The story thus far
I'll probably find ways to
keep making these "if you came in late" introductions shorter and shorter.
Here's the newest:
- Singapore Math has an extraordinary record as a successful curriculum for elementary through high school math, in a large number of international comparisons.
- Singapore Math is coming into the American school system via a number of pathways, including being included as one way to satisfy the Common Core goals, being adopted wholesale especially in STEM/gifted education, wide use in home schooling, and increasingly in other venues as well.
- My math tutoring focuses on kids with specific math blocks and barriers (as opposed to general cognitive problems/learning disabilities, or behavioral issues like procrastination or defiance). My college teaching is almost entirely with ADLs, Adult Disadvantaged Learners, many of whom hit a seemingly impassable barrier in math sometime in their childhood or early teens, and have not been able to progress beyond it. I have found Singapore Math tactics to be extraordinarily effective in unblocking both children and adults, freeing them to learn and progress in math. Children who haven't been beating their heads against the barrier too long sometimes even re-discover math as a favorite subject and move from the bottom to the top of the class.
- As I've become more familiar with American education and curriculum politics, it is becoming clear to me that the system -- not individual administrators, teachers, parents, voters, etc., but the system as a whole -- is very likely to throw away the once-in-a-generation opportunity to dramatically improve math instruction by making Singapore Math work.
- I've become convinced that if anyone can save Singapore Math, and with it the future of a few million kids who could like math and be good at it given this chance, it will have to be the parents.
- Hence: I'm working on a book called Singapore Math Figured Out for Parents.
- And one of the key things that parents need figured out is why Singapore Math is needed, which is to say why
- we couldn't just keep doing the same thing we'd always done,
- past reforms not only failed but were doomed to fail
- Singapore Math is not just different but better.
And here in this blog, I've been presenting excerpts of a sort of history of
American math teaching, from that critical standpoint.
Last week, I talked about traditionalproceduralism, what most Americans would think of as the most conventional way
of teaching math, and why it tended to produce early success at basic
calculation, followed by a wall of resistance and difficulty for students trying
to go beyond that. (The international comparisons show this very clearly: in
mathematics, America has top-notch fourth graders, mediocre eighth graders, and
abysmal high school seniors; we're rather like a distance runner with a great
takeoff and poor wind). All that's explained in much more detail in that post.
What the reformers tried to fix
Now, the reason why I spent so much of your time and mine
on the problems of traditional proceduralism is that today's problem, as we all
know, tends to be yesterday's solution. This is nowhere more true than in math
teaching; the various attempts to fix the problems with traditional
proceduralism led, at best, to different but just as serious problems.
Reformers, fixers, and
innovators have tended to focus on three things that are problems with
traditional proceduralism; that it is:
- Not fun: kids (and sometimes parents and teachers) don't see the point in the calculations and find the necessary memory training unpleasant.
- Not actually math: for most children and many adults, the patterned manipulations of symbols (algorithms) become so much the focus that they take over the mind-space that should be occupied by real math (number sense, ability to solve practical problems, understanding what numbers are and mean).
- Not empowering for further study. See that previous post for details, but the same process that produces early quick calculation programs kids to misallocate their attention and not to learn fundamentals they will need soon. As a result, they hit the wall far too early -- at some point like long division, fractions, or elementary algebra, years before the kinds of math they need for the STEM subjects that open doors to good jobs.
Nearly all reformers have tried to address all
three problems at once in some combination.
John Dewey: Prescription now, diagnosis later; the medicine is tastier but is it better?
The first serious attempt to
fix the problems of traditional proceduralism began just before Mary Boole
published her analysis of those problems, so in effect the medicine was
formulated about a decade before the disease was understood. John Dewey's The Psychology of Number (1895) (written with James A. McClellan but the ideas are pretty much pure Dewey) explored
how numbers become meaningful and useful to people, as applied to the problem
of teaching arithmetic.
