Sunday, April 12, 2015

Part III of what went wrong with American math teaching: How we all learned a lot from the New Math, but only a few of us learned any math from it



Another one of those story-thus-fars that regular readers can skip.

Let's see if I can review even more quickly this week:
  1. I'm working on a book called Singapore Math Figured Out ForParents.
  2. Singapore Math is the system of math teaching used in that small nation since 1981, in most of the top-performing-in-international-comparison Asian nations, and increasingly in many other countries.
  3. For about ten years it's leaking slowly into the United States via charter schools (especially for the gifted), homeschoolers, and some of the "college academy" schools that aim at increasing the college enrollment and success of low-income, first generation, and people of color.
  4. Singapore Math has been partially included in the Common Core standards, and many public school systems are considering moving to it as a way to meet those standards.
  5. Singapore math is genuinely better than other ways ofteaching math, but it absolutely requires teachers who really know what they're doing, which requires them to be adequately prepared and trained, which, historically, the United States has screwed up nearly every chance it got. So my guess is that my country is going to miss its best chance to become better at math, and this point in history is a spectacularly bad time to do that.
  6. As part of Singapore Math Figured Out for Parents, I've had to study a great deal of the history of math instruction, especially in the U.S. The main track of the book will be about how parents can make Singapore Math work out for their kids, but for a variety of reasons it seems like a good idea to have a section about how we got into this mess in the first place. 
  7. And so I've been blogging that history, about one sizable chunk per week. 
I began with the problems with traditional proceduralism (the "basics" that most "back to basics" want to get back to), which mainly involve a tradeoff: to achieve early computational proficiency (kids who are fast at basic arithmetic in grades 1-4), procedures are taught in a way that causes most people, later on, to run into a wall (that "horrible thing that math was easy up until" that is part of so many people's childhood misery in math -- the moment when it all stopped making sense, whether that moment was long division, fractions, decimals, or basic algebra, to cite the four most common). After that I looked at the JohnDewey-inspired "pragmatic" or "math for everyday use"approach, which essentially avoids the wall by going so slowly and staying so tied to very simple problems that most students never go far enough to hit it.

And so here we are at the most infamous of all math curriculum changes: the New Math.

New Math: the first try to fix math instead of avoiding it.

In the whole history of mathematics education in the USA, probably nothing began with more promise and higher hopes, or ended in more dismal results, than the School Mathematics Study Group's curriculum, known forever to everyone as New Math. 
In 1957, the Soviet Union orbited the first artificial earth satellite, Sputnik, beating the United States into orbit by what turned out to be almost four months. At the time, because the possibilities of artificial earth satellites had been covered heavily in the press, most American politicians, reporters, and educated people knew that a rocket big enough to orbit a 183-pound satellite was big enough to deliver an atom bomb anywhere in the world.
After that, events seemed to conspire with the Soviets to make it all extra-humiliating for the Americans. Sputnik-1, at 83 kg, went up in early October 1957, and Sputnik-2, at 500 kg (half a ton), went up a month later. Then in December, Vanguard, the first American attempt to launch a satellite (accurately described by Khruschev as "about the size of a grapefruit"), collapsed and exploded, having attained all of four feet toward orbit. Finally, on January 31, 1958, a hastily cobbled-together and restarted Army program managed to send up the American Explorer I, which was roughly the size and shape of a fencepost, weighing in at 13 kg (or about thirty pounds). So in a space of about four months, the Soviets showed that they were there first -- twice before we even got one to blow up on the ground -- and all of it much bigger and more impressive.
For years before the "Sputnik crisis" (as Eisenhower himself called it) in America's technological self-confidence, college mathematics professors had grumbled about the poor preparation of entering freshmen. Hard to imagine as it is today, the closest thing there was then to a "national curriculum" was the bestseller list of some of the largest textbook publishers (out of hundreds of textbook publishers, and even the biggest were nothing like the Big Three of today). A few schools of education at some state universities suggested curricula; most did not.
From school district to school district, what was taught, in what order, with what rationale, and with what quality of instruction varied immensely. A few kids from high schools like Bronx Science arrived at college at sixteen, already having completed a year of very theory-heavy calculus. Many more kids from Speedtrap, West Dakota, might only have had the option of three years repeating Practical Math (a common course at the time: a review of common arithmetic word problems) because there was never enough interest to put together an algebra or geometry class.
Furthermore, many instructional materials were frankly terrible. A retiring math teacher who needed to supplement a pension could self-publish a hastily-written summary of his or her lesson plans and call it a math textbook, which his or her old school district would then adopt. Math instructors at small colleges, often just former star students with only a BA from the same college, published similar textbooks via the small commercial publishers (whose sales people frequently bribed them into school districts with donations to athletic teams or price breaks on cleaning supplies).
Many of these textbook authors had never been particularly good at math, and were simply repeating or plagiarizing the work of similar predecessors, often with new errors in each generation of copying-without-comprehension. Furthermore, the World War II-Cold War frantic demand for engineers and technicians of all kinds guaranteed that anyone with decent math chops could usually make more (and find more interesting work) in industry than in public schools, so those confusing not-quite-right textbooks were often being explained by hastily re-allocated gym or home ec teachers.
Because traditional proceduralism had held out against Deweyan reforms in many areas, at least until after the Second World War, there were many proficient "computers" (in the sense of that era: people who were really fast and accurate at memorized algorithms) around. For example, here are the computers the Jet Propulsion Laboratory used for the Explorer I launch:

