Learning in Numbers

There is strength in numbers, but is there wisdom?

I’ve heard from multiple sources the claim that parenting, paradoxically, gets easier after the fourth child. There are several simple explanations for this phenomenon: people get more skilled at parenting after lots of practice; the older kids start helping out with the younger ones, etc.

But what if the phenomenon rests on something much more basic about human psychology–our desire to imitate others?

(Perhaps you don’t, dear reader. There are always exceptions.)

As Aristotle put it, man is a political animal–by which he meant that we are inherently social and prone to building communities (polities) together, not that we are inherently prone to arguing about who should govern North Carolina, though that may be political, too. In Aristotle’s words, a man who lives entirely alone is either a beast (living like an animal) or a god (able to fulfill all of his own needs without recourse to other humans.) Normal humans depend in many ways on other humans.

Compared to our pathetic ability to learn math (just look at most people’s SAT-math scores) and inability to read without direct instruction, humans learn socially-imparted skills like the ability to speak multiple languages, play games, assert dominance over each other, which clothes are fashionable, and how to crack a socially-appropriate joke with ease.

Social learning comes so naturally to people that we only notice it in cases of extreme deficit–like autism–or when parents protest that their children are becoming horribly corrupted by their peers.

So perhaps households with more than 4 children have hit a threshold beyond which social learning takes over and the younger children simply seem to “absorb” knowledge from their older siblings instead of having to be explicitly taught.

Consider learning to eat, a hopefully simple task. We are born with instincts to nurse, put random things in our mouths, and swallow. Preventing babies from eating random non-food objects is a bit of a problem for new parents. But learning things like “how to get this squishy food into your mouth with a spoon without also getting it everywhere else in the room” is much more complicated–and humans take food rituals to much more complicated heights than strained peas and carrots.

Parents of new children put a great deal of effort into teaching them to eat (something that ought to be an instinct.) Those with means puree fresh veggies, chop bits of meat, show a sudden interest in organics, and sit down to spoon every single last bit into their infants’ mouths. It is as if they are convinced that kids cannot learn to eat without at least as much instruction as a student learning to wield a welding torch. (And based on my own experience, they’re probably right.)

By contrast, parents of multiple children have–by necessity–relaxed. As a popular comic once depicted (though I can’t find it now,) feeding at this point becomes throwing Cheerios at the highchair as you run by.

Yet I’ve never seen any evidence that the younger children in large families are likely to be malnourished–they seem to catch the Cheerios on the fly and do just fine.

What if imitation is a strong factor in larger families, allowing infants and young children to learn skills like “how to eat” without needing direct parental instruction just by watching their older siblings? You might object that even infants in single parent households could learn to eat by imitating their parents (and they probably do,) but having more people around probably enforces the behavior more strongly, and having younger children around gives an example that is much more similar to the infant. We adults are massive compared to children, after all.

If basic learning of life skills proceeds more easily in an environment with more peers,(for infants or adults,) then what effects should we expect from our current trend toward extreme atomization?

I recently came across an essay about life in a trailer park vs sturdier housing:

To me, growing up in that trailer park meant playing until dark with neighborhood kids, building tree houses and snow forts. Listening out my bedroom window for the sound of my dad’s pickup truck leaving for work in the early morning. Riding my bike down the big hill at the top of the lot, avoiding potholes and feeling safe because there wasn’t much traffic and if I fell and skinned my knee, someone would come out on their front porch and ask if I was okay.

Some of the only happy memories I have of my childhood were from that time in my life, before my parents were thrust into insurmountable debt, before my mother was hospitalized, before I had to go live with my grandmother. Nana had a real house. She didn’t live in a trailer. But when she would scream at me or try to attack me as I squeezed by her and fled upstairs, I wished I had neighbors close by to hear her — to believe me, and to perhaps even help.

The most dysfunctional and unstable years of my life were spent in a real house, with four walls and a slanted roof — where fences went up between the houses so that no one ever had to feel responsible for what went on behind their neighbor’s front door.

This is more about atomization than learning, but still interesting. Is it good for humans to be so far apart? To live far from relatives, in houses with thick walls, as single children or single adults, working and commuting every day among strangers?

Certainly the downsides of being among relatives are well-documented. Many tribal societies have downright cruel customs directed at relatives, like sati or adult circumcision. But that doesn’t mean that the extreme opposite–total atomization–is perfect. Atomization carries other risks. Among them, staying indoors and not socializing with our neighbors may cause us to lose some of our social knowledge, our ability to learn how to exist together.

