Title: How Did It Ever Come to This?
1. What is “this”?
I speak of mathematics only, though there are obvious parallels in other subjects, especially science. The phenomenon is briefly described as a galloping anti-intellectualism, a “dumbing-down” of the curriculum for all students – but in the name of improved “understanding”. The pretentious phrase “higher-order thinking skills”, so prevalent in the education community today, is about as good a symptom as one can find: It permits, indeed insists on, the avoidance of mathematics itself. It pursues “real education”, stuff that goes with words like “today’s world” and “the new century”, replacing or adding to the progressive education cant of sixty years ago, where “vibrant” and “rich” were the adjectives describing an empty education program.
The new programs haven’t yet covered the entire nation but are spreading; and the propaganda is relentless. They have federal dollars, too, and state dollars, and local dollars, and they have votes. We haven’t invented “the math wars”; we are really faced with two cultures, both with the word “mathematics” in their names, but they are not the same. The education-school mathematics is in fact a form of anti-mathematics, as my fellow panelists will describe, and it is designed for the benefit of the people who are selling it, just as automobiles are designed for the benefit of the auto companies. But in the case of automobiles the public understands what it is getting, and has some alternate sources if dissatisfied. In education we don’t have that.
2. Where and why did it begin?
Well, I suppose it all began with “progressive education” -- John Dewey, Columbia Teachers College and all that. It has taken root first and most solidly in America, I believe, because it affects an egalitarian spirit which even Europe did not adopt until more recently. It has two facets: (a) a desire to have everyone succeed equally, whether or not he has learned equally, and (b) a desire to have schooling be a pleasure.
All this is painted as a reaction to the mores of the preceding era – Victorian rigidity. It is true that before 1900 nobody cared if a child didn’t learn, except perhaps his parents or friends. In most societies – even today -- people are born into a station and stay there. Education is by apprenticeship and imitation of parents, except for the aristocracy; so it is not a public problem.
Not so in 20th Century America. Here, especially with the great wave of immigration around 1900, education became visibly the way to get ahead, and legislators and educators wanted to make it universally available. This was America, after all; we cared publicly, and we put public money into it, so that little by little a public bureaucracy had to grow up to administer this unusual public policy, a very hard task, maybe impossible, and surely unprecedented in the history of the world. How to succeed?
Alas, though not everyone used it at first, there is actually an easy way to succeed: The way to guarantee an education available to everyone is, with enough public money, to make it easy, and to make it fun. Our educational theorists then discovered how to do this, by a species of inflation.
3. Inflation in Education
Making things easy leads to inflation. Between 1900 and 1950, education became increasingly valuable to those who had it. The high school diploma became the mark of a high school education, which employers had in earlier days taken as a sign of competence, discipline, prudence, literacy, and even skill in arithmetic. (Imagine!) A high school education had been uncommon before 1900 with good reason but became a real ambition by the end of World War I. So much so that its mere symbol, the diploma, became the thing on sale.
It is easier to print diplomas than to teach things to children and tell the losers they flunk. This development is quite analogous to what happens in economic policy where a populist ruler prints money to enrich the populace. It is easy to make people happy for a while if you are a good enough liar, or if, like other zealots, you yourself are so convinced of a false doctrine that you can induce others to follow. Progressive education, like Communism, has had some of each: some zealots, some liars.
Making things fun is harder. The theorists argued that learning could not be successful if the student didn’t feel a need for what he was learning. This looking for “felt needs” on the part of students led to the exaltation of “practical”, since apparently the kids should be attracted by what they saw around them, and not by abstractions. So the education people concentrated on that, along with the elimination of the difficult part of math.
4. Has this process been going on steadily?
No. I don’t believe such a policy can last for a century, and in America it got interrupted by World War II and its aftermath. Actually, only half of the progressivist program really took hold in the first half of the 20th century – the part about “felt needs”, that is, making math attractive by pointing to its practical value. Tradition was strong, too, and so the “discovery” part of progressivist mathematics, the removal of the transmission of knowledge in favor of exploration and “real life experience”, didn’t really become widespread until fairly recently. Thus school math in the first years of progressivist propaganda was actually mainly drill on practical things, some of them difficult relics of the Victorian age, pointless despite their alleged practicality. It was also very dull to many, so some parts of it -- -- the difficult parts – gradually deteriorated between 1900 and 1940. The progressivists had scored a partial victory, anyhow.
