I know there are some very good math people on this forum, so I am curious as to what some opinions are about this article.

Mine: this is the worst kind of educational theorist dribble. The arrogance, and black-and-whiteness behind a statement such as "the traditional approach of teaching mathematics simply does not work" (paraphrased), is so typical of those that drive educational reform. Too bad such misinformed "educators" are so reliant on jumping on anything new, and throwing out everything old, to make a living. Personally, I am a teacher: not an "educator". I make my living teaching kids directly, five days a week.

Beware the Common Core. It is simply the latest fad. That is, it is simply the latest "answer" advertised to solve all the problems of a very complicated situation. As is the case with all cyclical educational fads, it doesn't even do what it is advertised to do. (which is to make the curriculum less broad so that teachers can explore the subject more deeply. ) This is the lie that is the Common Core.

I've taught high school math for a very long time. I try to be open to learning new methods that I see work. I am fortunate to work with many talented and diverse teachers who do an excellent job teaching kids math. (No, not kids who just perform on standardized tests, but rather kids who graduate from school and then go on to use math in their lives in some meaningful way.)

My experience indicates that kids learn math best from teachers who:
1. know the math that they teach inside and out.
2. give the kids time to work on a question that has been posed during class. (i.e. from teachers who have the kids actively engaged in doing math during class.)
3. move around their classroom and engage with each individual student as much as possible in doing math, while encouraging students to communicate to each other about the question that has been posed.
4. are passionate and focused about math, and constantly strive for improving their craft.

As someone who got thru algebra II with lots of struggles, and even tried Trig (without success).

I finally learned something important that math was about things in the world. They weren't abstractions when someone finally showed how x and y graphs represent space, much like geometry, which I was good at.

Otherwise, I can unequivocally say I had terrible math teachers in jr and sr high school.

The best reasoning I've seen behind why other countries beat us at math is the structure of our language.

IIRC, this was laid out in a Malcom Gladwell essay in his book "Outliers". The gist being that the language structure of numbers in some languages is more conducive to learning and performing math and remembering sequences of numbers.

Here's an except:

In languages as diverse as Welsh, Arabic, Chinese, English and Hebrew, there is a reproducible correlation between the time required to pronounce numbers in a given language and the memory span of its speakers.

There is also a big difference in how number-naming systems in Western and Asian languages are constructed. … [In] China, Japan, and Korea, they have a logical counting system. Eleven is ten-one. Twelve is ten-two. Twenty-four is two-tens-four and so on.

[This allows] Asian children learn to count much faster than American children. Four-year-old Chinese children can count, on average, to 40. American children at that age can count only to 15, and most don’t reach 40 until they’re five. By the age of five, in other words, American children are already a year behind their Asian counterparts in the most fundamental of math skills.

The regularity of their number system also means that Asian children can perform basic functions, such as addition, far more easily. Ask an English-speaking seven-year-old to add thirty-seven plus twenty-two in her head, and she has to convert the words to numbers (37 + 22). Only then can she do the math: 2 plus 7 is 9 and 30 and 20 is 50, which makes 59. Ask an Asian child to add three-tens-seven and two-tens-two, and then the necessary equation is right there, embedded in the sentence. No number translation is necessary: it’s five-tens-nine.

Pedagogy is an entire field of study itself. In other words, there is nothing "easy" or straighforward about teaching. I've taught sciences and math (statistics/probability mostly) at the high school and college levels. There are many different learning styles, and accomodating all of those styles in a teaching lesson is a big challenge. Some kids are visual/spatial learners, some are verbal, some conceptual, others relative/analagous.

The best reasoning I've seen behind why other countries beat us at math is the structure of our language.

I disagree with the premise that other countries beat "us" at math. If the "us" are some of the students at my school, I would put them up against students from any country in the world, and would expect them to compare favorably.

I never said teaching was "easy". It most definitely is not easy. I said basic.

Pedagogy is an entire field of study itself. In other words, there is nothing "easy" or straightforward about teaching.

THAT must be why it's given one six credit class in the entire ed program at UW! Seriously, if you want to see why education in America sucks and is going more downhill, I would start with education programs at universities across the country. That is where education was undermined originally, and that is where it needs to be reformed the most, and for the greatest effect. We also need to equalize funding for all public schools already too for fuks sake. You would think this was the middle ages or something, with rigid and pre-determined "classes" of people.

Doesn't it ultimately come down to desire? I have a friend who grew up in
India in a house with packed earth floors and didn't speak on a telephone
until he was 18. He is about to earn his PhD and MD at the same time.

good on you for teaching math. i see very few talented math folks who have any desire to teach k-12. why would they?

folks with decent math aptitudes and skills-- and more importantly, the ability to help convey those skills and concepts to others --can go into tech or analytics or finance.

common core is just the latest attempt to get something for nothing.

pretty amazing to watch longtime and vocal posters here in the science threads demonstrate, over and over, that they lack the basic math concepts we would expect of 8th graders.

we need math literacy because these idiots vote on science and tech policy.

i notice common core also dropped the old, token "history of science" from the core elements of science ed.

There are many different learning styles, and accomodating all of those styles in a teaching lesson is a big challenge. Some kids are visual/spatial learners, some are verbal, some conceptual, others relative/analagous.

So true!
What also bothers me is that standards based core curriculum/testing does not account for students different developmental rates or cultural diversity.

Doesn't it ultimately come down to desire?

Bingo, and then there are those students (and parents) that just don't give a sh!t about their education making our jobs even tougher.

not sure if if i'd say 'americans' as specific, as i do not know how others fare, but i know how i do...

and i know, that for me, it is a hard chore, that if i am not 'drawing' or talking to myself, as i do it... it goes wrong...

*liked geometry, due to pictures, which is my lead in, to the rest of my post:

i like this quote OF elcapinyoazz:

Ask an Asian child to add three-tens-seven and two-tens-two, and then the necessary equation is right there, embedded in the sentence. No number translation is necessary: it’s five-tens-nine.
--
Pedagogy is an entire field of study itself. In other words, there is nothing "easy" or straighforward about teaching. I've taught sciences and math (statistics/probability mostly) at the high school and college levels. There are many different learning styles, and accomodating all of those styles in a teaching lesson is a big challenge. Some kids are visual/spatial learners, some are verbal, some conceptual, others relative/analagous.

this 'way' that the POSTER of the quote, described the ?asian? way, seems to me, like it would have helped me immenstly, as, i see and think, in pictures, shapes, and groupings, and when i have to think numbers, i really go blank--very odd... even when i do math now, i see, for example:

the seven, as a bent stick, that i count the TOP, BEND, AND BOTTOM, in a patter, of twice, and add one, for the fact that it is one line, and it MEANS seven to me... OR, i think a rhyme, SEVEN shoots off to heaven, as, it looks like an out of shape arrow...

fine way to keep SEVEN in the brain, when adding and such, huh, :O
and i have one for each number...

so by the time, i transfer all this info, and such, there i am way behind everyone... if i draw the numbers, and not do in my head, then, i SEE the pictures and it goes faster, but i have a few spots, where 'carrying over' in a column' make a GLITCH...

IF i could have learned in groups of what the quote said, i sure would seen and done it a lot faster, and EVEN understood that we are dealing ?essential? with THINGS that we are grouping up into
collected amounts that flow faster and ACTUALLY come out into
easier matches, as to our goal...

i can do this with objects, or shapes, a lot better... it is very strange with the numbers, sad to say for me... but i don't think this is why other americans, etc, may have trouble??

i DO think that if kids have troubles though, it IS DUE to teachers NOT being allowed or having TIME to teach each child according to HOW they learn, or, how their brain clicks... (though i DO feel that even if your b brain clicks/works ONE way, it is ALWAYS good after you learn the thing to practice the HARD ways to teach the brain NEW trails, etc, and help the person branch out)...

i like this part of the quote, that was IN the quote that i shared above--it was the poster's SEPARATE ADDED COMMENT:

There are many different learning styles, and accomodating all of those styles in a teaching lesson is a big challenge. Some kids are visual/spatial learners, some are verbal, some conceptual, others relative/analagous

SINCE i had trouble learning some things, and made my OWN way, (did good with words, though, as, i made them into rhythms and pictures, too, and enjoyed diagramming sentences, as i could SEE how they worked--the structure and ideas, etc) i KIND of reshaped info, so i could understand...

thus, i have taught and do know how to teach kids to read, that have troubles... there are a variety of ways for each brain and they are not really what the SCHOOLS do now...
and--i have taught and can teach DANCE, as, i can see what each brain is doing and what and how they need to adjust, to find success...

but see, THIS is teaching--taking the pupil and helping them see what is wrong, what is right, what leads to the next level of success, etc, and that THEY CAN DO IT--if--they keep trying the right key to solve their difficulty...

teaching is wonderful, but i know so many that do not like learning, and i think perhaps, there are many reasons:

-the limits put on teachers...

-the 'use only this set' system...

-lack of good basic homework, and--that when they TAKE it home, they SHOULD know the principles of HOW to do it, and not be lost...

-to many kids in room, perhaps a coach should be in the class with teacher...

-no desire to learn, as, folks at home never help the young babe at
home have a stimulated mind of inquiry... edit: so the child grows up having no desires, but for its OWN whims and for things that are SIMPLE and require no work, or challenges...

-the praise given for success, only, and not for those that MEET the challenge of the 'never give up'...

sure would have helped and caught my math trouble, early one, but:
may not have actually FIXED it all, but i'd sure not get brain-roof-caves-in-while floor-falls-down thing when i see all the numbers... :O

Our society has changed dramatically in the last several decades. We just don't have the level of commitment and willingness to work hard and to achieve . All sorts of cultural, and institutional, and political change has brought these shift in values about--- not just in education but in all areas of life.

That's been done in Colorado, Locker. Let's have this conversation a few years out, and let's see how that single legislative act has changed things.

Neebee, I appreciate your thoughts on this.

How can we expect to teach a subject as precise, demanding, and frankly challenging as math, to a society that spends its evenings watching reality TV.???

"How can we expect to teach a subject as precise, demanding, and frankly challenging as math, to a society that spends its evenings watching reality TV.???"...

Yes, no doubt TV is the main problem...

Turn them off and all of a sudden kids will start doing better at math...

Turn them off and all of a sudden kids will start doing better at math...

TV isn't the main distraction, it's the damn smart phones they all seem to be picking at in class these days. If I had a buck for every time I told a kid to put it away or confiscate it I could retire. I have also caught kids using them to cheat on tests.

I was answering sarcastically about the problem being TV's...

I talk about getting rid of Tenure and truly do believe that it needs to go...

But the problems do not start and stop there...

There is more to it and it's not that simple...

I know locker,
BTW, my 15 yr. tenure and teachers union didn't save my job when the economy tanked and the schools enrollment dropped to an all time low.
Last hired, first fired, we lost 4.
I have been sub teaching since (7yrs.) and really enjoy it, stress level has gone way down and I have more time to hang at the Taco Stand with clowns like you. ;-)
Teaching, like any profession has the good, the bad and the ugly.

.
neebee, It is interesting hearing how your mind works.... When you boil it down>> not everybody learns the same way, and all schools need to address that issue.

That is why i love the Kahn Academy and their method of teaching.. If you have a child struggling with math, please view the Kahn Academy videos....

I told a girlfriend about the Kahn Academy, and her child went from failing math to understanding math and getting much better grades..

So I studied mathematics in graduate school, taught while a graduate student (taught my own classes NOT a TA), briefly ran a small tutoring franchise, and even had a summer gig one year teaching "gifted and telented" 6th graders. From all this experience I've concluded that the problem lies with all the B.S math ed reform and all the culturial crap.

1 + 1 = 2 (base 10) no matter what color, creed, religion, etc you are.

No litte Johney shouldn't get partial credit if his attempts at a solution were not headied in a direction that would solve the problem.

I routinely had freshman algebra students that could not ad 1/2 + 1/4 without a calculator. The education system failed them. How did they graduate High School??? Never mind -1 -2 = ? or what is 1/0 =. The k-12 system fails....

On a more comical note, at a good second tier university with a respected mathematics dept. the mathematicians strongly disliked the math ed peeps. Infact the topologists would routingly vote for the weakest new prof hires when they were math ed types because they (and the rest of us) wanted the math ed dept to wither (remember they think that little johney should get credit and that religion and culture matter wrt (with respect to) 1 + 1 = 2).

I enjoy going to the deli counter at supermarkets and asking for 5 ounces of something only to see the employee's brain vapor lock as he (she) looks at the digital scale.

I tell them, "Remember back in the fifth grade when they said that you are going to need this stuff in real life? Well guess what? They were right."

It's interesting that such a complex question: how to effectively teach mathematics, devolves so quickly to overly simplistic explanations.

First, there is no agreement as to what the math curriculum should be. Let's take a radical stance, that we require children to be educated because our democracy requires "an informed public." While understanding Andrew Wiles proof of Fermat's Last Theorem, would be a marvelous cultural objective, it isn't directly applicable to those issues on which the public votes.

Yet just being able to do various arithmetic calculations is not sufficient, either. Yet even this is a challenge for some students.

More and more, the public has been involved in trying to understand the nature of quantitative information based on statistical arguments. People playing the state sponsored lottery might want to better understand the odds of winning, vs. using those dollars for some other investment. The efficacy of medical procedures, or cures, or alternative medicines. The likelihood of floods, or other weather events, and on and on...

While many mathematicians might argue whether or not statistics should be included in mathematics, few people would see statistics as anything other than mathematics. So having an educated voting public capable of actually understanding statistical arguments might be a worthy goal of public education.

Yet that would be a daunting task, some would say "impossible." And in the current political climate it would be impossible, however worthy. To accomplish it would require a huge investment in remaking the mathematics curriculum, way beyond what most of the public believe would be necessary.

As wbw also observes, ideally it would require teachers who thoroughly understand the math in order to be successful teachers. It is a generally held belief that anyone who possess such skills would do better, financially, doing something other than teaching.

Teachers as a class have always been disparaged in American society, "those who can't, teach." They are a very low paid profession, and the best teachers often find work elsewhere. For a long time teaching was a profession relegated to women, whose effort was greatly undervalued. Often we hear the statement that teaching is "a calling" to justify the low compensation, yet as a society we have not made a commitment to those taking the calling. Tenure provided job security to those who "answered the call" yet that system is now seen as a root cause for poor teachers.

Obviously, if we valued teachers we'd put our money where our collective mouths are and pay them a competitive wage, yet there are all sorts of forces aligned against letting "the market" set teacher compensation. This is most obvious in the pseudo-outrage over increasing college and university costs, and the increases in faculty salaries, which are market driven. Why not let it happen? Somehow collective activity is "evil" but limiting the compensation is ok (but we would never do that for corporate CEO's).

If you want the best to be teachers, you're going to have to compete with the private sector to attract those people. If you think giving education to the private sector will improve things without seeing increased costs, and probably unsatisfactory outcomes, you're delusional.

Americans are historically "practical" people, where practical has to do with commercial success. Education doesn't mean much to Americans as a whole except when education makes the students more successful, commercially. This is driving the American curriculum to be more a certification process for the private sector than what we'd traditionally refer to as a liberal education: "a philosophy of education that empowers individuals with broad knowledge and transferable skills, and a stronger sense of values, ethics, and civic engagement ... characterised by challenging encounters with important issues, and more a way of studying than a specific course or field of study"

This definition would be anathema in the current political setting.

What the current argument revolves around is just what are the "specific course or field of study" that is required to provide the best advantage to a student's economic prospects.

Why Americans stink at math is because they don't see how math helps those personal economic prospects.

When the second order Taylor series expansion of the coupled time dependent return on investment is viewed as such a complicated and obscure bit of knowledge that it can be used in defense of the financial professions being "duped" by a bunch of physicists and cause international economic calamity, one can only sigh in disbelief of the denial of the importance to understand mathematics, even if it doesn't effect you bottom line directly.

I sucked at math K thru 12. I was lazy and some how thought the math would come to me just by looking at the book. It wasn't until I started working the problems over and over that I began to do well with it in college. I think it had in my case far more to do with a work ethic than teachers not doing their job. I was lazy and would rather goof off and be an idiot than sitting down and working math problems.

As a math student, I found that the most important part of studying was just doing the problem sets.

A thought I have had that would increase the appeal of math is to show how powerful it is in solving conceptually difficult problems. I remember when I was first introduced to algebra and the idea of operating on an unknown value through the use of equations. This seemed extremely cool.

Obviously, if we valued teachers we'd put our money where our collective mouths are and pay them a competitive wage, yet there are all sorts of forces aligned against letting "the market" set teacher compensation. This is most obvious in the pseudo-outrage over increasing college and university costs, and the increases in faculty salaries, which are market driven. Why not let it happen? Somehow collective activity is "evil" but limiting the compensation is ok (but we would never do that for corporate CEO's).

If you want the best to be teachers, you're going to have to compete with the private sector to attract those people. If you think giving education to the private sector will improve things without seeing increased costs, and probably unsatisfactory outcomes, you're delusional.

Hard to disagree with this. I'd much rather have someone with an MS or a PhD in a science who's education was focused on the science as a teacher but it doesn't pay! Then (in California) there's the CA single subject credential requirment for teaching HS mathematics hoop (replace "mathematics" with science of your choice).

If teaching actually paid well and there weren't the hoops I probably would have followed that path post grad school.

i agree pay the teachers more. but no, its ceo's and some lawyers, doctors, realtors and politicians that make the fortunes.

i just read that american fast food eaters (already says something there) chose a 1/4 pound burger over a 1/3 pound burger at the same price because they thought 4 was bigger than 3 :-/ seriously.

If you have a child struggling with math, please view the Kahn Academy videos....

Word! Thanks Nita.
Some of us recommended this for Whitemeat a few months back when he was struggling in math. He reported back that it helped him understand the concepts he was struggling with.

....which protect teachers from financial political influence. Seriously. They also make it harder to get rid of shitty teachers, but the fact that they are what largely protects academic freedom among teachers makes them an overall bonus in my book. I'll take a few shitty teachers over enfranchising educational institutions to be able to say "if you don't like teaching the curriculum (handed down to us from god knows where as opposed to a gathering of teachers who all brainstormed at the latest national teachers union meeting), we will hire someone else." It's all about the core lessons which are taught and who gets to determine which lessons get taught. Should it be some "committee" of "educators" somewhere, or all the teachers who actually do the job? Seems like there is one obvious choice there for me.

The key to any learning is desire. How do you make it real and how do you make it possible?

Teachers who really understand the subject and really understand their students learning and social skills can readily prepare the vast majority of kids tfor real life math applications. Two trains travleing towards each other - no one cares. However, a real life lesson like here are your credit balances, interest rates, and minimum amount due, how should you prioritize your payments and how long will it take to be out of debt - those are far more real life problems that kids need to solve. My former A student in Math needed Mom to show her that she could dig out of debt with a better prioritization, and bankruptcy was not necessary.

I would not be a fan of the group grope for understanding. I was a shy and awkward kid who would have HATED math if it was a group guided discovery. I was also not a fan of memorization, though I will admit that it was terribly efficient not to be actively computing 12x12 every time.. In my mind, nothing replaces true understanding. That needs to start with the teacher.

Always hated the gross generalizations - Why Americans stink at math......I am American, and I am great at math. That is inaccurate characterization of an array. Much better stated as "Why many Anericans stink at math", "Why the mean mathemeatical literacy of Americans is lower than other countries".......

...can readily prepare the vast majority of kids tfor real life math applications.

what is "real life math applications"? those examples you provided are interesting, but not the end of "real life" applications. In fact, "application" is a very interesting word, it implies that something is being applied, in this case, mathematics, which by the same implication is not "applied" but "pure."

So without learning mathematics, you don't have anything to apply.

You could just teach those specific lessons, and have the students use those lessons by rote, to the specific "real life applications" but you can't anticipate all the different application.

However, teaching students how to apply the same mathematics to different applications would seem to be a goal. If you can't see debt rates and trains as applications of constant rate coupled equations to be solved algebraically, then you've missed the point.

It is easy to disparage word problems, but the idea of the word problem is to learn how to analyze the problem and set it up to be solved. Once you learn how to do that, you can apply it to trains, and to debt, or any other such problem, or the issues related to the national budget and the assumptions going into the arguments over default...

Teachers who really understand the subject and really understand their students learning and social skills can readily prepare the vast majority of kids tfor real life math applications. Two trains travleing towards each other - no one cares.

College freshman relate much better to a (albiet fictitious) drinking/exponential decay/DUI word problem than the traditional half life problem.

I think it had in my case far more to do with a work ethic than teachers not doing their job. I was lazy and would rather goof off and be an idiot than sitting down and working math problems.

Why is this a problem? because we don't actually understand how we were educated. Somehow, everyone thinks their an expert, and that their own experience is some self evident truth. Following this logic, it is not too difficult to see why teachers aren't respected and that learning about education is considered a waste of time.

If everyone is an expert, it should be no problem to teach our children. It is even easier to do that by telling the children that the teacher doesn't know how to teach and that the particular assignment is stupid and irrelevant and that the parent can testify that surviving in the "real world" doesn't require mastery of the subject.

The standards on what gets accepted and published would get you laughed out of any other department including the soft ones if submitted.

interesting assertion, perhaps you can actually support it with some real cases, or are you just passing on what you heard somewhere else?

Not that that sort of logic would get you laughed out of the STForum, since that sort of logic is pervasive.

And then to generalize it to state that College/University Education Departments are the root cause of bad teaching, well, you might want to shore up that conclusion a bit too.

It's not just math it's the entire teaching paradigm that needs to change. The greatest impediment are the university education departments. There's a national database with about every Masters and PHD thesis on education collected in it for the last 20 years or so called ERIC. The standards on what gets accepted and published would get you laughed out of any other department including the soft ones if submitted.

that's a bit colorful, but yes, the general rule is that professional schools generally--ed, med, law, and business--produce scholarship that on average is less rigorous than that in many of the disciplines. there are stacks of exceptions, obviously, but yes, that is the consensus.

until very recently, much research on education in k-12 was driven by a desire to produce quantifiable measurements of outcomes and efficiency. that came partly from dynamics internal to education (quantitative research looks more rigorous) but also in response to policy demands for cheaper public ed that could be measured in metrics so simple than even an average st poster could understand them.

one of the ironies of the standardized tests, is that in undergraduate education, education majors consistently rank in the lowest percentile of students by major. another irony is that business consitently ranks as another of the lowest performers.

put "business" and "education" together and you get NCLB and other horror shows.

Pedagogy is an entire field of study itself. In other words, there is nothing "easy" or straighforward about teaching. I've taught sciences and math (statistics/probability mostly) at the high school and college levels. There are many different learning styles, and accomodating all of those styles in a teaching lesson is a big challenge. Some kids are visual/spatial learners, some are verbal, some conceptual, others relative/analagous.

El Cap for the win. My daughter and son-in-law both teach math and statistics at the high school level, and both get consistently excellent reviews. Math was one of my undergraduate majors, and always came easily and intuitively to me, so I can't offer any personal stories about overcoming difficulty. I have, however, taught law and economics for over 25 years, and always got superior reviews from my students, despite having no formal training in education. Well, there is one exception. One Summer School economics class I taught at Fresno City College had a one-student review in Ratemyprofessors.com that gave me a "C," but that's an outlier. ;-)

Anyway, I think the idea that anyone who "knows how to teach" can teach anything simply does not fit my experience. I find no substitute for thorough knowledge of the subject matter. Unfortunately for teaching, math is one of the few undergraduate majors that offers a lucrative field with just a bachelor's degree, namely being an actuary. I calcualte that it costs my daughter and son-in-law a combined $150,000+/year to teach math, rather than to be employed as actuaries.

In California, however, the courts have ruled that it is illegal to base a teacher's pay on the subject matter taught, even though the opportunity cost differs radically for different subjects. My other daughter has an advanced degree in music composition, but her opportunity cost of teaching is zero. Why should she be paid more than my other daughter, who has only a bachelor's degree, but in a field with a very high opportunity cost?

I realize many people don't understand, appreciate, or even believe in opportunity costs, but they are, in fact, the only real costs in life. Consequently, a teacher with a master's degree in education and an undergraduate minor in mathematics is often paid more than one with a bachelor's degree in math but "only" an education credential. Our California teacher compensation system says, in effect, it's more important to know how to be an educator than it is to know your subject matter.

Be careful bashing Common Core, though. My favorite math prof at Berkeley, Hung-Tsi Wu, co-wrote an Op-Ed piece for the Wall Street Journal endorsing Common Core standards for math. Dr. Wu didn't exactly choose the most friendly forum, but his arguments had a great deal of force. While I have issues trying to teach "relevance," I realize that when I teach a class, I normally incorporate -- either as problems or examples -- so many of my "war stories" that I show and teach the relevance as a matter of course.

All of this is a very long-winded way of saying that I think the article makes some good points, particularly in warning about relying on the "new, best, way to teach math" to the exclusion of every other option. Teaching remains an art, and art generally remains incompatible with "best practices."

All math was created by the human mind.
You have a human mind.
You are capable of creating all math.

If anyone has a problem with math you find them chanting this: "I'm bad at math." It's the biggest lie in the world.

Repeating a negative suggestion such as this over and over is a form of self-hypnosis and it will be true until you just get off it.

If you go back and revisit your education in math you will find you started to be bad at math at the point when the math teacher used a word you did not understand, like "arithmetic" or "algebra."

Math teachers fail because they fail to clearly define the terms of mathematics.

