Homeschooling Corner: A Mathematician’s Lament, by Paul Lockhart

Paul Lockhart’s A Mathematician’s Lament: How School Cheats us of our Most Fascinating and Imaginative Artform is a short but valuable book, easily finished in an afternoon.

Lockhart’s basic take is that most of us have math backwards. We approach (and thus teach) it as useful but not fun–something to be slogged through, memorized, and then avoided as much as possible. By contrast, Lockhart sees math as more fun than useful.

I do not mean that Lockhart denies the utility of balancing your checkbook or calculating how much power your electrical grid can handle, but most of the math actual mathematicians do isn’t practical. They do it because they enjoy it; they love making patterns with numbers and shapes. Just because paint has a very practical use in covering houses doesn’t mean we shouldn’t encourage kids to enjoy painting pictures; similarly, Lockhart wants kids to see mathematics as fun.

But wait, you say, what if this loosey-goosey, free-form, new math approach results in kids who spend a lot of time trying to re-derive pi from first principles but never really learning algebra? Lockhart would probably counter that most kids never truly master algebra anyway, so why make them hate it in the process? Should we only let kids who can paint like the Masters take art class?

If you and your kids already enjoy math, Lockhart may just reinforce what you already know, but if you’re struggling or math is a bore and a chore, Lockhart’s perspective may be just what you need to turn things around and make math fun.

For example: There are multiple ways to group the numbers during double-digit multiplication, all equally “correct”; the method you chose is generally influenced by things like your familiarity with double-digit multiplication and the difficulty of the problem. When I observed one of my kids making errors in multiplication because of incorrect regrouping, I showed them how to use a more expanded way of writing out the numbers to make the math clearer–promptly eliciting protests that I was “doing it wrong.” Inspired by Lockhart, I explained that “There is no one way to do math. Math is the art of figuring out answers, and there are many ways to get from here to there.” Learning how to use a particular approach—“Put the numbers here, here, and here and then add them”–is useful, but should not be elevated above using whatever approach best helps the child understand the numbers and calculate the correct answers.

The only difficulty with Lockhart’s approach is figuring out what to actually do when you sit down at the kitchen table with your kids, pencil and paper in hand. The book has a couple of sample lessons but isn’t a full k-12 curriculum. It’s easy to say, “I’m going to do a free-form curriculum that requires going to the library every day and uses every experience as a learning opportunity,” and rather harder to actually do it. With a set curriculum, you at least know, “Here’s what we’re going to do today.”

My own personal philosophy is that school time should be about 50% formal instruction and 50% open-ended exploration. Kids need someone to explain how the alphabet works and what these funny symbols on the math worksheet mean; they also need time to read fun books and play with numbers. They should memorize their times tables, but a good game can make times tables fun. In short, I think kids should have both a formal, straightforward curriculum or set of workbooks (I have not read enough math textbooks to recommend any particular ones,) and a set of math enrichment activities, like tangrams, pattern blocks, reading about Penrose the Mathematical Cat, or watching Numberphile on YouTube.

(Speaking of Penrose, I thought the chapter on binary went right over my kids’ heads, but yesterday they returned all of their answers in math class in binary, so I guess they picked up more than I gave them credit for.)

YouCubed.org is an interesting website I recently discovered. So far we’ve only done two of the activities, but they were cute and I suspect the website will make a useful addition to our lessons. If you’ve used it, I’d love to hear your thoughts on it.

That’s all for now. Happy learning!

What IS “Social Studies”?

Sometimes you can’t see the forest for the trees, and sometimes you look at your own discipline and can’t articulate what, exactly, the point of it is.

Yes, I know which topics social studies covers. History, civics, geography, world cultures, reading maps, traffic/pedestrian laws, etc. Socialstudies.org explains, “Within the school program, social studies provides coordinated, systematic study drawing upon such disciplines as anthropology, archaeology, economics, geography, history, law, philosophy, political science, psychology, religion, and sociology, as well as appropriate content from the humanities, mathematics…” etc. (I’m sure you did a lot of archaeology back in elementary school.)

But what is the point of lumping all of these things together? Why put psychology, geography, and law into the same book, and how on earth is that coordinated or systematic?