The cure for difficulties in the programs, especially if they are difficult for the teachers themselves, newly educated in the rising teachers colleges, was to offer only what was comprehensible, and here the theorists did gradually gain. By 1940 the high school mathematics program was sadly diminished, even for “college-prep” programs. One could stop math in the 10th grade and still have a “college-preparatory” diploma. For those who wanted to go further, maybe to go to engineering schools, the more advanced courses were mainly watered-down versions of books of earlier times, written by the new experts in the schools rather than the serious scholars who had written in the preceding era. We had Euclid left over in degraded form from some 19th century “prep-school” models, algebra for no discernable purpose, and trigonometry suitable for 18th Century surveying and navigation. For students not going to college we had cooking, typing and interest-rate tables.
During this time, behind the scenes as it were, Columbia Teachers College was spreading the word of student-generated learning, “experiences” instead of “lessons”, and practical knowledge instead of theoretical. To the mathematics education community being educated by such schools in rich, vital and vibrant education, anything in mathematics beyond the needs of the dry-goods store was something to be avoided, unless of course the child demanded it. The reduced programs were convenient, too: Since widespread public education required a cadre of teachers far larger than could be produced by college education, the average schoolteacher was educated in a teacher’s college, or a “Normal School”, where mathematics is never a strong point. Even if the profession had wanted a rigorous mathematics program in the schools, we didn’t have the teacher corps necessary – as we discovered in the later “new math” era. Teachers were used to arithmetic. Of course this would not have lasted, and would have given way to something that was more fun, except for an unforeseen accident of politics.
World War II shook things up, since it showed dramatically how wrong the 1940 notion of “practicality” in science and mathematics had been. Radar, and atom bombs were in the newspapers, and electricians and machinists also needed math. Soldiers returning from the war with GI Bill money were much wiser to the ways of the world than Professor Kilpatrick on upper Broadway, and reform of math education got started in the universities, though in only a few projects, long before Sputnik and the “missile gap” encouraged the federal government to call it a national priority. And guess who was then asked – in 1958 -- to help? Mathematicians!
How naïve they were in Washington, to ask mathematicians about mathematics education. The Professional Education Bureaucracy, which for fifty years had been building up a political base, never got over the insult, or the really sad sight of a river of education money suddenly flowing from the NSF to the wrong people. From 1950 to 1975, they increasingly took advantage of every apparent failure of the new programs to pave the way for their political counter-attack. They even helped out with these failures by their initial over-enthusiastic promotion of what they did not understand, especially in the creation of commercial textbooks.
6. Death of the New Math
What really went wrong makes a story much more complex than telling about what appeared to have gone wrong. The effort of the 1960s had its faults, but on the other hand most students in the country were never exposed to anything that really looked like what the few solid federally-supported projects were offering. Most of the project-related federal money of the time was actually going into valuable teacher-training Institutes. The silliest excesses of “the new math” were performed by commercial writers recruited by publishers for the exciting new market that bought anything with “new math” in the title. To say that “the mathematicians gave us the new math”, as Chester Finn once put it briefly in a letter to Commentary, is to ignore a great deal of history (and I won’t have it!).
What is true is that at the end of that period our math in the schools was on the whole better than it had been at the beginning, but not in the popular perception. Washington was flooded with propaganda from such organizations as the teachers unions and the National Council of Teachers of Mathematics (NCTM), saying “newmath” had been a disaster and the cure was to pour more money into education research, school lunches, professional development, closing the racial gap, abolishing tracking and elitism, and removing “memory” as a feature of education altogether. The campaign succeeded: The federal legislation of the 1970s abandoned the curriculum and teacher-training programs in which mathematicians could have any part at all, and replaced them by grants financing the teachers colleges and federal “educational laboratories”, to make endless studies of how children actually learn, but with no competent attention to what they should or can learn. The country sent the mathematicians back to their blackboards, and Education took back the schools.
By 1975 there was federal money for education such as would have amazed any mathematicians of 1960, but it was earmarked for projects defined in such a way that there was no place for a mathematician or even his advice in them. Yes, NSF did finance mathematical research, and wonderfully well, but not in a way that would turn mathematicians towards the problems of the schools. From 1975 until today the world of math education has been essentially devoid of mathematicians, former mathematicians, or people who really believe that knowing something about mathematics should have something to do with its teaching in the schools.
