The End of American Hegemony?Bye-Bye, Copenhagen and Obama; Hello, Beijing!

Much of the discussion of the collapse of the Copenhagen Climate Summit has missed the real story: the collapse of the climate conference symbolizes a significant turning point in world history, not simply because of the issue of global warming, but rather because it represents the collapse of American leadership in the world.

I pointed out shortly before the conference began that the Chinese had clearly determined beforehand to reject the proposals of the United States and its allies (“a major offensive on rich nations at the Copenhagen conference on climate change,” to quote the exact words used by The Times of India ) and that the Chinese had put together an international coalition able and willing to advance their goals.

The course of the Conference was indeed controlled by the Chinese: the UK Prime Minister whined that the Western powers were “held to ransom by only a handful of countries” led by China.

And, the President of the United States was treated with contempt.

In a report by NBC News environmental correspondent Anne Thomspon on the final day of the Conference, December 18, 2009, President Obama referred to the Chinese proposal by saying, “That doesn’t make sense. It would be a hollow victory.”

Changhua Wu, China Director for the Climate Group, brusquely dismissed Obama’s statement by sharply declaring, “It’s not only an attack; it’s humiliating to a certain level.”

The full extent of the humiliation suffered by the American President was described by Ms. Thompson:

For thirteen hours, the President went from room to room, meeting with various world leaders, trying to figure out what they could agree on. Frustrations reached a crescendo when, waiting to meet with the Chinese Premier and Brazil’s leader, Obama found out they were already meeting and walked in uninvited.

The President of the United States was left waiting by the Chinese because they were meeting with the leadership of Brazil. And he managed to get in to the meeting only by barging in “uninvited.”

Thompson summed up the President’s experience by declaring:

The final agreement, which even the most optimistic environmental groups called insufficient, left the President clearly exhausted and dejected.

Exhausted and dejected indeed.

What does this have to do with homeschooling?

As I pointed out recently, we Westerners view and teach history as if the ascendancy of the West during the last several centuries was natural and inevitable.

The perspective from much of Asia is rather different: the period of Western ascendancy seems a peculiar but brief historical anomaly that is ending now that Asia is reasserting its natural supremacy.

The real issue here is not partisan politics, Barack Obama’s personal political skills, or even global climate. What we observed in Copenhagen is a reassertion of Asia’s sense of its natural superiority over the West, and the collapse of the United States’ position as the natural, and generally acknowledged, leader of the world.

Twenty years from now, the whole Copenhagen fiasco, possibly the whole issue of “global warming,” will be a distant memory, rather like earlier environmentalist frauds such as the supposedly disastrous “population explosion”: try reading the 1968 best-seller The Population Bomb by Stanford Professor Paul Ehrlich and his wife, and have a laugh over how ludicrously wrong their prophecies turned out to be (I myself actually read Ehrlich’s book a year after it was published at the urge of my high-school history teacher, so I can remember how serious people once took Ehrlich seriously).

However, while the Copenhagen conference may strike our grandchildren as ancient history, the collapse of American preeminence and the rise of Asia will strike our grandchildren as very real. At some level, this is no doubt inevitable: the physical size of Asia and the fact that the majority of the human race lives in Asia means that Asia was always destined to be of key historical importance.

Of course, the foolishly irresponsible financial, educational, political, and environmental decisions pursued by the American political and cultural elite make the American decline even more certain and, potentially, catastrophic.

But, however one views the decline of the United States of America and of the West in general, it is of historic importance, and it is clearly illuminated by the collapse of US influence at Copenhagen.

The news of the last few days makes a very good history lesson for us homeschoolers to discuss with our kids.

Climate Scientist Allegedly Threatens New York Times Reporter with the 'Big Cutoff'

Allegedly, Michael Schlesinger of the University of Illinois recently e-mailed the following message to New York Times science reporter Andy Revkin:

Andy:
Copenhagen prostitutes?
Climate prostitutes?
Shame on you for this gutter reportage.
This is the second time this week I have written you thereon, the first about giving space in your blog to the Pielkes.
The vibe that I am getting from here, there and everywhere is that your reportage is very worrisome to most climate scientists.
Of course, your blog is your blog.
But, I sense that you are about to experience the 'Big Cutoff' from those of us who believe we can no longer trust you, me included.
Copenhagen prostitutes?
Unbelievable and unacceptable.
What are you doing and why?
Michael

This, of course, is bizarre on several levels – the Copenhagen prostitute story is simply a silly story about Copenhagen politics that should not have worried Schlesinger or any serious scientist.

More importantly, this is apparently the second time in one week that Schlesinger tried to tell Mr. Revkin how to do his job as a reporter.

But, the most significant point is the threat to punish Revkin with the “Big Cutoff,” presumably denying him access to “superstars” like Mike Schlesinger.

Except… Mike Schlesinger is no superstar – I’d never even heard of him before this incident (although he does indeed seem to be senior faculty at U. of Illinois).

Being cut off from access to Mike Schlesinger is not exactly a catastrophe for the New York Times!

I’m sorely tempted to believe that this is just some bizarre practical joke… except that it fits in with this leaked e-mail from the CRU Team (note the “p.s.”):

At 17:07 27/10/2009, Michael Mann wrote:

Hi Phil,
Thanks--we know that. The point is simply that if we want to talk about about a meaningful "2009" anomaly, every additional month that is available from which to calculate an annual mean makes the number more credible. We already have this for GISTEMP, but have been awaiting HadCRU tobe able to do a more decisive update of the status of the disingenuous "globe is cooling" contrarian talking point,
mike
p.s. be a bit careful about what information you send to Andy and what emails you copy him in on. He's not as predictable as we'd like [emphasis added]

In fact, Revkin has not been noticeably hostile to the climate fraud folks. Note the complaint: he’s “not as predictable as we'd like.” Exactly what level of predictability did they expect from Mr. Revkin?

I'd planned on writing a post here about the serious issues of scientific method involved in the global-climate fraud and the need to discuss these issues with our kids: the fundamental problem with the global-climate modeling is that real science makes predictions that the scientists accept as make-or-break tests of their models. The global-science-modelers, on the contrary, treat failed predictions as simply an excuse for further tweaking of their models, not as proof that their science is simply and provably wrong.

They are not real scientists.

However, rather than going into the needed details on that issue, I felt compelled to address this current, truly bizarre behavior from Professor Mike Schlesinger.

Mike, let me give you some advice: it is not cool to threaten a reporter from The New York Times with the "Big Cutoff" in an e-mail since, given the current situation, that e-mail is likely to be published all across the Web.

Mike, you're nearly sixty-seven years old. Maybe, it is time to step aside in favor of a younger person who has some understanding of how the modern world works: you know, modern things like e-mail, the Web, The New York Times?

You're only making this whole scandal worse, much, much worse.

And, someone, please tell me that this is really some weird prank, and that senior faculty at the University of Illinois, once one of the country's great public universities, are not really this stupid!

Note added: A few minutes before posting this, I watched Anderson Cooper on CNN discuss the scandal. Representing the climate establishment was Bill Nye the Science Guy! Nye is not a scientist -- he has a Bachelor's in engineering and has been a successful children's entertainer, a younger version of "PeeWee Herman." Nye made a fool of himself. Was CNN really unable to find an actual Ph.D. scientist to represent the climate establishment? The critic of the climate establishment was Pat Michaels, a serious scientist whose book I recommended in a previous post. Perhaps by choosing Nye to represent the establishment view, CNN is trying to send the subtle message that the climate-alarmist view should only be taken seriously by children. The whole story just gets weirder and weirder -- Bonfire of the Vanities meets Waiting for Godot.

(For my earlier comments on Climategate, see here, here and here. Here are my comments a couple months before Climategate became public, in which I pointed out that those of us who are scientifically competent had known for some time that there was something seriously rotten within the media-governmental-scientific global-warming establishment.)

MIT Meteorology Professor on Climategate

Those of us scientists (and there are a lot of us) who have tried for years to warn of the scientific misconduct that has been occurring among global-climate modelers have often been dismissed on the grounds that we have not done work in the climate field ourselves or that we do not hold positions at sufficiently prestigious universities.

It was therefore with some glee that I saw that, on the same day that I initially published my own views here on the Climategate scandal, Richard Lindzen, the Alfred P. Sloan Professor of Meteorology at MIT (and a member of the National Academy of Sciences), published an op-ed in the Wall Street Journal on Climategate.

What makes me rather gleeful is that Professor Lindzen’s views so nicely confirm my own conclusions.

Lindzen’s key point is:

At this point, few scientists would argue that the science is settled. In particular, the question remains as to whether water vapor and clouds have positive or negative feedbacks.