http://www.nasa.gov/mission_pages/explorer/computers.html

And in Denise Kiernan's The Girls of Atomic City you can meet some of the computers who helped design the atomic bomb. (Along with many other women doing many other things at Oak Ridge during the war).
 So we had plenty of good "computers" and some of them were finding their way into college math classes. But a relatively small number of people had ever even been taught the concepts of real mathematics. To most people, including most mathematics teachers, math was carrying out those patterned manipulations you'd use to run Eb's General Store, just like in 1910, except that fifty years of Deweyan pragmatism had eliminated most of the difficult patterns.
Even in the relatively few (by today's standards) secondary schools that taught appropriately advanced mathematics, learning was impaired by math instructors who didn't know math very well teaching from textbooks by people who knew very little more. Many of them were teaching ideas that were not true, patterns of reasoning that were not valid, rules that didn't always work, and in short a load of mathematical nonsense that students would have to unlearn in college.
It would have been difficult to pretend that no one knew before Sputnik. Critical activists, most importantly the National Council of Teachers of Mathematics, but also the main university mathematician organizations, had been screaming from the late 1940s on.
Moreover, even before Sputnik, the public was having a pretty thoroughgoing reaction against Deweyan pragmatism. The method of teaching reading then known as look-say was taking a well-deserved beating in the conservative press and in books like Flesch's Why Johnny Can't Read (1955).
All of this was suddenly excruciatingly visible in the rocket's red glare. Less than a year after the Sputnik launch, Robert Heinlein was giving a painfully accurate summary of the consequences of Deweyan pragmatic education in Have Spacesuit,Will Travel (1958):
I simply stared. "Why, I'll graduate from high school, Dad. That'll get me into college."
"So it will. Into our State University, or the State Aggie, or State Normal. But, Kip, do you know that they are flunking out 40 per cent of each freshman class?"
"I wouldn't flunk!"
"Perhaps not. But you will if you tackle any serious subject—engineering, or science, or pre-med. You would, that is to say, if your preparation were based on this." He waved a hand at the curriculum.
I felt shocked. "Why, Dad, Center is a swell school." I remembered things they had told us in P.T.A. Auxiliary. "It's run along the latest, most scientific lines, approved by psychologists, and-"
"-and paying excellent salaries," he interrupted, "for a staff highly trained in modern pedagogy. Study projects emphasize practical human problems to orient the child in democratic social living, to fit him for the vital, meaningful tests of adult life in our complex modern culture. Excuse me, son; I've talked with Mr. Hanley. Mr. Hanley is sincere—and to achieve these noble purposes we are spending more per student than is any other state save California and New York."
"Well . . .what's wrong with that?"
"What's a dangling participle?"
I didn't answer. He went on, "Why did Van Buren fail of re-election? How do you extract the cube root of eighty-seven?"
Van Buren had been a president; that was all I remembered. But I could answer the other one. "If you want a cube root, you look in a table in the back of the book."