We might expect that physical atomization due to technological change (sturdier houses, more entertaining TV, comfier climate control systems,) could cause symptoms in people similar to those caused by medical deficits in social learning, like autism. A recent study on the subject found an interesting variation between the brains of normies and autists:

So great was the difference between the two groups that the researchers could identify whether a brain was autistic or neurotypical in 33 out of 34 of the participants—that’s 97% accuracy—just by looking at a certain fMRI activation pattern. “There was an area associated with the representation of self that did not activate in people with autism,” Just says. “When they thought about hugging or adoring or persuading or hating, they thought about it like somebody watching a play or reading a dictionary definition. They didn’t think of it as it applied to them.” This suggests that in autism, the representation of the self is altered, which researchers have known for many years, Just says.

This might explain the high rates of body dysmorphias in autism. It might also explain the high rates in society.

I remember another study which I read ages ago which found that people basically thought about “God” in the same parts of their brain where they thought about themselves. This explains why God tends to have the same morals as His believers. If autists have trouble imagining themselves, then they may also have trouble imagining God–and this might explain rising atheism rates.

Even our rising autism rates, though probably driven primarily by shifts in diagnostic fads, might be influenced by shrinking families and greater atomization, as kids with borderline conditions might show more severe symptoms if they are also more isolated.

On the other hand, social media is allowing people to come together and behave socially in new and ever larger groups.

For all their weaknesses, autists are probably better at normies at certain kinds of tasks, like abstract reasoning where you don’t want to think too much about yourself. I have long suspected that normies balk at philosophical dilemmas such as the trolley problem because they over-empathize with the subjects. Imagining themselves as one of the victims of the runaway trolley causes them distress, and distress causes them to attack the person causing them distress–the philosopher.

And so the citizens of Athens condemned Socrates to death.

But just as people can overcome their natural and very sensible fear of heights in order to work on skyscrapers, perhaps they can train themselves not to empathize with the subjects of trolley problems. Spending time on problems with no human subjects (such as mathematics or engineering) may also help people practice ways of approaching problems that don’t immediately resort to imagining themselves as the subject. On the converse, perhaps a bit of atomization (as seen historically in countries like Britain and France, and recently AFAIK in Japan,) helps equip people to think about difficult, non-human related mathematical or engineering problems.

Thoughts?

Homeschooling Corner: Math ideas and manipulatives for younger grades

Archimedes

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!)

Countables

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.

Fractions

Fraction blocks and fraction circles are both handy.

Games

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.

Books

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.)

 

Textbooks

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.

Anthropology Friday: Numbers and the Making of Us, by Caleb Everett, pt 3

Welcome back to our discussion of Numbers and the Making of Us: Counting and the Course of Human Cultures, by Caleb Everett.

The Pirahã are a small tribe (about 420) of Amazonian hunter-gatherers whose language is nearly unique: it has no numbers, and you can whistle it. Everett spent much of his childhood among the Piraha because his parents were missionaries, which probably makes him one of the world’s foremost non-Piraha experts on the Piraha.

Occasionally as a child I would wake up in the jungle to the cacophony of people sharing their dreams with one another–impromptu monologues followed by spurts of intense feedback. The people in question, a fascinating (to me anyhow) group known as the Piraha, are known to wake up and speak to their immediate neighbors at all hours of the night. … the voices suggested the people in the village were relaxed and completely unconcerned with my own preoccupations. …

The Piraha village my family lived in was reachable via a one-week sinuous trip along a series of Amazonian tributaries, or alternatively by a one-or flight in a Cessna single-engine airplane.

Piraha culture is, to say the least, very different from ours. Everett cites studies of Piraha counting ability in support of his idea that our ability to count past 3 is a culturally acquired process–that is, we can only count because we grew up in a numeric society where people taught us numbers, and the Piraha can’t count because they grew up in an anumeric society that not only lacks numbers, but lacks various other abstractions necessary for helping make sense of numbers. Our innate, genetic numerical abilities, (the ability to count to three and distinguish between small and large amounts,) he insists, are the same.

You see, the Piraha really can’t count. Line up 3 spools of thread and ask them to make an identical line, and they can do it. Line up 4 spools of thread, and they start getting the wrong number of spools. Line up 10 spools of thread, and it’s obvious that they’re just guessing and you’re wasting your time. Put five nuts in a can, then take two out and ask how many nuts are left: you get a response on the order of “some.”*

And this is not for lack of trying. The Piraha know other people have these things called “numbers.” They once asked Everett’s parents, the missionaries, to teach them numbers so they wouldn’t get cheated in trade deals. The missionaries tried for 8 months to teach them to count to ten and add small sums like 1 + 1. It didn’t work and the Piraha gave up.

