"A grain of discovery"

[Dictated into a smart phone. I tried to catch all of the homonyms, but, hell, it's late and I'm fading.]

Having opened up what will probably be a long thread on Common Core, we really need to bring George Pólya into the conversation.

Pólya was a first rate mathematician and an enthusiastic and dedicated teacher. so dedicated that he spent somewhere in the neighborhood of 70 years in front of a classroom and he spent a large part of those years thinking long and hard about how we learn mathematics.

I'll go into more detail later about his conclusions. In addition to being insightful and interesting in their own right, they provide a useful alternative when discussing both the rigorously axiomatic approach of 'new math' and the reductionist 'deliberate practice' being pushed by advocates of common core.

For now though, I think it's important to start with the underlying assumptions of Pólya's approach to mathematics: Humans are by nature curious and drawn to interesting problems; therefore, if taught correctly, math should be both stimulating and enjoyable (as are the many popular games and puzzles based on mathematics).

In other words, if my kids hate math, I'm not teaching it right.

This is how Pólya opens the preface to How to Solve It:

A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime.

Thus, a teacher of mathematics has a great opportunity. lf he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking. 

Here's the entire preface, preprefaced with more of my thoughts on the subject.

Diane Ravitch provides some historical perspective on international math rankings

A perverse problem with education reform is that, on a high enough level, almost everyone agrees it's a good thing. Unfortunately, it is easy to confuse the question of "is it a good idea to improve education?" with some far murkier questions such as: 

Why is it urgent now to change education radically?

What constitutes ideal educational outcomes?

What are the most effective steps for improving education?

These are difficult questions which interact with each other in highly complex ways. Discussing them is hard enough under the best conditions but it gets much worse when advocates for certain proposals dismiss criticism by saying the people on the other side simply don't care about children (a practice that far predates the current education reform debate).

As for the question of urgency, members of the education reform movement have almost always pointed to either our decline in international tests such as PISA or our race and class based achievement gaps. The second of these is very real and very worrisome, but it often has a rather indirect relationship with reform proposals like introducing common core or greatly reducing teacher tenure.

As for the first, education historian and movement reformer turned gadfly, Diane Ravitch offers the following context.

International testing began in the mid-1960s with a test of mathematics. The First International Mathematics Study tested 13-year-olds and high-school seniors in 12 nations. American 13-year-olds scored significantly lower than students in nine other countries and ahead of students in only one. On a test given only to students currently enrolled in a math class, the U.S. students scored last, behind those in the 11 other nations. On a test given to seniors not currently enrolled in a math class, the U.S. students again scored last.

The First International Science Study was given in the late 1960s and early 1970s to 10-year-olds, 14-year-olds, and seniors. The 10-year-olds did well, scoring behind only the Japanese; the 14-year-olds were about average. Among students in the senior year of high school, Americans scored last of eleven school systems.

In the Second International Mathematics Study (1981-82), students in 15 systems were tested. The students were 13-year-olds and seniors. The younger group of U.S. students placed at or near the median on most tests. The American seniors placed at or near the bottom on almost every test. The “average Japanese students achieved higher than the top 5% of the U.S. students in college preparatory mathematics” and “the algebra achievement of our most able students (the top 1%) was lower than that of the top 1% of any other country.” (The quote is from Curtis C. McKnight and others, The Underachieving Curriculum: Assessing U.S. Mathematics from an International Perspective, pp. 17, 26-27). I summarized the international assessments from the mid-1960s to the early 1990s in a book called National Standards in American Education: A Citizen’s Guide (Brookings, 1995).

The point worth noting here is that U.S. students have never been top performers on the international tests. We are doing about the same now on PISA as we have done for the past half century.

Does it matter?

