OUR FATHER

Our Father who art in heaven, hallowed be thy name. Thy
kingdom come. Thy will be done on earth, as it is in heaven.
Give us this day our daily bread, and forgive us our
trespasses, as we forgive those who trespass against us, and
lead us not into temptation, but deliver us from evil. For thine
is the kingdom, and the power, and the glory, forever and
ever. Amen

SUBJECT PHONOLOGY AND GRAMMAR

Homophone

This article is about the term in linguistics. For other uses, see Homophony (disambiguation).
A homophone is a word that is pronounced the same as another word but differs in meaning. The words may be spelled the same, such as rose (flower) and rose (past tense of "rise"), or differently, such as carat, caret, and carrot, or to, two, and too. Homophones that are spelled the same are also both homographs and homonyms. Homophones that are spelled differently are also called heterographs. The term "homophone" may also apply to units longer or shorter than words, such as phrases, letters or groups of letters that are pronounced the same as another phrase, letter or group of letters.
The word derives from the Greek homo- (ὁμο-), "same", and phōnḗ (φωνή), "voice, utterance". The opposite is heterophones: similar, but not phonetically identical words that have the same meaning

Homograph

For the typographical sense, see Homoglyph. For the geometrical sense, see Homography.

A homograph (from the Greek: ὁμός, homós, "same" and γράφω, gráphō, "write") is a word or a group of words that share the same written form but have different meanings. When spoken, the meanings may be distinguished by different pronunciations (in which case the words are also heteronyms) or they may not (in which case the words are also both homophones and homonyms). Homograph disambiguation is critically important in speech synthesis, natural language processing and other fields. Identically-written different senses of what is judged to be fundamentally the same word are called polysemes; for example, wood (substance) and wood (area covered with trees)

Homonym

For homonyms in scientific nomenclature, see Homonym (biology).
In linguistics, a homonym is, in the strict sense, one of a group of words that share the same spelling and the same pronunciation but have different meanings (in other words, are both homographs and homophones), usually as a result of the two words having different origins. The state of being a homonym is called homonymy. Examples of pairs of homonyms are stalk (part of a plant) and stalk (follow/harass a person), and left (opposite of right) and left (past tense of leave).
In a looser non-technical sense, the term "homonym" can be used to refer to words that share the same spelling irrespective of pronunciation, or share the same pronunciation irrespective of spelling – in other words, they are homographs or homophones. In this sense, pairs such as row (propel with oars) and row (argument), and read (peruse) and reed (waterside plant), would also be homonyms.
A distinction may be made between "true" homonyms, which are unrelated in origin, such as skate (glide on ice) and skate (the fish), and polysemous homonyms, or polysemes, which have a shared origin, such as mouth (of a river) and mouth (of an animal).

Minimal pair
In phonology, minimal pairs are pairs of words or phrases in a particular language, which differ in only one phonological element, such as a phone, phoneme, toneme or chroneme and have a distinct meaning. They are used to demonstrate that two phones constitute two separate phonemes in the language.
As an example for English vowels, the pair "let" + "lit" can be used to demonstrate that the phones [ɛ] (in let) and [ɪ] (in lit) do in fact represent distinct phonemes /ɛ/ and /ɪ/. An example for English consonants is the minimal pair of "pat" + "bat". In phonetics, this pair, like any other, differs in a number of ways. In this case, the contrast appears largely to be conveyed with a difference in the voice onset time of the initial consonant as the configuration of the mouth is the same for [p] and [b]; however, there is also a possible difference in duration, which visual analysis using high quality video supports.[citation needed]
Phonemic differentiation may vary between different dialects of a language, so that a particular minimal pair in one accent is a pair of homophones in another. This does not necessarily mean that one of the phonemes is absent in the homonym accent; merely that it is not present in the same range of contexts.

SUBJECT : EXPRECION AND CREATIVITY

 MUSICAL INSTRUMENT

A musical instrument is a device created or adapted for the purpose of making musical sounds. In principle, any object that produces sound can serve as a musical instrument—it is through purpose that the object becomes a musical instrument. The history of musical instruments dates back to the beginnings of human culture. The academic study of musical instruments is called organology.
The date and origin of the first device of disputed status as a musical instrument dates back as far as 67,000 years old; artifacts commonly accepted to be early flutes date back as far as about 37,000 years old. However, most historians believe determining a specific time of musical instrument invention to be impossible due to the subjectivity of the definition.
Musical instruments developed independently in many populated regions of the world. However, contact among civilizations resulted in the rapid spread and adaptation of most instruments in places far from their origin. By the Middle Ages, instruments from Mesopotamia could be found in the Malay Archipelago and Europeans were playing instruments from North Africa. Development in the Americas occurred at a slower pace, but cultures of North, Central, and South America shared musical instruments.
Definition
A musical instrument can be broadly defined as any device created or adapted for the purpose of making musical sounds. Once humans moved from making sounds with their bodies—for example, by clapping—to using objects to create music from sounds, musical instruments were born
Archaeology
In pursuit of understanding who developed the first musical instruments and when, researchers have discovered various archaeological evidence of musical instruments in many parts of the world. Some finds are as much as 67,000 years old, but their status as musical instruments is often in dispute. Consensus solidifies about artifacts dated back to around 37,000 years old and later. Only artifacts made from durable materials or using durable methods tend to survive. As such, the specimens found cannot be irrefutably placed as the earliest musical instruments.