Exactly. As I have said again and again, it is true that CO₂ produced by humans has made the globe at least slightly warmer than it otherwise would have been. But how much warmer? Will it be enough to be a real problem?

The only honest answer is that we do not know. Contrary to the lies in the mainstream media, the magnitude of global warming is not “settled science.”

Lindzen’s other key point is:

The answer brings us to a scandal that is, in my opinion, considerably greater than that implied in the hacked emails from the Climate Research Unit (though perhaps not as bad as their destruction of raw data): namely the suggestion that the very existence of warming or of the greenhouse effect is tantamount to catastrophe.

Yes, if the CRU crew did intentionally destroy their raw data to keep it out of the hands of their scientific critics, that is serious scientific misconduct (and perhaps a crime). But, the real scandal in the field of global climate change is the repeated claims by some prominent climate scientists, repeated incessantly by scientific illiterates in the American news media, that it is “settled science” that “the greenhouse effect is tantamount to catastrophe.”

The CRU gang’s misconduct is comparatively minor. The real fraud is the claim of settled scientific results that do not in fact exist, and the attempt to use that fraudulent claim to impose controls on the global economy that may not be at all necessary.

Let me reiterate what I have said before. Yes, the CO₂ we have dumped into the atmosphere will almost certainly make the world at least a bit warmer than it otherwise would have been. Yes, this might be a big problem.

But, it might not be a problem at all. It might even be beneficial if, perchance, we are entering a natural cooling period.

We just don’t know.

Lindzen also has a brilliant and fascinating, though much longer, paper, well worth reading, that discusses in more detail the science at issue here as well as the underlying sociological, political, and economic motives that have caused some climate scientists to engage in scientific fraud, aided and abetted by many politicians and by the mainstream news media. Again, I am in nearly complete agreement with Lindzen’s paper: to be specific, his points about how academic science can suffer severe dysfunctions as the result of government funding matches my own personal observations.

In particular, he makes a crucially important point about some scientists' discovering that dishonest fear-mongering was the way to gain funding and advance their careers:

It is my impression that by the end of the 60’s scientists, themselves, came to feel that the real basis for support was not gratitude (and the associated trust that support would bring further benefit) but fear: fear of the Soviet Union, fear of cancer, etc. Many will conclude that this was merely an awakening of a naive scientific community to reality, and they may well be right. However, between the perceptions of gratitude and fear as the basis for support lies a world of difference in incentive structure. If one thinks the basis is gratitude, then one obviously will respond by contributions that will elicit more gratitude. The perpetuation of fear, on the other hand, militates against solving problems.

Yes. If your job depends on convincing people that global warming is a massive threat to civilization, then you certainly do not want to publicize the conclusion that it may not be that much of a problem after all, now do you?

Note that this paper is dated a year before the Climategate scandal broke; yet, the paper explains in detail the underlying problems that have led to Climategate.

Let me make clear that I reached my own conclusions on all this, before I had ever heard of Professor Lindzen, based on my own personal knowledge of math, physics, and computer modeling, and my own observations as to how academic scientists behave. (Indeed, I posted here back in September, a couple months before the current scandal broke, a warning about the scientific and media misconduct in the area of global warming.)

But, since both Professor’s Lindzen’s knowledge and his prestige in this area are so much greater than my own, I am very gratified to see his essays confirming my own conclusions.

Let me also make clear that, of course, I am not claiming that Professor Lindzen is right on every single detail concerning global climate change: he would not claim that himself. But, he is a credible guy who has disagreed for a long time with the now-discredited “consensus.” That alone should have been enough to keep people from saying that the old fake “consensus” view was “settled science.” If serious people with excellent credentials, such as Lindzen, had serious, detailed scientific objections to the fraudulent “consensus” then the science was not at all “settled.”

Furthermore, as a serious and active climate scientist himself, Professor Lindzen has had a ringside seat to the long chain of misconduct, abuse, and chicanery, which many of us strongly suspected and which has now been publicly revealed thanks to the CRU whistleblower.

From now on, when the issue of global warming comes up, I plan to start by saying that I agree with the Alfred P. Sloan Professor of Meteorology at MIT.

Try it with your friends – that is, if you actually know anyone who still believes in the lies about global-warming.

(For my earlier comments on Climategate, see here and here. Here are my comments a couple months before Climategate became public, in which I pointed out that those of us who are scientifically competent had known for some time that there was something seriously rotten within the media-governmental-scientific global-warming establishment.)

The New “Classical Carnival of Homeschooling.”

Back in October, Ritsumei began a “Classical Carnival of Homeschooling.” I'm happy to say that my post on homeschooling world history and how we divide history into periods was mentioned in the first edition of the carnival (this is a belated announcement due to my vacation and illness).

The newest edition just became available today.

I hope the new carnival will be complementary to, not competitive with, the long established “Carnival of Homeschooling.” The CoH deals with homeschoolers of all varieties; I think it also does make sense to have a Carnival that centers on those of us who are focused on an academically classical approach to homeschooling.

I have provided cute little links somewhere over on the right of this page to both carnivals, and I hope to contribute form time to time to each of them.

And, Ritsumei, thanks for your efforts in setting up the new Carnival!

The Collapse of the Global-Warming Fraud?

The Times of India reports that the rising Asian powers have decided to reject the attempts by the United States and Europe to stifle the global economy in the name of the global-warming fraud:

In an unprecedented move, India on Saturday joined China and two other developing countries to prepare for a major offensive on rich nations at the Copenhagen conference on climate change next month.

The four countries, which include Brazil and South Africa, agreed to a strategy that involves jointly walking out of the conference if the developed nations try to force their own terms on the developing world, Jairam Ramesh, the Indian minister for environment and forests (independent charge), said.

“We will not exit in isolation. We will co-ordinate our exit if any of our non-negotiable terms is violated. Our entry and exit will be collective,” Ramesh told reporters in Beijing…
...
The developing nations will also not accept any pressure from developed countries to establish legally binding emission targets at Copenhagen.

No “legally binding emission targets” means, of course, that they will wait and see if global warming turns out to be a real problem (and, yes, that is possible) or if it turns out to be of little consequence (and, yes, that too is possible) before agreeing to any action of any real substance.

Until there is real scientific evidence – all we have now are the results of deeply flawed computer models – that is the only sensible approach.

The times they are a’changin’ – who’d have thought, a few decades ago, that India and China would be teaching the West the virtues of common sense and the dangers of over-regulation of market economics?

I also recommend a wonderfully fair and balanced discussion of Climategate on the Freakonomics blog: Steve Dubner points out that, quite aside from all the sound and fury over the whistleblower's publishhing of the CRU e-mails, “the central scientific issue here” is:

that the most prominent climate scientists’ computerized models may be neither as robust nor as predictive as many people think…

He goes on to quote from my fellow physicists Nathan Myhrvold (of Microsoft fame) and Lowell Wood, explaining why the computer models should not be trusted.

At The Atlantic, Clive Crook explains why he is more outraged now that he has waded through the leaked CRU e-mails than he expected to be:

The stink of intellectual corruption is overpowering. And, as Christopher Booker argues, this scandal is not at the margins of the politicised IPCC [Intergovernmental Panel on Climate Change] process. It is not tangential to the policy prescriptions emanating from what David Henderson called the environmental policy milieu [subscription required]. It goes to the core of that process.
...
I'm also surprised by the IPCC's response. Amid the self-justification, I had hoped for a word of apology, or even of censure. (George Monbiot called for Phil Jones to resign, for crying out loud.) At any rate I had expected no more than ordinary evasion. The declaration from Rajendra Pachauri that the emails confirm all is as it should be is stunning. Science at its best. Science as it should be. Good lord. This is pure George Orwell. And these guys call the other side "deniers".

Perhaps what Crook had particularly in mind was CRU head Phil Jones' allegedly declaring:

The two MMs have been after the CRU station data for years. If they ever hear there is a Freedom of Information Act now in the UK, I think I’ll delete the file rather than send to anyone...

The “two MMs” are mathematician Steve McIntyre and economist Ross McKitrick, two Canadians who have been trying to uncover the details of the global-warming fraud for the last several years

Now, reports indicate that the original raw data, absolutely vital to judging the scientific validity of CRU's global-climate work, have been either "lost' or, possibly, intentionally destroyed.

Obviously, the e-mail from Jones raises the possibility that he carried out his plan and actually did intentionally delete the vital information, perhaps to cover up scientific malfeasance.

If anyone ends up going to jail for all this, the destruction of data to illegally evade "freedom of information" requests may be the reason.

Anyone who wants to get their hands dirty actually digging through the leaked documents might start at the blog Shadow of the Olive Tree, which has kindly posted the infamous "Harry Read Me" file for everyone's enlightenment.