In my recent book, Reign of Error, I quote extensively from a brilliant article by Keith Baker, called “Are International Tests Worth Anything?,” which was published by Phi Delta Kappan in October 2007. Baker, who worked for many years as a researcher at the U.S. Department of Education, had the ingenious idea to investigate what happened to the 12 nations that took the First International Mathematics test in 1964. He looked at the per capita gross domestic product of those nations and found that “the higher a nation’s test score 40 years ago, the worse its economic performance on this measure of national wealth–the opposite of what the Chicken Littles raising the alarm over the poor test scores of U.S. children claimed would happen.” He found no relationship between a nation’s economic productivity and its test scores. Nor did the test scores bear any relationship to quality of life or democratic institutions. And when it came to creativity, the U.S. “clobbered the world,” with more patents per million people than any other nation.

Baker wrote that a certain level of educational achievement may be “a platform for launching national success, but once that platform is reached, other factors become more important than further gains in test scores. Indeed, once the platform is reached, it may be bad policy to pursue further gains in test scores because focusing on the scores diverts attention, effort, and resources away from other factors that are more important determinants of national success.” What has mattered most for the economic, cultural, and technological success of the U.S., he says, is a certain “spirit,” which he defines as “ambition, inquisitiveness, independence, and perhaps most important, the absence of a fixation on testing and test scores.”

Baker’s conclusion was that “standings in the league tables of international tests are worthless.”

Fairness and regulation

Dean Baker has some good points:

For the last three decades the government has pursued a wide range of policies that have had the effect of redistributing income upward. For example our trade policy, by deliberately placing manufacturing workers in direct competition with low paid workers in the developing world, has lowered the wages of large segments of the work force. By contrast, we have left in place the restrictions that protect doctors and other highly paid professionals from foreign competition, ensuring that their pay stays high.

and

The question at issue is not the amount of redistribution, the question is the direction of the redistribution. The Post seems to want readers to imagine that the upward redistribution of the last three decades was just a fact of nature, as opposed to being an outcome of government policy. That is a major distortion of reality.

This really is the heart of the argument about inequality today.  The notion that the wages earned by high status professions (medicine, finance) is a result of pure market forces and nothing to do with the regulatory environment is weak.  In the same sense, the huge returns to intellectual property have a lot to do how the laws protecting this property are implemented.  And it is not an accident that 401(k) style retirement plans are popular among companies that get to collect management fees.  This isn't to say all IP law is bad (it isn't) or all wages earned by finance gurus is unearned (also untrue). 

But it is a worthwhile starting point to realize that the web of laws and customs that we do have does benefit some groups at the expense of others.  So it is a very weak claim to argue fairness of economic outcomes in the face of differential regulatory treatment. 

Happy PISA Day, everybody!

Richard Rothstein and Martin Carnoy of the Economic Policy Institute have some strong words about how PISA data is released.

It is usual practice for research organizations (and in some cases, the government) to provide advance copies of their reports to objective journalists. That way, journalists have an opportunity to review the data and can write about them in a more informed fashion. Sometimes, journalists are permitted to share this embargoed information with diverse experts who can help the journalists understand possibly alternative interpretations.

In this case, however, the OECD and ED have instead given their PISA report to selected advocacy groups that can be counted on, for the most part, to echo official interpretations and participate as a chorus in the official release.* These are groups whose interpretation of the data has typically been aligned with that of the OECD and ED—that American schools are in decline and that international test scores portend an economic disaster for the United States, unless the school reform programs favored by the administration are followed.

The Department’s co-optation of these organizations in its official release is not an attempt to inform but rather to manipulate public opinion. Those with different interpretations of international test scores will see the reports only after the headlines have become history.

Such manipulation in the release of official government data would never be tolerated in fields where official data are taken seriously. Can you imagine the Census Bureau providing its poverty data in advance only to advocacy groups that supported the administration, and then releasing its report to the public at an event at which these advocacy groups were given slots on a program to praise the administration’s anti-poverty efforts? What if the Bureau of Labor Statistics gave its monthly unemployment report in advance to Democrats, but not to Republicans, and then invited Democratic congressional leaders to participate in the official release?

I know I've hammered this point before, but the education reform movement has been playing a very aggressive long game when it comes to lobbying and PR. Add to that a tradition of advocate research and a culture that tends to eschew firewalls and turn a blind eye to conflicts of interest, then lubricate the gears with a flood of government contracts and private grants. The result is a movement prone to all manner of problems and abuses.

