I have sworn upon the altar of God eternal hostility against every form of tyranny over the mind of man. -- Thomas Jefferson

Sunday, October 11, 2009

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.

Tuesday, October 6, 2009

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.

Sunday, October 4, 2009

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.

Thursday, October 1, 2009

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.