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.) 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. 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: Science (geology and geography)


I have yet to find any “science kits” that actually teach science–most are just science-themed toys. There’s nothing wrong with that, but don’t expect your kid to re-derive the principles of chemistry via a baking soda volcano.

Smaller kids aren’t ready for the kind of thinking required for actual scientific research, but they can still learn plenty of science the mundane way: by reading. So here are some of our favorite science books/activities:

We did geology over the winter, centered around Rocks, Rivers, and the Changing Earth. It’s a lovely book (reading level about second grade?) with instructions for many simple experiments (eg, put rocks, sand, water in a glass jar and carefully shake/swirl to observe the effects of different water speeds on riverbanks) and handily complements any nature walks, rock collecting trip, or expeditions to the seashore.

WARNING: This book was published before plate tectonics became widely accepted and so has a confused chapter or two on how mountains form. SKIP THIS CHAPTER.

We also tried making polished stones in a rock tumbler (verdict: not worth the cost.)

After geology, we transitioned to geography with A Child’s Introduction to the World: Geography, Cultures and People–from the Grand Canyon to the Great Wall of China. I admit that geography sounds more like social studies than science, but it flows so perfectly from our understanding of geology that I have to mention it here.

I like to read this with a globe and children’s atlas at hand, so I can easily demonstrate things like latitude and longitude, distances, and different map projections.

With spring’s arrival we also began a study of plants and insects.

If you’ve never started your own plants from seed, any common crop seeds sold at the store–beans, peas, corn, squash, and most flowers–will sprout quickly and easily. If you want to keep your plants indoors, I recommend you get a bag of dirt at the garden center. This dirt is supposed to be “clean”; the dirt found outside in your yard is full of bugs that you probably weren’t intending on studying in your living room.

Speaking of bugs, we bought the “raise your own ladybugs” and butterflies kits, but I don’t recommend these as real caterpillars are nowhere near as cute and interesting as the very hungry one in the story. I think you’re better off just collecting ladybugs in the wild and reading about them at home.

The Way Things Work (also by this author: How Machines Work: Zoo Break) This is a big, beautiful book aimed at older kids, maybe about 10+. Younger kids can enjoy it if you read it with them.

Super Science: Matter Matters is a fabulous pop-up/lift-the-flap book about chemistry. We were very lucky to receive this as a birthday gift. (Birthday hint: the homeschooling families in your life would always like more books.) The book is a little fragile, so not appropriate for younger children who might pull too hard on the tabs, but great for everyone else.

Magic Schoolbus anything. There are probably several hundred books in this series by now. Who Was Albert Einstein? We finished our math biographies, so on to science bios. Basher Science: Astronomy  This is cute, and there are a bunch in the series. I’m looking forward to the rest. Professor Astro Cat‘s Atomic Adventure (also, Space!)

Homeschooling Corner: Math ideas and manipulatives for younger grades


When you love a subject and your kids love it, too, it’s easy to teach. When you’re really not sure how to approach the subject or your kids hate it, it gets a lot trickier. (See: spelling.)

So I thought I’d make a list of some of our favorite math related materials–but please remember, all you really need for teaching math is a paper and pencil. (Or less–Archimedes did math with a stick and some sand!)


Little ones who are just learning to count and add benefit from having something concrete they can hold, touch, and move around when thinking about concepts like “two more” or “two less.”

You can count almost anything–pebbles, shells, acorns, pennies, Monopoly money, fingers–but having a box of dedicated, fun, colorful countables on hand is useful. My favorites:

Abacus. The abacus has the lovely advantage that all of its counters are on rods and so don’t get scattered around the room, stepped on and lost. I made my own abacus (inspired by commenter Dave‘s abacus) out of a shoe box, plastic beads, pipecleaners, and tape. You can count, add, subtract, multiply, divide, etc., on an abacus, but for your purposes you’ll just need to learn addition and subtraction.

Different abaci have different numbers and arrangements of beads. If your kids are still learning to count/mastering addition and subtraction up to ten (standard kindergarten goals,) I’d use an abacus with 9 beads per string. (Just like writing numbers, after you get to nine on the “ones” string, you raise up one bead on the “10” string.)

We adults tend to take place value for granted (“it’s obvious that we use the decimal system!”) but switching from column to column can be confusing for young kids. There’s no intuitive reason why 11 doesn’t = 2. The abacus helps increase awareness of place value (typically taught in first grade) because you simply run out of beads after 9 and have to switch to the next row.