Let me emphasize once again that of course the CO₂ we have dumped into the atmosphere will almost certainly make the world at least a bit warmer than it otherwise would have been. And, yes, this might be a big problem.

But, it might not be a problem at all. It might even be beneficial if, perchance, we are entering a natural cooling period.

We just don’t know.

The fraud comes not from those who claim that global warming might happen and that it might be a problem. The fraud is from the handful of scientists, and the large number of scientific illiterates in the mass media, who keep saying that it is “settled science” that global warming will be huge and hugely damaging.

That is not settled science. The fraud of global warming consists of the false claim that global warming will be a major problem when neither the empirical data nor the deeply flawed computer models are yet able to indicate how large global warming will actually be.

In a way, the CRU gang have become the fall guys for a much larger scandal: yes, we know from the published e-mails that the CRU guys played nasty little unprofessional games to silence their critics, that they were more concerned with protecting their turf than with advancing science, and that they are incredibly poor computer programmers.

But the real scandal is the larger group of climate modelers around the world who have falsely claimed to know how big a problem global warming will be when they do not really know at all but are simply relying on very dicey computer models.

Over my career, I have been involved with numerous computer simulations, ranging from elementary-particle physics detectors to satellite-communication systems. No responsible scientist fully trusts such simulations until they have been well validated by experimental data.

The global-warming simulations are more speculative, less embedded in accepted science, than the simulations I have worked on. Yet, the global-warming simulations have not been validated by making detailed, unambiguous predictions and then rigorously checking those predictions against reality.

This is not science: it is pseudo-science.

The real fraud in the area of global warming is the covering up of this fact by the mass media, by the political establishment, and by so many climate scientists themselves.

(See also my September post, published before the current scandal broke, explaining why the global-warming scam is fraudulent, and my previous post on the Climategate scandal.)

Global Warming Revealed As Fraud?A Chance for Teaching How Science Really Works

Reportedly, a hacker has gotten into the e-mails of a major research institute on global warming and publicly released e-mails that show that the whole global warming thing is just a scientific fraud.

What’s it all mean? Well… first of all, if this is legit, it does not really prove outright fraud, at least not from what I have seen so far.

What it does show is scientific politicking, scientific infighting, attempts to outflank one's scientific opponents, etc. And, the most recent news does suggest carelessness bordering on scientific incompetence (the researchers have, it seems, lost their original data, which means that other scientists cannot check their calculations but merely must accept them on faith. “Faith” is not a good basis for science.). A couple of scientists from within the climate-change establishment have had the honesty to concede that “Climategate” does show that there are real problems with the attitudes and procedures followed by the pro-global-warming camp.

Contrary to what many members of the news media and the public seem to think, all this is not, alas, behavior that is actually that unusual among scientists (who are, after all, simply human beings). Most scientists are not coolly objective, Vulcan-like pursuers of abstract truth: I speak from rather painful personal observations here. On the contrary, most scientists have fairly large egos, are eager to defend their own pet theories, often bear grudges against their scientific opponents, etc. A few (not the majority, happily) are outright liars. And, truth be told, a significant number of scientists are incompetents.

Does this mean that the theory of evolution is just a fairy tale pushed by scientists with an anti-religious agenda, that the theory of relativity is just a left-wing conspiracy imposed by those who shared Einstein’s political views, etc.?

Well… no. Over the long run, things balance out. In fact, they balance out largely because of the combative nature of scientists. As an adolescent, I myself thought I had found an error in Einstein’s theory of relativity. Alas, all I had actually found was a lack of clarity in the particular book I had been studying.

But… I would still love to achieve everlasting fame by showing Einstein was wrong, by showing that quantum mechanics is mistaken, etc. Most scientists have similar desires. The eagerness of the young whipper-snappers to prove that their elders got it all wrong does, almost always, correct scientific errors in the long run.

In the long run – that is the key. A century from now, the global-warming issue should all be nicely sorted out: present personal and political animosities will have been long since forgotten, and good science will have triumphed.

In the long run – but not yet.

I’ve been following the issue of global climate modeling since the late ‘60s. I’m not a climate modeler myself, but I do understand the underlying physics, the basic issues involved in computer modeling, etc.

And, I can testify that we are still in the “shake-out” period, when politicking, personal feuding, and the sheer difficulty of the scientific research make it impossible to really see what the final result will be.

Let me be clear: I am nearly certain that the earth is warmer than it otherwise would have been because of the massive amount of CO₂ we humans have dumped into the atmosphere.

But… exactly how much warmer? Is it even possible that the globe is entering a natural cooling period and that we need the anthropogenic CO₂ simply to maintain a stable climate?

No one really knows.

So, don’t take the current “scandal” as a sign that global warming is a monstrous conspiracy tied to the Illuminati, the Bilderbergers, etc.

But do take it as a sign that scientists are human beings, that it takes a long time to get the right answer to complicated questions, and that the real issue concerning global warming – how big a problem will it really be for human beings? – is still unanswered.

And, discuss the whole thing with your kids – this is an excellent chance to see how science is really done, to see all the uncertainties, personal conflicts, and grand debates involved in the process of scientific research, and, ultimately, over the next few decades, to see how everything will be sorted out in the end.

If you want intelligent, informed, non-political introductions to some of the scientific difficulties involved in climate change modeling, try reading Patrick Michaels' book, published earlier this year, Climate of Extremes and S. Fred Singer's book, published in an updated edition last year, Unstoppable Global Warming: Every 1,500 Years.

Please note: I am not suggesting that Michaels or Singer has all the right answers; I am merely suggesting that they show, contrary to what the scientific illiterates who anchor the network news claim, that the scientific debate is not yet over. The biggest problem in the global-climate debate is the extraordinary ignorance combined with unspeakable arrogance of the mainstream media on this subject – and I am including “conservative” pundits as well as “liberals.” The American people need to learn to stop listening to blowhards – whether Rush Limbaugh or Al Gore – who have no idea what they are talking about (cf. Gore's recent comment about the temperature of the earth's core).

Also see my post from early September discussing the fact that there apparently are simple technological fixes to global warming if it does turn out to be a problem, technological fixes almost never mentioned by the mainstream media.

UPDATE: In the comments, silvermine posted a link to a site with some interesting insights see the November 21-30, 2009 posts specifically.

A Thank-You Note to the Real Heroes of History

Well… I’m back. We left on a lengthy (almost three-week long) vacation to Hawaii in mid-October. The day before we returned, I came down with a bacterial infection (in three different areas of my body – no, the doctors don’t know how that happened): I spent several days in the hospital on an IV drip. Fortunately, I’m now pretty much recovered.

Anyway… when I get a chance I’ll blog about homeschooling on a Hawaii vacation – there are lots of things to do in Hawaii, many quite educational, besides swimming and snorkeling (yes, we did also swim and snorkel). And, I have a lot of other topics, from teaching complex numbers to kids to a possible scandal concerning global warming to the idea of natural rights that I want to talk about. I'll also try to reply to some of the comments I missed during my absence. So, back to my regularly scheduled blogging…

Oh, and I am really grateful to the people who invented antibiotics in the middle of the twentieth century. We tend to forget that infections that are now an inconvenient nuisance once used to kill people in very, very large numbers.

The real heroes of history? Those who invented anesthesia (imagine a root canal, much less abdominal surgery, without anesthetics!) and the folks who discovered antibiotics.

Math Interlude:Homeschool Math by Rotating Wheat Thins Boxes

I have a theory that if you cannot explain an idea in some form to a bright, attentive six-year-old, then that may be a sign that you do not really get the idea yourself.

So, here is a very simple math exploration that can be done even with a homeschooled six-year-old and that in fact connects to some quite advanced mathematics.

3-D Rotations Need Not Commute:

Get two identical boxes – we used a couple of Wheat Thins boxes.

Put both boxes on the table or floor in front of you facing towards you.

Now, the idea is to perform two rotations on the boxes, but in different orders.

First, take the box on the left and rotate it a quarter turn counter-clockwise towards yourself (i.e., 90 degrees counter-clockwise around the vertical axis): call this rotation Z.

Now, take the box to your right and rotate it a quarter turn so that the front face ends up face down on the floor (i.e., 90 degrees around an axis going from left to right): call this rotation X.

Now, let’s perform rotation X on the left box: i.e., rotate it so that the face which is now vertical and facing towards you is rotated forward and down onto the floor.

Finally perform rotation Z on the right box: i.e., rotate it counter-clockwise around the vertical axis a quarter turn.

In math we usually write transformations like this in reverse order: i.e., the first one performed in time ends up on being written on the right.

So, the left box ends up as X * Z * Box.

