Post-Slump Links

Since every hour that passes I’m more certain I’m not going to keep blogging, here are a few good links for your perusal:

Stuart Staniford of the Oil Drum explains carefully why the Saudi production decrease is due to peak oil rather than a voluntary reduction. The minutiae of the Saudi production curve are more consistent with a post-peak slump rather than with a voluntary reduction meant to give Saudi Arabia the power to flood the market at any given time.

C. L. Hanson notes that the two basic principles of relationships – that people have the right to say no to sex and that people shouldn’t sleep with anyone but their partners – are incongruous. As such, she talks about how cheating can save relationships.

Stentor rebuts market-based arguments against environmental legislation. He explains specifically that air pollution needs to be curbed collectively since air is naturally a shared resource. This isn’t an especially novel argument – the tragedy of the commons is a recognized market failure – but some libertarians’ hostility to it requires repeating it more than should be necessary.

Melissa Franklin, Harvard’s first tenured female physics professor, speaks at a conference about women in science that has just given her an award. She recounts experiences ranging from students’ crying because they couldn’t finish their problem sets to sexual assault.

More Carnivals

I’m not sure why I missed Help Us Help Ourselves yesterday, but I did. The carnival’s posted on the 1st of every month, so it’s not as if it was hard to remember.

Also, there’s a new carnival in town, dealing with issues of women in science (and engineering, and math, and technology): Scientiae. The first edition features many good posts about discrimination against women in the academia; the highlights are Am I a Woman Scientist?’s writeup about why women publish less than men, and Female CS Grad Student’s experience with being the only female in her graduate electrical engineering class. The next edition will be posted on 3/15 on Post Doc, Ergo Propter Doc, but the submission deadline is 3/12.

Carnivalia

The seventh installment of Philosophia Naturalis, the quadriweekly carnival of physics and technology, is up on Geek Counterpoint; the highlight is John Conway’s two-part series about the search for the Higgs boson. The next carnival will be posted on March 29th; the carnival webpage is here. Note that although the host is not yet publicly posted on the carnival webpage, the next available hosting opportunity is on 5/24.

The 33rd Carnival of the Liberals is up on Blue Gal in a Red State. The next edition will be posted on Brainshrub on 3/14.

The 108th Carnival of Education is up on Dr. Homeslice. No highlight is offered, although there were a few possible candidates. The 109th carnival will be hosted on 3/7 on What It’s Like on the Inside.

The Skeptics’ Circle is now up on The Second Sight, with a special numerology theme. The highlights are Hlynes’ takedown of yet another a scare story about genetically modified food and Phil Plait’s rant about the story of the creationist who got a Ph.D. in geoscience and made national headlines.

And, of course, remember that the Carnival of Mathematics is up on Michi’s Blog on 3/9 and the next Carnival of the Godless will be posted on Hell’s Handmaiden on 3/4.

The Teachers’ Union is the Source of All Evil in the World

Shelley finds a flowchart that documents how hard it is to fire a tenured teacher in New York, the idea being that if only the evil teachers’ union stopped demanding that teachers not be arbitrarily fired, the state of American education would be a lot better. Of course, as Mark Kleiman notes, in the South it’s already the case, and public schools stink even more than they do here…

Focusing on individual bad teachers misses the point. The point is that there’s a severe shortage of good teachers, which has gotten to the point that California has to accept teachers who flunk a tenth-grade-level reading test. Now, California’s schools are severely underfunded – per student funding in California is below national average even though housing prices are the highest of all states – but similar problems with teachers happen even with decent funding.

People who think the teachers’ unions are the source of all that’s evil in the world just focus on the wrong problem. There already exists a process for getting rid of bad teachers; it’s called not giving them tenure in the first place. And even if they’re fired, the state has to find an alternative teacher, typically a rookie who won’t necessarily be any better.

Look, you don’t need mega-pay to have good teachers. On average, schools in the US spent $8,300 per student in school year 2003-4, of which three fifths went to teacher pay and benefits. Stuyvesant’s per student spending is about the same (it was $8,200 in 2003 by a definition that leaves a small amount of spending out), so its teachers can’t be paid that much more, even though New York is hardly a cheap place to live in.

The American school crisis is mostly a low-income school crisis. Upper middle class suburbs like Westchester and Nassau Counties have non-selective public schools that do perfectly well. Part of it is because of an insane cash infusion, but that’s only true for some suburbs.

So it makes sense to ask how come low-income schools have teachers who stink. Is it because good teachers would rather get paid $40,000 a year to teach at a magnet school that produces Nobel Prize winners than get paid $40,000 to teach in a ghetto? Or is it because low-income schools naturally lack one of the most important control mechanisms, parental involvement (there’s a reason scripted learning works in low-income schools)? Or, is it really a funding question, with a few exceptions for glamorous places like Stuyvesant and Bronx Science.

Saturday Link Roundup

I wanted this roundup to be science-themed, but there’s been too few linkworthy science posts and too many political posts. Still, starting with the science, GrrlScientist reports about how sulfur particles cause some global cooling, which can be exploited to mitigate global warming. The only thing I have to say about that is to recall the Futurama episode where Fry says at a ski resort, “It’s a good thing global warming never happened.” Leela retorts, “It did, but the nuclear winter balanced it out.”

Orac writes about the dilemma of whether to allow individuals access to experimental drugs. He comes down strongly on the side of not allowing, explaining that,

The entire ruling also seems to rest on a misperception that there are “miracle drugs” out there that we will have to wait years for because the FDA is too slow to approve them. However, if there really were such a “miracle drug” that was amazingly effective compared to anything we have now, a large randomized phase III trial would not be necessary to detect its efficacy. Indeed, its efficacy would almost certainly show up in even a small phase I trial. There’d be examples of amazing tumor shrinkage or even outright cures. In reality, we don’t see these things in Phase I trials, because there are no miracle drugs, at least not yet. Because the effects of most new drugs against various tumors tends to be less than miraculous, we need Phase III trials to determine safety and efficacy.

Kevin Alexander Gray of Black Agenda Report skewers Obama as a bland, white-identified politician who’s not listening to the black community’s concerns. Obama happens to be black, but he’s not the black voters’ candidate; black voters prefer Clinton, who they’re backing by several percentage points more than whites do, while supporting Obama by no greater numbers than whites do. It could be due to unfamiliarity, but it could also be due to Obama’s failure to tap into traditional sources of black support.