Despite these difficulties, Everett insists that the Piraha are not dumb. After all, they survive in a very complex and demanding environment. He grew up with them; many of the are his personal friends and he regards them as mentally normal people with the exact same genetic abilities as everyone else who just lack the culturally-acquired skill of counting.

After all, on a standard IQ scale, someone who cannot even count to 4 would be severely if not profoundly retarded, institutionalized and cared for by others. The Piraha obviously live independently, hunt, raise, and gather their own food, navigate through the rainforest, raise their own children, build houses, etc. They aren’t building aqueducts, but they are surviving perfectly well outside of an institution.

Everett neglects the possibility that the Piraha are otherwise normal people who are innately bad at math.

Normally, yes, different mental abilities correlate because they depend highly on things like “how fast is your brain overall” or “were you neglected as a child?” But people also vary in their mental abilities. I have a friend who is above average in reading and writing abilities, but is almost completely unable to do math. This is despite being raised in a completely numerate culture, going to school, etc.

This is a really obvious and life-impairing problem in a society like ours, where you have to use math to function; my friend has been marked since childhood as “not cognitively normal.” It would be a completely invisible non-problem in a society like the Piraha, who use no math at all; in Piraha society, my friend would be “a totally normal guy” (or at least close.)

Everett states, explicitly, that not only are the Piraha only constrained by culture, but other people’s abilities are also directly determined by their cultures:

What is probably more remarkable about the relevant studies, though, is that they suggest that climbing any rungs of the arithmetic ladder requires numbers. How high we climb the ladder is not the result of our own inherent intelligence, but a result of the language we speak and of the culture we are born into. (page 136)

This is an absurd claim. Even my own children, raised in identically numerate environments and possessing, on the global scale, nearly identical genetics, vary in math abilities. You are probably not identical in abilities to your relatives, childhood classmates, next door neighbors, spouse, or office mates. We observe variations in mathematical abilities within cultures, families, cities, towns, schools, and virtually any group you chose that isn’t selected for math abilities. We can’t all do calculus just because we happen to live in a culture with calculus textbooks.

In fact, there is an extensive literature (which Everett ignores) on the genetics of intelligence:

Various studies have found the heritability of IQ to be between 0.7 and 0.8 in adults and 0.45 in childhood in the United States.[6][18][19] It may seem reasonable to expect that genetic influences on traits like IQ should become less important as one gains experiences with age. However, that the opposite occurs is well documented. Heritability measures in infancy are as low as 0.2, around 0.4 in middle childhood, and as high as 0.8 in adulthood.[7] One proposed explanation is that people with different genes tend to seek out different environments that reinforce the effects of those genes.[6] The brain undergoes morphological changes in development which suggests that age-related physical changes could also contribute to this effect.[20]

A 1994 article in Behavior Genetics based on a study of Swedish monozygotic and dizygotic twins found the heritability of the sample to be as high as 0.80 in general cognitive ability; however, it also varies by trait, with 0.60 for verbal tests, 0.50 for spatial and speed-of-processing tests, and 0.40 for memory tests. In contrast, studies of other populations estimate an average heritability of 0.50 for general cognitive ability.[18]

In 2006, The New York Times Magazine listed about three quarters as a figure held by the majority of studies.[21]

Thanks to Jayman

In plain speak, this means that intelligence in healthy adults is about 70-80% genetic and the rest seems to be random chance (like whether you were dropped on your head as a child or had enough iodine). So far, no one has proven that things like whole language vs. phonics instruction or two parents vs. one in the household have any effect on IQ, though they might effect how happy you are.

(Childhood IQ is much more amenable to environmental changes like “good teachers,” but these effects wear off as soon as children aren’t being forced to go to school every day.)

A full discussion of the scientific literature is beyond our current scope, but if you aren’t convinced about the heritability of IQ–including math abilities–I urge you to go explore the literature yourself–you might want to start with some of Jayman’s relevant FAQs on the subject.

Everett uses experiments done with the Piraha to support his claim that mathematical ability is culturally dependent, but this is dependent on is claim that the Piraha are cognitively identical to the rest of us in innate mathematical ability. Given that normal people are not cognitively identical in innate mathematical abilities, and mathematical abilities vary, on average, between groups (this is why people buy “Singapore Math” books and not “Congolese Math,”) there is no particular reason to assume Piraha and non-Piraha are cognitively identical. Further, there’s no reason to assume that any two groups are cognitively identical.

Mathematics only really got started when people invented agriculture, as they needed to keep track of things like “How many goats do I have?” or “Have the peasants paid their taxes?” A world in which mathematical ability is useful will select for mathematical ability; a world where it is useless cannot select for it.

