In an on-going discussion about Benezet on the math-teach@mathforum.org 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