The points of some other school subjects are obvious. Reading and writing allow you to decode and encode information, a process that has massively expanded the human ability to learn and “remember” things by freeing us from the physical constraints of our personal memories. We can learn from men who lived a thousand years ago or a thousand miles away, and add our bit to the Great Conversation that stretches back to Homer and Moses.

Maths allow us to quantify and measure the world, from “How much do I owe the IRS this year?” to “Will this rocket land on the moon?” (It is also, like fiction, pleasurable for its own sake.) And science and engineering, of course, allow us to make and apply factual observations about the real world–everything from “Rocks accelerate toward the earth at a rate of 9.8m/s^2” to “This bridge is going to collapse.”

But what is social studies? The question bugged me for months, until Napoleon Chagnon–or more accurately, the Yanomamo–provided an answer.

Chagnon is a anthropologist who carefully documented Yanomamo homicide and birth rates, and found that the Yanomamo men who had killed the most people went on to father the most children–providing evidence for natural selection pressures making the Yanomamo more violent and homicidal over time, and busting the “primitive peoples are all lovely egalitarians with no crime or murder” myth.

In an interview I read recently, Chagnon was asked what the Yanomamo made of him, this random white guy who came to live in their village. Why was he there? Chagnon replied that they thought he had come:

“To learn how to be human.”

Sometimes we anthropologists lose the signal in the noise. We think our purpose is to document primitive tribes before they go extinct (and that is part of our purpose.) But the Yanomamo are correct: the real reason we come is to learn how to be human.

All of school has one purpose: to prepare the child for adulthood.

The point of social studies is prepare the child for full, adult membership in their society. You must learn the norms, morals, and laws of your society. The history and geography of your society. You learn not just “How a bill becomes a law” but why a bill becomes a law. If you are religious, your child will also learn about the history and moral teachings of their religion.

Most religions have some kind of ceremony that marks the beginning of religious adulthood. For example, many churches practice the rite of Confirmation, in which teens reaffirm their commitment to Christ and become full members of the congregation. Adult Baptism functions similarly in some denominations.

Judaism has the Bar (and Bat) Mitzvah, whose implications are quite clearly articulated. When a child turns 13 (or in some cases, 12,) they are now expected to be moral actors, responsible for their own behavior. They now make their own decisions about following Jewish law, religious duties, and morality.

But there’s an upside: the teen is also now able to part of a minyan, the 10-person group required for (certain) Jewish prayers, Torah legal study; can marry*; and can testify before a Rabbinic court.

*Local laws still apply.

In short, the ceremony marks the child’s entry into the world of adults and full membership in their society. (Note: obviously 13 yr olds are not treated identically to 33 yr olds; there are other ceremonies that mark the path to maturity.)

Whatever your personal beliefs, the point of Social Studies is to prepare your child for full membership in society.

A society is not merely an aggregation of people who happen to live near each other and observe the same traffic laws (though that is important.) It is a coherent group that believes in itself, has a common culture, language, history, and even literature (often going back thousands of years) about its heroes, philosophy, and values.

To be part of society is to be part of that Great Conversation I referenced above.

But what exactly society is–and who is included in it–is a hotly debated question. Is America the Land of the Free and Home of the Brave, or is it a deeply racist society built on slavery and genocide? As America’s citizens become more diverse, how do these newcomers fit into society? Should we expand the canon of Great Books to reflect our more diverse population? (If you’re not American, just substitute your own country.)

These debates can make finding good Social Studies resources tricky. Young students should not be lied to about their ancestors, but neither should they be subjected to a depressing litany of their ancestors’ sins. You cannot become a functional, contributing member of a society you’ve been taught to hate or be ashamed of.

Too often, I think, students are treated to a lop-sided curriculum in which their ancestors’ good deeds are held up as “universal” accomplishments while their sins are blamed on the group as a whole. The result is a notion that they “have no culture” or that their people have done nothing good for humanity and should be stricken from the Earth.

This is not how healthy societies socialize their children.

