Friday, November 6, 2009
- By Douglas WeaverMathematics Coordinator, Taperoo High Schoolwith the assistance ofAnthony D. SmithComputing Studies teacher, Taperoo High School.
IntroductionOn the topic of mathematical symbols....."Every meaningful mathematical statement can also be expressed in plain language. Many plain-language statements of mathematical expressions would fill several pages, while to express them in mathematical notation might take as little as one line. One of the ways to achieve this remarkable compression is to use symbols to stand for statements, instructions and so on."Lancelot Hogben
Index
The factorial symbol n!
The symbols for similar and congruent
The symbols for angle and right angle
The symbol pi
The symbol for percent
The symbol for division
The symbols for inequality
The symbol for infinity
The symbols for ratio and proportion
The symbol for zero
The radical symbol
The symbols for plus and minus
The symbol for multiplication
The symbol for equality
The symbol for congruence in number theory
Complex numbers and the symbol i
The number e
The calculus symbols
List of ancillary symbols without explanation
APPENDIX --- Personalities
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The factorial symbol n!
The symbol n!, called factorial n, was introduced in 1808 by Christian Kramp of Strassbourg, who chose it so as to circumvent printing difficulties incurred by the previously used symbol thus illustrated on the right. (Eves p132) The symbol n! for "factorial n", now universally used in algebra, is due to Christian Kramp (1760-1826) of Strassburg, who used it in 1808. (Cajori p341) EVES, HOWARD "Great Moments in Mathematics - Before 1650", Mathematical Association of America 1983. CAJORI, FLORIAN "A History of Mathematics", The Macmillan Company 1926.
The symbols for similar and congruent
Our familiar signs, in geometry, for similar (on the left), and for congruent (on the right) are due to Leibniz (1646-1715.) (Eves p253) Leibniz made important contributions to the notation of mathematics. In Leibnizian manuscripts occurs this symbol (on the left) for “similar,” and this symbol (on the right) for “equal and similar” or “congruent.” (Cajori p211)EVES, HOWARD "An introduction to the History of Mathematics," fourth edition, Holt Rinehart Winston 1976 CAJORI,FLORIAN "A History of Mathematics", The Macmillan Company 1926
The symbol for angle and right angle
In 1923, the National Committee on Mathematical Requirements, sponsored by the Mathematical Association of America, recommended this symbol (on the left) as standard usage for angle in the United States. Historically, Pierre Herigone, in a French work in 1634, was apparently the first person to use a symbol for angle. He used both the symbol above as well as this symbol on the right, which had already been used to mean "less than." The standard symbol survived, along with other variants, as follows. These appeared in England circa 1750.During the 19th century in Europe these forms were used to designate the angle ABC, and the angle between a and b , respectively. This symbol, representing the arc on the angle, first appeared in Germany in the latter half of the 19th century.
The symbol for right angle
This symbol (on the left) for right angle was used as early as 1698 by Samuel Reyher, who symbolized "angle B is a right angle" as illustrated on the right, using the vertical line for equality.This commonly used symbol for right angle appeared in America around 1880 in the widely used Wentworth geometry textbook. (NCTM p362,364)THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, "Historical Topics for the Mathematics Classroom", National Council of Teachers of Mathematics (USA) 1969
The symbol for pi
This symbol for pi was used by the early English mathematicians William Oughtred (1574 -1660), Isaac Barrow (1630-1677), and David Gregory (1661-1701) to designate the circumference , or periphery, of a circle. The first to use the symbol for the ratio of the circumference to the diameter was the English writer, William Jones, in a publication in 1706. The symbol was not generally used in this sense, however, until Euler (1707-1783) adopted it in 1737. (Eves p99)Oughtred's notation was the forerunner of the relation pi = 3.14159..., first used by William Jones in 1706 in his Synopsis palmariorum matheseos. Euler first used pi = 3.14159... in 1737. In his time, the symbol met with general adoption. (Cajori p158)This symbol for pi was used by Oughtred in an expression to represent the ratio of the diameter to the circumference. Isaac Barrow, from 1664, used the same symbolism. David Gregory used pi in an expression to represent the ratio of the circumference to the radius in 1697. The first to use pi definitely to stand for the ratio of circumference to diameter was an English writer William Jones. He used it to symbolize the word "periphery." Euler adopted the symbol in 1737, and since that time it has been in general use. (Smith p312)The number pi is the ratio of the circumference of a circle to its diameter. It is also the ratio of the area of a circle to the area of the square on its radius. The adoption of the symbol for pi for this ratio is essentially due to the usage given it by Leonhard Euler from 1736 on. In the 1730's, Euler first used p and c for the circumference -to-diameter ratio, then adopted this symbol for pi. However, he is not the originator of the symbol.An actual ratio symbol as illustrated here on the right had been used by William Oughtred in 1647 and by Isaac Barrow in 1664 to indicate the ratio of the diameter of a circle to it's circumference or periphery.David Gregory, nephew of Scottish mathematician James Gregory (1638-1675), used this symbol on the left for the ratio of circumference to radius in 1697. In 1706 the English writer William Jones, in a work that gave the 100-place approximation of John Machin, first used the single symbol for pi. This computation of pi to a large number of places by means of various series representations was aided by the use of such relations as pi/4 = 4 arctan (1/5) - arctan (1/239), as given by Machin in 1706. (NCTM p148,152)EVES, HOWARD "An Introduction to the History of Mathematics," fourth edition, Holt Rinehart Winston 1976.CAJORI, FLORIAN "A History of Mathematics", The Macmillan Company 1926SMITH, D.E. "History of Mathematics" volume II. Dover Publications 1958THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, "Historical Topics for the Mathematics Classroom". National Council of Teachers of Mathematics (USA) 1969
The symbol for percent
Percent has been used since the end of the fifteenth century in business problems such as computing interest, profit and loss, and taxes. However, the idea had its origin much earlier. When the Roman emperor Augustus levied a tax on all goods sold at auction, centesima rerum venalium, the rate was 1/100. Other Roman taxes were 1/20 on every freed slave and 1/25 on every slave sold. Without recognising percentages as such, they used fractions easily reduced to hundredths. In the Middle Ages, as large denominations of money came to be used, 100 became a common base for computation. Italian manuscripts of the fifteenth century contained such expressions as "20 p 100" and "x p cento" to indicate 20 percent and 10 percent. When commercial arithmetics appeared near the end of that century, use of percent was well estasblished. For example, Giorgio Chiarino (1481) used "xx. per .c." for 20 percent and "viii in x perceto" for 8 to 10 percent. During the sixteenth and seventeenth century, percent was used freely for computing profit and loss and interest. (NCTM p146,147}In its primitive form the per cent sign is found in the 15th century manuscripts on commercial arithmetic, where it appears as this symbol after the word "per" or after the letter "p" as a contraction for "per cento." The use of the per cent symbol can be seen in this extract from an anonymous Italian manuscript of 1684 (Smith p250) The percent sign, %, has probably evolved from a symbol introduced in an anonymous Italian manuscript of 1425. Instead of "per 100," "P cento," which were common at that time, this author used the symbol shown. By about 1650, part of this symbol had been changed to the form shown on the right. Finally, the "per" was dropped, leaving this symbol to stand alone, and this in turn became %. (NCTM p147)The solidus form (%) is modern. (Smith p250) This symbol stands for "per thousand". (Hogben p92) It is natural to expect that percentage will develop into per millage, and indeed this has not only begun, but it has historic sanction. Bonds are quoted in New York using this symbol on the right, and so in other commercial lines. At present, indeed, the symbol above (Hogben) is used in certain parts of the world, notably by German merchants, to mean "per mill," a curious analogue to % developed without regard to the historic meaning of the latter symbol.(Smith p250)THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, "Historical Topics for the Mathematics Classroom", National Council of Teachers of Mathematics (USA) 1969SMITH, D.E. "History of Mathematics" volume II, Dover Publications 1958HOGBEN, LANCELOT "The Wonderful World of Mathematics", Macdonald and Company 1968
The symbol for division
The Anglo-American symbol for division is of 17th century origin, and has long been used on the continent of Europe to indicate subtraction. Like most elementary combinations of lines and points, the symbol is old. It was used as early as the 10th century for the word est. When written after the letter "i", it symbolized "id est." When written after the word "it", it symbolized "interest." If written after the word "divisa", for "divisa est", this might possibly have suggested its use as a symbol of division. Towards the close of the 15th century the Lombard merchants used it to indicate a half, along with similar expressions such as this one on the right.There is also a possibility that it was used by some Italian algebrists to indicate division. In a manuscript entitled Arithmetica and Practtica by Giacomo Filippo Biodi dal Aucisco, copied in 1684, this symbol stands for division, suggesting that various forms of this kind were probably used.The Anglo-American symbol (above top) first appeared in print in the Teutsche Algebra by Johann Heinrich Rahn (1622-1676) which appeared in Zurich in 1659. This symbol was then made known in England by the translation of Rahn's work by Dr. John Pell in London in 1688. (Smith p406)Around the year 1200, both the Arabic writer al-Hassar, and Fibonacci (Leonardo of Pisa), symbolised division in fraction form with the use of a horizontal bar, but it is thought likely that Fibonacci adopted al-Hassar's introduction of this symbolisation.In his Arithmetica integra (1544) Michael Stifel employed the arrangement 8)24 to mean 24 divided by 8. (NCTM p139)Michael Stifel (1486?-1567) was regarded as the greatest German algebrist of the 16th century. (Cajori p140)SMITH, D.E. "History of Mathematics" volume II, Dover Publications 1958THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, "Historical Topics for the Mathematics Classroom", National Council of Teachers of Mathematics (USA) 1969CAJORI, FLORIAN "A History of Mathematics", The Macmillan Company 1926
The symbol for Inequality
Thomas Harriot (1560-1621) was an English mathematician who lived the longer part of his life in the sixteenth century but whose outstanding publication appeared in the seventeenth century. He is of special interest to Americans, because in 1585 he was sent by Sir Walter Raleigh to the new world to survey and map what was then Virginia but is now North Carolina. As a mathematician Harriot is usually considered the founder of the English school of algebraists. His great work in this field, the Artis Analyticae Praxis was published in London posthumously in 1631, and deals largely with the theory of equations. In it he makes use of these symbols above, ">" for "is greater than" (on the left), and "<" for "is less than" (on the right.)They were not immediately accepted, for many writers preferred these symbols, which another Englishman William Oughtred (1574-1660) had suggested in the same year in the popular Clavis Mathematicae, a work on arithmetic and algebra that did much toward spreading mathematical knowledge in that country.Isaac Barrow (1630-1677), in a book Lectiones Opticae & Geometricae (London 1674), used these symbols as follows: this meant "A major est quam B"and this meant "A minor est quam B."These symbols to the right are modern and are not international.The symbol on the left means "is not equal to."The middle symbol means "is not less than."The symbol on the right stands for "is not greater than."In the 1647 edition of Oughtred's Clavis mathematicae these somewhat analogous symbols appear for "non majus" (on the left) and "non minus" (on the right) respectively. On the Continent these symbols, or some of their variants, apparently invented in 1734 by the French geodesist Pierre Bouguer (1698-1758), are commonly used. Bouguer was one of the French geodesists sent to Peru to measure an arc of a meridian. (Eves p251, Smith p413)EVES, HOWARD "An introduction to the History of Mathematics," fourth edition, Holt Rinehart Winston 1976 SMITH, D.E. "History of Mathematics" volume II, Dover Publications 1958
The symbol for infinity