Drawing of disputed flute by Bob Fink
In July 1995, Slovenian archaeologist Ivan Turk discovered a bone carving in the northwest region of Slovenia. The carving, named the Divje Babe flute, features four holes that Canadian musicologist Bob Fink determined could have been used to play four notes of a diatonic scale. Researchers estimate the flute's age to be between 43,400 and 67,000 years, making it the oldest known musical instrument and the only musical instrument associated with the Neanderthal culture.However, some archaeologists question the flute's status as a musical instrument German archaeologists have found mammoth bone and swan bone flutes dating back to 30,000 to 37,000 years old in the Swabian Alps. The flutes were made in the Upper Paleolithic age, and are more commonly accepted as being the oldest known musical instruments.
Archaeological evidence of musical instruments was discovered in excavations at the Royal Cemetery in the Sumerian city of Ur (see Lyres of Ur). These instruments include nine lyres, two harps, a silver double flute, sistra and cymbals. A set of reed-sounded silver pipes discovered in Ur was the likely predecessor of modern bagpipes.The cylindrical pipes feature three side-holes that allowed players to produce whole tone scales.These excavations, carried out by Leonard Woolley in the 1920s, uncovered non-degradable fragments of instruments and the voids left by the degraded segments which, together, have been used to reconstruct them. The graves to which these instruments were related have been carbon dated to between 2600 and 2500 BCE, providing evidence that these instruments were being used in Sumeria by this time.
A cuneiform tablet from Nippur in Mesopotamia dated to 2000 BCE indicates the names of strings on the lyre and represents the earliest known example of music notation.
History
Scholars agree that there are no completely reliable methods of determining the exact chronology of musical instruments across cultures. Comparing and organizing instruments based on their complexity is misleading, since advancements in musical instruments have sometimes reduced complexity. For example, construction of early slit drums involved felling and hollowing out large trees; later slit drums were made by opening bamboo stalks, a much simpler task.It is likewise misleading to arrange the development of musical instruments by workmanship since all cultures advance at different levels and have access to different materials. For example, anthropologists attempting to compare musical instruments made by two cultures that existed at the same time but who differed in organization, culture, and handicraft cannot determine which instruments are more "primitive". Ordering instruments by geography is also partially unreliable, as one cannot determine when and how cultures contacted one another and shared knowledge.
German musicologist Curt Sachs, one of the most prominent musicologist     and musical ethnologists] in modern times, proposed that a geographical chronology until approximately 1400 is preferable, however, due to its limited subjectivity.Beyond 1400, one can follow the overall development of musical instruments by time period.
The science of marking the order of musical instrument development relies on archaeological artifacts, artistic depictions, and literary references. Since data in one research path can be inconclusive, all three paths provide a better historical picture.
Primitive and prehistoric