Unfortunately, Dewey was no
mathematician. He fundamentally didn't like the very notion of math; he
certainly recognized that it was useful and necessary but he seemed to feel
about it the way many liberals do about the army or many conservatives do about
taxes.
Because John Dewey is one of
the most revered (and most hated) figures in American education, Deweyan
pragmatism has become entangled with the ideals of progressive education. Most
of the people for and against Dewey's "pragmatic" approach to math
teaching concern themselves more with whether or not it is progressive, and
whether or not that's a good thing, than they do with whether or not it is (or
can be) an effective way to teach math.
Programs and curricula based
on Dewey's ideas sell well to professional educators, who tend to revere those
ideas; unfortunately, they don't pay off in better math skills because very
often the goal is a Deweyan progressive-education goal, not mathematical
proficiency at all.
I'm going to try to stick
pretty close to the question of effectiveness here, but some context is in
order first. The problem with Dewey pragmatism isn't so much Dewey as it is
pragmatism.
Pragmatism and Math: What kind of a job you gonna get with that?
John Dewey certainly has his
haters, and he earned a great number of them. He was the quintessential
proclaimer-from-on-high with little practical experience of children.
Nonetheless, much of what he proclaimed was badly needed, long overdue, and
obvious to anyone who tried to look with any sort of clear vision, analytic
mind, or kind heart.
The schools of the 1880s and
90s, against which Dewey was rebelling, were horrible places that made kids
unnecessarily miserable, frequently for irrelevant purposes (unless you regard
sadism toward children as a relevant purpose). Many of the things that help to
make school bearable to kids were Dewey innovations. If sometime in your
experience of school you served on student council, enjoyed recess, researched
and wrote a skit, did any hands-on project, or took a field trip, you can thank
Dewey and his disciples. In contrast to much of the prevailing view of kids as
little sinner-monkey heathen to be ground down into factory-fodder, Dewey's
progressivism was a humane impulse. Many intelligent teachers embraced his
innovations not so much because they liked his philosophic reasons, but because
they opposed strapping small children into desks and making them labor for many
hours at incomprehensible unexplained tasks.
Dewey's ideas had more than
just kindness going for them. His training was as an analytic philosopher, so
he tended to think in precise terms. His intellectual allies were the Pragmatists,
who thought the key question for evaluating an idea was not whether it was true
but whether it was useful. "Pragma" is the Greek word for
"deed," or in more modern vocabulary, "action" or
"thing that is done," and the Pragmatists meant that in one of the
usual ways that ordinary people use it: in the
contrast between "deeds, not words," "action, not
talk," or "judge what people do and not what they say."
Pragmatists held that the
value of an idea was not so much in its truth or its adherence to moral values,
but in what people could use it to do. In preference to the more traditional
philosophic question, "Is the idea true (or good, or beautiful), and how
can we know that?" the pragmatists asked, "Is the idea a good
tool?" To pragmatists, it didn't matter as much that multiplication was
commutative as it did that if you treat multiplication as commutative, your answer
will be useful in the real world.
That pragmatic emphasis on
usefulness led Dewey toward a correct perception that theory and application
had to be brought together for math to be of any use. However misleading his
light, he was shining it in exactly the right place.
In The Psychology of Number, Dewey critiqued the purely theoretic way
of doing things inherited via medieval scholars from the Greeks. He saw such
pure theory as stranding mathematics so far away from ordinary life that it
could never be of any use.
It is less well-noted that he
also opposed the crude counting-and-stacking manipulatives (which is
math-teacher-ese for "objects the child can handle while counting"),
an idea which was then being used experimentally by a few educational psychologists.
Dewey didn't see much difference between counting disks or sticks and counting
on the fingers; to him, they were all a crutch for the incapable rather than a
tool for the capable. He was willing to concede that you could introduce an
idea that way, but some of the early educational psychologists were trying to
teach entirely with manipulatives,
believing that showing kids seven sticks, seven pennies, and seven sheep would
teach them all they needed to know about the idea of "seven."