Also, there is is a fundamental missing from math instruction:

There is a form of math more fundamental than simple addition and subtraction. It is "counting."

The skill of counting goes back to the street markets of ancient Egypt or even counting the number of animals in heard or members of a tribe that need to be fed. Counting should be well drilled in young children and practiced regularly throughout education until a person can look at a pile of apples and in less than a second or two, state precisely how many apples are present.

Counting is done on Sesame Street and many kindergarten classes but is not emphasized enough. High standards need to be set and enforced.

Counting being done well, the rest of math, with terms understood, follows forward very quickly.

Our education system should be producing people who don't just know formulas, but are aware enough to invent formulas as needed to solve problems in life.

I went back to school late in life and got my BS in my 40's I needed lower division units and since I was working for a school district at the time could take the classes to clear a Voc Ed credential for next to nothing so both the wife and I took the classes.

One of the requirements was to use ERIC as a research tool.

I was frankly astounded at the lack of rigor generally accepted in Masters and PHD work in Education departments. If I'd turned in work like that in any of my undergraduate Business Administration classes, I'd have flunked.

one of the most ridiculous things about the constant chant that we need to teach kids only what they can use on the job, is that the people who say that have created a curriculum that is truly useless. look at what we have the kids actually do --

if an alien visited an american classroom to observe the results of that philosophy, he'd conclude that most americans get paid to fill in little circles with number 2 pencils all day long.

So to elaborate on my experience with topologists....

So I had taught Trig, Freshman Alg, 1st and 2nd semester enginnering calculus
and so I got "picked" for Dr x's pet project. Dr X (a topologist) had become so frustrated with the quality of the undergraduate students as well as the math ed department he felt he needed to do something. So he studied the problem. He went to the TIMMS studys (world wide math/science studies) and learned that the Singaporeans placed very very high year after year. Dr X then went out and got the ciriculioum they used. He then (much to the chagrin of the math ed dept) wrestled control over the course entitled "Mathematic for Elementary Education Majors". Basically the elementary ed required math ed calss - typically sophmore and juniors. His goal was to teach the future teachers how to think and understand the ciriculium of the Singaporeans - a noble idea. Sadly I got stuck teaching this class for a few semesters. The reason I say sadly goes as follows: Although about 1/4 of the class put forth a reasonable effort and perhaps 1/2 of that 1/4 might turn out to be good teachers, the other 3/4 had the attitude that "Well I don't need to know this I just want to teach kindergarden or 1st grade - I just need a D to pass this class." Unfortunatlely that's not how getting your first job out of college works and they all (the ones who graduated) are certified for K-6.

I don't think it's a leap to see how this creates very very unperpaired bad teachers.

Good points, Kev. I have known a great many elementary school teachers that are functionally innumerate, and see no problem with that state of affairs.

To be more OT on something explicitly OT, I think that topology was my favorite branch of math, but Berkeley only taght it at the graduate level. We did have a "Measure and Integration" class, that could in theory be taken by sophomores, that was full of graduating seniors and grad students. The real analysis in that class got me hooked on real topology. I think it helped that John Kelley, who wrote the intro graduate topology text, taught that class as well. I think my enthusiasm rubbed off on my nephew, who got his Ph.D. in algebraic topolgy. Good stuff!

It's not just math it's the entire teaching paradigm that needs to change.

The greatest impediment are the university education departments.

There's a national database with about every Masters and PHD thesis on education collected in it for the last 20 years or so called ERIC.

The standards on what gets accepted and published would get you laughed out of any other department including the soft ones if submitted.

Agreed TGT. Never seen ERIC but the things taught to me in the ed program I went partway through left me wondering WTF ANY of it had to do with teaching kids to do math. I felt a good interactive teacher with a solid and confident knowledge of the subject who was engaging to students could work wonders with a chalk board. Instead I was supposed to think great math teachers tried something new every year, ensured a proper bell curve for every test/assignment/quiz, and did their best to differentiate between "good" and "bad" students. The only people they graduated from the program were people who portrayed themselves as devote christians and people who entirely kept their opinions to themselves, or expressed opinions in line with the department and vowed to practice their brand of "education". In short, the program was a complete joke and an insult to the institution of education. I see things like this as the major block to education in this country. It drives all the real teachers away, among a host of other problems.

BTW, most smart folks I know who would teach if things were not as they are were largely drawn by the holistic picture of teaching as being a fairly free-form and fulfilling life. They are not they types of people fulfilled by a big paycheck.

that's a bit colorful, but yes, the general rule is that professional schools generally--ed, med, law, and business--produce scholarship that on average is less rigorous than that in many of the disciplines. there are stacks of exceptions, obviously, but yes, that is the consensus.

Hmm, that's an interesting way to look at it; legal "scholarship" is fair game for criticism I'm sure, but I've never heard anyone really compare it to whatever they hell do in education curriculum (or medicine or business schools for that matter, but education just stands out as an apples-to-oranges comparison). Rightly or wrongly, I can assure that you high-brow law professors would be pretty surprised to hear themselves being lumped in with D.Ed types.

About teacher pay: I'm sure teachers would like more money, wouldn't we all? But is there a shortage of teachers, at current salaries? I haven't heard that. On the contrary, I've heard that newly minted teachers have a hard time finding jobs. That suggests that teacher pay (which after all, is set by the government, heavily influenced by teachers unions in support of higher pay) is higher than it "should" be in something like a free market, not lower.

And I question the job market for all these math majors--if you want to pretend that everyone with a BS in math could get a high paying job as an actuary, fine, but how many actuaries do you know?
Computer programmers, sure, but that's not really math, especially these days.

Also wanted to add that the assault on "ineffective teachers" and "obstructive teacher unions" comes from the massive federal spending to reduce classroom size (which was literally statistically unsupported at the time this was implemented, and remains statistically unsupported to this day), which unsurprisingly had no effect on education quality. Nonetheless somehow politicians now argue that it is the "bad" teachers fault the money had no effect and not that reducing classroom size has no effect. They are chasing the rabbit down the hole. It's a B.S. and invalid argument.

Anyway, most of the shitty teachers I had were the people burnt-out from struggling uphill and against the current in a B.S. public education system.

"Today there is such a shortage of highly qualified K–12 teachers that many of the nation’s 15,000 school districts4 have hired uncertified or underqualified teachers. Moreover, middle and high school mathematics and science teachers are more likely than not to teach outside their own fields of study (Table 5-1). A US high school student has a 70% likelihood of being taught English by a teacher with a degree in English but about a 40% chance of studying chemistry with a teacher who was a chemistry major.

These problems are compounded by chronic shortages in the teaching workforce. About two-thirds of the nation’s K–12 teachers are expected to retire or leave the profession over the coming decade, so the nation’s schools will need to fill between 1.7 million and 2.7 million positions5 during that period, about 200,000 of them in secondary science and mathematics classrooms.6

We need to recruit, educate, and retain excellent K–12 teachers who fundamentally understand biology, chemistry, physics, engineering, and mathematics. The critical lack of technically trained people in the United States can be traced directly to poor K–12 mathematics and science instruction. Few factors are more important than this if the United States is to compete successfully in the 21st century."

The problem is that the "educators" at the university level have never had a real world job or any experience teaching at a primary or secondary level.

The worst thing that happened was the nationalization of education and the expansion of the Federal dept of Ed.

Administrative staffs and expense in local school districts exploded during the Clinton administration and never has slowed down since. Focus became getting grants and checking all the right boxes to keep the federal dollars rolling in.

The most desired job in many districts has come to be that of grant /program administrator. Lots of power, no accountability, and with a six figure salary.

A US high school student has a 70% likelihood of being taught English by a teacher with a degree in English but about a 40% chance of studying chemistry with a teacher who was a chemistry major.

Hmm, that's an interesting way to look at it; legal "scholarship" is fair game for criticism I'm sure, but I've never heard anyone really compare it to whatever they hell do in education curriculum (or medicine or business schools for that matter, but education just stands out as an apples-to-oranges comparison). Rightly or wrongly, I can assure that you high-brow law professors would be pretty surprised to hear themselves being lumped in with D.Ed types.

it's been a regular item of discussion in the scholarship on r1s as well as a topic of conversations 'ive had with folks over the years in the professional schools. law is an especially obvious example: the prestige journals are edited by grad students in a 3-year program. no prestige journal in any serious research discipline can do that and still be taken seriously as peer review.

the shift in science funding, toward engineering and "applied sciences," like medbiotech and eecs, may change the way folks view the dynamic broadly.

disciplines also rise and fall over time-- the 20th century tradition of theoretical physics as the pinnacle of intellectual rigor is probably over-- certainly the eecs students today grew up with a world so different, that they view the old traditional pecking order as a historical curiosity.

it's not clear what things will look like in another 25 years.

MYTH: Teachers make just as much as other, comparable professions.

FACT: According to a recent study by the National Association of Colleges and Employers, the teaching profession has an average national starting salary of $30,377. Meanwhile, NACE finds that other college graduates who enter fields requiring similar training and responsibilities start at higher salaries:

Computer programmers start at an average of $43,635,

Public accounting professionals at $44,668, and

Registered nurses at $45,570.

Not only do teachers start lower than other professionals, but the more years they put into teaching, the wider the gap gets.

A report from NEA Research, which is based on US census data, finds that annual pay for teachers has fallen sharply over the past 60 years in relation to the annual pay of other workers with college degrees. Throughout the nation the average earnings of workers with at least four years of college are now over 50 percent higher than the average earnings of a teacher.

An analysis of weekly wage trends by researchers at the Economic Policy Institute (EPI) shows that teachers' wages have fallen behind those of other workers since 1996, with teachers' inflation-adjusted weekly wages rising just 0.8%, far less than the 12% weekly wage growth of other college graduates and of all workers. Further, a comparison of teachers' weekly wages to those of other workers with similar education and experience shows that, since 1993, female teacher wages have fallen behind 13% and male teacher wages 12.5% (11.5% among all teachers). Since 1979 teacher wages relative to those of other similar workers have dropped 18.5% among women, 9.3% among men, and 13.1% among both combined.

Teachers lost spending power for themselves and their families as inflation outpaced increases in teacher salaries last year, according to NEA Research. Inflation increased 3.1 percent over the past year, while teacher salaries increased by only 2.3 percent.

MYTH: Teachers are well-paid when their weekly or hourly wage is compared with other professions.

FACT: Teacher critics who make this claim use data collected by the U.S. Bureau of Labor Statistics (BLS) in its annual National Compensation Survey (NCS). But NCS data are based on employer surveys, and the NCS measures scheduled hours -- not the work teachers do outside the school day. Because teachers do not work the familiar full year and roughly 9-5 schedules that most professionals have, the comparison is one of apples to oranges.

Economic Policy Institute President Lawrence Mishel explains that in the NCS data "Teachers are measured by days worked (say 190 official school days divided by five, resulting in 38 weeks), while others are measured as days paid (work days plus paid time off: breaks, vacations and holidays)."

The bottom line: NCS data vastly understate the weekly hours of teachers and the weeks teachers work each year, and thereby significantly overstate the hourly wage or weekly wage for a given annual wage.

If you believe the NCS hourly pay data, then you believe that English professors ($43.50) make more per hour than dentists ($33.34) or nuclear engineers ($36.16).

MYTH: The school day is only six or seven hours, so it's only fair that teachers make less than "full-time" professionals.

FACT: Other professionals hardly have the monopoly on the long workday, and many studies conclude that teachers work as long or longer than the typical 40-hour workweek.

Six or seven hours is the "contracted" workday, but unlike in other professions, the expectation for teachers is that much required work will take place at home, at night and on weekends. For teachers, the day isn't over when the dismissal bell rings.

Teachers spend an average of 50 hours per week on instructional duties, including an average of 12 hours each week on non-compensated school-related activities such as grading papers, bus duty, and club advising.

When the Center for Teaching Quality studied teachers' workdays in Clark County, NV, it found that not only did most teachers work additional hours outside of the school day, but that "Very little of this time is spent working directly with students in activities such as tutoring or coaching; far more time is reported on preparation, grading papers, parent conferences, and attending meetings."

MYTH: Teachers have summers off.

FACT: Students have summers off. Teachers spend summers working second jobs, teaching summer school, and taking classes for certification renewal or to advance their careers.

Most full-time employees in the private sector receive training on company time at company expense, while many teachers spend the eight weeks of summer break earning college hours, at their own expense.

School begins in late August or early September, but teachers are back before the start of school and are busy stocking supplies, setting up their classrooms, and preparing for the year's curriculum.

MYTH: Teachers receive excellent health and pension benefits that make up for lower salaries.

FACT: Although teachers have somewhat better health and pension benefits than do other professionals, these are offset partly by lower payroll taxes paid by employers (since some teachers are not in the Social Security system), according to the Economic Policy Institute (EPI).

Teachers have less premium pay (overtime and shift pay, for example), and less paid leave than do other professionals.

Teacher benefits have not improved relative to other professionals since 1994 (the earliest data EPI has on benefits), so the growth in the teacher wage
disadvantage has not been offset by improved benefits.

The benefits of other workers would not have declined as much in recent years if they had the protection of a union, collective bargaining, and an independent voice on the job -- like public school teachers.

MYTH: Thanks to tenure, teachers can never be fired, no matter how bad they are.

FACT: Tenure does not mean a "job for life," as many people believe. It means "just cause" for discipline and termination, be the reason incompetence or extreme misconduct. And it means "due process," the right to a fair hearing to contest charges. Quite simply, any tenured teacher can be fired for a legitimate reason, after school administrators prove their case. That's similar to what American citizens expect when charged with violation of a law.

MYTH: Unlike other professions, teachers get automatic raises, regardless of how well they perform their work.

FACT: Name a profession in which people earn less each year! Through collective bargaining or state legislation, most teachers are placed on a salary schedule with pay "steps" or "increments" for seniority -- seasoning -- on the job and added professional development.

Teachers never have a chance to stand still or go stale. They are rigorously evaluated, face recertification requirements, deal with ever-more-complex state and federal standards, and are expected to advance to the master's degree level and beyond.

A well-constructed salary schedule rewards classroom experience, promotes continued professional learning, and promotes both retention and recruitment of quality staff.

MYTH: If schools were allowed to grant merit pay, good teachers would be well compensated .

FACT: The fundamental problem is low teacher pay, period. Merit pay schemes are a weak answer to the national teacher compensation crisis.

Merit pay systems force teachers to compete, rather than cooperate. They create a disincentive for teachers to share information and teaching techniques. This is especially true because there is always a limited pool of money for merit pay. Thus, the number-one way teachers learn their craft --learning from their colleagues -- is effectively shut down. If you think we have turnover problems in teaching now, wait until new teachers have no one to turn to.

The single salary schedule is the fairest, best understood, and most widely used approach to teacher compensation -- in large part because it rewards the things that make a difference in teacher quality: knowledge and experience.

Plus, a salary schedule is a reliable predictor of future pay increases. Pay for performance plans are costly to taxpayers and difficult to administer. In contrast, single salary schedules have known costs and are easy to administer. School boards can more easily budget costs and need less time and money to evaluate employees and respond to grievances and arbitrations resulting from the evaluation system. Worse yet, there is often a lack of dedicated, ongoing funding for merit pay systems.

Merit pay begs the question of fairness and objectivity in teacher assessments and the kind of teacher performance that gets "captured" -- is it a full picture, or just a snapshot in time? Is teacher performance based on multiple measures of student achievement or simply standardized test scores? Are there teachers who are ineligible to participate in a merit plan because their field of expertise (art, music, etc.) is not subject to standardized tests?

By November 2006, 50 Texas schools rejected state grants to establish merit pay programs for teachers, tied to higher student test scores. Many schools reported that teachers opposed the idea or that administrators were reluctant to decide who should get a bonus and who shouldn't. Teachers at schools opposed to merit pay said it was not worth the extra money to break up the team spirit among teachers and spend time filling out paperwork for the program. In Bellaire, Texas, fifth grade science teacher Tammy Woods voiced her paperwork concern to the Dallas Morning News. "Most of us felt our time would be better spent working with the kids than working on the incentive-pay plan," she said. "We also felt there would be hard feelings no matter what happened because not everyone who worked to accomplish our goals would be rewarded."

MYTH: Teaching is easy -- anyone can do it.

FACT: Teachers, like many professionals, including accountants, engineers, and registered nurses, are trained, certified professionals. They have college degrees in education or in the subject that they teach plus a teaching credential.

More than half (57%) hold master's degrees, and all have completed extensive coursework in learning theory and educational practice. Most find that teaching is a calling and a gift, which includes a love of children and an ability to engage them in the learning process.

Education is also complex, demanding work that requires high levels of creativity, adaptability, thoughtful planning and resourcefulness -- much of which is learned from cumulative classroom experience.

While there are many experts who excel in their fields, most do not have the ability to translate that knowledge into teaching strategies useful in a classroom setting.

MYTH: The rewards of working with children make up for low pay.

FACT: It is true that most educators decide to enter the teaching profession because of a desire to work with children, but to attract and retain a greater number of dedicated, committed professionals, educators need salaries that are literally "attractive."

The intrinsic rewards of an education career are often used as a rationale for low salaries. But low teacher pay comes at a very high cost.

Close to 50 percent new teachers leave the profession during the first five years of teaching, and 37 percent of teachers who do not plan to continue teaching until retirement blame low pay for their decision to leave the profession.

New teachers are often unable to pay off their loans or afford houses in the communities where they teach. Teachers and education support professionals often work two and three jobs to make ends meet. The stress and exhaustion can become unbearable, forcing people out of the profession to more lucrative positions.

But is there a shortage of teachers, at current salaries? I haven't heard that.

There certainly seems to be a shortage of math teachers. My daughter spent the last couple of years teaching at the middle school level, after four years teaching at the high school level. Her school district has a "job fair" for existing teachers who would like to transfer to a different school. My daughter got offers from three different high schools. Several other friends of mine, but in different disciplines (none involving math or science) got zero offers.

Admittedly, my daughter not only has her degree in math, not education, but also has a reputation already in the district as something of a star. Still, the openings for math teachers with undergraduate math degrees were rather plentiful.

And sorry to NEA fans, but I'd be wary of dealing with any of their statements. I'm not sure there's enough BS Neutralizer to deal with it all.

Teachers don't have summer off? They sure as hell aren't working. Yes, they're paid less, but intelligent people compare annual salaries and total work days. The NEA's argument is, to use Chouinard's words, "if not by, exclusively for half-wits and imbeciles."

Finally, I can't resist commenting (so to speak) on law reviews. I was invited to join the UCLA Law Review after my first year because of my grades, but my father died immediately before my finals, and I had things I needed to do in Fresno for my family that summer to deal with that, rather than spend the summer starting to write a comment. In the process of declining the invitation, I came across a quote from Learned Hand, one of the greatest 20th Century American jurists, in declining the invitation to join Harvard's Law Review: "I did not go to law school to edit a magazine."

I went and had a look at the common core standard for math, and I’m truly puzzled by the responses to them, or at least to the sample I looked at. I can’t imagine any informed scientist arguing against them. They seem to me to propose basic topics for understanding which are really beyond question. I’d be interested in having a common core critic single out some particular objective (or really, to merit blanket condemnation, a large set of objectives) and explain why they are inappropriate or unnecessary for the level of basic mathematical competence our society might hope for. The entire list is at http://www.corestandards.org/Math/ .

Now that I’ve glanced at them, I wouldn’t be surprised if it turned out that a large percentage of folks criticizing the common core don’t actually know what is in there. I saw nothing, for instance, about methods, only desirable outcomes.

My guess is that the real problems have occurred with implementation and that the critics are in the process of throwing the baby out with the bath. Implementation involves all kinds of political, commercial, and special-interest groups and pressures for immediate measurable results. For example, I can't understand why you would think it advisable to graft wholesale changes onto the the entire system at once. If you have something new, you start out with kindergartners and let the wave of change ride with them through the system. But then we have to wait twelve years for the first high-school results and I guess too many elections will have happened, too many apoplectic talk show know-nothings will have weighed in, and too many stocks will have traded for that kind of patience.

I've found Ed Hartouni's voice in this thread to be rational and authoritative. I don't want to repeat things he's already said about as well as they can be. So let me try for some other perspectives.

The first thing to understand, I think, is that elementary math isn't simple at all. The basic concept of number is something the human race has needed a long time to grasp. We do have certain routine processes for the most basic integer calculations, processes that may seem arbitrary and arcane to learners, precisely because they have been worked over and refined for thousands of years and are now in a highly evolved state.

For example, proper fractions can be considered as a representation of a number of parts selected from an equally divided whole. But they also can be represented as points on a number line. How are these two concepts related, how do these representations inform (or fail to inform) the arithmetic one does with the objects, in what ways do the different perspectives reinforce each other, and what are the pitfalls of using one perspective in the other context?

Another example: I think many kids learn that multiplication is a shorthand for repeated addition. But how does this interpretation apply to multiplication involving negative numbers and/or fractions? Is some extension of the concept of multiplication needed? What is it? Does anyone tell the kids about this or are they just presented with multiple processes called "multiplication" with no conceptual integrity to justify the common name?

One of the features of innumeracy is not the lack of facility with calculation, but rather an inability to select the appropriate operations to apply (and the correct order to apply them in). There seems to be a particular difficulty in recognizing what sorts of situations call for division, especially when fractions are involved. One might infer from this that too much time was devoted to the processes and not enough to recognizing when the processes are or are not called-for.

I think we tend to forget that the point of those processes is to free the mind to think about more complicated things. If you learn those processes by rote in way that obscures the underlying connections, and if you aren't helped in your education to confront complexity and progress in you ability to resolve difficulties, then I think it reasonable to consider your education at least a partial failure.

All this is the beginning of an argument that says that you have to know and understand a lot of mathematics to teach elementary school math. To the extent that the new standards really do promote critical thinking---whatever the hell that is---they demand a far higher level of knowledge and competence from the instructors, who have to be able themselves to deal with unexpected situations and novel ideas from their students.

An important aspect of critical thinking has to do with language skills, because without cogent explanations all you have are unsupported opinions, an environment that is fatal to anything that could possibly be defined as critical thinking. So the development of language skills is a fundamental aspect of science and math education, and some of those language skills ought to be developed in the context of math and science.

I think it is almost beyond question that our schools of education have failed to provide the kind of instruction required, but we should also recognize the tremendous resistance coming from college students who, by virtue of their own inadequate training, think they already know what the subject is "about." As institutions of higher learning model themselves more and more as consumer businesses, it will become harder and harder for anyone to tell the consumers that they will have to re-examine a host of things they "know" that just ain't true. And I might add that once tenure is eliminated, it will become increasingly difficult to do much beyond what the audience demands, a bizarre reversal in which the ignorant end up dictating what the experts teach.

One of the most exciting things about mathematics is the way in which it extends your mental capabilities, and I worry that we don't provide the kinds of experiences that would reveal this aspect. There doesn't seem to be enough of the really big and the really small. Why aren't more fifth graders calculating how many blades of grass there are in the front lawn or the number of pine needles in their christmas tree, i.e. things mathematics can do that they can't do without it. I know kids find such questions interesting. For example, when my daughter was ten or so, she was looking out a screen window and turned to me and asked how many squares there were in the screen. Of course we worked out an answer.

I have no idea whether there are any solutions to these extensive problems. You've got to have your head in the sand to believe that other countries aren't doing better, but whether we can or want to imitate their approaches (which in the article cited by the OP are our approaches, except that no one in the U.S. will listen) is a big open question. I'm positive that the Common Core is not a problem, although the way in which it is being implemented may very well be disastrous. I think that there is little hope for elementary math education until we recognize that the exceptionally daunting task of teaching all subjects is far too much to expect of any one individual. Elementary schools need specialized math/science teachers whose only job is to teach math and science.

And this brings me to my last point, which is that at least as far as I can tell, there is almost a total separation of elementary math and science, which means that two subjects that have grown up together and, in the real world, cross-fertilize each other are taught in complete isolation. Wouldn't it be better if some significant part of the math curriculum was taught as a response to issues arising from the science curriculum, which is to say in a way that recapitulates much of how the human race developed the subject?

I went and had a look at the common core standard for math, and I’m truly puzzled by the responses to them, or at least to the sample I looked at. I can’t imagine any informed scientist arguing against them. They seem to me to propose basic topics for understanding which are really beyond question. I’d be interested in having a common core critic single out some particular objective (or really, to merit blanket condemnation, a large set of objectives) and explain why they are inappropriate or unnecessary for the level of basic mathematical competence our society might hope for. The entire list is at http://www.corestandards.org/Math/ .

rich (and sully), i see common core largely as a hopeful attempt to address the problems of nclb and industrialized testing. as best i can tell, the biggest improvement comes mostly in english comp (which is also closer to my core competence),

if folks who care can use common core as a lever to make other changes, more power to them.

the basic problem with math and science ed in k-12 public schools, though, appears to me (and lots of other folks who do this for a living) less with "standards" and more with infrastructure: if we don't fund labs (and we don't) then there's no science. and if we can't hire math teaching talent, tweaking standards may well reinforce the poular american idea that any problem can be solved on the cheap by applying the proper technology. and banning teacher unions.

at k-8 math is probably the subject most readily assessed via standardized tests-- most knowledgeable folks i know see the difficulty of attracting teaching competent teaching talent as the major problem, rather than the old standards.

i'm sypmathetic to the op, who seems to have put in the years in the trenches, watching unfunded mandates come down from above while spending more and more time on spreadsheets.