John Wallis (1616-1703) was one of the most original English mathematicians of his day. He was educated for the Church at Cambridge and entered Holy Orders, but his genius was employed chiefly in the study of mathematics. The Arithmetica infinitorum, published in 1655, is his greatest work. (Cajori p183) This symbol for infinity is first found in print in his 1655 publication Arithmetica Infinitorum. It may have been suggested by the fact that the Romans commonly used this symbol for a thousand, just as today the word “myriad” is used for any large number, although in the Greek it meant ten thousand. The symbol was used in expressions such as, in 1695, "jam numerus incrementorum est (infinity)." (Smith p413)The symbol for infinity, first chosen by John Wallis in 1655, stands for a concept which has given mathematicians problems since the time of the ancient Greeks. A case in point is that of Zeno of Elea (in southern Italy) who, in the 5th century BC, proposed four paradoxes which addressed whether magnitudes (lengths or numbers) are infinitely divisible or made up of a large number of small indivisible parts. (Brinkworth and Scott p80)Wallis thought of a triangle, base length B, as composed of an infinite number of “very thin” parallelograms whose areas (from vertex to base of the triangle) form an arithmetic progression with 0 for the first term and ( A /(infinity)). B for the last term - since the last parallelogram (along the base B of the triangle) has altitude (A/(infinity)) and base B.The area of the triangle is the sum of the arithmetic progressionO + . . . . + (A/(infinity)).B = (number of terms/2). (first + last term)=(infinity/2).(0+(A/(infinity)).B)=(infinity/2).(A/(infinity)).B=(A-B)/2(NCTM p413)CAJORI, FLORIAN "A History of Mathematics", The Macmillan Company 1926SMITH, D.E. "History of Mathematics" volume II, Dover Publications 1958BRINKWORTH & SCOTT "The Making of Mathematics", The Australian Association of Mathematics Inc. 1994THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, "Historical Topics for the Mathematics Classroom", National Council of Teachers of Mathematics (USA) 1969
The symbols for ratio and proportion
The symbol : to indicate ratio seems to have originated in England early in the 17th century. It appears in a text entitled Johnson’s Arithmetick ; In two Bookes (London.1633), but to indicate a fraction, three quarters being written 3:4. To indicate a ratio it appears in an astronomical work, the Harmonicon Coeleste (London, 1651), by Vincent Wing. In this work the forms A : B :: C : D and A.B :: C.D appear frequently as being equal in meaning. (Smith p406) William Oughtred (1547-1660) was another English mathematician who wrote as follows: A : B = C : D as A B :: C D.He laid extraordinary emphasis upon the use of mathematical symbols; altogether he used over 150 of them. Only 3 have come down to modern times, and one of these is this symbol for proportion. His notation for ratio and proportion was later widely used in England and on the Continent. (Cajori p157).In his Clavis Mathematicae (1631) Oughtred used the dot to indicate either division or ratio, but in his Canones Sinuum (1657) the colon : is used for ratio. He wrote 62496 : 34295 :: 1 : 0 / 54.9- (Smith p 407)As this notation gained ground it freed the dot . for use as the symbol for separation in decimal fractions. It is interesting to note the attitude of Leibniz (1646-1715) toward some of these symbols. On July 29, 1698, he wrote in a letter to John Bernoulli thus ".... in designating ratio I use not one point but two points, which I use at the same time, for division; thus for your dy.x :: dt.a I write dy:x = dt:a; for dy is to x as dt is to a, is indeed the same as, dy divided by x is equal to dt divided by a. From this equation follow then all the rules of proportion.” This conception of ratio and proportion was far in advance of that in contemporary arithmetics. (Cajori p158)It is possible that Leibniz, who used : as a general symbol for division, took it from these writers, for he wrote in 1684 “x : y quod idem est ac x divis. Per y seu x/y.” The hypothesis that the ratio symbol : came from the symbol for division by dropping the bar has no historical basis. Since it is more international than the division symbol, it is probable that the latter symbol will gradually disappear. Various other symbols have been used to indicate division, but they have no particular interest at the present time. (Smith p407)Ratio - the quotient of two numbers or quantities indicating their relative sizes. The ratio of a to b is written a : b or a/b. The first term is the antecedent and the second the consequent. (Daintith and Nelson p274)The symbol :: for the equality of ratios, now giving way to the common sign for equality, was introduced by Oughtred circa 1628, for he later wrote "proportio, sive ratio aequalis ::" and a Dr. Pell gave it still more standing when he issued Rahn's algebra in English in 1668. The symbol seems to have been arbirarily chosen.This symbol for continued proportion was used by English writers of the 17th and 18th centuries. For example it was used by Isaac Barrow (1630-1677) in his Lectiones Mathematicae (London, 1683), where he wrote "The character is made use of to signify continued Proportionals." It is still commonly seen in French textbooks. (Smith p413)SMITH, D.E. "History of Mathematics" volume II, Dover Publications 1958CAJORI, FLORIAN "A History of Mathematics", The Macmillan Company 1926DAINTITH, JOHN and NELSON,R.D. "Dictionary of Mathematics", Penguin 1989
The symbol for zero
Although the great practical invention of zero has often been attributed to the Hindus, partial or limited developments of the zero concept are clearly evident in a variety of other numeration systems that are at least as early as the Hindu system, if not earlier. The actual effect of any one of these earlier steps in the full development of the zero concept - or, indeed, whether there was any actual effect - is by no means clear, however.The Babylonian sexagesimal system used in the mathematical and astronomical texts was essentially a positional system, even though the zero concept was not fully developed. Many of the Babylonian tablets indicate only a space between groups of symbols if a particular power of sixty was not needed, so the exact powers of sixty that were involved must be determined partly by context. In the later Babylonian tablets (those of the last three centuries B.C.) a symbol was used to indicate a missing power, but this was used only inside a numerical grouping and not at the end. (NCTMp49)Not to be overlooked is the fact that in the sexagesimal notation of integers the "principle of position" was employed. Thus, in 1.4 (=64), the 1 is made to stand for 60, the unit of the second order, by virtue of its position with respect to the 4. The introduction of this principle at so early a date is the more remarkable, because in the decimal notation it was not regurlarly introduced until about the ninth century after Christ. The principle of position, in its general and systemic application, requires a symbol for zero. We ask, Did the Babylonians possess one? Had they already taken the gigantic step of representing by a symbol the absence of units? Babylonian records of many centuries later -of about 200 B.C.-give a symbol for zero which denoted the absence of a figure, but apparently it was not used in calculation. It consisted of two angular marks as illustrated above on the right, one above the other, roughly resembling two dots, hastily written. About 130 A.D. Ptolemy in Alexandria used in his Almagest the Babylonian sexagesimal fractions, and also the omicron o to represent blanks in the sexagesimal numbers. This o was not used as a regular zero. It appears therefore that the Babylonians had the principle of local value, and also a symbol for zero, to indicate the absence of a figure, but did not use this zero in computation.Their sexagesimal fractions were introduced into India and with these fractions probably passed the principle of local value and the restricted use of the zero. (Cajori p5)When the Greeks continued the development of astronomical tables, they explicitly chose the Babylonian sexagesimal system to express their fractions, rather than the unit-fraction system of the Egyptians. The repeated subdivision of a part into 60 smaller parts necessitated that sometimes “no parts” of a given unit were involved, so Ptolemy’s tables in the Almagest (c. A.D. 150) included both of these symbols for such a designation.Considerably later, in approximately 500, Greek texts used this symbol, the omicron, the first letter of the Greek word ouden (“nothing”). Earlier usage would have restricted the omicron to symbolizing 70, its value in the regular alphabetic arrangement.Perhaps the earliest systematic use of a symbol for zero in a place-value system is found in the mathematics of the Mayas of Central and South America. The Mayan zero symbol was used to indicate the absence of any units of the various orders of the modified base-twenty system. This system was probably used much more for recording calendar times than for computational purposes. (NCTM p49)The Maya counted essentially on a scale of 20, using for their basal numerals two elements, a dot representing one and a horizontal dash representing five. The most important feature of their system was their zero, this character as illustrated, which also had numerous variants. (Smith p44)It is possible that the earliest Hindu symbol for zero was the heavy dot that appears in the Bakhshali manuscript, whose contents may date back to the third or fourth century A.D., although some historians place it as late as the twelfth. Any association of the more common small circle of the Hindus with the symbol used by the Greeks would be only a matter of conjecture. (NCTM p50)There is no probability that the origin will ever be known, and there is no particular reason why it should be. We simply know that the world felt the need of a better number system, and that the zero appeared in India as early as the 9th century, and probably some time before that, and was very likely a Hindu invention. In the various forms of numerals used in India, and in later European and Oriental forms, the zero is represented by a small circle or by a dot. Variations include these, as illustrated. (Smith p70)Since the earliest form of the Hindu symbol was commonly used in inscriptions and manuscripts in order to mark a blank, it was called sunya, meaning “void” or “empty.” This word passed over into the Arabic as sifr, meaning “vacant.” This was transliterated in about 1200 into Latin with the sound but not the sense being kept, resulting in zephirum or zephyrum. Various progressive changes of these forms, including zeuero, zepiro, zero, cifra, and cifre, led to the development of our words “zero” and “cipher.” The double meaning of the word “cipher” today - referring either to the zero symbol or to any of the digits - was not in the original Hindu. In early English and American schools the term “ciphering” referred to doing sums or other computations in arithmetic. (NCTM p50)The traditional Chinese numeration system is a base-ten system employing nine numerals and additional symbols for the place-value components of powers of ten. Before the eighth century A.D. the place where a zero would be required was always left absent. A circular symbol for zero is first found in a document dating from 1247, but it may have been in use a hundred years earlier. (NCTM p43)Interestingly enough, the forms of the modern Arabic numerals are not the same as the Hindu-Arabic forms of the western world. For example, their numerical representation for five is 0 and their zero is representated by a dot. (NCTM p49)This can be illustrated as shown; (Smith p70)The various forms of the numerals used in India after the zero appeared may be judged from this table. (Smith p70)This table illustrates some later European and Oriental forms. (Smith p71)The name for zero is not settled even yet. Older names and variations include naught, tziphra, sipos, tsiphron, rota, circulus, galgal, theca, null, and figura nihili.(Smith p71)THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, "Historical Topics for the Mathematics Classroom", National Council of Teachers of Mathematics (USA) 1969CAJORI, FLORIAN "A History of Mathematics", The Macmillan Company 1926SMITH, D.E. "History of Mathematics" volume II, Dover Publications 1958
The radical symbol
