I am specifically referring to Han Chinese from the People’s Republic of China (hereafter simply called “China,”) but wanted to keep the title to a reasonable length.
There are about a billion Han Chinese. They make up about 90% of the PRC, and they have some of the highest average IQs on the planet, with particularly good math scores.
Of the 56 Fields Medals (essentially, the Nobel for Math) awarded since 1936, 12 (21%) have been French. 14 or 15 have been Jewish, or 25%-27%.
By contrast, 0 have been Han Chinese from China itself.
France is a country of 67.15 million people, of whom about 51 million are native French. The world has about 14-17.5 million Jews. China has about 1.37 billion people, of whom 91.51% are Han, or about 1.25 billion.
Two relatively Chinese people have won Fields medals:
Shing-Tung Yau was born in China, but is of Hakka ancestry (the Hakka are an Asian “market-dominant minority,”) not Han. His parents moved to Hong Kong when he was a baby; after graduating from the Chinese University of Hong Kong, he moved to the US, where he received his PhD from Berkley. Yau was a citizen of British-owned Hong Kong (not the People’s Republic of China), when he won the Fields Medal, in 1982; today he holds American citizenship.
Terence Tao, the 2006 recipient, is probably Han (Wikipedia does not list his ethnicity.) His father hailed from Shanghai, China, but moved to Hong Kong, where he graduated from medical school and met Tao’s mother, another Hong Kong-ian. Tao himself was born in Australia and later moved to the US. (Tao appears to be a dual Australian-American citizen.)
(With only 7.4 million people, Hong Kong is doing pretty well for itself in terms of Fields Medalists with some form of HK ancestry or citizenship.)
Since not many Fields Medals have been awarded, it is understandable why the citizens of small countries, even very bright ones, like Singapore, might not have any. It’s also understandable why top talent often migrates to places like Hong Kong, Australia, or the US. But China is a huge country with a massive pool of incredibly smart people–just look at Shanghai’s PISA scores. Surely Beijing has at least a dozen universities filled with math geniuses.
So where are they?
Is it a matter of funding? Has China chosen to funnel its best mathematicians into applied work? A matter of translation? Does the Fields Medal Committee have trouble reading papers written in Chinese? A matter of time? Did China’s citizens simply spent too much of the of the past century struggling at the edge of starvation to send a bunch of kids off to university to study math, and only recently achieved the level of mass prosperity necessary to start on the Fields path?
Whatever the causes of current under-representation, I have no doubt the next century will show an explosion in Han Chinese mathematical accomplishments.
Lockhart’s basic take is that most of us have math backwards. We approach (and thus teach) it as useful but not fun–something to be slogged through, memorized, and then avoided as much as possible. By contrast, Lockhart sees math as more fun than useful.
I do not mean that Lockhart denies the utility of balancing your checkbook or calculating how much power your electrical grid can handle, but most of the math actual mathematicians do isn’t practical. They do it because they enjoy it; they love making patterns with numbers and shapes. Just because paint has a very practical use in covering houses doesn’t mean we shouldn’t encourage kids to enjoy painting pictures; similarly, Lockhart wants kids to see mathematics as fun.
But wait, you say, what if this loosey-goosey, free-form, new math approach results in kids who spend a lot of time trying to re-derive pi from first principles but never really learning algebra? Lockhart would probably counter that most kids never truly master algebra anyway, so why make them hate it in the process? Should we only let kids who can paint like the Masters take art class?
If you and your kids already enjoy math, Lockhart may just reinforce what you already know, but if you’re struggling or math is a bore and a chore, Lockhart’s perspective may be just what you need to turn things around and make math fun.
For example: There are multiple ways to group the numbers during double-digit multiplication, all equally “correct”; the method you chose is generally influenced by things like your familiarity with double-digit multiplication and the difficulty of the problem. When I observed one of my kids making errors in multiplication because of incorrect regrouping, I showed them how to use a more expanded way of writing out the numbers to make the math clearer–promptly eliciting protests that I was “doing it wrong.” Inspired by Lockhart, I explained that “There is no one way to do math. Math is the art of figuring out answers, and there are many ways to get from here to there.” Learning how to use a particular approach—“Put the numbers here, here, and here and then add them”–is useful, but should not be elevated above using whatever approach best helps the child understand the numbers and calculate the correct answers.
The only difficulty with Lockhart’s approach is figuring out what to actually do when you sit down at the kitchen table with your kids, pencil and paper in hand. The book has a couple of sample lessons but isn’t a full k-12 curriculum. It’s easy to say, “I’m going to do a free-form curriculum that requires going to the library every day and uses every experience as a learning opportunity,” and rather harder to actually do it. With a set curriculum, you at least know, “Here’s what we’re going to do today.”
My own personal philosophy is that school time should be about 50% formal instruction and 50% open-ended exploration. Kids need someone to explain how the alphabet works and what these funny symbols on the math worksheet mean; they also need time to read fun books and play with numbers. They should memorize their times tables, but a good game can make times tables fun. In short, I think kids should have both a formal, straightforward curriculum or set of workbooks (I have not read enough math textbooks to recommend any particular ones,) and a set of math enrichment activities, like tangrams, pattern blocks, reading about Penrose the Mathematical Cat, or watching Numberphile on YouTube.
(Speaking of Penrose, I thought the chapter on binary went right over my kids’ heads, but yesterday they returned all of their answers in math class in binary, so I guess they picked up more than I gave them credit for.)
YouCubed.org is an interesting website I recently discovered. So far we’ve only done two of the activities, but they were cute and I suspect the website will make a useful addition to our lessons. If you’ve used it, I’d love to hear your thoughts on it.
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.)
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.
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.
(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.
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.
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.
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.
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.