There are breakthrough moments in learning. Recall the Annie Sullivan/Helen Keller moment where a link was made between the sign and the object. I recall tutoring my siblings - and somehow it escaped all of them that x was a representation of a number or set of numbers. Working on algebra without first understanding the concept that variables are a representation of a quantity - useless. That is why I favor trying to make a practical link to the math being taught. It is just as important that this conceptual link be made from expressing quantities numerically - 2 and then expressing relationships with variables - x. Like most, it seems obvious what works for me, but other minds may not be able to make that leap from the same place.

It is mystifying how people can get pushed up through the grades when they clearly are missing a critical element of the foundation. Repeating a grade - doing the same thing the same way for a second time - that is probably not going to be successful. If I didn't understand you the first time, I won't do better when you repeat yourself louder. There has to be a different approach to fill in those foundational gaps before moving on to a more advanced topic. I don't take my Bunny Slope skiers onto the green or blue slopes until they have mastered stopping - and I have several different methods to teach that skill. I have seen kids make radical leaps when they have an opportunity to take a different approach to a learning objective.

Still many generalizations here - all teachers underpaid, or brilliant mathematical minds get paid more elsewhere. That makes many presumptions. As someone that works with a lot of very technical people, I can assert that brilliance is not the only factor for getting hired or being successful in Corporate America. Many brilliant people have not successfully negotiated the corporate HR hurdle or lasted through the business cycles. Not all success is measured in dollars. The most brilliant math teacher I ever had was a nun in a Catholic High School. She could find a way to help her students make a connection.

It is still fair game for us to be concerned about the best way to educate our young. The fallacy is in thinking that there is "a way" and "a solution".

I find this topic very interesting. I'm a lifelong scientist, and as such, I have often reflected that the significance of math is that it is the language of science.

And math is the gift given us by God to describe the natural world, particularly calculus. I'm biased toward biological systems, but I'll be the physicists feel the same.

How can we expect to teach a subject as precise, demanding, and frankly challenging as math, to a society that spends its evenings watching reality TV.???

Let's rephrase that:

How can we expect to teach people steeped in the modern world, using the skills and techniques of the 12th century?

I think that the best teachers know that they have to adjust the teaching to the student, that it is fairly difficult to try to change the student to the learning.

The issue of teacher pay keeps coming up, and it often does.

I'm not entirely sure why, when the starting salary for teachers in Ca is approximately the same as the AVERAGE pay for all working Americans. (around 40K), and that the average teacher in Ca makes a salary of about $68,000

On top of which, there is a huge market of people who have trained to be teachers, who cannot find a job. When there is a huge oversupply of workers for any job, it tends to suppress the pay.

I don't understand why educators think that everyone has to be good at math. Only a small percentage of people need to know anything more than basic arithmetic to do their jobs. Forcing everyone to learn advanced algebra, trigonometry, etc. just turns kids off to education in general.

Here is an enlightened intellect. Imagine if we tapped into education where it mattered and produced.

I have, however, taught law and economics for over 25 years, and always got superior reviews from my students, despite having no formal training in education

This is a huge point. I am in the same boat, >25 years as a medical school professor, but never, ever required to have any formal training in education.
(I got a considerable amount on my own).

This leaves the average student to the mercy of luck. Certainly the tenured professors are hired and promoted virtually exclusively on the basis of published research. Most don't care about teaching, and it shows.

I thought that the teaching skills were better in High School than University than graduate school than professional school. I think the experience may vary due to the progression in student LEARNING skills. By the time someone gets to professional school, they are professional students, and can probably teach themselves most things they might be interested in learning.

I read the article, and it does seem clear that the implementation of the standards in the examples is really bad. The actual standards, however, are

Question 1 on the first-grade test is based on the New York Common Core Standard, 1.OA4 Understand subtraction as an unknown-addend problem. Question 12 tests standard 1.OA6, which requires students to use the relationship between addition and subtraction to solve problems. Question 8 assesses Standard 1.OA 7 which requires students to determine whether addition or subtraction sentences are true or false.

It seems inconceivable that any curriculum would fail to address these issues. The author complains that they are "back-mapped" from skills needed for college and may be developmentally inappropriate, but these are just basic arithmetic properties that have to be addressed somehow in the curriculum. I don't know about age-appropriateness, but my intuition is that these standards could be implemented in an age-appropriate way.

By the way, the fact that parents might struggle with some of the questions may be because those questions are poorly posed, as is the case of the examples in the Post article, but it may also be because the parents are themselves victims of the kind of procedural approach the Common Core is trying to supplant.

This can happen with pretty highly-educated parents. Some of my daughter's friends used to come to me for math help. Their parents were engineers. The parents used all kinds of math, but couldn't explain any of it to their own kids, because they had little beyond (some pretty high level) procedural knowledge, and no good way of transmitting any sense-making. I can see the author of the Post article saying (she actually does say something like this), "my friends are engineers, and they can't do these problems." Maybe this is a valid condemnation of the questions, but maybe not.

...most knowledgeable folks i know see the difficulty of attracting teaching competent teaching talent as the major problem, rather than the old standards.

I tried to say that in a gentler way. As for the "old standards," were there in fact any?

i'm sypmathetic to the op, who seems to have put in the years in the trenches, watching unfunded mandates come down from above while spending more and more time on spreadsheets.

i'm open to persuasion.

No, I'm sympathetic too. I taught high-school math right after college and before grad school, and I enjoyed it, but I don't think I'd last a semester in today's atmosphere. And I think more and more students are starting to figure that out, as the hostility toward teachers (some of it on display here), the erosion of the perks that partially offset the low pay and lack of respect, and the imposition of high-stakes testing with inadequate input from the folks on the front lines and extreme consequences for personnel and institutions all make the job look less and less attractive.

As for the glut of teachers and hoardes of jobless graduates someone referred to, I don't see any of that. Every one of the math students who graduate from my little institution has found a teaching job, usually right away, and often quite a good job at that.

klk, the Common Core math test that my high school piloted had an impressive math assessment that I proctored with an admin.. Not only was it all online, it had writing portions for justifying answers. Additionally, there was a group task involving designing a playground using correct measurement, proportion, etc.. Common Core includes a real world and cooperative project angle beyond its online test.

delighted to hear that you think it improves on the older standards. i don't teach math, so don't have a professional judgment about its relation to the older standards.

given the current constraints on k12, it may well be an improvement. but i have a difficult time seeing the current stands as the primary problem for math and science.

i have higher hopes for the reading & comp. the test change there is huge.

What I don't understand about "common core" is I don't remember a time in my k-12 education when we were not measured against national standards once a year, and in my ed classes we talked a lot about national standards in terms of writing curriculum, so the whole idea of national standards seems to me to be one of the fundamental aspects of public education and has been for some time. How are the "common core" standards different?

I'm sympathetic too. I taught high-school math right after college and before grad school, and I enjoyed it, but I don't think I'd last a semester in today's atmosphere. And I think more and more students are starting to figure that out, as the hostility toward teachers (some of it on display here), the erosion of the perks that partially offset the low pay and lack of respect, and the imposition of high-stakes testing with inadequate input from the folks on the front lines and extreme consequences for personnel and institutions all make the job look less and less attractive.

the most talented folks i see here (who have both tech and communication skills) view k12, if they view it at all, as akin to the peace corps or working in a soup kitchen. lots of them do it for a year or three to pay their tithe, and then they step into triple-figures in the private sector. can't blame them.

that may change. i was just at a conference surrounded by talented science folks on their 2nd or 3rd post-doc and with little hope of academic placement. but most of them won't go into k12-- they'll prolly go write code.

It boils down to household dynamics. How parents run the home show. TV-VideoGm or quiet/study time and personal help. Don't blame the education system. Math takes after-hour practice/discipline. More for some. I figured out myself first time at Calculus. I'd give myself a 2X test everyday. By test time I was dialed. Like hang-dogging!

I'm biased toward biological systems, but I'll bet the physicists feel the same.

They do, Ken, and then some. In the words of Feynman's famous quip, "Math is to physics as masturbation is to sex."

I second Rich's recommendation of specific math and science teachers at the elementary school level. My main climbing partner for the last 47 years, Tim Schiller, is an MIT graduate. His daughter's third grade teacher (also a friend of mine and the wife of a friend of mine) basically told Tim that she was uncomfortable with arithmetic, and would not assign arithmetic homework. What happens to her third grade students? I just cannot accept the idea that an elementary student does not get competent instruction in age-appropriate math because the elementary school teacher is uncomfortable with the subject.

it's well-known that the PhD in education has been among the least-respected doctorates awarded at American universities. Lack of rigor, absence of theory, and the very worst attempts at empirical analysis will do that.

another difficulty is tying student performance to teacher accountability. The teacher is not responsible and cannot be responsible for the range of student preparation, capability and motivation. The variability of even a graduate class at the university could be quite large, and things that worked brilliantly one year fell flat another.

in addition, increasing the size of the class makes the ability of the teacher to compensate for the variability of the students difficult to impossible.

Interestingly, there is a great deal of value of studying fiction relevant to preparing a capable workforce. Generally speaking, fictional literature explores a range of human issues that can be a springboard to discussing cultural values, ethical behavior, and a host of issues that are more clearly drawn in a work of fiction than in the description of actual events.

Debbie (my wife) teaches various biological subject classes at the local community college. One of her major goals is to get her students to write a logically argued paper, apparently most of her students are not capable of doing that. The regional high schools are considered to be quite good, yet the students would not meet even a rudimentary measure of accomplishment in writing.

Often, this ties to their ability (or lack of ability) to read critically, and to think critically, skills that are common to learning any subject.

I guess one could question where in life does one have to write a critical paper, or think critically at all... I'm not a good judge of that, I get paid to think and to communicate the results of those thoughts, so it is rather important in my career.

But I am not at all sure I would lay the students' inability at the failings of their teachers alone. It seems a more general failure of our society.

The propaganda and mis-statements are so pervasive as to make any other post by this poster suspect to being agenda driven.

LOL,
Ken believe me I don't have an agenda,
merely posted the NEA's research based take on the matter of teacher pay.

You have the right to view it as "propaganda and mis-statements" and I'll respect your opinion.

What I don't appreciate or respect is you calling any of my future posts "suspect to being agenda driven" when you don't even know my opinion on the matter.

Next time don't accuse based on unfounded suspicions,
I don't know you, however I hope you're better than that.

Ed, most proposals I've seen about tying teacher pay to student performance proposed value added measurements. They can't rely on absolute level of students' test scores, precisely because of the variables you cite over which the teacher has no control.

Even value added creates measurement issues, particularly if the student is not proficient in English. Nonetheless, for all the measurement issues, it surprises me to hear opponents of "high stakes testing" contend that other options are better, without telling us how they measure outcomes.

the whole concept of "outcomes" is a very strange one, also... as outcomes are often measured in lifetimes, or at least in careers.

I wouldn't have helped most of my teachers if doing well on standardized tests was a measure of my academic achievement, I was a horrible test taker... I doubt that I'd be any better had I taken any of the standard tests today.

But I never saw education as taking standardized tests, either... many classes that I took and did not do well in inspired me to learn the material other ways, outside of class. The "outcome" was positive, but unmeasured.

How rotten is your product, if you can't give it away for free?

If the consumer doesn't like it, whose fault is that?

The kids and their families seem to not want the free education the taxpayers try to give them. Most public school students in L.A. won't graduate high school. Damn few public school students can pass a math and English test.

Yet families will camp out in line in an attempt to get their kids into a private school, willing to pay for the education private schools provide.

The kids don't eat the free public school food, either. They throw most of it away.

But they will stand in line to pay for In-n-Out or Rosa Maria's, and throw none of it away.

Why can't the public schools give away something the private enterprises are selling?

The dinner guests were sitting around the table discussing life. One man, a CEO, decided to explain the problem with education. He argued, “What’s a kid going to learn from someone who decided his best option in life was to become a teacher?” To stress his point he said to another guest; “You’re a teacher, Barbara. Be honest. What do you make?” Barbara, who had a reputation for honesty and frankness replied, “You want to know what I make? (She paused for a second, and then began…) “Well, I make kids work harder than they ever thought they could. I make a C+ feel like the Congressional Medal of Honor winner. I make kids sit through 40 minutes of class time when their parents CAN’T make them sit for 5 without an I Pod, Game Cube or movie rental. You want to know what I make? (She paused again and looked at each and every person at the table) I make kids wonder. I make them question. I make them apologize and mean it. I make them have respect and take responsibility for their actions. I teach them to write and then I make them write. Keyboarding ISN’T EVERYTHING. I make them read, read, read. I make them show all their work in maths. They use their God given brain, not the man-made calculator. I make my students from other countries learn everything they need to know about English while preserving their unique cultural identity. I make my classroom a place where all my students feel safe. Finally, I make them understand that if they use the gifts they were given, work hard, and follow their hearts, they can succeed in life (Barbara paused one last time and then continued.) Then, when people try to judge me by what I make, with me knowing money isn’t everything, I can hold my head up high and pay no attention because they are ignorant. You want to know what I make? I MAKE A DIFFERENCE. What do you make Mr. CEO? His jaw dropped, he went silent. -

i never understood algebra or anything aside from basic multiplication/ division very well until i took a calculus class recently. the teacher put everything into the perspective of speed, velocity and acceleration and it made much more sense. the physical contacts really helped to understand rates and functions. i think teachers focus too much on graphs and what they look like, when the real business is what they're visually representing

for me, one of the difficulties in learning higher level math was overcoming the idea that math is somehow self-evident.

in other words, a person might beat themselves up while trying to learn another language because it is difficult or it is a slow process, but in general i don't think people blame themselves for not immediately understanding a tongue that they didn't grow up with.

otoh, i think many who understand and teach math [including a generalized "society"] forget that math is as constructed as say the spanish language. the implication that math is "self-evident" can create for some of those trying to learn, including me personally, the idea that their not having immediate understanding is a personal failure rather than a natural part of learning any new language.

once i accepted that math had to be worked at like another language, and quit beating myself up for not immediately understanding or being able to apply new concepts, i was able to overcome a hump that presented itself as i tried to continue my mathematical education.

regardless of what the ultimate reasons for math difficulties are, it would behoove us as a society to more effectively teach math [especially at higher abstract levels] not because it necessarily has an immediate application but rather because it can foster particularly rigorous and deeply creative ways of thinking...

My pet peeve in this regard in when i hear other teachers say, "i'm not good at math", sometimes even to their students. They would not dream of saying "i''m not good at reading" or "i'm not good at writing".

I like the foreign language analogy above. Nice thread.

I didn't read the entire thread but wanted to share my own experience. Way back in the day, (60's) someone in their infinite wisdom came up with, "New Math", I cannot tell you what the Hell it was but it was not successful. I was horrid at math, tried as I could just did not succeed. As well as the so called new math the teachers did little to help us struggling students.
When I began college I tested so poorly in math I was put in, "dummy math", thank goodness 'cause it helped me to finally grasp the basic simple facts. The only way I was able to succeed in algebra and statics was using a tutor and I must say he was WAY more accomplished that the college prof. in explaining the concepts for me.
Because of all of the above, I swore my kids would not get left behind or struggle as I did. To prevent this, I spent extensive time with them, going over and over the basics (memorizing time tables, working out complex multiplication problems, ect.), this was a tremendous help for them and resultingly they did not fail like I did.
Teachers just do not have the time to go above and beyond to help their students, it is up to us parents.
However, the bottom line in my case is I do not have a math/science thinking mind, give me English any day!!!!! Far rather compose a poem than try to scratch out a damn equation.

How rotten is your product, if you can't give it away for free?

Chaz is right. They've NO PROBLEMS giving it away in Africa or Asia.
Hell, they even get away with charging for it over there. And kids will
walk miles to school, like some of us did. But we had to walk through
blizzards.

Even value added creates measurement issues, particularly if the student is not proficient in English. Nonetheless, for all the measurement issues, it surprises me to hear opponents of "high stakes testing" contend that other options are better, without telling us how they measure outcomes.

i don't know many competent folks who are entirely opposed to testing or assessment-- teaching involves constant evaluation of student performance, formal and informal.

folks oppose the expansion of standardized (typically multiple-choice) testing out of fields of study fro which it is appropriate and into fields where it isn't.

the basic problem is that competent assessment, like competent teaching, requires a high level of skill. someone who isn't competent in calculus can't design his or her own quiz or test, much less grade the results. standardized exams seem to work well enough in limited applications in basic math.

but they're close to useless for history, writing, and lab science.

the idea that you could end lab science, or hire teachers with no math competence, and then show improvement in outcomes by applying a standardized regime of constant tests in which you train kids for testing, is simply hopeless as pedagogy.

i don't know of a single, highly-ranked private school k-12 that follows that practice-- it's been roundly rejected by the free market. affluent americans spend lots of money sending their kids to schools that don't follow industrial practice.

Again, how is common core different from the standards which have been used for decades? Everyone here advocating for common core just keeps giving wonderful arguments for why we need well thought out standards for measuring learning and for deciding when it is appropriate to graduate vs. not graduate someone from one level to another, but standards have always been a part of what you refer to when coming up with a curriculum or deciding whether someone has earned graduation. So, again, how are the common core standards fundamentally different?

Sorry, lots of people here just seem to be under the assumption that common core is the first time we have seen the concept of standards in education, which is just not the case.

Many parents and teachers are seeing common core as just some newfangled way of teaching developed by academics who have no clue who teaching really occurs in the field.

That's an accurate description of what it is.

The problem with the federal intrusion isn't the establishment of minimum standards, it's all the bureaucracy that goes along with it and the unstoppable tendency for it to grow.

I've seen the local school districts headquarters staff explode to 5-10 times the level it was in the 80's for the same enrolment levels.

All of those people just shuffle state and federally required paperwork. they don't educate anyone.

Can anyone show that federal involvement has improved outcomes anywhere?

Most public school students in L.A. won't graduate high school.

you should look it up... probably a bullshit statistic... I'll do it later but you could also read this:
Public School Graduates and Dropouts from the Common Core of Data: School Year 2009–10

• Across the United States, a total of 3,128,022 public school students received a high school diploma in 2009–10, resulting in a calculated Averaged Freshman Graduation Rate (AFGR) of 78.2 percent (table 1). This rate ranged from 57.8 percent in Nevada and 59.9 percent in the District of Columbia to 91.1 in Wisconsin and 91.4 percent in Vermont. The median state AFGR was 78.6 percent.

• Across the United States, the AFGR was highest for Asian/Pacific Islander students (93.5 percent) (table 2). The rates for other groups were 83.0 percent for White students, 71.4 percent for Hispanic students,
69.1 percent for American Indian/Alaska Native students,
3 and 66.1 percent for Black students.

• A comparison of data from 2009–10 to data from the prior school year, 2008–09, shows a percentage point or greater increase in the AFGR for 38 states (table 3). The AFGR decreased by a percentage point or more for only the District of Columbia during that same time period.

• Across the United States, a total of 514,238 public school students dropped out of grades 9–12, resulting in a calculated overall event dropout rate of 3.4 percent in 2009–10 (table 4). New Hampshire and Idaho had the lowest event dropout rates at 1.2 and 1.4 percent, respectively, while Mississippi and Arizona had the highest at 7.4 and 7.8 percent, respectively. The median state dropout rate was 3.4 percent.

• Across the United States, the calculated dropout rates increased as grade-level increased (table 5). This pattern was also true for 24 states. The lowest dropout rate was for grade 9 (2.6 percent) while the highest grade-level dropout rate was for grade 12 (5.1 percent).

• Across the United States, the calculated dropout rate was the lowest for Asian/Pacific Islander students at 1.9 percent and White students at 2.3 percent (table 6). The dropout rates for American Indian/Alaska Native, Black, and Hispanic students were 6.7, 5.5, and 5.0 percent respectively.

• Comparisons between high school dropout rates in the 2008–09 and 2009–10 school years showed a decrease of a percentage point or more in Delaware, Illinois and Louisiana (table 7). An increase by the same margin or greater was also found in three states; Mississippi, New Mexico, and Wyoming.4

• Across the United States the dropout rate was higher for males than for females at 3.8 percent and 2.9 percent, respectively (table 8). The dropout rate was higher among males in every state. The male- female gap ranged from lows of 0.2 percentage points in Idaho to highs of 1.7 in Connecticut and Rhode Island.

So were you able to help your daughter's friends with their math homework?

Yes.

If you were the only one who could help, did the daily process of learning math for these children now involve regular trips to the college professor's house to get help on their elementary school problems?

I don't think I was even close to "the only one who could help," and my daughter's friends consulted me intermittently, not continually.

Here's how common core is actually working in practice: Kids come home with their homework, do what they can, ask mom and dad for help, mom and dad don't understand either, mom and dad start calling parents of other students to collaborate (or even call the teacher if they can), eventually they might find someone who gets whatever unusual approach the particular question is using.

I'm not saying the common core implementations are any good; it is clear that there is a tremendous amount of variation. I was reflecting on the fact that parents unable to do kids homework is not a priori a sign of the fact that there is something the matter with the homework, especially if the homework is supposed to repair educational defects the parents themselves may have suffered from.

All the while the kid sits at the table staring at the paper wondering how they will ever learn math if the parents can't do it either.

What the kids make of this is conditioned by how the parents treat the experience. I surpassed my father's knowledge of mathematics in high school; he used to say "you know more than I do now, I can't help you with any of this." But he said it with evident pride that his son was surpassing him, and that made me proud and excited too. It never occurred to me that his difficulties with the material might be mine.

The problem with my engineering friends is that they had a dysfunctional model for what it means to "do math" in spite of the fact that they used mathematics regularly. We might think of an experienced regular commuter who understands little or nothing about how their car works and is at a loss if anything goes wrong. The algebra topics my friends' children were dealing with were from an old-fashioned standard curriculum from before the days of reform. Their inability to explain things in a way the kids could understand was based on what I would call flaws their own (quite substantial) educations and not on any "defect" in the school curriculum, although nowadays the people in the same situation might well be tempted to blame the common core for their struggles.

My own secondary education was deeply influenced by what was then called "the new math." It had a profound effect on many in our class, which now includes a bunch of Ph.D. scientists and mathematicians, because it showed us that the rote nonsense of our elementary school experience could, in fact, be understood, and that collections of apparently arbitrary disparate rules and procedures could be traced to a few common principles that could then be easily extended to new situations. The sense of intellectual power and scope was heady and obviously shaped quite a few futures. I mention this because the "new math" is regularly cited as the first and primary example of a failed educational reform. Parents couldn't do their kids homework. Tom Lehrer, who is a mathematician, wrote a hilarious parody song about it. But it trained the generation that responded to Sputnik and leapfrogged the U.S. into a position of primacy in science that is only now starting to erode.

Obviously, I was blessed with teachers who understood that the "new math" was, in fact, just "math," and were able to convey it to us. I suspect that the new math failed because too many teachers weren't up to the task, not because there was anything intrinsically wrong with it. But nowadays the extreme pressures of the current situation, the polarization and politicization of the "stakeholders," the pervasive dysfunction in governmental decision-making, the abandonment of reasonable compromise and the embrace of simplistic solutions, all of this amounts to an atmosphere for teachers and students that is a far cry from my days in high school.

Jammer, standards were ushered in with nclb. Common Core streamlined standards to around fifteen because there were so many previously.

I'm pretty sure they were in place much longer than NCLB, which was being implemented a little less than ten years ago to my knowledge. Standards have been used as long as public education has been around in order to guide curriculum development.

One difference with common core and the new idea of standards that I understand is the idea that somehow kids these days, despite hugely variable home stability and support, despite funding for public schools being based on the schools surrounding property values, will all be able to pass some academically rigorous standard to graduate high school. This is a new idea. This wasn't in place before because under the current funding model it's laughably unrealistic.

This is why I think Daves kids maths problems are so cryptic at times: they are trying to find ways to reward kids with "other intelligence" so they can find some way to meet these ridiculous, out of touch, academically rigorous standards.

I predict, as TGT pointed out, that non-teacher jobs will start to grow, further diluting available funds for education especially for poor districts were they will likely decimate education funds entirely. This will completely f*#k the American education system further, providing a vile argument for politicians that both spending money on public education is the worst idea ever so it should be privatized further (more school vouchers for one of the private schools, something like 90% of which are religious BTW) or even have federal dollars outright cut, and that certain groups of people (as a result of the compiled data of recurrent failure to meet the same standards by underfunded schools) are just "less". Then, it will be further down the rabbit hole as we chase the culprit of America's failed education system, all the while ignoring the elephant in the room, which is the funding problem.

Question for you: How is a poor district, whose funding has all been channeled into the new education bureaucracy brought about by all the new standard makers/assesers (which is somehow far superior to teacher unions?) going to meet those standards, and how will their failure to do so benefit the world?

Also, how is this a better approach than just equalizing funding? Could an even better one be equalizing funding and implementing more academically rigorous standards, which would just basically happen in and of itself if we equalized funding?

How rotten is your product, if you can't give it away for free?

But it can still be free and very successful - consider that Singaporian students are consistantly, year after year, in the top 5 in the world wrt mathematics for K-8.

The first question to ask is what's the difference and how can/do we replicate their success.

After solving the above then we should ideally improve on it.