The ancient writers commonly wrote the word for root or side, as they wrote other words of similar kind when mathematics was still in the rhetorical stage. The symbol most commonly used by late medieval Latin writers to indicate a root was R , a contraction of radix, and this, with numerous variations, was continued in the printed books for more than a century. Thus it appears as such in the works of Boncompagni (1464), Chuquet (1484), Pacioli (1494), de la Roche (1520), Cardan (1539), Tartaglia (1556), Ghaligai (1521), and Bombelli (1572.) The symbol was also used for other purposes, including response, res, ratio, rex and the familiar recipe in a physician's prescription.Meanwhile, the Arab writers had used various symbols for expressing a root, including this sign on the right, but none of them seem to have influenced European writers.This symbol first appeared in print in Rudolff's Coss in 1525, but without our modern indices. It is frequently said that Rudolff used this sign because it resembled a small "r", for radix (root), but there is no direct evidence that this is true. The symbol may quite have been an arbitrary invention. It is a fact , however, that in and after the 14th century we find in manuscripts such forms as the following for the letter "r."It was a long time after these writers that a simple method was developed for indicating any root, and then only as a result of many experiments. French, English, and Italian writers of the 16th century were slow in accepting the German symbol, and indeed the German writers themselves were not wholly favourable to it. The letter l (for latus, side; that is, the side of a square) was often used. In the 17th century our common square-root sign was generally adopted, of course with many variants. The different variants of the root sign are too numerous to mention in detail in this work, particularly as they have little significance. By the close of the 17th century the symbolism was, therefore, becoming fairly well standardised, although there still remained some work to be done. The 18th century saw this accomplished, and it also saw the negative and fractional exponent come more generally into use.Some variations on the radical sign are as follows. The illustrations are the work of many different writers, including Stifel (1553), Gosselin (1557), Ramus-Schoner (1592), Rahn (1659), Stevin (1585), Vlacq, Biondini (1689) and Newton (1707).- for the square root - for the cube root -for the fourth root -for the fifth root -for the sixth root -for the eighth root (Smith p407 - p410)SMITH, D.E. "History of Mathematics" volume II, Dover Publications 1958
The symbols for plus and minus
The symbols of elementary arithmetic are almost wholly algebraic, most of them being transferred to the numerical field only in the 19th century, partly to aid the printer in setting up a page and partly because of the educational fashion then dominant of demanding a written analysis for every problem. When we study the genesis and development of the algebraic symbols of operation, therefore, we include the study of the symbols in arithmetic. Some idea of the status of the latter in this respect may be obtained by looking at almost any of the textbooks of the 17th and 18th centuries. Hodder in 1672 wrote "note that a + (plus) sign doth signifie Addition, and two lines thus = Equality, or Equation, but a X thus, Multiplication," no other symbols being used. His was the first English arithmetic to be reprinted in the American colonies in Boston in 1710. Even Recorde (c1510-1558), who invented the modern sign of equality, did not use it in his arithmetic, the Ground of Artes (c1542), but only in his algebra, the Whetstone of witte (1557). (Smith p395)There is some symbolism in Egyptian algebra. In the Rhind papyrus we find symbols for plus and minus. The first of these symbols represents a pair of legs walking from right to left, the normal direction for Egyptian writing, and the other a pair of legs walking from left to right, opposite to the direction for Egyptian writing. [Eves 1, p42]The earliest symbols of operation that have come down to us are Egyptian. In the Ahmes Papyrus (c1550 B.C.) addition and subtraction are indicated by these symbols on the left and right above respectively. The Hindus at one time used a cross placed beside a number to indicate a negative quantity, as in the Bakhshali manuscript of possibly the 10th century. With this exception it was not until the 12th century that they made use of the symbols of operation. In the manuscripts of Bhaskara (c1150) a small circle or dot is placed above a subtrahend as illustrated for -6, or the subtrahend is enclosed in a circle to indicate 6 less than zero.The early European symbols for plus are listed opposite. The word plus, used in connection with addition and with the Rule of False Position is not known before the latter part of the 15th century.The use of the word minus as indicating an operation occurred much earlier, as in the works of Fibonacci (c1175-1250) in1202. The bar above the letter simply indicated an omission. In the 15th century, this third symbol was also often used for minus, but most writers preferred the other variations.In the 16th century the Latin races generally followed the Italian school, using the letters p and m, each with the bar above it, or their equivalents, for plus and minus. However, the German school preferred these symbols, neither of which is found for this purpose before the 15th century. In a manuscript of 1456, written in Germany, the word "et" is used for addition and is generally written so that it closely resembles the modern symbol for addition. There seems little doubt that the sign is merely a ligature for "et", much in the same way that we have the ligature "&" for the word "and."The origin of the minus sign has been more of a subject of dispute. Some have thought that it is a survival of the bar above the three symbols for minus as listed above. It is more probably that it comes from the habit of early scribes of using it as a shorthand equivalent of "m." Thus Summa became Suma with the bar above the letter u, and 10 thousand became an X with ther bar above the letter. It is quite reasonable to think of the dash (-) as a symbol for "m" (minus), just as the cross (+) is a symbol for "et." Other forms of minus are here illustrated.There were other various written forms for plus and minus, as in piu (Italian), mas (Spanish), plus (French) and et (German) for plus and as in de or men (Italian), menos (Spanish), moins (French) for minus. Examples of such usage include:Pacioli (1494), Italian de or m for minusTartaglia (1556) and Catanes (1546), Italian, piu and menSanta-Cruz (1594), Spanish, mas and menosPeletier (1549), French, plus and moinsGosselin (1577), P and MTrenchant (1566), + and -The expression "plus or minus" is very old, having been in common use by the Romans to indicate simply "more or less". It is often found on Roman tombstones, where the age of the deceased is given as illustrated to indicate "94 years, more or less".These signs first appeared in print in an arithmetic, but they were not employed as symbols of operation. In the latter sense they appear in algebra long before they do in arithmetic.They appeared in Johann Widman's (c1460-?) arithmetic published in Leipzig in 1489, the author saying: "Was - ist / das ist minus...vnd das + das ist mer." He then speaks of "4 centner + 5 pfund," and also of "4 centner - 17 pfund," thus showing the excess or deficiency in the weight of boxes or bales. (Smith p395 to 399)Observe that Francis Vieta (1540-1603) employed the Maltese cross (+) as the shorthand for addition, and the (-) for subtraction. These two characters had not been in very general use before his time. The introduction of the + and - symbols seems to be due to the Germans, who, although they did not enrich algebra during the Renaissance with great inventions, as did the Italians, still cultivated it with great zeal. The arithmetic of John Widmann, brought out in 1489 in Leipzig, is the earliest printed book in which the + and - symbols have been found, and the facsimile shown is from the Augsburg edition of his work, dated 1526. The + sign is not restricted by him to ordinary addition; it has the more general meaning "et" or "and" as in the heading, "regula augmenti + decrementi." The - sign is used to indicate subtraction, but not regularly so. The word "plus" does not occur in Widmann's text; the word "minus" is used only two or three times. The symbols + and - are used regularly for addition and subtraction, in 1521, in the arithmetic of Grammateus, the work of Heinrich Schreiber, a teacher at the University of Vienna. His pupil Christoff Rudolff, the writer of the first text book on algebra in the German language (printed in1525) employs these symbols. So did Michael Stifel, who brought out an improved second edition of Rudolff's book on algebra Die Coss in 1553. Thus, by slow degrees, the adoption of the + and - symbols became universal. Several independant paleograhic studies of Latin manuscripts of the fourteenth and fifteenth centuries make it almost certain that the + sign comes from the Latin et, as it was cursively written in manuscripts just before the time of the invention of printing. The origin of the sign - is still uncertain. (Cajori p139)The first one to make use of these signs in writing an algebraic expression was the Dutch mathematician Vander Hoecke, who in 1514 gave this illustration (on the left) for radical three quarters minus radical three fifths, and for radical 3 add 5 he gave the sign as shown on the right.These symbols seem to have been employed for the first time in arithmetic, to indicate operations, by Georg Walckl in 1536. The illustration on the left indicates the addition of one third of 230, and the one on the right indicates the subtraction of one fifth of 460. From this time on the two symbols were commonly used by both German and Dutch writers, the particular signs themselves not being settled until well into the 18th century.England adopted the Teutonic forms, and Robert Recorde (c1510-1558) wrote (c1542) "thys fygure +, whiche betoketh to muche, as this lyn, - plaine without a cross lyne, betokeneth to lyttle". As symbols of operation most of the English writers of this period reserved the + and - signs for algebra. Thus Digges (1572) in his treatment of algebra: "Then shall you ioyne them with this signe + Plus", and Hylles (1600) says: "The badg or signe of addition is +," stating the sum of 3 and 4 as "3 more 4 are 7," and writing 10___3 for "10 lesse 3." (Smith p399-402)SMITH, D.E. "History of Mathematics" volume II, Dover Publications 1958EVES, HOWARD "An introduction to the History of Mathematics," fourth edition, Holt Rinehart Winston 1976 CAJORI, FLORIAN "A History of Mathematics", The Macmillan Company 1926
The symbol for multiplication
William Oughtred (1574-1660) contributed vastly to the propagation of mathematical knowledge in English by his treatises, the Clavis Mathematicae, 1631, published in Latin (English edition 1647), Circles of Proportion, 1632, and Trigonometrie, 1657. Among his most noted pupils are the mathematician John Wallis (1616-1703) and the astronomer Seth Ward. Oughtred laid extraordinary emphasis upon the use of mathematical symbols : altogether he used over 150 of them. Only three have come down to modern times, namely the cross symbol for multiplication, :: as that of proportion, and the symbol for "difference between." The cross symbol, on the left, occurs in the Claris, but the letter X, seen on the right, which closely resembles it, occurs as a sign of multiplication in the anonymous "Appendix to the Logarithmes" in Edward Wright's translation of John Napier's Descriptio, published in 1618. This appendix was most probably written by Oughtred.Leibniz (1646-1715) objected to the use of Oughtred's cross symbol because of possible confusion with the letter X. On 29 July 1698 he wrote in a letter to John Bernoulli : "I do not like (the cross) as a symbol for multiplication, as it is easily confounded with x; .... often I simply relate two quantities by an interposed dot and indicate multiplication by ZC.LM." Through the aid of Christian Wolf (1679-1754) the dot was generally adopted in the 18th century as a symbol for multiplication. Wolf was a professor at Halle, and was ambitious to figure as a successor of Leibniz. Presumably Leibniz had no knowledge that Harriot in his Artis analyticae praxis, 1631, used a dot for multiplication, as in aaa__3.bba=+2.ccc. Harriot's dot received no attention, not even from Wallis. (Cajori p157) The common symbol as illustrated was developed in England about 1600. It was not a new sign, having long been used in cross multiplication, in the check of nines, where Hylles (1600) speaks of it as the "byas crosse" in connection with the multiplication of terms in the division or addition of fractions, for the purpose of indicating the corresponding products in proportion, and in the "multiplica in croce" of algebra as well as in arithmetic.The symbol was not readily adopted by arithmeticians, being of no practical value to them. In the 18th century some use was made of it in numerical work, but it was not until the second half of the 19th century that it became popular in elementary arithmetic. On account of its resemblance to x it was not well adapted to use in algebra, and so the dot came to be employed, as in 2 . 3 = 6 (Europe) as well as in America. This device seems to have been suggested by the old Florentine multiplication tables; at any rate Adriaen Vlacq (c1600-1667), the Dutch computer (1628), used it in some of his work, thus:factores---- 7 . 17faci---------119although not as a real symbol of operation. In his text he uses a rhetorical form, thus; "3041 per 10002 factus erit 30416082." Christopher Clavius (1537-1612), a Jesuit of Rome, wrote in 1583 using the idea of a dot for multiplication, as in 3/5.4/7 for 3/5 X 4/7; and Thomas Harriot (1560-1621) in a posthumous work of 1631 actually used the symbol in a case like 2.aaa = 2a cubed. The first writer of prominence to employ the dot in a general way for algebraic multiplication seems to
Interview with Derrick Niederman, author of Number Freak
In this interview we sit down with author and mathematician Derrick Niederman to discuss his engaging, recently published book about the first two hundred natural numbers, ‘Number Freak: From 1 to 200, The Hidden Language of Numbers Revealed’.
