For those of you who just
came in: my major writing project at the moment is a nonfiction book called Singapore Math Figured Out for Parents,
which is just that. As I finish it I'm posting some sections that may be of
general interest to parents, math educators, and nerds here. The following is
an excerpt from Part Zero, part of a section of a brief and necessarily
incomplete history of math teaching in America.
To clarify things for
newcomers, especially those trying to get the hang of this new Singapore Math
stuff, Singapore Math was developed primarily under the direction of Dr. Kho
Tek Hong in Singapore in the early 1980s, when that very small nation was
realizing that their only hope of not being swallowed by more powerful
neighbors, either politically or economically, lay in developing a modern
high-tech workforce. It's a bit hard to
tell how much of the insight and work was Dr. Kho's directly and how much was
his management, because he's a modest person, in addition to being extremely
good at what he does.
Singapore Math is associated
with the elevated test scores of the half dozen nations that have adopted it
thoroughly, most of them in Asia. In the United States it is coming in
piecemeal via various charter schools (many nominally for the gifted and
talented), but some parts of it have been incorporated into the Common Core.
This gives me a more than uneasy feeling because one thing American taxpayers
don't like to pay for is the needed teacher training to make curriculum reforms
work; many great programs have succeeded beautifully in pilot programs and then
been rolled out in the form of a pile of textbooks and a superintendent's
orders to stay a week ahead of the kids, and have collapsed because learning a
genuinely new way of teaching takes time, energy, focus (and thus enough time
to think) and we almost never give our teachers enough of that.
Dr. Kho's great insight, as
developed by two generations of great math teaching (mostly in Asia), is that
preparation for the highest level of mathematics a student can reach (whether
that's a developmentally-delayed student learning to count change, or a whiz
kid starting calculus before puberty) requires a coordinated, integrated,
developmentally appropriate balance between fundamental skills and mathematical
understanding.
In Singapore Math classrooms
as set up and run by Dr. Kho and his colleagues, students actively pursue quick
and ready skills to comprehend more, and use their better comprehension to
sharpen and direct their skills. That virtuous circle works because it's not
just a means to some end like higher test scores, better job prospects,
increased self-esteem, or parents getting off teachers' cases. Singapore Math is fundamentally about starting and feeding
the cycle of correctly-related proficiency and understanding, spiraling upward
into confident mastery of as much mathematics as the student ultimately chooses
and can handle.
Bluntly, I think my own sloppy, often willfully dumb
country will probably screw up the introduction of Singapore Math. Teachers who
have a horror of drills and skills will try to teach only the "understanding"
part; teachers who like the old algorithms will find the parts of the book
where they can drill those and skip over "all that theory stuff";
many teachers won't see how the two parts are related because no one will pay
anyone to explain it to them. Our big sloppy stupid impatient country will then
blame everyone except our own unwillingness to spend the money, and more
importantly the patience and care, to do it properly. Something in the American
psyche thinks that if you can't win a game of tennis, the solution is not
lessons, not going back to fundamentals, not training, but buying a newer, more
expensive racket, and then busting it over your knee in a fury when you're
still a crappy tennis player.
So one auxiliary part of the
book is my history of past reforms and why they didn't work; and this first
part is a history of why reforms were needed in the first place. I suppose you
could call it, as I do above, so what was wrong with American math
teaching in the first place?
This week we'll look at what's
wrong with the oldest American tradition, traditional proceduralism, the
"basics" that some people always want to go "back to," because to understand how reasonably smart,
honest people made their mistakes, we need to understand what earlier mistakes
they thought they were going to fix.
Traditional proceduralism led
us into digging a hole we can't afford to stay in, and the time has come to
climb out of it, for the future of our kids and our country. And as common
sense tells us, you can't get out of a hole by digging. Unfortunately, many
subsequent reformers tried to get out of it by pretending it's not a hole,
believing they could fly, or holding the shovel upside down. But we'll pick on
them in the weeks to come; for the moment, let's see where Grandpa's way went
wrong.