The generation in charge today has always known the answer it had waiting for the ills of school mathematics education, but had been sidetracked by the war and the Cold War for a while. Now, after that unfortunate period of waiting for power, they can say it straight out: What mankind has learned in the way of elementary mathematics in the past few thousand years is out of date, and to learn it is “rote-learning”, necessarily dull and impractical, and is not a vibrant education at all. They portray the 1940 world of arithmetic as the one their opponents dream of returning to, as if there were only two choices, and they know which choice is being made by a profession – their profession -- which, unlike the mathematicians, really understands children. And they have a propaganda machine that cannot lose, for who can be against money for education?
7. Rise of the Know-Nothings
In 1980 the NCTM published An Agenda for Action, a fairly short manifesto but one in which the entire program for the next 25 years (or forever) was prefigured. In 1989 it published its famous “Standards”, which are not really standards but a “vision” of what school mathematics should be. Scenarios are given there, of utopian classrooms, children discovering new mathematics for themselves every day. The NCTM hired a publicity organization to make sure the country took their Standards to heart, and its regional and national meetings featured only papers pressing and praising their progressivist doctrine. (It is called “constructivism”, by the way, and sometimes gets linked with other post-modernist nonsense, but the link is strained, and generally invisible to the teacher at the grass roots.) NCTM publishes journals, such as The Mathematics Teacher and the Journal for Research in Mathematics Education, printing only papers in support of official doctrine. Their journals differ from each other as much as Pravda and Izvestia did in the good old days. The era of the New Math, the 1960s, when there was a link between university mathematicians and the world of K-12 education, had featured a lively debate, within the mathematics and education communities as well as in the newspapers, over curriculum and teaching both, and some of this debate even made its way into NCTM publications; but while debate is not downright illegal since the 1989 NCTM Standards, it sure doesn’t appear in the NCTM journals.
An iron conformity has descended upon the world of mathematics education in the United States, heavily supported by federal money, and via devices of considerable political intricacy by state and local money as well.
Today’s doctrine is much more widespread than ever was any part of the New Math, because the educational bureaucracy has found its way into the national treasury to finance projects by which the official word is spread to teachers and principals, using commercial text material also developed with NSF money in the 1990s by the same people who now govern the teaching of teachers and future teachers. Some of the worst of these texts were officially anointed as “exemplary” or “promising” by our Department of Education in 1999, via a political ruse too complicated to describe here, and despite an open public protest, signed by a long list of mathematicians. The conflict is rightly called a war, though for the first few years there was only one side with any troops. However, though it took a while for some mathematicians and many more parents to discover, during the 1990s, what had been happening to their children, we are finding our voice.
Footnote: What I would have said if there had been time for a longer lecture.
Since the discussion given above is rather abstract it might clarify the terms of the math warfare to give an example of what I have been calling “today’s doctrine”, a piece of what NCTM considers exemplary of true “mathematical understanding”. The Year 2000 revision of the 1989 Standards (called PSSM for Principles and Standards for School Mathematics) mentions the division of fractions only once, and recommends against use of any algorithm for performing that operation. The traditional algorithm, briefly described as “invert and multiply”, is widely derided by those education experts who oppose the teaching of “paper and pencil arithmetic” by quoting the sarcastic rhyme, “Yours is not to question why; just invert and multiply.” The idea is that the old dispensation taught routines without meaning while the new teaches something deeper. What NCTM actually says about this in PSSM is found on page 219:
Although “invert and multiply” has been a staple of conventional mathematics instruction and although it seems to be a simple way to remember how to divide fractions, students have for a long time had difficulty doing so. Some students forget which number is to be inverted, and others are confused about when it is appropriate to apply the procedure. A common way of formally justifying the “invert and multiply” procedure is to use sophisticated arguments involving the manipulation of algebraic rational expressions – arguments beyond the reach of many middle-grades students … If students understand the meaning of division as repeated subtraction, they can recognize that 24 divided by 6 can be interpreted as “How many sets of 6 are there in a set of 24?” This view of division can also be applied to fractions, a seen in Figure 6.5 …”
In Figure 6.5 the problem is given,
If 5 yards of ribbon are cut into pieces that are each ¾ yard long to make bows, how many bows can be made?