Once kids have the basic idea, you can switch to a more advanced abacus like the Soroban. The top bead on the Soroban is worth 5, so students count 1-2-3-4, then click the 5 bead and clear the unit beads, then add unit beads to the five to count 6-7-8-9, then click one bead in the tens column and clear all of the beads in the unit and five column. My apologies if it sounds complicated; it really isn’t, it’s just a little tricky to put into words.

You can get abacus workbooks; I have not used any so I cannot review them but they look fun. Rather, I just use the abacus as a complement to the other math problems we are already doing. (I have read Mr. Green’s How to Use a Chinese Abacus, which was the only book my library had on the subject. It is a very good introduction aimed at adults.)

Counting Penguins

There is nothing magical about penguins; I just happen to like them. The set has 100 penguins in ten sets (distinguished by color) plus ten “ice bars” that hold ten penguins each. (Besides addition and subtraction,) I find these useful for introducing and visualizing multiplication , eg, 3 rows of 5 penguins = 3×5.

Counting Cubes

For bigger numbers, we have a bag of 1,000 interlocking cubes. Kids will want to just plain build with them, like Legos, which is fine–a fun treat after hard work. You can easily use these to represent 1s, 10s, and 100s (it takes a while to assemble a full 1,000 cube,) and to represent operations like 3x3x3, helping bridge both place value and multiplication. Legos work for this, too, though you’ll probably want to sort out ones that are all the same size and shape.


Pattern Blocks

(I think I’ve been incorrectly calling these tangrams, though the principles are similar.)

These pattern blocks are a family heirloom, sent to me by my grandmother upon the birth of my first child. I played with them when I was a child; my siblings played with them; now my children play with them. Someday I will pass them on to my grandchildren… but you can also get them on Amazon. (We use these with a book of pattern block activities that hails from the 80s; I am sure there are many good books of a similar nature published within the past couple of decades.

Apparently there are workbooks with pattern block activities aimed all the way up to 8th grade, but I have not read them and cannot comment on them.

Cuisenaire rods

We didn’t use cuisenaire rods when I was young, but I think I would have liked them. Similar to the tangrams pattern blocks, there are lots of interesting workbooks, games, and other activities you can do with these.

Building toys

Open-ended building toys (Legos, Tinker Toys, blocks, magnetic tiles) come in almost endless forms and can be used to build all sorts of geometric shapes.


Fraction blocks and fraction circles are both handy.


Almost any kids’ board game can be transformed into a math game by adding cards with math problems to be solved before completing a turn or using math dice. Your local games shop can help you find dice with numbers higher than six, or you can just tape paper onto an existing cube to make a custom die of your liking (like an + and – die). There are also tons of fun logic games; I pull these out whenever kids start getting restless.


There are so many great math books, from Sir Cumference to Penrose, that I can’t hope to list them all. I encourage you to check out your library’s selection. Here are a few of my favorites:

The Adventures of Penrose the Mathematical Cat (plus sequels) makes a very pleasant enrichment portion of our daily maths. Each day we read one of Penrose’s stories (on subjects like Fibonacci numbers, primes, operations, etc) and do a short, related math activity.

Penrose is probably most appropriate for kids in mid to late elementary, not little ones just learning to count and add. (Note: the first story in the book was about binary, which flew over my kids’ heads.) Sir Cumference is more appropriate for younger learners.

Mathematicians are People, Too: biographies of great mathematicians. I’m not keen on the title, but my kids liked the chapter on Archimedes.

Balance Benders These workbooks come in different levels, from beginner to expert. Each puzzle presents students with a drawing of a balance with shapes on either side, and asks them to figure out, from a choice of answers, which statements about the shapes are true, eg “One circle equals two squares” after viewing a balance with two circles and four squares. (We also do logic puzzles and picture sudoku.)



I am not recommending any textbooks because I don’t have any idea which is the best. We don’t use a pre-packaged curriculum, because they tend to be expensive–instead I’ve just picked up a whole bunch of different math texts at the second hand shop and been gifted some lovely hand-me-downs from relatives. At this point I might have too many math books… I use 3 or 4 interchangably, depending on exactly which concepts we’re covering and whether I think the kids need more practice or not. I recently lucked into a volume of the “What your X Grader Needs to Know” series, and it gives a very nice overview of grade-level math expectations (among other things.)