§
Traditional proceduralism: abusive psychology meets obsolete
economics, or how to succeed right till you hit the wall
One tradition for which many
Americans retain a great deal of affection is traditional proceduralism. That's
the version that Cobb and Edwards milked nostalgia from in their 1907 song
"School Days (When We Were A Couple o' Kids)":
"readin' and 'ritin' and 'rithmetic,
taught to the tune of the hick'ry stick"
taught to the tune of the hick'ry stick"
That song was a big hit in
1907, the year my grandfather was born, and in it, an old couple are
reminiscing about their pre-pubescent schoolkid mutual crush, presumably
several decades before in the 1840s or 50s. Given that I'm a grandfather
myself, that's about nine generations of nostalgia.
"School Days (When We
Were A Couple o' Kids)" gave us that illiterate phrase the "Three
Rs." I do not think it's any accident that in addition to celebrating that
literally "old school" curriculum, it also celebrates writing with
chalk on slates, kids coming to school without shoes, and teachers beating kids
with sticks.
Among our citizens, one
strain in thinking about education seems to dislike kids and feel that pain,
privation, shame, and misery, especially if pointless, is what the little scum
deserve. It's an impulse that never seems to be far below the surface in
present-day conservative critics of education, a part of the impulse behind
some charter schools and homeschooling, a notion that often underlies
"back to basics" and "in my day..."
So let's begin by dismissing
the nostalgic attraction of that little red schoolhouse. It's a story about as
meaningful and true as George Washington's cherry tree, or the depictions of
happy slaves in the cotton fields. The one-room schoolhouse steeped in rural
poverty and staffed by the only single woman in the county who could read
hasn't really existed since about World War One. Joe and Nelly in "School
Days" would have been a couple o' kids well before the Civil War. We might
as well look for contemporary American fashion and activities in Currier and
Ives prints.
Traditional proceduralism
began in the methods first used to train the children of the merchant and
shopkeeper class in the Renaissance for reliable bookkeeping. Kids memorized
step-by-step procedures, usually organized as patterns of manipulating symbols.
This was thought to train and develop the mind, just as throwing straight
punches into a heavy bag and doing lots of pushups will train the triceps.
The same principle later
underlay memorizing long vocabulary lists, state capitals, chemical elements,
and so forth. The mind was mostly memory, memory was a muscle, muscles needed
developing, and muscles are developed by exercise. The harder the exercise, the
more development: no pain, no gain. The mind developed just as well regardless
of whether what it was working at something useful or just working, just as it
didn't really matter if you were throwing hay bales in the field, or doing
bicep curls in a gym.
Perhaps an example will make
this clear: before 1890, nearly all public schools in the U.S. required eighth
graders to be able to extract square and cube roots with pencil and paper. As
late as World War II, a large number of kids had to learn it.
Chances are that no one
taught you how. If you try to learn, you'll find that there are many more, and
more complicated, steps than you ever encountered in long division. (You might
say it's all the difficulties of long division, squared and cubed). Your
three-times-great-grandfather probably did it for an hour a day for several weeks
to learn it in sixth grade, if the records of nineteenth century schools are to
be trusted. According to surveys taken in the 1930s (admittedly, by education
researchers who were trying to release children from extracting cube roots by
hand), the great majority of students were never told how or why the method
worked, and a substantial proportion of your great-grandfathers who could
extract roots had no idea what a root was.
Is there anybody who feels
that you are missing something by not being able to extract roots by hand? Do
you wish you'd spent more of your childhood learning a lengthy, complicated
process to produce a number that you might or might not understand? And do you
wish you had been taught it under threat of being paddled or slapped (which
would not be permitted in boot camp today)? Please notice I'm not asking
whether you think it was good for kids; would you personally like to have a
memory of being spanked for not knowing your cube root extraction algorithm by
heart?
So to begin with, harsh, uncompromising meaninglessness partly
provoked the original reactions against traditional proceduralism, and might
partly explain some of its enduring attraction. Some people just like to wallop
kids.
But that is not and cannot be
the whole story. Nowadays, traditional proceduralism is taught with immensely
more patience and kindness than it used to be. The teacher doesn't make the
kids who failed the quiz over the sevens table face the class to be laughed at.