The right box ends up as Z * X * Box.

(By the way, the reason for the reverse order is that it seems natural, at least to mathematicians, to put the operation that operates first on the box to the immediate left of the word “Box.” Why does the word “Box” have to go on the right? It doesn’t, of course, but it is usually done that way.)

You’ll see that the boxes end up in very different positions.

In short, X * Z * Box is not equal to Z * X * Box.

So what?

Well… first, this is pretty weird. I would have thought they would end up the same! That such a simple geometry experiment gives unexpected results is rather a surprise.

Second, this invites various other experiments. What if we rotate by half-turns instead of quarter turns? What if we let X be a quarter turn and Z a half-turn. (By the way, I chose “X” and “Z” because the axes we are rotating around are what are usually called the “x-axis” and the “z-axis,” but I did not need to use those particular letters.)

Third, kids nowadays are expected to learn the “commutative laws” of addition and multiplication in early grade school. It tends to be hard for kids to see why these are really a big deal: how could things not commute!

Well, rotating Wheat Thins boxes by quarter turns is something even young children can do, and yet these operations do not commute. Commutativity can fail in fairly simple ways.

Finally, this ultimately connects with some quite advanced math, that is of interest both in pure mathematics and in applied fields ranging from computer graphics and robotics to elementary-particle physics.

Rotations are normally represented by matrices, but they can also be represented by “quaternions,” invented by the nineteenth-century mathematician William Rowan Hamilton: the fact that rotations can fail to commute is therefore a sign that matrices and quaternions will also have to exhibit this kind of non-commutativity.

Hamilton’s invention of quaternions (and their generalization to “octonions”) is an interesting story all by itself, and it connects to another simple math demonstration: the fact that you can rotate a teacup (with tea in it) by two full turns, holding it rigidly in your hand, without spilling a drop and without dislocating your shoulder (this is known variously as the “Philippine Wine Glass trick,” the “plate trick,” etc., but it is not magic, but a simple fact of mathematics).

More broadly, the group of rotations in three-dimensional space is what is knows as a “Lie group” (after the nineteenth-century mathematician Sophus Lie), and most Lie groups have this same property, i.e., that most members of the group fail to commute.

In physics, this failure to commute is one of the most important differences between the strong nuclear force and the electromagnetic force: the electromagnetic force is due to a commutative Lie group, the strong nuclear force to a non-commutative Lie group.

In short, there is a whole lot of math and science hidden behind a couple of Wheat Thins boxes!

So, what does all this have to do with homeschooling?

Well, this is about as simple a homeschool project as you can get in terms of necessary equipment and preparation time.

But, more than that, it illustrates a central point I am trying to make in this blog: ideas that are usually considered very advanced and complex in math, science, etc. can actually be introduced at a very early age.

Young kids cannot of course understand everything (indeed, neither can adults), but they can understand at least a bit about most things.

More than that, nobody can grasp complex ideas in one huge gulp: the idea in American schools – whether public schools or universities – that you can grasp algebra or calculus (or Lie groups) in just one nine-month period is a horrible mistake.

(In fact, I myself recently learned something about Lie groups – a simple proof of a theorem called the Baker-Campbell-Hausdorff theorem, which shows how the violations of the commutative law are almost the only thing that really makes Lie groups complicated. If not for the violation of the commutative laws, Lie groups would turn out to be rather like the surface of doughnuts – hyper-tori, as mathematicians say.)

This belief in teaching subjects in one huge gulp is connected to the “developmentalist” fallacy: i.e., the belief that kids are not ready to learn anything about many subjects until they reach a certain “developmental” level, and then, all of a sudden, the whole huge subject can be shoved down their throats.

Human beings do not learn that way.

One of the greatest advantages of homeschooling is that we can dump this dogma of “developmental appropriateness.”

We can talk to our kids about black holes, or have them see that rotations do not commute, in first grade. They can read about knights and castles, pharaohs and mummies, fossils and plate tectonics, early in grade school.

They will not grasp everything, but they will grasp much more than the dogmatic disciples of “developmental correctness” claim they can grasp.

So, get a couple of Wheat Thins boxes (or Cheerios boxes, or whatever you have in the pantry) and show your kids how simply rotating simple objects is much stranger than it looks.

And, tell them that understanding this strangeness is not only useful in robotics and computer graphics but that it also helps explain what holds protons and neutrons together inside the nuclei of atoms.

Slicing Up History When We Homeschool: Beyond Ancient/Medieval/Modern

The traditional approach to dividing up human history is:

• Ancient 3000 BC - 500 AD (3500 years)
• Medieval 500 AD - 1500 AD (1000 years)
• Modern 1500 AD - 2000 AD (500 years)

The disparities in the lengths of time encompassed by each of these periods is striking: the Ancient Period is approximately seven times as long as the Modern Period!

It is natural that we are more interested in what happened in 1700 AD than in what happened in 1700 BC. But to make the Ancient Period seven times as long as the Modern Period risks badly biasing our understanding of the past: surely human life and thought must have changed rather significantly between 3000 BC and 500 AD.

The elapsed time between the building of the Pyramids and the death of Caesar is greater than the time from Caesar to ourselves. To lump Caesar in with the Pyramids as "ancient" is deceptive.

Of course, the origin of this framework goes back to early modern times, and is really simply a division among:

• Greeks and Romans
• The Feudal Period
• The Renaissance and Post-Renaissance Age

I.e., this way of dividing human history comes from Early Modern Westerners who were largely ignorant both of history prior to the Greeks (early Egypt, Mesopotamia, etc), and of non-Western history (China, India, etc.).

A more balanced division of history is:

• Archaic Period 3000 BC - 1000 BC (2000 years)
• Ancient Period 1000 BC - 500 AD (1500 years)
• Period of Barbarian Conquests 500 AD - 2000 AD (1500 years)

It is not simply that this division gives periods that are more equal in duration; this division also helps us think more clearly about long-term historical processes.

When we talk about the “Bronze Age” vs. the “Iron Age,” we are already implicitly recognizing some such division: the Bronze Age world of the Pharaoh Khufu or the Great King Sargon of Akkad was a very different world from the Iron Age world of Qin Shi Huang Di and Julius Caesar, of Confucius, Buddha, Socrates, and Jesus.

The Archaic Period could be described as the period of “temple-states,” where the gods were largely the affair of the rulers and their dependent priests. During the Ancient Period, that system was replaced by wide-ranging thought and speculations about the nature of human life and reality. The Greek philosophers, the competing schools of thought in late Zhou China, the creation of the Upanishads and Buddhism in India, the creation of Judaism and then Christianity in Palestine – none of this has much precedent during the Archaic Period. (Because the central religious and philosophical perspectives that are still widespread today arose during the Ancient Period, the central part of this period has been labeled the “Axial Age” by the German philosopher Karl Jaspers.)

Belief systems during the Archaic Period tended to focus on the relationship between the gods and the entire community as mediated through the rulers and the priests. The transition during the Ancient Period to systems of thought and belief that focused on the individual and his relationship to the gods and to reality is a dramatic shift in human thought.

It's not that all of these revolutionary thinkers – from China to Greece – had similar thoughts. Quite the contrary! It is rather the diversity of thought in the Ancient Period that is so startling. Is the goal of life to be part of a society structured like a family (Confucianism)? Or is the goal of life to escape the cycle of rebirth through extinction of the self (Buddhism)? Or should we strive to live up to our potential as rational beings (Aristotle)? Or must we atone for our innate sinfulness by accepting the ultimate sacrifice made by Christ?

These are incommensurable goals for human life.

Why the “Period of Barbarian Conquests”?

To Westerners, the big news of 500 AD to 10000 AD is the collapse of the Roman Empire in the West, the loss of classical culture, and the Dark Ages.

But, from the perspective of most of the Old World civilizations, the fate of the Western borderlands was of limited interest. The real news of those years was the explosive expansion of Islam due to Arab barbarians who conquered the original heartlands of human civilization in the Near East and beyond.

Similarly, the real news of 1000 AD to 1500 AD was the repeated irruptions of the Turkish and Mongol peoples out of Central Asia, overpowering, at one time or another, almost all of the civilized areas of the Old World, except of course the Western borderlands of Europe.

And, after 1500 AD, the conquest of the rest of the world by the West was, from the perspective of most of the civilized world, simply one more wave of barbarian conquests.

One of the key characteristics of the Period of Barbarian Conquests is the dominance of much of the world by the two religious systems descended from ancient Judaism: the sister religions of Islam and Christianity.

Christians and Muslims may see each other as the ultimate heretics, but from the view of much of the world, and from the perspective of pre-Christian pagan culture, the two monotheistic religions are remarkably similar, not only in their beliefs but also in their relentlessly expansionist, missionary, imperialist activities.