            It was a very open secret: in math instruction, and in many other subjects, the 1950s was the heyday of emphasizing socializing the kids at the expense of dumbing down, when in many states a single "health" class (don't pick your zits, puberty is normal, eat the food pyramid) was being counted as the mandatory "science" for a high school diploma. 
And why not? 
It was, after all, the era when boys in Detroit referred to lifelong employment at GM as "thirteenth grade," and when social promotion from grade to grade had become the norm because, well, why not?
Suddenly, Sputnik.
Suddenly, we'd been beaten out of the gate in the race to the next frontier. Suddenly, the country of Edison, Westinghouse, and Oppenheimer,  had been skunked by the country of steam tractors and cheap suits.
All at once, the people who had been complaining about the Deweyan pragmatic reduction of school to a day camp for future housewives and assembly-line workers (because "you could get a job with that") had a much bigger audience. The quiet of 30s, 40s, and 50s in math education was exposed as the silence of the empty chicken coop, where rocket-propelled commie chickens now roared in to roost.
Teenagers in the tech-labor-force pipeline were unprepared or underprepared. It was painfully clear to millions of Americans that they couldn't trust the math teachers to teach math. Worse yet, what was being taught was usually watered down and sometimes actually wrong. Where could America turn for the math whizzes to beat those Russians right now?
Strangely enough, we had a large share of the world's math talent; we just weren't doing much in K-12 to develop more of it. For generations, the university mathematicians had been mostly ignoring what was going on in the schools, except to arrange private tutoring or secure one of the few good high schools for their own kids, and to complain about the incoming freshmen who appeared never to have seen an x in an equation before. Now, in the post-Sputnik panic, a small group of public-spirited and civic-minded, genuinely excellent university mathematicians organized themselves into the School Mathematics Study Group (SMSG). In the urgency of the moment, with money flying out of Washington and the state capitals, they secured National Science Foundation funding to tackle the problem: what should the K-12 math curriculum look like?

New Math and Old Myth: Scraping off some of the accumulated barnacles of self-justification

Because what followed was a disaster, and history is re-written by the culpable, it's important to avoid swallowing all the myths whole. New Math did not come from nowhere and it was not an obviously bad idea; it was an emergency attempt to deal with a perceived gigantic problem, and before it was tried, there were many good reasons to think it could and would work. In those consensus-on-the-Cold-War days, the New Math had the political fingerprints of everyone from the deepest conservatives to the most blazing liberals all over it.
Far from creating New Math or conspiring to foist it on the public, the educational establishment and its attendant bureaucracy (like any establishment or bureaucracy confronted by a demand for immediate radical change) hunkered down to resist it, actively or passively. Though individual teachers out in the schools tried to make New Math work (and sometimes succeeded), New Math was crammed down the choking and gagging throats of the mostly non-teaching educators in the teacher's unions and the schools of education, who resented and feared the loss of control over a major area of school curriculum.
Furthermore, New Math was not concocted by eggheaded theory addicts for the greater glorification of theory. The SMSG sought to prepare students for some of the best jobs then available, at least white male middle-class students. (In the climate of the time, a claim that women, non-whites, or poor people had "special needs" would have been regarded as a thinly veiled statement that they couldn't hack it, and as a basis for discrimination.) The objective was to replace a curriculum that did not prepare people for science, engineering, and high-level technical work with one that thoroughly prepared them.
Later, after New Math had crashed and burned, some conservative politicians and some teachers' organization leaders would try to paint the pre-New Math era as one of idyllic contentment in which the kids just quietly learned their math. Then, their story ran, (conservative version) evil teachers unions and (teachers union version) wicked ivory tower mathematicians, in cahoots with the brutal text book industry (both versions, and they did have a point) set out to destroy American mathematics, and using their eerie totalitarian powers of mind control, seized control of mathematics in America and destroyed it.
That tale wasn't true. Again, math instruction had been deteriorating for decades before Sputnik triggered the formation of the SMSG. Again, right and left, labor and business, everyone from the hippest beat to the grayest square in a gray flannel suit had been complicit as math programs decayed for decades before Sputnik, and supported the New Math at it inception. The most prominent opposition group before New Math was adopted was actually the "professional educator" bureaucracy, who felt quite correctly that they had been denied a seat at the table.
Moreover, the SMSG actively sought practicality; that was the whole reason to introduce a far more rigorous math curriculum, reversing decades of dumbing down. The story that New Math was a bunch of ivory tower educrats replacing real math with a bunch of theoretical nonsense is a bogus rationalization formulated long after the fact, mainly to support fundraising in conservative parent groups and to justify the counterrevolution of the classroom teachers and their professional leadership.
Unfortunately, though an ill-advised alliance of emotional supporters of traditional proceduralism and self-interested supporters of Deweyan pragmatism were the people who carried out the counter-revolution against New Math, the obstinate fact is that New Math was a disaster, and it really did have to be scrapped. The energy to get rid of it may have been supplied by Sister Mary Hick'ry Stick and Curriculum Director Homer "Get a job" Simpson, but it had to go anyway.

