cohen despises math - it explains his allegiance to Marxist economic theories. He wants students to concentrate on courses that can be used to indoctrinate them.

Now, the real fun part is watching Greg struggle with the conflict of responding to the implied disconnect between his ideology and his profession, rather than finding yet another way to work in a diatribe about the Iraq situation.

11:42 AM, March 07, 2006  
Blogger Anne said...

Thank you so much for promoting the carnival here.

As for the algebra, I have only developed an appreciation for it as an adult as I homeschool my 8th grader. He hates algebra, but the light has finally come on for me. I realize that my dislike of it all these years was really just fear. Other subjects came easily to me, but math was always a struggle. I didn't like what I wasn't good at, and that really is "egocentric".

I don't have any brilliant ideas on how to interest kids in math. I'm just hoping that the light comes on for my son before he's my age! :-)

11:51 AM, March 07, 2006  
Anonymous Anonymous said...

Hi "Dweeb":

You cannot want to have another circular logic filled Kuperfest, I hope. I would much rather focus on this divide between liberal arts and sciences---specifically, this anti-science bias to which you refer.

At the rest of yet another source of Kupermania, the Republicans get a bad rap for being anti-science, but I find that the only science that the Democrats get excited about seems to revolve around issues related to abortion rights, global warming, and other politicized issues.

Here is the take home lesson. I know many scientists who are ashamed that they don't know more literature. I know scads of, say, sociologists who appear proud that they don't know more physics and math.

By the way, I do think that teaching students practical applications of math helps---how trig is used in surveying for example.

But I agree with you dweeb. Funny world we live in, huh? So far as the Left and science, read a bit about Trofim Lysenko and the Stalin regime in the old Soviet Union.

Politics and science never mix.

12:01 PM, March 07, 2006  
Blogger DRJ said...

During my early school years, math and science were subjects that didn't excite me that much, probably because I was better at the liberal arts and I knew it. I would do what it took to make an A in my math and science courses, but I would go the extra mile in my other courses.

Then I had a biology teacher who made the sciences come alive because of his passionate love of the topic and because he made us work hard in his class. I came away from that class with 2 things: A realization that math and science can be exciting subjects, and the knowledge that I would have to work hard to learn them but I could do it.

So, in my view, the answer to teaching math and science is teachers who have enthusiasm for their subjects and want to share that passion with their students. I think colleges have made a grave mistake in creating education departments that teach prospective teachers how to be administrators but don't require them to master - let alone become passionate about - their subjects.

12:20 PM, March 07, 2006  
Blogger Henry Cate said...

One of the issues resolving around Algebra is that it is hard work. For some people mastering the basics of Algebra is very hard work. Our society has become more lazy. Many parents haven't taught their children to work hard. There seems to be a feeling that if it is hard, there is something wrong.

12:31 PM, March 07, 2006  
Anonymous Anonymous said...

Whenever I see comments like "who needs algebra?" I am always reminded of a long past family dinner, where my grandmother claimed that Math is too confusing and she could never do it. My uncle asked her "When you sew a blouse from a pattern, do you have to figure out how much cloth you need for a sleeve? You just solved for an unknown amount. That's algebra."

12:42 PM, March 07, 2006  
Anonymous Teri Lester said...

Grandma probably also did some pretty sophisticated algebra when she adjusted her recipes for more servings, less servings, a bigger chicken than usual, etc.

Whenever my daughter claims that she will never use algebra for anything, I invite her to help me cook dinner - and show her that all the relationships and conversions and proportions of tablespoons to cups to pounds of meat are really algebraic formulas.

Anybody who claims to never use algebra must have never made a big glass of chocolate milk. They tell you how much powder to put in with 8 ounces of milk - have you seen an 8 ounce cup in an American kitchen any time in the last 30 years?

12:49 PM, March 07, 2006  
Anonymous Cyn said...

I can't tell you how much I hate the idea, oft stated by otherwise intelligent people, that certain people "just can't do math." No one gets away with saying they "just can't do English." It's an excuse for laziness.

I am unsurprised that a journalist said this. The innumeracy and poor understanding of probability and statistics among their field is astounding.

12:59 PM, March 07, 2006  
Anonymous Anonymous said...

Exactly, Ms. Lester. Maybe we should have a contest, listing all the ways regular people do use higher math in their daily lives.

I started out in college as a Psych major, but changed my major out of fear of the required statistics course. I put myself on the road to Law School, rather than Med School like my Grandfather, or Business School like my Father, because of my distaste of Math and the resultin innumerancy. Guess what? I still have to use algebra every time I have to compute the accrued interest on an unpaid debt as part of litigation damages, e.g.

Excel is nothing more than an algebra engine, but it is useless without knowing the proper formula to enter.

1:10 PM, March 07, 2006  
Anonymous cassandra said...

Another thing going on here is the pervasiveness of whole-math, new-new math, "constructivist" or "fuzzy math" curricula in our K-12 public schools. I mean, first-graders using calculators...no more memorization of multiplication tables, no more long division because it's "too boring" and technology has made it obsolete...right.

This educrap is everywhere, and has made it even more difficult for students to handle algebra in the 8th or 9th grade. Once they start getting behind in fractions or whatever, they are lost without special intervention.

1:11 PM, March 07, 2006  
Anonymous Anonymous said...

Sorry, that should be innumeracy, not innumerancy. I forgot to subtract n.

n, I wish I new how to quit you.

1:14 PM, March 07, 2006  
Anonymous Anonymous said...

Crap, that should be knew, not new. Clearly I didn't switch my major to English. Or preview.

I should quit while I'm still not ahead.

1:17 PM, March 07, 2006  
Anonymous Anonymous said...

That's ok..Helen can't spell "mathematician" either.

1:19 PM, March 07, 2006  
Anonymous grad03 said...

It occurs to me that a useful way to teach certain mathematical concepts to high schoolers would involve taking an everyday news source such as USA Today or Time and working through the fallacies that underlie many of their graphs and charts.

If we could teach students how to read study results during the years of mandatory education, they'd be a lot more suspicious of printed statistics. But then, maybe that's why some journalists hope we all find math useless.

1:29 PM, March 07, 2006  
Blogger Greg Kuperberg said...

It seems that just about the entire blogosphere, from Daily Kos at one end to this site at the other end, hates Richard Cohen's block-headed column about high school algebra. The blogosphere is correct, of course, as it often is with easy targets. So if I'm going to say anything as a professional mathematician, I should try to make it at least slightly original.

First, Helen only very briefly mentioned the key to the whole question, the crucial thing that she learned from her dad. (Her dad, by the way, got his PhD from Berkeley, just as I did 25 years later.) It's not just that math is important — everyone except Richard Cohen knows that. It's common advice even from bad math teachers, except that it comes across as the eat-your-peas type of advice, which children see through about as quickly as "Why Mommy is a Democrat".

No, the crucial extra is that math is beautiful. That is what is largely missing from traditional American culture (the "Scots-Irish" kind that Glenn Reynolds describes). It's a very different scene in Israel and in many Eastern European countries. For example, take a look at last year's winners of the Putnam, which is the annual college-level mathematics contest for the United States and Canada. Of course these are not students who do math just because it's useful, nor are they cyborgs who multiply large numbers just for the fun it. The Putnam is based on mathematical ideas, not long calculations, and the students who do well on it love mathematical ideas. Anyway, the thing to notice is that three of the top 15 names are Romanian, two are Russian, and one is Ukranian. Only about 2 or 3 percent of Americans and Canadians are from any Eastern European country, yet they were 40% of the top students on the Putnam last year. This shows you the huge cultural interest in mathematics in those countries.