I think the most important aspect of the general perspective I am proposing here is that it sheds some light on where we stand today.

First, is the Age of Barbarian Conquests now at its end, and are China and India re-asserting their position as dominant civilizations?

Second, will Christianity and/or Islam dominate human systems of belief in the future or is a new pattern of belief arising?

Of course, it is not possible to answer those questions with certainty.

But it is clear that the rise of science during the last five hundred years is radically changing the old games of civilization: physics is the same in Beijing as in London; chemistry is the same in New Delhi as in New York. Is it possible that the wild and lush diversity of systems of thought created during the Ancient Period are now being replaced by a single unified system of belief – natural science?

How does all this affect homeschooling?

I know of no textbook that does an adequate job of restructuring the narrative of human history along the lines I am suggesting here. Some high-school and college texts try to present a more balanced view of world cultures that moves beyond the narrow Greece-Rome/feudalism/Modern-West framework, but I know of none that adopt the broader framework I have sketched out here.

But we homeschoolers do not need to use a single textbook. There are a number of excellent books, at an upper-grade-school through high-school level that discuss separate civilizations and cultures. Homeschoolers can selectively use those books to create a broad curriculum that explores the pattern of human history that I have laid out.

Time-Life published two very nice series, The Emergence of Man and Lost Civilizations, that include a number of useful volumes: e.g., we have used The First Farmers and The First Cities in the former series and Early Europe: Mysteries in Stone in the latter series. Lost Civilizations, incidentally, is not about silliness such as Atlantis, but rather focuses on archaeological discoveries relating to ancient humans. Other Time-Life series, such as Time-Frame and The Great Ages of Man are also worth checking out. (All of these are long out of print, but generally available through public libraries and used book sources on-line.) It is important to cover the prehistoric agricultural revolution and the urban revolution: the two books I mentioned from The Emergence of Man series are helpful in doing that.

Lucent Books, in its World History Series, has a number of well-written books at a middle-school level: we have used, for example, Don Nardo's Ancient Mesopotamia.

Dorling-Kindersley's beautifully produced Voyages Through Time series, all written by Peter Ackroyd, is much better written then most DK books, but of course still has the wonderful illustrations that DK is famous for. We have gone through most of the volumes in this series: these books are our kids' favorites among all the history/social-studies books we have used up till now in our homeschooling.

At a beginning grade-school level, I recommend Anne Millard's and Particia Vanags' Usborne History of the World (this is the “white” book, not the “Internet-linked” book), which uses cartoons to give a nice, brief overview of world history up to 1900. The book is admirably neutral – not pro-Christian nor anti-Christian, not “politically correct” nor hiding past atrocities, but just a nice description of the broad course of world history at an early to mid-grade school reading level.

None of these books explicitly lays out the picture of history I have described here. But they do provide detailed and interesting factual narratives that can serve as the basis for a broad view of human history.

It is then the homeschooling parent's job to present the broad picture of history over the last five millennia, and explain how all of these historical details fit into that broad picture.

How we slice up history, how we categorize the past, can of course never tell us why past events occurred as they did, much less predict the future. In the end, one needs not simply a broad framework but also knowledge of detailed historical facts.

But, thinking about the periods of the past more clearly, and fitting all of the details of history into a broader narrative, can help us ask better questions not only about the past but also about the future.

Why Taking Classes is a Lousy Way to Learn

I suppose that most homeschoolers get questions of the sort “But where will she take algebra class?’ or “Won’t he at least take the normal classes in high school?”

Most American adults’ own experience has ingrained in them the idea that “taking classes” is the natural way to learn “academic” material.

From a “human engineering” viewpoint, I find this bizarre.

Suppose that you were starting from scratch to design a system that would do the best possible job of teaching serious, challenging material to children (or, indeed, adults).

On the face of it, would the best way to do this be to have one teacher simultaneously servicing a very large number of students? Or would it be better to have a single tutor dealing one-on-one with each student throughout much of the day, meaning of course that the tutor could only have a handful of students?

Would it be best to toss a couple dozen students together and have them move at the same pace despite their varying abilities? Or would it be best to let a student move at his own pace, depending on his innate abilities and the ease or difficulty he is currently having with the material?

Is it best to learn a subject from a single textbook expressing the style, perspective, and knowledge of a single author? Or is it better to read several books on a subject to see different perspectives and approaches?

Do most subjects naturally fit neatly into a three-month or nine-month time frame (i.e., a school quarter or school year)? Or should the natural, logical structure of the subject itself, and the needs of the student, determine the time-frame during which the subject is covered?

Those questions readily answer themselves.

A normal classroom situation, with one teacher presenting the material to dozens of students at once, with the students moving in lockstep together through the material, in a time-frame determined not by the logic of the subject but by the constraints of the school-year, is quite obviously not the optimal way to learn.

For most subjects, there are really only two reasons for using traditional classes as the framework for learning.

First, treating children as if they were interchangeable parts, subject to a factory model, is easy and cheap for adults. One teacher for thirty students is a lot cheaper than one teacher for two or three students. And, if a teacher has thirty students, it is certainly convenient for her to pretend that all those students can proceed at the same pace, that they should read from a single textbook, etc.

The second reason is “socialization”: all of us who went through traditional schools know that the traditional schools certainly do not “socialize” all students to be kind, trustworthy, or tolerant. But traditional schools do serve to accustom all students to the idea that the way society works is like an assembly line.

To put it bluntly, taking traditional classes trains you to accept the hassles of dealing with the DMV (Department of Motor Vehicles), the IRS, etc.

Are there no situations where group learning makes sense?

There are a few activities that are inherently group activities: I don’t see how you can learn to play football without being part of a team or to play a symphony without being part of an orchestra. Also, there may be situations where equipment is so expensive (e.g., lab equipment) that it has to be shared among a group of students.

So, yes, there are times when something like a classroom situation might make sense. But, these are rarer than one might think, simply because the disadvantages of group learning are so great.

For example, three years ago, our kids took a weekly Chinese class: I had been teaching them Chinese, and, since I am not fluent in Chinese, it seemed obvious that the class would work better.

It didn’t. Even though the other kids came from homes where Chinese was spoken in the home, and even though our kids were the youngest in the class, our kids read Chinese characters better than any of the other students. The teacher was a native Chinese speaker, but almost all of the speech one actually heard in the class came not from the teacher but from the other students, who spoke Chinese even worse than I do. Our kids’ learning of Chinese slowed to a crawl.

So, even in this case, where there were some obvious shortcomings to homeschooling, the disadvantages of group learning proved so great that homeschooling still turned out to be better.

Let me make clear that I am not arguing for an education that is lacking in structure or planning. You are not likely to master French or calculus simply by random, casual reading (although random, casual reading might be the initial impetus that got you interested in French or calculus). Nor are you likely to master French or calculus by having good intentions to learn those subjects “someday,” without any planned time-frame (though the process of actually learning the subjects may cause you to revise the time-frame that you originally laid out).

Nor am I suggesting that kids (or adults) should simply kick back and forget about serious learning once they are free of “classroom discipline.”

On the contrary, my point is that serious learning means not slipping in to some pre-fabricated one-size-fits-all classroom situation but rather proactively working out a means to teach each individual student in the most efficient, thorough way one can find for that individual student.

Getting an “A” from the teacher in a classroom is an easy way for a student (and her parents) to be convinced she has achieved something.

Too easy.

The real question should be: what have you really learned? What skill or knowledge do you now possess that you did not have before you studied all this?

Does this mean all kids should be homeschooled?

If taken literally, “homeschooled” is perhaps too narrow a term. Kids can be and should be, to some degree, “library-schooled.” Almost all adults are, to some degree, “on-the-job schooled.” And, you learn theater by being “on-the-stage schooled,” sports by being “on-the-sports-field schooled,” etc. If finances allow, it may make sense to hire a tutor outside the home (an obvious example would be a piano teacher) to tutor a “homeschooled” child one-on-one in a particular subject.

So, no, I am not saying that it is optimal for all education to be within the walls of the home, carried out solely by the parent.

But, if “homeschooling” means schooling primarily under the family’s control and not carried out within a traditional classroom format, yes, in that sense, it would be a good thing if all children were “homeschooled.”

A classroom environment is not an effective means of using a child’s time and energy to enable him or her to develop his intellectual capability to the fullest.

We place very little value on children’s time. Although we know that the period from birth to their early twenties is, for most people, the last period of their life when they will have the time and energy to devote much of their effort to learning, adults do not generally care whether children are learning efficiently or whether the time and energy they put into schooling is largely wasted.

They’re “only” kids.