For want of a kindergarten teacher, algebra was lost ...

The crash that followed the SMSG's optimistic and well-meaning start had many causes. First of all, the professional educators did have a major point: almost no classroom teachers and no educational psychologists at all were involved in the curriculum design. Given the state of American math education at the time, that probably seemed like a good thing, a kind of spiteful spurning of a profession which had done so badly for so long.
But commitment to a set of bad ideas does not necessarily mean that people are ignorant of everything important, and commitment to do better does not create needed information out of nowhere. The SMSG had no one to tell them that first and second graders do not really have a sense that things could have gone differently than they have, so it makes little sense to try to teach them probability. There was no voice to point out that most kids don't really see how a proof can be true in abstract, or grasp the distinction between valid and true, until they are about ten or eleven.
Rather than using known, well-established information about average children in ordinary schools, the SMSG seems to have drawn on a small population of former children who were in command of advanced concepts at a very young age -- that is, themselves, kids who grew up to be mathematicians. So, based on their own experiences as bored prodigies having their time wasted in dumb-to-them classes, the mathematicians naturally preferred theoretical rigor and ignored developmental appropriateness, and there was no one to point this out.

If the cardinality of the set of stools=2 and is restricted to integers, how do we define the location that is between two stools?

It also didn't help that much of the American mathematics community was pretty far toward the systematization pole in the eternal cycle between pioneering and systematization. In the long history of math stretching back through several different cultures and many millennia, some mathematicians (like Galois, Cantor, Riemann, Noether, or Godel) have always been subversive pioneers, finding the problems and issues that no one else had looked at and proposing solutions to them. Other mathematicians (like Euclid, al-Kharizmi, Newton, Gauss, or Hilbert) have been systematizers, who construct the proofs and build the chains of logic that tame those frontiers and regularize them into orderly branches of mathematics. Mathematics needs both, and most research mathematicians are themselves a mixture of pioneer and systematizer.
But at different times, places, or branches of mathematics one or the other tendency will be dominant. To use everyone's favorite metaphor, it's a pendulum: systematizers lock things down into rigid structures. Pioneers begin to look at the foundations and discover incompleteness and weaknesses, eventually undermining or transforming the structure. That creates work for the systematizers who build it up again, better and more complete before.
In the late 1950s, American mathematics was passing from a long generation of capable, brilliant systematizers into a younger generation of iconoclastic pioneers. Many senior mathematicians had spent most of their working lives in an environment where math research tried to resolve questions with proofs and to pave simple, clear, straightforward roads into the wild country that the mathematicians of 1850-1910 had opened up. The systematizers' dream of having a definitive list of axioms and rules of proof, from which all of mathematics could be constructed in one vast derivation, seemed closer to possible then than it has at any time since. Although Whitehead and Russell had failed at one of the most ambitious attempts to unify all math in a common set of axioms, and Kurt Godel and Alan Turing had shown that it could never be done completely, the vision lingered on in older mathematicians that somehow, one day, there would be a list of axioms, and a very long single proof ending with "QED Math!"
At the same time, there were a host of marvelous cracks and odd spots in the foundations of traditional mathematics. Whole new branches and subjects like game theory and discrete mathematics were opening up, and fields that had been moribund for decades like computation and number theory were moving again. Younger mathematicians were finding immense, fruitful new areas to work in. It was an exciting time in math research, with the frontiers advancing in many directions all at once.
So when the SMSG set out to derive a math curriculum, they were looking for two different contradictory goals:
  1. a rigorous structure in which math would be built up from fundamental axioms, via logic, into complex deep concepts, thus ensuring that mathematically talented students would arrive into university classrooms with a deep understanding of the basis of math itself.
  2. a mind-broadening freewheeling experience of the many exciting faces of modern mathematics, of how much there was yet to be discovered just for the sheer joy of knowing it, and applied to sciences and human affairs in ways not yet dreamt of.