If you are using a pre-packaged curriculum, it should be reasonably easy to check whether the makers hold similar values as yourself. If you use a more free-form method (like I do,) it gets harder. For example, YouTube* is a great source for educational videos about all sorts of topics–math, grammar, exoplanets, etc.–so I tried looking up videos on American history. Some were good–and some were bad.

*Use sensible supervision

For example, here’s a video that looked good on the thumbnail, but turned out quite bad:

From the description:

In which John Green teaches you about the Wild, Wild, West, which as it turns out, wasn’t as wild as it seemed in the movies. When we think of the western expansion of the United States in the 19th century, we’re conditioned to imagine the loner. The self-reliant, unattached cowpoke roaming the prairie in search of wandering calves, or the half-addled prospector who has broken from reality thanks to the solitude of his single-minded quest for gold dust. While there may be a grain of truth to these classic Hollywood stereotypes, it isn’t a very big grain of truth. Many of the pioneers who settled the west were family groups. Many were immigrants. Many were major corporations. The big losers in the westward migration were Native Americans, who were killed or moved onto reservations. Not cool, American pioneers.

Let’s work through this line by line. What is the author’s first priority: teaching you something new about the West, or telling you that the things you believe are wrong?

Do you think it would be a good idea to start a math lesson by proclaiming, “Hey kids, I bet you get a lot of math problems wrong”? No. Don’t start a social studies lesson that way, either.

There is no good reason to spend valuable time bringing up incorrect ideas simply because a child might hold them; you should always try to impart correct information and dispel incorrect ideas if the child actually holds them. Otherwise the child is left not with a foundation of solid knowledge, but with what they thought they knew in tatters, with very little to replace it.

Second, is the Western movie genre really so prominent these days that we must combat the pernicious lies of John Wayne and the Lone Ranger? I don’t know about you, but I worry more about my kids picking up myths from Pokemon than from a genre whose popularity dropped off a cliff sometime back in the 80s.

“We are conditioned to think of the loner.” Conditioned. Yes, this man thinks that you have been trained like a dog to salivate at the ringing of a Western-themed bell, the word “loner” popping into your head. The inclusion of random psychology terms where they don’t belong is pseudo-intellectual garbage.

Updated values chart!

The idea of the “loner” cowboy and prospector, even in their mythologized form, is closer to the reality than the picture he draws. On the scale of nations, the US is actually one of the world’s most indivdualist, currently outranked only by Canada, The Netherlands, and Sweden.

Without individualism, you don’t get the notion of private property. In many non-Western societies, land, herds, and other wealth is held collectively by the family or clan, making it nearly impossible for one person (or nuclear family) to cash out his share, buy a wagon, and head West.

I have been reading Horace Kephart’s Our Southern Highlanders, an ethnography of rural Appalachia published in 1913. Here is a bit from the introduction:

The Southern highlands themselves are a mysterious realm. When I prepared, eight years ago, for my first sojourn in the Great Smoky Mountains, which form the master chain of the Appalachian system, I could find in no library a guide to that region. The most diligent research failed to discover so much as a magazine article, written within this generation, that described the land and its people. Nay, there was not even a novel or a story that showed intimate local knowledge. Had I been going to Teneriffe or Timbuctu, the libraries would have furnished information a-plenty; but about this housetop of eastern America they were strangely silent; it was terra incognita.

On the map I could see that the Southern Appalachians cover an area much larger than New England, and that they are nearer the center of our population than any other mountains that deserve the name. Why, then, so little known? …

The Alps and the Rockies, the Pyrennees and the Harz are more familiar to the American people, in print and picture, if not by actual visit, than are the Black, the Balsam, and the Great Smoky Mountains. …For, mark you, nine-tenths of the Appalachian population are a sequestered folk. The typical, the average mountain man prefers his native hills and his primitive ancient ways. …

The mountaineers of the South are marked apart from all other folks by dialect, by customs, by character, by self-conscious isolation. So true is this that they call all outsiders “furriners.” It matters not whether your descent be from Puritan or Cavalier, whether you come from Boston or Chicago, Savannah or New Orleans, in the mountains you are a “furriner.” A traveler, puzzled and scandalized at this, asked a native of the Cumberlands what he would call a “Dutchman or a Dago.” The fellow studied a bit and then replied: “Them’s the outlandish.” …