Two Aztec slit drums, called teponaztli. The characteristic "H" slits can be seen on the top of the drum in the foreground
Until the 19th century AD, European written music histories began with mythological accounts of how musical instruments were invented. Such accounts included Jubal, descendant of Cain and "father of all such as handle the harp and the organ", Pan, inventor of the pan pipes, and Mercury, who is said to have made a dried tortoise shell into the first lyre. Modern histories have replaced such mythology with anthropological speculation, occasionally informed by archeological evidence. Scholars agree that there was no definitive "invention" of the musical instrument since the definition of the term "musical instrument" is completely subjective to both the scholar and the would-be inventor. For example, a Homo habilis slapping his body could be the makings of a musical instrument regardless of the being's intent.
Among the first devices external to the human body considered to be instruments are rattles, stampers, and various drums. These earliest instruments evolved due to the human motor impulse to add sound to emotional movements such as dancing Eventually, some cultures assigned ritual functions to their musical instruments. Those cultures developed more complex percussion instruments and other instruments such as ribbon reeds, flutes, and trumpets. Some of these labels carry far different connotations from those used in modern day; early flutes and trumpets are so-labeled for their basic operation and function rather than any resemblance to modern instruments. Among early cultures for whom drums developed ritual, even sacred importance are the Chukchi people of the Russian Far East, the indigenous people of Melanesia, and many cultures of Africa. In fact, drums were pervasive throughout every African culture One East African tribe, the Wahinda, believed it was so holy that seeing a drum would be fatal to any person other than the sultan
Humans eventually developed the concept of using musical instruments for producing a melody. Until this time in the evolutions of musical instruments, melody was common only in singing. Similar to the process of reduplication in language, instrument players first developed repetition and then arrangement. An early form of melody was produced by pounding two stamping tubes of slightly different sizes—one tube would produce a "clear" sound and the other would answer with a "darker" sound. Such instrument pairs also included bullroarers, slit drums, shell trumpets, and skin drums. Cultures who used these instrument pairs associated genders with them; the "father" was the bigger or more energetic instrument, while the "mother" was the smaller or duller instrument. Musical instruments existed in this form for thousands of years before patterns of three or more tones would evolve in the form of the earliest xylophone.Xylophones originated in the mainland and archipelago of Southeast Asia, eventually spreading to Africa, the Americas, and Europe[Along with xylophones, which ranged from simple sets of three "leg bars" to carefully tuned sets of parallel bars, various cultures developed instruments such as the ground harp, ground zither, musical bow, and jaw harp.
Antiquity
Images of musical instruments begin to appear in Mesopotamian artifacts in 2800 BC or earlier. Beginning around 2000 BC, Sumerian and Babylonian cultures began delineating two distinct classes of musical instruments due to division of labor and the evolving class system. Popular instruments, simple and playable by anyone, evolved differently from professional instruments whose development focused on effectiveness and skill.Despite this development, very few musical instruments have been recovered in Mesopotamia. Scholars must rely on artifacts and cuneiform texts written in Sumerian or Akkadian to reconstruct the early history of musical instruments in Mesopotamia. Even the process of assigning names to these instruments is challenging since there is no clear distinction among various instruments and the words used to describe them  Although Sumerian and Babylonian artists mainly depicted ceremonial instruments, historians have been able to distinguish six idiophones used in early Mesopotamia: concussion clubs, clappers, sistra, bells, cymbals, and rattles.Sistra are depicted prominently in a great relief of Amenhotep III,and are of particular interest because similar designs have been found in far-reaching places such as Tbilisi, Georgia and among the Native American Yaqui tribe.The people of Mesopotamia preferred stringed instruments to any other, as evidenced by their proliferation in Mesopotamian figurines, plaques, and seals. Innumerable varieties of harps are depicted, as well as lyres and lutes, the forerunner of modern stringed instruments such as the violin

DRUM

The drum is a member of the percussion group of musical instruments, technically classified as the membranous. Drums consist of at least one membrane, called a drumhead or drum skin, that is stretched over a shell and struck, either directly with the player's hands, or with a drumstick, to produce sound. There is usually a "resonance head" on the underside of the drum. Other techniques have been used to cause drums to make sound, such as the thumb roll. Drums are the world's oldest and most ubiquitous musical instruments, and the basic design has remained virtually unchanged for thousands of years.
A few types of drums such as timpani are always tuned to a certain pitch. Often, several drums, other than timpani drums, can be arranged together to create a drum kit.
Construction


Drum carried by John Unger, Company B, 40th Regiment New York Veteran Volunteer Infantry Mozart Regiment, December 20, 1863
The shell almost invariably has a circular opening over which the drumhead is stretched, but the shape of the remainder of the shell varies widely. In the western musical tradition, the most usual shape is a cylinder, although timpani, for example, use bowl-shaped shells. Other shapes include a frame design (tar, Bodhrán), truncated cones (bongo drums, Ashiko), goblet shaped (djembe), and joined truncated cones (talking drum).
Drums with cylindrical shells can be open at one end (as is the case with timbales), or can have two drum heads. Single-headed drums normally consist of a skin which is stretched over an enclosed space, or over one of the ends of a hollow vessel. Drums with two heads covering both ends of a cylindrical shell often have a small hole somewhat halfway between the two heads; the shell forms a resonating chamber for the resulting sound. Exceptions include the African slit drum, also known as a log drum as it is made from a hollowed-out tree trunk, and the Caribbean steel drum, made from a metal barrel. Drums with two heads can also have a set of wires, called snares, held across the bottom head, top head, or both heads, hence the name snare drum.
On modern band and orchestral drums, the drumhead is placed over the opening of the drum, which in turn is held onto the shell by a "counterhoop" (or "rim), which is then held by means of a number of tuning screws called "tension rods" which screw into lugs placed evenly around the circumference. The head's tension can be adjusted by loosening or tightening the rods. Many such drums have six to ten tension rods. The sound of a drum depends on several variables, including shape, size and thickness of its shell, materials from which the shell was made, counterhoop material, type of drumhead used and tension applied to it, position of the drum, location, and the velocity and angle in which it is struck.
Prior to the invention of tension rods drum skins were attached and tuned by rope systems such as that used on the Djembe or pegs and ropes such as that used on Ewe Drums, a system rarely used today, although sometimes seen on regimental marching band snare drums.
Sound of a drum