But I still find myself stumped by the "methods" being taught to ten-year-olds.

perhaps because you don't have the mental development of a 10-year-old anymore...

you certainly have the ability to think about problems in quite different ways, binary arithmetic operations, as you point out... and you most likely look at division very differently than what is taught... does that mean that the way it is taught is incorrect for a 10-year-old?

klk, there is a very vocal group of teachers in Fresno who oppose "high stakes testing" in any form, and who get an inordinate amount of publicity about it here. They never propose an alternative, though, so I doubt many pay much attention to them, despite the attempt of certain local media to stir the pot.

I, too, am wary of multiple-choice, standardized tests, although I was always very good at them. I was almost alone among the economics instructors at Fresno City College in avoiding the scan-tron, even though it automated test grading. I wanted my students to demonstrate that they could convey their reasoning verbally.

Multiple-choice tests, of course, have limited usefulness in areas where answers can differ, such as much of law. The California State Bar Exam has a mulstiple-choice portion, called the Multi-state Exam, but it cannot provide comprehensive coverage of legal understanding because the answer to so many legal issues depends on the choice of law. For example, in contracts, it tests almost exclusively on the Uniform Commercial Code (largely the same in 49 states), rather than on common law contractual issues, because so many of the latter have differing outcomes depending on each states's common law. Then there's Louisiana, with its civil law foundation. . .

I did note a pretty good correlation between the SAT II U.S. History exam and the AP U.S. History exam when I was in high school in the 1960's, though. My class was the second to take an AP test. On the AP U.S. History exam that year, another student and I were the only ones in our school to score 5's in spring of our junior year, and we also had the two highest grades in the school (795 and 789) on the SAT II U.S. History exam the next fall. The two exams could not have been more different, however. The AP exam allowed us to demonstrate our understanding of how one's perspective affects interpretaion of historical events. How do you design a multiple-choice exam to do that?

Interestingly, my daughters both got 5's on AP U.S. History their junior years, but only scored in the 600's on the SAT II version their senior years. They had crammed (with lots of help from me) for the AP exam, and forgotten much of what they learned over the summer. The difference between their AP and SAT U.S. Hisotry results validated (to my mind) my observations -- and, much more importantly, preserved my now limited bragging rights with them.

since 1989 the average graduation rate has been 61%

which is greater than 50% thus, chaz's claim that: "Most public school students in L.A. won't graduate high school. '
is demonstrably false... chaz, you do understand fractions and percents, right? where'd you learn that first?

over that same period only 49% of black males, and 46% of hispanic males graduated. Asian women had the highest graduation rate over that same period 97%

these statistics don't include movement into or out of the districts, it is calculated by the number of students graduating to the number of freshmen enrolled 4 years earlier.

you certainly have the ability to think about problems in quite different ways, binary arithmetic operations, as you point out... and you most likely look at division very differently than what is taught... does that mean that the way it is taught is incorrect for a 10-year-old?

Ed and Dave, I think much of the criticism of Common Core has little to do with the meat of Common Core -- namely the standards of what the students should learn -- but the methods of getting there. Too often, I've encountered mandates about how a teacher must teach a subject. One thing that always interests mathematiciams is proving the existence and uniqueness of various concepts. I am unaware of the existence, much less the uniqueness, of the "proper" way to teach a ten-year-old.

I know that in law, we argue our points in many different ways, because we never know which one the judge or jury will find convincing. I think teachers need that same flexibility, because students' methods of learning differ.

I'd like to turn the question around: Where is the evidence that these new approaches are actually better?
Adding It Up: Helping Children Learn Mathematics (2001)

Jammer, I work in a poor, agricultural district. Teaching close reading and critical writing is not that expensive. We have a college-going culture that is implementing Common Core and it's working. Plus, second language student programs throughout the nation have been given more government $. I have taught eld. Have you? Pm me with more questions. I know my sh#t.

One example does not make a rule. It sounds like the district you are in has a pretty good proportion of students who come from homes with two caring parents (i.e. "a college going culture"), which is the primary predictor of success.

Contrasting that district with an inner city school is the kind of B.S. compilation of data I am talking about. What do you think the real-world political implications of this will be when the contrast gets made? That is a serious question.

I maintain my prediction, now please answer my question(s) :).

since 1989 the average graduation rate has been 61%

Ed, the state changed methodology a few years ago, in a way that they now can track student movement. According to the new methodology, almost 2/3 of LA Unified students graduate on time.

I am aware of criticism of the new methodology. Some contend that the data start too late in the students' lives, so we already miss some dropouts, but I am unaware of any good statistics on this. The big thing is that data from more than about three years ago form a different time series from the most recent data.

good questions, John, you're saying that no data is relevant?

the link I provided is the way the state is presenting the data... and the table is calculated with the definition I provided above... from 1989 to the present for various groups of students by gender and by diversity category...

"...I suspect that the implementation of the process did not take into account the entire educational system."

a suspicion you could easily verify by doing a little work with Google. It was relatively easy to find the basis for the standards, you might look first at the Wiki entry, then track down the references...

Engineers have a "dysfunctional model for doing math?"

Engineering, by definition, is the application of math toward a beneficial purpose.

If engineers aren't doing functional math ... then who is?

Well, first of all, I said "a dysfunctional model for doing math" and you changed that to something completely different, "doing functional math." Second of all, I didn't mean to tar the entire engineering profession with the dysfunctional model brush, just my two friends. Thirdly, the engineers in question were using math but in strictly procedural ways, with an arguably tenuous grasp of underlying issues. This works as long as it works, maybe for a whole career depending on what the challenges are, but it put them in a position of only being able to explain procedures to their kids. If the kids, for some reason, were taught a different (but equivalent) procedure for the same underlying result, they couldn't adapt.

Perhaps we might make an analogy based on your own experience. A person who understands place-value can do addition and subtraction, almost immediately, in any base, as I'm sure you can. But a person who doesn't understand place-value and is perfectly competent in base 10 arithmetic will be utterly lost if they are presented with a base 5 arithmetic problem, nor will they ever notice the parallels between, say, polynomial evaluation and number representation, and they surely won't recognize problems that are naturally expressed in terms of other bases, such as the Cantor subset of the interval from 0 to 1.

The latter type of knowledge is strictly procedural and leads nowhere beyond its immediate application to adding and subtract ordinary numerals. If something about the procedure is forgotten, there are no mental props for recovering it. If some different procedure is proposed, there is again no mental infrastructure for deciding whether or not the proposal is equivalent to the known method and, if not, under what circumstances the proposed method is valid.

The former type of knowledge is far more general, allows for reasoning about the algorithms for addition, allows the framing of appropriate questions about them, allows for understanding what kinds of modifications will or will not be valid, and opens entirely new mental pathways for either using or recognizing the presence of representations in other bases. For an engineering example, it seems highly unlikely to me that anyone unfamiliar with different base representations would have thought to investigate ternary logic as more efficient model for computer chips.

Our education system, at all levels, often doesn't address the distinction between the merely procedural and something else that we call understanding. Some of the reasons for this are obvious; it is easy to teach and test procedures and hard to even define, much less test, understanding.

When it comes to elementary and perhaps also secondary education, there are very reasonable concerns about intellectual readiness for the kinds of thinking that might be involved. This might mean, as the Washington Post article suggests, that some of the features of understanding in the common core are inappropriate for the age levels that are supposed to be learning them, which leads to a very old claim that we should teach 'em how to do it and later they'll understand it. I'm guessing that's exactly how you arrived at your current state of expertise, but the counter argument is that this process doesn't work for a large number of people, including some engineers, who really do need it to work for them.

I suspect that the general attitude will be more like, "if English was good enough for Jesus, it's good enough for me."

Good one, Richie!

I also have my personal foreign language analogy. When I was a grad student in economics at UCLA in the 1970's, we had a language requirement for a Ph.D. The doctoral candidate had to know French (for Walras, primarily), Russian (under the theory that Marxist theory would probably be most developed in the USSR) or math, because math is a foreign language to economists.

My job title is "Senior Research Engineer" (i must be getting old),
and my education is in mathematics....

Guess what I just spent 3 hours doing at work - finding a slick way to solve an equation that had people stumped. Next step is to write it up as a proof.

Engineer is sort of a vague word (remember the good ol' sanitation engineer, f/k/a garbageman). Even assuming we're talking about ones who have legitimate engineering degrees, I'm sure there's lots of variation regarding how much math they use in their jobs, if any.
Still, as the holder of an engineering degree from a prestigious university, I'm surprised that anyone who calls himself an engineer is stumped by elementary-school math problems (I don't write this to insult anyone; it's just genuine surprise).

Can anyone post an example of a problem that has the engineers scratching their heads?
Maybe that will shed some light on whether the problem is with the engineers, the modern curriculum, both, or neither.

"In 23 years of teaching, I've met very few teachers that fit that description."

I have to agree completely with Wbw. I'm not a teacher but I've got two kids, one about to be a junior in high school and one entering his second year of middle school. That adds up to a lot of teachers I've come to know pretty well, and from kindergarten until now I've seen exactly three teachers in that whole bunch who probably put tenure above the the needs of their students. So, three bad teachers out of 45, and the majority of that remaining 42 were very good to excellent, truly dedicated teachers. Our community strongly supports its public schools and teachers (Mira Mesa, in San Diego Unified) to a great degree (and I'm not talking about $ here), something that isn't true for virtually every school in this large district that has problems. You're either part of the solution or...

Thanks, very interesting. I have to admit, at least on my cursory inspection, I have little if any idea of what the questions are getting at.
More straightforwardly, I imagine I would "flunk" the assignment.

I can sorta see how the Glenn Becks of the world see this sort of stuff as a vast liberal conspiracy, seems sort of intended to make sure that parents are "outsiders" to the curriculum. Even if that's not the intent, that's the clearly foreseeable result.
As someone posted above, at some not-super-high level, a good student is going to surpass what a typical parent knows about math and science.
But I think most of use would be surprised when this happens at second grade.

The Huffington post article is interesting. Of course, as I've suggested above, it suggests an implementation problem.

But frankly, I see something else. Someone with a BS in engineering---electrical engineering at that, the most mathematical of the engineering majors---who can't immediately grasp what method is being suggested and can't immediately see what went wrong, much less is incapable of puzzling it out, is a classic example of what I was speaking of when I said that the people who have been trained to replicate procedures may end up with very limited mathematical abilities.

We can argue all day about whether the process illustrated is a good way to conceptualize subtraction or, if it is, whether it is appropriate for the audience it is intended for. But when someone with a degree in electrical engineering can't grasp what is happening in that picture and can't explain it to someone else, eg a kid, then we are looking at an example, and I'm afraid not too uncommon an example, of a substantial failure of the the entire classical education edifice.

Whether this type of mathematical disability can be addressed by this or that new curriculum or set of standards is an open question, but as a symptom of the kinds of problems the current education system manages to produce, the example is stark.

OK after reading rgold's response, I have to change my response slightly and sort of agree with him. Part of my initial problem is that I ignored the handwritten numbers on the numberline (I found them hard to read). Feeling spurred on, challenged, and/or insulted by rgold, I took a closer look and agree that the subtraction method that Jack was supposed to use does seem to be relatively clearly suggested, and perhaps understanding that method does give a better "feel" for what subtraction means and what base 10 numbers mean than doing it the "standard" way that the parent wrote in his letter (which could be done, I suppose, mechanistically in such a way that the person performing the subtraction doesn't really have any idea what the process means).
But Jack's seeming errors are so random and nonsensical (how does he get from 127 to 107?) it seems silly to make that a standard math question instead of some sort of strange intelligence test (which apparently the parent and I did poorly on, at least the first time around).

Jack forgot to subtract 10, first subtracting 3 100s then 6 ones, that's only 306

the handwritten numbers are the engineer father not figuring out what is going on... and playing around with it to try to get it to work out...

that's a pretty standard behavior among first year physics students if you know the answer (maybe it's in the back of the book) and you just play around with combinations until you get the answer. Of course you can't explain it... and you don't get any points for your solution.

here is a resource for teachers that helps them find resources to develop lesson plans
maybe it would be useful for interested parents.

I didn't mean to insult anyone. We are speaking of educational failures, not personal ones.

The handwritten numbers in the example come from the confused engineer. Just look at the printed numbers.

Jack was supposed to subtract 316. That means he needed to "back off" by three 100's, then by one 10, and then by 6 ones. But Jack forgot to back off by the 10; the rest is ok.

All the engineer dad can do is to illustrate, with absolutely no explanation, how to get the right answer with the standard subtraction algorithm. He can't figure out what Jack was supposed to do, he can't see what Jack did right, and he can't tell Jack what he did wrong.

Let's take this out of the context of the engineer dad and suppose this guy was the class teacher. He's got a student Jack who, I would say, understands the problem fine but has just overlooked the operation with the ten. What Jack gets from the "teacher" is that his method is long, confusing and will get him terminated from his job when he grows up, and here is the right way to do with with no explanation and no connection to what Jack had in mind.

I think this nightmare scenario probably plays out in homes and classrooms across the country over and over again. Kids who have begun to make sense of arithmetic are told they are all wrong when they are not, and are asked to substitute meaningless (to them) manipulations for the beginnings of understanding they have begun to formulate. Then we're surprised when people say the most logical of subjects is obscure and confusing. We're surprised to hear about math anxiety after we attack children's valid attempts at sense-making and replace them with catechistic incantations with no intuitive content.

You'd have to know something about what went on in class to decide whether this question was appropriate for the audience it was addressed to, but I don't see anything the matter with it as a question at all. The fact that it is being bandied about as some sort of failure of the common core strikes me as powerful evidence for the failure of what came before.

EDIT
I guess I was typing this as the same time as that other ivory tower denizen Ed was posting. I'm so surprised to get this from Kos, who surely understands exactly what the Huffington post problem is driving at, and just how much of a problem it is for the education system that a guy with a BS in electrical engineering can't grasp what the question is about. I'm even more surprised that Kos pretends to divine base motivations for my opinions and brands me simultaneously arrogant and pathetic.

Ah well, but its the internet right? What was I thinking.

Could the fact that we don't negotiate when buying things at markets diminish our ability to utilize basic math on a daily basis have any impact on the original question?

Math as an element in everyday things like going to market does shape how the basics get re-enforced. Make no mistake, nimble thinking is essential in providing food in other nations.

I wrote a long explanation but decided in the end that Dave Kos would just see it as supporting his contention that the "only objective of" my "posts is to demonstrate that you are smarter than everyone else. It's arrogant."

It reminds me of a story that involved my wife, Debbie, who had gone back to school and was taking undergraduate physics. She rarely asked me anything about her assignments (maybe she had me check her homework before she submitted it) but one time.

At breakfast I was reading the NYTimes, which is routine, and she was hovering around waiting for me to finish the front section. I folded the paper and "bam" she put her Physics text in front of me and said "how do you do problem 35?!"

I had a reputation, in my household, of trying to lecture on the subject of problems rather than "just doing it" from both Debbie and our daughter. Debbie wasn't in a pleasant mood, due to this particular problem, so I decided to "just do the problem."

On a clean piece of paper, I wrote down each step, sequentially, explaining briefly the logic of the solution for each step, inexorably arriving at the conclusion that matched the answer in the back of the book exactly.

She burst out in tears and wailed "it's so easy for you!" which was not the response I was trying to achieve...

I reminded her that I had been studying physics since I was 10, and that I taught at the University for 10 years, and was employed as one, and that it pretty much better be easy for me, at least compared to someone who had been doing it for less than 3 months. She never asked me again, I like to think she understood that she could figure it out, it wasn't impossible.

If the solution to the problem seems easy to me (and to rgold) and you believe that we are arrogant because we do it to demonstrate how smart we are, you've really lost control of yourself over this issue, STForum is not the place I come seeking professional affirmation. And not only that, you're letting your ego get in the way of a child's education.

That collective ego is that 2nd graders can't possibly understand math that their parents can't.

The tragedy is that the educational system in the country has produced multiple generations of parents whose math understanding is less than what the Core Curriculum expects from a 2nd grader.

If that ego can't deal with that, then yet another generation of students will be let down by their education. While there will still be students that will achieve an understanding and use that to their great advantage in the future workplace (actually it is the workplace of today) a great majority of students won't have the opportunity.

All because some dad couldn't do his kid's homework?

[remember a couple of generations ago most of the parents couldn't do their kid's homework...]

it would seem that it should be relatively easy for that dad to learn what is being taught and support the teacher and the lesson... what lesson do you think that poor kid learned?

Sully, you make some really compelling arguments for re-assessing the currently used standards to reflect contemporary expectations in terms of general computer knowledge and things like that. How is having a central agency determine the standards a step up from having teacher unions handle it? Why can no one answer this question? You all can't possibly believe that THAT was the problem with education, teacher unions...

So, three bad teachers out of 45, and the majority of that remaining 42 were very good to excellent, truly dedicated teachers.

This was apparently in a high-performing school. You might accept that in a lower-performing school, that ratio might be higher.

Here is another thought, since we are talking engineering. You have a car that has only 45 parts, but 3 don't work. How well does your car work?

Only one bad teacher can have a devastating effect if it comes at the wrong time.

But the converse it true: If you talk to most successful people, they can name the teacher(s) that were responsible for the person they became, and generally speaking, those teachers were few.

I do not understand the argument that a superb teacher is irrelevant and meaningless.

I imagine that there is a normal distribution of teaching skills. A few bad, a bunch average, and a few superb.

Why would we not want to have a system that tends to push the average towards the superb column? We seem to have a system that protects mediocre performance, and protects the terrible from termination in any rational timeframe.

Here is another thought, since we are talking engineering. You have a car that has only 45 parts, but 3 don't work. How well does your car work?

Which is better, a car with 100 parts and three not working (and not really fixable), or a car with 3 parts and one not working (and not really fixable)? Considering a perfectly working car is not a real-world possibility, which of these options do you choose?

How is having a central agency determine the standards a step up from having teacher unions handle it? Why can no one answer this question?

what is this question?

is it:
why shouldn't educational standards be determined at the local level?

If we presume that the educational systems being independent is a way to experiment with many different approaches, then we can benefit from looking at the outcomes from all the experiments and find new ways to educate our children.

But no single educational entity can afford to do that kind of analysis and synthesis. Who would pay for it?

Now that sounds like the role of some larger organization, perhaps the states, and then the federal government to do it across the states.

At some point we have synthesized the most effective lessons from all the various ways of educating students.

Further, while teacher groups, the private sector, etc., may have specific agendas, the larger government organizations can provide guidance on larger issues, important at the state and federal level. These governmental organizations can embark on research to determine what standards are required for future needs of the state and the country, and from those studies propose those standards.

Changing teaching methods costs money, so those organization can also offer support to the schools seeking to change their curriculum to achieve the goals of the new standards.

Right now, it seems that that has been left as a choice to the various states and local government and to some extent to the local school districts, maybe even to local schools and to individual teachers themselves.

I am shocked that California, having adopted the Common Core has not made available support for teacher training to implement those standards, the state is currently running surplus budgets and can afford some level of support. Shame on the legislative bodies!

Now perhaps we can get rid of any outside influence of local schools, letting them get on teaching the three R's, after all, that was good enough for the parents, wasn't it? Why do more?

We've found that there are better practices, discovered in the vastness of the diversity of schooling. That students who were lucky enough to be educated in those ways did better. Is it ethical not to share that? It is every bit like some local hospital finding a miracle cure and it not being shared, only the local people are allowed to benefit from that knowledge. It is not so outlandish to equate the "miracle cure" to a superior education, they both effect the lifetimes of individuals.

The California Governor's budget for 2014-2015 proposes 42.4% to K-12 education. There is funding available for Common Core teacher training (enabled by Proposition 98), but that isn't available now... The total budget is something like $77 billion. This reflects a shift in the way education had been funded, with the passage of proposition 13 in 1978 which limited the rate of local property tax increases.

That Proposition 13 was a response to the findings of Serrano v. Priesthttp://en.wikipedia.org/wiki/Serrano_v._Priest that California's local funding of schools was in violation of the equal protection clause of the US Constitution 14th Amendment is another move away from local control to state control (in this case). Generalizing that, the federal government has an interest in seeing those same protections afforded to citizens in all states.

This is a long way from how schools were funded in the times of Ichabod Crane, were the townspeople of Sleepy Hollow foraged up a way to support a teacher for their children, those that felt an education was important for their children. Ichabod Crane is not a very sympathetic character, and in the end he looses out to the town rowdy in his bid for the hand of the Katrina, the only daughter of the wealthiest man in Sleepy Hollow.

Somehow teachers don't get a very sympathetic hearing in the United States, and funding education seems to always have been contentious. But given the global competition for a well educated and competent workforce, doing anything less seems to be ceding the future. And worse, we are making these decisions for the children, decisions that have the important legacy of determining their future. It's not something that should be taken lightly.

If we presume that the educational systems being independent is a way to experiment with many different approaches, then we can benefit from looking at the outcomes from all the experiments and find new ways to educate our children.

But no single educational entity can afford to do that kind of analysis and synthesis. Who would pay for it?

Ed, thank you. I believe this "analysis and synthesis" was traditionally handled anecdotally (due to the real-world affordability constraints you point out) by teacher unions. I would say that is better since the teachers doing the evaluation are actually in touch with who they are evaluating. Whatever that has been replaced with is now completely disconnected. I would say that is bad.

As for some kind of federal agency which oversees all the individual districts, do you mean to suggest that in the previous model there was zero curriculum guidance from some central agency who got info/data from people such as economists in order to generally inform where to direct education, so that there were qualified individuals to meet the current demands? Do you really mean to suggest that this kind of idea just now came about?

Again, how do you make everyone meet the same standard if they don't even get the same tools in public school (same funding), let alone at home. Why is this such an elephant in the room?

Sully, just learn to use the politics of embarrassment on those people. They are not going anywhere, and overall that will be much more effective. I would rather my educators not have to worry about their potential job security every two to four years than have some especially easy means of firing them, even if that means is truly implemented to do nothing more than get rid of vile individuals.

Which is better, a car with 100 parts and three not working (and not really fixable), or a car with 3 parts and one not working (and not really fixable)? Considering a perfectly working car is not a real-world possibility, which of these options do you choose?

I reject your premise.

why are you deliberately choosing failing options? Rather than "commitment to excellence", you appear committed to the least that can get by. No wonder education is in trouble.

This was the EXACT same premise that got American industry in such trouble, and pushed the Japanese to the front.

the case study taught to me in business school had to do with tolerances in production. The American company knew that the part could only be manufactured to specs 95% of the time. So they contracted to pay for only 95% of the product received. The Japanese manufacturer was confused by this, and they enclosed a note with the shipment:

"we do not understand why you have contracted for 5% out of spec parts, nor what use you will make of them, but we have included them in a separate box, so that you will not accidentally mix them up."

That was when the lines crossed in American and Japanese manufacturing (thanks to Deming)

Ken M, I agree with you completely! As good and as dedicated as the majority of my kids' teachers have been to date, which of those teachers will be the one, (or if we're lucky "ones") that makes that huge difference in opening up an entire world, even a passion for, a subject or endeavor that will shape and inspire the rest of their lives? I suspect there will be one or two, but you're right that that is a far rarer thing than it should be in a perfect world. As for that one bad teacher at the wrong time, all too true that it can have a devastating effect and--at the very least--shut down an entire subject and countless fields of endeavor as future options for a kid. That's tragic, yet I see no system and no possible way that you can ever root out every single teacher capable of causing that train wreck in a child's educational journey. Yes, I've seen a couple of those teachers, my oldest son has actually had two, though only one he won't entirely recover from. It's going to take a truly great teacher to now bring back his interest in learning Spanish. On the other hand, his other truly bad teacher (clearly "tenure first, students second") was a Geometry instructor, and full recovery in that subject happened the very next year (the school year just ended) in the hands of a truly great math teacher. We're so lucky that he landed in that class on the heels of his only truly bad math experience. Yep, the great teachers are a priceless commodity and a system that finds, nurtures and inspires more like that is the best possible investment we can ever make.
But my original point was simply that most teachers, in my experience and YES at our undoubtedly higher performing schools, are good and dedicated. And I'm talking about public schools here, schools in a huge district that has some significant problems overall, but in our neighborhood the public schools have tremendous community support and backing and it makes a difference. I wouldn't call Mira Mesa a poor area of town, but it's certainly a far cry from an affluent community here. Largely working class, its public schools feed into a high school that puts more kids into UC schools than any high school in the district. My point is only that it can be done because I've seen it done here.

This was the EXACT same premise that got American industry in such trouble, and pushed the Japanese to the front.

not what I think... the premise was that "American industry" was some how exceptional, and that it was superior because it was American.

It sort of looked past the devastation visited on most of the world by World War II, certainly all of the "first world" was affected, with the exception of North America. From a commercial point of view the United States benefitted greatly, its competition from foreign activities was literally destroyed. The universities benefitted from the emigration of the ousted European intelligentsia, and the fact that these universities were intact and functioning served as a beacon for the world.

In this greatly tilted playing field the United States somehow thought that that exceptional quality was something else...

...now, with a world recovered, the United States faces stiff competition from abroad, and is searching to find out what happened, why don't our assumption of "American exceptionalism" work any more? Did we loose some essential "American" quality?

Oddly, the rather enlightened post-war policies, to help rebuild the world after the devastation of war, helped put the international partners on a footing to be able to compete with the United States. It was thought, and probably correctly, that binding the international community in mutual dependence would prevent the types of conflict that resulted in world war.

Yet we now bemoan the lost domination of the United States competitive edge. The response should be to take up the challenge and to become more competitive. Upping our educational standards is part of the response.

why are you deliberately choosing failing options? Rather than "commitment to excellence", you appear committed to the least that can get by. No wonder education is in trouble.