Matthew Yglesias turns his attention to Iran. Scott McLemee has an entirely misguided column on Inside Higher Ed that accuses liberals of not caring about Iranian democracy. Matt Yglesias notes that he has no idea what he’s talking about. After all, American conservatives want to bomb Iran, a surefire way to cement support for the regime, while the liberals are letting the regime crumble under its own weight.

Via Ars Mathematica I found a long article in the New York Magazine about praise and self-esteem. The two-line conclusion is that praising children’s intelligence will only hurt them by making them complacent and causing them to view failures as embarrassments, while praising their effort will make them work harder. In addition, praise needs to be specific – e.g. “It’s good that you can concentrate for so long” – or else it will be perceived as disingenuous. Draw your own conclusions about education.

Carnival of Mathematics: Inaugural Edition

I’ve gotten plenty of submissions that span the entire gamut of math-blogging: education, pure math, applied math, debunking bad math – it’s all there. Only the gender distribution could be made slightly more equal (and that’s an understatement). I’m linking to the posters in roughly increasing order of mathematical difficulty, but don’t let my opinions deter you from reading the posts closer to the bottom.

First, Denise of Let’s Play Math has a long collection of quotes exhorting people to study mathematics. George Washington, Martin Gardner, Henri Poincaré, Abraham Lincoln, Galileo Galilei, Bertrand Russell – they’re all there.

Barry Leiba of Staring at Empty Pages uses modular arithmetic to solve a digit puzzle. The puzzle is based on writing any positive integer – say 8293 – and jumbling its digits and subtracting: 8293 – 3982 = 4311. Then delete a nonzero digit. Based on the resulting number, it’s possible to tell which digit you deleted.

P. Sternberg of Discreet Math links to various companies producing non-orientable material, like Möbius shoes and Klein bottle wool hats.

Heath Raftery writes about a probability paradox. Two dice are rolled, of which one is a 4. Given that, what is the probability that the sum of the two dice is a 7? Without further qualification – for example, “At least one die is a 4” – it’s obviously 1/6. And yet Heath’s professor maintained that the answer was 2/11.

Charles Daney of Science and Reason gives a brief history of algebra from the Middle Ages till the Renaissance. Whereas most accounts begin with a relatively modern mathematician, like Fermat or Euler, Charles begins far earlier, with Al-Khwarizmi, who invented the word “algebra” and after whom the word “algorithm” is named.

In the bad math department, Tyler DiPietro has a long rant fisking the arguments of Sal Cordova, an intelligent design proponent who makes dishonest arguments from information theory and computability theory in order to try refuting evolution.

JD2718 has a puzzle in plane geometry. A parallelogram is defined as a quadrilateral where two opposite pairs of sides are parallel. But there are other equivalent definitions, and other conditions that look like they define parallelograms but in fact don’t.

Science Pundit Javier Pazos tells an anecdote about John von Neumann and the bee problem. The bee problem involves two trains initially spaced 40 km apart and approaching each other at 20 km/h, and a bee that flies at 40 km/h, starting from one train, flying toward the other, turning around when it meets it, flying toward the first train, etc. How far does the bee fly until the two trains meet?

Suresh Venkatasubramanian of GeomBlog has a long primer for the game theory of author ordering. When publishing a scholarly paper, each contributor wants to be as close as possible to first author in order to get more credit; Suresh explains which ordering strategies are stable and which appear the most utilitarian.

He also notes that the theory of algorithms is undergoing fundamental changes as computing power reaches saturation. In the past, it was sufficient for researchers to say that the run time of a process is proportional to some simple function, like x^2; but now, the constant that accompanies that function is becoming increasingly important.

John Kemeny introduces spigot algorithms, which can be used to systematically generate digits of some irrational numbers, including e and pi.

Jeffrey Shalit of Recursivity writes about the prime game. The initial observation is that the decimal representation of every prime p can be reduced to one of 26 primes by scrubbing off digits. That leads to a more general investigation of the concept of subsequences.

Eric Kidd programs infinity into Haskell, in order to solve such expressions as dividing infinity by 2 or adding 1 to infinity. The trick is to teach Haskell about the finite natural numbers first, and then write into it something strong enough to code for infinite numbers.

David Eppstein of 0xDE writes about subgreedy algorithms for Egyptian fractions. An Egyptian fraction is a representation of a fraction p/q as 1/n1 + 1/n2 + … + 1/n(k). A greedy algorithm is one that tries getting the best-looking result in one step without regard for efficiency in later steps; a subgreedy algorithm is one that is almost greedy, but tends to get far better results in the long run.

He also writes about a local version of the central limit theorem, which differs from the regular theorem in that it says repeated distributions look approximately normal in a small neighborhood of 0 rather than globally; and about coloring tilings in such a way that on the one hand the colors display a regular pattern, just like the tiling, but on the other no two adjacent tiles have the same color.

And finally, Mikael Johansson has multiple great posts about algebraic topology and related subjects; choosing three was a fairly hard decision. Of the three that made it into this edition, the easiest is the Borsuk-Ulam theorem, which roughly states that any given time, there’s a pair of antipodal points on Earth with the same temperature.

Also, A_∞ for the Layman explains algebraic topology, beginning with simple definitions and ending with an overview of homological algebra. For the braver souls, there’s his post about carry bits and cohomology; group cohomology is an indispensable tool in algebra, and Mikael applies it to addition and multiplication modulo 10, or in other words carry digits.

The next edition will be posted in two weeks, on 2/23, on Good Math, Bad Math. Send submissions to Mark CC, or to me so that I’ll forward them to him.

On a final note, I should remind everyone I’m still looking for someone talented enough to make a decent logo or banner for the carnival.

Generic Issues

1. I intended to dedicate this day to Giuliani, not McCain. So here it goes: Giuliani had nothing to do with the drop in crime in New York. The blue dot is when Giuliani’s predecessor, Dinkins, assumed office. The red dot is when Giuliani assumed office.

(The data comes from here)

2. Tyler blogs about a right-wing nut who complains that Israel is recognizing gay marriages performed in other countries: “The pro-family official’s concern, he explains, is that Israel’s acceptance of same-sex marriage will give ammunition to its Islamic enemies and fuel their propaganda.” Tyler notes that this is just the Dinesh D’Souza strategy of saying that conservative values are good because the terrorists hate liberal values.

My own comment on that is that the greatest number of Palestinian terrorist attacks is on settlers, who are fairly religious, and on targets in Jerusalem, a conservative city. Attacks on liberal Tel Aviv are the most spectacular, but while half of Israel’s Jewish population lives in Tel Aviv metro, far fewer than half of Palestinian attacks are on Tel Aviv metro.