            They weren't thinking very much about whether either of these approaches would work for a student of average ability trying to learn from a teacher who didn't understand it very well, assisted by parents who didn't see any particular need for either perfect logical rigor or the discovery of more esoteric math, especially not when they mainly expected their kids to learn to "put down two, carry the one."
Furthermore, because mathematicians are unsurprisingly drawn from kids who were good at math and had little trouble with it in early grades, the SMSG was made up of people with a deep disdain for drill. They had mostly been that seven-year-old who had the multiplication table in two days. That had been painfully frustrating when teachers kept drilling it for three months, chanting out "two times ... three times ... four times ... " That was, after all, the curriculum around the time of World War I, when senior mathematicians of the 1950s were in grade school.
Back then, no matter how much the seven-year-old mathematicians-to-be really wanted to ask about the main diagonal forming the set of perfect squares, or some rudimentary idea of prime numbers, they quickly learned not to ask about it; classroom order was still being enforced with paddles and dunce stools. So the math professor of 1958 learned to survive, in 1918, by plodding through the drills, but he didn't learn to like it. By the end of the second grade, most of them still loved math, hated math class because it was all drill, and probably despised the inept classmates and teacher who forced the drill on them.
Forty years later, trying to figure out an elementary school math curriculum, they thought a great deal about how dull that drill was for that talented kid. And they weren't about to invite some second-grade teacher or educational psychologist into the room to explain what drill did for the average kid (whether the average kid liked it or was aware of it or not).
So on one level, New Math was designed to be what the top 10% of the students in every math class had always wished math class would be. Which was great if you were in that top 10% -- and so was your teacher. For everyone else, it was a worse version of what had been going on for a long time: the teacher didn't know much math and many kids didn't acquire much of what s/he did know.
 But that was only the first stool. On another level, New Math was also supposed to get kids excited about all that great math they could learn, or would be learning shortly, in all those hot new fields.  It was rather like being trapped in a room with a pack of math nerds, all of whom are anxious to tell you why their math is the coolest math there is. That's a tough enough experience for an evaluator from the National Science Foundation, let alone for a third-grader.
At the very basic level, the branches of mathematics seem far apart and as if they have little to do with each other. At the high end, everything you can say in mathematical logic you can say in set theory, which is why set theory is a powerful tool designing computer algorithms; at the low end, one is about whether or not Socrates is mortal and the other is about overlapping circles. A first-grader can handle a little bit of rudimentary set theory, logic, arithmetic, number theory, and geometry, but what a first-grader can handle just doesn't overlap enough to make deep connections.
So instead, teachers and students tended to concentrate on whatever was easiest to understand at that grade level. First-graders took time off from addition facts to learn geometric definitions that they might not use again for a year or more; fourth graders escaped from long division to talk about functions; everyone started every year with set theory because, although it has few applications at the most basic level, it's relatively simple.
Naturally this reinforced the student tendency to learn math "in the moment" or "in the unit," only remembering what they need to do the work in class today. Young students often like the pretty patterns and rhythms but seldom see why they might want to remember it; "learning math" is a possibly fun activity for now, not a cumulative process leading into the future. That's a large part of why traditional proceduralism sends kids into the wall: the patterns are fun, so they learn the patterns and are rewarded for it, but no one insists that they also get the ideas that they will need. Then one day they run out of pattern and need an idea -- and don't even know what one is.
New Math managed to make that process even more devastating to young math students than traditional proceduralism had been. Those long breaks to visit "other cool math" were interrupting the already-difficult process of building math up from basic bedrock concepts to advanced conceptual structures tied together with cables and beams of logic and proof. Either the goal of building math like a mathematician from unifying concepts or the goal of giving the students mathematical breadth by introducing them to many different fields would have been hard enough by itself. Trying to do both of them at once would probably have failed with the best mathematics teachers in the world.