Incidentally, the local public school math expectations appear to be:

Kindergarten: Reliably add and subtract the numbers 0-10; add small numbers to numbers between 10 and 20; be able to write all of the numbers from 0-20; count to 100.

1st grade: Place value; add and subtract one and two digit numbers with no regrouping.

2nd grade: Add and subtract multiple two an three-digit numbers.

I think they only explain regrouping in third grade.

In my experience, kids can do a lot more than that. These aren’t the standards I use in my classroom. But if you’re struggling to get your kindergartener to concentrate on their math worksheets, just remember: professional teachers don’t actually expect all that much at these ages. (And my kids don’t like doing a bunch of worksheet problems, either.)

Don’t sweat it. Do a few problems every day, if you can. Try teaching the same material from different angles, if necessary. Don’t be afraid to pull out pencil and paper and just make up a few problems and work through them together. Make patterns. Play games. Relax and have fun, because math at these ages really is beautiful.

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.

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.

Homeschooling Corner: Erdos, Fibonacci, and some Really Big Numbers

One of the nice things about homeschooling is that it is very forgiving of scheduling difficulties and emergencies. Everyone exhausted after a move or sickness? It’s fine to sleep in for a couple of days. Exercises can be moved around, schedules sped up or slowed down as needed.

This week we finished some great books (note: I always try to borrow books from the library before considering buying them. Most of these are fun, but not books you’d want to read over and over):

The Boy who Loved Math: The Improbable Life of Paul Erdos, by Deborah Heligman, was a surprise hit. I’ve read a bunch of children’s biographies and been consistently disappointed; the kids loved this one. Improbable, I know.

I suppose the moral of the story is that kids are likely to enjoy a biography if they identify with the subject. The story starts with Erdos as a rambunctious little boy who likes math but ends up homeschooled because he can’t stand regular school. My kids identified with this pretty strongly.

The illustrations are nice and each page contains some kind of hidden math, like a list of primes.

Professor Astro Cat’s Frontiers of Space, by Dominic Walliman. This is a lovely book appropriate for kids about 6-11, depending on attention span and reading level. We’ve been reading a few pages a week and recently reached the end.

Minecraft Math with Steve, by Steve Math. This book contains 30 Minecraft-themed math problems (with three sub-problems each, for 90 total.) They’re fairly simple multiplication, subtraction, division, and multiplication problems, probably appropriate for kids about second grade or third grade. A couple of sample problems:

Steve wants to collect 20+20 blocks of sand. how much is that total?

Steve ends up with 42 blocks of sand in his inventory. He decides that is too much so drops out 12 blocks. How many blocks remain?

A bed requires 3 wood plank and 3 wools. If Steve has 12 wood planks and 12 wools, how many beds can he build?

This is not a serious math book and I doubt it’s “Common Core Compliant” or whatever, but it’s cute and if your kids like Minecraft, they might enjoy it.

We are partway into Why Pi? by Johnny Ball. It’s an illustrated look at the history of mathematics with a ton of interesting material. Did you know the ancient Greeks used math to calculate the size of the Earth and distance between the Earth and the moon? And why are there 360 degrees in a circle? This one I’m probably going to buy.

Really Big Numbers, by Richard Evan Schwartz. Previous books on “big numbers” contained, unfortunately, not enough big numbers, maxing out around a million. A million might have seemed really good to kids of my generation, but to today’s children, reared on Numberphile videos about Googols and Graham’s number, a million is positively paltry. Really Big Numbers delivers with some really big numbers.

Let’s Estimate: A book about Estimating and Rounding Numbers, by David A. Adler. A cute, brightly illustrated introduction. I grabbed notebooks and pens and made up sample problems to help the kids explore and reinforce the concepts as we went.

How Big is Big? How Far is Far? by Jen Metcalf. This is like a coffee table book for 6 yr olds. The illustrations are very striking and it is full of fascinating information. The book focuses both on relative and absolute measurement. For example,  5’9″ person is tall compared to a cat, but short compared to a giraffe. The cat is large compared to a fly, and the giraffe is small compared to a T-rex. My kids were especially fascinated by the idea that clouds are actually extremely heavy.