1. Some of our readers are likely familiar with your work, but could you tell us more about yourself and your mathematical background?
I majored in mathematics as an undergraduate at Yale, from which I graduated in 1976. I think I even won a couple of math prizes, but I have to confess that I wasn’t the top mathematician in my class. That distinction would surely have gone to Jonathan Rogawski, who last I knew was a professor of mathematics at UCLA. (Notice that I just created the impression that I was the second-best mathematician in my class. I don’t know whether that’s true, but I’ll take it.)
Anyway, I went on to get a Ph.D. in mathematics at M.I.T. and have remained in the Boston area ever since. I went into the investment business in the early 1980s, based on the assumption that quantitative expertise would be a good match. But the truth is that I got progressively more qualitative as time went by, going from securities analyst to investment writer. I don’t know whether that transition made complete sense, but it ultimately gave me the opportunity to write some books – first about investments and then about numbers, including several volumes of puzzle books.
2. What inspired you to write Number Freak?
I was asked by a publisher to come up with a concept that would do for mathematics what a slightly different concept did for the natural sciences. The idea I came up with was more of a coffee-table book than the sized-down version I now have in my hands, but that effort was considered too expensive. I subsequently cast a wider net for the project, and was fortunate enough to attract publishers in the U.S., the U.K., and Australia.
3. The book is chock-full of interesting facts about the first 200 natural numbers. What did you learn in the process of writing this book that you didn’t know before?
Well, I guess the pat answer is that I learned how little I actually knew. Some of the work on planar tilings was new to me, even though it probably shouldn’t have been – for example, the Archimedean and Laves tilings I discuss in #11 are quite beautiful but I hadn’t been aware of their categorization and duality. And I wasn’t familiar with the work of mathematicians such as Erich Friedman of Stetson University, somebody who surely could have pulled off a book like this: I was only too happy, for example, to include “Friedman numbers” such as 127.
In self-defense, I wasn’t a complete neophyte. One big advantage I had in writing the book – apart from doing it in the Internet age, which gave me an abundance of material – was that I have a good memory for mathematical and pop culture trivia. For example, I enjoyed reaching back and remembering that the ultra-high security “D” block at Alcatraz prison had precisely 42 individual cells, something that meshed quite nicely with the picture of the “magic cube” I displayed elsewhere in the discussion of #42.
4. Having read this book I feel that it’s accessible to virtually anyone. Who do you feel is the ideal target audience for the book?
Boy is that a good question. My answer is that it’s for absolutely anyone, but if that’s too mealy-mouthed a reply, I guess I would say that I’d be especially pleased if parents bought Number Freak to (successfully!) introduce their kids to the world of numbers in a way that maybe, just maybe, is friendlier than what those kids were getting elsewhere.
5. Was there anything that you wish you could have included in the book but didn’t?
Another good question, and I’m afraid a painful one. The book was originally slated to go from 1 to 300 — as in a perfect game in bowling, among other things — but the editorial powers-that-be eventually whittled that down to 200. Too bad, as my discussion of the infamous 256th level of Pac-Man was worth the price of admission. (Say, that’s a topic I didn’t know about when I started the book!) I also lost some precious photos, charts and diagrams along the way. And you can imagine how I felt when a friend berated me for not mentioning “77 Sunset Strip,” when of course my original manuscript mentioned the show – and I have a photograph of Efrem Zimbalist, Jr. to prove it! (Those of my vintage – I’m 54 – will remember the show’s catchy theme song, but not many are aware that 77 was a particular good choice for the street address because it is the smallest integer whose English pronunciation requires five syllables.)Other than that, I deliberately went easy on the cult surrounding the number 23, for example, and left a bunch of numerology and religious interpretations for somebody else to ponder. That’s another book all by itself.
6. What’s the answer to life, the universe and everything?
Why it’s 42, of course. You know, I had already answered question #3 above before I saw this one!
7. What’s your favorite number and why?
When I started the book, 17 had the edge. First of all, “At 17” by Janis Ian is probably my favorite song of all time. It came out in 1975, which was my favorite music year of all time. (Perhaps I should have written it in 1975.) But 17 is famous in mathematics for Carl Friedrich Gauss’s famous straightedge-and-compass construction of a regular 17-gon, for the 17 “wallpaper” symmetries of the plane, and for the fact that if you connect 17 suitably spaced dots with a segment of red, blue, or green, you will automatically create a “monochromatic” triangle whose three vertices are among the original 17 dots. And nobody has yet created a solvable Sudoku puzzle with fewer than 17 original entries. How about that?
But by the time I finished Number Freak, my favorite number had become 36. What happened is that while doing research for the book I came across a conjecture from the 18th century called the 36 Officer Problem. I had never heard of it before (yet another example!), perhaps because the problem was resolved in the early 20th century and then ceased to be of interest. But there was a three-dimensional wrinkle to the problem that hadn’t been explored, and I used that wrinkle to design a puzzle with a gray base and 36 towers of various colors. I went to Toy Fair and showed the puzzle to ThinkFun, a great game and puzzle company out of Alexandria, Virginia. And guess what? They made me a deal for the puzzle and after a year tinkering with the basic model, they launched it as “36 Cube” in the fall of 2008—many months before Number Freak came out! I was thrilled that the lessons of the book came to life in such a tangible way, so I’d be lying if I didn’t admit that 36 holds a very special place in my heart.
Thank you very much, Derrick, for your insightful answers. And to our readers, if you haven’t already done so, check out his book.
Bookmark and share:
Related Articles:
Ten Must Read Books about Mathematics
I’m reading a fantastic mathematical novel
An accessible Calculus book and Benjamin Franklin’s secret passion
The most enlightening Calculus books
The nicest math book I own
If you enjoyed this post, then make sure you subscribe to our RSS Feed.
1. Some of our readers are likely familiar with your work, but could you tell us more about yourself and your mathematical background?
I majored in mathematics as an undergraduate at Yale, from which I graduated in 1976. I think I even won a couple of math prizes, but I have to confess that I wasn’t the top mathematician in my class. That distinction would surely have gone to Jonathan Rogawski, who last I knew was a professor of mathematics at UCLA. (Notice that I just created the impression that I was the second-best mathematician in my class. I don’t know whether that’s true, but I’ll take it.)
Anyway, I went on to get a Ph.D. in mathematics at M.I.T. and have remained in the Boston area ever since. I went into the investment business in the early 1980s, based on the assumption that quantitative expertise would be a good match. But the truth is that I got progressively more qualitative as time went by, going from securities analyst to investment writer. I don’t know whether that transition made complete sense, but it ultimately gave me the opportunity to write some books – first about investments and then about numbers, including several volumes of puzzle books.
2. What inspired you to write Number Freak?
I was asked by a publisher to come up with a concept that would do for mathematics what a slightly different concept did for the natural sciences. The idea I came up with was more of a coffee-table book than the sized-down version I now have in my hands, but that effort was considered too expensive. I subsequently cast a wider net for the project, and was fortunate enough to attract publishers in the U.S., the U.K., and Australia.
3. The book is chock-full of interesting facts about the first 200 natural numbers. What did you learn in the process of writing this book that you didn’t know before?
Well, I guess the pat answer is that I learned how little I actually knew. Some of the work on planar tilings was new to me, even though it probably shouldn’t have been – for example, the Archimedean and Laves tilings I discuss in #11 are quite beautiful but I hadn’t been aware of their categorization and duality. And I wasn’t familiar with the work of mathematicians such as Erich Friedman of Stetson University, somebody who surely could have pulled off a book like this: I was only too happy, for example, to include “Friedman numbers” such as 127.
In self-defense, I wasn’t a complete neophyte. One big advantage I had in writing the book – apart from doing it in the Internet age, which gave me an abundance of material – was that I have a good memory for mathematical and pop culture trivia. For example, I enjoyed reaching back and remembering that the ultra-high security “D” block at Alcatraz prison had precisely 42 individual cells, something that meshed quite nicely with the picture of the “magic cube” I displayed elsewhere in the discussion of #42.
4. Having read this book I feel that it’s accessible to virtually anyone. Who do you feel is the ideal target audience for the book?
Boy is that a good question. My answer is that it’s for absolutely anyone, but if that’s too mealy-mouthed a reply, I guess I would say that I’d be especially pleased if parents bought Number Freak to (successfully!) introduce their kids to the world of numbers in a way that maybe, just maybe, is friendlier than what those kids were getting elsewhere.
5. Was there anything that you wish you could have included in the book but didn’t?
Another good question, and I’m afraid a painful one. The book was originally slated to go from 1 to 300 — as in a perfect game in bowling, among other things — but the editorial powers-that-be eventually whittled that down to 200. Too bad, as my discussion of the infamous 256th level of Pac-Man was worth the price of admission. (Say, that’s a topic I didn’t know about when I started the book!) I also lost some precious photos, charts and diagrams along the way. And you can imagine how I felt when a friend berated me for not mentioning “77 Sunset Strip,” when of course my original manuscript mentioned the show – and I have a photograph of Efrem Zimbalist, Jr. to prove it! (Those of my vintage – I’m 54 – will remember the show’s catchy theme song, but not many are aware that 77 was a particular good choice for the street address because it is the smallest integer whose English pronunciation requires five syllables.)Other than that, I deliberately went easy on the cult surrounding the number 23, for example, and left a bunch of numerology and religious interpretations for somebody else to ponder. That’s another book all by itself.
6. What’s the answer to life, the universe and everything?
Why it’s 42, of course. You know, I had already answered question #3 above before I saw this one!