Several American Indian-style drums for sale at the National Museum of the American Indian.
Several factors determine the sound a drum produces, including the type, shape and construction of the drum shell, the type of drum heads it has, and the tension of these drumheads. Different drum sounds have different uses in music. Take, for example, the modern Tom-tom drum. A jazz drummer may want drums that are high pitched, resonant and quiet whereas a rock drummer may prefer drums that are loud, dry and low-pitched. Since these drummers want different sounds, their drums will be constructed a little differently.
The drum head has the most effect on how a drum sounds. Each type of drum head serves its own musical purpose and has its own unique sound. Double-ply drumheads dampen high frequency harmonics since they are heavier and they are suited to heavy playing. Drum heads with a white, textured coating on them muffle the overtones of the drum head slightly, producing a less diverse pitch. Drum heads with central silver or black dots tend to muffle the overtones even more. And drum heads with perimeter sound rings mostly eliminate overtones (Howie 2005). Some jazz drummers avoid using thick drum heads, preferring single ply drum heads or drum heads with no muffling. Rock drummers often prefer the thicker or coated drum heads.
The second biggest factor affecting the sound produced by a drum is the tension at which the drum head is held against the shell of the drum. When the hoop is placed around the drum head and shell and tightened down with tension rods, the tension of the head can be adjusted. When the tension is increased, the amplitude of the sound is reduced and the frequency is increased, making the pitch higher and the volume lower.
The type of shell also affects the sound of a drum. Because the vibrations resonate in the shell of the drum, the shell can be used to increase the volume and to manipulate the type of sound produced. The larger the diameter of the shell, the lower the pitch. The larger the depth of the drum, the louder the volume. Shell thickness also determines the volume of drums. Thicker shells produce louder drums. Mahogany raises the frequency of low pitches and keeps higher frequencies at about the same speed. When choosing a set of shells, a jazz drummer may want smaller maple shells, while a rock drummer may want larger birch shells. For more information about tuning drums or the physics of a drum, visit the external links listed below.

GUITAR

The guitar is a plucked string instrument, usually played with fingers or a pick. The guitar consists of a body with a rigid neck to which the strings, generally six in number, are attached. Guitars are traditionally constructed of various woods and strung with animal gut or, more recently, with either nylon or steel strings. Some modern guitars are made of polycarbonate materials. Guitars are made and repaired by luthiers. There are two primary families of guitars: acoustic and electric.
Acoustic guitars (and similar instruments) with hollow bodies, have been in use for over a thousand years. There are three main types of modern acoustic guitar: the classical guitar (nylon-string guitar), the steel-string acoustic guitar, and the archtop guitar. The tone of an acoustic guitar is produced by the vibration of the strings, which is amplified by the body of the guitar, which acts as a resonating chamber. The classical guitar is often played as a solo instrument using a comprehensive fingerpicking technique.
Electric guitars, introduced in the 1930s, rely on an amplifier that can electronically manipulate tone. Early amplified guitars employed a hollow body, but a solid body was found more suitable. Electric guitars have had a continuing profound influence on popular culture. Guitars are recognized as a primary instrument in genres such as blues, bluegrass, country, flamenco, jazz, jota, mariachi, reggae, rock, soul, and many forms of pop.


History


Illustration from a Carolingian Psalter from the 9th century, showing a guitar-like plucked instrument.
Before the development of the electric guitar and the use of synthetic materials, a guitar was defined as being an instrument having "a long, fretted neck, flat wooden soundboard, ribs, and a flat back, most often with incurved sides"he term is used to refer to a number of related instruments that were developed and used across Europe beginning in the 12th century and, later, in the Americas.These instruments are descended from ones that existed in ancient central Asia and India. For this reason guitars are distantly related to modern instruments from these regions, including the tanbur, the setar, and the sitar. The oldest known iconographic representation of an instrument displaying the essential features of a guitar is a 3,300 year old stone carving of a Hittite bard.
The modern word guitar, and its antecedents, have been applied to a wide variety of cordophones since ancient times and as such is the cause of confusion. The English word guitar, the German gitarre, and the French guitare were adopted from the Spanish guitarra, which comes from the Andalusian Arabic قيثارةر qitara,itself derived from the Latin cithara, which in turn came from the Ancient Greek κιθάρα kithara, and is thought to ultimately trace back to the Old Persian language. Tar means string in Persian.
Although the word guitar is descended from the Latin word cithara, the modern guitar itself is not generally believed to have descended from the Roman instrument. Many influences are cited as antecedents to the modern guitar. One commonly cited influence is of the arrival of the four-string oud, which was introduced by the invading Moors in the 8th century.Another suggested influence is the six-string Scandinavian lut (lute), which gained in popularity in areas of Viking incursions across medieval Europe. Often depicted in carvings c. 800 AD, the Norse hero Gunther (also known as Gunnar), played a lute with his toes as he lay dying in a snake-pit, in the legend of Siegfried. It is likely that a combination of influences led to the creation of the guitar; plucked instruments from across the Mediterranean and Europe were well known in Iberia since antiquity.
Two medieval instruments that were called "guitars" were in use by 1200: the guitarra moresca (Moorish guitar) and the guitarra latina (Latin guitar). The guitarra moresca had a rounded back, wide fingerboard, and several soundholes. The guitarra latina had a single soundhole and a narrower neck. By the 14th century the qualifiers "moresca" and "latina" had been dropped and these two cordophones were usually simply referred to as guitars.
The Spanish vihuela or (in Italian) "viola da mano", a guitar-like instrument of the 15th and 16th centuries, is widely considered to have been a seminal influence in the development of the guitar. It had six courses (usually), lute-like tuning in fourths and a guitar-like body, although early representations reveal an instrument with a sharply cut waist. It was also larger than the contemporary four course guitars. By the late 15th century some vihuelas were played with a bow, leading to the development of the viol. By the sixteenth century the vihuela's construction had more in common with the modern guitar, with its curved one-piece ribs, than with the viols, and more like a larger version of the contemporary four-course guitars. The vihuela enjoyed only a short period of popularity in Spain and Italy during an era dominated elsewhere in Europe by the lute; the last surviving published music for the instrument appeared in 1576. Meanwhile the five-course baroque guitar, which was documented in Spain from the middle of the 16th century, enjoyed popularity, especially in Spain, Italy and France from the late 16th century to the mid 18th century.[9][10] Confusingly, in Portugal, the word vihuela referred to the guitar, whereas guitarra meant the "Portuguese guitar", a variety of cittern.