Dewey's idea was that there
needed to be some balance between throwing first-graders into the deep end of
theory and leaving sixth-graders paddling around in the shallow pool of pure
material manipulation, and he went looking for a rule for striking that
balance. Again, there's little to complain about in his quest; it's what he
mistook for the grail that was the matter.
Dewey argued that thought,
theory, consciousness, and the mind -- and communication about them -- were
what made human activities human. After all, crows can count up to about seven,
bees do geometric dances, geese solve problems in long distance navigation, but
they don't do those things in a way we recognize as human.
Somehow, Dewey saw, for math
to be a fully human activity, ideas
would have to get into it. Kids needed to encounter the real world, but they
needed to be thinking about abstract ideas and concepts while they did so. Kids
needed to develop a habit, method, and rule for going through the gateway to
theory.
That would require more than
what Dewey's disciples to this day tend to call "drill and kill,"
i.e. meaningless-to-the-kid repetitions of tables, procedures, steps, and
patterns. Somehow there had to be a way to guide kids from "three groups
of seven sheep each" to "three times seven is twenty-one." From
there, they had to be steered to the notion that three, seven, and twenty-one
applied not just to sheep (or dollars or pencils or stacks of wooden disks) but
to groups, piles, and finally to things that had no physical existence at all,
via relationships that became harder and harder to put together out of
manipulatives.
Yet at the same time Dewey
also keenly felt the justice in the accusation that math was already much too
high and inaccessible in its ivory tower. If the mover wanted to know if the
couch would go into the van and through the door, what would he gain by knowing
that the doorway-fit and van-fit problems were the two and three dimensional
expressions of "finding a diagonal," and that to find the diagonal he
needed to extract the square root of the Pythagorean sum? Why not just cut a
stick or a rope to the length and measure?
The reasonable answer to that
hypothetical furniture mover's reasonable question is that a solution that
works for many different kinds of problems is a more valuable thing to know
than one trick that works once. But instead of building from that answer, Dewey
took up the side of Homer Simpson, Peter Griffin, and their millions of
real-life equivalents -- all of whom, educationally, are pragmatists, whether they
know it or not, because their basic question is, "What kind of a job can
you get with that?" When someone tells you they try not to think above
their pay grade, you're hearing pragmatic philosophy.
The problem, obviously, is
that those fictional characters are clowns created to mock idiocy, and Dewey
was supposed to be a genius and a public intellectual, not an advocate for
self-satisfied ignorance.
Looking in the right place and finding the wrong answer: Dewey focuses on the link between theory and application
The pragmatist ideal was not
that people would learn all they could manage, which would only make the
proficient learners needlessly smug, prideful, and anti-democratic. Rather,
students would learn what they needed (including that they only needed that
much and no more). If thinking above your pay grade is useless (and hence, to a
pragmatist, of no meaningful value), why torment kids trying to force them to
do that?
Dewey's pragmatism saw ideas
as tools to be applied to problems. Thus math education had to both give the
students the tools (which traditional proceduralism tried to do, mostly
succeeding with the simplest tools) and a way to apply them to problems (which
traditional proceduralism largely neglected, leaving students to learn applications
of math in the farm, shop, mill, or store). Dewey focused tightly on linking
the theoretical idea to the real-world process: theory and practice needed to
be fused, right from the start of math instruction. In his eyes this made the
difficulty and complexity of theory into necessary evils, to be strictly
limited to what he thought most people would use in practice.
That determination to limit
theory, holding it down to the bare minimum needed for applications, has the
undoing of curricula and programs based on Deweyan pragmatism. The dialogue
between doing math and understanding math is vital, but neither doing nor
understanding should exist only to serve the other. They are two equal feet,
and if you insist on walking mainly on one of them, you will go in circles, or
you will limp.
The shortcut through theory (or Elfland)
The real power of mathematics
in the real world is its ability to take a shortcut through theory. Applied
mathematics -- real world problem solving -- is a process of abstracting information
from the real world, processing those abstractions into an answer in pure
theory, and then returning to reality to apply the answer.
