Saturday, April 11, 2009

Memorizing vs. Rote Learning & Drilling: More Ways In Which Anti-Reformers Get It Wrong

In an on-going discussion about Benezet on the list that unfortunately is shedding limited light on the subject, the suggestion was made that "promoters" of Benezet and other reformers (which in this context means "progressive education, student-centered learning, humanistic, and constructivist types") are opposed to memorization. This sort of false accusation is typical of the muddying of the waters by certain fanatic critics of anything outside the neo-Prussian school of how to teach mathematics to children.

On Apr 10, 2009, at 1:53 AM, Paul A. Tanner III wrote:

[T]here is much written by reformers against this necessary part. This includes all that anti-rote rhetoric which serves as backdoor anti-memorization rhetoric.
My Response

Wrong again, Paul. I know of no one who opposes memorization (which by the way is NOT the same as rote learning. More on that in a moment). What many people oppose or question (again, not the same thing; I feel it is necessary to bore the people who read fluently when addressing ANYTHING to you, because you are SO adept at misreading, misconstruing, misinterpreting, garbling, or otherwise distorting or changing what people actually say and believe into nifty little straw ideas you can "decimate" with your famous logic "traps"), is MINDLESS rote learning of things that can be learned effectively, possibly MUCH more effectively, in other ways.

I don't claim to be a mnemonist, but I am quite experienced with and knowledgeable of memory training methods. I know quite well the methods that were used by Bruno H. Furst in his courses (published in a number of forms, one of which was a printed course called STOP FORGETTING) and later Harry Lorrayne (with Jerry Lucas) in THE MEMORY BOOK, and on his own in REMEMBERING PEOPLE. None of the methods Furst or Lorrayne taught were original to them, as far as I could determine from researching this field around 1980 when I was asked to do so for a company in New York City called The College Skills Center. The methods are, however, extremely easy to understand and put into practical use for a wide range of applications.

One thing that became quite clear to me at the time was that these methods pay off in inverse proportion to the arbitrariness of the material being memorized: what that means is that when faced with , say, a random or arbitrary list of items, dates, facts, etc., the more random the list, the less conceptual links or "common knowledge" might be involved, the more a person using mnemonics would gain from using these techniques, because otherwise the main option would be. . . {wait for it} some variation on pure rote.

However, less time was needed for memorizing information with more structure, because the "inherent logic" or interconnectedness of the information helped one memorize. Arbitrary or random information (and that's a subjective thing in many cases: what might seem unstructured to me might be very clearly structured to someone more knowledgeable about the things in question) is damnably difficult to hold onto. Getting more that 7 +/- 2 bits of information into short-term memory is challenging for most people, as researchers in the field have known for a very long time. Getting such bits into long-term memory usually requires apply structure where no such structure may have been obvious (or even exists until an individual human mind creates it).

Mathematics already is based on logical and conceptual links. Hence, it is often the case that what needs to be "memorized" in the sense mentioned above is minimal and arbitrary. What sorts of things would be arbitrary in mathematics? Well, things like Order of Operations, which consists of conventions, not something that simply HAS to be. Terminology. Notation. Axioms. Things that do not follow from first precepts.

Even going beyond that, it is undoubtedly true that we need to "memorize" certain fundamental relationships and identities in specific areas of mathematics in order to not have to tediously look them up for every single instance in which they arise. In trigonometry, for example, understanding the definition of sine, cosine, and tangent in right triangle trigonometry is a key "fact" that one does much better to have at one's mental fingertips than not. Hence, Big Chief SOHCAHTOA. However, knowing that tangent = sin/cos reduces three things to two. Knowing about the reciprocal functions and cofunctions reduces a lot of other facts to far fewer. Knowing the angle sum and difference formulas for sine and cosine allows one to figure out sine and cosine and other trig functions for a bunch of "non-standard" angles if one knows and gets the derivation of the quadrant angles and the angles of the 30-60-90 and isosceles right triangles, and how to rotate them through the unit circle. One can also derive the double angle formulas for sine and cosine (etc.) with the angle sum formulas.

My point in the above is that the amount that "must" be memorized is often far smaller than one originally believes (or is falsely led to believe by dull or incompetent instructors), because of underlying relationships and concepts that create natural connections among a smaller set of facts. Anyone who is led to believe that it all needs to be memorized by drill or rote is being mistaught.