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FLAUTE

The flute is a musical instrument of the woodwind family. Unlike woodwind instruments with reeds, a flute is an aerophone or reedless wind instrument that produces its sound from the flow of air across an opening. According to the instrument classification of Hornbostel-Sachs, flutes are categorized as Edge-blown aerophones.
A musician who plays the flute can be referred to as a flute player, a flautist, a flutist, or less commonly a fluter.
Aside from the voice, flutes are the earliest known musical instruments. A number of flutes dating to about 40,000 to 35,000 years ago have been found in the Swabian Alb region of Germany. These flutes demonstrate that a developed musical tradition existed from the earliest period of modern human presence in Europe
History
Further information: paleolithic flutes and prehistoric music.

Chinese women playing flutes, from the 12th-century Song Dynasty remake of the Night Revels of Han Xizai, originally by Gu Hongzhong (10th century)
The oldest flute ever discovered may be a fragment of the femur of a juvenile cave bear, with two to four holes, found at Divje Babe in Slovenia and dated to about 43,000 years ago. The authenticity of this fact, however, is often disputed. In 2008 another flute dated back to at least 35,000 years ago was discovered in Hohle Fels cave near Ulm, Germany.The five-holed flute has a V-shaped mouthpiece and is made from a vulture wing bone. The researchers involved in the discovery officially published their findings in the journal Nature, in August 2009 .The discovery is also the oldest confirmed find of any musical instrument in history.[6] The flute, one of several found, was found in the Hohle Fels cavern next to the Venus of Hohle Fels and a short distance from the oldest known human carving.On announcing the discovery, scientists suggested that the "finds demonstrate the presence of a well-established musical tradition at the time when modern humans colonized Europe". Scientists have also suggested that the discovery of the flute may help to explain "the probable behavioural and cognitive gulf between" Neanderthals and early modern human.
A three-holed flute, 18.7 cm long, made from a mammoth tusk (from the Geißenklösterle cave, near Ulm, in the southern German Swabian Alb and dated to 30,000 to 37,000 years ago) was discovered in 2004, and two flutes made from swan bones excavated a decade earlier (from the same cave in Germany, dated to circa 36,000 years ago) are among the oldest known musical instruments.

Panflute players. Cantigas de Santa Maria, mid-13th century, Spain
Playable 9000-year-old Gudi (literally, "bone flute"), made from the wing bones of red-crowned cranes, with five to eight holes each, were excavated from a tomb in Jiahu in the Central Chinese province of Henan.
The earliest extant transverse flute is a chi (篪) flute discovered in the Tomb of Marquis Yi of Zeng at the Suizhou site, Hubei province, China. It dates from 433 BC, of the later Zhou Dynasty. It is fashioned of lacquered bamboo with closed ends and has five stops that are at the flute's side instead of the top. Chi flutes are mentioned in Shi Jing, compiled and edited by Confucius, according to tradition.
The Bible, in Genesis 4:21, cites Jubal as being the "father of all those who play the ugab and the kinnor". The former Hebrew term refers to some wind instrument, or wind instruments in general, the latter to a stringed instrument, or stringed instruments in general. As such, Jubal is regarded in the Judeo-Christian tradition as the inventor of the flute (a word used in some translations of this biblical passage). Some early flutes were made out of tibias (shin bones). The flute has also always been an essential part of Indian culture and mythology, and the cross flute believed by several accounts to originate in India as Indian literature from 1500 BCE has made vague references to the cross flute.