Standing between two stools is even harder when there's no support

And that wasn't who was trying to teach it. With a whole new mathematics curriculum coming in, much of it unfamiliar to teachers already out in the schools, there were three alternatives the school boards of America had for meeting their mandates to move kids over to New Math:
  1. What no one did: Retrain or replace most of the teachers in the lower grades, at a large cost, and begin New Math with the incoming first, second, and third graders, working your way upward into the higher grades as you obtained more teachers who really understood it. It would also have helped to supply supplementary material and sessions to explain things to parents so that they could stay involved and learn along with their children. This would have cost a great deal, but probably would have worked as well as anything could, given the damage already caused by split focus and developmental mistakes.
  2. What the SMSG recommended: since the hard way was clearly out of the question, the SMSG suggested that school districts should begin by introducing New Math via high school advanced classes, where bright students could quickly catch up on the pieces they were missing, and where teachers tended to be more mathematically proficient, so that retraining would be easier and quicker. This would allow time to introduce New Math into teacher training curricula in the teacher's colleges. Then the schools could gradually spread the concepts downward (from Grades 10-12 through junior high, middle grades, and primary grads) and outward (from advanced to standard academic to general and remedial classes) as older teachers retired and better-prepared ones moved in. This meant a more or less constant retraining budget for about ten years, working down from 12 to K, giving the maximum time to the less math-oriented lower grade teachers (and allowing some graduates to come back around the cycle and enter the system in elementary education). The estimate was that in about ten years the whole school system could be converted to the new way, with most of the (large) expense falling in years 4-7 of the process.
  3. What they actually did: For just the cost of replacing the textbooks (and remember, that's money for kickbacks and new football uniforms), hand all the teachers at all levels the new textbooks in the last couple weeks of summer break and tell them to look these over and try to stay a chapter ahead of the students. If any teacher absolutely insists, send them to one of the cheap or free SMSG summer math seminars, from which they can return either as confused wet blankets, or as unpopular know-it-alls, either way helping to ensure nobody else would want to go (and ask for travel money) the next year.
Math is for everyone but being a mathematician is not
With all those forces taken all together, New Math would have been doomed even if it had been a genuinely great way to teach math. But even if we could have willingly afforded double-Ph.D. math/Ed.D. child psych students in every classroom from first grade on, the primary problem with New Math is that although seeing how all of mathematics hangs together is central to a mathematician's understanding of math -- and often seems to be the moment when they really realize that they are mathematicians and that mathematics is what they want to do with their time -- it isn't the way in which most people who need to get math get it.  The developmental path for nearly everyone is drastically different from the path of formal logical exposition that mathematicians mostly build for themselves in retrospect.
That last point is important enough to elaborate a little: most mathematicians themselves learned basic math by a developmental path, that is, one that introduced concepts and ideas in long-form, slow ways that a kid's mind can grasp, and gradually introduced more advanced concepts first as abbreviated or short-hand versions of the basic ones, and then as extensions to them. Later, the extensions become a quick, clear explanation and take the place of the developmental experience.
For example, developmentally, most people naturally learn addition as "counting on": you have three pieces of candy, someone gives you two more, and you first learn that to "add" the three and the two, you start at three, and then count two more: 1. "four", 2. "five." Since counting on always gives the same correct result no matter what you're counting, be it galaxies or bumblebees, you memorize "addition facts" so that you don't have to spend all your time counting. Then you extend further to learn that among the countable things are numbers themselves: three tens and two tens make five tens, but we can abbreviate that as 3x10+2X10=50, and so on through multiplication.
Now, if you're a mathematician, one day, when you are starting to see that the way addition grows out of multiplication, and exponentiation out of multiplication, is something much more subtle and powerful than just "grouping" or "repeating," you are ready to learn to talk about all of them as related operations which form a closed group on the integers.
The mathematician quite possibly went over that same road, though further than you did. But looking back, with fully developed mathematical eyes, s/he sees a completely different definition of addition, one that has nothing to do with counting. It will begin with a definition of the integers (the union of the set of whole numbers and their additive inverses) and then give five brief equations (the first one is n+0=n) which can be chained together to define the solution of any addition of integers no matter how complicated, and finish by noting that the group of integers is closed under addition (meaning if you add two integers, you always get another integer).
To a professional mathematician, a computer scientist, and sometimes to a physicist or chemist, that sets-operations-groups definition of addition is actually more useful than the plain-old ordinary regular one. Indeed, it's useful to the mathematician or the computer scientist to think of the sets-operations-groups definition as the thing that you use to confirm that the everyday, developmental one works for all the possible integers.
But it is not necessarily best for a first grader to try to start out with sets, operations, or groups. Intuition about what numbers are and what they mean is vital for the first few years of mathematics, particularly for learning to see the world mathematically, and intuition, or "number sense," develops out of correctly interpreted specific and simple experiences.
The way of the pure mathematician is to start somewhere in theory and explore from there, and be only vaguely aware that someone, someday, may tunnel in from the applied world and find a use for the highway you've built. It is a good way for mathematicians to learn, discover, or explore the middle part of the journey -- getting around in theory. But finally, it's neither what most people need to do, nor the way to make them clear on what needs doing and confident in doing it. Sadly, for most people, traditional proceduralists were right (the time would be more profitably spent learning multiplication tables) and so were the Deweyan pragmatists ("you can't get no job with that.")