I'm not deliberately choosing failing options, I'm just a realist. I seem to be one of the only ones here. I would address the vast majority of the holes in the old system by just equalizing funding and writing newspaper articles about "touchy" administrators and the like. It just sounds more reasonable and actually sounds implementable, which is more than I can say about almost all the current trends in education, all of which are the kinds of things which drive otherwise competent, caring, able teachers away. The pay is kind of lame, but that actually filters out the parasites IMO. The HUGE positive nudge you get, with all the social respect that comes with that, in addition to the lower-middle-class income you make, is payment enough for most people truly interested in the job.

I might also point out that "raising math ability" inherently implies raising the mean math ability, which is likely best accomplished by bringing the bottom schools up through adequate funding and reasonable standards, which is exactly the opposite of what common core and NCLB accomplish. I have absolutely no clue whatsoever how it's proponents think it's going to work.

At first I also found the huffington post problem confusing but then I realized that the handwritten text were not included in the problem.

I really don't see any problem with that exercise. It shouldn't really be that difficult to understand and solve for someone that know how to subtract numbers with the help of a number line.

I can't solve many of those problems because I don't know what for example a number disc is (but I can guess).

But is that really a problem? I am sure that the kids learn about those concepts in school and that it shouldn't take a parent much time to learn them.

I don't know if this way of teaching math (arithmetic) is good or bad but it clearly is about trying to learn the kids the basics of math. So that the kids don't need to use a computer to do calculations or use a basic rule that they don't understand.

Should this not be allowed because the parents learned it in another way?

This was the EXACT same premise that got American industry in such trouble, and pushed the Japanese to the front.

not what I think... the premise was that "American industry" was some how exceptional, and that it was superior because it was American.

Comrade Ed, I didn't mean to imply that what I'd cited was the only wrong premise....as you point out, this is somewhat complex.

However, I have been a believer in American Exceptionalism. Just not in the way most people mean it. I do not believe there is anything exceptional about Americans. I'm not positive of what it is. I have thought that it might be the nature of our higher education.

BTW, I've attended a number of lectures about the progress of the Chinese, and am now fairly convinced that they are a very long way from competing with us on the creative level. The requirement of uniformity and non-deflection from the mean, greatly limits creativity, which I think underlies exceptionalism.

I would not underestimate the creative ability of any people...

...the cultural connections that prevent moving into the future are not limited to the Chinese, they are alive and well in our very own country, and on this same topic, math education, in fact, all education...

Thanks for posting that, Ed. I found this part VERY enlightening:

One of the most vivid arithmetic failings displayed by Americans occurred in the early 1980s, when the A&W restaurant chain released a new hamburger to rival the McDonald’s Quarter Pounder. With a third-pound of beef, the A&W burger had more meat than the Quarter Pounder; in taste tests, customers preferred A&W’s burger. And it was less expensive. A lavish A&W television and radio marketing campaign cited these benefits. Yet instead of leaping at the great value, customers snubbed it.

Only when the company held customer focus groups did it become clear why. The Third Pounder presented the American public with a test in fractions. And we failed. Misunderstanding the value of one-third, customers believed they were being overcharged. Why, they asked the researchers, should they pay the same amount for a third of a pound of meat as they did for a quarter-pound of meat at McDonald’s. The “4” in “¼,” larger than the “3” in “⅓,” led them astray.

But our innumeracy isn’t inevitable. In the 1970s and the 1980s, cognitive scientists studied a population known as the unschooled, people with little or no formal education. Observing workers at a Baltimore dairy factory in the ‘80s, the psychologist Sylvia Scribner noted that even basic tasks required an extensive amount of math. For instance, many of the workers charged with loading quarts and gallons of milk into crates had no more than a sixth-grade education. But they were able to do math, in order to assemble their loads efficiently, that was “equivalent to shifting between different base systems of numbers.” Throughout these mental calculations, errors were “virtually nonexistent.” And yet when these workers were out sick and the dairy’s better-educated office workers filled in for them, productivity declined.

The unschooled may have been more capable of complex math than people who were specifically taught it, but in the context of school, they were stymied by math they already knew. Studies of children in Brazil, who helped support their families by roaming the streets selling roasted peanuts and coconuts, showed that the children routinely solved complex problems in their heads to calculate a bill or make change. When cognitive scientists presented the children with the very same problem, however, this time with pen and paper, they stumbled. A 12-year-old boy who accurately computed the price of four coconuts at 35 cruzeiros each was later given the problem on paper. Incorrectly using the multiplication method he was taught in school, he came up with the wrong answer. Similarly, when Scribner gave her dairy workers tests using the language of math class, their scores averaged around 64 percent. The cognitive-science research suggested a startling cause of Americans’ innumeracy: school.

Most American math classes follow the same pattern, a ritualistic series of steps so ingrained that one researcher termed it a cultural script. Some teachers call the pattern “I, We, You.” After checking homework, teachers announce the day’s topic, demonstrating a new procedure: “Today, I’m going to show you how to divide a three-digit number by a two-digit number” (I). Then they lead the class in trying out a sample problem: “Let’s try out the steps for 242 ÷ 16” (We). Finally they let students work through similar problems on their own, usually by silently making their way through a work sheet: “Keep your eyes on your own paper!” (You).
Continue reading the main story

By focusing only on procedures — “Draw a division house, put ‘242’ on the inside and ‘16’ on the outside, etc.” — and not on what the procedures mean, “I, We, You” turns school math into a sort of arbitrary process wholly divorced from the real world of numbers. Students learn not math but, in the words of one math educator, answer-getting. Instead of trying to convey, say, the essence of what it means to subtract fractions, teachers tell students to draw butterflies and multiply along the diagonal wings, add the antennas and finally reduce and simplify as needed. The answer-getting strategies may serve them well for a class period of practice problems, but after a week, they forget. And students often can’t figure out how to apply the strategy for a particular problem to new problems.

Thinking back to math class (I SUCKED at math in elementary school, wiped the floor with the remedial stuff in high school, and now hold a B.S. in maths), it was most definitely the context of the classroom that did it for me, but not in the sense of the lesson being "boring" because the kids who were good at math clearly LOVED math. I think math is the kind of thing where different people are "ready" to learn different things about math at different times. As a result, most kids have the experience at some point of being presented with math they are not "ready" to learn yet, so they don't get it, and they are told it is a problem with them, that they just "don't get it". This just makes their brains turn off when it's math time. At least that is how it was for me.

I think all the new approach really does is make kids realize they are not alone in their lack of understanding, and make them realize that in fact they all understand different aspects of the maths well. As a result their brains don't turn off during math class, and the results are much better.

Again, I don't really see how teacher unions necessarily rule out this approach, and thus how we need to completely re-vamp all of education just to teach math in this way. I also don't see how punishing the schools which have more failing students by taking away their funding (99% of the time they are failing due to lack of funding) as per NCLB helps either.

Thinking back to math class (I SUCKED at math in elementary school, wiped the floor with the remedial stuff in high school, and now hold a B.S. in maths), it was most definitely the context of the classroom that did it for me, but not in the sense of the lesson being "boring" because the kids who were good at math clearly LOVED math. I think math is the kind of thing where different people are "ready" to learn different things about math at different times. As a result, most kids have the experience at some point of being presented with math they are not "ready" to learn yet, so they don't get it, and they are told it is a problem with them, that they just "don't get it". This just makes their brains turn off when it's math time. At least that is how it was for me.

My experience as well jammer.

I was reluctant to join this convo until you posted that.

I sucked at math in school. As soon as I could stop taking those classes, I did.

In fact, as soon as I could stop taking any classes at all, I did. I have no post HS education, save a few writing and ethics classes at a community college.

I now find myself in a math intensive career.
I'm amazed by math! I'm learning.

I realize now that I didn't suck at math, I sucked at school.

"I realize now that I didn't suck at math, I sucked at school."

Great point. It wasn't my experience, but it was damn sure the experience of a couple of my friends and my sister in law. All very bright, just not good at K-12 school, all in careers now that require an extensive amount of math, which they learned on the job. For them, the school environment didn't trigger their inherent abilities but something about the workplace opportunities did. I see kids that don't do that well in school but who have an excellent work ethic otherwise. I don't worry about those kids (not that you can't develop the solid work ethic after your school years), but I do worry about the kid who is good in school but lacks that work ethic.

Or it is actually in fact the case that who gets rewarded in school is arbitrarily decided or decided by means other than an objective assessment of potential? A stint in the education program I went partially through may leave you with this impression...

...It also might be the case I was told I would not graduate said program because I told them I wouldn't do things like make every assignment I gave have scores which perfectly fit a bell curve...maybe...and brought in Bad Religion for my favorite music which contrasted literally every other gospel music selection of my classmates.

I think math is the kind of thing where different people are "ready" to learn different things about math at different times (jammer)

I can't seem to find a reference, but I once read (Piaget?) that the concepts of calculus were not appropriate for most children until after the age of fifteen. There are, of course, spectacular exceptions.

I first encountered and taught the "new math" to freshmen at Murray State in Kentucky during the mid 1960s. The first chapter in the college algebra text we used was more an introduction to mathematical thought, describing the axiomatic structure of the subject, providing examples, then requiring students to develop several common features of algebra. The top ten percent or so of my classes - those students who were blessed with a native mathematical and reasoning talent - seemed to enjoy the exercises, but the remainder of the class lost interest quickly and found this systematic, theoretical approach both incomprehensible and repellent. I like to think I did my best and that I was not an inferior teacher, but perhaps I didn't use appropriate strategies and tactics. My fellow instructors had similar experiences, however.

My own secondary education was deeply influenced by what was then called "the new math." It had a profound effect on many in our class, which now includes a bunch of Ph.D. scientists and mathematicians (rgold)

I seem to recall you went to a very elite high school, Rich. Am I remembering correctly?

I seem to recall you went to a very elite high school, Rich. Am I remembering correctly?

A well-regarded NYC private high school, yes.

The kids in my class where more than ready for something beyond a diet of unexamined recipes for how to do things without any real explanations for why anything did or did not work and how various aspects of the curriculum were related.

Whether this diet was only for an "elite," or whether it held out the promise of finding the hidden elite in less selective environments, or whether in fact it would have benefited all students is something I don't think we know, because as I suggested, the national teaching corps may not have been up to the task and of course even then there was resistance from parents who couldn't do their kid's homework. I personally was fortunate to have marvelous teachers, people who set me on my life's course, and parents who celebrated their inability to do my homework as a sign of the kind of progress they fervently hoped for their children.

As for your experience with the large spectrum of approaches labeled "new math" John, I think that there was at least one problem with some of those curricula, and it was that the people who designed the curriculum confused the mathematical concept of "proof" with the developmental notion of "understanding." They thought that if you developed the subject on a strictly logical basis, the way it might be done in a book on the foundations of mathematics, then you would have conveyed an understanding of the field.

Since those days, developmental psychologists have fleshed out a much more nuanced model for understanding, one in which strictly logical justification plays a far less central role---something mathematicians have known probably since Euclid started the whole axiomatic business.

With young children, as you mentioned, work started by Piaget (who, as it turns out, was not exactly right about a number of things) suggests that kids go through various stages of logical development (Piaget made these stages analogous to the acquisition of the axioms for a group). Children don't reach his version of full logical maturity until their teenage years, which would mean that approaches based on at least certain aspects of logical reasoning would be doomed to failure with younger learners, who Piaget showed are capable of accepting contradictory observations without any sense that something is amiss.

When I started out as an assistant professor at the U of So Colo in 1971 we used a similar text, to my annoyance. However, there was a humorous aspect to that experience: we had three retired army colonels on staff who taught beginning courses and one day early in the semester one of them stormed into my office and said "What the hell is this stuff!?"

Samuel Butler is reported to have said in debate, "Sir, I have found you an argument; but I am not obliged to find you an understanding."

It seems to me that mathematical proof has, in many cases, this quality. The logic, written so that it is verifiable, tells you that the conclusion is valid. Whether even carefully following the prover's logical steps confers an understanding of the result is an open question, one that some of the new math curricula begged.

Of course part of the difficulty has to do with understanding what is meant by understanding.

It is amazing that we have gotten this far without the following classic take on the subject by Tom Lehrer in 1959. Sadly, I couldn't find a video that actually showed the performance, but the voice is his.

I went to Claremont High School, while remarkable in some respects, a public high school in a college town, perhaps the most remarkable aspect when I atended was the "college scheduling" of classes... our only attendance requirement was to be on campus from the time of our first class to the end of our last... and we didn't have a day's worth of solid classes...

that left a lot of time to talk to each other, students, and teachers when you could find them... our days were not totally structured, and certainly not totally in class, so we had time to explore various interests, to discuss "stuff" with each other, and to pursue various academic interests.

But I would not generalize from my own education. First, my parents somehow put in my mind that my education was largely my own responsibility, and second that I was to respect my teachers, they were an important resource to my education.

All aspect of math were fun to me, and an exploration, an adventure, into some really strange place. But that didn't mean I found math easy, it was a difficulty I was willing to endure because of the amazing things that came from mastering it...

I can't attribute my interest to an individual teacher, or even my parents (my mother had a high school education and my father wasn't really any help, for some reason last time I recall asking for help on homework the 3rd grade).

The point is that you have to work at it to get it, it's not easy. I can't imagine that I'd have had any motivation to even try if my parents hadn't supported my teachers, and supported me. There was some vague belief that education was a key to a future career, but even that was oddly abstract and indefinite. I did want to be a physicist, and I did become one... but all that has happened in that journey was totally unanticipated.

Probably not a good example... so I don't hold myself up as one.

But that didn't mean I found math easy, it was a difficulty I was willing to endure because of the amazing things that came from mastering it...

It never flowed serenely into my head, either. That made it an attractive challenge all the way from elementary school to PhD. During my career at USC (asst prof 1971, prof 1980, retired 2000) I tried to keep that fascinating challenge functional by always having a modest research project to balance with teaching. I still have little projects in my dotage.

I was hired as the analyst and almost immediately given responsibility for the dreaded senior year advanced calculus two-semester course, then other courses: intro to complex variables and intro to topology, plus any analysis courses required in a master's program shared with the physical sciences. The first few years there were a small number of NSF-sponsored math ed majors required to take advanced calculus; they were excellent students and received A's. As time passed this funding expired and the quality of math ed (ME) majors declined to the point that for much of my tenure there were ongoing arguments for and against these majors being required to take this course, with one of my colleagues with a PhD in ME from Michigan pushing for the requirement, while another with an EdD trying to reduce difficulty levels for an undergraduate degree. In general ME majors were weaker, sometimes dramatically so, than regular math majors. There were a few, however, who were quite proficient and were very successful as HS math teachers. One became superintendent for the whole district. We also had some good physics majors taking these senior-level courses - one heads up a lab somewhere in CA.

Nobody who pursues math far enough finds it easy, any more than anyone who is serious about bouldering, sport climbing, trad climbing, alpine climbing, etc. finds them easy.

Math isn't easy, and I think that it can only be made "easy" by obscuring from the students' view all the things that are really there that make it hard. At which point, one could argue we aren't teaching math, we are just programming the audience, who, as it turns out, will need to be deprogrammed later on if they are to successfully pursue advanced studies.

The fact of the matter is that humans don't seem to be interested in things that are easy. If they were, the baskets in basketball would be five feet off the ground and three feet in diameter, and I could go on through every sport and game, changing things so that all of them were easy, in which case almost no one would be interested. The Owen-Spaulding route would be the only route on the Grand Teton, and there would be no routes on El Cap.

I suspect that one of the deep problems with elementary math instruction is that many parents and teachers think it is supposed to be easy when it isn't, probably by mistaking familiarity for simplicity.

When people learn to climb, they learn an expanding collection of strategies for dealing with difficulty. We don't seem to have managed very well in doing the analogous thing in math and science education.

John, I think one of the problems with upper-level math courses for Math Ed. majors is that the courses were, by and large, originally designed to prepare the students for graduate study leading to research careers in mathematics, and they haven't evolved to reflect the fact that the audience is not going on and will be teaching things far less complex.

Personally, I would be and have been against "diluting" the content, but I do think one ought to ask why secondary teachers should know these things and then make sure the course illuminates the importance and value of the content for that audience. That is a different argument than should they or shouldn't they take this course I think.

I had a reputation, in my household, of trying to lecture on the subject of problems rather than "just doing it" from both Debbie and our daughter.

Thank you for that comment, Ed. I had the same reputation with my two daughters. Now one is a math teacher married to a math teacher, and the other heading toward a doctorate in music (as she puts it, so Adele and I can have a purpose for the rest of our lives, i.e. supporting her).

And Rich, thank you for your insightful comment about difficulty. I guess it should not surprise us that climbers -- who tend to push things to or beyond the point where they challenge us -- would do the same with academics.

I guess it should not surprise us that climbers -- who tend to push things to or beyond the point where they challenge us -- would do the same with academics.

Yes, but my point is that the interest in and pursuit of difficulty seems to be a fundamental human trait, not at all restricted to climbers but observable in a vast range of human activities.

The flip side is that if we are searching for something hard to do, that search is conditioned by some inner self-confidence that we will, some of the time, succeed. And so we gravitate to this or that activity, not because it is easy, but because it is hard but we have some sense that we can manage it anyway. The mystery of human nature is that we get pleasure from succeeding at hard things.

If you buy this, then one of the jobs of education is to help to build that sense of possibility in the face of difficulty. We do this through subject matter of course, but how exactly does the subject matter, which is important in its own right, also become a vehicle for conveying that expanding range of strategies analogous to what climbers acquire?

These musings, which in one sense are the idle ramblings of an old man trying to make sense of what he's spent his life on, are still not far from the original topic. There is a lot more going on, and a lot more at stake, than the most efficient rote way to subtract one integer from another.

John, I think one of the problems with upper-level math courses for Math Ed. majors is that the courses were, by and large, originally designed to prepare the students for graduate study leading to research careers in mathematics, and they haven't evolved to reflect the fact that the audience is not going on and will be teaching things far less complex

To some extent I would agree, Rich. However, I have always felt that one should be knowledgable of material a minimum of one step above one's teaching level, and in this case I would argue that a HS math teacher who might be expected to teach an elementary calculus course should be aware of the foundations of real analysis and the theory behind elementary calculus (e.g., theory supporting the Riemann integral and the Fundamental Theorem) which is the direction I steered AC, rather than simply more "regular" calculus ( we had a junior-level course for that). Although AC is a traditional stepping stone for grad work, it also provides a foundation for secondary school instruction.

However, topology and complex variables are a bit of a stretch in this regard.

Yes, but my point is that the interest in and pursuit of difficulty seems to be a fundamental human trait, not at all restricted to climbers but observable in a vast range of human activities. . . . If you buy this, then one of the jobs of education is to help to build that sense of possibility in the face of difficulty.

Rich, you make an excellent point. I've known too many students who didn't make the connection between their ability to overcome initial difficulty in one area of their life, and doing the same thing in other areas of their lives. Your post made me realize that I focused a lot of my effort particularly one-on-one with individual students, helping them to make that same connection. One of my greatest joys is when former students report back on the difference that made.

I made my post about climbers because most I know already made that connection, and recognize their attraction to difficulty.

The question posed in the OP is interesting. In recent global rankings the US does poorly compared to other countries in terms of the math skills of high school students (something like 30th in the most recent PISA rankings I'm aware of). The question becomes an exercise in cultural comparison: what differences are there between the places in this list http://www.businessinsider.com/pisa-rankings-2013-12 with scores over 500 and the US, that leads to high school students who have significantly better math skills. Since the US already spends more money than most countries in education, just throwing more money at the problem doesn't seem to be the simple solution. One possible factor is aleady considered in PISA study: relatively speaking US students don't think math is very important. I imagine this is a general cultural phenomenon and (if so) won't be easily changed just by modifying the way math is taught in schools: New Math, etc., all by itself, won't fix the problem.

When it comes to the differences in the way math is taught in (at least some of) the countries which perform better, I thought the book "Count Down: The Race for Beautiful Solutions at the International Mathematical Olympiad" gave an interesting possible comparison. The author suggests that instead of teaching math as a series of formal step-by-step procedures that must be mastered in succesive increments (the US method of education) math can be taught in terms of a problem solving approach: directing students to attack an assortment of challenging problems which can be solved by a variety of methods. I don't know for sure if this characterization is really spot-on, at least in terms of getting to the root of an important cultural difference, but it does jive with the following article from NPR:

Math is certainly difficult, challenging and frustrating. I think the most important ingredient that is missing with respect to US (and Argentine) education is the push (the drive) to learn the subject that comes from the student. I'm not very convinced that a bureaucratic top-down curriculum change can provide the solution. But you never know!

I went and had a look at the common core standard for math, and I’m truly puzzled by the responses to them, or at least to the sample I looked at. I can’t imagine any informed scientist arguing against them. They seem to me to propose basic topics for understanding which are really beyond question.

So while you folks were thinking about why we suck, I've just returned from a a very rainy week at your very lovely Iceberg Lake. Interesting responses; I thought it would just be a cursory and predictable "Americans suck" comment from Werner, and that would be the end of this thread.

To return back to the article I originally read, (Ed, my original post was in response to the NYT article you cite), I'll tell you what I have thus far experienced with the Common Core as a teacher.

*The other countries, in particular in Asia and northern Europe, do better at math because their curriculums are not the traditional inch-deep-and-mile-wide of those used in the US. Because these other curriculums give teachers a chance to explore fewer topics more deeply, the students in these countries have a deeper and more profound understanding of mathematics. As a teacher, I am personally all over this idea. I crave having that time with my students so that we can explore the greater depth where the beauty of mathematics truly exists. Reality of the Common Core: The number of topics to be taught may have been consolidated on paper, but teachers are required to teach at at least as much content as before the Common Core. In Algebra 2, for example, a few minor subjects were removed from the traditional curriculum, and then a whole unit of statistics that includes topics one would encounter in a full stats course were added. Net result: more stuff for students to study at a superficial level. This is personally my single biggest pet peeve about the lie that is the Common Core.

**My school district is on a mission to educate parents about the Common Core. The district math dept. sends out frequent emails attempting to help parents learn how to support their kids with the new curriculum, and warning them that for a period of time they might see their kid's achievement in mathematics decrease. Part of the flaw of the Common Core is that at the elementary level, kids must show many ways to solve a problem. Reality of the common core: Because of the requirement that a problem or operation be solved (or carried out) in many different ways, which is certainly desirable in a theoretical sense, methods that are absolutely insane (inefficient and incomprehensible, even to mathematically confident parents) are employed that confuse everyone. I was told by my daughter's 4th grade math teacher, who was not confident in her own mathematics, that traditional methods for addition and division did not yield true understanding of those operations. The Common Core methods might strive for understanding, but the true result is that some of the methods used in elementary school confuse everybody, including the teacher! As a high school math teacher, I am often hindered in my attempts to teach concept, by students' inability to accurately calculate. Math operations are tools for engaging in math. Turning them into concepts that have to be looked at in multiple ways by all students in the class simply has the effect of more mistakes being made in the class. Ironically when that would happen, my daughter's teacher would simply take the class to a computer lab for a session of mind-numbing drill-and-kill.

A big part of implementing the Common Core in mathematics is the use of technology. I was on the committee for adopting new textbooks in my district a couple of years ago. I reviewed a lot of textbooks and supporting technology by different publishers of textbooks. We spent
$1.4 million on new textbooks for the district, a substantial portion of which was spent on technology support that came with textbooks. Most of the technology that I saw was really targeted to make teachers' lives easier (for example providing a pre-generated multiple choice test that the teacher does not have to grade). Reality of the Common Core: The new wave of textbooks written to support the Common Core look very similar to the old textbooks in many cases, and the technology that is provided to support greater understanding on the part of students, is as far as I can tell an attempt to sell teachers on the Common Core by reducing the time they spend on assessing student understanding.

My point of the original post was not to get into the discussion of pay for teachers, or tenure for teachers. My point is that the Common Core is not the answer to greater and deeper mathematical understanding on the part of students. The Common Core is a lie, conceived by politicians to be one thing, and then developed by the educational establishment to be something completely different.

Personally I think this Common Core stuff is a huge step in the wrong direction. I think counting/addition/subtraction/multiplication/division are elementary operations and every single human being is more than capable of entirely mastering them. IMO, the fact that some people fail to is a reflection on the philosophy of education I encountered in the program I went partway through, which is to always have the grades for every assignment fit a bell curve. Any teacher following this philosophy must by necessity tell people, for who knows what reason (personal bias perhaps?), that they are failures at elementary things like counting.

I can say that for myself, the first time I "got math" was the time the teacher treated me like I could and should succeed. It was downhill after that.

Because of the requirement that a problem or operation be solved (or carried out) in many different ways, which is certainly desirable in a theoretical sense, methods that are absolutely insane (inefficient and incomprehensible, even to mathematically confident parents) are employed that confuse everyone. I was told by my daughter's 4th grade math teacher, who was not confident in her own mathematics, that traditional methods for addition and division did not yield true understanding of those operations. The Common Core methods might strive for understanding, but the true result is that some of the methods used in elementary school confuse everybody, including the teacher! As a high school math teacher, I am often hindered in my attempts to teach concept, by students' inability to accurately calculate. Math operations are tools for engaging in math. Turning them into concepts that have to be looked at in multiple ways by all students in the class simply has the effect of more mistakes being made in the class. Ironically when that would happen, my daughter's teacher would simply take the class to a computer lab for a session of mind-numbing drill-and-kill.