A kid who gives a dumb answer is not rapped on the knuckles with a ruler, and a
kid who asks a dumb question is not made to sit in a corner; getting math wrong
or not understanding it is no longer assumed to be willful criminality or moral
shame. We don't declare that on Friday every student will work a long division
problem on the board and those who cannot do it will receive three hard whacks
on the buttocks with a hick'ry stick.
Traditional proceduralism
with kindness, patience, and explanations demonstrably works much better at
developing children's math abilities than the old traditional proceduralism via
conformist brutality did. So let's stipulate that nothing would stop us from
teaching it kindly and gently. Without that aggressive, kid-hating emotional
understructure -- without the stick -- without the spirit of "because my
great-grandpa could calculate sines by hand, that's why!" -- what positive
benefits does traditional proceduralism offer? What strengths make it so
persistent among people who show no signs of hating kids? And are those
benefits and strengths adequate to our present needs?
The answers are more
complicated, interesting, and ambiguous than many education professors and
reformers would like to talk about. Let's begin right at the heart: what does
traditional proceduralism say that mathematics is, and how does it say we
should learn it?
In traditional proceduralism,
a student learns to copy and memorize patterns of manipulating the symbols for
numbers, with relatively little immediate attention paid to understanding how or
why it works or why s/he is expected to perform these procedures.
In your own grade school experience, the teacher was
teaching and using traditional proceduralism if you were taught to multiply two
three-digit numbers by doing something like this (boxed text for example only,
skim!):
a.
write the two numbers one above the other,
matching columns.
b.
multiply the third digit of the top number by the
third digit of the bottom number.
1. If
the result is less than ten, write it in the rightmost column of the first
subtotal.
2. If
the result is ten or greater, write the right hand digit in the rightmost
column of the first subtotal, and the left hand digit as a small note to the
upper right of the second digit of the top number. That small digit is called
the "carried" number for the next step.
c.
Now multiply the second digit of the top number
by the third digit of the bottom number, and add the carried digit to them.
1. If
the result is less than ten, write it in the second-from-the-right column of
the first subtotal.
2. If the
result is ten or greater, write the right hand digit in the
second-from-the-right column of the first subtotal, and the left hand digit as
a small note to the upper right of the third digit of the top number. That
small digit is called the "carried" number for the next step ...
I hope you've been skimming;
that's why I boxed that section, as I'm not trying to teach you to multiply (if
you need that, there's help in Part 2), but to see the basic approach. Notice
that the student is not supposed to be thinking about what the top and bottom
numbers are -- to the procedure they are just collections of digits -- and that
the first subtotal is also purely a set of symbols that the student is building
up from right to left (the opposite direction from the way we read a number). A
well-trained chimp or a well-programmed machine could do this, and neither the
chimp nor the machine -- nor the kid -- has to have any understanding of what's
happening. It's about putting symbols in the right boxes, rather like crosswords,
sudoku, or filling out tax forms.
If the process of multiplying
two three-digit numbers is written out in this fashion, there are actually
eighteen total steps to the process, ending with
p. add any carried digits to the
leftmost digit in the third subtotal, and write the result in the leftmost
column of the answer.
q. count the total number of places
to the right of the decimal in the top and bottom numbers.
r. take the number of decimal places
you just counted and counting right to left, put the decimal point that many
places to the left of the rightmost digit in the answer.
Now, obviously though you (or your great-grandfather) did learn to do something like that, you
didn't learn by sitting down and
memorizing procedures, certainly not in such complex language.
What you actually did was
much easier than that. The pattern was simply:
1.
a 2-step set up,
2.
repeating a familiar 3-step pattern (multiplying
the top 3-digit number by a single digit from the bottom 3-digit number) 3
times,
3.
executing the familiar pattern for adding
multiple-digit numbers, and then
4.
positioning the decimal point with a counting
rule.
If your school was really
traditional-proceduralist you might have learned it as a simple visual pattern:
Kids learn more complicated
patterns than that all the time when jumping rope, braiding hair or lanyards,
playing clapping and chanting games, knitting, tying shoes, or playing
tic-tac-toe. Such patterns are the underpinning of folk tales, fairy tales, and
plain old jokes in every culture: the troll meets three billy goats gruff, the
prince dances with three sisters, and Goldilocks messes up the possessions of
three bears; the big goat, Cinderella, and "just right" are the
"solutions."