The problem isn't that most of the people in the movement are uncaring and insincere; it's that they aren't. The typical movement reformer cares deeply about kids and genuinely feels that our education system is in a state of crisis. Given that mindset, it's easy to understand the decision to feed the data to groups that will give it the 'correct' interpretation in those first few days when the narrative starts to set.

But to understand the behavior is one thing. To condone it is another. Official data needs to be presented so that all sides start on an equal footing, even if that means your side may lose.

* The Alliance for Excellent Education, Achieve, ACT, America Achieves, the Asia Society, the Business Roundtable, the Council of Chief State School Officers, the College Board, the National Board for Professional Teaching Standards, and the National Center on Education and the Economy.

"Up until about a month ago, we all knew what merit pay meant."

Dana Goldstein has a good write-up on a recent education study:

In 10 cities, including Los Angeles, Miami, and Houston, researchers at Mathematica identified open positions in high-poverty schools with low test scores, where kids performed at just around the 30th percentile in both reading and math. To fill some of those positions, they selected from a special group of transfer teachers, all of whom had top 20 percent track records of improving student achievement at lower poverty schools within the districts, and had applied to earn $20,000 to switch jobs. The rest of the open positions were filled through the usual processes, in which principals select candidates from a regular applicant pool.

If a transfer teacher stayed in her new, tougher placement for two years, she’d earn the $20,000 in five installments, regardless of how well her new students performed. In public education, $20,000 is a whopping sum, far more generous than the typical merit pay bonus of a few hundred or a few thousand dollars.

In the process, a remarkable thing happened. The transfer teachers significantly outperformed control-group teachers in the elementary grades, raising student achievement by 4 to 10 percentile points—a big improvement in the world of education policy, where infinitesimal increases are often celebrated.

 but I have one big problem with the way she presents the findings.

That’s why the results of a new study, the Talent Transfer Initiative, financed by the federal government, are so important. Surprisingly, this experiment found merit pay can work.

Up until about a month ago, we all knew what merit pay meant. Under these systems of compensation, your pay would vary based on certain performance metrics from the previous period.

The initial rationale for applying this type of compensation to education was based on the assumption that teachers would work harder and do a better job if their performance (defined by their students' test scores) was tied to their pay . This played in very closely with the sub-narrative that problems in American education were largely driven by lazy, tenured teachers.

Unfortunately, when this idea was was tried in various pilot programs, it failed to show any substantial effect. It almost appeared as if teachers were already, for the most part, trying to do their best even when their pay didn't vary with performance.

This failure of concept led to a wave of revisionism from movement reformers such as Jonathan Chait. They quickly came up with a new claim. Under the revised history, rationale for merit pay had never been about incentives; push had always been about selection and retention of the best people.

There were always significant problems with this new rationale. Unless you were to assume that being a good teacher was strongly correlated with being bad at mathematics, the size of the bonuses would have to be very large to compensate for the deferment and variability of compensation. Even more troubling, the metrics proposed to evaluate teachers had been shown to be wildly unstable, thus teachers would have very little idea of what their annual take-home pay would be from year-to-year.

The flaws in that second rationale were so obvious, it almost suggested that the proponents were simply looking for a face-saving claim to make before backing away from the issue.

That face-saving quality is even more prominent in the coverage of the recent Chicago experiment in hiring bonuses for teachers. As generally presented by Goldstein and others, this seems like a partial win for both sides. Movement reformers can point to the findings and say that merit pay works while counter-reform advocates can point out that the teachers who got these great results were highly experienced and certified.

The problem with that story is that the first part of the claim is based on a complete 180-degree redefinition of merit pay. The bonus had nothing to do with metrics of success; instead it was solely contingent on those teachers serving out a specified term in the position. In other words, it used the exact same rationale that movement reformers have always objected to when applied to tenure and pay raises for seniority. (You can see Diane Ravitch making some of the same points here.)

I understand the benefits of keeping a debate civil. We should constantly make it clear that the vast majority of people on both sides of this debate share the common goal of improving education. We cannot, however, let the desire for civility become an excuse for dishonesty, particularly not when we are reporting on research that affects open policy questions.