Blockhead: The Life of Fibonacci, by Joseph D’Agnes. If your kids like Fibonacci numbers (or they enjoyed the biography of Erdos,) they might enjoy this book. It also takes a look at the culture of Medieval Pisa and the adoption of Arabic numerals (clunkily referred to in the text as “Hindu-Arabic numerals,” a phrase I am certain Fibonacci never used.) Fibonacci numbers are indeed found all over in nature, so if you have any sunflowers or pine cones on hand that you can use to demonstrate Fibonacci spirals, they’d be a great addition to the lesson. Otherwise, you can practice drawing boxes with spirals in them or Pascal’s triangles. (This book has more kid-friendly math in it than Erdos’s)

Pythagoras and the Ratios, by Julie Ellis. Pythagoras and his cousins need to cut their panpipes and weight the strings on their lyres in certain ratios to make them produce pleasant sounds. It’s a fun little lesson about ratios, and if you can combine it with actual pipes the kids can cut or recorders they could measure, glasses with different amounts of water in them or even strings with rock hanging from them, that would probably be even better.

Older than Dirt: A Wild but True History of Earth, by Don Brown. I was disappointed with this book. It is primarily an overview of Earth’s history before the dinosaurs, which was interesting, but the emphasis on mass extinctions and volcanoes (eg, Pompeii) dampened the mood. I ended up leaving out the last few pages (“Book’s over. Bedtime!”) to avoid the part about the sun swallowing up the earth and all life dying at the end of our planet’s existence, which is fine for older readers but not for my kids.

Hope you received some great games and books last month!

Homeschooling Corner: Math Philosophy

Music is a hidden arithmetic exercise of the soul, which does not know that it is counting.–Gottfried Leibniz

Fibonacci Spiral

You may have noticed that I talk a lot more about math than reading or writing. This is not because I dislike the language arts, but because they are, once learned, not very complicated. A child must learn to decode symbols, associate them with sounds, and then write them–tricky in the beginning, but most children should have the basics down by the age of 6 or 7. For the next several years, the child’s most important task is simply practice. If a child has a book they love to read, then they are already most of the way there and will probably only need some regular instruction on spelling and punctuation.

Math, by contrast, is always advancing. For every new operation or technique a child masters, there is another waiting to be learned.

I don’t hold with the idea that mathematical concepts must be taught in a particular order or at particular ages–I introduced negative numbers back in preschool, they’ve learned about simple logarithms in elementary, and they seem none the worse for the unusual order.

Count on Math gives the logic behind Particular Order:

Developmental sequence is fundamental to children’s ability to build conceptual understanding. … The chapters in this book present math in a developmental sequence that provides children a natural transition from one concept to the next, preventing gaps in their understanding. …

When children are allowed to explore many objects, they begin to recognize similarities and differences of objects. When children can determine similarities and differences, they can classify objects. When children can classify objects, they can see similarities and difference well enough to recognize patterns. When children can recognize, copy, extend and create patterns, they can arrange sets in a one-to-one relationship. …

This developmental sequence provides a conceptual framework that serves as a springboard to developing higher level math skills.

This logic is complete bollocks. (Count on Math is otherwise a fine book if you’re looking for activities to do with small children.)

Humans are good at learning. It’s what we do. Any child raised in a normal environment (and if you’re reading this, I assume you care about your children and aren’t neglecting them) has plenty of objects around every day that they can interact with, observe, sort, classify, etc. You don’t have to dedicate a week to teaching your kid how to tell “similar” and “different” in objects before you dedicate a week to “classifying.” Hand them some toys or acorns or rocks or random stuff lying around the house and they can do that themselves.

Can you imagine an adult who, because their parent or preschool skipped straight from”determining similarities and differences” to “making patterns,” was left bereft and innumerate, unable to understand fractions? If the human mind were really so fragile, the vast majority of people would know nothing and our entire civilization would not exist.

More important than any particular order is introducing mathematical concepts in a friendly, enjoyable way, when the child is ready to understand them.

For example, I tried to teach binary notation this week, but that went completely over the kids’ heads. They just thought I was making a pattern with numbers. So I stopped and switched to a lesson about Fibonacci numbers and Pascal’s triangle.

Then we went back to practicing addition and subtraction with regrouping, because that’s tricky. It’s boring, it’s not fun, and it’s not intuitive until you’ve really got base-ten down solid (base 10, despite what you may think, is not “obvious” or intuitive. Not all languages even use base 10. The Maya used base 20; the Babylonians used base 60. There are Aborigines who used base 5 or even 3; in Nigeria you’ll find base 12.) Learning is always a balance between the fun stuff (look what you can do with exponents!) and the boring stuff (let’s practice our times tables.) The boring stuff lets you do the fun stuff, but they’re both ultimately necessary.