Pattern-mastery evolved to activate most intensely in the
first decade of life, when kids are learning physical skills, language,
customs, and so many other pattern-based behaviors. Consider how fast young
kids learn songs and nursery rhymes, and how many of them decorate their
belongings with repeating patterns. Because it's the age where they quickly and
easily pick up patterns, it's the ideal age for learning a musical instrument,
rhymed and metered poetry, and games like jacks or hopscotch.
You may have heard that "when you have a hammer,
every problem looks like a nail." Young kids learn patterns so easily and
quickly that if a procedure can be expressed as a pattern, kids will use it to
get through their math homework, and not just because they want to get back to
playing video games or snaffling up Disney products (more pattern-based
activities). Many kids, maybe most, are also wired to find fun in seeing how
quickly and neatly they can run through the pattern of symbols. These are often
the kids who later in life will say they "loved math but hated word
problems" (because they liked to run the pattern, not match it up with all
those messy words) or that they were "good at math till nth grade."
That "till nth grade" is the wall that we're
going to be looking at in detail: the place where suddenly math stops being
easy and fun, and never is again. Typically nth=somewhere between 4th and 7th,
far short of what they're going to need.
Sadly, the patterning that allowed
kids to sail through math up till the wall suddenly rams them into it.
Teaching a procedure as a
pattern -- that is, as an algorithm -- and evaluating kids on their neatness,
precision, and accuracy in following it is fun, or at least tolerable, for the
kids who are quick to get it. The rest can be pressured and hassled into tolerating
it (usually without the hick'ry stick). Even kids who hate having to memorize
the patterns do get a sense of accomplishment once they have done it.
Furthermore, because the
patterns themselves are traditional, parents and teachers feel a great deal of
comfort with them -- whether they understand them or not. It's a bit like
growing up in a religion whose holy language you don't actually speak; knowing
the pattern is comforting all by itself.
But the day of reckoning comes for many students when
their memories can no longer hold patterns that are too big, complicated, or
hard to apply. The kid who has only learned patterns, and not the math behind
them, has no other way to learn more. That's the kid who says, "I
can't," and gives up, and that's the phenomenon of hitting the wall.
"Hitting the wall" and "good at math": their
underlying connection
Pattern-mastery is very
unevenly distributed to the population. Some people barely recognize a song
they've heard a thousand times; natural champions at "Name that Tune"
know thousands of pop songs well enough to recognize them from 1-3 opening
pitches. Some people can't learn the base step for the foxtrot; Broadway
dancers can acquire and retain a hundred moves in one long rehearsal. Some
people struggle with remembering the rules to "Go Fish," whereas a
professional poker player will not only know dozens of variations on poker but
can easily acquire another one from a few-second announcement at the beginning
of the game. In those examples, the natural champions, Broadway dancers, and
professional poker players are of course "good at" song-recognizing,
dance, and poker; "some people" are obviously hopeless doofuses who
"just aren't good at it." They might have started off able to
remember a few nursery songs, the macarena, or basic solitaire, but at some
point they ran into patterns that overwhelmed them, and went from being
"good at it" to "not good at it."
In traditional proceduarlism, which area of math will be
the wall for any given kid depends on how much of a natural patterner s/he is.
The most common "I don't get it" points for people who were
"good at math until ..." are places where patterns get too long,
become too complicated, or stop being relevant.
Fractions can be processed by regular procedural patterns, but students
have to keep track of numbers that are expressed as relations of numbers. Long
division can have any number of steps at all, and introduces a branching
procedure, i.e. students have to make decisions about which of the possible next
steps to execute. Exponents and radicals require knowing how and when to insert
the PEMDAS rule into a procedure. All of these are far more complicated and
require far more decision-making than the early-grades down-the-column or
make-an-x patterns for basic addition, subtraction, and multiplication; many
people who can remember a simple pattern with no branching decisions become
hopelessly lost in a complex, branching one.