3. Hat-tip to JD2718: Ray of Education and Technology rips into the Wall Street Journal, which decided to resurrect the meme that public school teachers work 7 hours a week. Based on calculating the number of hours they spend teaching, it arrived at an hourly pay figure of $34/hour. Based on calculating the actual number of hours they work, including teaching, staff meetings, and grading, $15/hour is closer to reality.

Also due to JD2718, the NYC Department of Education’s fetish for small schools combines the worst features of small schools and large schools. Ordinary small schools have their own buildings and are self-contained enough to teach 400-500 students independently. New York’s small schools share the same building with other schools, so they have to coordinate things like bells; a better way to describe them would be large schools supervised by committee.

4. Wal-Mart’s dreadful history of discriminating against women is finally resulting in a trial. Wal-Mart isn’t even denying that the discrimination exists, but instead tries weaseling out of a class action lawsuit and says individual women should sue individual stores.

In a way this is significant beyond Wal-Mart, because the company’s ridiculous claim that it “did not have a policy of discriminating against women” can help underscore a strict liability doctrine in civil rights cases. A corporation is responsible for making sure it’s an equal opportunity employer; if it isn’t aggressively punishing managers who discriminate, it’s exposed to class action lawsuits. It’s just how states are responsible for making sure their militaries don’t murder civilians when occupying a foreign country.

5. Hamas and Fatah are negotiating in Mecca. They’ve been negotiating for a while; while their leaders are talking to each other about how to forge a unity government, their foot soldiers are killing each other as well as any civilians who happen to be at the wrong place at the wrong time.

Teaching Math by Discovery

Robert and Ellen Kaplan’s latest, Out of the Labyrinth, describes the joys of pure discovery learning as applied to mathematics. The current system emphasizes the power of aptitude and centers around a slow accumulation of knowledge that takes place over 12 years. Instead, they recommend their own Math Circle, where children who were not screened for prior knowledge or aptitude are nonetheless caused to invent irrational numbers at the age of 5.

Most of the middle of the book is theoretical and explains analogies for math learning; as is the usual for books that weave together theory and practice, the theory can be safely skipped. The important parts of the book are the parts at the beginning that explain the setup of the Math Circle, the chapters on the problems with aptitude-based teaching, and the chapters toward the end about math curricula.

Chapter Three, “The Myth of Talent,” advocates ideas interesting and enlightening enough that it could be expanded into several books. The Kaplans don’t quite claim that all people are equally talented at math, which would be just plain wrong. Instead, they show how the American educational system emphasizes aptitude too much.

Much later in the book, they write about stereotype threats, which cause women (and minorities) to perform worse on tests when they’re told that women (or minorities) perform worse than men (or white people). I’ve heard that this research has been generalized to tiered learning: students in general perform at about the level they’re expected to, so that tiering causes lower-ranked students to underperform.

For sure, the USA’s problems in math education aren’t purely the result of fourth-grade aptitude tests. Singaporeans are segregated into tiers beginning in first grade and into different schools beginning in seventh, and yet routinely get the highest TIMSS scores in the world. The difference is mainly that Singaporean students are told, “You’re a failure, so you have to study hard,” which reduces performance less than the American version, “You’re a failure, so math is too hard for you and you needn’t care about it.”

So it’s partly cultural. However, the educational system can’t change culture, so focusing on telling people that test scores say very little about mathematical talent and deemphasizing standardized tests early on will work better. The Kaplans don’t talk about that entirely, but it’s the natural conclusion one draws from reading Chapter Three.

The Math Circle is designed to counter that problem. It’s not selective, although there’s a limited element of self-selection among its students. It stimulates discussions among students, which then promote the (re)discovery of mathematical facts, such as the existence of irrational numbers. It gets students to think for themselves, thereby reaching proofs of theorems on their own. And it has no tests or graded assignments.

Once the foundation of that is laid down, the instruction proceeds extremely rapidly; a whole topic takes ten weeks to cover. Among the list of topics for children ages 14 to 18 is algebraic geometry, which is usually only taught in the first year of graduate school to students repeatedly selected for interest in and talent for math.

Although within each topic the instruction is fairly linear, in general the topics don’t really depend on earlier topics. This counters the problem in traditional education, wherein students who have a problem with one area of instruction can’t keep up afterward.

Where the book’s thesis breaks down is toward the end, where the Kaplans generalize from their overwhelmingly positive experience with the Math Circle to general curricula. Standardized curricula are designed to be “Teacher-proof,” they say, requiring teachers to teach to the national average rather than to the class. Direct instruction should be replaced with more discovery, they imply.

One of the main problems with current education is that it’s developed and evaluated by teachers, who by and large were diligent A students in school. Not surprisingly, the system works well for diligent A students. Everyone else – creatives, uninterested students, students with specialized interests, B-F students – gets shafted.

Likewise, one of the main problems with educational reformers is that by and large they’re extraordinary teachers. Attempts to use the Math Circle’s principles in a normal classroom setting routinely confound most teachers.

Michel Thomas could teach people a language in a week – or at least make them believe they knew the language after a week, considering his record of creativity with the truth. He never explained how his system works, even when UCLA contacted him in order to use his method to teach their language courses. When it was finally reverse-engineered and fitted for use in a classroom setting, it no longer performed any better than other intensive language courses.

The same principle applies to discovery learning. In science education, studies have shown that direct instruction is superior to discovery learning in teaching not only scientific facts but also experiment design. In math education, American schools are increasingly using discovery learning hand in hand with overemphasis on calculators, without any improvement in results.

In other words, the Kaplans can make students excited about mathematics when they teach by discovery. The other couple hundred thousand math teachers evidently can’t. Not surprisingly, in low-income schools, which aren’t under immense pressure from parents to be decent, teacher-proof methods increase test scores.

It’s neither obvious nor shown in the book that the Math Circle requires discovery learning to be successful. For all I know, an equally enthusiastic teacher with a different teaching philosophy could achieve the same results by proving theorems on the blackboard and only asking students to think of ways of generalizing the results or the method of proof.

Not surprisingly, the apparent rarity of teachers who can teach in ways similar to those of the Math Circle is why application to schools is so limited. In elementary school, I had a series of classes that were supposed to teach creative thinking and worked in ways that were similar to the Math Circle. But I only had access to those classes because I was in a gifted class, which got perks other classes didn’t.

It’s entirely possible sending the same teachers to more classes but at less frequent intervals, or at the same frequency but to a regular class, would’ve helped even more. But there were neither the funds nor the number of teachers required for universal coverage, so instead, the city allocated these enrichment programs to just one class.