What else we’ve been up to:

Fractions, Decimals, and Percents, by David A. Adler. A brightly-colored, well-written introduction to parts of numbers and how fractions, decimals and percents are really just different ways of saying the same thing.

It’s a short book–28 pages with not much text per page–and intended for young children, probably in the 8 to 10 yrs old range.

I picked up Code Your Own Games: 20 Games to Create with Scratch just because I wanted to see what there was outside the DK Workbooks (which have been good so far, no complaints there.) So far it seems pretty similar, but the layout is more compact. Beginners might feel less intimidated by DK’s larger layouts with more white space, but this seems good for a kid who is past that stage. It has more projects than the shorter DK Workbooks but they’re still pretty simple.

I also happened across a Singapore Math Workbook, which seems fine. Sample problem:

Emily and Jasmine had the same number of stamps. After Emily gave Jasmine 42 stamps, Jasmine had twice as many stamps as Emily. How many did Jasmine have at the end?

At a movie, 1/4 of the people in the theater were men, 5/8 were women, and the rest were children. If there were 100 more women than children, what was the total number of people in the theater?

Our recorders arrived, so now we can play music.

Finished reading The Secret Garden, planted seeds, collected and identified rocks. Nature walk: collected fall leaves and pressed flowers. Caught bugs and observed squirrels for Ranger Rick nature workbook. Read about space and worked with cuisenaire rods. Etc.


Homeschooling Corner: Flying Kites

We had a lovely, windy day, so we grabbed the kites, invited the neighbors, and headed out to the park.

Homeschooling does put additional responsibility on the parents to help their kids socialize. That doesn’t mean homeschooled kids are necessarily at a disadvantage viz their typically-schooled peers when it comes to comes to socializing (I went to regular school and still managed to be terribly socialized;) it’s just one more thing homeschooling parents have to keep in mind. So I am glad that we’ve had the good luck recently to make several friends in the neighborhood.

I’ve been looking for good, educational YouTube channels. Now I haven’t watched every video on these channels and I make no guarantees, but they seem good so far:

Welch Labs:

Welch Labs also has a website with a free downloadable workbook that accompanies their videos about imaginary numbers. It’s a good workbook and I’m working through it now.

TedEd, eg:

VSauce, eg:

Numberphile, eg:

The King of Random, eg:

We finished DK’s Coding in Scratch Projects Workbook and started Coding in Scratch: Games Workbook, which is slightly more advanced (longer projects.)

The Usborne Times Tables Activity Book is a rare find: a book that actually makes multiplication vaguely fun. Luckily there’s no one, set age when kids need to learn their multiplication tables–so multiple kids can practice their tables together.

In math we’ve also been working with number lines, concept like infinity (countable and uncountable,) infinitesimals, division, square roots, imaginary numbers, multi-digit addition and subtraction, graphing points and lines on the coordinate plane, and simple functions like Y=X^2. (Any kid who has learned addition, subtraction, multiplication and division can plot simple functions.)

We started work with the cuisenaire rods, which I hope to continue–I can’t find our set on Amazon, but these are similar. We’re also using Alexander Warren’s book You can Count on it: A Mentor’s Arithmetic Patterns for Elementary Students for cusienaire activites.

If you’re looking for board game to play with elementary-aged kids, Bejeweled Blitz is actually pretty good. Two players compete to place tiles on the board to match 3 (or more) gems, in a row or up and down. (A clever play can thus complete two rows at once.) We play with slightly modified rules. (Note: this game is actually pretty hard for people who struggle with rotating objects in their heads.)

Picture Sudoku is fun for little kids (and probably comes in whatever cartoon characters you like,) while KenKen and magic squares and the like are good for older kids (I always loved logic puzzles when I was a kid, so I’d like to get a book of those.)

I’ve found a website called Memrise which seems good for learning foreign languages if you don’t have access to a tutor or know somene who speaks the language you want to learn. They probably have an app for phones or tablets, so kids could practice their foreign langauge on-the-go. (Likewise, I should stow our spelling book in the car and use car rides as a chance to quiz them.)