This is solved by dividing 5 by ¾. The solution is revealed pictorially in Figure 6.5: The author draws a (purple) picture of a 5-yard ribbon and a scale or ruler of its exact length just below it, marked off 0,1,2,3,4,5, each division representing a yard, and each yard subdivided into fourths, making it easy to recognize a length of ¾ on the ribbon pictured just above the scale. The ribbon itself is marked off into as many pieces of length ¾ as it will take in, and they are then counted! Sure enough, we end with 6 (purple) pieces of the desired length and a bit left over, easily seen when matched to the scale to be worth 2/3 of a ribbon. Answer: 6 and 2/3 ribbons. A prodigy of reasoning, straight out of the Stone Age! (How lucky the roll of ribbon from which the bows were cut was not a couple thousand yards long! In real life factories they are, you know.)
Now the “invert and multiply” algorithm, deemed too difficult to demand of middle school children, would give the solution as follows: 5 divided by ¾ is equal to 5 multiplied by 4/3, so 5/1 X 4/3, or 20/3, or 6 and 2/3 ribbons. This works as easily for ribbons of length 3291 yards, to be divided into bows of length 13/16 of a yard each, by calculating 3291/(13/16) = (3291x16)/3, the rest being familiar computation with integers. To be able to do such things is not considered “understanding” by PSSM, however, which despairs of making the algorithm itself either memorable or understandable by the student. But this last consideration, so elaborately given as justification for abandoning mathematics in favor of a bull headed reversion to first principles which only seem convenient because of the triviality of the example used, is in fact mistaken.
The “invert and multiply” algorithm is easily understood and easily recovered if forgotten, by a student who has been taught something essential about fractions before getting to the question of their division at all. It is only PSSM that is ignorant of how easy and understandable it is. That essential information, which must be in any program that would lead to algebra in any case, is the idea of equivalent fractions.
Two fractions are equivalent if they represent the same number. Thus ¾ is equivalent to 15/20, for example, or to 6/8. Anyone who uses fractions at all must understand these equivalences, and the very idea of “percentage” is of course a matter of finding, for a fraction, an equivalent fraction with denominator 100. The way this is done is by multiplying numerator and denominator by the same thing. Now, well before introducing “invert and multiply”, the problem 5 divided by ¾ can be solved with the use of equivalent fractions: 5, or 5/1 if you like, is equivalent to 20/4, and the result of dividing 20/4 by ¾ may be construed as a fraction with numerator 20/4 and denominator ¾, which is equivalent to 20/3, or 6 and 2/3. A middle school student should, in any good program, be given a good work-out in finding and recognizing equivalent fractions, in multiplying fractions (which is easily reduced to multiplication of integers, with a resulting single fraction), and then in the use of these techniques to change a fraction with fractional numerator and denominator to a fraction with integral numerator and denominator. When the time comes to speak of fractions more generally, “a/b” instead of some particular numerical example, it will be discovered without effort that this procedure, already understood from earlier numerical examples, produces the computation
(a/b)/(c/d) = (ad/bd)/(cb/db) = ad/cb,
which is the result of “invert and multiply”, that is, (a/b)/(c/d) = (a/b)x(d/c), or ad/cb. This result is a theorem, something to prove once and for all and then put away for future mindless use, the way we end up memorizing and using 6x9 = 54 without having each time to draw a 6 by 9 rectangle and count squares.
This method is not, to be sure, discoverable by a child, which is probably one reason it is disfavored by NCTM. Indeed, all the philosophers of Europe didn’t have access to such methods until the Renaissance. It is real mathematics, and needs to be taught, whereas the new dispensation wants a species of understanding that needs no teaching and is the opposite of real mathematics. Yet this method (among so many others suppressed by NCTM) is utterly necessary, not for ribbons and bows but for all extensions of arithmetic to algebra and its applications to science. It is as necessary as knowing the names for the integers themselves, no matter how many calculators or computers you own. Ignoring all this, or perhaps simply ignorant of the implications of what it is doing, the NCTM scenario pictures willful ignorance as real understanding.
Mathematics is intended to make hard things easy by accumulating the discoveries of our ancestors and passing them on to the young in organized form. Today’s doctrine in the schools denies this, by requiring a constant return to first principles as evidence of “understanding.” What turns out to be true is that such teaching makes mathematics difficult in the long run, not easy; but school is not the long run. This particular NCTM middle school prescription has made things look easy by omitting the truly important knowledge that makes them truly easy – except for the unrealistically simple examples they trot out for ribbon cutting specialists.
Après nous le déluge. Arrange the curriculum to include only trivialities; nobody will notice. Inflate the grades, change the exams. They have been working on this for twenty-five years now, with a good measure of success.