Nor is the love of mathematics evenly distributed in the United States. If you look either at the Putnam or at the professional ranks, you'll certainly find some people from Nebraska or wherever, but you will find a disproportionate number from the New York metropolitan area. That's because New York City has some of the love of mathematics that Romania has, above and beyond just knowing that it's useful. Even if you think that New York is a crazy city, you should admire the romantic way that its best schools teach mathematics. (Or at least used to; conditions may have changed.)

Second, Richard Cohen may be called a journalist, but he's really a pundit. I like some pundits, but I don't trust pundits as a group. That applies equally to the political blogosphere, which is mostly a large amateur pundit pool, despite what the "Pajamas Media" people say. My sort-of favorite newspaper pundit is Paul Krugman, who has ample respect for mathematics, even if it is not quite adoration. My favorite blogger is Robert Park (you can Google him), who is a physicist. All three of these guys are Democrats, and it doesn't matter to me whether silly old Richard Cohen is also a Democrat. On that note, there was a Pew Poll last year of various groups of opinions leaders, and Bush's approval rating among members of the National Academy of Sciences was about 5%. Although I haven't said much about that, because the moment that the subject turns from silly Democrats who hate math to wise Democrats who love math, it invites the charge of elitism. Certainly no one should decide who to vote for just by imitating people with PhDs; that's not the point. The point is that advanced training in mathematics and voting Democrat aren't remotely incompatible.

Third, it does irritate me when politicians ride roughshod over basic mathematics. Both Republicans and Democrats do this all too often, but in particular President Bush does it. The central "achievement" of his presidency is his tax cuts. I have to put "achievement" in quotes because he didn't do the first thing to match his tax cuts with spending cuts; on the contrary, he stepped on the spending gas pedal. But the really irritating, unmathematical side of it is that the tax cuts are just too large to match with spending cuts, even if Bush wanted to. It's like the guy who vows to run a four-minute mile, and there he is at 3/4 of a mile after 3 minutes and 30 seconds, and he says, "Hey I never said it would be easy, but I stick to my promises! I have resolve! I am determined to succeed!" The politicians cross the line from believable optimism to outright denial. Mathematical haziness has a lot to do with it.

1:56 PM, March 07, 2006  
Anonymous "Eric Blair" said...

Grad03---that is a great idea, that usually gets pooh-poohed by education people. I think that the solution is to call it "critical thought."

Leave all the politics out and put all the analysis in. That might fly.

Drj: quite right---find an exuberant, enthusiastic teacher who is willing to do the work necessary to make the subject come alive. But then such people have to survive the teachers bureaucracy.

There should be a "no teacher left behind" program, I think. Don't get me wrong, the only reason I have a Ph.D. today is because of primary school teachers who both tamed me socially and taught me critical thinking. But the bureaucrats are taking over, friends.

That is why "school choice" seems so dangerous to the status quo.

1:59 PM, March 07, 2006  
Anonymous "Eric Blair" said...

By the way, what a delightful post from Professor Kuperberg! This is his speciality, his love, and his prose is almost lyrical.

I recognize the "syndrome" because my wife is a mathematician as well.

Just two things. There is a world of difference between simple arithmatic and real mathematics. Anyone can learn to appreciate and use the former, but the latter is, in my opinion, like art. It takes talent. I am a scientist myself, but one who doesn't use mathematics beyond algebra and a bit of trig. But I recognize the beauty that is beyond.

The other thing is political. Please don't paint GW Bush with such a broad brush. I have for years had to listen to people whine about "cuts" in programs that are actually decreases in a rate of increase from year to year.

Scientists care about truth (mostly). Politicians care about getting elected or re-elected.

I think that a course in critical reasoning should be in every high school. The problem, of course, is that the political types would hijack it.

2:22 PM, March 07, 2006  
Anonymous Anonymous said...

We could try to make math textbooks more interesting, like this one:

http://www.netcomuk.co.uk/~jenolive/index.html

2:57 PM, March 07, 2006  
Blogger tcobb said...

We could try connecting the abstract to the real. If its just taught in a vacuum, math is very abstract and dry. It seems like a silly game. But if you connect it to solving real problems that people encounter, then one tends to grasp its utility. I remember when I was in high school and I was introduced to the concept of i --the square root of negative one. I thought the whole concept was useless--until I found out that certain things, including modeling the flow of air around an airplane's wings, couldn't be done without it. That made all the difference in the world to me at the time.

The very fact that this girl couldn't see HOW she would ever need algebra is an indictment of the people who were entrusted with teaching her math. If its just taught as an abstract game with no practical application to the real world you can't really blame students for considering it to be useless. For them, it is useless.

3:26 PM, March 07, 2006  
Anonymous O.B. said...

Pardon my bluntness, but people who say they don't need math, or that it doesn't matter, know that's not true. It's just a defense mechanism to disconnect the idea that they can't do it with the idea that they're not smart. "See, I just choose not to be good at math ... it's not that I'm lazy or stupid."

Actually, it's near certain that you're one or both. High-school math just isn't that hard.

3:30 PM, March 07, 2006  
Anonymous O.B. said...

I should say that the above comment is meant to apply to adults such as Cohen, not to young children who indeed might not know better (but will surely learn, whether they admit it or not).

3:34 PM, March 07, 2006  
Blogger Greg Kuperberg said...

Some other odds and ends: First, there is a difference between what mathematics you should learn and what mathematics you will actually use. It is important to pass algebra not because you will often need algebra in daily life, but because it solidifies your understanding of what came before. I wouldn't trust someone who only had high-school algebra to do real-life algebra problems. But I might trust such a person with the basic arithmetic that you need to understand loans and taxes. Likewise, I wouldn't trust a good calculus student to do a stress analysis of a skyscraper. But I might well trust a good calculus student to do the basic algebra that arises all the time in finance, engineering, and science. That is the main reason that so many university majors require calculus. It's not because of extensive use of calculus itself, but because they want to make sure that the students completely understand the previous thing.

Part of the point is that the mathematical way of thinking, not particular mathematical facts, is what is useful in real life. And in my personal view, not just useful but part of the complete human experience. Even to understand baseball, it is nice to see immediately that if the four bases make a 90 foot square, the distance from home plate to second base is therefore 90*sqrt(2) = 127 feet. It's not something that you have to know; but then, you don't iave to think about baseball at all — it's just a game. On the other hand, there are a lot of particular mathematical details that arise in real life from time to time. If you can't think mathematically, you won't even know what you're missing, usually. For example, you can quarrel all week over money without ever realizing the germane arithmetic that should end the quarrel immediately. One of the things that I like about being married to a physics PhD is that we can have logical conversations about our personal spending and our taxes.

This is also a problem with the typical American school district, from most of the teachers to most of the parents. Usually they don't even know what mathematics is missing from the school. In particular, they don't know what makes mathematics beautiful. They have trouble telling the teachers who like math from the ones who don't. There has been a lot of discussion about how to teach math. Different methods to teach math can be useful, depending on the method, but the most serious problem is to decide what to teach, and who is to teach it. Most teachers who like mathematics and who know what to teach can find adequate teaching methods with some practice. Teaching methodology is partly a matter of personal style and shouldn't be micromanaged.

Similarly with school choice. School choice could be useful. But it also might not be useful, if the parents do not know what mathematics is missing from the schools. It is therefore important to have good state math standards, and it is important for the system to find math teachers who like math and who know the math that they are supposed to teach.

3:35 PM, March 07, 2006  
Anonymous "Eric Blair" said...