You may be muttering that you
don't remember ever doing any theory when you do basic everyday arithmetic.
Let's see how that works in practice (putting on Dewey costumes for the moment;
didn't you always love doing skits in school? Come on, you know you did. You
either got to be the smart kid and get all the attention or you got to coast on
the smart kid's work!).
Take two problems:
- Sheep and lambs. You have four sheep; you loan two ewes to your friend for the season; they produce four lambs, and by your agreement, your friend returns the two ewes plus two of the four lambs. How many sheep do you now have?
- Dollars and lemonade. You have four dollars; your friend borrows two for lemonade ingredients, sells six dollars worth of lemonade, and gives you your two back, plus two dollars of the four-dollar profit. How many dollars do you end up with?
The real math
is not in the adding and subtracting that you just did in your head, and that
decades of experience have trained you to do, assuming you're old enough to be
a parent. The real math is in understanding that it's the same problem whether it's dollars, sheep, electrons, or
bandersnatches.
4-2+4=6 is theory.
4-2+4=6 is theory.
In math and science, theory
doesn't mean a guess, but an idea or group of ideas accepted as true and valid
because it can be used to explain many specific cases. The reason Newton had a theory of gravitation was not that he
guessed there must be some fall-downy stuff called gravity. It was because he
realized that one equation,
could explain and predict the trajectory of an apple falling from a tree, a bullet fired from a gun, the moon orbiting the Earth, and the Earth orbiting the sun. Thermodynamic theory isn't idle speculation that heat flows from hot objects to cold ones; it's a set of abstract relations between concepts like temperature, entropy, heat, work, and efficiency. Those relations explain and predict the behavior of any engine, refrigerator, pump, turbine, or propeller now existing or yet to be invented. (Incidentally, if creationists really understood what a theory was, evolution would probably upset them a lot more than it already does; evolution is not a guess about where life came from but modern biology's fundamental explanation of what life does and is, which happens to imply an explanation about its origins and development).
Meanwhile, now that we know what a theory is, back at
those sheep and dollars:
So, 4-2+4=6 is a small, highly specific part of the
theory of how numbers relate to each other. It's an abstract statement in which
it doesn't matter what you're counting or what order you count in. If you're a
bit more advanced than your first-grader, you probably saw at once that you
could group -2+4 into another abstract concept, "net change," solve
it as +2, and simplify the whole problem to 4+2=6. If you have a slightly more
theoretical bent of mind, you might even realized that any three numbers
expressing
original amount+net change=new amount
would work in the same way;
it's a very basic part of the theory of bookkeeping.
What Dewey correctly saw was that the thing that makes
math really math, is a specific three-part move:
1. from practical reality: specific, concrete dollars
or sheep being traded in specific arrangements between actual people,
2. to abstract theory: a set of abstract numbers
processed by the abstract operations of addition and subtraction
3. and then back to practical reality: into
the real world of sheep and dollars.
The most basic idea in applied mathematics -- math with
its work clothes on, the math that is something you can get a job and do a job
with -- is that correct theory applied to a correct understanding of the
problem leads to the same answer that running the experiment in the real world
would give you. That's an idea that first graders are only beginning to grasp
and sixth graders need to be secure in. It's the notion that stands behind that
common phrase, "Do the math," meaning "This prediction is not
just my opinion about how it works or what will happen; this prediction is
certain because the facts are true and the theory is correctly applied and
sound."
Traditional proceduralism didn't teach that idea
directly, relying on most kids to gradually come to understand the idea from
day to day life. Dewey saw that many of them were not getting it that way, and
set out to prepare a more certain and reliable way to bring them to that basic
concept.