And of course we’re still reading Professor Astro Cat/working in the workbook, which involves plenty of writing.

For Social Studies we’ve been reading about fall holidays.

Hope you all have a lovely October! What are some of your favorite educational videos?


Homeschooling Corner: Introducing Mr. Poop & Custom Dice

I happened to have a poop-shaped pinata sitting around (Why? Look, sometimes these things just happen) of the pull-the-flap-on-the-bottom variety rather than the smash-it-with-a-bat kind, so I decided to add a little fun to our day by filling Mr. Poop with school-related ideas written on strips of paper. Give Mr. Poop a shake and a scrap of paper flutters out–today’s idea was to design your own game, which the kids are working on now.

I’ve decided to incorporate the Cub Scout handbooks–which have lots of useful information about subjects like first aid, water safety, civics, history, etc.–into our rotation. (The Cub Scouts have a different handbook for 1st, 2nd, 3rd, and 4th graders.) Today we learned about knots–mostly square knots–complemented with The Camper’s Knot Tying Game. Knots are practical for anyone, but also good practice for kids with fine motor difficulties.

Over in Professor Astro Cat, we’re collecting space dust, keeping a moon journal (the eclipse was well-timed for this) and made impact craters in the sandbox. The book recommends spreading out newspaper indoors and using flour or cocoa powder, but sand, outside, is much easier to clean up. (Walmart sells beautiful colored sand for like $4 a bag. I sprinkled some green on top of the regular brown sandbox sand to simulate Earth’s surface.)

Custom Dice

There are lots of interesting dice–math dice, fraction dice, letter dice, place value dice, etc. Customized dice are easy to make: just take a cube (you probably have a building block or letter cube or some Legos lying around,) cover it with paper, and write whatever you want on the faces. (Note it is probably best to write on the paper before applying tape, as many pens won’t write properly on tape.) I have a custom die with +,-, <, and division signs on it that I use along with custom “numbers larger than six” dice for math games. (“Looks like you rolled 5,000,000,000 divided by 7,000!”) (For smaller kids, you may want to stick to + and -.)

I’m still trying to work out good ways to teach history. I’ve got some rudimentary ideas, but I’ll save them for later.

Homeschooling Corner: What is Educational?

Many years ago, I worked at a toy store. (It was a lot of fun–I like feeling helpful.) One day I was helping a grandmother pick out a gift for her young granddaughter, a  6-month old whom she proudly informed me was really at the developmental level of a 9-month old. She asked me to recommend something “educational” for the child.

Being, (I confess) not very good at this, I responded in confusion that pretty much everything is educational for a baby. Babies are learning all the time.

One of my co-workers helpfully jumped in and found her a stuffed dog that’s supposed to help babies learn to read.

Many years and a great deal of IQ and education research later, I stand by my original position: “educational” toys for babies probably aren’t. It would be great if we could find some magical technique–say, playing Classical music to your fetus–that could reliably make people into geniuses, but so far we haven’t actually found any.

Does that mean you shouldn’t play Classical music to your fetus? Of course not. There’s no evidence that Classical music hurts babies. It’s just highly likely that being the kind of person who would play Classical music to your fetus is a bigger factor in how your kids turn out than the actual music. But if you hate Classical, don’t sweat it.

If you’ve been hanging around the HBD-osphere for a while, you probably already know that adult IQ appears to be about 50% genetic and 50% random chance. That doesn’t mean you shouldn’t try to teach your kids things (you should definitely teach your kids things.) It just means that it’s more important to be good and kind to your kids than to buy them any particular toy or splurge on baby genius classes.


The youngest child has been really into rocks and crystals lately; at this age, they’re still fascinated by freezing cups of water to make ice. A little food coloring (rainbow ice!) fresh mint leaves, flowers, whatever you want to add makes the ice extra interesting; add salt for a lesson about icebergs and the ocean. I don’t know if this is really educational, but it’s fun.

The rest of our time has been focused primarily on regular old rote material–times tables, multi-digit addition and subtraction, handwriting, typing, spelling, etc. Luckily these skills are pretty flexible and so can be taught to multiple kids at different levels. Competitive multiplication games (try to call out the answer first!) work well in our household.

We also learned about If-Then statements in code and the Apollo 11 moon landing, which resulted in them coding a short animation about the moon launch.

I do hope that once I feel a little more secure about their basic skills, we can move outside for some active learning/PE/map-making type activities.