That is why the majority of American children grow up so woefully uneducated. We adults are willing to waste their childhood, their best opportunity to become educated human beings. Why should the kids themselves value learning when we adults seem to care so little about whether they are provided with an optimal framework for learning?

The traditional classroom approach for educating children is certainly convenient for adults. But it is almost never the optimal way for a child to learn.

How We Homeschool: Some Nitty-Gritty Details

A friend who is seriously considering homeschooling recently asked how we actually do our homeschooling, on a day-by-day basis.

And, of course, I had trouble giving a coherent answer: like so many homeschoolers, we do not clearly demarcate our homeschooling from dance and piano practice, from our vacations (we do try to make our vacations “educational,” but not in the sense of studying textbooks), from going swimming at a friend’s house (isn’t that “physical education”?), etc.

But I’ll try here to be a bit more coherent for the sake of our friend and anyone else interested in the general approach we try to take to our homeschooling.

We homeschool seven days a week, twelve months a year.

But that does not mean we actually do homeschooling all 365 days of the year.

The kids officially get their birthday and Christmas off. For obvious practical reasons, we also do not do “official” homeschooling (textbooks, workbooks, etc.) when we are out of town on vacation (I tried a few times taking textbooks with us on vacations – the only result was to strengthen my arms carrying the heavier luggage).

And, naturally, in the course of our lives, things happen.

If we spend a couple hours one day at a friend’s house or at a musical or theatrical performance, we can still get in a fair amount of schoolwork. But, if we spend the whole day at a friend’s, or go to some day-long event such as the state fair, that day is pretty much dead for “official” schooling.

So, realistically, I estimate we do about 240 days a year of “official” homeschooling.

That is still about thirty percent more than the number of schooldays for traditional public-school students.

Except… traditional students do often have homework on weekends. All of our schoolwork is “homework,” so if you include public-school students’ “homework days,” there may not be a dramatic difference here.

In short, I think we may have more “official” schooldays than traditional public-school students, but not dramatically more.

The big advantage we do have is flexibility in our schedule: we can go to Hawaii in October, Legoland in March, etc., without worrying about officially missing school.

How many hours a day do we do school?

Well… I consider piano practice (each of our kids practices between half an hour and a hour and a half a day), dance and piano class, and going to the library to be part of school. But, kids in traditional schools do those things too, and they are not counted as “school.”

So… excluding such activities, I estimate our kids average from six to eight hours a day on “official” schooling (more on some days, less on others, depending on the other activities that are occurring on that day). That may seem to be a bit more than kids in traditional schools; however, kids in traditional schools have homework, not counted as part of their schoolday, and, as I said, our “homework” is part of our schoolwork (in fact, by definition, homework is all of our schoolwork).

So, our kids might spend slightly more time each day on schoolwork than traditionally schooled kids, but not dramatically more.

Did I mention that our official homeschool time includes time for daydreaming, staring out the window, and squabbling with each other?

No. I didn’t mention that to the kids either, but these are kids – it happens. (They do have “official” permission to go to the window and watch an interesting new bird when one happens by.)

I do try to keep them relatively focused, especially when we are working together, but my main concern is that they are actually getting significant work done over the course of the day – I know they will not be focused every single minute.

While we do have times of the day that are officially “schooltime,” this varies somewhat from day to day, depending on our schedule. The kids have had a pretty broad choice from the beginning as to exactly when they do each particular subject, how many pages they complete in a particular workbook or reading book each day, etc. Within reason, they choose for themselves where to do their work: they're not confined to the kitchen table, the desk in their bedroom, or whatever.

I make sure they are not simply skipping a subject day after day, and, if it becomes clear that their progress has slowed to a snail’s pace in some subject, I encourage them to focus more on that subject.

What do they do the rest of the day when not “officially” doing schoolwork?

They have an hour or so after they get up to do “serious” unassigned activities – recreational reading, writing fantasy stories on the computer (which they love to do), writing computer programs (they are just beginning this), etc.

Since they are not rushing off to school in the morning, they can usually get as many hours of sleep as they need and can also eat a leisurely breakfast (sometimes a bit too leisurely!). They have various non-academic activities that they have chosen to commit to, such as piano and dance class.

And, of course, they play and goof off.

Their television viewing tends to be limited to the news and, occasionally, science or nature specials or the Food Network (how can any homeschooler not love Alton Brown?). We have no video games, and computer gaming and Web browsing are severely limited.

I hope this does not sound utopian (or dystopian!): I think they have a fairly normal childhood, except, like most human children throughout history, they learn at home, rather than at some official site outside the home with dozens of other children who are the same age.

Is there any way then in which this differs dramatically from traditional schooling, except that, since we are at home, I know in detail how they are doing in their schoolwork?

Yes, our “curriculum” differs rather dramatically from traditional schools’.

Since they started kindergarten, we have not yet used a single traditional textbook designed primarily for the American public schools. We may bend a bit on this as they reach high-school level (I like the BSCS biology “Blue Book,” for example). But, by and large, textbooks designed for the American public-schools are high in meaningless glitz and color and very low in interest and content: they encourage mindlessness and short attention spans and breed boredom.

I’m not real keen on workbooks, but I find them sometimes necessary – I am not willing to spend my time making up a bunch of math word problems or grammar exercises. But, I try to be selective about workbooks, and try to find ones that will encourage thinking rather than simply repetitive drill.

In math, we’re using Singapore Math and Life of Fred and trying out the “Art of Problem Solving” series. Our main “language-arts” book has been the Editor in Chief workbook series, and we will soon start sentence diagramming. They are also working through Julie MacIntosh Johnson’s Basics of Keyboard Theory workbooks.

I try to get them to write a couple book reports a month, with my helping them to correct organization, grammar, punctuation, etc. (they are on their own for the first draft). And, they write fiction stories (basically to amuse each other) outside of my supervision.

We are seriously trying to learn Chinese and dabbling in some other languages – I’m not sure if they will attain fluency in anything except Chinese, but a bit of knowledge of Latin and Ancient Greek (starting with Karen Mohs’ grade-school workbook series, Hey, Andrew! Teach Me Some Greek! and Latin’s Not So Tough!) is both fun and, I think, worthwhile. I’ve written a computer program to help drill them on language vocabulary (it’s basically an automated flashcard program).

The biggest respect in which our curriculum differs from traditional schools is that we put a heavy emphasis on serious science and history.

I consider science and history to be the real core of our curriculum: science is the sum of the knowledge we have of the natural world; history is the sum of the knowledge we have of the human past.

That covers pretty much everything.

As I said above, we are not using any US public-school textbooks in those areas: science textbooks below the high-school level are often factually wrong. Even at the high-school level, many are disasters (check out the reviews from the Textbook League ). And history texts for US public schools tend to be utterly boring and bloodless: how they manage to transmute the reality of history – heroes and villains, nobility and murder most foul – into stunningly unappetizing pabulum is a great mystery.

So, our science and history texts are generally books we get from the library, published by non-textbook publishers: Dorling-Kindersley, Usborne, Lucent, Benchmark/Cavendish, Rosen Publishing Group, etc. – much more interesting and much more accurate than public-school texts.

My own direct interaction with the kids is focused on whiteboard work on math (I teach them stuff not in their books or material that they have not yet reached in their books), on working together on Chinese, and, most importantly, on science and history.

We try to spend ten to fifteen hours a week together reading science or history books out loud or discussing what the kids have read on their own.

My own total time spent directly interacting with the kids therefore tends to range from fifteen to twenty hours a week – not counting my time planning things, choosing books, reminding them to do their work instead of staring off into space, etc.

Our friend asked me how carefully I plan our long-term schedule.

I’d say it is a “conceptual plan”: I have in mind (often in hand) books I know I want us to get to in the next year or so; I know what topics I want to teach them in math that are not in their books; etc. Our plan is subject to revision on the fly if we find a new and better book, if we find some other topic that deserves some real study, or if we find that we are going faster or slower than anticipated.

Our kids have been fairly consistently testing at twice their grade level in the “three Rs” (according to tests administered by a local school district, not by me). That gives me some flexibility: I don't really need to worry about their performing at “grade level” – I can focus simply on what is worth learning and on what makes logical sense in terms of what they have already learned.

E.g., we have never formally done spelling, but, as advanced readers who learned phonics, they test well on spelling.

Perhaps the single most important goal of our curricular approach and of my planning is to be radically “developmentally inappropriate” – to steal E. D. Hirsch’s phrase.

Learning should be rewarding, but it is hard work. It is made unnecessarily hard if basic concepts are kept hidden until, all at once, they are unveiled and the student is expected to grasp them immediately.