Applied math -- or word
problems, to use the ordinary person's term for it -- is about moving problems
over from the real world, where they are messy and confusing, into the realm of
abstract theory. Pure math or theoretical math, as you might expect, is about
the operations you do in that realm of theory. And it becomes applied math
again when you bring it back to the real world and "put a unit on
it," i.e. decide whether your answer is in miles, pounds, horsepower,
bathroom tiles, light-years, watts, bags of flour, seconds, or dollars.
Applied math is all about the
shortcut through abstraction: First you clear out everything that doesn't
matter in pure math. Then you apply a few operations that transform those pure
numbers (data) into other pure numbers (results). Then you take those results,
stick the units back onto them, and voila!
you know what will work and how things will come out.
In a sense, it's magic.
Instead of plodding your way to seeing how things come out in the future, along
the long dusty road through concrete reality, you grab up just what you need
and step through the gateway into the magic realm of theory. In the magic
realm, you make your trip quickly and easily and perhaps acquire other valuable
things along the way. At the end of that trip, you step through another gate
right next to the now-fixed problem in the real world. Applied math is about
finding the doors in and out of Elfland; pure or theoretical math is about
finding your way around Elfland.
"But don't go near the water": okay, kid, go through Elfland, but stay on the path and no looking around!
What Dewey wanted to do was
to stay as close to the doors as possible and make sure no one spent any time
on the scenery in between. His solution was that students should never work
with numbers that were not measurements. Dewey thought that literally to keep
it real, students needed to do immense quantities of word problems referring to
their real life, learning just enough abstraction to be efficient calculators.
Math would ideally not be a separate subject at all but an auxiliary to the
things he really wanted his contentedly-following worker bees to be happy with:
cooking, sewing, carpentry, to some extent sports, and other real-world
matters.
Thus, Dewey declared, every
number should have a unit of measure attached to it, and kids would learn only
as much theory as they needed to modify a recipe, tile a bathroom floor, or
track inventory.
What's the problem with that?
Why do we want people doing math above their pay grade? How does that answer
Homer Simpson's question?
If you can't go through Elfland, there's a dead end waiting for you
I had one student, who I'll
call Willard, in a developmental (polite word for remedial) college algebra
class. He was trying to move up to being a construction foreman after a lot of
years spent carrying boards and tightening bolts. The contractor would have
been happy to promote Willard except that it was painfully apparent that when
confronted by a situation calling for simple math, Willard was usually right,
but sometimes extremely wrong, and quite literally didn't seem to know what he
was doing.
In class I discovered that he
insisted he was completely bewildered by and did not understand "none of
them Xs and Ys, they just don't make no sense," and that any arithmetic
beyond basic addition and subtraction was iffy, but "you give me a
calculator and if it's real I can do it," Willard said, with honest pride.
Except, he honestly and shame-facedly admitted, sometimes he'd come up with
answers like only needing one sixteen-foot two by four for a whole house, or
needing ten four-by-eight ply sheets to put a floor into a ten-by-ten laundry
room. Usually he caught those and re-did the problem, but he really didn't know
why it came out wrong the first time or two, or why it would suddenly be right.
I discovered that I could
control how well Willard did by manipulating the word problems I gave him. He
was usually right if the scale of the result was fairly self-evident within his
experience (for example, if he could immediately guess that it was going to be
somewhere between ten and a fifty feet, thirty and eighty miles per hour, or
nine and twelve dollars), and there
was only one arrangement of numbers into ordinary arithmetic that would give a
result in that range. In that case, Willard would set it up properly and get
it, although sometimes with weird stumbles into alternate arrangements of
numbers along the way. The same group of numbers arranged the same way on the
page, without units or reference to any word problem, would baffle him.
Furthermore, if there were several ways of sticking the numbers in a problem
together, Willard would nearly always choose the simplest arrangement of
numbers that gave a plausible answer.