It gets even worse with algebra, because, while there are
many patterns a student can acquire, the heart of algebra is making decisions
about which patterns to apply, or where a new approach for which there is no
pattern needs to be invented. For example, take this quadratic equation:
6x2+12x+48
= 3(4x+48)
"Given
this equation, solve for the roots" would be typical of assignments
somewhere between seventh and ninth grade for students on a regular track.
A
student proficient at mental math and with a good head for strategy (two things
Singapore Math excels at developing) doesn't even need to write anything down.
S/he sees the 12x and the 3(4x) on opposite sides of the equation and realizes
that the x terms go away, then sees that all the remaining terms divide by 6,
reducing it to x2+8 = 3(8), which is easily resolved to x2=16,
so x=4 or x=-4, the two roots of the equation.
Other
students will do various other kinds of expansion and reduction until they can
factor a solution by the difference of squares method. An exceptionally
un-intuitive student with a large set of memorized procedures might even
laboriously convert it to standard form and then grind all the way through the
quadratic formula, or even awkwardly kludge through the steps of completing the
square.
Any
of those might yield the correct two roots, so there's no one right way.
However, there are certainly better and worse ways; the one I began with takes
just seconds if you have practice and intuition, whereas the quadratic formula
might take a sheet of paper and a couple of minutes.
The
fastest, easiest strategy for one problem may be a wretched, tangled, confusing
mess for another. There's no pattern for choosing strategies, and not every
strategy is expressible as a pattern, and even if there is a pattern-based
strategy no one can possibly memorize all the possible patterns and the
conditions under which they apply. And at that point, many students, who have
been trained for years to look for a pattern of "just write this here,
then write this, then combine this way," are unable to start.
The same natural gift for
pattern that makes many small children "good at math" without
thinking in math leaves them stranded when they do have to think about
math.
This connection between
traditional proceduralism and hitting the wall has been known and analyzed for
more than a century. Just before 1900, Mary Everest Boole, the widow of the
great mathematician George Boole and a very formidable mathematician herself,
began to explore the problem of teaching mathematics to children. She published
her results in 1903, in Lectures on the
Logic of Arithmetic.
Boole found that for
generations, good math teachers had intuited a connection between memorized
procedure in the early grades and hitting the wall in later ones. Reverend
Thomas Vowler Short, in his 1840 book on how to teach fractions, discussed the
fraction barrier and even pointed to teaching arithmetic as a system of
memorized rules as the cause: Short noticed that compared to the basic four
operations on integers, the rules for manipulating fractions were much longer.
They also referred to more abstractions -- you had to remember which was the
numerator, which was the denominator, whether it mattered that a fraction was
reduced or improper and what that meant, etc. Most importantly, the
"higher" fraction rules were less intuitive than the lower arithmetic
tricks of borrowing, carrying, and cancelling. For example, most adults who are
proficient at arithmetic use the "flip the divisor, then multiply"
rule for dividing fractions without thinking about (and often without remembering)
why it works.
Reverend Short even suggested
the cause (notice that this was 175 years
ago; so much for any idea that in the good old days they knew how to teach
math!):
To those who are at all acquainted with the usual method of teaching
Arithmetic in England, it can be no wonder that the mass of the children so
taught are not fond of calculations. The rules are synthetically laid down, in
such language as is not very intelligible even to those who are fully able to
employ them, and which to the scholar, when he first enters on the task, must
be utterly unintelligible. They are so numerous, that in a work which I have
now before me, there are thirty rules preceding Vulgar Fractions, whereas, if
properly taught, Fractions ought to follow immediately after the first four
rules, as applied to whole numbers.
Reverend Thomas Vowler Short
(1840) p. 3
Mary Boole, with the benefit
of having spent years promoting and explaining the work of her husband, one of
the great logicians of all time, was able to do more than just observe that
many pattern-proficient kids hit a wall at a point where their memories simply
couldn't handle the tasks at hand anymore. Summarized in modern terminology,
Boole's findings were that many kids who learn elementary-level math by
treating written numbers as arbitrary symbols -- the same thing Reverend Short
called synthetic -- internalize the idea that mathematics is nothing but
knowing what pattern to do when. This is a clerk's viewpoint.