7. What’s your favorite number and why?
When I started the book, 17 had the edge. First of all, “At 17” by Janis Ian is probably my favorite song of all time. It came out in 1975, which was my favorite music year of all time. (Perhaps I should have written it in 1975.) But 17 is famous in mathematics for Carl Friedrich Gauss’s famous straightedge-and-compass construction of a regular 17-gon, for the 17 “wallpaper” symmetries of the plane, and for the fact that if you connect 17 suitably spaced dots with a segment of red, blue, or green, you will automatically create a “monochromatic” triangle whose three vertices are among the original 17 dots. And nobody has yet created a solvable Sudoku puzzle with fewer than 17 original entries. How about that?
But by the time I finished Number Freak, my favorite number had become 36. What happened is that while doing research for the book I came across a conjecture from the 18th century called the 36 Officer Problem. I had never heard of it before (yet another example!), perhaps because the problem was resolved in the early 20th century and then ceased to be of interest. But there was a three-dimensional wrinkle to the problem that hadn’t been explored, and I used that wrinkle to design a puzzle with a gray base and 36 towers of various colors. I went to Toy Fair and showed the puzzle to ThinkFun, a great game and puzzle company out of Alexandria, Virginia. And guess what? They made me a deal for the puzzle and after a year tinkering with the basic model, they launched it as “36 Cube” in the fall of 2008—many months before Number Freak came out! I was thrilled that the lessons of the book came to life in such a tangible way, so I’d be lying if I didn’t admit that 36 holds a very special place in my heart.
Thank you very much, Derrick, for your insightful answers. And to our readers, if you haven’t already done so, check out his book.
Bookmark and share:
Related Articles:
Ten Must Read Books about Mathematics
I’m reading a fantastic mathematical novel
An accessible Calculus book and Benjamin Franklin’s secret passion
The most enlightening Calculus books
The nicest math book I own
If you enjoyed this post, then make sure you subscribe to our RSS Feed.
Friday, October 9, 2009
Discuss a problem a teacher could experience if he or she used instructional software with students without evaluation.
Software Evaluation
Evaluation of teaching/learning software consists of two types: formative and summative. Formative methods of evaluation are used when a projectÃs outline has been decided and work has begun on the design and development of the various parts. It can be deliberate and consist of a series of methods to determine whether the project can work as planned or it can be so ad hoc that it consists mainly of obtaining the opinions of passers-by as to the visual effectiveness of a series of screens. As the first researchers of software development methods under the former CAUT (Committee for the Advancement of University Teaching)-funded grants, Hayden and Speedy (1995) found that, although formative evaluation was a project requirement, many grantees either paid lip service or simply ran out of time before they could implement it. These authors suggest that the grantees did not understand the main purpose of such evaluation and so considered it an ëadd-onÃ. Alexander and Hedberg, in noting the level of academic effort which goes into developing educational software, state
Given the high expectations of technology to provide more cost-effective learning and to improve the quality of the learning, together with the need to gain recognition for academics undertaking such development projects, the time has come for a re-examination of the role of evaluation in the development and implementation cycles. (Alexander & Hedberg, 1994:234)
Yet Moses and Johnson, in their review of CAUTÃs National Teaching Development Grants, found that ë...some projects were funded despite [added italics] the proponentsà lack of expertise in evaluation and of knowledge about learning theories and practicesà (1995:36). Thus, although formative evaluations should occur during the process of developing a teaching program, whether it consists entirely of software or also has other components, it appears that many projects reach completion without the benefit of data which have the potential to inform and improve them.
Northrup, writing about formative evaluation for multimedia, states that it is ë...an ongoing process conducted along every step of program developmentà (1995:24) and finds that, if a first draft or version of a product is created before a formative evaluation is conducted, then major modifications will not occur even when they appear to be required. Too much money, effort and time will have gone into the product to allow a major rework to take place. To help prevent this unfortunate situation, she offers guidelines for the development team which include the need for all the stakeholders to be involved and support for, and enforcement of, formative evaluation at all stages. She also discusses how data can be collected and used. The only aspect Northrup does not address is the recognition of students or other potential users as stakeholders. However, Biraimah (1993), Barker & King (1993), Reiser & Kegelmann (1994) and Henderson (1996) all agree that learners are stakeholders and that they should help carry out the formative evaluation in a number of ways, even if they function mainly to check for biases of gender and race or to see if the program will actually load. Indeed, Reiser and Kegelmann (1994:64) note that student evaluation of software is necessarily subjective and should be supplemented by that of subject matter experts, media specialists and administrators.
In comparison, summative evaluations can be much wider in scope. They occur when the finished product is examined and can benefit from hindsight. Thorpe, an open learning specialist, defines evaluation as ë...the collection, analysis and interpretation of information about any aspect of a programme of education and training, as part of a recognised process of judging its effectiveness, its efficiency and any other outcomes it may have (1988:5). She notes that a number of characteristics go with this definition, such as inclusiveness, the search for both intended and unintended effects and the capability of the activity to be made public. She emphasises that evaluation is not synonymous with assessment.
Teaching approach
Although both types of evaluation are important, and should be conducted at appropriate times throughout the life cycle of any educational program, they are less effective for the stated purpose when they occur in isolation from the evaluatorÃs teaching philosophy and preferred methods. Although some software examples may be considered to be more ësophisticatedà than others because, for example, the screens are more visually interesting or require more student input, those programs may actually be examples of the reproductive/transmitting method of teaching.
Bain and McNaught examined the ways in which academic faculty view student learning. They suggest that academics hold certain views on the ways in which students learn and therefore tend to adopt one of the following teaching approaches:
a reproducing/transmitting/expository conception which tends to encourage...reproductive learning
a pre-emptive orientation...sensitive to past student learning difficulties.focuses on explanations
a conversational or transformative conception.understanding is constructed by the student with the assistance of the teacher (Bain & McNaught, 1996:56).
All three of these approaches can be found in educational software. For example, the reproducing/transmitting/expository conception can be found in software which provides drill-and-practice or the short explanation, selection of readings and student input-to-exercises model used in many Web-based subjects or in some examples of electronic books or simulations. The pre-emptive orientation, in which the academic knows much about the learning difficulties past students have exhibited, can be found in interactive multimedia as well as in games, simulations and problem-solving courseware. The conversational approach may be found in multimedia exploration-of-a-microworld examples and in simulations and games where students interact with both software and people to construct knowledge and receive feedback on their thinking.
Barker states that
People design ëlearning productsà in order to meet some perceived learning or training need. We therefore define ëlearning designà as the overall effects of the cognitive activity that takes place within the...design team during the conception and formulation of a learning product... produced to meet some pre-defined pedagogic requirement. (1995:87)
Therefore, it is hardly surprising that some software is only treasured by its developers. When its advocates leave teaching, the product is no longer used.
Approaches to evaluation
A simple question for any educational software should be, ìCan this product actually teach what it is supposed to?î It is a simple question to ask, but often is difficult to answer because the product may have so many beguiling features. It requires the evaluator to recognise his/her own view of the ways in which students learn, to relate that view to the learning objectives of that portion of the course and to determine how and whether those objectives are carried out in the software.
Evaluation of teaching/learning software consists of two types: formative and summative. Formative methods of evaluation are used when a projectÃs outline has been decided and work has begun on the design and development of the various parts. It can be deliberate and consist of a series of methods to determine whether the project can work as planned or it can be so ad hoc that it consists mainly of obtaining the opinions of passers-by as to the visual effectiveness of a series of screens. As the first researchers of software development methods under the former CAUT (Committee for the Advancement of University Teaching)-funded grants, Hayden and Speedy (1995) found that, although formative evaluation was a project requirement, many grantees either paid lip service or simply ran out of time before they could implement it. These authors suggest that the grantees did not understand the main purpose of such evaluation and so considered it an ëadd-onÃ. Alexander and Hedberg, in noting the level of academic effort which goes into developing educational software, state
Given the high expectations of technology to provide more cost-effective learning and to improve the quality of the learning, together with the need to gain recognition for academics undertaking such development projects, the time has come for a re-examination of the role of evaluation in the development and implementation cycles. (Alexander & Hedberg, 1994:234)
Yet Moses and Johnson, in their review of CAUTÃs National Teaching Development Grants, found that ë...some projects were funded despite [added italics] the proponentsà lack of expertise in evaluation and of knowledge about learning theories and practicesà (1995:36). Thus, although formative evaluations should occur during the process of developing a teaching program, whether it consists entirely of software or also has other components, it appears that many projects reach completion without the benefit of data which have the potential to inform and improve them.
Northrup, writing about formative evaluation for multimedia, states that it is ë...an ongoing process conducted along every step of program developmentà (1995:24) and finds that, if a first draft or version of a product is created before a formative evaluation is conducted, then major modifications will not occur even when they appear to be required. Too much money, effort and time will have gone into the product to allow a major rework to take place. To help prevent this unfortunate situation, she offers guidelines for the development team which include the need for all the stakeholders to be involved and support for, and enforcement of, formative evaluation at all stages. She also discusses how data can be collected and used. The only aspect Northrup does not address is the recognition of students or other potential users as stakeholders. However, Biraimah (1993), Barker & King (1993), Reiser & Kegelmann (1994) and Henderson (1996) all agree that learners are stakeholders and that they should help carry out the formative evaluation in a number of ways, even if they function mainly to check for biases of gender and race or to see if the program will actually load. Indeed, Reiser and Kegelmann (1994:64) note that student evaluation of software is necessarily subjective and should be supplemented by that of subject matter experts, media specialists and administrators.
In comparison, summative evaluations can be much wider in scope. They occur when the finished product is examined and can benefit from hindsight. Thorpe, an open learning specialist, defines evaluation as ë...the collection, analysis and interpretation of information about any aspect of a programme of education and training, as part of a recognised process of judging its effectiveness, its efficiency and any other outcomes it may have (1988:5). She notes that a number of characteristics go with this definition, such as inclusiveness, the search for both intended and unintended effects and the capability of the activity to be made public. She emphasises that evaluation is not synonymous with assessment.
Teaching approach
Although both types of evaluation are important, and should be conducted at appropriate times throughout the life cycle of any educational program, they are less effective for the stated purpose when they occur in isolation from the evaluatorÃs teaching philosophy and preferred methods. Although some software examples may be considered to be more ësophisticatedà than others because, for example, the screens are more visually interesting or require more student input, those programs may actually be examples of the reproductive/transmitting method of teaching.
Bain and McNaught examined the ways in which academic faculty view student learning. They suggest that academics hold certain views on the ways in which students learn and therefore tend to adopt one of the following teaching approaches:
a reproducing/transmitting/expository conception which tends to encourage...reproductive learning
a pre-emptive orientation...sensitive to past student learning difficulties.focuses on explanations
a conversational or transformative conception.understanding is constructed by the student with the assistance of the teacher (Bain & McNaught, 1996:56).