You begin your response with a quote from me which you never actually argue against, so I assume this means you agree with it. You appear to agree that fundamental intellectual strategies such as exploring multiple approaches to a problem are desirable "in principle." So it does seem that, as usual in such things, the devil is in the implementation.

Other pieces I've read have argued forcefully that the implementation phases of at least some Common Core programs were carried out without appropriate collaboration with experienced and knowledgeable classroom teachers.

You also make what I think is a critical observation, which is that progress at "higher" levels requires facility with the routines used at "lower" levels, otherwise the learner's intellectual edifice never rises above ground level because they are busy reinventing the wheel every time they need to take a journey. At some point you have to be able to jump in the car and drive away without mentally recapitulating the entire theory of the internal combustion engine.

So on the one hand we do want our children to be facile with the basic operations of arithmetic and algebra, but on the other hand we don't want them to have learned them in a manner so mindless that they don't understand which operation to choose and what the scope and limits of the operations are, something that is for a very large contingent of students a sad reality. And make no mistake---the consequences of this sad reality is that arithmetic and algebra remain forever in a quarantined compartment of the mind, unavailable for any application that wasn't on the exam, which is to say almost everything.

We as a society and we as individual parents are fully justified in hoping for much more than this sad reality, which as I described it is no more than basic "clerical" competence in the realm of mathematics and its associated domains. We surely ought not to be satisfied with less, and that includes the status quo.

If we could ever get down to the essence of the matter, I would say that mathematics and science have to be taught as subjects that are accessible to human reason, because they are the product of human reason. If this sounds obvious, it has not, in general, worked out that way.

The danger of reactions to the Common Core is that they end up throwing out the baby with the bath, and push us back to the bad old days when the "mind-numbing drill-and-kill" referred to above was in fact the entire content of elementary mathematics schooling. In today's politicized atmosphere, there are plenty of voices advocating for just such a regression. They may even win, but it won't be any kind of victory for our children.

Much of what is said here assumes all students are capable of developing or enhancing skills in critical thinking. As a retired college teacher my impression was students seemed either to have these skills or not when they reach college age. Those of you posting here might address this issue regarding elementary/secondary-age students. Nature or nurture? Both probably but in varying degrees. Like mathematical or musical ability, is there a latent talent in critical thinking?

I've skimmed the thread and believe most of you are missing the real problem entirely.

Much of what is said here assumes all students are capable of developing or enhancing skills in critical thinking.

Yes - what kind of students are showing up to class these days, where did they come from, what are their examples at home?

My take is simple - more and more we are breeding a lazy, dull, consumption driven youth. The kids would rather Facebook and eat Twinkies than actually have to think. Our systems promote and reward a deep need for instant gratification and consumption. The strain of learning math isn't compatible with these systems.

Rgold, I'm not against a national curriculum or national standards. I guess if pressed, I do agree with you that it is the implementation of the Common Core that is problematic. Just out of curiosity, what was it in the Common Core that impressed you that wasn't previously written into other math curriculums? What part of the Common Core is your baby in the bathwater?

I am however, against a national curriculum that is supposed to accomplish reform by drastically changing the old curriculum, when in fact it is simply more of the same: too much to teach/learn in a meaningful way in the allotted time.

Common Core mathematics, as I understand it was the result of a commitment by politicians to produce a new national curriculum, that would allow teachers the time to actually get kids thinking about mathematics in a more meaningful way. Predictably, "educators" couldn't resist writing so much content into the curriculum, so then it became this thing that justifies all these "new" methods of teaching math, particularly at the elementary level. . . because after all, "traditional methods of teaching math simply do not work" (the premise of the original NYT article I cited).

But make no mistake, the Common Core is an inch deep, and a mile wide, just like the curriculum it is supposed to replace. The Common Core has become the current trendy flashpoint for math education reform, while at the same time it contains the inherent flaws of the old curriculum.

School teacher is the #1 profession in my family. My dad, step brother and his wife, my cousin, her husband, and her daughter are teachers. I might be forgetting someone.

So when we get together, I hear a lot of shop talk.

From what I gather, teachers mainly want two things ( except for my step-brother's wife, she wants an SEIU shirt in a 6X, because 5X is tight on her ).

They want to be able to eject poor and disruptive students, like I was ejected at the end of my academic career. Having to explain things two, three, four, five times cuts instruction time in half, by two-thirds, three-fourths, etc. for the students who are bright enough to grasp the material the first time around.

At some point, those who want an education need to be separated from those who don't. The sooner the better for everyone involved. Second or third grade isn't too early. I knew in the second grade my mission in life was to do whatever it took to get out of school, but it took the school until I was a junior in high school to finally concur.

And they want their administration to back them up on obvious issues. For example, if a student tells his mom his teacher called him a "poopy head", and mom complains to the principal, the proper thing for the principal to do is to laugh in the parent's face and send her on her way. NOT call the teacher on the carpet to address the issue as if it actually happened.

As far as math goes, my cousin teaches math at the same Junior High I attended. She is head of the math department because she's the only math teacher there who can do long division.

wbw, it is entirely possible that "the curriculum" per se isn't the problem. Since I only skimmed the common core, I don't think it wise for me to start elevating particular examples as models of right-headedness. My observation was that I didn't see anything I thought a scientist would disagree with.

The real question is, as I said before, are we going to try to teach mathematics as a subject accessible to human reason, or are we going to insist, quite possibly under the pressures of high-stake exams mandated by partially if not wholly clueless administrations, to make it into a set of mystical catechisms, uttered reflexively in response to predetermined stimuli? Of course, I'm being hyperbolic here for rhetorical effect, but at heart this is the issue.

John, I think some of the difficulties with "critical thinking" we see are actually the side-effect of an education system that has in some cases promoted mathematics as a field not accessible to critical thinking. After twelve or thirteen years of that for some kids, it may well be too late to convince them that the subject is very different from what they have come to believe. The school mentioned above that doesn't include word problems in the mathematics curriculum is a stunning example of how math education can jump the tracks and end up not being about anything recognizable as mathematics.

Part of the process of critical thinking in any scientific context and probably in almost all contexts involves multiple attempts, incorrect formulations, and periods of clarity and confusion. Much of our education system, at all levels, hides these processes in favor of communicating only the polished end results. Most students know that it takes years of practice to learn athletic skills, and along the way there will be many failures, many plateau periods were no progress seems to be made, and times of regression when things actually seem to get worse. But when it comes to mathematics, we show them the perfect-10 uneven bar routine but none of what went into producing it, and then we're surprised when a huge majority of students conclude that they can't do it.

One of the things we seem to be terrified of is confusion. Heaven forfend the children---or the teacher, or the parent---should ever be confused. But confusion is an integral component of intellectual progress. You have to learn how to deal with it and ultimately resolve it in order to be any kind of scientist, but you can't learn that if everyone is trying at all moments, not only to banish confusion from the process, but to suggest that the existence of confusion is some kind of failure. Now you don't go about confusing kids on purpose, but somehow there needs to be opportunities in which confusion occurs, is worked through, and resolved. From that comes pleasure and the self-confidence to confront the next difficulty.

One of the things we hear with great frequency is that lessons based on the common core are confusing. I'm not saying all confusion is good, but I am saying that the existence of some confusion is not, by itself, necessarily a fatal flaw. The real question is what happens next. And I might add that when adults go ballistic because they are confused by Johnny's homework, you can be sure that what Johnny really learns is that uncertainty frightens the daylights out of the grown-ups and is certainly to be avoided at all costs. And there goes Johnny's critical-thinking opportunities down the toilet.

I am not knowledgeable about primary and secondary education. I don't know if there is any hope for motion in the direction I mentioned. The NCTM has been trying to promulgate many of these ideas for years, apparently with little more than pockets of success. The wonderful teachers I had in high school were able to do the kinds of things I mentioned; there is nothing new or even remotely radical about my mentioning such things more than sixty years later.

The question of technology has been raised. I use it every day, but I don't think it is likely to have any impact on developing the kinds of intellectual skills I mentioned, so to some extent it is a very expensive distraction. I also think it isn't being used properly. We should be using it not to replace ordinary arithmetic with small numbers, but rather to do calculations that involve very big and very small numbers, calculations that are not practical for hand arithmetic. As I asked earlier in the thread, why aren't kids computing how many blades of grass are on the school lawn?

Of course, when technology enters the picture, the users need more knowledge, not less. The black box is perfectly happy to give you garbage out for the garbage you put in.

John, I think some of the difficulties with "critical thinking" we see are actually the side-effect of an education system that has in some cases promoted mathematics as a field not accessible to critical thinking. After twelve or thirteen years of that for some kids, it may well be too late to convince them that the subject is very different from what they have come to believe (rgold)

The problem seems, in general, to cross academic lines. I recall a study done several years ago that implied there was very little if any improvement in critical thinking skills in all areas between the freshman college year and graduation. I wonder if studies have been done on freshman to senior levels in high schools. It might be there is a window of opportunity at a very early age for a child to develop this skill . . . or not, so that home environment may be crucial. I haven't a clue. Maybe someone else on this thread does.

That being said, I think there can be critical thinking in one area and not another. I've known people who have excellent skills in the humanities, but are flops in math and science. The opposite seems true as well on occasion, for there are technical types who have limited language skills. (a phenomenon not seen on this site, with Ed and rgold and others superb at both ends of the spectrum)

so all you math teachers out there, just how does a computer do division? what's the algorithm?

and how do you think they came up with that?

how many times do you do long division every day?
how many times do you do division by estimating?
and then refining the estimate?

what are all the ways you do division?

and why are we, as a culture, fixated on doing long division precisely and accurately?
what's the point?

as has been stated many times in this thread, we confuse the flawless execution of a specific algorithm for a computational exercise as mathematics.

More interestingly, can you describe what the algorithm is doing? can you implement it in an arbitrary base (binary long division? can you do it? hexidecimal? vigesimal? sexagesimal?... ) can you divide one polynomial by another using long division? why?

Can you do it only by multiplying?

what are all the different ways?

just what is division, anyway? and why is it important that we understand it at all?

I think Ed asks a lot of important and appropriate questions.

For someone who is going to go on in mathematics at the undergraduate level or above, there are a host of concepts related to "long division," or more accurately to the nature and possibility of carrying out long division, they will have to master.

But even the algorithm itself is important (well, many people are not going to think what I say next is at all important). There are various types of real numbers, and in particular every real number is either rational or irrational. The human recognition of the existence of irrational numbers is typically ascribed to the Pythagoreans, and so is something like 2300 years old at least. The ancient Greek philosophers were far more sophisticated about the concept of number and its implications than most of our current population, and so the labored to produce a logical system capable of embracing what we now call irrational numbers.

Nowadays, one way of describing the distinction between rational and irrational numbers is in terms of their decimal expansions. (Yeah...infinite decimals...what exactly do they mean and in what sense can they possibly denote "numbers?") The rational numbers are precisely the ones with terminating or infinite repeating decimal expansions, whereas the irrational numbers are precisely the ones with infinite non-repeating decimal expansions. How do we know this? The only argument I know for a rational number having a terminating or infinite repeating decimal expansion comes from an analysis of the algorithm for long division.

I mention this because it is not at all atypical; various topics in elementary mathematics are gateways to more advanced results.

Chaz, I think devoting a large segment of the eigth-grade curriculum to Roman numerals is a waste of time. Maybe a little bit so that the person encountering them has an idea how to decode them, since they do still get used. One reason to at least mention Roman numerals is to illustrate what a miracle place-value notation is. You had to be a genius to multiply two three-digit numbers if your only framework for representing them is Roman numerals, and in principle no finite table of products would suffice for all possible calculations. With decimal notation, you learn 55 multiplication facts and you can handle anything! Division with Roman numerals---fuhgettaboutit.

when I encountered them (and it is strange that they were taught in math) I did try to figure out how to divide with them, it wasn't a part of the course, but it was an interesting question in my whatever-the-grade-was mind...

I didn't come up with a solution (not surprisingly) but I had a lot of fun playing around with it... and multiplication too...

One reason to at least mention Roman numerals is to illustrate what a miracle place-value notation is.

the power of zero!

Chaz, given your story, why do you feel compelled to participate in this thread? Common Core cannot be even a slight concern of yours.

Boy, who would think long division would come under such fire. As Rich mentioned the algorithm is fundamental in mathematics.

How much of critical thinking ability is genetic? Is there a crucial window of opportunity? How much of mathematical ability is genetic? (the great breakthroughs are most frequently seen in practitioners under the age of thirty-five). Is it like musical ability - prodigies like Mozart writing serious music at age 11? Idiot savants can multiply enormously large numbers or do similar things without any real understanding of the theory. How does all of this fit together in a curriculum designed to bring all students up to an acceptable level of proficiency? Is this even possible in the age of Facebook, Twitter, Tweets, instant electronic connections. How many of the posters here have had children go through the educational system? Have watched as thinking skills materialize and mature? Questions, questions, questions.

How many of the posters here have had children go through the educational system? Have watched as thinking skills materialize and mature?

My daughter was educated in the Arlington (NY) public school system and as far as I'm concerned got a great education. Was every teacher fantastic? Of course not, but the overall effect was excellent, all without the Common Core by the way, but before the general rejoicing starts the New York State Regents did have standards that were tested by exams and those results were part of the student's final grade in the course, and the AP courses she took were also governed by standards-based exams.

With the very few crappy teachers she had, we were sympathetic, up to a point, but also took the point of view, "how will you make the best of this?" We didn't complain to the school, and we didn't bad-mouth the teacher in front of our daughter. Life isn't perfect and you have to learn how to make lemonade when what you've got is lemons.

She then went to Boston University and double-majored in Philosophy and Musicology, but her goal from a very young age was, I think, to become a professional musician and she never took her eye off that prize.

I never thought much about "thinking skills" at the time and looking back on it, I don't think I have any conveyable sense of "development." There are perhaps two things we did that might be considered unusual. The first is that we didn't turn on the TV until the end of middle school. She thought the TV was a device for playing videotapes until, finally, we turned it on so she could watch the olympics. Lots of luck trying to keep technology at bay nowadays, but we thought it was important at the time and I haven't learned anything since to change my mind. I am beyond grateful that cell phones were not part of her growing up, and that the internet became a presence only later in high school.

The second thing is that we enrolled her in a gymnastics class quite early. Not for the reasons you might think though. It had nothing to do with athletics; I have a pretty negative view of organized athletics programs for kids. We did it for reasons that are echoed in my comments about mathematics above. It seemed like the best experience a young child could have that would show them that they could, through hard work, lots and lots of practice, yes frustration and confusion too, manage, over time, to do amazing things they could not initially have conceived of as possible. And have fun along the way. I couldn't think of anything else that would give them the experience of looking back and saying, "a year ago this was unimaginable."

There was a third thing but I can't take credit for it. At around seven, maybe a little earlier, she begged for a piano, and that began what has become her career. That was all her. But the fact that she, of her own accord, got up at 6AM every morning before school to practice, right through high school---I'd like to think that came from what she had learned from gymnastics, the confidence that you could master things with work that didn't seem even remotely possible, the ability to persevere, and the fortitude to withstand the frustrations that are an inevitable component of learning anything hard.

I have to say that I also worried about gymnastics. The awful body-image pressures, the injury potential. And so I was more than relieved when, after maybe six years of it, she said she was done. (She then moved on to basketball, to the horror of her piano teacher, who rightfully worried about finger injuries.)

IMO, Majority of Americans are programmed not to remember anything in general and not just math. There may be political reasons behind this however, not knowing math makes it easier to f*#k up in savings and to do daily calculations on where you are financially.In short sentence, you are always broke therefore you must always work till the day you die.

Go to any street in Asia, ME , Africa or even Europe and ask couple of kids that you want to add this plus that and they know the answers but here in America, even behind the cash register, you buy a burger for $5.01 and you pass a $10 bill ,you got to wait a minute for machine to figure out the $4.99 change before cashier boy can give your change back.

True story from SLC, Utah that took place (I think) back in the late eighties: on the way home we went through a drive-in because a friend, who was with us in the car, wanted to grab something to eat. As I recall, he ordered a hamburger (around $2) some french fries (about $1) and a coke (also about $1). When we got to the window the kid working there said: "That'll be $8" (I don't remember the exact amount here, but it was something like double what the order should cost). So my friend says: "No way, can't cost that much". Then the kid looks back down at the cash register and, shaking his head, says: "Sorry sir, but the order costs $8". So then my friend says: "Look, a hamburger costs about $2" and the kid nods in agreement, "a french fries costs about $1" which elicits another nod of agreement "and a coke is about $1. So the order should be about $4". Now the kid was getting kind of nervous and he starts looking back and forth from the cash register to my friend and then he says: "I'll have to go get the manager". After a few moments the kid comes back with the manager and the manager asks: "What's the problem sir?" And my friend explains how 2+1+1 should equal 4, not 8. So the manager looks back down at the cash register, then starts talking quietly with the kid and after a few moments answers back: "but there's also tax on the order". This response evokes audible guffaws from the driver (me) and causes my friend to totally lose it. Keep in mind, sales tax in Utah at this time was about 10%. "There's NOT $4 of tax on a $4 order" screams my friend, his arms waving wildly in the air.

Anyways, I can't recall exactly how much more discussion it took to finally get things straightened out, but somehow the anecdote seemed to be relevant to this thread.

My daughter grew up going to public schools, New York (Westchester Co.), Massachusetts (Amherst) and eventually college at UCDavis...

yes, it was an amazing thing to know someone from their very beginning to becoming an adult, a real privilege, and to have helped out along the way an honor.

While she is troubled, as the rest of us are, with coming to grips with what we aspire to be and what we are, she's a great person.

What does this have to do with "Why Americans Stink at Math"?

To infer that mastery at math is something only people with access to elite institutions of learning could achieve seems a very tired theme. While not everyone will become a mathematician, everyone benefits from learning mathematics. And it is learning that is becoming more and more important.

To which I would add, we don't know who among our young children might or will become a mathematician, scientist, etc. Assuming (correctly) that only a minority will do this but that we don't know who that minority is, should we create an educational system that prepares no one for advanced careers? The genius of America's attempt at a non-elitist education system is that we assume, at least in principle, that everyone might become a rocket scientist (etc etc). We don't say, look, here are the little rocket scientists, we'll give you an enriched education, and here are the little burger flippers, you'll never have any use for the rocket-scientist stuff so we won't tax you with any of it---we don't do this because we know that neither group is going to end up where we predicted.

The price we pay for giving everyone a chance is that some people will look back and say they've gone through life and never needed any of it. With apologies to those who were afflicted, I think this is preferable to any of the alternatives.

The genius of America's attempt at a non-elitist education system is that we assume, at least in principle, that everyone might become a rocket scientist (etc etc).

America does NOT have a non-elitist education system though. Otherwise funding would be equalized for all students across public schools, which is in no way the case. Besides rhetoric and idealism, what leads you to believe we actually strive for a non-elitist education system? Seriously.

This is the biggest disconnect: so many people don't make the correct assessment of the situation to begin with, so no progress can be made.

Again, if we want our mean or median math scores to improve, which is what we mean when we say we want Americans to be better at math, the best strategy, and the one likely followed by all the countries kicking our ass at math, is to find a way to turn the low scores into high (or higher) scores.

How many of the posters here have had children go through the educational system? Have watched as thinking skills materialize and mature? Questions, questions, questions.

My daughter (who is a dual citizen) is going through the Argentine public school system, which is at least as bad as the American version. I don't believe this is necessarily a bad experience for her. This year (in March) she started 7th grade and has her first real math teacher, a trained accountant, who is not half bad (math, in the grade school she went to, was basically non-existent). There are about 40 kids in her grade and the level of math among this group sets the bar pretty low. She is certainly one of the best (if not the best) in her grade in math and seems to like the subject, which already makes her a bit of an outsider, culturally speaking. I believe it can be a good experience for her to develop some emotional strength in establishing her independence from the group like this.

When I read the article from the NY Times in the OP I tried asking her that simple question that so many Americans apparently got wrong: which is bigger 1/4 or 1/3? I was a bit suprised she didn't immediately know the answer (she hasn't worked much with fractions in school). After she hesitated for a few moments, I said: "Try thinking about it this way: imagine the table there divided into four equal pieces and then into three equal pieces". She immediately answered that 1/3 was bigger than 1/4.

Yesterday she was doing some kind of interesting exercises for class with prime numbers. One of the questions asked if the sum of two prime numbers was ever prime. She was patient with Dad when he said we should really think about that one for awhile. We figured out together that if neither of the primes was 2 then the sum was necessarily even, so it couldn't be prime. Then we looked at some examples of "twin" primes. We saw that 3, 3+2=5 and 3+4=7 were all prime and I asked her if she thought this sort of thing could happen again, for bigger primes than 3. We checked it with some examples: 11 and 13 are prime, but 15 is divisble by 3. 17 and 19 are prime but 21 is divisible by three. Then we figured out together the reason that if p is a prime bigger than 3 and p+2 is prime than p+4 is divisible by 3. We did all this without writing anything down. Finally I asked her if she thought the amount of "twin" primes was infinite. She said she didn't know. I told her that this was a famous and extremely difficult problem that best mathematicians hadn't been able to solve for hundreds of years, but in the last few years mathematicians were getting very close. I told her the story about Terence Tao and we looked at some pictures of the boy genius on internet and I also told her about some of the very recent and extraordinary work that "almost" solves the twin prime conjecture. She seemed pretty interested and in terms of the "critical" thinking aspect was able to construct the arguments we developed with very little help from me.

I think it woukld be great if her school and culture at large offered much more opportunity where she could engage in activities like the one I just described, but I'm afraid that just aínt gonna happen.

PS rgold: We have "Netflix" in the house and I occasionally buy movies from itunes (Amazon is unavailable in Argentina) but other than that, we don't have TV in the house and Ceci doesn't seem to miss it. I also feel she has a pretty good "nuclear" peer group with her closest friends being more studious (i.e. valuing "education" more) than the norm, and therefore also finding themselves a bit outside the mainstream.

I believe that at some point earlier in the thread it was questioned if undergraduate mathematics education majors to need to math beyond what they're going to be teaching. I think that this type of thinking is an example of why we suck at math. The fundmental lack of people getting why this is important (especially for teachers) and troubling.

Ed, you left out another long division twist : long dividing two infinite series to get another

Yesterday she was doing some kind of interesting exercises for class with prime numbers. One of the questions asked if the sum of two prime numbers was ever prime. She was patient with Dad when he said we should really think about that one for awhile. We figured out together that if neither of the primes was 2 then the sum was necessarily even, so it couldn't be prime. Then we looked at some examples of "twin" primes. We saw that 3, 3+2=5 and 3+4=7 were all prime and I asked her if she thought this sort of thing could happen again, for bigger primes than 3. We checked it with some examples: 11 and 13 are prime, but 15 is divisble by 3. 17 and 19 are prime but 21 is divisible by three. Then we figured out together the reason that if p is a prime bigger than 3 and p+2 is prime than p+4 is divisible by 3. We did all this without writing anything down. Finally I asked her if she thought the amount of "twin" primes was infinite. She said she didn't know. I told her that this was a famous and extremely difficult problem that best mathematicians hadn't been able to solve for hundreds of years, but in the last few years mathematicians were getting very close. I told her the story about Terence Tao and we looked at some pictures of the boy genius on internet and I also told her about some of the very recent and extraordinary work that "almost" solves the twin prime conjecture. She seemed pretty interested and in terms of the "critical" thinking aspect was able to construct the arguments we developed with very little help from me.

Funding can still be equalized and some kids (whos parents work extra hard to have the extra time to raise their kids) will still have an advantage. As a result, I am not sure why so many parents insist on public school doing everything it can to ensure certain students are given an advantage.

To infer that mastery at math is something only people with access to elite institutions of learning could achieve seems a very tired theme. While not everyone will become a mathematician, everyone benefits from learning mathematics. And it is learning that is becoming more and more important.

Why not convey to our children the benefit?

When will we ever use this? Ah, the question that often comes from the student hoping against hope, that this question that distracts the teacher may just get them out of the homework assignment the teacher planned.

Even though I am portraying this from a jaded point of view (it's meant to be humor, in case that's not obvious), and the question can throw some math teachers back on their heels, here is my response. Note: I rarely tie my response to how a specific math concept will lead to wealth and happiness.

Studying math is taking a journey. What you learn today may very possibly never be used in your daily life, but if one engages in the journey, I guarantee it will be beneficial. Engaging in mathematical thinking teaches one to organize one's thoughts, think critically, systematically and to reject unreasonable solutions to problems. Engaging in math is a workout for the brain. One's ability to grasp complex ideas is greatly increased as one gets further into the journey. It's not about what we're learning today in math class. It's about what one gains from a journey that takes many years, and at times may be quite frustrating.

I also tell my students that if they have never experienced the joy of solving a difficult problem in math class, which I believe is accessible to all students, I hope they have that experience in my class.

I took those hard math classes in college that Professor Gill refers to. Did I understand everything in my real analysis class? Definitely not. But having that experience of thinking, beating my head against the wall in frustration, and persevering so that at least I had some level of understanding absolutely helps me to be a better math teacher for high school students. (And I also found the process to be very rewarding.)

Getting back to the Common Core, if it is true that many (if not most) math concepts are simply part of an important journey, and not in-and-of themselves critical to one's education, why can professional "educators" not resist the urge to pack so much into a curriculum, that a classroom teacher has to rely on more superficial methods to teach math??