Everett may still be correct that you wouldn’t be able to count if you hadn’t been taught how, but the Piraha can’t prove that one way or another. He would first have to show that Piraha who are raised in numerate cultures (say, by adoption,) are just as good at calculus as people from Singapore or Japan, but he cites no adoption studies nor anything else to this end. (And adoption studies don’t even show that for the groups we have studied, like whites, blacks, or Asians.)

Let me offer a cognitive contrast:

The Piraha are an anumeric, illiterate culture. They have encountered both letters and numbers, but not adopted them.

The Cherokee were once illiterate: they had no written language. Around 1809, an illiterate Cherokee man, Sequoyah, observed whites reading and writing letters. In a flash of insight, Sequoyah understand the concept of “use a symbol to encode a sound” even without being taught to read English. He developed his own alphabet (really a syllabary) for writing Cherokee sounds and began teaching it to others. Within 5 years of the syllabary’s completion, a majority of the Cherokee were literate; they soon had their own publishing industry producing Cherokee-language books and newspapers.

The Cherokee, though illiterate, possessed the innate ability to be literate, if only exposed to the cultural idea of letters. Once exposed, literacy spread rapidly–instantly, in human cultural evolution terms.

By contrast, the Piraha, despite their desire to adopt numbers, have not been able to do so.

(Yet. With enough effort, the Piraha probably can learn to count–after all, there are trained parrots who can count to 8. It would be strange if they permanently underperformed parrots. But it’s a difficult journey.)

That all said, I would like to make an anthropological defense of anumeracy: numeracy, as in ascribing exact values to specific items, is more productive in some contexts than others.

Do you keep track of the exact values of things you give your spouse, children, or close friends? If you invite a neighbor over for a meal, do you mark down what it cost to feed them and then expect them to feed you the same amount in return? Do you count the exact value of gifts and give the same value in return?

In Kabloona, de Poncin discusses the quasi-communist nature of the Eskimo economic system. For the Eskimo, hunter-gatherers living in the world’s harshest environment, the unit of exchange isn’t the item, but survival. A man whom you keep alive by giving him fish today is a man who can keep you alive by giving you fish tomorrow. Declaring that you will only give a starving man five fish because he previously gave you five fish will do you no good at all if he starves from not enough fish and can no longer give you some of his fish when he has an excess. The fish have, in this context, no innate, immutable value–they are as valuable as the life they preserve. To think otherwise would kill them.

It’s only when people have goods to trade, regularly, with strangers, that they begin thinking of objects as having defined values that hold steady over different transactions. A chicken is more valuable if I am starving than if I am not, but it has an identical value whether I am trading it for nuts or cows.

So it is not surprising that most agricultural societies have more complicated number systems than most hunter-gatherer societies. As Everett explains:

Led by Patience Epps of the University of Texas, a team of linguists recently documented the complexity of the number systems in many of the world’s languages. In particular, the researchers were concerned with the languages’ upper numerical limit–the highest quantity with a specific name. …

We are fond of coining new names for numbers in English, but the largest commonly used number name is googol (googolplex I define as an operation,) though there are bigger one’s like Graham’s.

The linguistic team in question found the upper numerical limits in 193 languages of hunter-gatherer cultures in Australia, Amazonia, Africa, and North America. Additionally, they examined the upper limits of 204 languages spoken by agriculturalists and pastoralists in these regions. They discovered that the languages of hunter-gatherer groups generally have low upper limits. This is particularly true in Australia and Amazonia, the regions with so-called pure hunter-gatherer subsistence strategies.

In the case of the Australian languages, the study in question observed that more than 80 percent are limited numerically, with the highest quantity represetned in such cases being only 3 or 4. Only one Australian language, Gamilaraay, was found to have an upper limit above 10, an dits highest number is for 20. … The association [between hunter-gathering and limited numbers] is also robust in South America and Amazonia more specifically. The languages of hunter-gatherer cultures in this region generally have upper limits below ten. Only one surveyed language … Huaorani, has numbers for quantities greater than 20. Approximately two-thirds of the languages of such groups in the region have upper limits of five or less, while one-third have an upper limit of 10. Similarly, about two-thirds of African hunter-gatherer languages have upper limits of 10 or less.

There are a few exceptions–agricultural societies with very few numbers, and hunter-gatherers with relatively large numbers of numbers, but:

…there are no large agricultural states without elaborate number systems, now or in recorded history.

So how did the first people develop numbers? Of course we don’t know, but Everett suggests that at some point we began associating collections of things, like shells, with the cluster of fingers found on our hands. One finger, one shell; five fingers, five shells–easy correspondences. Once we mastered five, we skipped forward to 10 and 20 rather quickly.