As a foretaste, in the three and a half miles crossing Little House and Big House mountains, one ascends 2,200 feet, descends 1,400, climbs again 1,600, and goes down 2,000 feet on the far side. Beyond lie steep and narrow ridges athwart the way, paralleling each other like waves at sea. Ten distinct mountain chains are scaled and descended in the next forty miles. …

The only roads follow the beds of tortuous and rock-strewn water courses, which may be nearly dry when you start out in the morning, but within an hour may be raging torrents. There are no bridges. One may ford a dozen times in a mile. A spring “tide” will stop all travel, even from neighbor to neighbor, for a day or two at a time. Buggies and carriages are unheard of. In many districts the only means of transportation is with saddlebags on horseback, or with a “tow sack” afoot. If the pedestrian tries a short-cut he will learn what the natives mean when they say: “Goin’ up, you can might’ nigh stand up straight and bite the ground; goin’ down, a man wants hobnails in the seat of his pants.” …

Such difficulties of intercommunication are enough to explain the isolation of the mountaineers. In the more remote regions this loneliness reaches a degree almost unbelievable. Miss Ellen Semple, in a fine monograph published in[Pg 23] the Geographical Journal, of London, in 1901, gave us some examples:

“These Kentucky mountaineers are not only cut off from the outside world, but they are separated from each other. Each is confined to his own locality, and finds his little world within a radius of a few miles from his cabin. There are many men in these mountains who have never seen a town, or even the poor village that constitutes their county-seat…. The women … are almost as rooted as the trees. We met one woman who, during the twelve years of her married life, had lived only ten miles across the mountain from her own home, but had never in this time been back home to visit her father and mother. Another back in Perry county told me she had never been farther from home than Hazard, the county-seat, which is only six miles distant. Another had never been to the post-office, four miles away; and another had never seen the ford of the Rockcastle River, only two miles from her home, and marked, moreover, by the country store of the district.”

When I first went into the Smokies, I stopped one night in a single-room log cabin, and soon had the good people absorbed in my tales of travel beyond the seas. Finally the housewife said to me, with pathetic resignation: “Bushnell’s the furdest ever I’ve been.” Bushnell, at that time, was a hamlet of thirty people, only seven miles from where we sat. When I lived alone on “the Little Fork of Sugar Fork of[Pg 24] Hazel Creek,” there were women in the neighborhood, young and old, who had never seen a railroad, and men who had never boarded a train, although the Murphy branch ran within sixteen miles of our post-office.

And that’s just Appalachia. What sorts of men and women do you think settled the Rockies or headed to the Yukon? Big, gregarious families that valued their connections to society at large?

Then there are the railroads. The video makes a big deal about the railroads being funded by the government, as proof that Americans weren’t “individuals” but part of some grand collectivist society.

Over in reality, societies with more collectivist values, like Pakistan, don’t undertake big national projects. In those societies, your loyalty is to your clan or kin group, and the operative level of social planning and policy is the clan. Big projects that benefit lots of people, not just particular kin networks, tend not to get funded because people do not see themselves as individuals acting within a larger nation that can do big projects that benefit individual people. Big infrastructure projects, especially in the 1800s, were almost entirely limited to societies with highly individualistic values.

Finally we have the genocide of the American Indians. Yes, some were definitely killed; the past is full of sins. But “You’re wrong, your self-image is wrong, and your ancestors were murderers,” is not a good way to introduce the topic.

It’s a pity the video was not good; the animation was well-done. It turns out that people have far more strident opinions about “Was Westward Expansion Just?” than “Is Pi Irrational?”

I also watched the first episode of Netflix’s new series, The Who Was? Show, based on the popular line of children’s biographies. It was an atrocity, and not just because of the fart jokes. The episode paired Benjamin Franklin and Mahatma Gandhi. Gandhi was depicted respectfully, and as the frequent victim of British racism. Franklin was depicted as a buffoon who hogged the spotlight and tried to steal or take credit for other people’s ideas.

It made me regret buying a biography of Marie Curie last week.