This study suggest taking something like the Canadian approach to managing and compensating teachers. In other words, roughly the opposite of what those advocating merit pay have called for. You can question the quality of this study. You can call it impractical and question whether the results can be scaled up to a useful level. What you cannot do is claim that the data supports your position because you have changed your definitions mid-argument.

Weekend blogging -- save this one for late in the day

As a former creative writing major, I probably shouldn't admit this but I wasn't really familiar with Paul Verlaine and I had no idea that Clair de lune was inspired by his poem of the same name.

Nothing more to add in the way of comment except the recommendation that you wait till late in the evening before pushing play.




Clair de Lune by Paul Verlaine

Your soul is a chosen landscape
Where charming masqueraders and bergamaskers go
Playing the lute and dancing and almost
Sad beneath their fanciful disguises.

All sing in a minor key
Of victorious love and the opportune life,
They do not seem to believe in their happiness
And their song mingles with the moonlight,

With the still moonlight, sad and beautiful,
That sets the birds dreaming in the trees
And the fountains sobbing in ecstasy,
The tall slender fountains among marble statues.

Putting Arne Duncan's remarks in context

“It’s fascinating to me that some of the pushback is coming from, sort of, white suburban moms who — all of a sudden — their child isn’t as brilliant as they thought they were and their school isn’t quite as good as they thought they were, and that’s pretty scary,” Arne Duncan said. “You’ve bet your house and where you live and everything on, ‘My child’s going to be prepared.’ That can be a punch in the gut.”

To understand why Duncan hit such a nerve, you need to consider the long and complicated role that racial politics have played in this debate.

The public face of the education reform movement has always been pictures of eager young African-American and Hispanic children. Not only has the movement been sold as a way of helping these children but people who object to parts of the reform agenda have often been accused, implicitly or explicitly, of not wanting to help children of color. This naturally has caused some resentment by those, such as myself, who disagree with many of the proposals and who have actually taught in places like Watts and the Mississippi Delta, but there are more serious sources of tension.

For starters, with certain notable exceptions, the leaders of the reform movement tend to be white or Asian (for example, "2012 members of TFA are 62 percent white and only 13 percent African American"). By comparison, the tenured and/or unionized teachers who have paid the highest price in terms of policy changes and school closures have been disproportionately African-American. Under these circumstances, you can imagine the reaction when education reformers make statements like “I think the best thing that happened to the education system in New Orleans was Hurricane Katrina.”


(as a side note, Jay Altman is one of the best paid administrators in the city)

Even more troubling is the disconnect between the marketing and the actual focus of the reform movement. Though the defining image of the movement is of a reformer surrounded by a happy group of African-American or Hispanic elementary school students in brand-new charter school uniforms, almost none of the major reform initiatives are specifically targeted at helping these particular kids. Initiatives like Common Core and teacher accountability are being proposed for all schools. Sometimes reformers will argue that though these changes affect all students they will have their greatest impact on disadvantaged kids. Other times, they simply let their photo ops do the talking for them.

Even TFA, which was held up as the definitive program for helping kids in poor neighborhoods, is now focusing more on developing leaders and administrators and is actually providing teachers for areas like Chicago and even more notably Huntsville that have a surplus of highly qualified instructors applying for the jobs.

Perhaps people did read too much into Duncan's comments but, considering recent history, you can see how some might react badly to his suggestion that race was a factor in people's decision to criticize his proposals.

So what am I missing?

This conclusion of a piece by Megan McArdle was confusing:

Those victims should not be abandoned -- no American should be allowed to starve in retirement. But the federal government should not step in to guarantee those false promises, any more than it should attempt to re-create the vulnerable housing developments that were washed away by the storm.

The context is that of the Detroit bankruptcy, where retirees are getting 16 cents on the dollar.  There is also issues with health care that I am not clear about, but I am presuming that these people are eligible for Medicare (they are not eligible for Social Security since they did not participate in the contributions).

What I don't understand is what the path forward being proposed is?  We cannot recover the money from past administrations, and the issues with the pension are complex.  Fault is very hard to assess.  So the choices seem to be: short other creditors, default on pension obligations, or find another source of funds.  Unless the suggestion is that the state should backstop these obligations, I am unclear what the source might be? 