All three of these approaches can be found in educational software. For example, the reproducing/transmitting/expository conception can be found in software which provides drill-and-practice or the short explanation, selection of readings and student input-to-exercises model used in many Web-based subjects or in some examples of electronic books or simulations. The pre-emptive orientation, in which the academic knows much about the learning difficulties past students have exhibited, can be found in interactive multimedia as well as in games, simulations and problem-solving courseware. The conversational approach may be found in multimedia exploration-of-a-microworld examples and in simulations and games where students interact with both software and people to construct knowledge and receive feedback on their thinking.
Barker states that
People design ëlearning productsà in order to meet some perceived learning or training need. We therefore define ëlearning designà as the overall effects of the cognitive activity that takes place within the...design team during the conception and formulation of a learning product... produced to meet some pre-defined pedagogic requirement. (1995:87)
Therefore, it is hardly surprising that some software is only treasured by its developers. When its advocates leave teaching, the product is no longer used.
Approaches to evaluation
A simple question for any educational software should be, ìCan this product actually teach what it is supposed to?î It is a simple question to ask, but often is difficult to answer because the product may have so many beguiling features. It requires the evaluator to recognise his/her own view of the ways in which students learn, to relate that view to the learning objectives of that portion of the course and to determine how and whether those objectives are carried out in the software.
Category & Discussion
Quality of end-user interface design
Investigation shows that the designers of the most highly-rated products follow well-established rules & guidelines. This aspect of design affects usersà perception of the product, what they can do with it and how completely it engages them.
Engagement
Appropriate use of audio & moving video segments can contribute greatly to usersà motivation to work with the medium.
Interactivity
Usersà involvement in participatory tasks helped make the product meaningful and provoke thought.
Tailorability
Products which allow users to configure them and change them to meet particular individual needs contribute well to the quality of the educational experience.
Excerpted from Barker & King (1993) p309.
The technical approach to evaluation used to be very important. For example, many papers were written in the 1980s about the importance of ëdebuggingà software and ensuring it would run as intended. Students were said to be frustrated with technical problems and to complain that these interfered with their learning. Technical evaluations of software are still of significance even though students of the 1990s are accustomed to computer crashes and often know how to address them. Technical difficulties often arise with authoring products produced in an educational organisation. With some echoes of the well-known MicroSift courseware proformas, Squires and McDougall (1994) provide a helpful series of lists for technical evaluations of software.
Barker and King (1993) have developed a method for evaluating interactive multimedia courseware. They provide four factors which their research suggests are of key importance to successful products. They state that several other factors should be considered as well, although their importance is seen as somewhat less than the four listed in Table 1. These secondary factors are: appropriateness of multimedia mix, mode and style of interaction, quality of interaction, user learning styles, monitoring and assessment techniques, built-in intelligence, adequacy of ancillary learning support tools and suitability for single user/group/distributed use (Barker & King, 1993:309).
Although Barker and KingÃs factors do make substantial contributions to the ëlook and feelà of successful products, some of them need more explanation. For example, the ëmode and style of interactionà impacts on how a user navigates through the product. Difficulty in choosing an appropriate navigation method may arise if the designer and the academic hold different views of the ways in which users will learn from the software.
Young (1996) points out that, if students are allowed to control the sequence and content of the instruction, they must acquire self-regulated learning strategies for the instructional experience to be successful. YoungÃs research concerned students in seventh grade -- a learning stage which might be construed as ënaiveà and at which students may hold beliefs which are poorly thought out. Lawless and Brown, in surveying research on navigation and learning outcomes, find that learners who are ë...limited in both domain knowledge and metacognitive skillsà (1997:126) may not benefit from a high degree of learner control and may get lost in the environment. They also note that such students may be beguiled by special features not central to the instruction and fail to acquire the information important tothe section. This finding is supported by Blissett and Atkins (1993), who find that the sophistication of the multimedia environment may prevent some students from taking time to reflect on what they have just learned. Yildiz and Atkins suggest that students do not cope well with multimedia if they lack ë...advance organizers or mental frameworks on which to hang the surrogate experience...they therefore had difficulty in making personal, meaningful sense of what they saw and did...à (1993:138). Laurillard (1993: 30-31) notes that university students may hold naive beliefs and that university lecturers may make erroneous assumptions about their studentsà grasp of prerequisite concepts. Unless software is specifically designed to expose naive beliefs and support the construction of more accurate knowledge, it is likely that at least some users can navigate through a product without recognising that they hold erroneous ideas.
The importance of context
Administrators of tertiary institutions in which there is an increased use of educational software to supply some of the teaching may hold a perception that the desired learning has taken place if the assigned work is accomplished. Ramsden thinks that the context of learning is very important and remarks on unintended consequences of planned educational interventions which can result in an increase in superficial learning (1992:62-63) rather than the opposite. He suggests that assessment methods may have a negative effect on student learning. If these effects are true, then an outcome of multimedia teaching could be superficial learning, just as with more traditional methods.
Some multimedia proponents suggest that experiential, visually accurate, interactive software will help users attempt to solve problems in the ways that experts would. Henderson, after a careful long-term examination of mature studentsà work with multimedia packages, states that ë...knowledge acquisition is essentially and inescapably a socio-economic-historical-political-cultural processà (1996:90) and that studentsà mental processes depend on context specificity. Thus, students from cultures different from that in which a software product is developed are likely to experience difficulties in using that product. Baumgartner and Payr state
Learning with software...is a social process in at least two ways: first, it takes place in a certain social situation (in the classroom, at work, at home) and is motivated by it. Secondly any relevant learning process has as its goal the ability to cope with the social situation (professional or everyday tasks, etc.) The evaluation of interactive media has to satisfy three conditions: 1. It has to take into account the social situation in which the media are used, and must not be limited to the media themselves; 2. It has to take into account the goal of dealing with complex social situations and must not limit itself to the isolated individual learner; and 3. It must take into account the specific forms of interaction between the learner and society. These interactions range from the passive reception of static knowledge to the active design of complex situations that characterizes the ëexpertÃ. (1996:32)
RamsdenÃs concern with context in traditional classrooms is thus seen to be of relevance to software developers or evaluators who want educational products which will fulfil a variety of teaching/learning needs.
Conversation
Laurillard (1993) notes that ëconversationà about oneÃs perception of an instructional sequence is an important part of learning and gives examples of ways in which conversation can be carried out by instructors and students face-to-face (102-104) or via intelligent tutoring systems. Blissett and Atkins (1993) advocate a strong teacher role in pursuing conversation about multimedia experiences and in promoting student reflection on their learning, in part due to their finding that students may not have acquired knowledge at a deep level from that experience. Collis (1996), in agreement with these authors about the importance of conversation about multimedia learning, advocates the provision of computerised communication opportunities among the lecturer and students and among the students themselves. She believes that what she terms ëtelelearningÃ, even for lecturers who use reproductive/transmitting teaching methods, forces the introduction of more communication into the ëinstructional balanceà (Collis, 1996:299). Her suggestions of supplying IRC-type group discussion facilities and email communication with the instructor as part of each computer-mediated instructional event would offer students the opportunity to engage in conversation about what they are learning even while it happens. It is much more likely that learners would then stop and think about their learning if an opportunity to share it with others were offered than if they are simply carried along by a multimedia experience and not ëanchoredà to the active, cognitive world.
Summary
A number of approaches to formative and summative evaluation have been touched upon above, supported by a set of references which should help beginning evaluators check out this time- and resource-intensive area further. It is no wonder that software developers may wish to avoid formative evaluation at every step of their project, especially if they have commenced it in a wave of enthusiasm, as Hayden and Speedy have noted. Designers and developers working on large-scale projects in Europe such as the DELTA (Barker & King, 1993) have had to turn away from the fun of carrying out innovative ideas and instead establish criteria whereby the work can stand up to evaluation which, as Thorpe states, is capable of being a very public thing. The work of software evaluation is very necessary but it is also expensive if done properly. The time and money required for this aspect -- whether formative evaluation or an evaluation to see if a produced program is effective -- should be a budget item whenever software development or use is considered.
Discuss a problem a teacher could experience if he or she used instructional software with students without evaluation.
Using bibliotherapy to teach problem solving.
Students with high-incidence disabilities (e.g., specific learning disabilities, behavioral disorders, mild mental retardation) can benefit from using bibliotherapy by learning how to become proactive problem solvers. Often students with high-incidence disabilities are characterized as inefficient in recognizing and solving problems. By learning a problem-solving strategy and applying it to children's literature titles, students with disabilities can learn to become independent and effective problem solvers.
After lunch Mr. Jones was reading aloud to his third-grade resource room students from The Meanest Thing to Say, by Bill Cosby (1997), when suddenly Kenyan raised his hand. He enthusiastically exclaimed, "I am going to try that the next time my brother calls me a name. I think saying `so' will get him to stop." Kenyan, a student with a specific learning disability, identified with the literary character and discovered a new solution to his challenging problem. The identification and insight Kenyan attained from the literary character is often termed bibliotherapy.
What Is Bibliotherapy?
Have you ever read a book for self-help or to find answers to your difficulties, such as how others dealt with a loss, learned to become self-assured, or overcame a hardship? If you responded "yes," then you have used bibliotherapy. Bibliotherapy is simply defined as "the use of books to help people solve problems" (Aiex, 1993, p. 1). Most people have read books to determine how others have approached a delicate issue. Teachers can use children's literature to help students solve problems and generate alternative responses to their issues.
Using books to solve problems is not a new idea but one that has received increased attention recently. Aiex (1993) identified nine potential reasons a teacher may choose to use bibliotherapy with students:
to show an individual that he or she is not the first or only person to encounter such a problem,
to show an individual that there is more than one solution to a problem,
to help a person discuss a problem more freely,
to help an individual plan a constructive course of action to solve a problem,
to develop an individual's self-concept,
to relieve emotional or mental pressure,
to foster an individual's honest self-appraisal,
to provide a way for a person to find interests outside of self, and
to increase an individual's understanding of human behavior or motivations.
This list highlights some of the many potential benefits of using the bibliotherapy approach to problem solving with students. Sridhar and Vaughn (2000) reported that additional benefits from bibliotherapy include improving students' self-concept and behavior.
Bibliotherapy is helpful for students with high-incidence disabilities who are experiencing difficulties or who may be likely to encounter problems similar to those discussed in the literature (McCarthy & Chalmers, 1997). In addition, all children can benefit from being taught a literature bibliotherapy lesson because students are likely to encounter similar issues during their school years. For example, a student may not be confronted by a bully or teased today but may experience similar problems later.
The situations most teachers are exploring with students when using bibliotherapy are types of everyday life problems such as anger, teasing, bullying, and issues of self-concept. These types of problem-solving issues are best accomplished through small-group or whole class readings and discussions of the topic. Doll and Doll (1997) described this approach as "developmental bibliotherapy" because it focuses on helping children cope with developmental needs rather than relying on a clinical or individualized approach to bibliotherapy. Through this developmental process, students will likely experience identification with the main character in the story, experience a catharsis and release of emotion, and develop insight to solve their problems. Developmental bibliotherapy includes the steps of selecting materials to use with students, presenting the materials, and building students' comprehension of the issue.
