Math II (562417), страница 15
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In an unrelated line of studies, ethologists demonstrated, starting with birds, that animals have at least some innate sense of number. The experiments of Otto Koehler (1889-1974) showed that pigeons, for example, could distinguish the differences between numbers of objects (or actions) only to five, after which they became confused. Ravens and parrots could get to seven. Humans do better because they can count aloud or to themselves in words, "one, two, three,..." To determine the human ability excluding language, Koehler flashed different numbers of objects on a screen too briefly to allow a count. Under these circumstances, people's abilities were remarkably similar to those of birds. Some human subjects could distinguish only to five, and few reached as far as eight. (10)
More recent studies have used the window into nonverbal cognitive capacities afforded by attention time. (11) It is now well established that animals and human infants will gaze at unexpected scenes longer than at expected ones. Researchers can thus investigate expectations ("thoughts," "built-in beliefs") without language. If, for example, a caged monkey pays more attention to an object that freefalls on a slant (controlled by invisible wires, say), than one that falls straight down, then one can conclude that the monkey expects the straight drop. This means that the monkey has a kind of prelinguistic grasp of the effects of gravity.
Using attention-time experimental designs, researchers have shown that children develop some abilities much sooner than Piagetian research had shown. For example, such studies reveal that human infants as young as four months expect ("think") that one and one will equal two. A typical experiment goes like this: build a miniature stage with a movable curtain. With the curtain open, show the infant a single doll approaching, entering into, and then appearing in the stage area. Close the curtain. Now show a second doll doing the same thing. Open the curtain. If the stage now has two dolls where only one had been before, the infant quickly loses interest. If, however, the stage (which has an unseen back door through which the experimenter can "alter reality," so to speak) now shows only one doll, the infant will gaze at the stage for a longer time. In this way the experiment demonstrates that the infant expects one and one to make two.
From such clever experiments it has now been shown that certain arithmetical abilities are clearly prelinguistic, probably innate, and at least that they develop without known instruction. These abilities include at least subitizing, the ability already noted to distinguish precisely between small numbers, the adding and subtracting of very small numbers, and approximating, the ability to make reliable distinctions between larger collections of quite different size (such as, say, between 20 and 12).
These modern studies, then, focused on how mathematical ability relates to factors other than language.
A More Recent Trend
More recently, however, language has come to the fore. Investigators have begun to address in earnest the effect that language has on mathematical development. I'll mention five books that illustrate this trend. Eleanor Wilson Orr's 1987 book, Twice as Less, focused on the effect that non-standard English usage might have on learning mathematics. (12) It specifically asked on the dust jacket, "Does Black English stand between black students and success in math and science?" It's not an easy book to read, because it deals with algebraic word problems, many people's worst dread. It suffers also from its narrow concentration on one group. For example, it attributes some usages to Black English (such as, "two times smaller than") that are also found in the scientific writings of astronomers, microbiologists, and computer scientists. That the book arrived while some people were pushing Black English (Ebonics) didn't help, and it has apparently not shaken its unfortunate status as politically incorrect.
The Mathsemantic Monitor's own book Mathsemantics: Making Numbers Talk Sense appeared in 1994. (13) It said there's a field available for scholarly study where math and the meanings of ordinary language interact, but which both disciplines have neglected, and it dubbed this area "mathsemantics." It described the dimensions of this field as involving both "the math side and the semantics side, childhood beliefs and stages, errors of all types, education and math anxiety, linguistic and cultural differences, evolution and history, math notation and number-memory, games and sports, childhood exercises that develop mathsemantic savvy, physical science and its philosophy, money and jobs, politics and the media, business and the professions, population and the environment, estimating and accounting, punctuality and time frames, gender differences, statistics and surveys, the future, what we can do about it, and so on; you name it." It wouldn't be fair, perhaps, to cite it, by itself, as particularly indicative of any trend.
Stanislas Dehaene's book The Number Sense: How the Mind Creates Mathematics appeared in 1997. (14) It presents neurological and other studies of innate mathematical abilities, the so-called "primitive module," and also the developmental effects of language.
Though mathematical language and culture have obviously enabled us to go way beyond the limits of the animal numerical representation, this primitive module still stands at the heart of our intuitions about numbers.
Dehaene asks, "How did Homo sapiens alone ever move beyond approximation?" and answers, "The uniquely human ability to devise symbolic numeration systems was probably the most crucial factor." He then traces some of the development. Languages distinguish through inflection the differences between one and two (in English, singular and plural), "but no language ever developed special grammatical devices beyond 3." Counting depends on the body, so that "in countless languages ... the etymology of the word 'five' evokes the word 'hand'"; "children spontaneously discover that their fingers can be put into one-to-one correspondence with any set of items"; and in New Guinea the counting by body parts has evolved so far that "the word six is literally 'wrist,' while nine is 'left breast.'" Try counting with your right hand on your left hand, follow through on your wrist and arm, and you'll see how this works quite naturally.
Keith Devlin's The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip appeared in 2000. (15) It quickly disavows the idea of a single math gene, saying it is a purely metaphorical expression for the idea that one has "an innate facility for mathematics." Devlin then makes language the central player.
My argument that you possess the math gene -- i.e., that you have an innate facility for mathematics -- is simply this: your genetic predisposition for language is precisely what you require to do mathematics.