I had to laugh at a couple of comments from Professor Kuperberg (no worries---in a good way):

He wrote:

"One of the things that I like about being married to a physics PhD is that we can have logical conversations about our personal spending and our taxes."

Friends, this perpetuates a myth that all scientists (and mathematicians are scientists, folks) are logical and common sense-oriented regarding everyday life! It might be true in some cases, like Professor Kuperberg's, but it ain't where the smart money bets.

Scientists are just people, and as people, they are good at compartmentalizing parts of their lives. This is why many intellectuals and scientists are so utterly clueless about politics.

The other business has to do with school choice. So long as we underpay teachers and overpay administrators, we will have horrible schools. So long as we at universities give students who consider a career in teaching in the primary and secondary schools the idea that it is a "second class" career choice (which many academics transmit very clearly to students), we will have horrible schools. And most of all, so long as we allow our schools to follow "political fashions" in education, we will have horrible schools.

Look: when students knew more math in 1966 at a given age than they do in 2006, it is time to ask why. And if something isn't working, we need to go back to what we KNOW works.

I don't mean to come across all butch on this subject, but I have two young sons and I worry endlessly about their schooling.

3:56 PM, March 07, 2006  
Blogger Greg Kuperberg said...

By the way, if anyone wants to understand better how mathematics is beautiful (not just useful), I can recommend a celebrated book,
The Mathematical Experience, written by two mathematicians. It is even better with its old cover, which has unfortunately been replaced in new printings, but it is still great.

Also a look at the Putnam problems might give you some idea of at least the tone of good mathematics. Here is one of the questions:

Let H be an n n matrix all of whose entries are 1 and whose rows are mutually orthogonal. Suppose H has an a by b submatrix whose entries are all 1. Show that ab <= n.

And one of the solutions for this problem begins:

Suppose without loss of generality that the a by b submatrix occupies the first a rows and the first b columns. Let M be the submatrix occupying the first a rows and the last n-b columns...

The point is not to try to understand either the question or the solution. It's abstract stuff and it requires training. My point is that this is not a long eat-your-peas calculation. The solution is an explanation written in English, with just a few symbols here and there. That is the real style of mathematical thinking. Even though this question is college-level material, the principle applies at all levels. It is both wonderful and useful to be able to suppose something "without loss of generality", not necessarily using those words, but just to be able to think in those terms.

4:11 PM, March 07, 2006  
Blogger BarbO said...

Helen, you might want to investigate Kumon mathematics for your daughter. When I was in junior/senior high school, I also hated math. I earned my first college degree in Russian. Fortunately, a stint in the military afterwards showed me the potential of technology, so I went back to school for a degree in electrical engineering--and promptly fell in love with math. I ended up getting my doctorate in engineering. Many of my engineering friends are from China and India, and they understand the poor state of education in math in this country (by 12th grade, US students are two to three years behind a typical Chinese student in math). I noticed my friends taking their children to a local Kumon center. So I put my own children in Kumon. My girls are not natural math whizzes or anything, but because of the extra practice they've gotten over the years, they are now VERY proficient in math. Especially if your daughter doesn't like math (at least at present), I highly recommend the Kumon program. You'll be very pleasantly surprised at how she will change.

4:16 PM, March 07, 2006  
Blogger Helen said...

anonymous,

Thanks for the spelling tip--you wouldn't know that I won a spelling bee in 6th grade!

Barbo,

Thanks for the advice. I would hate to see my daughter become discouraged at this early age.

4:22 PM, March 07, 2006  
Blogger Greg Kuperberg said...

The main difference between Kumon and any other solution is that it is far more widely marketed than any other solution. That doesn't necessarily make it better or than if it weren't widely marketed. iPods are widely marketed and they are good; Pepsi is widely marketed and it isn't very good.

It's not quite my style to buy into widely marketed fads, even when they are good. In particular, I prefer European cultural appreciation of mathematics, and the American version to the extent that it exists, to Asian approaches like Kumon. I have nothing against Asian culture, but I am just less used to it. If nothing else, it's harder for me to know which Asian teachers truly love mathematics, and which ones love to teach, perhaps, but don't love the material.

A book like The Mathematical Experience may aim too high for junior-high school students who are not sold on mathematics. It is great for adults and advanced high school students. Another celebrated book is Mathematics: A Human Endeavor, by Harold R. Jacobs. Jacobs also wrote textbooks on algebra and geometry in the same vibrant style. Any book by Jacobs would be a good antidote to the sterile mathematical environment that you see in many grade schools.

A promising quote about the book that I cited:

Harold R. Jacobs originally wrote Mathematics, A Human Endeavor for students who had failed algebra and disliked math.

This is from a Christian home schooling site. That is also not my style, but Christian home schoolers, just like Asian educators, can certainly be right sometimes.

5:55 PM, March 07, 2006  
Anonymous Anonymous said...

Geometery could be taught by adding in a little history: geo = earth, meter = measure, geometery is how you measure the earth, first developed by the ancient Egyptians (Nile flooding), still used by surveyers and astronomers, manufacturers, and animators.

Teach a little of the history of the field, its uses, i.e., provide context.

6:17 PM, March 07, 2006  
Anonymous Denise said...

Does anyone else see the irony of a man who says he can't understand percentages a) being a journalist, and b) using words like "most" and "countless"? Not to mention in his paragraph on reasoning. He says that the proof of math not teaching reasoning is that a girl who was good at math didn't know where the Sahara Desert was, and pointed to the Gobi Desert instead. Now, let's jump past the complete non sequitur-ness of this "proof" and look at it. I'm assuming that this poor, illogical, unthinking soul was looking at a map. Let's think, umm, logically and reasonably.
1) The Sahara? That's a big desert, and I'm pretty sure it's not in the US, or I'd know.
2) Ergo, I should look at the rest of the world. Let's see, some deserts in South America, a giant one in Africa, and there's a pretty good size one in Asia too. I think the Sahara is on the other side of the world, so let's just pick one...
And she picks the Gobi. Not too bad for "not being able to reason".

7:18 PM, March 07, 2006  
Anonymous denise said...

PS-- my husband, on reading my last post, made a good point. he said, "Actually, I can understand him using "countless" a lot. It perfectly captures his inability to count."

7:28 PM, March 07, 2006  
Blogger BarbO said...

Greg, actually there is a big difference between Kumon and most other widely marketed methods. I've written some academic papers about Kumon's strategies and effectiveness--(the papers are online at http://www2.oakland.edu/users/oakley/Research%20and%20Publications.htm). Most people are unaware that for roughly the past twenty years, the National Council of Teachers of Mathematics (NCTM) has strongly discouraged practice and repetition to assist in learning mathematics. The position of the NCTM has been that it is more important to understand the concepts behind mathematical operations than to memorize facts--that repetition kills creativity. Virtually any scientist, engineer, or mathematician, however, shakes his or her head at that perspective. Those who USE mathematics know that absolute mastery of fundamental facts related to mathematics (that is, that a person instantly knows that 7 times 3 is 21, rather than having to punch it in on a calculator or place beans together to figure it out) is critical to being able to advance to higher levels of mathematical thought. In fact, latest neurological findings are supporting that practice of mathematics is extremely important in being able to learn mathematical concepts. Learning math, in fact, is a lot like learning a language or a musical instrument. Who would ever think they could learn the guitar by just "understanding" theory instead of actually practicing? Modern K-12 education in the US has very little practice of mathematics. This, coupled with convoluted textbook explanations such as those promulgated by the Chicago Math series, has really done a disservice to American students. The proof is in the pudding. Every day, I see Indian, Chinese, or Middle Eastern engineering students in my classroom who far outshine American students when it comes to anything mathematical. These students rarely had anything like the privileged education of standard Americans, and aren't usually brilliant (the brilliant ones go to Harvard or at least U of M). But their grasp of mathematics is superior. In my discussions with these foreign students, they are often completely dismissive of American K-12 mathematics preparation. An excellent blog related to mathematics K-12 education in this country can be found at http://www.kitchentablemath.net/twiki/bin/view/Kitchen/WebHome

8:48 PM, March 07, 2006  
Blogger Kathy said...

