Math (Несколько текстов для зачёта), страница 2

To divide fractions

The method was not to invert and multiply but to find the product of the numbers joined by each of the first pair of arrows.

(15/16) / (2/3) = 45/32 = 1 13/32

(15/16) / (3/2) = 32/45 = 1 13/32

THE OPIE COLLECTION

Those two books represent the mathematics typically learned by middle-and upper-middle-class teenagers. The reader might ask about younger readers and more advanced mathematics students. For the answer, we turn to the Opie Collection, a special collection in the Bodleian Library, Oxford University. When it was presented to the university, the Opie, as it is known, was dedicated by Prince Charles, and it will soon be available for North American viewers through the UMI microfiche collection. See www.umi.com/hp/Support/Research /Files/220.html.

The 20 000 items in the collection include fairy tales, nursery rhymes, games, comic books, and coloring books, as well as game boxes and other educational items. Early American children's books, especially those reprinted in London, are part of the Opie. They include Tommy Thumb's Song Book (1794), which is thought to be the earliest known surviving edition of what may have been the first English nursery rhyme book. Mother Goose had already been printed in Boston. Then, and now, rhymes with counting were considered to be a child's first, and possibly best, introduction to arithmetic.

For the youngest students

At 3 3/4 by 2 1/4 inches, the size is the first thing that one notices about A Compendium of Simple Arithmetic; in which the First Rules of That pleasing Science are made familiar to the Capacities Of Youth, a book for elementary-school-age children. These books were "little books for little people." Indeed, the Opie has books so small that they can scarcely be held between the thumb and index finger. They typically begin with writing and spelling the counting numbers. Wallis's Compendium (1800), the title page of which is shown in figure 5, then goes to great length to explain the advantages of the "cypher," or place value, and the "decadary" system.

Wallis writes--for young children--that "neither a Euclid nor an Archimedes with all their wonderful mechanical powers" was able to extricate their number system from a "labyrinth of confusion." As in other titles of this decade, addend, minuend, and subtrahend are explained, but products are composed of "factors." The checking of subtraction and division is called the "PROOF" in bold letters. Definitions appear, for example, "Simple division is the finding how often one simple number is contained in another. The calculation is written as

Dividend Quotient

Divisor 3) 12 (4

or, for a longer problem, as follows:

833) 3104679 (3727 88/833

6056

2257

5919

88

Another definition is, "Reduction is the conversion of numbers from one name to another, but still retaining the same value." Although it was written for young children, this tiny book, like all titles in this article, contains tables for wine measure, as well as ale and beer measure. See figure 6.

For more advanced students

Three particular qualities of mathematics of this era should be noted:

1. For British students, advanced mathematics was synonymous with geometry, and most students studied an edition of the first six books of Euclid's Elements. The most popular edition of that time was the one by Robert Simson, of the University of Glasgow. The obvious advantage the student using later editions by Simson is that the Euclidean propositions were each followed by a proof. Moreover, each book of Euclid was accompanied by sample examination questions.

Another edition of Euclid was written by John Playfair, of the University of Edinburgh. It contains his Axiom 12, now known as Playfair's axiom, which states, "Two straight lines that intersect one another cannot both be parallel to the same straight line." This statement, and its implied deviation from earlier editions of Euclid, evolved into the largest controversy of nineteenth century British mathematics.

In his preface (1795, pp. iv-v), Playfair remarks that Dr. Simson had been "the most successful" modern editor and had left "very little room for the ingenuity of future editors to amend or improve the text of Euclid or its many translations." Playfair wrote that Simson's objective was "to restore the writings of Euclid to their original perfection, and to give them to modern Europe as nearly as possible in the state wherein they made their first appearance in ancient Greece." Playfair praised Simson by stating that he knew languages, was profoundingly skilled in geometry, and was an "indefatigable" researcher. To "restore" Euclid was a perfect mission for Simson. Playfair, however, believed that despite Simson's endeavors to remove corruptions, something was "remaining to be done." Playfair wrote that "alterations might be made that would accommodate Euclid to a better state of the mathematical sciences," and thus the Elements would be "improved and extended," more than at any "former period."

2. Until the American Revolution, one book--a single copy--was typically shipped across the Atlantic and then carefully used by an instructor to lead advanced mathematics students through a course of study. The Revolution brought about a change. In 1803, for example, an edition of Simson was printed in Philadelphia. Copies of Simson's later editions are still available in several older libraries.

3. Although mathematics journals existed, scant exchange occurred between German mathematicians and French or English mathematicians. However, the Opie collection does contain a fine translation from the University of Paris of Selected Amusements in Philosophy and Mathematics proper for agreeable exercising of the Minds of Youth (Despiau 1801). The introductory material is similar to that in the English books previously described in this article, but it ends with a discussion of topics that are now associated with probability. It includes factorials, permutations, combinations, Pascal's triangle, and various types of "gaming" odds--all topics that were highly developed in France. Actuarial tables on expected length of life include corrections for the large number of deaths that occurred in the first year of life. See figures 7, 8, and 9.

FOR THE UNIVERSITY STUDENT

The British Library has a copy of the "most important parts" of the arithmetic and algebra examinations required of candidates for an "ordinary" bachelor of arts degree from Cambridge in the early nineteenth century. A Cambridge or Oxford degree did not--and still does not--have "breadth" requirements. Unlike in American universities, one who "reads maths" studies no other subjects. The undergraduate degree is given at the end of three years. Mathematics majors must successfully write examinations that include only mathematics questions.

The arithmetic problems in the early 1800s required computational skills, conversion of measures and money, extraction of square and cube roots, and applications to business, especially interest and discount. Most of the algebra is commonly taught in high school today. However, some problems are unusual, whereas others are surprisingly familiar. Consider, for example, the following:

5. What will be the price of carpeting a room of 13 feet 4 inches long, and 12 feet 6 inches broad, at 4 shillings 6 pence a square yard?