No one can fully grasp algebra or calculus, chemistry or relativity, world history or the history of life, in one huge gulp swallowed over one nine-month period during one school year. So, I started explaining evolution in kindergarten, black holes in first grade, calculus in fourth grade, etc.

No, they did not fully “get” those ideas at those ages. But, then, few adults fully “get” those ideas either.

But my kids did start being “eased in” to those ideas at an early age. As the years roll past, we re-visit all of that, they understand more and more, and, by the time they are ready for college, I am confident that they will have a real mastery of calculus, world history, etc.

Most of us adults know a lot about pop music and entertainment, the historical events that transpired during our own lifetime, etc. Yet, we did not study all that in one huge marathon effort: we simply lived through it.

I am trying to see to it that science and history are subjects that my kids have “lived through” since kindergarten, so that they can no more forget who Oliver Cromwell or Erwin Schrödinger or Felix Mendelssohn was than most Americans could forget who Oprah or Michael Jordan or O J Simpson is.

Yeah, quarks, genes, and the curvature of spacetime are more complicated than basketball, rock music, and the Oprah show. All the more reason to start learning about them from an early age!

I’ve given a pretty exhaustive description here, and, yet, I still have left out a lot of details as to textbooks, math, etc. I hope eventually to post much of that in my “Math Interludes” on this blog, in lists of all the books we have used on my central (and currently empty) Website, etc. I hope other homeschoolers will do the same.

Anyway, I think many homeschoolers are following a course not that different from ours. On the whole, for over half a decade so far, it has been fun and rewarding. Of course, we do have our days… but the storms come and pass.

Again, I think the most important point that is distinctive about our approach is the emphasis on teaching significant content about science and history as early and as fully as possible. This would be very hard in the public schools because of the “urge to test.” Someday, I suppose, my kids may have to take a test on black holes, but they did not need to take a quiz after we first discussed black holes in first grade. They could focus on grasping the idea and slowly improving their understanding of the subject, and I could probe, through direct personal interaction, their understanding of the subject and help them correct misconceptions.

And, for that reason, if and when they do take a test in college on black holes (and evolution and world history and molecular biology and all the other things that we started casually discussing early in grade school), I am confident that they will be very well prepared.

Besides, black holes are fun.

Math Interlude*: Lagrange Interpolation as Self-Checking Algebra Practice

My kids started algebra last year, and, while they seem to get the basic concepts, they need practice – practice using the distributive law correctly to simplify algebraic expressions, practice dealing correctly with all those negative signs, etc.

Of course, I could just give them a huge number of polynomials to multiply, but, aside from being boring, that would require me to work out the answers myself in order to check their answers!

I’ve found a simple alternative that goes back to the great eighteenth-century mathematician Lagrange. “Lagrange interpolation” is actually interesting and useful in itself (although even well-educated technical people seem often to be ignorant of it nowadays), it happens to require quite a lot of multiplying of polynomials, checking of signs, etc. so that it is good algebra practice, and, best of all, it is automatically self-checking.

Here in a nutshell is how it works:

You are given a table of values for the variables x and y, and you want to find a polynomial that gives exactly the correct values of y when you plug in the values for x.

Suppose, for example, you are given the following values:

x y
1 1
2 4
3 9
5 9

and you want to find a polynomial

y = a x³ + b x² + c x + d

that goes through those points.

There are various ways to solve this problem -- for example, you can use linear algebra if you view (a,b,c,d) as a vector in a four-dimensional space.

The method published by Lagrange uses a much simpler idea.

What we do is find four separate polynomial, each of which vanishes at all but one of the values of x.

For example, expression A:

(x - 2) * (x - 3) * (x - 5)

obviously vanishes when x is 2, 3, or 5, but obviously does not vanish when x is 1.

What is the value of expression A when x equals 1? Well, just plug 1 in for x and you find that the value is -8.

Now, when x is 1, according to our table, we need y to have a value of 1, not -8. So, we will just divide expression A by -8 and multiply it by 1, getting expression B:

1 * (x - 2) * (x - 3) * (x - 5) / (-8)

If you do the same thing for the case where x is 2, you get expression C:

4 * (x - 1) * (x - 3) * (x - 5) / (3)

Run the same trick for x equal to 3, and you get expression D:

9 * (x - 1) * (x - 2) * (x - 5) / (-4)

Finally, for x equals 5, you get expression E:

9 * (x - 1) * (x - 2) * (x - 3) / (24)

Now, expression B gives the right value for y when x is 1, and, it is created so that it will vanish at the other three values of x, so it will not mess up the values of y there. Similarly, expression C is created so that it gives the right value for y when x is 2, and it is zero at the other three values of x.

So, if we simplify expressions B, C, D, and E by multiplying each one out, and then add them all together, combining like terms, we will get a polynomial that gives the right values of y for each of the four values of x.

I don't need practice on algebra, so I had my kids do this. Their answer is expression F:

y = (-2/3) x³ + 5 x² + (-22/3) x + 4

How do they (and I) know that they did the algebra right?

Simple – they plugged into expression F the values 1, 2, 3, and 5 for x. They should find that the values of y will then be 1, 4, 9, and 9 as planned. If they do not get the right values of y, they need to find their algebra error!

Note that Lagrange interpolation always works: you can choose any real numbers, positive, negative or zero, integral or fractional, for x and for y. (You can even use complex numbers if you wish.)

In particular, the values of x do not have to be evenly space: in my example, I “skipped” 4, and it worked fine. Nor do the values of y have to be in any pattern: 1, 4, and 9 in my example seemed to be starting a pattern, but I wrecked that pattern by using 9 twice instead of using the “obvious” choice of 16.

The values of x do have to all be different (although, you can play some interesting tricks by letting two values of x get “infinitely close” – basically, you can then control the slope at x as well as the value of y).

You can in fact prove that this is the only polynomial of degree three or lower that goes through our four points. (In general, if you have n points, with different values of x, there is always a unique polynomial of degree n-1 or lower that goes through those points.)

The more points you use, the more complicated the algebra gets. I'd start with only two or three points for someone who is just learning algebra.

Isn't this too hard for kids in first-year algebra? No, my kids have learned it without too much trouble: there is no division of polynomials here, no quadratic formula, no trig functions, etc. This really is just first-year algebra.

But, isn't it complicated?

A little.

But this is the kind of complication that you get in real math applied to real problems. No one in real life (not even in science or engineering) ever faces the problem of multiplying ( x - 2 ) times ( x- 3) just for the fun of it. And, it is very, very rare that anyone ever faces the familiar textbook sort of algebra problem: “A train leaves Albuquerque going towards Santa Fe at 70 mph and a train leaves Santa Fe...” or “Jane had five times as many dolls as Ginger, but after Jane got two more dolls....”

Algebra is an abstract science; traditional algebra is about understanding the abstract properties of the four basic arithmetic operations: what does and can happen when you use the operations of addition, subtraction, multiplication and division in a systematic way? (Modern university algebra is about the properties of more general systems of mathematical operations that can operate on elements very different from ordinary real numbers.)

Mathematics is not really about balancing your checkbook or calculating the amount of tile you need to re-tile the kitchen – we have electronic calculators to do that for us.

Mathematics is about the possible abstract structures that can logically exist.

Those structures are often based on arithmetic and geometry, so you do need to know traditional math to understand modern mathematics.

But really learning math means trying not just to learn to get the right answer but actually exploring the universe of mathematics much as traditional explorers explored newly-discovered continents.

Lagrange interpolation is a very simple example of such exploration. Actually graphing the polynomials you get through Lagrange interpolation can also be enlightening: while the method always works, it can give some pretty “snaky” curves if you fit more than three points (for three points, the resulting curve is much nicer).

Is it too complicated? Well, someone unwilling to tackle topics such as Lagrange interpolation is not really learning math. If you want to know what math is really about, rather than just working you way through the watered-down, over-simplified picture of mathematics portrayed in American public-school textbooks, you need to try to wrap you mind around ideas such as Lagrange interpolation.

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* From time to time, I plan on posting a “Math Interlude,” in which I’ll try to explain some significant idea in math not known to most educated American adults, that I have in fact taught to my own kids in grade school or middle school, and that a bright middle-school student should be able to grasp.

Socialization and the Carnival of Homeschooling

The current Carnival of Homeschooling links to an insightful (and truly hilarious!) post on the homeschooling socializtion issue.

It is mentioned in the same paragraph as my recent post on a Chinese-American perspective on the issue of whether homeschooled kids are not independent enough. I had thought my post was pretty decent, but the post from The Learning Curve wins out. Check it out.

Are Homeschooled Kids Too Dependent? Chinese vs. American Perspectives

The question of whether homeschooled children are too “dependent” on their parents is related to the oft-encountered “socialization” question, but the “dependency” question seems to get less attention among homeschoolers.