Unfortunately, Paul, you never seem to see such distinctions, or if you do, you seem to steamroller over them in your fervor to turn yet another issue that could be reasonably debated into something absolute. In this case, it is, for you, either one hates, loathes and refuses to allow students to engage in memorization of any sort, or one very reasonably indulges in and encourages it.

But such a dichotomy is, quite frankly, utter nonsense. It's impossible to get through mathematics without certain facts. It's also impossible to actually DO mathematics without a lot of concepts and habits of mind. The two areas are clearly interrelated, and only a fool would utterly proscribe learning facts. The key questions in this regard when it comes to mathematics however, are not quite as simple as it might seem. They are: 1) What exactly MUST be known, and 2) How exactly MUST what must be know be learned?

I don't know if you honestly think that everything in the first category must be learned through rote. But I do know that there is a great deal that need not be learned that way, and I think that what a lot of us who find your views off-putting think is that you don't seem to grasp the power for students of NOT memorizing, but rather understanding the linking ideas so that anything not used frequently (for practice is absolutely essential for recalling "raw, arbitrary facts," as anyone who know about memory training will tell you; and that means not only initial practice during the learning, but periodic revisiting/refreshing of previously learned material if one isn't using it regularly through the natural course of one's life) will deteriorate.

On the other hand, that which is understood conceptually has a better chance of lasting, and can be more readily recreated through the concepts even if the "at one's fingertips" recall has been weakened or extinguished. Most people are well aware of this through personal experience. Perhaps you are unique or one of the few who has not figured this out or had it happen (in which case, I can more readily understand your viewpoint, even if I think it's not one that applies to very many people, and in fact need not apply to most people unless they have some actual organic damage that would prevent them from using their memories the way most folks can do naturally.

So while it's lovely of you to talk about "all those anti-memorization reformers," I think they are essentially a myth, more straw people for you to demolish. But the argument isn't "Pro-memorization vs. anti-memorization." It is pro-rote & drill vs. those who wish to minimize the time spent UNNECESSARILY on such an approach where it is not appropriate and where other methods may have far larger payoffs in the long run, especially for the time invested.

Very little in life worth knowing gets memorized with zero effort as we get older (though kids manage to absorb inordinate amounts of information, STAGGERING amounts, in fact, with little or no conscious effort at all). This fact is not unrelated to the Benezet-Kamii arguments, of course. On their view (and mine) traditionalists prematurely try to treat little kids as if they're older, more rigid learners. Stuffing one "absolutely right and best" way of doing arithmetic down little kids' throats is, according to Kamii's research, both unnecessary and for many of them a practice with a negative expected value. I know you, Wayne, and a few others can't accept this, in part at least because that likely SEEMS to run counter to your own personal childhood experiences. The problem is, of course, that first, you don't know how things might have gone had you been taught less traditionally, and you certainly don't know that what worked for you automatically is best for everyone else. You seem absolutely wedded to the notion that everyone SHOULD learn identically, and that that way they should identically learn happens to be the way you did or believe you did, or still do.

How absolutely egocentric of you (collective you here and above, for the most part). How selfish. And how utterly wrong-headed. Ditto when it comes to the issue of memorization through primarily or exclusively rote methods. That just is NOT a sufficient repertoire and it is clearly a huge turn-off, road block, and ultimate mistake for a lot of kids. I realize that it offends you deeply to call rote learning what it is: drill. But it offends a lot more educators to be told that rote is a better way to teach subjects that are fraught with already existing structures that students can use to build understanding (and hence have a framework upon which to hang facts as they acquire them). I know, too, that you will move not one single angstrom away from your entrenched views, no matter how many studies you might be shown, no matter how clear or eloquent my arguments and examples might be, no matter how many experts in cognition and memory might be called forth to explain to you that you just have it wrong for the most part. But some other people here may benefit from reading what I've written, and so I have taken the trouble to discuss this as if I thought you might indeed learn something

Friday, April 3, 2009

LEARNING MATH BY THINKING - Hassler Whitney, Louis P. Benezet, and how many more wasted lives and decades will it take?

It was 1986, folks (or perhaps 1929), for those keeping score at home. Twenty-three (or eighty) years later and the same arguments are going on, the same mistakes are being made, as if nothing at all has been said like what Louis P. Benezet or Hassler Whitney offered. As if Constance Kamii's work has never been done or published.