Everett proposes that some numeracy was a necessary prerequisite for agriculture, as agricultural people would need to keep track of things like seasons and equinoxes in order to know when to plant and harvest. I question this on the grounds that I myself don’t look at the calendar and say, “Oh look, it’s the equinox, I’d better plant my garden!” but instead look outside and say, “Oh, it’s getting warm and the grass is growing again, I’d better get busy.” The harvest is even more obvious: I harvest when the plants are ripe.

Of course, I live in a society with calendars, so I can’t claim that I don’t look at the calendar. I look at the calendar almost every day to make sure I have the date correct. So perhaps I am using my calendrical knowledge to plan my planting schedule without even realizing it because I am just so used to looking at the calendar.

“What man among you, if he has 100 sheep and has lost 1 of them, does not leave the 99 in the open pasture and go after the one which is lost until he finds it? When he has found it, he lays it on his shoulders, rejoicing.” Luke 15:3-5

Rather than develop numbers and then start planting barley and millet, I propose that humans first domesticated animals, like pigs and goats. At first people were content to have “a few,” “some,” or “many” animals, but soon they were inspired to keep better track of their flocks.

By the time we started planting millet and wheat (a couple thousand years later,) we were probably already pretty good at counting sheep.

Our fondness for tracking astronomical cycles, I suspect, began for less utilitarian reasons: they were there. The cycles of the sun, moon, and other planets were obvious and easy to track, and we wanted to figure out what they meant. We put a ton of work into tracking equinoxes and eclipses and the epicycles of Jupiter and Mars (before we figured out heliocentrism.) People ascribed all sorts of import to these cycles (“Communicator Mercury is retrograde in outspoken Sagittarius from December 3-22, mixing up messages and disrupting pre-holiday plans.”) that turned out to be completely wrong. Unless you’re a fisherman or sailor, the moon’s phases don’t make any difference in your life; the other planets’ cycles turned out to be completely useless unless you’re trying to send a space probe to visit them. Eclipses are interesting, but don’t have any real effects. For all of the effort we’ve put into astronomy, the most important results have been good calendars to keep track of dates and allow us to plan multiple years into the future.

Speaking of dates, let’s continue this discussion in a week–on the next Anthropology Friday.

*Footnote: Even though I don’t think the Piraha prove as much as Everett thinks they do, that doesn’t mean Everett is completely wrong. Maybe already having number words is (in the vast majority of cases) a necessary precondition for learning to count.

One potentially illuminating case Everett didn’t explore is how young children in numerate culture acquire numbers. Obviously they grow up in an environment with numbers, but below a certain age can’t really use them. Can children at these ages duplicate lines of objects or patterns? Or do they master that behavior only after learning to count?

Back in October I commented on Schiller and Peterson’s claim in Count on Math (a book of math curriculum ideas for toddlers and preschoolers) that young children must learn mathematical “foundation” concepts in a particular order, ie:

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.

When children can match objects one to one, they can compare sets to determine which have more and which have less.

When children can compare sets, they can begin to look at the “manyness” of one set and develop number concepts.

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

The Count on Math curriculum doesn’t even introduce the numbers 1-5 until week 39 for 4 year olds (3 year olds are never introduced to numbers) and numbers 6-10 aren’t introduced until week 37 for the 5 year olds!

Note that Schiller and Everett are arguing diametrical opposites–Everett says the ability to count to three and distinguish the “manyness” of sets is instinctual, present even in infants, but that the ability to copy patterns and match items one-to-one only comes after long acquaintance and practice with counting, specifically number words.

Schiller claims that children only develop the ability to distinguish manyness and count to three after learning to copy patterns and match items one-to-one.

As I said back in October, I think Count on Math’s claim is pure bollocks. If you miss the “comparing sets” day at preschool, you aren’t going to end up unable to multiply. The Piraha may not prove as much as Everett wants them to, but the neuroscience and animal studies he cites aren’t worthless. In general, I distrust anyone who claims that you must introduce this long a set of concepts in this strict an order just to develop a basic competency that the vast majority of people seem to acquire without difficulty.

Of course, Lynne Peterson is a real teacher with a real teacher’s certificate and a BA in … it doesn’t say, and Pam Schiller was Vice President of Professional Development for the Early childhood Division at McGraw Hill publishers and president of the Southern Early Childhood Association. She has a PhD in… it doesn’t say. Here’s some more on Dr. Schiller’s many awards. So maybe they know better than Everett, who’s just an anthropologist. But Everett has some actual evidence on his side.

But I’m a parent who has watched several children learn to count… and Schiller and Peterson are wrong.