THE ENVIRONMENT

INTRODUCTION

T he environment is everything that surrounds it and that we cherish and that we must take care to keep clean our city, school, home,. etc. in short everything that we can be, so we did research about the environment.

 

CONCEPT OF ENVIRONMENT

Environmental abiotic itemset (solar energy, soil and air) and biotic (living organisms) that make up the thin soil layer called the biosphere, household livelihoods and living beings.

COMPONENTS OF ENVIRONMENT

The atmosphere, which protects the earth from excessive ultraviolet radiation and allows the existence of life is a gas mixture of nitrogen, oxygen, hydrogen, carbon dioxide, water vapor, other elements and compounds, and dust particles. Heated by the sun and water vapor, other elements and compounds, and dust particles. Heated by the sun and the radiant energy of the earth, the atmosphere circulates around the planet and changing the temperature differences. As regards the water. 97% are found in oceans, 2% is ice and the remaining 1% is fresh water from rivers, lakes, groundwater and atmospheric humidity and soil.

Soil is the thin mantle of matter that sustains life on Earth. Is a product of the interaction between climate and rocky or bedrock, and glacial moraines and sedimentary rocks, and vegetation.

MONTESSORI METHOD

The Montessori Method of education that it obtained from its experience, has been successfully applied to all children and is very popular in many parts of the world. Regardless of the criticism of his method in the early 1930s-1940s, has been applied and obtained a revival.

In 1907 he established the first Montessori Children's House, 'Casa dei Bambini', in Rome. Since 1913, there was intense interest in her method in North America, interest later waned. (Nancy McCormick Rambusch revived the method in America, setting the American Montessori Society (American Montessori Society) in 1960.) Montessori was exiled by Mussolini to India during the Second World War, largely because he refused to compromise his principles and turning children into young soldiers. Montessori lived out the rest of his life in the Netherlands, a country which is the headquarters of the AMI, or Association Montessori Internationale. Died in Noordwijk aan Zee. Her son Mario headed the A.M.I. until his death in 1982.

Numerical Mechanisms and Children’s Concept of Numbers

Numerosity and Ordinality of Infants

Children, just like animals and adults, are quite accurate with very small numbers, and they can compute approximately with larger numbers. Piaget suggested that infants are born with no understanding of numerosity, which is the ability to discriminate arrays of objects on the basis of the quantity of items presented—for example, being aware of that a quantity of two is different than a quantity of three. Early Piaget experiments (Piaget 1942) described infants' lack of numerosity as a poor perception of quantity conservation. However, recent experiments have shown that infants between ages of 4 and 7 ½ months are able to discriminate two items from three items, but not 4 items from 6 items (Starkley et al. 1983). In particular, 7-month old infants were presented with two photographs of two or three items accompanied with two or three drumbeats. The infants looked longer at the photos with the number of items matching the number of drumbeats, indicating intuition of quantities up to 3 or 4 and suggesting at the same time that this ability for numerosity abstraction is neither visually nor auditory based (Geary 1994). Other experiments, conducted independently, showed that 10 to 12-month old infants could discriminate 3 from 4 items and, sometimes, 4 to 5 items (Strauss & Curtis 1981).
Nevertheless, the experimental results mentioned above did not indicate that infants perceived that 2 is more than 1 or 3 is more than 2. That awareness of ordinal relationships between numerosities (ordinality) is developed slowly across small values (up to three or four) in the first 18 months of life (Geary 1994). At this age, infants are sensitive to small changes of small quantities, for example they seem to understand the result of addition 1+1=2 or subtraction 2-1=1. The ability to understand even small quantities (numerosity) from the first months of life therefore indicates that there is an innate mechanism for number sense which can provide the seed for further development of numerical skills and abilities.

Numbers, Number Words and Counting

The notions of numbers and counting dates back to prehistory, and all tribes or societies, however simple, have some system of counting. With the invention of writing, symbols were found to represent the numbers. Different methods of representing numeric symbols were invented, but the most common one was the division in groups of ten. The numeric systems invented vary across time and place, and there is no doubt that the properties of such a system can facilitate or impede the development of children’s mathematical understanding. Chinese (and Asian languages based on ancient Chinese) are organized such that the numerical names are compatible with the traditional 10-base numeration system. So spoken numbers correspond exactly to their written equivalent: 15 is spoken as "ten five" and 57 as "five ten seven." Most European systems of number words are irregular up to 100. For example in French, 92 is said as "four twenty twelve," corresponding to 4*20 + 12. The more complicated the number word system is, the harder it is for children to learn the counting sequence. An interesting system is that of the Oksapmin (Saxe 1982), a horticultural society in Papua New Guinea, where counting and numerical representations are mapped onto 27 body parts (Figure 1).