How Do I Teach Using Bibliotherapy?
The sample lesson plan presented in Appendix A is based on a teaching framework for bibliotherapy and problem solving and contains the four elements of
1. prereading,
2. guided reading,
3. postreading discussion, and
4. a problem-solving/reinforcement activity.
Prereading
The element of prereading contains two steps; the first is selection of materials. Careful selection of material is important so that students can identify and relate to the real or fictional literary character. The school or public library media specialist is an excellent resource to consult when selecting materials because he or she will have a comprehensive knowledge of children's literature rifles.
Friday, September 11, 2009
Porposal Using Technology...!
Amateur radio is a technology and activity that offers great potential when integrated into academic or vocational curricula. Programs with electrical, electronics, and electromechanical content can benefit from the use of amateur radio, and can also enhance language and communications skills. The biggest value of amateur radio may lie in its ability to enhance a student's motivation and self-esteem. In addition to its specific vocational and technical applications, amateur radio can assist in teaching basic skills and in reducing the isolation of students and teachers as it promotes interdisciplinary education and cultural awareness. Amateur radio is distinctly different from citizens band, as it is a noncommercial service. Ham operators do not need an electronics background, although technical knowledge and skills are helpful. Several examples of the educational use of amateur radio illustrate its potential for academic and vocational education. (Contains 23 references.) (SLD)
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The use of touch-sensitive displays and digital writing tablets to augment computer resources in media equipped classrooms will lead to dramatic improvements in classroom lectures. Multi-media equipment with digital writing input will allow teachers to interactively present lessons while facing the class and will enhance and accelerate old style chalk and talk lectures. In additions to classroom presentations, this technology will automate production of digitized class notes for web access by students
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SAMPLE PROPOSAL FOR CLASSROOM TECHNOLOGY
This proposal is adapted from one written by Allison Lide, a high school
physics teacher and modeler from Ohio who now teaches in Kathmandu Nepal.
Allison submitted it to a local educational foundation and to her parents'
organization (PTA). She was awarded 4 complete computer workstations,
including the computers! Her school later added more workstations. Her
proposals included giving an oral presentation to the PTA.
Description of Proposal
This proposal is requesting funds to partially equip a science lab with lab
interfacing equipment in order to implement an innovative, highly
successful physics instruction methodology called the Modeling Method.
Traditional physics instruction generally consists of lectures, and
memorization of reams of formulas that are often meaningless to students;
students rarely achieve more than a minimal understanding or appreciation
of physics concepts or the science of physics.
A much more effective way for students to gain a deeper
understanding of physics concepts is to target the students' misconceptions
about the rules governing physical phenomena. By giving students the
technological tools for investigating and exploring, students can develop a
model of a phenomena. In the Modeling Method, very simply, students design
an experiment, and then use the computer interface to gather and analyze
the data from the experiment. The computer interface allows the taking of
data such as distance, force, velocity, time and much more, all in
real-time, through the use of sensors/probes that take the measurements.
This kind of instant feedback results in students developing a much deeper,
more realistic understanding of the physics involved.
From this data-taking and analysis, students construct an accurate
mathematical model of the phenomena. In this method of instruction,
students are dynamically engaged in their own learning, resulting in a
thorough understanding of the concepts of physics. Student involvement is
the guiding principle behind the Modeling Method, a teaching method that
has been developed and researched at Arizona State University (ASU) for the
last fifteen years. In 2000 and 2001 Modeling Instruction was designated an
exemplary K-12 science program and a promising K-12 educational technology
program by the U.S. Department of Education.
In summer 2001 I learned the Modeling Method by participating in an
intensive __-week Modeling workshop in ------ for ---- University graduate
credit. [Ex. The university, my school, and -----] funded the Modeling
Workshop. However, participants must acquire the computer hardware and
software necessary for successful implementation. The hardware
requirements entail one computer workstation for every three students, to
ensure accessibility for all students. A workstation is comprised of one
computer and a lab interfacing system that includes at least the following
probes: 2 photogates, a motion detector, and a force probe.
I am implementing the program and am pleased at my students'
enhanced learning. The program is well-supported, since follow-up and
listserv teacher networking are integral parts of the program, both in the
summer and throughout the school year.
This proposal is requesting funds to purchase [ex. 7 motion
detectors] to implement Modeling Instruction in mechanics.
Benefits
This instructional approach (using technology to develop an
understanding of concepts) has been proven to be highly successful, and
results in deeper student comprehension and greater enthusiasm. Much of
this success is due to the high level of student investment in the learning
process, and the student-centered approach in instruction, as contrasted
with the lecture method, which is teacher-centered and results in
alienating many students from the sciences.
Infusing technology into a science curriculum is imperative for
helping students become technology-literate beyond word processing. This
method of technology-driven instruction ensures that students will be
comfortable with other innovative uses of technology, and will be
well-prepared for collegiate computer-based science labs. In fact, using
this technology in high school will give many students an advantage in
college, as they will already be accustomed to taking and analyzing data
with computers.
Evaluation of the Project
Evaluation of the Modeling Method is built into its implementation,
since --- University encourages extensive evaluation of the effectiveness
of the Modeling Method by means of pre- and post-instruction assessment of
students' conceptual understanding of science.
///////////////////////////////////////////////
The use of touch-sensitive displays and digital writing tablets to augment computer resources in media equipped classrooms will lead to dramatic improvements in classroom lectures. Multi-media equipment with digital writing input will allow teachers to interactively present lessons while facing the class and will enhance and accelerate old style chalk and talk lectures. In additions to classroom presentations, this technology will automate production of digitized class notes for web access by students
//////////////////////////////////////////////
SAMPLE PROPOSAL FOR CLASSROOM TECHNOLOGY
This proposal is adapted from one written by Allison Lide, a high school
physics teacher and modeler from Ohio who now teaches in Kathmandu Nepal.
Allison submitted it to a local educational foundation and to her parents'
organization (PTA). She was awarded 4 complete computer workstations,
including the computers! Her school later added more workstations. Her
proposals included giving an oral presentation to the PTA.
Description of Proposal
This proposal is requesting funds to partially equip a science lab with lab
interfacing equipment in order to implement an innovative, highly
successful physics instruction methodology called the Modeling Method.
Traditional physics instruction generally consists of lectures, and
memorization of reams of formulas that are often meaningless to students;
students rarely achieve more than a minimal understanding or appreciation
of physics concepts or the science of physics.
A much more effective way for students to gain a deeper
understanding of physics concepts is to target the students' misconceptions
about the rules governing physical phenomena. By giving students the
technological tools for investigating and exploring, students can develop a
model of a phenomena. In the Modeling Method, very simply, students design
an experiment, and then use the computer interface to gather and analyze
the data from the experiment. The computer interface allows the taking of
data such as distance, force, velocity, time and much more, all in
real-time, through the use of sensors/probes that take the measurements.
This kind of instant feedback results in students developing a much deeper,
more realistic understanding of the physics involved.
From this data-taking and analysis, students construct an accurate
mathematical model of the phenomena. In this method of instruction,
students are dynamically engaged in their own learning, resulting in a
thorough understanding of the concepts of physics. Student involvement is
the guiding principle behind the Modeling Method, a teaching method that
has been developed and researched at Arizona State University (ASU) for the
last fifteen years. In 2000 and 2001 Modeling Instruction was designated an
exemplary K-12 science program and a promising K-12 educational technology
program by the U.S. Department of Education.
In summer 2001 I learned the Modeling Method by participating in an
intensive __-week Modeling workshop in ------ for ---- University graduate
credit. [Ex. The university, my school, and -----] funded the Modeling
Workshop. However, participants must acquire the computer hardware and
software necessary for successful implementation. The hardware
requirements entail one computer workstation for every three students, to
ensure accessibility for all students. A workstation is comprised of one
computer and a lab interfacing system that includes at least the following
probes: 2 photogates, a motion detector, and a force probe.
I am implementing the program and am pleased at my students'
enhanced learning. The program is well-supported, since follow-up and
listserv teacher networking are integral parts of the program, both in the
summer and throughout the school year.
This proposal is requesting funds to purchase [ex. 7 motion
detectors] to implement Modeling Instruction in mechanics.
Benefits
This instructional approach (using technology to develop an
understanding of concepts) has been proven to be highly successful, and
results in deeper student comprehension and greater enthusiasm. Much of
this success is due to the high level of student investment in the learning
process, and the student-centered approach in instruction, as contrasted
with the lecture method, which is teacher-centered and results in
alienating many students from the sciences.
Infusing technology into a science curriculum is imperative for
helping students become technology-literate beyond word processing. This
method of technology-driven instruction ensures that students will be
comfortable with other innovative uses of technology, and will be
well-prepared for collegiate computer-based science labs. In fact, using
this technology in high school will give many students an advantage in
college, as they will already be accustomed to taking and analyzing data
with computers.
Evaluation of the Project
Evaluation of the Modeling Method is built into its implementation,
since --- University encourages extensive evaluation of the effectiveness
of the Modeling Method by means of pre- and post-instruction assessment of
students' conceptual understanding of science.
Saturday, September 5, 2009
Mathematics Teaching Tools
Mathematics Teaching Tools
Using Smart Boards With Computers
© William De Salazar Jun 21, 2007
Using Smart Boards with Computers to enhance student participation as well as using the technology as an aid in presenting and saving lessons for students.
One of the more useful tools available to mathematics teachers available is the use of a “Smart Board". There are several manufacturers with different models but the important objective is making use of the board.
For example, as one teaches algebra 2, solving equations using the Algebraic Field Properties is very important for students to master the material. The teacher can present a topic, assign homework, and then review the homework the next day with the students. Using a Smart Board which is attached to a computer and an LCD display panel used to project a "virtual chalkboard" unto a screen. However, now there is no chalk but an electronic writing pen which allows one the opportunity to write on the screen just as if one were writing on a dry erase or chalkboard.
One of the advantages of the SmartBoard technology is that everything that is written is also captured as an image, and or audio file so that students that are absent as well as information the teacher wants to present again is saved as a file in the computer. And, whatever content was reviewed can be transferred to a website, a printed hard copy, or transferred anywhere as an electronic document.
The other advantage of use of the Smart Board in Mathematics is that today’s students like technology. So, they will enjoy using the "smart electronic pen" and write their work on the board to use the technology. So, in algebra 2, the students will be motivated to get out of their desks and show one their work using the SmartBoard. And, the teacher can have several students do the same or similar problems on different pages of the Smart Board. The students using the electronic "smart" pen can use different colors with different thickness and create their own individual style of writing to show the other students. In this manner, the Smart Board has now enabled students to change from being spectators but now become active participants in working out and showing the work solving those algebraic equations.Read more: http://technological-teaching-aids.suite101.com/article.cfm/mathematics_teaching_tools#ixzz0QIKsibDn
Using Smart Boards With Computers
© William De Salazar Jun 21, 2007
Using Smart Boards with Computers to enhance student participation as well as using the technology as an aid in presenting and saving lessons for students.