Eventually he confessed to
his process: Willard was shuffling numbers around through the arithmetic he
knew until his calculator result matched up with his visualization of the
answer. His intuition and experience were good enough to know he needed about
three or four sheets of plywood for that ten by ten room; initially he had
added 10+10 (rather than multiplying) because he knew he needed the
"size" of the room and didn't really grasp "area." Then,
since "20" looked like too big a number, he had divided by 8 (the
length of a sheet,) to get 2½, and multiplied by 4 (width of a sheet) because
that looked too small. Since 10 was too much, he started over, combining the
numbers in different ways.
With considerable pain and
difficulty, I walked Willard all the way back to the idea that a abstract
operations like adding, subtracting, multiplying, and dividing corresponded to
everyday, real-life processes like piling up, making change, area, and putting
into equal sized groups. After that, he and I fought our way together to the
idea that one abstract operation might underlie thousands or millions of different
real life processes. From there we made a final assault on the concept that
elementary algebra depends on: that an unknown number will behave exactly like
a known number. (Willard, at first, did not see how we could know that 2x+3x=5x
if we didn't know what x was, and could also clearly see that we couldn't
possibly perform the experiment of trying all the infinite possible values of x
to make sure; nor did he see that we wouldn't have to do that for every
possible equation).
Now, in personal interactions,
Willard came across as a pretty smart guy; he read nonfiction for pleasure,
followed science news and sports statistics, and could fairly coherently
explain how simple gadgets worked. But like illiterate people who only eat in
restaurants where there are pictures of the food, he was severely limited in
his options, both by things he just could not do at all and by the work-arounds
and fakes he had to use constantly. Somewhere in school, he became proficient
enough at his guess-and-plug-in method for word problems to survive with
passing (though atrocious) grades, and they turned him loose to go as far as he
could down the dead-end street of pragmatic math you can get and do a job with.
(Remember, he'd been quite successful as a worker who didn't have to do much
math).
There are many Willards out
there, and many of them originate in the Dewey-inspired curricula. Sadly, it is
often very easy to reduce the complexity and difficulty of theory, and to make
the answers more and more obvious, so that student scores go up. Such curricula
and programs not only make it easy to dumb them down, they provide a rationale
for it -- "most people don't need to solve a quadratic equation to get a
job," or "you can learn trig if you ever get that far, meanwhile most
people just need this table."
The moral of the story: Not just the math you absolutely need right now, but the math you need to access the math you'll need later.
And that is why approaches
that derive from "just the math they need for daily life or a starting
job" just can't keep up with Singapore Math: their goals are limiting.
They are more concerned with the math they are not going to teach than with the
math they are. It's very much an easy sell to parents, school boards, and even
not-very-mathematical teachers, because most of them are not particularly good
at math themselves, and it promises a way of avoiding math, or at least
controlling its presence in their lives. It appears to directly answer the
perennial complaint of small business owners that they can't find high school
graduates who can reliably do simple calculations; "you want a kid who can
figure out a purchase order, track inventory, and make change? We'll give you
one that can't do anything else."
Politically, socially, and
psychologically, Deweyan pragmatism is very appealing as an alternative to
traditional proceduralism. Unfortunately, it doesn't actually teach math very
much or very well. Worse still, because it steers kids away from the deeper
concepts and theory needed to advance further, it is quite likely to create a
higher and more difficult wall -- though that may be less noticeable, because
it also discourages teachers and kids from even trying to get over or through
it.
It's also still highly
influential in American mathematics teaching, and the politics that surround
it, today. Sadly, this explains a great deal: textbooks like Everyday Math, notorious for its simple
word problems, late introduction of key ideas, oversimplifications, and
presentations of the bare minimum of theory without proof or much discussion. It
also explains something you can occasionally see on the job: the person who is
rearranging or redescribing the situation in order to be able to apply the math
they know, rather than the math that works. Ultimately, it's about making
the student comfortable camping out permanently on the bad side of the wall (and
perhaps never even seeing that the wall is there).
There were other approaches,
and I'm sure many of you are saying, "Well, then why don't we just teach
them where the gates are, or how to climb over walls?"
That's exactly what the SMSC,
the body behind New Math, tried to do.
And we'll take up that story in our next episode.