Figure 1: Oksapmin counting system, based on 27 body parts (Saxe 1982).
Linguistic aspects of numeration systems not only can affect the speed of learning the counting sequence, but also influence the children’s understanding of base structure, place value (units, tens, etc.) and associated arithmetical computations. For example, Asian children have a better understanding of the 10-base concept than their first grade American peers. The system of spoken numbers affects the child’s cognitive representation of numbers. The speed at which numbers can be pronounced also has an affect on the child’s memory span for numbers. The ability to keep more number words in short-term memory seems to influence early mathematical skills which require counting, for example in problems of simple addition. Important is that children first learn number words ("one," "two," "three"…), and then couple them with their quantity concepts. How quickly those associations happen vary across different cultures as they are influenced by linguistic factors.

Counting Principles

It should be emphasized that there is no reason to require a child to use conventional count words in the conventional order. It can be safely assumed that there is a need for a set of unique tags to tick off the items in a collection, during the counting process, using these tags in a fixed order. The set of number words meets these criteria, but then so do other sets of tags, like the alphabet. It is noteworthy that many languages have used the alphabet as count word tags, for example, Greek and Hebrew. Tags need not even be verbal. They may be idiosyncratic entities, including short-term memory bins (Gelman and Galistel 1978). In any case, independent of the kind of counting tags, whether they are number words, the alphabet or other child-dependent sequence, five principles govern and define counting. The first three deal with rules of procedure, or how to count; the fourth with the definition of countables or what to count and finally the fifth involves a composite of features of the other four principles. We will mention briefly the five counting principles that are based on the influential work of Gelman and Galistel (Gelman and Galistel 1978):
The one-to-one principle
This principle emphasizes the importance of assigning only one counting tag (number word, alphabet element, or other) to each counted object in the array. For example, the child should never state "one, two, two." To follow that principle, a child has to coordinate two processes, partitioning and tagging. This simply means that every item being counted needs to be transferred from the to-be-counted category o the counted category (partitioning) while a distinct tag must be set aside, not to be used again in the counting sequence (tagging). Children employ many strategies to facilitate the coordination of partitioning and tagging, pointing to the objects and stating at the same time the associated number word is a common one.
The stable-order principle
Counting involves more than the ability to assign arbitrary tags to the items in an array. The counting tags chosen must be arranged in stable (i.e. repeated) order. For example, the child might count three objects stating "one, three, four" and four objects by stating "one, three, four, five."
The cardinal principle
This principle reflects the child’s understanding that the last number word of an array of counted items has a special meaning: it represents the set as a whole and the numerosity of this set of items. It seems likely that the cardinal principle presupposes the one-to-one principle and the stable-order principle and therefore should develop after the child has some experience in selecting distinct tags and applying those tags in a set.
The abstraction principle
The realization of what is counted is reflected in this principle. A child should realize that counting could be applied to heterogeneous items like toys of different kinds, color, or shape and demonstrate skills of counting even actions or sounds! There are indications that many 2 or 3 year olds can count mixed sets of objects.
The order-irrelevance principle
The child has to learn that the order of enumeration (from left to write or right to left) is irrelevant. Consistent use of this principle does not seem to emerge until 4 or 5 years of age (German and Galistel 1978).
Although children at the age of 3 seem to understand the basic principles of how to count, Piaget's experiments indicated that counting proficiency, and mature number sense do not emerge until the age of 8. It seems that the innate, primitive mechanism of number understanding and counting needs constant refinement through practice and experience. Minsky states that younger children possess adequate knowledge about amounts and numbers. However, they lack knowledge about their knowledge or "they have not acquired the checks and balances required to select or override their hordes of agents with different perceptions and priorities." (Minsky 1985).

Early Arithmetical Skills

Starkey (1992) showed that very young children could represent numerical quantities without the use of language. Even more importantly, they could understand that addition increases the numerosity of the set of items, while subtraction does the opposite. Starkey used a box where a child could search for tennis balls without being able to look inside. Children were shown a small set of balls put into the box, and then asked to retrieve that set of balls. An assistant, in anticipation of every child’s retrieval, secretly put balls inside the search box so that their number remained constant. 36 to 42 month-old children were able to perceive numerosities of up to 4. When children were shown an additional placement or removal of 1 to 3 balls and then asked to search for the set of balls, the question was whether they would retrieve the number of balls placed originally, or whether they would search for the number of balls after the addition or subtraction. Nearly all 18-24 month-old children searched for the set of balls after the addition or removal, signifying that they could understand the result of a simple addition or subtraction of up to 4. The above experimental results do not contradict Piaget's experiments that suggest poor number sense and arithmetical efficiency of children until the age of 7: the Starkey experiments did not rely at all on visual cues—or, at least as much as Piaget’s! Moreover, the above experiment did show a deficiency for more complex addition and subtraction problems, at the preverbal stage of children life.
Baroody and Ginsburg (1986) and others suggested that children adapt their already-existing counting skills and knowledge to problems requiring addition and subtraction. Adaptation occurs during the development of verbal counting, after children have learned the number words of their language and the strategies involved depend on each culture’s counting system. This remark is very important if we consider that children recognize and use counting as an addition/subtraction problem-solving strategy before formal education in school. Verbal counting seems to be a very reasonable arithmetical strategy, given that basic nonverbal skills of children appear to be applicable to only small values (Geary 1994).