Because I married into a Chinese family (my parents-in-law were born in mainland China), I try to think about such issues from a “multicultural” perspective: how does our American idea of children’s “independence” compare to Chinese ideas of “independence”?

Chuansheng Chen, a professor at the University of California at Irvine, has done research in the area of Chinese vs. Western adolescence, and has noted:

Feldman and Rosenthal (1991) found that U.S. and Australian adolescents had earlier expectations for autonomy than did Hong Kong adolescents. The largest cultural differences were found for behaviors that would fall into the category of misconduct (e.g., smoking and drinking alcohol) and those related to peers (e.g., "attending boy-girl parties," "dating," and "preferring to do things with friends than with family")…

For example, peer factors play a less important role in Chinese adolescents' misconduct than in American adolescents' misconduct because Chinese adolescents spend less time with their peers (Chen et al., 1998).

Does the fact that Chinese have “less expectations of autonomy” than Americans imply that Chinese kids are more “dependent” than American kids?

My wife has explained to me that it is more subtle than that.

Consider which of the following sorts of children should be considered truly dependent:

• Those kids who spend a lot of time with their parents and whose parents work hard to instill mature values and an understanding of the consequences of one’s decisions, so that, when the children finally become adults, they can independently make intelligent and informed decisions

or

• Those kids who lack an adult’s perspective on values and consequences and are therefore, in reality, heavily dependent on the ill-formed judgments of their adolescent peers.

The American concept of a child’s independence tends to be that the independent child makes decisions on his own, even though those decisions may show no regard for his parents' values or judgments or what the parents have tried to teach him.

The Chinese concept of independence is that a child shows independence when he has properly internalized the parents’ teaching and is able to make a judgment similar to the judgment that the parents would have made had the parents been present and privy to all the relevant information.

The American concept of independence means ignoring the value of adult (especially parental) judgment. The Chinese attitude is that true independence comes when the child has acquired and accepted adult standards of judgment.

Most importantly, when a child makes an “independent” decision as a result of bowing to peer pressure, Americans still see this as a sign of the child’s independence, even if the decision is obviously unwise.

To Chinese, this is a sign of a very unhealthy form of dependence: dependence on the ill-informed and immature opinions of other children.

To Americans, a child’s spending more time with his peers is therefore a sign of his growing independence. Chinese have a different perspective.

I think this helps illuminate the “dependency” question we homeschoolers face.

If “independence” means kids’ making decisions without having internalized an adult’s understanding of values and consequences, then, yes, our homeschooled kids are less “independent” than many American kids.

At least, I hope so.

But, if “independence” means that a child is not dependent on peer pressure and can make mature decisions because he has acquired and internalized an adult perspective on decision-making, then I think that homeschooled kids may often be more independent than typical American adolescents.

Carnival of Homeschooling

My post on “Should Homeschooled Kids Study Philosophy?” is mentioned in the September 7, 2009 Carnival of Homeschooling.

I did not submit anything to this week’s Carnival , and I will probably only submit something once every month or two. I do think the Carnival is a good way for homeschoolers to hear about other homeschoolers’ blogs.

I notice that my blog seems to be less personal, less about family activities, than most of the homeschooler blogs. I’m not sure if this is because I am a male and most homeschool bloggers are females, or, perhaps, as a physicist, I am less inclined to focus on personal matters.

One blog that is a bit more similar to mine is Kitchen Table Math : this is not specifically limited to homeschoolers, but does have a large homeschooler presence. Kitchen Table Math also has a lot of nice links that are good for learning about math education.

Tyranny vs. Chaos: The False Dichotomy of “Progressive” vs. “Traditional” Education

Alfie Kohn has a revealing essay* posted on Education Week, currently available for free to the public.

Kohn is a well-known advocate of “progressive” education and his article exhibits nicely the basic error that “progressives” make, as well as the opposite error made by far too many advocates of “traditional” education.

On the one hand, Kohn criticizes the traditional teacher-as-lord-and-master approach as consisting of “mandates handed down from on high… where test scores drive the instruction and students are essentially bullied into doing whatever they’re told.”

I think that is a fair rap.

“To divide fractions, invert and multiply.” (Why?)

“Democracy is the best system of government.” (Then why not carry out open-heart surgery democratically?)

“Humans are descended from fish.” (How do we know this?)

To simply order children to believe the “right” answer causes them to accept that one can only learn the truth from authority, and that they, and humans in general, lack the ability to reliably determine the truth for themselves.

And, if kids ever start to wonder how the “authorities” learned the right answer, since the authorities themselves also are mere humans, the kids may fall into a naïve skepticism, thinking that no human can ever really know any truths at all.

But, sadly, Kohn offers, as a false alternative, the old “progressive” solution of giving kids the “opportunities to discover answers to their own questions,” i.e., the “constructivist” approach where kids have to create knowledge for themselves rather than systematically being taught what humans have discovered, at enormous effort, during the last three thousand years.

That really would be a swell approach – if kids had a spare three thousand years to work out everything, and if all kids were as bright as Euclid, Einstein, etc.

So, how to avoid the false choice of “progressive” vs. “traditional” education?

The answer should really be obvious from everyday life.

In real life, we explain to kids that you need to brush your teeth because you will otherwise get cavities, you need to wash your hands because there are germs on your hands that can make you sick, etc. We give explanations.

The idea of giving rational explanations, as opposed to the false dichotomy of either issuing irrational commands or forcing the students to discover everything for themselves, is really not that complicated!

E.g., why “invert and multiply” to divide fractions?

Well, division is the inverse operation to multiplication: if you multiply by some fraction and then wish to undo the multiplication, inverting and multiplying will indeed undo the original multiplication.

Liping Ma, in her brilliant Knowing and Teaching Elementary Mathematics, a “must-read” for all homeschooling parents, goes into much greater depth on this issue of dividing fractions: this is one of the toughest things to explain clearly in elementary mathematics.

But it can be explained. It is unrealistic to expect kids to discover for themselves how to divide fractions, or to fully understand on their own why it works, even if they do stumble upon it. However, it is also not necessary to teach the standard algorithm as an arbitrary rule imposed, for some mysterious reason, by adults.

As Ma explains, if one wishes to use division as the inverse of multiplication, if one wants division to be a means of carrying out repeated subtractions, if one wants the “cancellation law” to apply to division, one has no choice: there is only one right answer, the one given by the standard algorithm.

Ma advocates a “profound understanding of fundamental mathematics”: i.e., both a serious conceptual understanding of elementary math, as well as a practical mastery of the elementary math facts and the standard algorithms.

That indeed should be the goal in all academic (and non-academic) subjects: a conceptual understanding of American history combined with detailed factual knowledge of dates and historical events; an understanding of the experimental bases for scientific theories as well as detailed knowledge of the important scientific facts; etc.

As a practical matter, it is sometimes necessary to say to a student, “We will see the justification for this next month or next year.” Sometimes, one cannot fully understand the evidence for a theory until one has grasped exactly what the theory is. And, it would be foolish to slavishly imitate all the false starts and errors made in the historical development of scientific theories, in the historical creation of various concepts in economics, in the historical discovery of various methods in mathematics, etc.

A student’s learning need not and should not recapitulate the historical process by which knowledge was originally discovered. The whole point is to make it easier for the student than it was for the original discoverer.

Often, the historical experiments or reasoning that led to a discovery are relevant: this is true, for example, of Rutherford’s discovery of the nucleus. But the important thing is to present the best proof and justification we possess today for a particular piece of knowledge.

The other error to avoid, which tends to be shared by both “progressive” and traditional approaches to education, is the false belief that kids have to wait until they are mature to be told of discoveries that the human race only stumbled upon in the last century or so.

It may have taken humans a long time to discover the Big Bang, the fact that humans are descended from fish, etc. But a six-year-old can grasp those ideas – they are not that complex. Even the basic evidence for those facts – e.g., the fossil record, the fact that the galaxies are expanding outward – can be explained at a simple level to six-year-olds.

That the human race had to be “mature” to discover such things does not mean that young kids cannot understand those discoveries.

An educational approach based on giving rational explanations, as opposed to the false dichotomy of either issuing irrational commands or expecting the students to discover everything for themselves, is really not that hard to grasp. Both progressivism and the traditional approach to education are wrong.

We need to teach our kids that humans can and do have a rational understanding of reality. Neither the “progressive” nor the “traditional” approach to schooling really achieves that.

Our kids deserve a "content-rich" approach that teaches them, at an early age, the marvelous and amazing facts that human beings have discovered about reality.

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* Thanks to Barry Garelick at Kitchen Table Math for bringing Kohn's essay to my attention.