Helen, look at www.livingmath.net. The site has a whole math course of "living" books, which you could use to supplement what your daughter is getting at school.

We're going to use material from that site, but primarily Ray's Arithmetic, a mental arithmetic course focusing on word problems. It was very popular in one-room schools in the 19th century!

9:33 PM, March 07, 2006  
Blogger Greg Kuperberg said...

BarbO: Virtually any scientist, engineer, or mathematician, however, shakes his or her head at that perspective.

Yes, I am one of those mathematicians who shakes his haead at the concepts-only perspective. Of course drill is important. But I also shake my head at the drill-and-kill perspective, which I personally suffered in third grade. In fact, by now I shake my head at all of the mantras about teaching methods in mathematics, because I think that what really matters is the content.

After all, as you say, who would ever learn guitar by "understanding" it without practicing it? And who would ever learn guitar by practicing it without understanding it? That is, by drilling a lot of chords and scales without ever combining them to make music. What is the fun in that? My interest in guitar, if I were to learn it, is that I love guitar music. (Seriously, we have a CD of classical Spanish guitar music which I think is fabulous.)

Just like I learned mathematics because I love that too. I don't particularly like arithmetic; I was never very fast or reliable at it. In any case there is more to life than just eating your peas. I love mathematical ideas, because I learned from my mathematician parents that mathematics is beautiful, just as Helen learned it from her mathematician father. I think that most reasonably bright students, not just the Harvard-bound ones, can learn to love mathematics at least a little bit. Otherwise I don't see how they can be any good at it, no matter how much they drill the algorithms or fluff the concepts.

It is very frustrating to be caught in an interminable partisan quarrel between two groups of math educators: the Little Endians who believe in drill but not concepts, and the Big Endians who believe in concepts but not drill. They are both half right, yet they both miss the point. I wish that they would both go blow and let working mathematicians, scientists, and engineers set the content standards, then let the teachers choose the best mix from a variety of methods.

12:03 AM, March 08, 2006  
Blogger Greg Kuperberg said...

Also, as it happens both of my children have used the "Chicago Math" series, otherwise known as "Everyday Mathematics", in some of their grades. My daughter is going through one of the workbooks this year. I don't see what is so terrible about this book series. I won't claim that it's perfect, but I think that it has the basic mix of drill, concepts, and explanations that a mathematics curriculum ought to have. I don't understand the position that explanations sabotage learning because they might be confusing. Are students supposed to plunge into drill without explanations?

As I said, I don't know know a whole lot about Kumon and it might be okay. But if the reason that it's so great is that it pursues only one teaching method, drill uber alles, then that already makes me wonder. That and the fact that Kumon is a tutoring brand and not just a textbook series. Not that I have anything against for-profit companies; the point is that their main goal is to retain high-paying tutees rather than to satisfy a larger base of grade schools.

12:33 AM, March 08, 2006  
Blogger Greg Kuperberg said...

Sorry to post three things in a row, but I can concede a partial correction to the previous comment. It seems that "Everyday Mathematics" is on the concepts side of the war between the drill and concepts camps. The mathematicians who entered into this discussion in the late 1990s, and who did a great job with the California State math standards, don't like this series. One article suggested that it is the best of a bad bunch, which may be why I didn't realize that it was in a bad bunch. The favorite of the concepts camp is "MathLand", which I have also seen and which could be the very worst one.

Well, I am not an expert which particular textbook series are the best ones or which ones fit whatever state math standards. I am particularly unknowledgeable about what textbooks are best for math teachers who didn't major in math. The California State Math standards were written by people who I trust, and who agree with me about general principles. The approved list of textbooks that came from these standards is here.

1:17 AM, March 08, 2006  
Blogger quadrupole said...

Greg... your last post sort of begs the question... why the hell do we have people without degrees in math as math teachers? What get's taught in high school and non-technical majors in college is basically the alphabet of numeracy. No words, no grammar, just learning the letters. We would be scandalized at the notion of having people who couldn't read, but who did know the alphabet teachiner reading, why are we not scandalized at people with analogous levels of mathematical background teaching math?

5:24 AM, March 08, 2006  
Blogger BarbO said...

Greg, you made good points about killing through drill. I'm afraid in my previous post, by emphasizing the importance of practice, I made it sound as if I don't think understanding of mathematics is important. Nothing could be further from the truth! I absolutely agree that both practice and understanding are necessary for true mastery of mathematics to take place. One of the reasons I'm so impressed with Kumon is how cleverly designed it is. Yes, it gives practice, but it is always gradually ramping up the level of learning needed. A cursory look might make you think it's just a bunch of rote materials, but it isn't that at all. (I spent ten years with my girls going through it, so I know the curriculum pretty well.) Kumon is valuable purely as a supplement, however. It can't replace a regular math curriculum, and indeed, isn't meant to do so.
I've taught in China, and have noticed how difficult it is for those with a standard Chinese education to think "outside the box." This is true even, to some extent, with mathematics. However, I think this has as much to do with traditional Chinese teaching methods (I talk--you listen and don't ask questions) as anything else. I think we could best teach our students here in the US by continuing to emphasize our strong point of questioning and understanding the material, but also give our students additional cleverly designed practice that goes far beyond mind-numbing rote. Kumon, to my mind, is the ideal supplement that can join the best of the East to the best of the West.
Incidentally, Quadrupole mentioned the poor background in mathematics that teachers often have in the US. I believe academic politics plays a role in this. It is to the best interests of any school of education at any university to maximize how many courses are taken from that school (ie--in the school of education), and to minimize what is taken outside that school (ie, in the professional subject, such as math, that will be in the teacher's area of competency). Certainly I see at my own university that schoolteachers-in-training have to take an extraordinary number of courses in "how" to teach--but far fewer in the subject matter of "what" they are to teach.

7:33 AM, March 08, 2006  
Blogger dadvocate said...

Greg - Do you have any books, etc. that you would recommend for a parent to help their children with math/algebra/geometry/etc.? I have children in school in 4th,7th and 11th grades. My basic math skills are solid enough that I can help them with any thing they've brought home so far but I'd like to be able to do a little extra with them at home, especially over the summer.

Thanks.

10:47 AM, March 08, 2006  
Blogger Greg Kuperberg said...

BarbO: From the sound of it, the logical function of Kumon is to help students catch up in mathematics with extra drill. This is a very reasonable form of damage control for students who hate mathematics, get very little practice in it, and have no feel for it. But, even if it is an effective antidote to math failure and math phobia, it won't do much to get people to really like mathematics and move ahead. (Except that they will move ahead of other students with math phobia.) You say that it is only intended as a supplement; if that is way that it is really used, then there is nothing wrong with it.