Implementing the easy reforms, such as making math classes less calculator-dependent, is likely to only make math education less dismal. Something more fundamental is needed to make it satisfactory, let alone good. And by all means, experimenting is good, as is supporting good enrichment programs like the Math Circle. Getting children excited about what the school system turns them off of is always a positive thing. But the Math Circle’s principles are as generalizable to the school system as the USA’s victory in Korea was to Vietnam.

Dreadful Links

Just Dreadful is quickly becoming my favorite stop on the blogosphere, so it gets first few links of this edition.

Jenny expands on Dinesh D’Souza’s delineation of the difference between liberals and Islamists. In an interview on Townhall, D’Souza complains that although liberals are the “polar opposites” of Islamists, they don’t support killing large numbers of innocent people in order to destroy a regime that had nothing to do with Islamism. Jenny translates

So, one wants to impose its fundamentalist ideals on everyone, and the other wants to let people decide for themselves who to marry and when to have children.

(…)

So, even though the “cultural left” opposes fundamentalist extremism, they won’t get behind the invasion of a secular country that had nothing to do with 9-11? WTF? And as if that weren’t bad enough, they’re also against the erosion of our civil liberties, warantless wire-tapping, and other war-mongering activities!

Jessica notes that female athletes are paid horrendously little, anti-discrimination laws or no anti-discrimination laws. In the US, the situation got so bad that female soccer players went on strike. She explains,

We don’t hear much about female athletes, and if we hear anything about them at all, we only hear about tennis, golf, soccer, basketball and boxing. For the most part. Soccer is my thing, so that’s what’s up with all the soccer posts. What I say about soccer you can apply to almost any sport in which women compete (or try to compete) professionally. The US women’s national soccer team was one of the best in the world, yet becuase of a lack of funding, there is no longer a women’s major league (it used to be the WUSA). Yeah we can argue that Americans just don’t care about soccer, except for the fact that David Beckham is getting paid $1 million a week to play for the Los Angeles Galaxy.

Women always do the least appreciated jobs. It goes both ways: jobs that become predominantly female, as secretarial work did early last century, become underappreciated, and jobs that are underappreciated can become predominantly female. Globally, soccer is very popular, so it’s reserved for men, but in the US men play baseball and football. Now that Americans start caring about soccer, as seen in the Beckham situation, it’s likely American female soccer players will get shafted even more.

Jessica also rants about parents who want to genetically ensure their children are disabled. CNN quotes Slate as saying, “Old fear: designer babies. New fear: deformer babies.”

Ok, so parents want thier kid to look like them and be able to relate to them, but do they realize what the hell they are doing? As kids, they (the parents) probably had to endure teasing, feeling out of place, ect. And now they want to force that on thier child? I don’t know about you, but if I grew up constantly feeling out of place and being taunted and came to find out that my parents were responsable for all of that, I would be pretty damn pissed.

My own take on it is that disability isn’t race. It’s not something to be normalized; it’s a medical condition to be fought. Parents who overdose on identity politics and deliberately cripple their children are no different from parents who overdose on religion and chain their children to their bed until they repent.

Moving on to other blogs, Stentor quotes an ABA article on the problems ex-convicts face in the US.

There’s a nice — albeit too short — article (“Run-on Sentences”) about “collateral consequences” of being convicted (or sometimes just charged) with a crime.

By and large, people with felony convictions are banned from enlisting in the U.S. military. Fifteen states bar convicted drug offenders from recieving welfare or food stamps. In various states, people with convictions are excluded from public housing, barred from recieving educational loans, and denied driver’s licenses. In New York, for instance, a man who had learned to cut hair in prison was denied a barber’s license when he got out.

(…)

What’s more, if you’re one of those crazy people who think that the goal of the criminal justice system should be to reduce crime, these “collateral consequences” make no sense. It’s absurd to hold a strict “personality trait” theory of crime (that crime is solely the result of the perpetrator’s internal dispositions). Yet any theory that allows for situational influences would have to admit that taking away opportunities for a person to become better integrated into, and invested in, society will tend to increase crime.

Ann writes about maternity centers, where conservatives send unmarried women who get pregnant to give birth, give up the baby forever, and return to normal society.

So you know how, in the pre-Roe years, young women who found themselves with unwanted pregnancies were often sent away by their parents to deliver their babies in maternity homes?

Well, these homes still exist. And on Tuesday, three pregnant teens staged a jailbreak from the New Hope Maternity Center in rural Utah. They hit the director of the home with a frying pan, tied him up with electrical cords, and made off in a stolen van. Whoa. I know they’re “troubled teens,” and I’m not trying to justify their violent behavior, but things must have been pretty bad for them to resort to these tactics.

Jason Rosenhouse of EvolutionBlog is looking for math blogs. Many of the blogs listed appeared in my old “Where are all the math blogs?” post, but some don’t – Growth Rate n lg n, The n-Category Café, Recursivity, and Antopology, in particular.

Shnakepup, who was kind enough to whore my blog on EvolutionBlog, has a good post about Weinberg’s review of Dawkins’ book. The bone of contention is that Weinberg said scientists should be allowed to comment on philosophy and religion, and John Lynch said it was no better than theologians’ commenting on science. Says Shnakepup,

I’d like John to explain to me exactly how scientific expertise is qualitatively equal to non-scientific expertise.

In order for this to be true, one must assume that all forms of expertise are equal. Would my expertise in, say, the union attendance contracts at my job be qualitatively equal to, say, PZ’s Ph.D. in Biology? I would hope not.

Speaking of Growth Rate n lg n, Tyler assails the traditional American method of teaching math, which is very atomistic with classes specializing in algebra, geometry, trigonometry, etc.

The way mathematical concepts are communicated is fundamentally hierarchical. We teach it in a way that gives the impression that you have subjects with a successive level of difficulty. You start with arithmetic, move up to elementary algebra, then to geometry, then to more advanced algebraic concepts. Then the smarter kids who specialize in math move on to super advanced subjects like trigonometry, calculus and statistics. The kicker is this: I think this hierarchical communication of math is bad idea, and we should find a better way to do it.

The first problem with the hierarchy implicit in the method I described above is that it is entirely mythical. It is actually counter-intuitive to non-math students when I say that, overall, I find proofs in calculus and analytic geometry to be far easier than proofs in abstract algebra, which in turn I find overall a bit easier than problems in combinatorial logic (when they have any idea what the latter is). These misconceptions exist because math concepts were taught to them in a way conveyed the idea that algebra was an intermediate step between common arithmetic and advanced concepts like calculus and statistics. They would be highly surprised to learn that many of the problems in higher arithmetic are enough of a pain in the ass to make abstract algebra look like child’s play. Consider the quandary presented by Fermat’s last theorem:

Lindsay explains why she has absolutely no problem with Pelosi’s embrace of motherhood politics: politicians always project an image to help themselves get elected and stay popular.