Devlin defines mathematical thought more narrowly than most people would. He excludes subitizing, counting, adding, subtracting, and approximating, whether these are the products of innate abilities or aided by language. For him, mathematics requires a level of abstraction beyond ordinary language.
As he sees it, language is the third level of abstraction, the first one that permits off-line thinking. (16) (To follow Devlin's arguments, one must regard language not as communication but as a representational system.) It is found only in humans, permits them to have "imaginary versions of real objects," which is " "to all intents and purposes, equivalent to having language."
What Devlin says distinguishes the fourth (or mathematical) level is that "mathematical objects are entirely abstract; they have no simple or direct link to the real world, other than being abstracted" from it. The simplest example is algebra, where letters (such as, a, b, x, y) stand for numbers, not particular numbers, but variable numbers, numbers in the abstract.
Devlin then discusses how particular characteristics of language, as given mainly in the works of Bickerton (17) and Chomsky (18), enable the further step to mathematics.
A Korzybskian could with good reason dispute some of Devlin's distinctions. Nevertheless, one should applaud his attempt to show that mathematics is an outgrowth of ordinary language abilities.
Another book published in 2000, the fifth and final book I wish to mention here, is Lakoff and Nunez's Where Mathematics Comes From. (19) The authors are, respectively, a linguist and a psychologist. They claim with some justification that their book is the first real attempt to ground mathematics totally in the human brain, body, and language.
It starts by reviewing the by-now familiar territory of innate abilities but quickly moves to more fundamental embodied-mind abilities, such as basic motor control, which gives rise to the source-path-goal schema. What makes the schema important are the distinctions it involves: a starting point, a moving trajector, an intended destination, an intended route, the actual trajectory, position at a given time, direction at that time, and the actual final location.
From this and simpler schemata (such as "container"), the book argues, each human creates categories and basic logical relations. The creations arise through metaphorical mappings. For example, take the "categories are containers" metaphor, which maps things true of containers onto categories. If one puts an item in container A, and puts container A in container B, then the item is in container B. Change "container" to "category," and you have the metaphorical mapping. The argument proceeds with laudable care.
From such beginnings, with the critical addition of the ability to use symbols (language), the book argues that humans have created mathematics by a long (and unended) series of metaphorical mappings (such as of "object collection" and "motion along a path" onto arithmetic). A critical metaphorical mapping was that of "iterative processes" ("John jumped and jumped again, and jumped again") onto the unending series of integers used for enumeration. This permitted a numerical characterization (but not the only one) of mathematical infinity.
The metaphorical mappings grow ever more complicated until they encompass algebra, symbolic logic, sets and hypersets, real numbers, transfinite numbers, infinitesimals, and the discretization program (in which innate geometry goes out the window, calculus and space become motionless arithmetic, and "points" are not "in" space; rather the numbers for the points define space). It's a veritable tour de force.
The book, as the authors say, is not mathematics; it's a cognitive study of mathematics. It tries to trace mathematics back to human experience, an embodied mind. In a way it's like Ernst Mach's program of showing that abstract concepts draw their ultimate power from sensuous sources, but with a sophistication Math might never have imagined.
Now, what these five books have in common is an emphasis on the relationship of mathematics and ordinary language. (20). Orr shows how non-standard English can disrupt mathematical learning. MacNeal identifies the area where mathematics and ordinary meanings mix and names it "mathsemantics." Dehaene says we share innate, primitive, mathematical abilities with many animals that language permits us to transcend. Devlin says mathematical ability (in his more abstract sense) depends on a further abstraction from ordinary language. Lakoff and Nunez detail the progression from innate math and language, via schemata and metaphor, to algebra and higher math.
Perhaps at no other time have so many books appeared that investigate, each in its own way, the relationship of math to ordinary language.
A Current Research Project
The Mathsemantic Monitor's book addressed a combined math language overlap field and asked a question outside the purview of most present-day math and English teachers. It is a question, however, that readers of this journal might see as central; for it is in the Korzybskian tradition. The question is this: In what ways does an unconscious reliance on the meanings implicit in ordinary language disturb mathematical understandings that conscious training could avoid? The book presents examples showing many kinds of disturbance at work, which reviewers have generally applauded. (21)
That much done, the next question would seem to be this: Where should the necessary conscious training take place? The answer is not easy, for neither of the usual candidates, math and English departments, has jurisdiction over mathsemantics. It's apparent that what's needed is some kind of alliance of Englishmath interests. Sadly, until a short time ago, the Mathsemantic Monitor had had no success in generating any such alliance. (22)
The turning point was an e-mail from a Dr. Kurtis H. Lemmert, Associate Professor of Mathematics at Frostburg State University (FSU), the westernmost institution of the University System of Maryland. Dr. Lemmert fired off his e-mail as a shot in the dark. He wanted to know if the author of Mathsemantics had any ideas that he (Lemmert) might pursue on a sabbatical he intended to seek. The answer was yes.
A few e-mails later, Dr. Lemmert had proposed a program of mathsemantics research for his sabbatical, and, through friends in the FSU English Department, had found a Dr. Glynn Baugher, who had recently become Professor Emeritus of English. Dr. Baugher was interested. Math had been his undergraduate major. The three of us now constitute a mathsemantics research team representing English, math, and business.
Our first objective is to determine where mathsemantics might be taught. The research plan approaches this by a survey that asks (a) whether, and in what courses, respondents have studied mathsemantic problems in their own classroom education and Co) whether, and in what courses, they think such problems should be addressed in formal education.
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