However, drill-and-kill supplements increasingly miss the point from algebra (often assigned to 9th grade) onward. From this point on, the concepts become more important, and harder for the teachers too. The concepts-only crowd does not steer teachers away from the default drill-only mentality at this level, although they might create a worst-of-both-worlds curriculum without drill or concepts. The overriding problem becomes sheer lack of teacher preparation in the subject matter (although as you say, they are stuffed with preparation in teaching methods), and an absence of cultural enjoyment of advanced mathematics. The one bright spot is AP courses, which cannot be trashed too badly because they are defined by AP exams.

quadrupole: Yes, the core problem is definitely that so many math teachers in grade school hate math themselves. No matter what methods they prefer, they end up teaching the children their attitudes. There was a survey, mentioned in the Notices of the American Mathematical Society, that only about half of elementary school math teachers in America can name a number between 3 and 3.1. (This is impromptu; maybe with some studying they could do it.) But it is a mistake to a declare a scandal, even though it is scandalous, because the sad truth is that people in education declare scandals and crises all the time. Declaring a crisis has been abused, just like the color-coded terrorism warning system has been abused.

10:59 AM, March 08, 2006  
Anonymous Timothy said...

The Best Answer: Bring Back Square One.

11:48 AM, March 08, 2006  
Anonymous dweeb said...

Greg, I've found that with every musical instrument I've learned, it was necessary to begin with mechanics and practice of the techniques, before I could make music enjoyably. It's the same for many other endeavors. You can't read great literature until you've mastered the mundane skill of decoding the written word. You can't have a meaningful discussion of great movements in history if the participants don't know the timeline of events, and you can't appreciate the beauty of math if you can't do arithmetic. I had plenty of drill and kill math, and the beauty part came largely from applying it. When Newton created calculus, he did so because he needed a tool to describe the mechanical behavior of the universe, and for me, calculus became a thing of beauty not when I learned about derivatives, but when, in my physics class, I learned sa that this tool described the relationship between velocity and acceleration.

By the way, the marketing materials for "Everyday Math" are adamantly opposed to learning standard algorithms like long division. If your kids weren't damaged by Everyday Math, then the school wasn't following it (and based on the sales agreement our district got, they're in breach of the sales contract) All the math literate people I know in the district spend hours each day re-teaching their own kids math. Perhaps your experience with it is somewhat unique - I wonder if a child raised by an involved parent with your love for and knowledge of math could be negatively impacted by any even the worst math curriculum.

Oh, and I'm pleasantly surprised that you didn't take the bait, although you did manage to work in your mandatory anti-Bush diatribe.

12:32 PM, March 08, 2006  
Blogger Greg Kuperberg said...

dweeb: First, "Everday Math" was revised in 2002 precisely in response to pressure from California. Around 1998, California adopted first-rate math standards that were essentially written by some Stanford math professors. (Whatever else I may say about Pete Wilson's politics, I'm very grateful that he screwed the education establishment in such an enlightened way.) The standards are clear and traditionalist, but, even better, in later grades they have some of the romantic view of math that I have described.

But anyway, "Everyday Math" was revised. It still fell short of the state standards, which rather disgusted some of the California experts, but it is definitely less bad than it was before. It does teach long division and it has a plausible amount of drill, if maybe too little for many students. You have guessed the truth on one point though. My daughter and another student are on a special track and they have their own workbook, but no textbook. My wife and I provide the explanations, and it may be true that our explanations are better than those in the textbook. It is also true that my daughter needs less drill than some.

As for "anti-Bush diatribes", you were the one who politicized the discussion, not me. You "asked" whether (or rather, claimed that) my profession conflicts with my politics in this case. The answer is no, it doesn't. I don't much care what Richard Cohen has to say on any issue. Cohen is not a journalist and he also isn't consistently pro-Democrat. And the plain truth is that the vast majority of working mathematicians are disgusted by Bush's number sense. Even most mathematicians who voted for him have nothing positive to say about it, only negative things to say about Democrats. The very Stanford mathematicians who wrote the standards that defeated Gabriela in LA are on my side of this, I'm sure.

1:18 PM, March 08, 2006  
Blogger DRJ said...

Greg - Are you and your wife effectively home-schooling your daughter in math?

1:25 PM, March 08, 2006  
Anonymous "Eric Blair" said...

Oh, well.

Dr. Kuperberg, I was so enjoying your discussions of mathematics. Lyrical and nice. My wife is a combinatorist and graph theorist, so I know well how you feel and appreciate how well you express that love of mathematics.

I also know how it feels NOT to be able to "do" higher maths.

I have followed your posts for many months, and the vast number of your comments about politics (and your posts usually involve politics) are anti-Bush. You are, with all due respect, far, far, FAR from being nonpartisan---despite your protestations. All anyone has to do is look over your prior posts across lots of blogs, including this one.

And that is okay. Really it is. We truly get it.

I was just hoping for more positive solutions about teaching math, not more "Bush/Republicans/nonacademics is/are awful" meme building. After all, the scores keep going down and the skills keep going down, no matter who is in the White House, or who gets elected to Congress. Good Lord, look at the SAT!

But someone like YOU could make a difference, precisely because you are not a politician.

You should be writing editorials about the short sightedness and anti-science bias the Cohen column reveals. That editorial---about the beauty of mathematics, and how it needs to be shown to schoolchildren---could end up in your Aggie at UC Davis, or the BEE, or even a nationally based magazine. You have the "union card" two ways: as a mathematician and as a parent of children taking mathematics.

And that contribution would mean so much more, and have so much more impact than any amount of the political wrangling.

Sorry to go on. But again, YOU could make a difference this way.

1:36 PM, March 08, 2006  
Blogger Greg Kuperberg said...

DRJ: The truth is in between. She is very independent and she does most of the exercises at school. She does them with the other student. Just like me, one of her methods of learning is to explain the material to someone else. But yes, when she doesn't understand something, we often explain it to her. The teacher is reasonable, but she doesn't have much time for two special-case students.

dadvocate: My simple answer (and again, it is not an expert answer) is three names: Saxon, Jacobs, and Gelfand. Just find the high-school algebra and geometry books by these people. Buy from all three (you won't go broke) and see which you like best. Saxon provides the Japanese-style drill, Jacobs the American-style fun, and Gelfand the Russian-style love of mathematics. (Gelfand is by the way a towering figure in research as well. He was to Russian mathematics what Feynman was to American physics.)

I have less advice for the earlier grades, but for a start you could look at the California approved textbook list that I already linked above.

You should also buy "The Mathematical Experience" for yourself.

1:37 PM, March 08, 2006  
Blogger BarbO said...

Greg, Kumon isn't at all a program only for those who have fallen behind. When a student begins the program, he or she is tested to see his or her ability level--high, low, or par-for-the-course. The student is then placed at a level slightly below ability level. As a result, the student initially advances very quickly. This inspires confidence, particularly for the math phobic. But it is just as inspiring for the math superstar. Kumon offers a lot of different awards, and children find it really satisfying to jump up to a higher level. As a result, I see a LOT of exceptionally gifted children from Indian and Chinese families using Kumon to help slake their child's thirst and feed the child's desire to learn more. Kumon works best because it is a nicely structured program that a working Mom and Dad can use on a daily basis to help their child, while simultaneously having experts on hand during the once or twice a week visit to the center to assist with their child's progress.
My older daughter in particular was often well in advance of her regular classroom math curriculum. It was a struggle for her when she first saw the concepts through Kumon, but when she saw the concepts again through her regular class, it reinforced everything nicely and she really enjoyed what she was learning. (It's like taking a second trip through a foreign country, but this time you know more of the lingo.)
Anyway, I've gone on a bit about Kumon on this thread just because once Americans hear that a math program has a rote component and is broadly marketed, it must be mundane and, at the very least, suitable only for the less gifted students. But Toru Kumon, the teacher who developed the method, was a good friend of Shinichi Suzuki, who developed the Suzuki method for violin. Kumon's methods are exceptionally well-thought out and, as I can attest after watching my daughters' development, very effective.