I am hoping that further information makes this situation seem less dire. 

"As God as my witness..." is my second favorite Thanksgiving episode line

If you watch this and you could swear you remember Johnny and Mr. Carlson discussing Pink Floyd, you're not imagining things. Hulu uses the DVD edit which cuts out almost all of the copyrighted music. .

As for my favorite line, it comes from the Buffy episode "Pangs" and it requires a bit of a set up (which is a pain because it makes it next to impossible to work into a conversation).

Buffy's luckless friend Xander had accidentally violated a native American grave yard and, in addition to freeing a vengeful spirit, was been cursed with all of the diseases Europeans brought to the Americas.

Spike: I just can't take all this mamby-pamby boo-hooing about the bloody Indians.
Willow: Uh, the preferred term is...
Spike: You won. All right? You came in and you killed them and you took their land. That's what conquering nations do. It's what Caesar did, and he's not goin' around saying, "I came, I conquered, I felt really bad about it." The history of the world is not people making friends. You had better weapons, and you massacred them. End of story.
Buffy: Well, I think the Spaniards actually did a lot of - Not that I don't like Spaniards.
Spike: Listen to you. How you gonna fight anyone with that attitude?
Willow: We don't wanna fight anyone.
Buffy: I just wanna have Thanksgiving.
Spike: Heh heh. Yeah... Good luck.
Willow: Well, if we could talk to him...
Spike: You exterminated his race. What could you possibly say that would make him feel better? It's kill or be killed here. Take your bloody pick.
Xander: Maybe it's the syphilis talking, but, some of that made sense.

Equality and Adam Smith

Bill Gardner from the Incidental Economist (in comments):

As you know, Adam Smith was a moral philosopher. You might want to ponder this quote from the Wealth of Nations:

“No society can surely be flourishing and happy, of which the far greater part of the members are poor and miserable. It is but equity, besides, that they who feed, clothe, and lodge the whole body of the people, should have such a share of the produce of their own labour as to be themselves tolerably well fed, clothed, and lodged.”

I think that this really is the piece that modern objectivists (Randians) miss.  Even in Atlas Shrugged, the hyper-capitalist society was clearly defined as being better for the average worker.  In fact, the story of the company becoming socialist suggested that socialism under-mined values like generosity. 

The notion that an equitable society is a better society is very important and yet seems to be increasingly lost in the rhetoric.  When we have a high unemployment rate, it cannot merely be laziness that prevents work and it seems that we are having trouble with the most expensive part of the formulation (lodged --it is housing that is the big expense these days). 

Now before the argument that this inequality was necessary for progress comes into being, remember that Adam Smith celebrated a nation of shopkeepers, not of corporations. 

Teacher incentives

Dana Goldstein writes about a program that gives a $20,000 bonus to teachers who transfer to "high poverty, low test score" schools.  These are all established mid-career teachers, not ivy-leaguers parachuted in to help.  They showed benefit in elementary schools but not middle school.  All of these transfers were inter-urban.

I have two thoughts.  One, huge merit pay bonuses are the precise opposite of revenue neutral.  There is no evidence that cutting everyone's pay to afford these bonuses for some teachers would be a cost-effective strategy.  Its an average causal effect applied to a population already getting an underlying rate of compensation.  So what this really has to be judged against are other interventions that would spend the same money in different ways. 

Two, the political feasibility seems low.  So long as education is funded via local taxes, it seems challenging to build support for a program to pull good teachers out of well funded schools and push them into high poverty area schools (almost certainly with lower funding).  Just how one might build the political coalition to do this (without encouraging mass abandonment of existing public schools) seems to be a open question.

You will never guess who I'm quoting...

And, God help me, she pretty much nails it:

There’s much more to the fight than simple left-right divisions. The Common Core peddlers include meddling, Fed Ed Republicans from Jeb Bush and Mike Huckabee to progressive billionaires Bill and Melinda Gates to Newscorp. media giant Rupert Murdoch and dozens of educational corporate special interests that stand to gain billions from the Common Core testing/textbook/data-mining boondoggle.