On a gut level, I’m not crazy about the mommy schtick. Yet, as a feminist and a partisan Democrat, I’m not going to complain. As Amanda argued several week ago, Nancy Pelosi’s in-your-face parenthood seems to be reaching a lot of women who might otherwise feel alienated by Democrats.

Electoral politics is about symbolism, not syllogism. It’s like the Village People. Everyone needs a character.

Gordo writes about the United States’ gloriously anti-authoritarian Attorney General:

I wasn’t going to post anything this weekend, because my friend’s computer is difficult for me to work with, and because I don’t have much online time while I’m here in Portland, but I just have to comment on this video of Attorney General Alberto Gonzales’ testimony. In it, Gonzales says that the Constitution doesn’t guarantee the right of habeas corpus to individual Americans.

That’s the right the right to demand that the government present evidence before locking you up.

If you’re not reading Appletree, you don’t know what you’re missing. Even though Gordo’s not posting that much lately, the discussion threads there tend to be livelier than here. There’s even a very thoughtful conservative regular, Dana of Common Sense Political Thought.

The Box Problem

Zenoferox has a very hands-on way to teach his students a math problem: suppose you have a rectangular piece of paper, a*b centimeters on the side. You cut out four x*x squares at each corner, and fold the margins you get to get a box with one face missing. How do you choose x to maximize the value of the box?

For the exact answer in terms of a and b, ask Dr. Math. What I’m thinking of is more about teaching methods for maximum/minimum problems.

The creative exercises, like the box problem or the window problem (maximize the area of a window shaped like a semicircle sitting on a rectangle, given a perimeter a), are a good way of applying principles already learned. I still think the best way to teach the differentiation rule for maximum/minimum problems is to start with a rectangle of a given perimeter and ask to maximize its area. In that case there are enough alternative formulas to help create the intuition that the area is maximized when the rectangle is a square.

Education Methods

Katie has a top-notch rant about the sorry state of public education in Canada. She explains what is exactly wrong with the teaching of most subjects – for example, in math, it’s the overemphasis on computation:

I would often ask my math teachers “Why?”. …I needed to know WHY we were to simply carry something over, WHERE a screwy/random number was coming from. I had to know. I needed to know the theory behind it – this is why I failed grade 10 math (that and not showing up…) because there was never any reason as to why we were doing the math. It didn’t help that the teachers didn’t care if we really knew the math – as long as we passed and they passed their grade quota.

Everyone can “get” math. People always tell me the just don’t “get” math. I was one of those people who didn’t “get” math. But really – its a matter of finding a way to make it work for you. The way it could have worked for me was the philosophical/theoretical approach to it. Reading Alon’s advanced math books, I understand more of some of those than I do of the grade 12 calculus book. But high school takes the easy way out, they prepare kids to be able to apply math to physics, or just give them formulas.

Having read the number theory book Katie’s talking about, I can add that it’s not a particularly comprehensible one. The same thing that happened to her is happening to me to some degree: I can do computations in Riemann surfaces, but I don’t understand the deeper meaning behind, say, sections of line bundles, or anything involving smooth functions, which means I do these computations worse than I do things in algebra, which I do understand.

More to the point, in the late 1950s, when Sputnik forced Americans to come to terms with the fact that their educational system stank, the US had to undergo serious educational reforms, a lot of which were a watered down version of what Katie’s recommending, at least with respect to having more theory in math and science. New Math was successful, in the sense that it improved math education in the US; the problems with it were that it went too far with set theory.

You don’t need to teach set theory to talk about the why of mathematics. The things in math that have reasons too deep for even the current high school curriculum can be skipped, since high school students don’t really need to know that e^pi*SQRT(163) is within 1/10^12 of an integer. Talking about her old math teacher’s daughter, Katie says, “They give her 13 bunnies, and tell her to make 3 equal piles. When she is left with one she understands why… that that bunny doesn’t fit into those piles so he has to start a new family.” Ideally it should be done way before high school, so that in high school they can explain why it is that quadratics where b^2 – 4ac < 0 have no real roots, but it illustrates the idea pretty well, I think.

The same applies to science teaching. The best way to make students hate science for the rest of their lives is to put them in high school science classes, where they learn all of biology is memorizing names of phyla, all of chemistry is memorizing the periodic table, and all of physics is memorizing names of particles. In contrast, Pharyngula and Cosmic Variance don’t attract tens of thousands of lay readers per day because they give you guides to how to memorize facts. Everything is easier when you only need to explain the general ideas – PZ wouldn’t attract so many readers if he structured his science posts as a textbook (for evidence, compare my traffic to Mark CC‘s) – but still, teaching the meat and potatoes of science should still be possible while offering a greater context.

One idea I’ve seen that is so farfetched it might actually work is hands-on learning. Mark Rosenfelder explains,

There’s more: every programmer knows, I think, that you can’t learn a programming language by reading a book. The book may give you a feel for it, but you don’t know it till you’ve written programs in it.

Now, I simply think that kids are the same way, if not more so. Illinois, where I grew up, requires classes on the Constitution in 8th grade. That’s utterly absurd— there’s zero practical use for that information at that age, so almost everyone is simply going to forget it.

The apparent success of schools is an illusion, due to several factors:

• The effectiveness of schools isn’t evaluated by kids (who are usually well aware of how poorly the system works), but by teachers and parents, who’ve bought into the system.
• A few kids, of a very abstract and systematic mindset, do learn something in school.
• Teachers tend to be of the same mindset. They love to see knowledge carefully set down in schematic order, and they assume everybody else does too.
• The system creates makework— i.e. tests— to focus on. The difficult larger goal (making kids learn things permanently) is replaced by a much easier one (making them learn things temporarily), and everybody pretends that they’re the same.

You might guess that I didn’t do well in school and resented it. Not true; I did very well. But 80% of it was a waste of time. The knowledge that’s stayed with me, from languages to geography to history to writing to programming, is all stuff I learned on my own.

How would I educate kids instead? Through learning by doing. School is supposed to prepare kids to live in a technological capitalist society. So a high school could be organized as a company to develop, produce, market, and sell real products. Accounting needs mathematics; marketing needs language skills; production uses all kinds of skills. Developing a video game, for instance, would require algebra and trigonometry.