My thanks to G. S. Chandy for pointing me to this article. It was published while I was in the process of taking undergraduate mathematics courses in NYC and slowly gravitating towards changing fields from literature to mathematics education. I'd never heard of NCTM or any of the other organizations involved in mathematics teaching and research, the Math Wars hadn't officially started yet, and had I read this piece at the time, I would have naively wondered how anyone could be on the other side from people like Benezet and Whitney. Having suffered through a K-12 mathematics education that was about as inspiring as a dead fish in the gutter, it is remarkable to me even today that I took it upon myself in my thirties to go back to school just to prove to myself that I could indeed learn more mathematics, from calculus to where I was led. Didn't plan to go into the field, and it was fortune, more than anything, the attention of one of my instructors, that led me to start teaching remedial mathematics and, eventually, to do graduate work in math education at the University of Michigan in the 1990s.

Yes, there has been some progress since then, but the entrenchment of traditionalists is fiercer than ever. The lies, distortions, selective quotations, meaningless and carefully culled data, shifting criteria for what "counts" when it comes to evaluating programs, teachers, schools, kids, materials, etc., and many other shady tactics continue unabated, fueled by a hatred for innovation and purveyed by politically-motivated, educationally conservative and reactionary pundits, think-tanks, and foundations, all fiercely determined to see to it that mathematics teaching and learning in this country remain in the hands of a smug, patronizing elite. As long as they are successful in reducing us to a standardized-test crazy culture, as Whitney so accurately describes below, the country as a whole and millions of children will suffer unnecessary torment and boredom when it comes to mathematics. And a populace that is mathematically ignorant is a populace that is far easier to lead by the nose.

Are Benezet's and Whitney's ideas really just those of a couple of isolated cranks, as the anti-progressives from groups like Mathematically Correct would have us believe? Consider for a moment the following anecdote about the great mathematician, Augustin Louis Cauchy:

A mathematical friend of Cauchy's father, Lagrange, recognized the young boy's precocious talent and commented to a contemporary, 'You see that little young man? Well! He will supplant all of us in so far as we are mathematicians.' But he had interesting advice for Cauchy's father. 'Do not let him touch a mathematical book till he is seventeen.' Instead, he suggested stimulating the boy's literary skills so that when eventually he returned to mathematics he would be able to write with his own mathematical voice and not one he had picked up from the books of the day.

It proved to be sound advice. Cauchy developed a new voice that was irrepressible once the floodgates protecting Cauchy from the outside world had been reopened." (Marcus du Sautoy's THE MUSIC OF THE PRIMES pp. 65-66)

No, the analogy to Benezet's experiment is not perfect by any means. But it does suggest that there has long been an awareness that some aspects of formal instruction as they become institutionalized can be stifling to creativity and originality.

Does this mean I am advocating for the destruction of public (or any) formal education and schooling? Not quite. What I am advocating for is a coming to sanity on the part of educators in this country when it comes to mathematics teaching (if nothing else). We are destroying our children, en masse, with the most stultifying approaches imaginable to learning and doing real mathematics, substituting instead a phony "school mathematics" that serves no one truly well, and from which only a small minority emerges able to actually do mathematics, in spite of, rather than because of, the way the subject is taught in most instances.

Few American K-12 teachers have the smallest idea what mathematics is, what it means to do mathematics, or what it means to be a professional mathematician. And what these teachers wind up doing, consciously or not, is to guarantee that very few students will ever find out.

Every time I bump into a piece like the one about Whitney and Benezet below, I am both amazed and sickened: amazed that I hadn't seen this before (though I've been aware of Benezet for over a decade now); sickened that the same lies rise like a foul smokescreen every time Benezet's name or any idea that sounds even vaguely like his, is presented. How much longer do our children have to be tormented by meaningless mathematics education? When will it be time for real mathematics to be taught and learned, in a manner suitable to how children are? If not now, when? If not us, who?

Published: June 10, 1986

SCHOOL reformers, business executives and politicians are demanding more mathematics for American children. Schools are responding, at least in terms of the hours given to math. Not all mathematicians are cheering. They worry that pressures for more hours of mathematics may hurt rather than help, unless mathematics is taught differently.

Dr. Hassler Whitney, a distinguished mathematician at the Institute for Advanced Study in Princeton, says that for several decades mathematics teaching has largely failed. He predicts that the current round of tougher standards and longer hours threatens to ''throw great numbers, already with great math anxiety, into severe crisis.''

Dr. Whitney has spent many years in classrooms, both teaching mathematics and observing how it is taught, and he calls for an end to what he considers wrongheaded ways.