I have to give credit where credit is due, even if it costs us at least one loyal reader.

Education blogging -- Common Core, accountability and the cost of deadwood

I've started digging into the Common Core standards and one of the things that hit me was the large amount of what I would consider deadwood, topics of limited value that take up valuable class time (my favorite example is synthetic division, but there are plenty of others).

The damage caused by deadwood is not that great when teachers are allowed some leeway in deciding what to focus on, but in an age of standardized tests and fetishized accountability, teachers are forced to make difficult decisions. Math teacher Gary Rubinstein has a truly depressing example.

When teachers have to teach too many topics, they do not have time to cover them all in a deep way.  The teacher, then, has to choose which topics to cover in a meaningful way, and which to cover superficially.  It would be as if an English teacher was told to cover fifty novels with her class.  Not being able to have her classes read all fifty books, she would pick some to read fully while having her class read excerpts or even summaries of the other ones.

I got to witness an extreme example of this decision making when I graded the Geometry Regents at the centralized grading center this past June.  A huge part of Geometry, in my mind the most important part, is deductive proofs.  I’d say that over half of a ‘true’ Geometry course would involve proving different theorems.  Well, on the Geometry Regents these proofs are not a large percent of the test, less than ten points out of eighty.  So on the June Regents the last question on the test, a six point question, was the proof question and I was assigned to grade about 200 papers from a school, I won’t say which one, to grade.  As I graded I noticed that many of the students left the proof blank.  By the end of my grading I realized that out of 200 papers, all that could have received up to 6 points for the proof — a total of 1,200 potential points to have been earned on this question, I had awarded only two points total.  That’s two points out of a possible 1,200.  I asked around and the consensus was that teachers, knowing that proofs would take months to cover but be only worth less ten percent of the points on the test, would be too risky to teach.  All the time spent on this tough topic would only, at best, get the students a few extra points while they would lose all that time they could use to teach some of the easier topics that were more likely to be on the test.

Of course, we could have a long discussion on whether proofs belong in HS math classes (I tend to agree with Rubinstein on this one, at least when it comes to geometry), but it's important to realize that's not what happened here. There was no discussion. No arguments were made. No supporting data was gathered. The people who wrote the curriculum simply dodged the question of what to leave out.

When you overstuff a curriculum you guarantee that certain topics will be skimmed or skipped entirely and when you apply tremendous pressure on teachers to raise test scores, you force teachers to make the kind of choices you've seen here.

Antibiotics: an ever evolving story

I really think that the issue of antibiotic resistance gets too little play, so I am happy to see it discussed -- even if the discussion errs on the alarmist side.  In particular, I am unclear as to why we can't rotate antibiotics, why we even imagine that antibiotic soap is a good idea, and would be interested to hear a good reason for the routine use of antibiotics in raising cattle. 

The last is the strangest -- we are subsidizing meat production by not making farmers pay for the externality of antibiotic resistance.  I have no trouble with meat consumption, but it is unclear to me that it is an ideal target for subsidy given the high energy costs of that food source. 

However, it is true that we would still have options post-antibiotics.  Alcohol, heat treatments, and, surprisingly, silver remain effective despite antibiotic resistance.  Having said that, there is no reason we couldn't be doing more to make use of these techniques and rely less on the medications to which germs become resistant with time. 

Ed reform background reading -- three from Wikipedia

I know this is rich coming from a blogger but too many people are joining in on the ed reform debate without having taken the time to learn the basics (Frank Bruni being a veritable poster child), particularly when it comes to curriculum reform and Common Core. Below I picked three topics that are important and generally well known among people who have been in the weeds of the education debate but which seldom show up in the standard coverage. If you're interested in this debate, they're worth checking out.

The first is one of the biggest education reform initiatives to predate the current era. It had striking parallels to many of the current initiatives and was often supported by almost identical rhetoric but it seems to have dropped down the memory hole.