Actually, on a somewhat personal note, I have a very abstract mindset, and learned nothing in school except how to get out of phys ed classes. But anyway, the idea of learning by doing is a really good one, although for the broadest and most theoretical applications, it appeals mostly to geeks.

The best way of learning about any social science is to have to write a fantasy or science fiction series that has world creation that wasn’t pulled out of Asimov’s ass. Mark himself notes that his fantasy setting required reading tons of historical linguistics, anthropology, and geography to be constructed right. The problem is that people who aren’t immersed in geekdom usually won’t care much for that kind of activity, while the people who are usually got that way by liking the dreck that passes for speculative fiction and that almost uniformly has substandard world creation.

Making math attractive to non-geeks that way is almost impossible, I think. Most people don’t have a business mentality, and at any rate, imitating businesses is a doomed project since the real world’s most powerful feedback, profitability, is absent (the same applies to model politics, by the way). It’s easier with science, because it is possible to imitate industrial production with a Soviet-style production quota replacing profitability.

But still, there are two ideas at odds here, which could work if combined. On the one hand, teaching people the why of things, especially math and science, will make them understand the concepts more fully. And on the other, devoting significant time not to drills and word problems but to more creative pursuits will make them concentrate on what interests them the most.

One of the problems of learning by doing is that when the curriculum gives totally free choice, it leads to overspecialization. In an environment where a significant number of people will go on to higher learning, where their interests might change, it’s not a good thing. It leads to a system that forces Katie to do social science not because she wants to but because she didn’t take enough science and math in high school. Ideally, making schools work better those ways should enable compressing 16 years of learning into 12, but in practice it’ll probably be a lot less spectacular than that.

Still, in a lot of contexts, learning by doing is the best way to go. I read a while ago about a freshman composition professor who taught his students syntax by making them design their own languages, complete with their own grammars. This is where the idea of learning by conworlding comes from: presumably, making people create their own worlds, and giving them more realistic feedback than what I had when I was 10, should stir them toward knowing enough about how societies in general work.

Math Education

Abbas’s top-notch post about innumeracy in the US led Lindsay to fish an article in the Washington Post about reforming math teaching (though in a comment on Appletree, SLC beat Lindsay to the link by 24 minutes).

Maryland math leaders meet today — and D.C. math educators gather tomorrow — to discuss Curriculum Focal Points, a new document from the influential National Council of Teachers of Mathematics that could profoundly influence math instruction in the region and nationwide.

It says the typical state math curriculum runs a mile wide and an inch deep, resulting in students being introduced to too many concepts but mastering too few, and urges educators to slim down those lessons.

Some scholars say the American approach to math instruction has allowed students to fall behind those in Singapore, Japan and a dozen other nations. In most states, they say, the math curriculum has swelled into a thick catalogue of skills that students are supposed to master to attain “proficiency” under the federal No Child Left Behind mandate.

The Focal Points document can be accessed from here; the article only gives a teaser of the entire curricular reform idea. In a nutshell, it’s very much like New Math, in the sense that it orders a lot of things in the way professional mathematicians think about them rather than in the way kids do. That’s not necessarily a bad idea – New Math’s problems were in its excesses, which Focal Points has none of – but using the word “cardinality” in the section on counting for kindergarteners doesn’t inspire much confidence.

Whereas the primary goal of New Math was to introduce rigor and pure math into schools, which had taught math using only applications up to that point, the primary goal of Focal Points is to sequentialize math teaching more. One of the greatest problems of math curricula in the US is that they tend to introduce too many concepts too early – for example, they introduce probability before the kids have even learned about fractions.

At that goal, Focal Points succeeds. It simplifies the multitude of goals of state curricular standards while barely stepping over the line of oversimplifcation, and, more importantly, creates something approaching unified national standards. Since one of the problems in the US is the incoherence resulting from non-uniform standards, centralization alone would make things better – for a start, a federal Board of Education will have an easier time saying “no” to TI instead of letting its calculators take over the entire curriculum.

However, Focal Points ends up dumbing things down too much. Traditional math education had people knowing how to multiply in first or second grade and memorizing the entire multiplication table by third grade; Focal Points doesn’t even introduce multiplication and fractions until third grade, and doesn’t expect mastery until fifth or sixth grade. Nowhere does the document mention percentages, which merit at least a mention in a section on fractions or decimals. While teaching probability in first grade makes no sense, neither does waiting until eighth grade to start talking about averages and until high school to teach probability.

Powerpoint and Math

PZ has criticized Powerpoint lectures in the past for being obfuscating and substituting annoying special effects for content. Now he clarifies that Powerpoint can be used well, by linking to a lengthy primer on Archaeoastronomy about how to properly use PPT slides in lectures.

My take on it is that what can be appropriate in archeoastronomy need not be appropriate in math. Math lectures tend to proceed along a theorem-proof sequence. I’ve tried suppressing that traditional sequence as much as possible in my math posts, and still it’s glaring. Mark Chu-Carroll‘s math posts do a better job at suppressing it, but they’re more of an overall survey, which I think is too general even for seminars, which in my experience prove fewer things than lectures.

In non-mathematical subjects, blackboards are used mostly for writing down the main points, so that PPT can be a good substitute, at least when used intelligently. In math, professors fill the entire blackboard with the details of a proof or examples, then go back to the beginning of the blackboard and erase it to write more, lather, rinse, repeat.

I haven’t seen any math professor at Columbia use any technological power tool; but in Singapore, my professors often used an overhead projector with text-saturated slides, or even projected a PDF file and scrolled down to reveal entire proofs. The theorem-proof sequence apparently works better with ordinary math text. I presume that just like a good PPT presentation in biology is just a typed up version of blackboard bullet points, so is a good PDF presentation in math a typed up version of the proofs the lecturer would write on the blackboard.

At least, that’s what my experience with math is. Your mileage may vary.

Saturday Evening Links

Avedon rebuts the obnoxious argument that violations of civil liberties are acceptable on the grounds that “the Constitution is not a suicide pact.” She says,

A terrorist might be able to abuse the freedoms America holds dear (and used to at least honor in law if not always in full implementation), to hurt people, and therefore we need to give up our freedoms so that terrorists won’t have them, too. This is, allegedly, to “protect our way of life,” but since our way of life has always been based on the idea that we do have those freedoms, that doesn’t actually make any sense.

Make sure you read the comment thread, which offers even better explanations, and in particular explains where the term “not a suicide pact” comes from and why it doesn’t apply to the War on Terror.