If your children are too young to read first-hand ethnographic accounts of Appalachia and the frontier, what do I recommend instead? Of course there are thousands of quality books out there, and more published every day, but here are a few:

A Child’s Introduction to The World

The Usborne Book of Living Long Ago: Everyday Life Through the Ages

What Your [X] Grader Needs to Know So far I like these, but I have not read them all the way through.

DK: When on Earth?

More important than individual resources, though, is the attitude you bring to the subject.

 

Before we finish, I’d like to note that “America” isn’t actually the society I feel the closest connection to. After all, there are a lot of people here whom I don’t like. The government has a habit of sending loyal citizens to die in stupid wars and denying their medical treatment when they return, and I don’t even know if the country will still exist in meaningful form in 30 years. I think of my society as more “Civilization,” or specifically, “People engaged in the advancement of knowledge.”

Homeschooling Corner: The Well-Trained Mind, by Susan Bauer and Jessie Wise

Today we’re reviewing Susan Wise Bauer and Jessie Wise’s The Well-Trained Mind: A guide to Classical Education at Home. (H/T to commentator Jefferson for the recommendation.)

The Well Trained Mind is not the sort of book that lends itself to quoting, so I won’t. It is, however, an extremely practical guide to homeschooling, with specific advice for each year, from pre-K through highschool, including information on how to write highschool transcripts, grades, and prepare your kids for the academic paperwork portion of applying to college. It is a kind of homeschooling reference book. (There are multiple editions online; I purchased the one in the photo because it was cheaper than the newer ones, but you might want the most recently updated one.)

By now I’ve probably read about a dozen books on homeschooling/education, everything from Montessori to Waldorf, Summerhill to Unschooling, math and science curriculum guides for preschoolers, and now The Well-Trained Mind.

The data on homeschooling is pretty good: homeschoolers turn out, on average, about as smart as their conventionally schooled peers. (I forget the exact numbers.) They tend to be better than average at reading and writing, and a bit worse than average at math and science. Unschooled kids (who receive very little formal instruction in anything,) tend to turn out about a year behind their peers, which isn’t too bad considering all of the effort that goes into conventional schooling, but I still can’t recommend it.

The Well-Trained Mind is an excellent staring point for any parent trying to get their feet under themselves and figure out the daunting task of “OMG How do I do this?” It lays out a subject-by-subject plan for every year of schooling, down to how many minutes per day to spend on each part of the curriculum.

If that sounds too detailed, remember that this is just a guide and you can use it as an inspirational jumping-off-point for your own ideas. It’s like arranging all of the colors of paint in a nice neat circle before you paint your own masterpiece.

If you need a curriculum–either because your state requires it, or it requires you to cover certain topics, or you would just feel better with a curriculum to guide you before you leap in unsupervised, this is a very good guide. If you already have your curriculum and you feel secure and confident in what you’re doing, you might find the information in this book superfluous.

Bauer and Wise lay out what’s known as the Trivium: grammar, logic, and rhetoric.

Elementary school is the “grammar” stage. At this age, students are learning (mostly memorizing) the mechanical rules they need for education, like letter sounds and times tables. At the logic stage, children begin applying what they know and trying to figure out why things happen. Rhetoric is for the highschoolers, and since I don’t have any highschoolers I didn’t read that part of the book.

The curriculum for the younger grades is straightforward and easy to use: 10 minutes a day of alphabet/phonics for the preschoolers, increasing over the years to include spelling, grammar, reading, and math. The authors particularly encourage reading history (they have a specific order) and children’s versions of classic novels/myths.

Their approach to writing is interesting: in the lower grades, at least, children do very little generative writing (that is, coming up with and writing down their own ideas,) and focus more on copy work–trying to accurately and neatly write down a few sentences their parents give them, and otherwise expressing themselves out loud.

This stands in stark contrast to how writing is taught in the local schools, where even kindergarteners are expected to start writing little stories or at least sentences of their own devising.

This works great for some kids. My kids hate it. I think the combination of tasks–hold the pencil properly, now form the letters, arrange them into a word, spell the word properly, oh, and come up with an original idea and a specific sentence to write about the idea was just overwhelming.

So Bauer’s approach, which breaks the mechanics and creative work into two different parts, is a welcome alternative that may work better for my family.