New Math

New Mathematics or New Math was a brief, dramatic change in the way mathematics was taught in American grade schools, and to a lesser extent in European countries, during the 1960s. The name is commonly given to a set of teaching practices introduced in the U.S. shortly after the Sputnik crisis in order to boost science education and mathematical skill in the population so that the perceived intellectual threat of Soviet engineers, reputedly highly skilled mathematicians, could be met.
...
Mathematicians describe interesting objects with set-builder notation. Under the stress of Russian engineering competition, American schools began to use textbooks based on set theory. For example, the process of solving an algebraic equation required a parallel account of axioms in use for equation transformation. To develop the concept of number, non-standard numeral systems were used in exercises. Binary numbers and duodecimals were new math to the students and their parents. Teachers returning from summer school could introduce students to transformation geometry. If the school had been teaching Cramer's rule for solving linear equations, then new math may include matrix multiplication to introduce linear algebra. In any case, teachers used the function concept as a thread common to the new materials.

Philosopher and mathematician W.V. Quine wrote that the "rarefied air" of Cantorian set theory was not to be associated with the New Math. According to Quine, the New Math involved merely..."the Boolean algebra of classes, hence really the simple logic of general terms."

It was stressed that these subjects should be introduced early. The idea behind this was that if the axiomatic foundations of mathematics were introduced to children, they could easily cope with the theorems of the mathematical system later.

Other topics introduced in the New Math include modular arithmetic, algebraic inequalities, matrices, symbolic logic, Boolean algebra, and abstract algebra. Most of these topics (except algebraic inequalities) have been greatly de-emphasized or eliminated in elementary school and high school since the 1960s.

The second is a widespread though perhaps fading approach to running a business. Outside of various questionable theories of incentives, it might be the most influential set of private sector ideas in the reform movement. (For a more detailed account of the relationship, check out this article by Shawn Gude)

Scientific Management

Its development began with Frederick Winslow Taylor in the 1880s and 1890s within the manufacturing industries. Its peak of influence came in the 1910s; by the 1920s, it was still influential but had begun an era of competition and syncretism with opposing or complementary ideas.

Although scientific management as a distinct theory or school of thought was obsolete by the 1930s, most of its themes are still important parts of industrial engineering and management today. These include analysis; synthesis; logic; rationality; empiricism; work ethic; efficiency and elimination of waste; standardization of best practices; disdain for tradition preserved merely for its own sake or to protect the social status of particular workers with particular skill sets; the transformation of craft production into mass production; and knowledge transfer between workers and from workers into tools, processes, and documentation. 

Scientific management's application was contingent on a high level of managerial control over employee work practices. This necessitated a higher ratio of managerial workers to laborers than previous management methods. The great difficulty in accurately differentiating any such intelligent, detail-oriented management from mere misguided micromanagement also caused interpersonal friction between workers and managers.

The third is rather specific and it's perhaps more up-and-coming than big, but it has some powerful supporters and is already having a having a major impact, particularly in mathematics education.

Deliberate practice

Psychologist K. Anders Ericsson, a professor of Psychology at Florida State University, has been a pioneer in researching deliberate practice and what it means. According to Ericsson:

"People believe that because expert performance is qualitatively different from normal performance the expert performer must be endowed with characteristics qualitatively different from those of normal adults." "We agree that expert performance is qualitatively different from normal performance and even that expert performers have characteristics and abilities that are qualitatively different from or at least outside the range of those of normal adults. However, we deny that these differences are immutable, that is, due to innate talent. Only a few exceptions, most notably height, are genetically prescribed. Instead, we argue that the differences between expert performers and normal adults reflect a life-long period of deliberate effort to improve performance in a specific domain."

One of Ericsson's core findings is that how expert one becomes at a skill has more to do with how one practices than with merely performing a skill a large number of times. An expert breaks down the skills that are required to be expert and focuses on improving those skill chunks during practice or day-to-day activities, often paired with immediate coaching feedback. Another important feature of deliberate practice lies in continually practising a skill at more challenging levels with the intention of mastering it.[4] Deliberate practice is also discussed in the books, "Talent is Overrated," by Geoff Colvin,[5] and "The Talent Code," by Daniel Coyle,[6] among others.