One of the more useful tools available to mathematics teachers available is the use of a “Smart Board". There are several manufacturers with different models but the important objective is making use of the board.
For example, as one teaches algebra 2, solving equations using the Algebraic Field Properties is very important for students to master the material. The teacher can present a topic, assign homework, and then review the homework the next day with the students. Using a Smart Board which is attached to a computer and an LCD display panel used to project a "virtual chalkboard" unto a screen. However, now there is no chalk but an electronic writing pen which allows one the opportunity to write on the screen just as if one were writing on a dry erase or chalkboard.
One of the advantages of the SmartBoard technology is that everything that is written is also captured as an image, and or audio file so that students that are absent as well as information the teacher wants to present again is saved as a file in the computer. And, whatever content was reviewed can be transferred to a website, a printed hard copy, or transferred anywhere as an electronic document.
The other advantage of use of the Smart Board in Mathematics is that today’s students like technology. So, they will enjoy using the "smart electronic pen" and write their work on the board to use the technology. So, in algebra 2, the students will be motivated to get out of their desks and show one their work using the SmartBoard. And, the teacher can have several students do the same or similar problems on different pages of the Smart Board. The students using the electronic "smart" pen can use different colors with different thickness and create their own individual style of writing to show the other students. In this manner, the Smart Board has now enabled students to change from being spectators but now become active participants in working out and showing the work solving those algebraic equations.Read more: http://technological-teaching-aids.suite101.com/article.cfm/mathematics_teaching_tools#ixzz0QIKsibDn
Friday, August 21, 2009
Pedagogy in Mathematics
PEDAGOGY IN MATHEMATICS
This field covers studies related to theories and practices in mathematics education for the enhancement of mathematical understanding. Amongst the approaches to be investigated, are constructivism, mastery, collaborative, contextual learning and problem-based learning. Other pedagogical perspectives include cognitively-guided instruction, zone of proximal development, expert-novice paradigm, postmodern pedagogy, critical pedagogy, development of mathematical thinking, mathematical values and beliefs, policy issues and current issues related to pedagogy of mathematics.
Teachers typically assume, according to the sources, are so prone to 'teach the same way they were taught in school.'Academic subjects such as civics, human civilisation and cultural studies require students to explore the world in search of their own knowledge. Mathematics, though base on facts and being thought in a procedural manner by teachers in school, can provide changes. For example, teachers could require students to look out for the history of how theorems are derived such as Pythagoras theorem. This not only makes the subject more interesting and more meaningful to any student as they can appreciate for themselves how such theories were derived a long time ago and still applicable today.
Technology pervades current life and has influenced our educational institutions including the manner ofinstruction and the design of curricula. Such change needs to be evaluated in terms of the impact uponboth teaching and learning. For example, the low cost, highly portable graphics calculator has become afeature of secondary school classrooms, yet are they being used mainly by teachers? How does thegraphics calculator affect the student’s comprehension and understanding of concepts? Is the use ofgraphics calculators more effective compared to the traditional chalk-and-talk methods?This paper discusses how the use of the graphics calculator changes the teaching and learning ofquadratic functions in the secondary classroom. Firstly, the paper compares teaching with the graphicscalculator to the traditional chalk-and-talk method. The second part presents and analyzes the perceptionof students to the graphics calculator in their learning of mathematical concepts. It further discusses thefindings based on data including students’ interviews that reveals how the use of graphics calculatorenhanced their learning. Finally, the paper advances several recommendations for pedagogicalstrategies when using graphics calculators.
The primary purpose for teaching mathematics is to enable students to learn and appreciatemathematics in the best way possible. With creativity, passion, and resources available to them,teachers are able to implement various techniques and strategies in the classroom to make learningmore meaningful and interesting to their students. In many classrooms, the usual way of teaching isthe chalk-and-talk method. Teachers give the input verbally or write on the board and the studentsfollow their instructions. However with the entry of technology into the classroom, the teaching ofmathematics is changed (Simonson & Dick, 1997). Technology, specifically graphics calculator,has been widely adopted by academic institutions and has influenced the pedagogy in theclassroom. For example, while graphics calculators were designed as personal tools, research byCavanagh (2005) reported that students tended to use them as a shared device. He found graphingcalculators played an important role in group activities as a kind of conversation piece for sharingmathematical ideas and making thought processes publicly available in the classroom. Thetechnology facilitated social interaction in the classroom because it acted as a common point ofreference for students as they discussed their ideas and results. Other researchers such as White(2004) have claimed that the graphics calculator has the potential to be a pedagogical Trojan Horse,subtly influencing a change in the usual teaching practices.
REF: www.google .com.my
(Abdullah & Saleh, 2005, P. 254).
Tuesday, August 18, 2009
BLACKBOARD
Blackboard!!!Blackboard!!!Blackboard !!! What is blackboard? In sence of 21st centery (new era) all people will dont know, what is that blackboard but they are know about Electronic Blackboard or E-Blackboard.ok. Blackboard and E-blackboard is totally defferents. Blackboard is normally a large,smooth, usually dark surface of slate or other material on which to write or draw with chalk or chalk board..! BUT E-Blackboard is used computer or internet (ICT) in teaching and learning process. Its explores ways in which it can be used in teaching and learning and looks at developing effective courses such as using project in classroom teaching. We can also access Blackboard from the right hand menu on the blue Intranet screen.Further use is evolving of electronic Blackboard, accessed through the School's web site, to enhance communication with the students such as e-mail,computer conferencing, voice mail and so on. E- Blackboard also has many things we can do such as;
- Teachers could hit a trouble maker in the middle of the forehead with a blackboard rubber without fear of missing or being sued!
- On our latest visit we decided to choose from the blackboard menu.
- The base should be painted matt black using blackboard paint
- Remember that uploading a file creates a copy of the file on the Blackboard server.
- Your personal Welcome page Your Welcome page will contain links to all of your Blackboard courses once you have enrolled in them.
- Let me have your own recommendations; or post them to the duo board [ Our module Blackboard site ] .
Teachers and students perspective
The study deployed a survey method to collect basic data on the current practice of ICT in theteaching of Science and Mathematics at secondary schools, and to investigate teachers’ needs fortraining and support in relation to the effective use of ICT. The study focus on the Science andMathematics teachers who are currently teaching at 21 government secondary schools inKuching, Sarawak. 250 copies of questionnaires were randomly distributed to Science andMathematics teachers from 18 government schools located in Kuching, and 212 filledquestionnaires were returned. This gives a response rate of 85%.
Teachers’ perceptions on the use of ICT in classroomsRespondents’ attitudes towards the use of ICT were examined in the survey by series ofstatement reflecting positive and negative attitudes towards ICT to which respondents indicatedtheir agreement and disagreement. In general, the respondents broadly agreed that utilization ofICT makes them more effective in their teaching (75%), and more organized in their work(80%), rely less upon textbooks (37%), and better able to meet the varying needs of students(48%). While 39.2% of the respondents broadly agreed that with the uptake of ICT they needlonger blocks of time for instruction, 43.4% of them disagreed that they give up too muchinstructional responsibility with the use of technology. In general, respondents broadly agreedthat with the use of internet and technology, their lesson plans are richer (55%), and the way theyorganize classroom activities has changed (56%). A further positive sign is 85% of themindicated that they would like to integrate more ICT applications into their teaching.Table 4: Teachers’ perception on the effects of using ICT for professional tasksNR – No ResponseResponse (%) Daily Weekly Monthly Occasionally Never NRTeaching and instructional support 41.0 34.0 10.8 13.7 0.5 0.0Classroom management activities 18.9 29.7 26.9 21.7 2.8 0.0Communications 7.5 17.9 12.7 37.7 21.7 1.9Personal development 2.8 9.4 9.4 39.6 36.8 1.9Response (%) Daily Weekly Monthly 1 – 2 times a year Never NRTeaching Courseware 30.2 42.9 14.6 9.0 2.4 0.9Presentation tools 20.3 22.6 22.6 17.5 14.2 2.8Online demos 0.9 5.7 9.4 10.4 69.3 4.2Graphical visualizing tools 7.1 16.5 14.6 19.3 37.3 5.2Spreadsheets 8.0 23.6 30.7 17.0 13.2 7.5Internet browsing 17.0 35.8 26.4 12.3 7.5 0.9Hypermedia / Multimedia 7.1 14.6 23.1 19.8 29.7 5.7Simulation programmes 4.2 9.4 22.6 21.7 36.8 5.2Response (%) Strongly Agree Agree Neutral Disagree Strongly Disagree NRUsing ICT makes me more effective in teaching. 12.3 62.7 17.0 4.7 1.9 1.4ICT helps me to organize my work. 12.7 67.0 15.1 3.3 0.9 0.9Lesson plans are richer with information from internet. 5.7 49.5 33.5 8.5 0.0 2.8I have changed the way I organize classroom activities. 2.8 53.3 34.9 6.1 0.0 2.8I rely less upon textbooks. 3.3 34.0 35.4 22.6 1.9 2.8I am better able to meet the varying needs of students. 2.4 45.8 39.2 9.0 0.0 3.8I would like to integrate more ICT into my teaching. 17.0 67.5 10.8 3.8 0.0 0.9I need longer blocks of time for instruction. 5.7 33.5 33.5 22.2 1.4 3.8I give up too much instructional responsibility. 0.9 14.6 37.3 38.2 5.2 3.8
Overall, a high 87% of the respondents perceived ICT as important tool to accomplish theirprofessional tasks, and 69% of them felt that amongst the various stakeholders, teachers (asclassroom practitioners) should have a greater voice or say in how ICT is being used in schools.Table 5: Teachers’ perception on the importance of ICT to accomplish their jobVery Important Somewhat Important Neutral Somewhat Unimportant Unimportant at all NRResponse (%) 34.0 52.8 9.0 1.4 0.5 2.4Table 6: Teachers’ view - who should have a greater voice in how ICT is used in schools?Teachers Students Principals Parents District administrator NRResponse (%) 68.9 9.0 9.0 2.8 5.2 5.2Another positive development is observed when 64.2% of the respondents stated themselves tobe either confident or very confident in engaging students with technology in class.Table 7: Level of confidence in engaging students with technology in classVery Confident Confident Neutral Not very confident Not confident at all NRResponse (%) 8.5 55.7 18.4 12.3 0.9 4.2Obstacles faced and training needsRespondents were also asked to indicate their reasons for not using a broader range of ICT inclassrooms. 205 teachers responded to this question, many of them citing more than oneobstacle. The numbers of survey responses for each item are as follows:Table 8: Obstacles faced in the use of ICT in schoolsLack of technical support when things don’t work 122Lack of time in school day 118Limited knowledge on how to make full use of ICT 81Limited understanding on how to integrate ICT into teaching 69Lack of software or websites that support state standards .
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