When that clerk-viewpoint
student reaches algebra, he or she is looking for patterns that often are not
there and can't be there. Their whole do-this-next concept of math is too small
to hold the things they need to learn. (Maybe an analogy will help: Suppose a
student has only learned traveler's French and thinks it's all the French there
is, believing that the French spend all their time conversing about the way to
the post office, what time the next bus for Lyons departs, the price of bread
from a sidewalk stand, and the like. S/he will have a bad time trying to read
Sartre, not just because of lacking the vocabulary but also because s/he will
not even be looking for anything Sartre is interested in talking about.) A
reductionist view of a subject is far more damaging than just not being able to
recall a pattern for the immediate problem; it's being permanently unaware of
even the possibility of genuine understanding.
Boole also observed that the
few bright kids who stayed "good at math" frequently had a habit of
playing with numbers, thinking about how numbers worked, and focusing on the
patterns in the numbers rather than in the synthetic manipulation of arbitrary
symbols.
Eventually, Boole's insight
was exploited by Dr. Kho Tek Hong; Singapore Math teaches, encourages, and
emphasizes the kind of number-play in the head that most "good at
math" people naturally start doing as small children (and most people
never really discover because they're stuck in the patterns).
Most kids who learn that math
is pattern-memorization will only be able to go as far as they can memorize
patterns. Especially in a modern world where kids and parents are more
resistant to mindless drudgery, for which there's not much economic advantage
anyway, pattern memorization will not take kids very far. Many will only touch
and not master the more complex patterns like long division and fractions.
Furthermore, they will have to painfully relearn everything when they cross
over into algebra (where one must choose
a pattern, and many problems have no pattern at all). The early comfort of feeling
"good at math" based on being quick about
"put down one, carry the two"
actually makes it harder for
many kids to advance to
"x is monotonic between x1 and x2,
so under the constraint x1<x<x2,
any solution for x has to be a corner solution."
so under the constraint x1<x<x2,
any solution for x has to be a corner solution."
Teaching pattern rather than meaning produces
self-satisfied little clerks at eight and frustrated math-phobics at eighteen.
Traditional proceduralism's deceptive quality of enabling kids to feel
"good at math" (while they're not actually doing math) actively
damages the ability of many students to pursue the kinds of higher mathematics
that would enable more productive and conscious lives in our present world.
Nonetheless, traditional
proceduralism lingers on. In part, this is because while it doesn't actively help most students through the conceptual wall of algebra,
it does facilitate bright students finding their own way. It wastes much
low-level talent but at least it doesn't block high-level talent. (And it is
the high-level talent that then survives to write the math standards and define
proficiency).
Then, too, the relatively few
teachers in lower grades who had a genuine mathematical gift themselves have
often been able to add enough material about the concepts that underlay the
processes. Reverend Short and Mary Boole both dedicated books to math teachers
like that, and I've seen some teachers of Saxon Math who always maintained a
strong conceptual focus, with extremely good results. Traditional proceduralism
doesn't damage everyone, and it's not terrible
for a smart student with a very smart
teacher; the right student and teacher matchup can slide right over the
difficulties and pitfalls.
But it lingers on in more
than the nostalgic memories of sadists and math wizards. Many ordinary people
of average math talent who struggled with traditional proceduralism and know
its strange miseries first-hand are still fans of it. Some, as noted above, may
simply be the sort of sadist who thinks "if I had to, my kids have
to," but surely that is far from all of them. Why would average students
who had mediocre teachers, and hit that terrible wall of fractions, long
division, algebra, or trig, politically support the system that creates those
problems?
What strengths does
traditional proceduralism offer for students who are not brilliant enough to break the wall?
Back when school leaving age
was third, sixth, or eighth grade, and society and the economy were very
different from what they are today, traditional proceduralism made a certain
kind of sense: you taught kids to add numbers in neat columns exactly the same
way you taught them to eat with knives and forks, wear matching socks, call God
Thou, or spell words unphonetically:
arbitrarily and synthetically, in a spirit of "Shut up and do it the
right, grown-up way."
This is because historically,
most people didn't really need to understand math; they just needed practical
everyday arithmetic. The traditional algorithms are still the arithmetic you
use if you're the assistant manager at a fast-food place. You use them to
second-guess the cash register, figure inventory of burger buns, decide how
much freezer space is needed for the frozen fish fillets, maybe work out the
square footage of the floor to order disinfectant.