7:10 PM, March 08, 2006  
Blogger BarbO said...

I should mention that I've tried extra math books and such to help assist my children. Those are nice, but I frequently found myself missing days of supplemental tutoring because I got caught up in other things (I'm the working spousal half who focuses on making sure the childrens' education is going well). Kumon is great because it gives the children (and I) a structured set of materials to do each day. With Kumon, it wasn't as if I could be left feellng guilty about missing a day of math tutoring--my daughters would dig the materials out for themselves, because they didn't want to miss keeping up on their daily packets. They bugged me about the packets only if they needed help.

7:25 PM, March 08, 2006  
Blogger Greg Kuperberg said...

I have to be skeptical that the same curriculum that rescues students with math phobia is somehow inspiring for the math "superstar". Actually, I don't like the term "superstar". I don't think that it's healthy for children to be called "superstars", or for children to hear other children described that way.

I have seen some very advanced students over the years. Maybe the most important intellectual transition for advanced mathematics students is to learn how to compose proofs. I do not mean the Keystone Kops version of proofs that are taught in 10th grade geometry. Rather, I mean the thrilling version (at least, thrilling for people who like math) that most math students see in upper-division college classes, or in graduate school. Some high school students are also ready for it. For example, the famous Ross Summer mathematics program at Ohio State University introduces high school students to proofs.

I am looking now at the Kumon curriculum on its web site, and I do not see proofs anywhere. Which is just as well for behind students, or average students, or even moderately advanced students. In fact it's sound pedagogy: You shouldn't teach proofs too early, just like you shouldn't teach Arabic when Spanish would already be too hard.

However, if there is no mention of proofs anywhere in the entire Kumon curriculum, that just confirms that it is a factory product and not an "inspiration" for "superstars". It could be a good factory product. After all, public schools are also a factory product, often a very bad one. Kumon may be clearly better than that. I don't think that it's appropriate to describe it as mathematical salvation for all children.

I can only say the same thing about the Suzuki violin curriculum. My wife saw Suzuki and my children have seen Suzuki (at least the music books, if not the full learning system). Just like Panasonic TVs and Toyota cars, Suzuki violin books are a great factory product. But that's all that they are. They won't get you to Carnegie Hall, nor even necessarily to like the violin.

9:41 PM, March 08, 2006  
Blogger dadvocate said...

Greg - thanks for the three authors. I'll check them out.

10:40 PM, March 08, 2006  
Blogger DRJ said...

I've been reading your comments for awhile so I know you are a caring parent, and I'm not surprised that you take an active role in your daughter's education. Good job.

11:15 PM, March 08, 2006  
Blogger DRJ said...

Sorry. My last comment was for Greg K.

11:16 PM, March 08, 2006  
Anonymous Anonymous said...

Drj is right on, as usual. Sometimes---many times---GK comes off as snippy, snobbish, elitist, and supercilious.

But look at his comments about mathematics and his comments about his children.

All we ever see of each other is posts. There is a great deal more--good things--- to GK than hating HalliBushChimphitler.

11:05 AM, March 09, 2006  
Anonymous dweeb said...

Greg,

Our district purchased Everyday Math in 2003, so it's probably the revised version, and if so, it's still an abomination. My colleague spoke to his son's teacher, who also hates it, but is unable to do anything about it. Our schools keep old textbooks around for students to take home for independent work, but when he asked for one of the old math books, he was informed they were all collected by the administration, because the sales contract for Everyday Math required that they "trade in" the old books, and that EM be the only materials used.

I'm a little confused - I seem to recall you saying your daughter using EM, but you say that even revised, it didn't meet CA standards, and you live in CA. Care to clarify the apparent conflict?

I pretty much made it obvious I was poking at you with the politics part, and you did manage to partially bite on the hook. As others have observed, this is the first thread you've revealed any facet of yourself other than Bush hatred, and I like to think that so blatantly baiting you created an incentive for you not to act predictably, and allowed us all to know a more interesting side of you.

By the way, you already admitted that the opinion of a bunch of math professors is not a basis for a voting decision, so why do you keep harping on their opinion of Bush? In fact, why don't we just leave Bush out of this entire thread (I didn't bring him up, you did.)

By the way, for a 9th or 10th grade geometry student, those elementary proofs lay the groundwork for more interesting ones, and at that stage, they can provide the same excitement, not to mention a desire to explore further. We had a Calculus teacher, who, when we questioned the importance of spending so much time on seemingly unimportant concepts, would respond by proving something absurd by ignoring the concept in question.

I've talked to a lot of education majors about all the novel methods they're learning about. There's this tendency to condemn the classic classroom lecture wit drilling method as some sort of sinister conspiracy, but it's really the natural evolutionary result of the drive for universal education - to educate everyone with the limited resources available, students are going to have to adapt to a certain degree to a one size fits all set of methods. You can't accommodate all learning styles without limiting the availability of education, just like, if we all had to wear custom tailored clothes, half the population would be naked, and if there were no factory farms, only the rich would have eggs. Even with the classic Prussian model of teaching, enough people fell in love with math to put humans on the moon. What bothers me most about all these attempts to re-invent education is the very endeavor of seeking them takes onus and responsibility away from the individual student, and places it upon the institution. Teacher isn't making it exciting for you? Too bad, because someday, your boss sure isn't going to feel obligated to do so. Cowboy up, and learn this stuff because it's your job. For thousands of years, people with sufficient knowledge of math, literature, history, and any other subject have discovered the romance in it without being led by the nose, and I see no reason to believe that this generation is less perceptive.

12:17 PM, March 09, 2006  
Blogger Greg Kuperberg said...

dweeb: Our district purchased Everyday Math in 2003, so it's probably the revised version, and if so, it's still an abomination.

I don't completely disagree with this, but it can be a mistake to get too polemical about just the textbook. There are crushingly horrible textbooks that all but prevent learning, but Everyday Math is not that bad. The most serious shortcoming is that it would require an unusually prepared math teacher (at least, unusual for elementary school) to make good use of it. It also covers the material more slowly than California wants it to. This does argue that it isn't a very good choice, but it is something of a compromise and you could work with it as one.

I seem to recall you saying your daughter using EM, but you say that even revised, it didn't meet CA standards, and you live in CA.

It seems that the state's approved list is not binding on districts. This shows you how local control can cut both ways. (Note that the local districts do not have nearly as much independence as charter schools do.)
The children still have to take the annual state test, called STAR, which is written to the state standards.

By the way, you already admitted that the opinion of a bunch of math professors is not a basis for a voting decision, so why do you keep harping on their opinion of Bush?

Because you (and others) claimed outright that Richard Cohen's inane opinion column was a contradiction between my profession and my politics. So I was explaining how it wasn't. If you don't want to talk about national politics, don't make characterizations.

By the way, for a 9th or 10th grade geometry student, those elementary proofs lay the groundwork for more interesting ones, and at that stage, they can provide the same excitement, not to mention a desire to explore further.

I suppose that they do, sometimes. I always hated statement-reason tables. I was brought up to think of a proof as an explanation, and not a formula or a catechism. (As you might guess, now I come up with them for a living.) They should be almost like political debate. You try to prove something that you care about, except that, unlike in politics, one side should win the debate conclusively. Also, the other side of the debate is the devil's advocate in your own head, not usually a separate person who genuinely disagrees.

I have the feeling that most high-school math teachers do see proofs more as catechisms that should be memorized. There is usually no passion that what you are trying to prove might be surprising or untrue.