In Colorado, there was a law passed in 2010 that basically takes away the "tenure" for public school teachers who cannot demonstrate measurable growth for their students. Thus far, the only practical way to measure that growth is through standardized tests.Everyone is so freaked out that their students won't show that growth if they don't tick all of the huge number of items on the curriculum, that many just don't find the time to go deeper into concepts where real thinking takes place. Again, the promise of the Common Core has completely been thrown out by the folks that wrote it, and in my mind that is a huge missed opportunity.

wbw, on the (unlikely) off-chance that you don't know about it, the MAA has many resources on "when you'll use this," see http://www.maa.org/careers

Some interesting tidbits: CareerCast ranked "mathematician" the best job in the country in 2014. PayScale found the top 15 highest-earning college degrees have mathematics as a common element.

As someone who embraced mathematics for its own sake and not for what it might or might not do for my earning potential (just like climbing!), I am always a little hesitant to fall back on vocational justifications. After all, we study art, music, poetry, history, and a host of other subjects without inquiring about their immediate application to future earnings, so why does mathematics get singled out for this type of question?

But there is an answer. Mathematics is a gateway subject. A gateway to a host of professions will beyond the top 15 highest-earning college degree careers. Professions most of the kids and some of their teachers may not even have heard about. Close that gate and you've shut off a vast array of future opportunities. And that array is growing, as more an more fields become "mathematicized." A somewhat recent example is biology.

rgold, I really would be interested in your answer to this question: Besides rhetoric and idealism, what leads you to believe we actually strive for a non-elitist education system?

I posit that the fact that we do not strive for a non-elitist education system in any real or meaningful way is the very reason we "struggle" compared to other countries academically.

We still turn out some of the best high-end students, and we still lead, or are at least at the front of the pack, in terms of new technology and innovation.

As a result, I think the debate concerning "raising test scores" in this country is fundamentally about giving every student an equal chance, and about giving a real, honest, actual shot at creating a non-elitist education system. I feel the best place to start is equalizing public education funding, and I am absolutely baffled why no education reforms are of this nature. It makes the whole system a transparent joke IMO.

What would it take to prevent parents from taking their kids to Mexico for a month or more at Christmas?

I've heard reports sometimes 1/3 or more of the class is absent for the first few weeks of the calendar year, because they're visiting Los Abuelos down in Mexico.

Parents used to schedule vacations around the school calendar. Most parents still do, but a growing percentage just don't care.

How much extra funding would it take to prevent parents from taking their kids to Mexico for a month or more at Christmas?

I've heard reports sometimes 1/3 or more of the class is absent for the first few weeks of the calendar year, because they're visiting Los Abuelos down in Mexico.

Lamest, most closet-racist like excuse I have heard yet. Did you know that the people who don't have too many kids happen to have a near perfect correlation with the people who received a good education? Neat how that works...

Chaz, I have more attendance issues with the kid that is in Europe or Costa Rica for an extended vacation when school is in session, than the kids that are (as you say) visiting Los Abuelos.

I posit that the fact that we do not strive for a non-elitist education system in any real or meaningful way is the very reason we "struggle" compared to other countries academically.

I think if you look at those international math tests that we allegedly don't do well on as a country, you'd find that the countries that do the best have a much more exclusive sample size that they draw on to take the tests.

In other countries, students get weeded out from an academic track in their education and get re-directed to trade schools. (Personally, I don't think this is necessarily a bad thing, but it certainly creates a more exclusive education system.)

In Colorado, poor (generally rural) school districts get a bigger share of the state funding pie (relatively speaking) than wealthier districts.

As a public school teacher, I truly believe that all students can learn, and it is my goal that *every* student in my class do so. Even though this is a goal to aspire to, I spend a lot of time in pursuit of this goal. Most of my colleagues that I have known over the years feel the same way.

Short of a socialist system (some might argue that the public education system in the US is the closest thing we have to socialism in our country), I don't know too much more we can do as a society to make education a less elitist system than what we have. Of course, it is the most motivated students (and those that come from families that support their education in some meaningful way) that rise to the top and become the elites.

Jammer, I would like to know more about why you say we don't strive for a non-elitist education system. What more might we do??

Colorado is the exception to the rule then. What about inner city districts compared to rural and suburban districts? Remember, we are talking total available funding for each student.

In most parts of the U.S., the property values which surround the school determine the available funding, and the higher the property values the more money the school gets. The fact that this is the dominant funding model is the very reason I say we do not strive for a non-elite education system in this country.

Jammer, as I understand it this is the result of a law that was passed in CO in recent history. I don't think rural districts have an advantage over poor, inner-city districts, but I can't honestly say I know the formula that determines how that state money is distributed.

In Boulder, we tend to get a lot of ballot measures for increased funding from Boulder taxpayers because with this law, we have budget shortfalls most years. These ballot measures are on top of high property taxes that I pay in Boulder.

wbw, it's good to know that CO, with it's disproportionately high number of educated people per capita, has it's ducks in a row. States like CO will be the ones which bring those math scores up :).

Jammer, I would like to know more about why you say we don't strive for a non-elitist education system. What more might we do??

Jammer's Jamm'in, but I'll add some too.

"What more might we do?" How about putting the monies in a general fund, like they do with the Parks?

Los Angeles Co. is buying iPads for every student at the tune of 6 mil bucks. Meanwhile in San Bernadino Co. here at Joshua tree elem. my daughters class has 9 old computers which they have to share with other classes.
And in Placer Co., Rocklin High got a new football stadium at around 3mil, while Placer High has dead grass and beat up old wood bleachers for the home team and the visitors gotta stand.

You may call it fairness through taxation. I call it Elite ism.

rgold, I really would be interested in your answer to this question: Besides rhetoric and idealism, what leads you to believe we actually strive for a non-elitist education system?

Rhetoric and idealism, but in the opposite order. That and the fact that we do not in general determine career trajectories from exams given early in the educational process.

I must be using a different use of the word "elitist" then. What I mean by our education system being "elitist" is that it actively promotes and maintains generational legacy, and hence favors the children of the "elite", and so is "elitist".

Your example of an objective test to determine career paths as opposed to the current system of having ones parents "legacy" largely determine which doors are open is then by my definition less "elitist".

Your example of an objective test to determine career paths as opposed to the current system of having ones parents "legacy" largely determine which doors are open is then by my definition less "elitist".

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There seems to be little question that parents' education and social status confers a significant educational advantage on their kids. Some but not all of this might be cured by more equitable funding, but there is always going to be an advantage there that will never be completely leveled.

The existence of a career path test won't help at all because the less privileged kids will not do as well for many of the reasons they aren't currently doing as well and so be disproportionately shuttled out of the academic track for good.

We are far better off hoping that some of them will beat the odds, and in fact this happens with regularity, although maybe not on a scale to make much of a dent in national statistics. My institution, for example, is justifiably proud of its record with first-generation college students. Every year we graduate people who weren't supposed to make it this far, and in Mathematics I know several who now have PhD's. In the alternate exam-based career tracking, it is unlikely that any of them would have been able to do this, because their academic "coming of age" would have been too late for those filters.

As for "why Americans stink at math," I think the issues go far beyond equitable funding, and it is not hard to imagine equitable funding making little difference in the absence of other changes.

My anecdotes had a very specific limited purpose. They are pedestrian walking anecdotes. Whatever small vestigial wings they possess are not sufficient for flight. As such they may or may not have anything to do with whatever you think constitutes the "truth."

When I speak of elite education, I mean prestigious institutions like Harvard or Stanford or Williams as well as the larger universe of second-tier selective schools, but I also mean everything that leads up to and away from them—the private and affluent public high schools; the ever-growing industry of tutors and consultants and test-prep courses; the admissions process itself, squatting like a dragon at the entrance to adulthood; the brand-name graduate schools and employment opportunities that come after the B.A.; and the parents and communities, largely upper-middle class, who push their children into the maw of this machine. In short, our entire system of elite education.

As a financial-aid kid whose life-prospects were significantly bolstered by attending an elite school, this subject is very personal for me, too. I come from a family of construction workers and laundry-owners in Brooklyn, the descendants of Italian and Chinese immigrants, respectively. My father is a laborer and my mother a human resources worker; they’ve both changed jobs across the years, owing to the recession and family circumstances. We don’t occupy an enviable financial situation by any means, and I’d hate to think our unsteady progress from working- to middle-class somehow makes me, as Deresiewicz puts it, “an entitled little sh#t.” He may have sleepwalked into college, but it's wrong to assume we all did.

I don't have any statistics available, but I would be willing to bet cold, hard cash (say odds 2 to 1) that for the vast majority, the US education (in math) is significantly worse than average (worldwide), but that for the top, say, 5% in terms of socio-economic class, US education is significantly better than the average (worldwide). I don't know if this means that US education is "elitist" but it does mean that it is is hugely favored towards the people who come from the highest socio-economic class. Of course in any set of data it will be difficult to tell what effects come directly from the educational system and what effects come from other cultural advantages that are related to membership in a higher socio-economic class.

So how's the trad climbing in Minneapolis treating you these days, ms55401? And while you're at it, got any interesting anecdotes of your own to add to the ongoing list?

Direct and indirect effects of socioeconomic status (SES) and previous mathematics achievement on high school advanced mathematics course taking were explored. Structural equation modeling was carried out on data from the National Educational Longitudinal Study: 1988 database. The two variables were placed in a model together with the mediating variables of parental involvement, educational aspirations of peers, student’s educational aspirations, and mathematics self-concept. A nonsignificant direct effect of SES on course taking suggests the lack of an ‘automatic’ privilege of high-SES students in terms of course placements. The significant indirect effect of previous mathematics achievement tells that it needs to be translated into high educational aspirations and a strong mathematics self-concept to eventually lead to advanced course taking.

As for "why Americans stink at math," I think the issues go far beyond equitable funding, and it is not hard to imagine equitable funding making little difference in the absence of other changes.

So the top schools in America turn out sub-standard students compared to other countries? Like Yanqui, I don't have any data, but I HIGHLY doubt this is true. Something tells me our top performers have about as much knowledge crammed into their heads as you could cram into a high school kids head.

edit: Ed, I call out your study. For one, all it does is apparently show there is no correlation between SES and advanced math course taking.

Also, at the bottom of page 29 they say they recoded the socioeconomic class data by parental occupation, meaning that the parents occupation was used to either lower or enhance the effect that would otherwise be credited solely to socioeconomic class. It's hard to say that there was data manipulation there, but it most certainly introduces the possibility. Without looking at the model they used, and comparing to an otherwise identical model where the socioeconomic class was not recoded, it is impossible to determine this however. They did not discuss having done this as far as I could see.

I'm also a bit baffled why they don't just give the raw correlation coefficient between SES and advanced math course taking. If there was no correlation, this could be clearly demonstrated with much simpler statistics and a much simpler argument than the one presented. The fact that they did not choose that path is quite telling IMO.

Abstract
Previous studies have shown that both student and school socioeconomic status (SES) are strongly associated with student outcomes, but less is known about how these relationships may vary for different students, schools and nations. In this study we use a large international dataset to examine how student SES, school SES and self-efficacy are associated with mathematics performance among 15-year-old students in Australia. We found that increases in school SES are consistently associated with substantial increases in achievement in mathematics and this phenomenon holds for all groups, regardless of their individual SES. Furthermore, our findings show that the association of school SES with maths achievement persists even when subject-specific self-efficacy is taken into account. However, our findings also suggest modest differences among student groups disaggregated by these factors. In particular, the association between maths achievement and school SES appears moderately stronger for students with higher levels of self-efficacy compared with their peers with lower self-efficacy. Furthermore, among students with similar levels of self-efficacy, the association between maths achievement and school SES tends to be stronger for lower SES students than for their more privileged peers. From these findings, we highlight the importance of the Australian case for comparable systems of education, and provide a discussion of policy implications and strategies for mitigating the influence of school socioeconomic composition on academic achievement more generally.

Yeah, the first study Ed posted had something interesting to say, but it is not a study of "the American educational system" or how elitist it is, since, for example, data from private schools is not even considered. What the study tries to do is look at certain predictors for why a kid in public school takes advanced math classes. The fact that the study finds, for example, that parental values (expectations is the word used) are more important than socio-economic class when it comes to predicting participation in advanced math classes in public schools does not suprise me so much. Of course, as the study says, parental expectations are indirectly related to socio-economic class, so some care must be taken to separate these two. The study also found that previous success in mathematics courses was another important predictor for taking more advanced classes, again no huge surprise. Strangely enough, after the study reports that parental values are an important predictor, it then suggests that schools can somehow fill in the gap, for disadvantaged students. I don't see much in the study to suggest that this might be true.

I don't have any statistics available, but I would be willing to bet cold, hard cash (say odds 2 to 1) that for the vast majority, the US education (in math) is significantly worse than average (worldwide), but that for the top, say, 5% in terms of socio-economic class, US education is significantly better than the average (worldwide). I don't know if this means that US education is "elitist" but it does mean that it is is hugely favored towards the people who come from the highest socio-economic class.

While I suspect this is true, I think a similar statement could be made about a lot of other things, such as the US health care system. I, like Yanqui don't know if this makes our education system elitist. And even if it is, there are various degrees of "elitist". I'll give an example (or anecdote if you prefer):

I suppose one could say the high school where I teach is "elite". Mostly affluent, mostly white (with a large population of Asian kids, relatively small group of black kids, and a growing population of Latino kids), very high performance on standardized tests (we had something like 28 National Merit semi-finalists last year), lots of kids go on to elite colleges. . . In the last few years, the Latino population has shown the most improvement on math scores on standardized tests, of any group. (I know standardized tests don't measure everything, but they do measure something.) We now have Latino students open enrolling into our school (perhaps as much as 30% of our kids come from out of the school's attendance area through open enrollment), same as other groups. They take our advanced math courses, maybe not at the same rate as the Asian kids who generally kick ass in math, but nonetheless this is a growing and positive phenomenon. Last year I recommended a female student for AP Calculus, whose parents are from Mexico and do not speak English. She hopes to be a doctor someday. (You have to admit MS55401, that is an uplifting anecdote.)

What does that say about my school and its status as an "elite" high school? I'm not sure, but I am very proud that kids from families that would not be described as "elite" in any socioeconomic sense, have access to an "elite" education and in many cases are thriving at my school. This would not occur in a truly "elitist" education system.

I suppose one could say the high school where I teach is "elite".

If you all have the same books and are teaching the same information, what makes for a good school vs. a bad school? Is it the character of the school, or the teachers, or the style of teaching, etc. that motivates the children to want to learn more?

I wonder why my daughters school is a #2 on the Calif. learning chart, while 20miles down the road another school is a #9? A #10 being the best.

It seems patently obvious that socioeconomic status affects both the school's readiness to teach and the student's ability to learn. Also the participation of the parents has a huge impact too.

In fact it takes a village to raise a child. If one lives in a shithole expect shithole schools. There will be exceptions of course, but if the citizens of a given school district simply don't give a sh#t, and there are millions of such people, then the schools will reflect that attitude right back to the citizens.

I would say that failures in American education system are not due to government nor teacher malfeasance. There is no single point of failure, no one thing you can point to and say, "ah HA, there's the problem!"

It is a lack of will and a lack of priority.

My only stake in this debate is that of a reasonably intelligent parent with two high school grad children. Oh and the fact that I or their mother were at most every parent/teacher meeting since kindergarten through 12th grade. In kindergarten, a lot (not all) of parents show up for school participation in various forms. By 8th grade parental participation seems to have dropped off significantly, but it also seems High School is where a lot of parents stop participating at all.

So for parent/teacher night in the 12th, when we visited each teacher for a few minutes of what they had going on, etc. and there were maybe 5-7 parents attending each meeting, for class sizes of 30 or more.

Normal, I suspect and my parents weren't much different in this realm. I was a good student but I strayed in High School and they were asleep at the switch.

I think involved parents who set high expectations of their kids, make the key difference and the lack of involvement, for whatever reasons good or bad, leads to a degradation of the school system itself.

Parents..... and their kids. It all starts there. A public school system cannot fix a bad parent through the child. Of course a good educator can make a huge difference even there, one kid at a time. But a school system can't do that.

That's the spirit wbw. IMO public education needs teachers with your kind of attitude.

Thanks Yanqui. That is a very nice compliment.

I think one of the ideas behind the Common Core (at least as it is playing out in my district) is that kids will have the same book, and that all classes are at more-or-less the same place, so that a student could move from one school to another, and get the same academic experience at any time. But this defies Common Sense.

At my school, we have an International Baccalaureate program, and have for years. That program pulls in hundreds of kids from out of the school attendance area. Because it is such a challenging program, most of those kids have done well before open enrolling into my school. This has the effect of helping my school to attract and retain good teachers, because the parent population demands it and because teachers want to teach students like this. This has an additional effect of attracting (mostly) good administrators. I have seen poor administrators run out of my school by teacher pressure. We have an administration that is very creative in problem solving, and actually supports creative methods in the classroom, even though many of us are more traditional in our methods than not.

Some in my district are critical of my school, saying that we "steal" their best kids from other schools.

Well, to that I respond that you can sit on your ass and wallow in mediocrity, or you can get up and get to work on giving kids the best possible education that you can, however you can. At my school, it is considered cool to be smart amongst the kids, so this also has a very broad and positive effect on the learning environment.

Dingus, you hit the ball out of the park on that last post. Parents and their appropriate involvement in their kid's education has BY FAR the biggest positive effect on their kid's education. BY FAR . . .

I believe the first study says that access to more math classes is the strongest correlative to success in advanced math. They also observe that students learn math at school, not at home. Parental expectations correlate, not parental tutoring.

The second study correlates student performance with the school'a SES and finds that "poor" schools indicate poor student performance. The first study also makes the observation that in poor schools the number of math classes available to the students is less than "rich" schools.

At least one of the ideas is to bring the curriculum to a standard for all schools. Presuming local, state and federal support for that standard which includes funding.

I believe both studies are optimistic regarding the remedy, which is adequate support for schools to offer more math classes.

Although I have been arguing that at least some aspects of the common core seem to make sense, the nature of its implementation seems to be a burgeoning problem, and much (but not all) of the frustration expressed here relate to the destructive side-effects of an evaluation-driven culture.

Yes, we still have the problem that some people, even and sometimes especially those with technical backgrounds, really don't understand mathematics and seem incapable of fathoming the difference between rote algorithmic competence and anything recognizable as understanding. That said, the destructive effects of current education policies on good teaching are becoming increasingly evident. The following letter of resignation from a social studies teacher captures many of the issues:

Mr. Casey Barduhn, Superintendent
Westhill Central School District
400 Walberta Park Road
Syracuse, New York 13219

Dear Mr. Barduhn and Board of Education Members:

It is with the deepest regret that I must retire at the close of this school year, ending my more than twenty-seven years of service at Westhill on June 30, under the provisions of the 2012-15 contract. I assume that I will be eligible for any local or state incentives that may be offered prior to my date of actual retirement and I trust that I may return to the high school at some point as a substitute teacher.

As with Lincoln and Springfield, I have grown from a young to an old man here; my brother died while we were both employed here; my daughter was educated here, and I have been touched by and hope that I have touched hundreds of lives in my time here. I know that I have been fortunate to work with a small core of some of the finest students and educators on the planet.

I came to teaching forty years ago this month and have been lucky enough to work at a small liberal arts college, a major university and this superior secondary school. To me, history has been so very much more than a mere job, it has truly been my life, always driving my travel, guiding all of my reading and even dictating my television and movie viewing. Rarely have I engaged in any of these activities without an eye to my classroom and what I might employ in a lesson, a lecture or a presentation. With regard to my profession, I have truly attempted to live John Dewey’s famous quotation (now likely cliché with me, I’ve used it so very often) that “Education is not preparation for life, education is life itself.” This type of total immersion is what I have always referred to as teaching “heavy,” working hard, spending time, researching, attending to details and never feeling satisfied that I knew enough on any topic. I now find that this approach to my profession is not only devalued, but denigrated and perhaps, in some quarters despised. STEM rules the day and “data driven” education seeks only conformity, standardization, testing and a zombie-like adherence to the shallow and generic Common Core, along with a lockstep of oversimplified so-called Essential Learnings. Creativity, academic freedom, teacher autonomy, experimentation and innovation are being stifled in a misguided effort to fix what is not broken in our system of public education and particularly not at Westhill.

A long train of failures has brought us to this unfortunate pass. In their pursuit of Federal tax dollars, our legislators have failed us by selling children out to private industries such as Pearson Education. The New York State United Teachers union has let down its membership by failing to mount a much more effective and vigorous campaign against this same costly and dangerous debacle. Finally, it is with sad reluctance that I say our own administration has been both uncommunicative and unresponsive to the concerns and needs of our staff and students by establishing testing and evaluation systems that are Byzantine at best and at worst, draconian. This situation has been exacerbated by other actions of the administration, in either refusing to call open forum meetings to discuss these pressing issues, or by so constraining the time limits of such meetings that little more than a conveying of information could take place. This lack of leadership at every level has only served to produce confusion, a loss of confidence and a dramatic and rapid decaying of morale. The repercussions of these ill-conceived policies will be telling and shall resound to the detriment of education for years to come. The analogy that this process is like building the airplane while we are flying would strike terror in the heart of anyone should it be applied to an actual airplane flight, a medical procedure, or even a home repair. Why should it be acceptable in our careers and in the education of our children?

My profession is being demeaned by a pervasive atmosphere of distrust, dictating that teachers cannot be permitted to develop and administer their own quizzes and tests (now titled as generic “assessments”) or grade their own students’ examinations. The development of plans, choice of lessons and the materials to be employed are increasingly expected to be common to all teachers in a given subject. This approach not only strangles creativity, it smothers the development of critical thinking in our students and assumes a one-size-fits-all mentality more appropriate to the assembly line than to the classroom. Teacher planning time has also now been so greatly eroded by a constant need to “prove up” our worth to the tyranny of APPR (through the submission of plans, materials and “artifacts” from our teaching) that there is little time for us to carefully critique student work, engage in informal intellectual discussions with our students and colleagues, or conduct research and seek personal improvement through independent study. We have become increasingly evaluation and not knowledge driven. Process has become our most important product, to twist a phrase from corporate America, which seems doubly appropriate to this case.

After writing all of this I realize that I am not leaving my profession, in truth, it has left me. It no longer exists. I feel as though I have played some game halfway through its fourth quarter, a timeout has been called, my teammates’ hands have all been tied, the goal posts moved, all previously scored points and honors expunged and all of the rules altered.

For the last decade or so, I have had two signs hanging above the blackboard at the front of my classroom, they read, “Words Matter” and “Ideas Matter”. While I still believe these simple statements to be true, I don’t feel that those currently driving public education have any inkling of what they mean.

I took math up to Vector Calculus. I rarely even use algebra. I use recursion every now and then in programming. Why do we need to learn all this math anyway? If I need to use it, I look it up on the internet. College isn't even necessary these days. Skip the debt, learn what you need to know on the job or taking online classes. I'm way ahead of my friends who went all in for the phd. They are still living in grass huts and complaining about privilege.

You never know when you might want to go back to school and get a degree in
physics or rocket science, do you? So who the hell is taking all this math?
Clearly a lot more are taking it than getting it. Most Americans are still
completely mystified by compound interest.

My favorite math prof when I was an undergrad at Berkeley, Hung-Hsi Wu, has co-authored a couple of opinion pieces for the Wall Street Journal in support of the Common Core math standards, so it's been hard for me to dismiss them. Unfortunately, as you point out, Richie, the implementation seems to leave much to be desired.

The letter you copied hits hard, but it ignores a lot of the symptoms that led to the destructive trends about which its writer complains. In particular, employers outside (and, I suspect, even inside to a certain extent) academia find educational credentials meaningless. A high school or college diploma carries with it no reasonable expectation of any particular knowledge or skill.

Frustrated employers started demanding some evidence that graduates demonstrate certain minimum skills and knowledge. Those same employers concluded that neither grades nor graduation gave the desired assurance. The standardized tests and curricula were a ham-handed attempt to convey the desired assurance.

I find it particularly interesting that many professions with demanding postgraduate educational requirments still administer tests outside of those administered in school. Lawyers must deal with bar exams, physicians with board certifications, etc. Why don't I hear law or medical faculty decrying the influence of those tests?

In contrast, I see school districts misusing test results to judge value - when we should measure educators by their value added - and teachers railing against outside testing rather than the misuse of those tests. I find it quite telling that the writer complains that the unions don't do enough to fight the imposition of those outside standards and measurements, but makes no demand that the unions propose something better.

When teachers organize to promote better teaching, rather than to oppose burdensome or insulting requirements, we can hope for real progress. Until then, I'm not holding my breath for better American performance in math, or in education generally.

Now wait a minute, Werner! Just because mere arithmetic differs from "real" math is no reason to criticize mathematicians.

I must admit, though, that one undergraduate linear algebra midterm at Berkeley had four numeric answers, all of which I got wrong because I was too hasty and sloppy in my arithmetic, and the prof still gave me 100% because my methods were right.

I teach Common Core high school standards daily, preferring them to the former CA state standards. It's exciting planning new units and authentic assessments with others. I am forced to teach new works, while getting to revive and share what I did in the 90s.

Not many weighing in here are actually in the trenches. You are college teachers, lawyers...

My union kicks ass, John E.. Countless union leaders have retired in my area, so a new crop is getting its sea legs. Hence the wobble.