In the old days, even manual
root extraction had a function: finding diagonals. If you wanted to know if
Mrs. Smith's couch would fit into your moving van and then through the door of
her new apartment, before you tried it and got stuck, you multiplied and added
and found square roots and had your answer. If you needed to know how big to
make the boxes so that nine tons of coal fit exactly into five cube-shaped
boxes, and order just enough lumber to build those boxes, you dusted off your
root-extracting skills. If the navy needed gunnery tables or navigational
almanacs, they hired a bunch of proficient root-extractors and
series-summarizers; ditto if a large bank needed an amortization table.
Traditional proceduralism was
robotic and factory-like, but back when mass education was starting in the
United States, the business world desperately needed very large numbers of
people to do routine, repetitive math accurately. Since most people's work was
dull and repetitive anyway, it made sense to do arithmetic in a way similar to
packing kits in a factory, following the corporate recipe in a chain
restaurant, or maintaining a rifle in the army.
As late as the 1930s the U.S.
Department of Commerce was forecasting bright prospects for
"computers." In those days, that didn't mean the machines that a few
top-level mathematicians, scientists, and engineers were working their way
toward inventing. It meant people who could do hundreds of long divisions,
trigonometric calculations, and square root extractions perfectly every day,
with pencil and paper and perhaps, in an advanced shop, the help of adding
machines, log or trig tables, and slide rules.
Of course it was also a
culture where large numbers of American children younger than ten were digging
coal, chopping cotton, assembling machine parts, and sewing piecework. The
traditional elementary school math for which so many people feel so much
nostalgia is aimed at producing many people who would make great clerks at Old
Eb's General Store, Haberdashery, and Pharmacy in East Podunk in 1914, and a
favored few who could do the steam table calculations for Marks's Standard
Handbook for Mechanical Engineers.
But whether you want your kid
to rise higher than retail clerk, or just doubt that there will still be jobs
for retail clerks throughout your kid's working lifetime, for those great STEM
jobs, or to be able to manage STEM workers, or just to understand most of the
news of the 21st century, they need to really understand real math.
The first step, as a nation,
to becoming genuinely good at math, is to lose the myth of a
mathematics-proficiency Golden Age. There was not a time in the Wonderful Past
when everybody was good at math. There was only a time when a larger fraction
of people could calculate quickly and reliably -- a good thing, and a
mathematically useful thing as Dr. Kho points out because it's almost
impossible to maintain enough concentration to learn higher math if your mental
calculation isn't quick and reliable.
So, sorry, teachers and
parents who are hoping the new and better way will mean that your kids are off
the hook for all that dull drill, practice, and memorization. The boring
memory-and-procedure loading parts are essential for the proficiency that makes
learning Singapore Math possible, and in fact kids in the nations where Singapore
Math is succeeding spend many hours in intensive drill. Higher order skills are
made up of lower order skills, and sometimes to cook, tango, or do math, your
wrist just has to know how to flip a pancake, your torso just has to know how
to lead a boleo, and you have to look
at "eight times nine" and say "seventy-two" faster than you
just read it. Kids learning Singapore Math, or any good system devised in the
future, will still need to know the multiplication table, but they also really
do need something more.
But equally sorry, hick'ry
stick fans. "Shut up and do it right" wouldn't work even if we still had
a culture that supported it. The number sense required to do higher level math
has to be introduced, nourished, and fed right along with the most basic
arithmetic; the seeds of calculus are there in how a child first learns to add
(if she learns it well, as in Singapore Math), and the seeds of "I hate
math and I will never use algebra" are there too (and our traditional way
of teaching fed and watered them).
So traditional math is an
accelerated path -- into a wall. It
produces some capable people and a great number of mathphobics and partial
innumerates. It had to be replaced, and
this was apparent by Mary Boole's time.
Unfortunately, John Dewey's
pragmatist math, the first great attempt to fix things, was at best a huge step
into irrelevance, and at worst a direct embodiment of the idea that if people
never move at all, they can't hit a wall.
I'll pick on that next week.