Of all sad words of tongue or pen
(and also calculation)
The saddest are these--it might have been
(but for the implementation).

Why "sadly"? Because there was a baby in that bathwater: the real understanding that leads to using math for clearer thinking, better jobs, and ultimately a more successful life does matter. In places where the New Math was implemented well and properly, there was in fact a bumper crop of well-prepared students who went off to great success in college and life.  If they hit the wall at all, they hit it later, more gently, and with more resources for getting past it.
The concepts to get around in theory-land, and to understand how and why to go in and out of it, are genuinely essential. As breathtakingly poor as the pedagogy of New Math was, the content was in fact just what the doctor would have ordered, if he hadn't been a quack. Unfortunately, the first real medicine that showed up was so unpalatable, and so poorly thought out, that it was virtually impossible to get it into anyone.
That disastrous lost generation also highlights a few other lessons for anyone seeking to improve math instruction: first of all, that parents are going to make it or break it, for reasons that make sense to the parent. If the parent doesn't know what this is about, or how to help their kid with it, there's going to be a revolt, not soon, but right now. If there's no resource for the parent to turn to, all those problems are going to be much greater.
Secondly, individual classroom teachers are going to be a mixed blessing for any kind of math reform; the ones who are eager to learn and use it will be the reason it succeeds, but not every teacher is going to want to do that, and during the transition, there will be teachers telling the children that "this stuff in the book doesn't make any sense" and urging parents to complain about it. If parents know what the program's about, and agree with it, the resistant teacher can't do much damage; if they don't, well, that's what killed the New Math.
Finally, the New Math saw the first "indignation entrepreneurs" -- people who began to make a living, and sometimes quite a good one, by organizing against a curriculum. They weren't major players in the 1970s as the New Math was being abandoned, because the movement was so large and diverse already that it didn't need much more recruitment or leadership. But the discovery that parental anxiety and dislike of schools was a potential source of money, votes, and indirectly of prestige and power was not lost; it would play an increasing role in both the advocacy and the defeat of every reform (and every counter-reform) from then on.
Speaking of which, in the next few weeks I'll take up several of those. Next up is what is variously known as discovery or "inquiry-based" math, which, like New Math, demonstrates that the same idea can be foolishly and unfairly maligned at exactly the same moment when it is also being wisely limited or even justly rejected. And for those of you who are really not into math teaching, I have some hopes of having something nonmathematical up later this week.

Thursday, April 9, 2015

Part II of what went wrong in American math teaching: The first try at dealing with the wall: Deweyan pragmatism, Homer Simpson as your curriculum director, and keeping the kids out of Elfland.


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:
  1. Singapore Math has an extraordinary record as a successful curriculum for elementary through high school math, in a large number of international comparisons.
  2. 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. 
  3. 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. 
  4. 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.
  5. 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.
  6. Hence: I'm working on a book called Singapore Math Figured Out for Parents.
  7. And one of the key things that parents need figured out is why Singapore Math is needed, which is to say why  
  8. we couldn't just keep doing the same thing we'd always done,
  9. past reforms not only failed but were doomed to fail
  10. 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:
  1. 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?
  2. 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.
            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.