As for the letter quoted above, the fellow could have used a change in school sites a few times.

i just recently got curious, about an abacus... as, i had seen one at someones house, as a kid...

somehow, i 'ran' into one, one the web and looked up more about it, to read-up on them...

such an old time, 'always been around' hardy thing, :) as to basics, :))

so, i practiced a bit online and then went and got one from amazon...
for me, that goes 'blank' at math, and transposes various combo letters or numbers... THIS was just beads... so i thought i'd give a go...

oddly, it is working out nice (though i HAD to put a few markers, scratches, on it, so i do not 'flip' my vision of it)...

well, it is helping me sort the thoughts in my head... expecailly when i have to subtract... i need to REDO the beads...
have to take out ONE pretty bead and put back TWO pretty beads on the other stick, so that i will have ENOUGH neat little beads to subtract with ...

once in a while i get stuck on which pole to adjust, but, i reckon, so far, as FAR left, as you need to...

i practice on my check book, which is already figured out, as to what i have taken out, etc... for to see if i get the beads right...

onward and upward, ... not sure how much it is helping my MATH but it sure is helping my brain and i do see in pictures, anyway, so i 'don't panic' at the numbers... :O

wish my daddy was still alive, somehow, i think he'd have enjoyed this
adventure of mine... :)
:(

Neebee's story about the abacus reminds me about the origins of calculus. Not the subject, the name. All you medical types know that a calculus is a stone (usually or perhaps always a kidney stone). And that is the original meaning of the word, stone or pebble.

Although Asians were clever enough to put beads on a stick, Greek and Roman merchants reckoned with pebbles manipulated in grooves in the dirt in the same way as the abacus is used. The same fundamental concepts of place value that eventually led to Arabic numeration were present in those pebble systems. And the manipulation of pebbles, of calculi, became associated with having a system for obtaining mathematical results, so much so that it seems only the medical world continues to use calculus in something like its original sense. For the rest of us, calculus and its associated term calculate are mathematical notions.

In the spirit of a return to ancient times, one apparently embraced by the American electorate, I propose that henceforward the Stonemasters be referred to as the Calculusmasters.

All these so called math guys who think they're smart are actually stooopid. Blah blah ... all they know is one dimensional

Mathematicians may be stooopid, but at least they're smart enough to know that some of the most interesting questions in analysis actually occur in infinitely many dimensions.

When teachers organize to promote better teaching, rather than to oppose burdensome or insulting requirements, we can hope for real progress.

I'm not a big union guy, but when teachers organize to oppose burdensome requirements, don't you think that might just be because burdensome requirements hinder better teaching? Your statement makes very little sense, and wreaks of the typical arrogance that many other professionals such as lawyers or investors have towards teachers. I got news for you: we're not all idiots.

The letter written by the teacher in Syracuse hits the nail on the head. I think what he is trying to say is that for his entire working life he has dedicated himself to being the best teacher he can, and because people perceive that ALL education in the US sucks, he's being forced teach a poorly designed curriculum (Common Core) that is 100% owned and operated by Pearson, Prentice-Hall (which is true), in the name of educational reform.

The Common Core demands that I teach more content in my classes than ever before. (Even though advocates that don't actually know will claim otherwise.) The kids I teach are missing more class than ever for these stupid tests such as CMAS and PARCC, and its all technology-based testing. They are getting tested to death, when they would rather be in a classroom with one of the many excellent teachers at my school.

With the new evaluation system here in CO, I am definitely spending more time proving my worth as a teacher by documenting inane aspects of my job, and it is taking valuable time away from the time I spend trying to actually improve my craft. All the serious, good teachers at my school feel this way.

wbw, my daughter and son-in-law both teach high school math, and they're no idiots. What they think of some of their particularly gung-ho-union colleagues is a different matter entirely.

Of course burdensome requirements can get in the way of teaching. What I ask, however, is that the union-types offer something else that actually helps deal with the problem the rest of the world sees, viz. poorly-trained students with diplomas.

For those wondering what is wrong with the Common Core, there are some very interesting comments made in this article by a math professor who sat on the Governors' Advisory Board for the Common Core.

WSW: When you are so far behind, comparing the United States with better-performing countries through the incredibly narrow lens of standards doesn’t make a lot of sense. I think Common Core is in the same ball park, certainly not up there with the best of countries, but Common Core isn’t up there with the best state standards either, and what does that mean? Look at California’s standards for example. They are great standards and have been unchanged for over a decade, but many in math education hate them. They think they are all about rote mathematics, but I think such people have little understanding of mathematics.

So, let’s just pretend for a moment that Common Core is just as good as the very best. Who, in education circles, will agree with that enough to put it all in practice? The standard algorithm deniers will teach multiple ways to multiply numbers and mention the standard algorithm one day in passing. Korea will say “no calculators” in K–12, a little extreme perhaps, but some in the U.S. will say “appropriate tools” means calculators in 4th grade. We, in this country, are still not on the same page about what content is most important, even if everyone says they’ll take Common Core. Without a unified, concerted effort to teach real mathematics, there isn’t much chance of catching up.[

John wrote: ...In particular, employers outside (and, I suspect, even inside to a certain extent) academia find educational credentials meaningless. A high school or college diploma carries with it no reasonable expectation of any particular knowledge or skill.

Frustrated employers started demanding some evidence that graduates demonstrate certain minimum skills and knowledge. Those same employers concluded that neither grades nor graduation gave the desired assurance. The standardized tests and curricula were a ham-handed attempt to convey the desired assurance.

I don't recall when education became a certification process for the private sector. In particular, given the general reluctance to pay local taxes to support the education system, why would the concerns of the private sector be relevant to the discussion?

As I've said many times in the past, the four years I spent as an undergraduate, and eight years at graduate school was not an "investment" in my future, at least I didn't see it as such at the time and I had little belief that I would be stamped "certified" and then accepted in some private sector job. I didn't quite know what job I would do, but certainly those 12 years were an intellectual adventure for me. It turns out that I did make a career in science, and I'm lucky that it was a very good career, at least as successful as some others I might have pursued.

I do understand the standpoint of the value proposition implied by investment in education, but as others have pointed out in this discussion, going to school isn't the only path to a successful life.

In particular, employers outside (and, I suspect, even inside to a certain extent) academia find educational credentials meaningless. A high school or college diploma carries with it no reasonable expectation of any particular knowledge or skill.

Wow, John, that's quite an overstatement.

Frustrated employers started demanding some evidence that graduates demonstrate certain minimum skills and knowledge. Those same employers concluded that neither grades nor graduation gave the desired assurance. The standardized tests and curricula were a ham-handed attempt to convey the desired assurance.

I'm not at all sure efforts at reform were driven by employers, though they may have piled on after the fact. Indeed, if the point of education is to train the citizenry for employment, then why isn't education funded by the industries that benefit from the training? Why are we paying tax dollars to fund employee training programs?

I don't think employee training was ever supposed to be the point of public education.

I'm sure the educational world has been swept by reform movements from its inception. The first one I was aware of is probably Plato's Academy, which said on the entrance, "Let no one ignorant of geometry enter here." It is worth thinking about this a bit, because Plato was not in the geometry business at all, and no doubt he had some students who said, "we never use geometry here, so why did I waste all that time learning it?" Three thousand years later we hear the same echo today from people who don't seem to realize how their education may have shaped their abilities.

The first reform I experienced was the Sputnik crisis, when the USSR put the first satellite into space. Life magazine ran an article purporting to compare the lives of typical students in the US and the USSR. The US students were cheering for football teams and preparing for proms, while the USSR students were staying up all night studying.

Government, not private, money flowed into what are now called the STEM fields, and presto, we landed first on the moon. The mathematics reform at the time was called New Math. Tom Lehrer wrote a funny song about it. It was part of an educational enterprise that created a generation of scientists who fueled American primacy in science and engineering for years, but it was deemed a failure and is still referred to as an example of bad educational reform. Many of the complaints were the same: the parents didn't understand it and some of the teachers didn't either, features which, if you think about it for moment, might just as well be signs of quality too.

I'm willing to wager many of the principles of the New Math can be found in the Core Curriculum Standards. The difference then and now is that having promulgated the ideas, the reformers trusted the teachers to implement them. The current system seems to begin with an implicit assumption of teacher incompetence and then tries to make those benighted souls support standards the system implicitly assumes they are too incompetent to implement. Is it really any great surprise that this is not working well?

I find it particularly interesting that many professions with demanding postgraduate educational requirments still administer tests outside of those administered in school. Lawyers must deal with bar exams, physicians with board certifications, etc. Why don't I hear law or medical faculty decrying the influence of those tests?

I'd guess there are several reasons. One is that the tests are administered once at the end of years of education, rather than the current moment-to-moment intrusions that distort the everyday fabric of the classroom. The content of those tests is made available to students to study, and it is the responsibility of the students to learn the material on the test. In particular, the material on the test may not be what was taught in those postgraduate courses, it being assumed that the courses provided the skills and knowledge that would enable the students to master new material of importance. This is particularly true of Bar exams, which are tailored to state law. The whole situation is about as far from the testing programs in elementary and secondary education as it is possible to get.

Another is that, at least for now, faculty in these demanding postgraduate fields are not held personally responsible for their student's performance on the exams (although this might change). The role of student effort and preparation in passing the exams is recognized as a primary ingredient in success, and no one thinks that the faculty is somehow responsible for forcing or motivating those efforts.

And another is, I think, that by and large the faculty are either in charge of designing the exam questions or at least have considerable trust in those who do.

I don't think any of these conditions apply in the case of elementary and secondary educational testing.

When teachers organize to promote better teaching, rather than to oppose burdensome or insulting requirements, we can hope for real progress. Until then, I'm not holding my breath for better American performance in math, or in education generally.

I think teachers have been organizing to promote better teaching for years. In mathematics, there is, for instance the NCTM, the National Council of Teachers of Mathematics and I'm pretty sure there are analogous very active groups in most of the major fields. I'm also pretty sure that the unions have invested many resources into improving teaching. I know this to be true in New York State and assume it to be the case elsewhere.

But I also think it fair to point out that the US system, in marked contrast to at least some of the Asian systems, makes it almost impossible for teachers to meet and collaborate and so advance their professional standards and abilities. US teachers are almost prisoners in their classrooms with little or no ability to interact on a professional basis with other teachers, and little if any useful mentoring. New teachers are basically thrown into the deep end to see if they sink or swim, in a system that devises ever more ways of sinking while providing inadequate support for swimming.

I took math up to Vector Calculus. I rarely even use algebra. I use recursion every now and then in programming. Why do we need to learn all this math anyway? If I need to use it, I look it up on the internet. College isn't even necessary these days. Skip the debt, learn what you need to know on the job or taking online classes. I'm way ahead of my friends who went all in for the phd. They are still living in grass huts and complaining about privilege.

Tone is difficult to express on the two-dimensional Internet, so I'll take your post as tongue-in-cheek.

In the event that you are actually serious, math contributes to developing your brain muscle (analytic skills), doesn't matter if you don't use specific calculus equations in your daily life.

I think it was Yvon Chouinard who said something to the effect that he never had any use for algebra. I see other comments like this in this thread and hear them all the time. I don't say much to these people, because it's pointless - they don't like math, there is no point trying to convince them that they should like it. Needless to say, selling overpriced garments is not rocket science, and Patagonia is never going to resemble Google.

On the original subject of this thread, "why Americans Stink at Math" - it is not just an American problem. There are many other countries which do not do well in math. Economic wealth is not related to strength in math. You cannot buy mathematics, you have to earn it by hard work. The situation is similar to climbing. You can have all the newest, most expensive climbing gear, but this does not propel you up an El Cap route - you have to earn it with patience, blood, sweat and tears. There is no royal road to geometry.

In my opinion, most pedagogical problems in math stem from lack of knowledge and enthusiasm of teachers (at the high school and lower levels). If I had my way, teachers of math would be required to have PhDs and would be paid accordingly. I would drop a lot of the purely pedagogical training (education departments) because it is largely BS.

Doing math seriously is akin to climbing El Cap or climbing a difficult alpine route. There is no room for mediocrity here. On an alpine route, you will die if you are incompetent. The same thing happens in mathematics - the laws of logic are as immutable in mathematics as the law of gravity is in climbing. Quacks and posers are quickly weeded out in math, so they typically migrate to easier subjects. This is okay. Climbers make up a very small fraction of the overall population, so do mathematicians.

In the event that you are actually serious, math contributes to developing your brain muscle (analytic skills), doesn't matter if you don't use specific calculus equations in your daily life.

Excellent comment. Question: "How am I going to use this?" Honest answer: "You're going to use it everyday because it's gonna make you smarter. Duh." I don't know why so many people don't get this.

Andrew, you make some great comments.

I can just hear John thinking to himself about his children that are teachers. . . that's such a cute career for kids, teaching. Maybe when they grow up they'll get a real career like law. Again, typical arrogance from a profession that takes people to the hoop every chance that they get.

One hears the "I never needed to use this" complaint at all levels of education right through grad school. And of course it is true in some cases, in other cases not, although I'd guess that it is very rare that someone "needs" (whatever that really means) anywhere near everything they learned in any field. But mostly the claim misunderstands the point of education, perhaps even highly vocational training, and tries to impose a strictly individual perspective on an enterprise that is obligated to look well past individual pathways.

Here's an imperfect analogy: I start a rock-climbing school. Naturally, I intend that my degree certifies that a student is competent in all climbing genres, so I teach them about everything. Now someone graduates, moves to NYC, and has a long climbing career consisting primarily of weekends in the Gunks, with trips to the Tetons, Wind Rivers, Tuolumne Meadows, Red Rocks, etc. Now it happens that during this long and productive career, the graduate never has to do any real offwidth climbing and certainly nothing technical requiring either refined technique or just the applications of more basic techniques to a very sustained pitch. And so they complain that the very substantial amount of time in my program devoted to offwidth was wasted, because they never had to use any of that stuff.

So how does my school respond to this criticism? Naturally, we say we could not peer into the future of any one of our students and figure out what types of climbs they were going to choose, that our program was intended to make its graduates competent to confront any and all of the difficulties they might encounter in a climbing life, and the fact that this or that individual never used some of the skills we taught is actually irrelevant. We prepared them. They sought us out precisely because we would prepare them. What they eventually did with that preparation is their business, but does not reflect on the appropriateness of what we did.

And let's say our graduate moves on to some other outdoor sport and never really does any climbing. Then of course they say the never needed any of what we taught, and point to people in their newly-chosen field who have done well without any climbing education. Why exactly are those observations relevant to what we do in our climbing school?

Now the analogy is imperfect, as I'm sure people will be quick to point out. I do think it works pretty well as the collegiate level, where people choose majors and degree programs. My hypothetical climbing students also choose to go to a climbing school. But in our primary and secondary system we require students to go to school and learn certain subjects, so in some sense we are now teaching climbing to everyone, whether they have any inclination to do it or not.

Surely this has drawbacks and benefits. Among the benefits, we get some climbers who would never have known about it or engaged in it otherwise. Among the drawbacks, we get people who hate everything about climbing and were forced to endure a lot of instruction they found distasteful and irrelevant.

Now to continue the analogy, we would have to stipulate that climbing is actually a basis for a wide spectrum of life pursuits, so that failing to teach young people to climb would close them out of many modern opportunities. This would be bad public policy and a bad approach to the development of the highest possible potential in our population, and most authorities, going back thousands of years, have understood that some dissatisfied customers are a price that has to be paid in the pursuit of an ultimately greater good. This does not make the dissatisfied customers feel any better, but it also doesn't mean that their afflicted cries should be taken as a call to stop preparing everyone equally for the future.

I am an engineer in the semiconductor industry and I find that find my education useful to do my day to day job and to have the foundation to teach myself new concepts and ideas as my role has changed over the past 20 years in industry.

I thought the Common Core was like the new math (sets in the 70's) in that it will make a huge amount of money for publishers of text books. I had one daughter who went through trad math and one through new math. The trad math gave a better result (I helped them on their homework when asked). Kids need to learn operations for which not much thinking applies. After that they get into math which is descriptions of reality, insofar as it is.

The concept of calculus is extremely important to your way of thinking, and it opens the door to exapnding how you view the processes of the universe. It is more than just "math."

People in the USA stink at math because they sit in front of a TV or computer game 10 hours a day instead of studying.

I thought the Common Core was like the new math (sets in the 70's) in that it will make a huge amount of money for publishers of text books. I had one daughter who went through trad math and one through new math. The trad math gave a better result (I helped them on their homework when asked).

Pearson Prentice-Hall has bought up nearly every small textbook company that exists in the last few years. Some of these small companies have excellent books, but can't compete with the Pearson giant. Pearson will profit enormously from the Common Core.

Pearson pours tons of money into technology for education, that is a joke. It is very superficial (this to support the Common Core that allegedly goes deeper), and when one attends an in-service with a Pearson rep to learn how to use the technology, the Pearson rep cannot even manipulate it in the way they advertise. Pearson also bought the rights to the GED recently, which is a different but equally disturbing issue.

The Common Core at the 6th grade level is befuddling. I work with my daughter (6th grade) who is good at math, and she is generally not too confused by the quirkiness of the Common Core standards. For kids that struggle in math, it is totally confusing trying to solve a problem 4 different ways.

Also, instead of providing a broad foundation in math, it is more like an Advanced Placement course approach, in that it is targeted to solve specific types of problems. The skills are supposed to be learned in the context for which they are needed for a specific problem. Younger math students need a broad, consistent foundation in order to access the advanced math classes taught at my high school. We are already seeing kids (en masse) that show up to 9th grade math class very confused and with poor skills. This is because of the Common Core.

You're right, and I apologize. I made an error that I usually advise others against. I overstated an unnecessarily divisive position. I should simply have expressed my frustration when undergrads in my intoductory economics classes couldn't perform the most basic algebraic manipulations, or graduate students who allegedly passed multivariate calculus couldn't understand its need or application in calculating the least-squares estimators for parameters and statistics of fit for multiple regressions

It was in that sense that I stated that academic credentials proved meaningless. Of course they aren't. They still make their holders more likely to have knowledge and skills related to their fields of study than those without those credentials. I can see now how my exaggeration led to obfuscation of my point, so I'll try to re-state my position.

When the politicians demand standards and measurement, they express what a large segment of the voting public requests. I, too, didn't go to college, grad school, or even law school for a job. I went because of intellectual curiosity, but the public that's paying for education isn't satisfied when we say education forms its own reward. They expect -reasonably in my opinion - that graduates will be proficient in a certain minimum set of skills and subjects.

Wbw's quote about the need for a concerted effort to teach real mathematics rings true for me. I think the Common Core math standards form a part of determining what constitutes "real mathematics." Their propenents intended to set forth the minimum set of mathematical knowledge and skills in which students must demonstrate proficiency. I know enough about the people who put it together to believe that they did so for altruistic, not remunerative, reasons.

The reaction to Coomon Core, and particularly the arguments I hear from the rather unusual coalition of teachers and Tea Partiers, reminds me, too, of the gripes I heard about "new math" 51 years ago when I was a student learning it. I shudder to consider the boredom I would have faced if I had to continue learning the "old math."

I aimed (rather poorly, I see) my swipe at teacher organizing at the unions and others who don't want to be bothered teaching something new, claim that Common Core won't help, but fail to offer an alternative set of standards or method of measurement. I didn't mean to disparage the true professional organizations. Once again, my rhetoric overstated my intent. My bad.

John, read the article I posted upthread on this page. There are very smart people who advised on the Common Core, and then when it came time to sign off in agreement chose not to. Look at your own standards in CA as an alternative model that is better than the Common Core.

What bothers me more than the Common Core is this farce that it somehow improves math education in the US.

And the opposition is not coming only from the far right. Please read up on the issue before posting your exaggerated opinions.

I read the article, wbw, and agree that California's standards, and those of several other states, exceed those of Common Core. The reaction to those rather minimal Common Core standards tells us how far we have yet to go if we intend to improve math education in the US.

Incidentally, my daughter taught math under an IB curriculum to seventh and eighth graders the last two years. She's now back teaching high schoolers, though, because, as she put it, "middle school students are just too fragile." (She has a sarcastic streak that her high school students love, but her middle school [and particularly seventh grade] students weren't so sure about).

I should simply have expressed my frustration when undergrads in my introductory economics classes couldn't perform the most basic algebraic manipulations, or graduate students who allegedly passed multivariate calculus couldn't understand its need or application in calculating the least-squares estimators for parameters and statistics of fit for multiple regressions

Oh yeah, I feel your pain.

We see these kinds of problems on a daily basis. But we also see some excellent well-prepared students who learn and grow in their mathematical sophistication. I do think there is a tendency for the poorly prepared and poorly motivated students to loom larger in our consciousness as we try to figure out how to do the things we are supposed to be doing with people who are several levels removed from being ready. And it does seem unfathomable how they got this far without someone pulling the plug.

One partial answer I've seen in action: they learn things and then just totally forget them. This is an experimentally-verified artifact of cramming behavior. But it is also, I think, a byproduct of the perception, advanced here several times, that none of the stuff matters and education generally is just a big fraternity initiation rite in which people who were obliged to do unpleasant things to join now, as members, inflict analogously meaningless pain on the newcomers. So you suffer through that multivariable calculus course, viewing it as the intellectual equivalent of swallowing goldfish, and purge it from consciousness as soon as possible, secure in the faulty knowledge that you'll never need that shite again.

Yesterday I read an article in the Press Democrat which said Sonoma County is undergoing a critical shortage of substitute teachers. The Common core is at the core of the uncommon problem. Subs don't want to do it or have to pay to learn how to do it, so one third or less of the normal applications have been received. Unintended consequences.

One partial answer I've seen in action: they learn things and then just totally forget them

Bingo. If you don't have sufficient interest in a subject to review it in your mind within a certain period of time after class chances are it will quickly fade, unless you are truly exceptional. Doing homework many hours later will be more difficult and understanding the material will be impeded, leading to increased frustration with, and dislike of, mathematics.

I seem to recall Feynman saying something like this about physics.

Nice presentation by Elly. Math is like climbing in that a project that is dispatched quickly and easily is not as rewarding as one requiring time and effort.

However, there should be a balance between memorization of algorithms and problem-solving, critical thinking. A famous mathematician once said that his ability to rapidly and accurately remember and apply rules of algebra and calculus without effort was an important key to solving difficult problems in research. Without that facility one may see the light at the end of the tunnel, but be unable to reach it in a reasonable time, getting bogged down with trivia.

And then, of course, the question: is critical thinking a product of nature or nurture or both?

One partial answer I've seen in action: they learn things and then just totally forget them. This is an experimentally-verified artifact of cramming behavior. But it is also, I think, a byproduct of the perception, advanced here several times, that none of the stuff matters and education generally is just a big fraternity initiation rite in which people who were obliged to do unpleasant things to join now, as members, inflict analogously meaningless pain on the newcomers. So you suffer through that multivariable calculus course, viewing it as the intellectual equivalent of swallowing goldfish, and purge it from consciousness as soon as possible, secure in the faulty knowledge that you'll never need that shite again.

True. Ironically, in my first textbook on undergraduate set theory (Halmos, Naive Set Theory) The Foreword essentially told us: Here it is. Learn it. Play with it. Forget it.

You studied that as an undergraduate; it was the text in my first course as a graduate student at the U of Alabama in 1962. All entering grad students were required to take it. It was an eye-opener for me as I had never studied set theory. We were given worksheets to develop all sorts of mathematical structures based on the Peano Axioms. I still take it off my shelves from time to time and enjoy Halmos's delightful writing.

I still take it off my shelves from time to time and enjoy Halmos's delightful writing.

I'm glad to see I'm not the only one, John. My wife thinks I'm nuts, and she and my younger daughter make motions as if they're slashing their wrists if I actually talk about the beauty of mathematics.

John Kelly's General Topoloy was my graduate set theory source, but I defected to economics and law too soon. I was particularly sorry I hadn't stuck around when I learned in 1973 that the Banach-Tarski paradox depended on the formulation of the Axiom of Choice. By then, it was too late to go back.

In 1935, my father, who had recently completed an MA in math at Alabama (winning the Comer Medal), decided he would pursue a PhD in pure math, and would begin at a summer session at Michigan. His first course there was intro to topology. At the end of the session he returned to the South and in 1950, after working at several non-academic jobs, got his doctorate at Texas in economic statistics. He told me that topology was too far out for him.

I was fortunate to have a great professor for my first topological adventure, and years later enjoyed teaching the course to our senior math students.

The axiom of choice bridges the gap between the real and the unreal, in my mind. But that's just me!

Here in Pueblo County, district 60 - the in-town district, with several pockets of poverty - has chosen to reduce the math requirement for graduation in the least academic track, while keeping the requirement for college-bound students. The district is under notice that unless general test results improve the state will come in and take over. Results have been steadily declining for several years. One tack the district is taking in desperation is to seek a less rigorous accreditation organization.

District 70, which includes Pueblo West, where I live, has done considerably better, with test results exceeding state standards. The suburban areas in the county are more prosperous and vote to give more than adequate financial support to the schools.

I suppose I have mixed feelings about reducing math requirements for students who have not expressed a desire to go to college. On the one hand many may never need the more advanced material that is to be eliminated, but of course this sort of tracking makes it more difficult for students to change their minds about higher education once they complete the vocational curriculum. In that instance, however, most colleges have remedial programs for students who lack basic credentials, as well as those who have taken the more advanced secondary offerings but show little evidence of remembering concepts and techniques. I saw a lot of that when I was teaching.