Zeno jokes about a candidate for Oklahoma state superintendent of education, who suggested using bulky books to stop bullets. Zeno has an even better suggestion: use gigantic calculus books to build a fort in case of a university shooting.

Finally there’s a use for these overpriced, verbose monstrosities that displace perfectly readable lecture notes that have one tenth as many pages (if anyone wants to start a Boycott Stewart movement, I’m all ears).
Matt Yglesias complains that Democrats and liberals care about Max Cleland too much.

The infamous anti-Cleland ad was legitimately scummy, presenting a seriously distorted and underhanded view of the issues at hand. That said, what does Cleland’s triple-amputee status have to do with it? Saxby Chambliss wasn’t attacking Cleland’s personal bravery, he was attacking Cleland’s policies. Democrats over and over again seem to think that biographical qualities either are or out to somehow immunize nominees from political attacks based on national security issues and they keep getting burned. They need to get over it — the world doesn’t work that way and the world shouldn’t work that way. This is on a par with whining that Republicans are politicizing national security. Well, guess what, national security is a political issue. The Democratic Party is full of politicians. They need to learn to do politics — the whining just looks weak and pathetic.

I suppose that when the Democratic blogosphere is too shrill, and the liberal blogosphere either joins in the squalls or doesn’t give a damn, the best place to look for an incisive response is the centrist blogosphere. It’s certainly better than continuing to beat dead horses that nobody outside the online Beltway cares about.

Ali Eteraz has a top notch post reproducing a letter from an Iranian man whose mother was stoned for prostitution when he was 14.

My mother used to tell me that she had become a sex-worker in order to feed us and to support us. She used to command us in being real men. She used to tell us to stand on our own feet and to never lose our hope in Ali (the first imam in shiasm).

(…)

I never forget the last words of my mother’s Islamic judge:

“I issued a verdict for stoning this woman to death so that other individuals learn a lesson from her doomed fate and to avoid sins of such nature. To execute by shooting would not have made her suffer enough!”

But he warns people who would use this as an excuse to promote war. Ahmadinejad’s approval rate is lower than Bush’s. The focus should be not on external regime change in Iran, which will only give the hardliners an excuse to paint the liberals as soft on America. Rather, it should be on supporting local democratic movements that agitate for free elections, freedom of speech, and separation of mosque and church.

Over at Orcinus, Sara Robinson refutes the notion that liberals have to be nice to religious fundamentalists to appeal to them, and in particular must never make fun of religious nuts or ask hard questions.

One of the gravest errors liberals have made over the past 40 years is our ongoing failure to ask our conservative friends the hard questions about their beliefs. We wanted to be inclusive. We wanted to respect their religious views. We didn’t want to make them squirm. We were being oh so tolerant.

Well, damn it — sometimes, people who are in error should be made to squirm a little. They should be called to account for their views, and queried thoroughly on what their agenda is for the rest of us. There comes a time when politeness has to take a back seat to the larger interests of the country — and we passed that moment way back in the early Reagan years.

Despite the overall shrill tone, Sara has a good point. Brown-nosing potential Evangelical converts to liberalism won’t do anything but move the overall liberal movement to the right on critical issues. Reasoning with any kind of radical – and psychologically, Evangelicals are more like communists than like traditional conservatives – is impossible unless it proceeds from inside The Movement; but once you’re within Evangelism, you’re no longer a liberal.

Thursday Night Links

Steve Gilliard links to a Green Party nut who wants the Republicans to win elections. I can understand not caring about certain races, but wanting a Republican wingnut – not even a Chafee – to win? Steve also talks a little about how out of touch with reality the Greens are.

All these fuckers, from Nader on down, imagine themselves great, selfless leaders. I’ve met 19 year olds who were more selfless, better leaders than they could ever be. They don’t have the ability to deal in the real world. They dream, and they plot and they don’t have any realistic plan to do anything. What has been the biggest green issue this decade? Single payer health insurance? Fairly funding education? Weekend elections? You know, issues which people might actually rally around. Nope. Instant Runoff Voting. That was the issue these people thought would mobilize the masses.

First, IRV is a bad voting system. Second, Steve is of course right: the first thing a party needs to do is to sell its agenda to the public. Having a more sensible voting system than plurality will help a third party gain power, but unless it shows the people that it should gain power, it makes no sense. The Greens’ emphasis on IRV is therefore as repugnant as the Democrats’ repeated assertions that the most important thing in the world is for the Democrats to win elections.

Lindsay takes on the onslaught of conservatives who deny the Lancet’s numbers on the flimsiest of excuses. The funniest has to be Instapundit’s, which combines a WW2 comparison with false numbers, the argument from incredulity, and the usual complaint that it’s election time so there ought to be a moratorium on inconvenient facts.

Timothy Shortell wrote an article for The Advocate about the conservative attack on the academia. Most of his article is personal, concerning a politically motivated attack on him after he was elected chair of Brooklyn College’s sociology department. But he does a tremendous job at connecting the personal to the political, his own anecdote to the observation that

The pundits and bloggers of the right are targeting anyone who dares dissent. Professional hysterics such as David Horowitz see much profit in a campaign against the academy, an institution that remains mostly independent of neo-con hegemony.

The anti-academy forces have a plan: faculty autonomy is to be eliminated just as journalistic autonomy has been.

He also praises me in another post of his, using language that makes it look like he’s never heard of Lindsay, even though she’s on his blogroll.

The puritan brigade is trying to Random Bird Jill for saying that she’s a fun feminist. Designated attack dog Molly rants,

Fun feminism, apparently, is feminism without all that pesky “the personal is the political” stuff attached. It’s “choice feminism” re-framed as a brilliant and new idea that just happens to coincide with going along with the patriarchy.

The radical attacks on the concept of choice aren’t out of the ordinary. What makes the radfem attack on choice feminism special is that the term choice feminism is double-booked. In its narrow sense, it refers to a conservative movement that pretends to be feminist and is represented by the Independent Women’s Forum; the term “choice” refers to that movement’s opposition to various governmental programs like maternity leave and daycare on the grounds that having children is a choice. It’s that sense that got the term bad rap.

But just like libertarians try hijacking the word “freedom” in order to capitalize on its positive connotations, so do radical feminists try hijacking the term “choice feminism.” To people like Molly, the problem isn’t that childbearing is seen as a choice; it’s that women have a choice of engaging in radical-unapproved personal practices. The idea that women should get more choices isn’t choice feminism – it’s feminism, plain and simple; it was consensus in the feminist roundtable at Yearly Kos, and it seems to be consensus on the feminist blogosphere, minus a small puritan segment.