Even with the classic Prussian model of teaching, enough people fell in love with math to put humans on the moon.

This was true in a different way than you mean it, because Wernher von Braun was German of course. Okay, that's dodging the point, because at least most of the Apollo engineers were American. More seriously, the United States is a vast country, and what you may not realize is that actually a too-large fraction of the best American engineers worked on Apollo. At the peak it was 5% of the federal budget. Part of the reason that human spaceflight is going to pot these days is the national shortage of technically literate Americans. The economy makes up for this shortage the same way that it makes up for the oil shortage...

1:00 PM, March 09, 2006  
Blogger WhidbeyIslander said...

As to how to get young people to become interested in math:

I am reminded of William F. Buckley's prescription: Put a math section on the test for a 16-year old to get a driver's license and they will burn the midnight oil. Have you seen 15-year olds memorizing your state's driving law manuals?

2:51 PM, March 09, 2006  
Anonymous Anonymous One said...

Dr. Kuperberg wrote (with a bit of a snippy tone, I'm afraid):

"Because you (and others) claimed outright that Richard Cohen's inane opinion column was a contradiction between my profession and my politics. So I was explaining how it wasn't. If you don't want to talk about national politics, don't make characterizations."

Apparently, this was in reference to this comment by dweeb:

"Now, the real fun part is watching Greg struggle with the conflict of responding to the implied disconnect between his ideology and his profession, rather than finding yet another way to work in a diatribe about the Iraq situation."

Funny. Here I thought that dweeb was making the excellent point that Dr. Kuperberg would find a way to defend a personal political viewpoint from a nonpolitical trained profession opinion. True to form, Dr. Kuperberg did, with his statement that:

" The point is that advanced training in mathematics and voting Democrat aren't remotely incompatible."

The deeply, deeply sad part is that Dr. Kuperberg is so blinded by his view that his partisanship is due to his extreme intelligence that he misses the REAL point:

Advanced training in mathematics and voting Republican are not remotely incompatible.

Therein lies the rub. Most intellectuals genuinely believe that they are correct in some "large C" cosmic sense, when their opinions on politics or everyday life are NOT seated in their trained critical reasoning, but in their everyday preferences.

A physicist doesn't choose a sandwich based on the absorption spectrum of kippered beef. She or he bases it on personal feelings and preferences.

Dr Kuperberg has tried to say this before about politics and academia, but is stymied by the genuine facts of academe: it is a Democrat-only social club.

Sure, there are exceptions, but as has been said before, let folks like Dr. Kuperberg (who did indeed claim that politics has nothing to do with hiring or tenure) stand up and say something remotely anti-Democratic Party on campus. Just to prove that there is no "anti-Republican" bias on campus, I mean.

I notice that he will not. And for good reason.

The poster who enjoyed Dr. Kuperberg's writing on mathematics was spot on: there was a subject where Kuperberg's opinion actually is more valuable than (for example) mine. His training and talents are there.

Otherwise, his opinions have just as much value as anyone else's.

I would like to think that Dr. Kuperberg would agree with that sentiment, but I suspect his elitism (double barreled as Harvardian AND European) will prevent him from doing so.

It's like the Leo Szilard story. When he was made a US citizen during WWII, the journalists (at that time, fairly patriotic) asked him what he thought of democracy, coming as he did from Hungary.

He said that he thought it was a fine idea, but one thing bothered him about it. The journalists pressed him for more information.

"Vell," he replied "I understand that the vote of a vise man is the same as the vote of a fool. Vat bothers me is that two fools outvote a vise man."

Sorry for the long post. It's okay that Dr. Kuperberg is knee-jerk partisan. Perhaps he will be equally contrarian when Democrats run things; I hope so, as I sense the contrarian in his nature (based on his posts). What is also true is that he writes wonderfully of mathematics, and he should do much more of that.

Not because he should be silent on issues political, but because his voice about matters mathematical is much more needed today.

5:27 PM, March 09, 2006  
Anonymous "Eric Blair" said...

In the interest of a short posting to this comments section, I noticed that greg kuperberg earlier wrote:

"It seems that just about the entire blogosphere, from Daily Kos at one end to this site at the other end, hates Richard Cohen's block-headed column about high school algebra."

I agree that Cohen is a blockhead, and a partisan blockhead at that, but I hardly think that this blog is as far to the rightwing as Daily Kos is to the leftwing.

I haven't seen one poster here celebrate the death of a political opponent. Koskids do that with regularity.

It does not make them, nor their cause (kause??) very attractive. Folks here are generally pretty even tempered.

6:12 PM, March 09, 2006  
Anonymous Anonymous said...

"Kause". Good one, Eric Blair.

7:46 PM, March 09, 2006  
Anonymous "Eric Blair" said...

Thanks, Anonymous 7:46. I am appalled by some of the things Ann Coulter has written (meaning comments about the death of political opponents). We cannot condemn the hateful bile of many Kos posters while allowing that.

Civility seems not to exist anymore. Wit is confused with mere humor. And it is possible to vehemently (even amusingly) disagree while still being civil.

Give me the clever days of Disraeli and Gladstone. Just a sample:

(Gladstone: "I do not know if my opponent will die of the pox"---by which he meant syphilis---" or the hangman's noose"

Disraeli: "That would depend, good sir, on whether I embrace your politics or your mistress."

Compare that to what goes on Congress (or the White House Press Room) today.

11:21 PM, March 09, 2006  
Anonymous dweeb said...

Because you (and others) claimed outright that Richard Cohen's inane opinion column was a contradiction between my profession and my politics.

Greg, I was teasing you; loosen your tie, and go have a few drinks to kill the bug up your behind. It's becoming apparent that, although you're obsessed with hating Bush (and I can think of my own reasons to hate him) you're not a complete party line liberal, and you don't actually share the planks that don't mix with math, which is why most people here don't despise you, even though they disagree with you. The teasing seems to be the thing that made it possible to learn this about you. Now don't go and spoil it.

You try to prove something that you care about, except that, unlike in politics, one side should win the debate conclusively.

Do you really care about the concept you're trying to prove, or the insight to be gained either way? In politics, people want their side of the debate to be right, but I would think a good mathematician would be just as satisfied to prove himself wrong, if doing so advanced knowledge.

I have the feeling that most high-school math teachers do see proofs more as catechisms that should be memorized. There is usually no passion that what you are trying to prove might be surprising or untrue.

I've NEVER heard of any teacher having students MEMORIZE proofs. Everyone I've ever spoken to had to generate proofs on their own. Now, of course, the propositions they were proving were either already proven elsewhere, or they were of no consequence, but there was never memorization required. I suppose someone could memorize a proof and parrot it on an exam, but no one knew what they woud be expected to prove until the exam.

Part of the reason that human spaceflight is going to pot these days is the national shortage of technically literate Americans.

And my point was that Apollo had an adequate pool of technically literate Americans in a generation subjected to the "evils" of traditional Carnegie/Prussian methods of education. Now, with all the wonders of modern pedagogy, from open classrooms to constructivist methods, the current generation does NOT have enough technically, culturally, or just plain literally literate Americans.

12:11 PM, March 10, 2006  
Anonymous dweeb said...

I am reminded of William F. Buckley's prescription: Put a math section on the test for a 16-year old to get a driver's license and they will burn the midnight oil. Have you seen 15-year olds memorizing your state's driving law manuals?

I've often said that I'd like to see a system where a high school diploma required one to

-show fluency in math up to trigonometry
-diagram sentences
-demonstrate converancy with literature at least 4 centuries old
-explain the concepts of American govt. - separation of powers, federalism, how a bill becomes law, etc.
-show quantitative competence in at least two natural sciences.
-demonstrate conversancy in a foreign language

AND that same diploma would be a prerequisite to drink, drive, vote, own guns, or breed.