Bauer and Wise are strong advocates of phonics instruction (which I agree with) and make an interesting point about emphasizing what they call parts-to-whole instruction and avoiding whole-to-parts. In the example they give, imagine giving a child a tray of insects (presumably fake or preserved,) and showing them five different kind of insect legs. The child learns the five kinds, and can then sort the insects by variety.

Now imagine handing the child the same tray of insects and simply asking them to take a good look at the bugs, figure out what’s the same or different between them, and then sort them. Well, children certainly can sort objects into piles, but will they learn much in the process? Let the children know what you want from them, teach them what you want them to learn, and then let them use their knowledge. Don’t expect them to work it all out on their own from scratch with a big pile of bugs.

I’ve noticed that a lot of children’s “educational” TV shows try to demonstrate the second approach. The characters have some sort of problem and the try to think about different ways to solve it. This is fine for TV, but in real life, kids are pretty bad at this. They struggle to generate solutions that they haven’t heard of before–after all, they’re only kids, and they only know so much. This doesn’t mean kids can’t have great ideas or figure stuff out, it just means they have sensible limits.

This is the same idea that underlies their approach to phonics–not that it’s wrong to memorize a few words (sew does not rhyme with chew, after all,) but that kids benefit from explicit instruction in how letters work so they can use that knowledge to sound out new words they’ve never seen before.

Whole language vs. phonics instruction isn’t quite the controversy it used to be, but there’s something similar unfolding in math, as far as I can see. Back in public school, they didn’t teach the kid the “algorithm” for addition and subtraction until third grade. My eldest was expected to add and subtract multiple two-digit numbers in their HEAD based on an “understanding of numbers” instead of being taught to write down the numbers and add them.

Understanding numbers is great, but I recommend also teaching your kids to write them down and add/subtract them.
AND FOR GOODNESS’S SAKES, WRITE EQUATIONS VERTICALLY. Always try to model best practice.

Many kids acquire number sense through practice. Seeing that 9+5=14 whether they are in the equations 9+5 or 5+9, 45+49 or 91+52, helps children develop number sense. Give children the tools and then let them use them. Don’t make the children try to re-invent addition or force them to use something less efficient (and don’t teach them something you’ll just have to un-teach them later.)

The authors recommend teaching kids Latin. I don’t recommend Latin unless you are really passionate about Latin.  IMO, you’re better off teaching your kids something you already speak or something they can use to get a job someday, but that’s a pretty personal decision.

 

Here’s how our own schedule currently looks:

After all of the holiday excitement and disruption, I feel like we’re finally settling back into a good routine. What exactly we do varies by day, but here’s a general outline:

2 Logic puzzles (I’m not totally satisfied with our puzzle book, so I can’t recommend a specific one, but logic puzzles come in a variety of difficulty levels)

2 Tangram puzzles (I like to play some music while the kids are working)

1 or 2 stories from Mathematicians are People, Too: Stories from the Lives of Great Mathematicians (Warning: Pythagoras was killed by an angry mob, Archimedes was killed by an invading soldier, and Hypatia was also killed by an angry mob. But Thales and Napier’s chapters do not have descriptions of their horrible deaths.) This is our current “history” book, because I try to structure our history around specific themes, like technology or math.

Math: multiplication tables and/or fractions

A game of some sort, like Mastermind, Fraction Formula, or Chess. (No-Stress Chess is  good teaching set.)

Science and/or social studies reading (the subjects often overlap.) I happened across a lovely stack of science, math, and social studies texts at the local used book shop the other day. When I got home, I realized they’re from India. Well, math is math, no matter where you’re from, and the social studies books are making for an interesting unit on India. In science we’ve just started a unit on Earth science (wind, water, stones, and dirt) for which I am well-prepared with a supply of rocks. (Come spring we’ll be growing plants, butterflies, and ladybugs.)

Free reading: my kids like books about Minecraft or sharks. Your kids like what they like.

Grammar/spelling/copywork: not our favorite subjects, but I’m trying to gradually increase the amount we do. Mad Libs with spelling words are at least fun.

I never manage to do as much as I want to do.