Ans. £4. 3s. 4 d., or 4 pounds sterling, 3 shillings 4 pence.

12. Extract the square root of x[sup 4] + 8x[sup 3] - 64x + 64.

Ans. x[sup 2] + 4x - 8.

13c. Solve the equation

1/x + a + 1/x + 2a + 1/x + 3a = 3/x.

Ans. -11 +/- Square root of 13/6 a.

15. Expand

1/Square root of a - x

to 4 terms by the binomial theorem.

Ans. 1/a[sup 1/2] + x/2a[sup 3/2] + 3x[sup 2]/8a[sup 5/2] + 5x[sup 3]/16a[sup 7/2] + &c.

The answer in Arithmetic and Algebra (Wallis 1835 p. 327) is incorrect. The answer should be

1/a[sup 1/2] + x/2a[sup 3/2] + 3x[sup 2]/8a[sup 5/2] + 5x[sup 3]/16a[sup 7/2] + &c.

See Anton (1992, p. 730).

16. Insert 6 arithmetic means between 1/2 and 2/3.

Ans. 1/2, 11/21, 23/42, 4/7, 25/42, 13/21, 9/14, 2/3,

Find the sum of the series.

Ans. 4 2/3.

17. Define a logarithm; and shew that log N[sup p] = p log N. Having given log[sub 10][sup 2] = .30103 and log[sub 10][sup 3] = .4771213, find log[sub 10]36 and log[sub 10].018.

Ans. 1.5563026 and 2.2552726.

The answer 2.2552726 represents the centuries old notation and "tables" answer of characteristic + mantissa, or (-2) + (.2552726), and is equivalent to the contemporary calculator answer of (-1.7447275).

CONCLUSION

These publications furnish a record of the skills thought to be essential at the turn of another century. These mathematical records illustrate the continued need to develop good materials and tests. The era that gave us the legendary names of Trafalgar, Waterloo, Nelson, and George III was preparing its young for the increasingly complex global society.

In Britain today, parents are just as concerned as Americans with the education of their children. Specific topics debated in Parliament and discussed in the media are uncannily similar to those in the United States. Testing for teacher competence in mathematics and English, meeting standards, reducing class size, overcoming the shortage of qualified teachers, finding after-school care, and censuring underachieving schools are discussed at least as much-or more--in the United Kingdom than they are in the United States. The BBC and the government broadcast professional commercials in which a famous person, for example, Paul McCartney, reminisces about a favorite teacher. The government rates schools, and the ratings appear in newspapers. Being scrutinized and meeting standards are accepted as part of the system.

THE EVOLUTIONARY CHARACTER OF MATHEMATICS

Source: Mathematics Teacher, Nov2000, Vol. 93 Issue 8, p692, 3p

Author(s): Davitt, Richard M.

In her article "The Changing Concept of Change: The Derivative from Fermat to Weierstrass," Grabiner (1983) notes the following:

Historically speaking, there were four steps in the development of today's concept of the derivative, which I list here in chronological order. The derivative was first used; it was then discovered; it was then explored and developed; and it was finally defined. That is, examples of what we now recognize as derivatives first were used on an ad hoc basis in solving particular problems; then the general concept lying behind these uses was identified (as part of the invention of calculus); then many properties of the derivative were explained and developed in applications to mathematics and to physics; and finally, a rigorous definition was given and the concept of derivative was embedded in a rigorous theory.

As Grabiner observes, the historical order of the development of the derivative is exactly the reverse of the usual order of textbook exposition, which tends to be formally deductive rather than intuitive and inductive. Grabiner's article contains a number of other well-articulated historical and pedagogical messages, and I strongly encourage every mathematics instructor to read it in its entirety. However, this article emphasizes only her use-discoverexplore/develop-define (UDED) paradigm to describe the derivative's evolution. This model is extremely useful for constructing accounts of the evolution of numerous mathematical concepts and theories in addition to the derivative. In various courses that I teach, I often ask my students to use UDED to compile their own accounts of the evolution of mathematical entities. Occasionally, I have also required students to report their findings to the class, but the final, structured account is usually intended for the individual student's benefit alone.

Such assignments have many advantages. By encouraging my students to refer to such reputable histories of mathematics as those cited in the bibliography in constructing their accounts, I introduce them to the history of mathematics in a manner that is not overwhelming. This same exercise helps students understand that because most historical accounts are somewhat subjective, students need to justify their historical claims by citing reliable sources. For example, by using the UDED paradigm, students can learn to appreciate the basis that an author uses to assert that Isaac Newton and G. W. Leibniz invented calculus, that Girolamo Cardano was the first to solve the general cubic equation, that Carl F. Gauss, Janos Bolyai, and Nikolai Lobachevsky invented non-Euclidean (hyperbolic) geometry, and the like. Furthermore, as Grabiner observes, students learn that creating mathematics is often incremental, inductive, and exciting and that our modern versions of mathematical theories are polished diamonds that started off as rough pieces of carbon.

When I heard a colleague in the physics department describe the scientific method as "the development of knowledge from observation of specifics to conjecture to experiment to theory," it dawned on me that the UDED paradigm is essentially nothing more than using the scientific, or experimental, method to describe how mathematical theories and concepts evolve. Fuzzy foreshadowings, false starts, and dead ends have occurred in developing scientific models before such modern theories as those of the atom, light, heat, electricity, evolution, and the cosmos have crystallized and have been accepted as legitimate scientific theories. Students need to see this connection of shared modi operandi in the evolution of both mathematics and the natural sciences.