I believe we'd have 75% passage of the exam within a decade, and a safer, better run country.

It's sort of like Heinlein's concept of earned citizenship in Starship Troopers, except citizenship would be based on competence, rather than service.

12:18 PM, March 10, 2006  
Anonymous "Eric Blair" said...

You go, dweeb. Count me in.

I want to live in the world you suggest. The Great Society types had their go at things. I never want to hear another word about defense spending, after I understood what Johnson and his pals did.

Again, when you try something new and it doesn't work...you don't continue to try the same failed approach. Even if it *ought* to work.

12:41 PM, March 10, 2006  
Blogger Greg Kuperberg said...

Do you really care about the concept you're trying to prove, or the insight to be gained either way?

Both! If you work on a hard problem, then you have to adopt an opinion in favor of one side or the other long before you have a solution. It doesn't feel good at all to be tripped by a counterexample, especially if someone else provides it. Of course if you are serious, you should still be satisfied to prove yourself wrong. But not just as satisfied, not after you've gone so far out on a limb. That would be shameless, like Rodney Dangerfield.

An example question: Is it true that a triangle always gains area if you lengthen all three sides? (Not that this is such a hard question, but it will do.) The question immediately invites an opinion. Either you are Mr. Proof and you want it to be true, or you are Ms. Counterexample looking for your namesake. One of the two is going to be sorely disappointed. As it happens, Ms. Counterexample is going to win this one. This particular small point famously tripped up a mathematician who needed it to be true for his claimed proof of a 300-year-old open problem. (Unfortunately, he was particularly arrogant too. The profession is perfectly happy to forgive simple mistakes, but not whitewashed, dowplayed, repeated errors.)

I've NEVER heard of any teacher having students MEMORIZE proofs.

I suppose that I overstated the point. There certainly are completely uncreative 10th grade geometry teachers who have no sense of either the direction or variety of a mathematical proof. They may not openly say that the proof from the book should be memorized. But on the other hand the homework problems are small variations of what is proven the book, and they do not realize that an original proof may also be valid. The students take the hint.

And my point was that Apollo had an adequate pool of technically literate Americans in a generation subjected to the "evils" of traditional Carnegie/Prussian methods of education.

I think that it is too polemical to brand the entire mathematical experience of any entire generation of Americans as "evil", or a complete failure, or a complete success. The story is much more complicated than that, of course. Despite all of the shortcomings of the current system, many of the world's best mathematicians are still nth generation Americans. But still, Apollo wasn't a great example. They had enough talent for a moon shot because they paid dearly for it, far more than NASA can or should pay now.

Even in the 1960s, America imported skilled workers. To the extent that it didn't, highly qualified Americans had to share that much more workspace with mediocrities.

1:06 PM, March 10, 2006  
Blogger Solomon2 said...

I needed algebra a good bit as an engineer, but as a physicist I needed matrix arithmetic much more. The need for mastery of matrix arithmetic is much greater now than before. High school algebra courses should spend two months on this nowadays rather than a week; that way, kids will at least be able to read their textbooks when they get to college.

1:21 PM, March 10, 2006  
Anonymous dweeb said...

Both! If you work on a hard problem, then you have to adopt an opinion in favor of one side or the other long before you have a solution. It doesn't feel good at all to be tripped by a counterexample, especially if someone else provides it. Of course if you are serious, you should still be satisfied to prove yourself wrong. But not just as satisfied, not after you've gone so far out on a limb. That would be shameless, like Rodney Dangerfield

Or stoicly objective. I think there is much to be gained from checking egos at the door. It seems to me that one should be just as satisfied to settle a previously undecided issue.

There are, of course, lousy geometry teachers, just like there are lousy plumbers and lousy surgeons.

I wonder if we could assemble a team of natural born citizens like those in the Apollo program, at any price. Have you seen the recent studies on the reading ability of current college graduates? Of course, while they may not be able to read at a 9th grade level, at least their self esteem id strong.

1:20 PM, March 13, 2006  
Blogger Greg Kuperberg said...

dweeb: I think there is much to be gained from checking egos at the door.

Well, yes, but not completely. You are right that it is typically just as good to disprove a standing conjecture that you were trying to prove. However, a more common situation is that you are trying to prove a conjecture that you came up with yourself, and that may only be useful if the answer is "yes".

Anyway, my larger point is that even though mathematics is an objective discipline, almost no one wants to completely erase human feelings. It's a human endeavor and mathematicians do care about being right, just like everyone else. Some famous ones care all too much, in fact. Most of those who care too little fade into obscurity.

I wonder if we could assemble a team of natural born citizens like those in the Apollo program, at any price.

I'm sure that you could, except for the fact that it would be expensive and that you wouldn't want to. Skilled workers are busy with an enormous variety of important things and it is a net loss to the nation to gather them together to work on One Big Thing. That was a point that Richard Feynman made about the Manhattan Project — it crippled research in basic physics during World War II. That may have been necessary measure for that period, but it was not healthy. And Apollo was unhealthy for the same reason.

Anyway if you really doubt that American mathematical talent still exists, go back and look at the Putnam winners. It is true that many of them are first-generation immigrants, but most are still American-born.

7:39 PM, March 13, 2006  
Blogger mathdad said...

Some comments about Kumon, for whetever it is worth.
If,like me, you are an expariate Indian in USA, you get to know many Kumon parents and children. That is a part of the expatriate social scene.

I can see that Kumon can be of some use for patents who are unable to help children with math in any way for some reason. It does help speed up basic arithmetic. The graduated worksheets are comfortably paced for easy transition and the kids love the accolades, gifts, and rewards constantly bestowed on them by Kumon. They are featured in community newspaper advertisements as "So and so is so many years ahead of his/her grade" with personal photos, and I can imagine how gratifying it must be for the parents and the kids.

But the problem our kids face is not that they are slow in applying well-practiced techniques. Their weakness is in solving multi-step word problems where you have to do some thinking to figure out how to proceed. Kumon does not help much in that area. Kumon can lull kids into overestimating their math capability until they are surprised how mediocre their performance is in local and national math competitions where problems are often out-side-the-box and require flexible thinking. I have seen this happening to Indian parents in USA all the time. So much so that now I have a group of ex-Kumonites coming over weekly for exploration of math concepts and challenging problems.

I do not necessarily urge parents to pull out their kids from Kumon. There is something to be said for proficiency in basic arithmetic skill. But it is equally important to develop mental flexibility and a range of strategies. If your kid is in Kumon, why not let him or her try some past problems from contests such as Math Olympiad for Elementary and Middle Schools (http://www.moems.org/sample.htm), AMC8 (http://www.unl.edu/amc/b-registration/b1-archive/2005-2006/05-amc8/05bro8jpgs.html),
MathCounts (http://www.mathcounts.org/webarticles/anmviewer.asp?a=510&z=54), or Math League (http://www.mathleague.com/contests.htm).
Even if your kid is in Kumon, you need to expose your child to such problems which are more fun than worksheets and hone true mathematical thinking skills.

11:24 AM, March 29, 2006  
Blogger mathdad said...

..oh oh ... how can I forget the Math Kangaroo contest (http://www.mathkangaroo.org/) - fun problems that make kids think. :)
Click on the "Tests & more" button on the left panel and then choose a year to view that year's problems. The problems are for grade 3 to grade 12.

11:36 AM, March 29, 2006  
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10:38 PM, June 07, 2009  

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