Long before school, he says, very young children ''learn in manifold ways, at a rate that will never be equaled in later life, and with no formal teaching.'' For example, they learn to speak and communicate, and to deal with their environment. Yet the same children find much simpler things far more difficult as soon as they are formally taught in school.

Learning mathematics, Dr. Whitney says, should mean ''finding one's way through problems of new sorts, and taking responsiblity for the results.''

''This has been completely forgotten'' in most schools, he finds. ''The pressure is now to pass standardized tests. This means simply to remember the rules for a certain number of standard exercises at the moment of the test and thus 'show achievement.' This is the lowest form of learning, of no use in the outside world.''

Dr. Whitney, in a recent report in The Journal of Mathematical Behaviour, recalled an experiment begun in 1929 by L. P. Benezet, then superintendent of schools in Manchester, N.H. Mr. Benezet was distressed over eighth graders' poor command of English and their inability to communicate ideas.

''In the fall of 1929,'' he wrote in 1935, ''I made up my mind to try an experiment of abandoning all formal instruction in arithmetic below the seventh grade and concentrate instead on teaching the children to read, to reason and to recite'' by reporting on books they had read and on incidents they had seen. The children were no longer made to struggle with long-division. ''For some years,'' Mr. Benezet went on, ''I had noticed that the effect of the early introduction of arithmetic had been to dull and almost chloroform the child's reasoning faculties.''

Over the years numbers crept into the children's experience, Mr. Benezet said. They learned to deal with ''halves'' and ''doubles,'' with estimates of size, with a natural development of multiplication tables and slowly, with formal arithmetic.

Mr. Benezet concluded that children who had not been dragged into early but only dimly understood mathematics eventually outdistanced those who had. Literacy in English and a capacity to think independently and to speak and write clearly helped many to do well in mathematics, too.

Dr. Whitney points to that experiment as he looks at today's mathematics teaching. He cites the responses to a problem on a recent test given by the National Assessment of Educational Progress: John and Lewis are planning a rectangular garden 10 feet long and 6 feet wide, and they want to put a fence around it. Ignoring such real matters as the need for a gate, the question was simply how many feet of fencing was needed.

Of the 9-year-olds who took the test, 9 percent chose 32 feet; 59 percent, 16 feet; 14 percent, 60 feet, and the remaining 18 percent gave other answers. Of the 13-year-olds taking the test, 31 percent said 32 feet; 38 percent, 16 feet; and 21 per cent, 60 feet, with 10 percent giving other answers that apparently did not use any arithmetical formulas.

''Why did not all the children get the correct answer?'' Dr. Whitney asks. ''If they were involved in it as a real problem they could have drawn a picture or made it real in some way, and looked to find the answer.'' Instead, he said, they did it ''the school way,'' guessing at what kind of ''operation'' to use - multiplying or adding the numbers.

Numbers, Dr. Whitney says, become a tool when you use them for a purpose. In a class of 6-year-olds, he recalls, the teacher explained how to find the sum of 3 plus 5 by drawing ducks on the board, not noticing a boy in the back of the room saying to another, ''Yesterday I gave you 10 cards; now you gave me 7, so you still owe me 3.''

In the traditional school climate, Dr. Whitney writes, children's natural thinking ''becomes gradually replaced by attempts at rote learning, with disaster as a result.'' In high school, students increasingly say, ''Just tell me which formula to use,'' a way of saying, ''Don't ask me to think.''

Because teachers must ''cover the material,'' Dr. Whitney adds, there is less time to think. When students are called on, they must answer instantly. Wrong answers are not discussed.

''Students and teachers are all victims'' as national commissions clamor for more mathematics without realizing, Dr. Whitney warns, that they may create less knowledge and more anxiety. He says it is crucial to stop just learning the rules.

Dr. Whitney's views are controversial, as were Mr. Benezet's in 1935. Some mathematics teachers and other experts may denounce them as soft on mathematics, but others may welcome relief from demands that turn youngsters off mathematics. Of course, some teachers, ignoring the demands of the moment, actually do teach in the Benezet-Whitney fashion.

However controversial his views, Dr. Whitney deserves a hearing. Present attitudes, he writes, ''lead to the lowest of goals, passing standardized tests,'' instead of encouraging the kind of thinking ''essential for true progress in science, techology and elsewhere.''

The mathematics teaching Dr. Whitney talks about makes children want to know the answers in situations that are real to them. It makes mathematics come alive for them as they do their own thinking and take control over their work, not for tests but for themselves.