Katie (who I must hat-tip here) puts it best:

This isn’t some competition to see who can be the harriest and ‘most stereotypical’ feminist. This is a fight for equality, and for the recognition of woman’s rights. We’re not in this as individuals shaving, or not shaving… we’re in this as a collective. A group of people with a focus of bringing equality and empowerment to women.

Finally, via Appletree, I’ve found that the Pope is Catholic:

In some of the deadliest fighting in recent weeks, Israeli ground troops raided a village in the Gaza Strip and the air force carried out strikes today that killed 8 Palestinians and wounded about 20, according to Palestinian medical workers.

At least four militants from Hamas were among the dead, but a teenage boy and a young girl were also killed, the medical workers said.

Required Religion Courses

Hat-tip to PZ (who got it from Jim Downey): Harvard considers requiring all students to take a course about religion. While Jim strongly disagrees, and PZ snarks that it’s like a 19th-century institution requiring a course on cholera, I don’t think it’s that bad an idea. The Crimson says,

[Link] The requirement that all students take a course focusing on the interplay between reason and faith—whether in wars of religion or debates over stem cell research—is unique among Harvard’s secular peer institutions. Columbia, which requires students to read parts of the Bible and Koran in its great books program, comes closest.

The so-called “Reason and Faith” requirement emerged early in the discussions that led to the new report, said Bass Professor of English Literature Louis Menand, one of the chairs of the six-professor committee that drafted the proposals this summer.

“Religion turns out to be an enormously important phenomenon in the world, which 30 or 40 years ago we didn’t think we had to deal with,” Menand said.

While the Reason and Faith category is unlike anything that Harvard mandates today—marking a clear break from the more philosophical focus of most present-day Moral Reasoning offerings—the report notes that the course catalogue already includes many options that would fulfill the new requirement.

For example, Social Studies 98ic, “Why Americans Love God and Europeans Don’t,” and Human Evolutionary Biology 1355, “Darwin Seminar: Evolution and Religion,” would both satisfy the Reason and Faith requirement, according to the report.

I’m not sure about the merits of this specific proposal – for one, if I were designing a curriculum, I’d probably put more emphasis on comparative religion so that students have to be exposed to non-Western religions – but I don’t think it’s pernicious. I submit that in the Cold War, every educated person should have known about the foundations of both capitalism and communism and been exposed enough to the theories of both to take an intelligent stance on any capitalist/communist debate.

Realizing that religion is becoming increasingly influential in the world doesn’t equal religious brainwashing. A hardcore atheist might turn not to the Bible or the Qur’an but to Michelle Goldberg, Sam Harris, Ibn Warraq, and Irshad Manji. It’s up to modern universities to make sure students get exposed to the basic tenets of each major religion, the political movements that have sprung from it, and its contemporary status.

Learning about religion can be more subversive than PZ thinks. Part of learning about modern American Christianity is learning about the precise nature of Dominionism, which is after all an immensely influential political movement. Similarly, everyone should know the historical and political roots of Islamism and the basic difference between a Jihadi and a run-of-the-mill Islamist.

Two Personal Notes

1. The University has just remembered it’s supposed to charge me hefty rent, which means I’m $3,600 down.

2. I managed to solve the two most difficult questions in this week’s commutative algebra homework set, 15-20 minutes before the homework was due. Now I need to hope my solutions are correct…

Notes from New York

Living in this neighborhood/borough/city/country for a month and going to grad school for two weeks has already taught me some valuable things.

1. Sliced bread tastes like plastic and brings me memories of previous eons, in which I still ate at McDonald’s.

2. Everyone knows universities sometimes forget when they owe you scholarships, but the flip side of the coin is that they sometimes forget to charge you rent.

3. It’s apparently impossible to find cottage cheese with more than 4% fat around here.

4. Manhattan may be the central borough of New York, but the central borough of New York bloggers seems to be Brooklyn.

5. A pessimist would peek at my complex analysis class and say “He’s teaching undergrad-level material!”. An optimist would say “He’s teaching the bulk of the two one-semester complex analysis classes I took in undergrad, in 3 weeks.”

6. The window I write posts in can couple as a WYSIWYG web page editor for when my teaching of mathematics professor assigns homework that involves page creation (you can verify that there’s no post here with ID number 107…).

Gender Discrimination in the Academia

Hat-tip to Jessica: a researcher at the University of East Anglia has made the shocking discovery that there is gender discrimination in the academia.

[Link] Sara Connolly, of UEA’s school of economics, has undertaken research that reveals for the first time what proportion of the pay disparity is due to women being younger, more junior or employed in different types of institution or subject areas. Her preliminary results suggest that almost a quarter (23%) of the pay gap is “unexplained” and may be due to discrimination against women.

(…)

Dr Connolly also found:
· an average pay gap between men and women academics working in science, engineering and technology of £1,000, rising to more than £4,000 for professors
· women only represent 29% of permanent academic staff in the sciences (despite women being employed in increasing numbers in universities and gender equality existing among science students)
· the gender gap widens with seniority – with women accounting for just 16% of professors in the sciences.

The mere surprise of seeing these figures caused my brain to hallucinate about having seen countless studies demonstrating conclusively that the academia has serious discrimination against women.

What I’d really like to see is a study that looks at the exact breakdown of how sexism works. The last link says,

Women in computer science must overcome two sets of obstacles to success, Roberts said. Some are impediments in most scientific and technical fields:

• Overt sexism, unwanted attention and sexual harassment create hostile working conditions.
• A lack of role models for women in technical fields is discouraging. “When faculty members are looking for the next person to win a Turing Award, which is computer science’s Nobel Prize, they tend to look for people like the last ones who won such awards. This usually involves looking in the mirror,” Roberts said.
• The number of women in technical fields is not large enough to establish a resilient, supportive peer community.
• Failures in the mentoring process contribute to women’s lack of interest. Men advance academically because the faculty push them to do research and excel.
• Expectations are different for male and female students. “Many women find that they get pats on the head for merely doing well in courses,” he said. As a result, they are not adequately prepared to engage challenges at higher stages of the academic pipeline.

There’s a very well-documented trend of the proportion of women’s decreasing as seniority increases. I think there needs to be a study that looks at each of the main contributing factors and their levels of importance. The academia is getting less sexist, so women who get their Ph.D.s later suffer from less discrimination. There’re attrition due to direct sexism, attrition due to child-bearing, lack of progress due to lack of rolemodels… I once read a very detailed internal study at Stanford, which I unfortunately can’t find, and which documents the exact way inequality works, but for some reason can’t say which forms of inequality are the most critical.