Arithmetical Development

As the child learns the culture’s number words and associates these words with sets of objects, for example, five with all of the fingers on one hand, the manipulation of quantities larger than those that the child can perceive innately becomes feasible. The basis for children arithmetical development seems to be formed initially through simple counting, using fingers or objects so that the child doesn’t lose track of what it has already counted. Later, after the child has gained some linguistic competence, verbal counting (thinking with number words) shapes the children's mathematical development.
In early preschool years, object counting using manipulatives is common. Children solve simple addition and subtraction problems by counting the whole set (for the case of addition) or the remaining set (in the case of subtraction) of objects. Manipulatives can be our fingers or other body parts, like in the case of the Oksapmin. They help keeping track of what has already been counted and involve special techniques for problem solving. For example, a student can solve "10-3" by raising the fingers of the two hands, folding three and counting the rest. Verbal counting is a more mature technique, where the child uses neither fingers nor objects, but monitors the process using only short-term memory. For example, in order to solve "5+3," he can count mentally "5 6 7 8," and then, using the principle of cardinality, infer that the result is 8.
Counting is an important exercise for children. It helps them explore the relationships between numbers. Reflecting on number ordinality and realizing that smaller numbers are included within bigger numbers helps them modify their problem solving strategies. For example, in order to solve "3+19," they should start counting from 19 and progress to "20 21 22" instead of starting from 3 and progressing to 19, because in the latter case, they would have to count much more, increasing the possibility of errors. This pattern of calculation therefore has its origins in the early counting explorations of every child. Because of its great importance and heavy use, the counting problem-solving strategy is an established, prominent skill even for adults.
Skillful counting as well as gradual understanding of the numerical system improves children's number sense. The structure of the number words plays a significant role, as explained previously. Children with a better number sense are able to decompose numbers into smaller groups, usually around powers of 10 or 5, depending on the kind of the problem, or regroup them later, simplifying their problem solving strategies. Number regrouping and decomposition (derived facts) accelerate problem solving and improve number understanding. Practice and success in arithmetical operations form experience in terms of long term memory storage of basic facts about numbers. Thereinafter, the solution of simple arithmetical problems involves direct memory retrieval of "hard-wired" facts (or according to Minsky’s theory, K-lines): for example, to solve "5+3," the child after a certain age will answer directly 8 without having to count.
It should be noted that children do not first solve simple numerical problems exclusively by means of finger counting, then exclusively through verbal counting, and finally through memory retrieval of facts about numbers, known from similar problems solved in the past. Rather, children have available to them a variety of all the above problem-solving strategies. As more arithmetical operations are performed, the available strategies are modified, some are complemented, some are abandoned, while some new ones are constructed from bits and pieces of existing procedures, depending on the new goals the child has set (goal-directed behavior). As the child gradually masters arithmetical operations, the variety of problem solving strategies changes, illustrating a shift from general reliance on finger/object counting, to verbal counting and finally, to more complex strategies like number-decomposition (for example, derived facts like 132 = 1x100 + 3x10 + 2x1) or memory retrieval based strategies. Again, it should be underlined that we are talking about a general shift, not a complete sacrifice of simple, primal problem-solving strategies (like finger counting) in favor of newer, more sophisticated ones (like direct memory retrieval).

Arithmetical Development and Education

It is apparent that the child’s concept of numbers and arithmetic gradually changes, affecting the observable skills. The strongest influence on arithmetical development is formal education, which can lead to the development of skills that would not have emerged in a more natural environment, without formal instruction. We emphasize the importance of education in arithmetical (and therefore mathematical) development, as it is claimed by recent neuropsychology research that "every human being is endowed with a primal number sense, an intuition about numerical relations. Whatever is different in adult brains is the result of successful education, strategies, and memorization"(Dehaene 1997). However, if formal education is so important for arithmetical-mathematical development and achievement, why do so many schoolchildren fear mathematics? Numbers and arithmetic (and therefore mathematics) are part of our everyday life, but why do they seem de-contextual from commonsense and human life when they are taught in classrooms? Moreover, how can modes of teaching affect ways of thinking about numbers? How can different teaching approaches improve children’s arithmetical problem solving strategies?
Traditional approach
Traditional arithmetic curricula focus on the acquisition of basic numerical skills such as number order and counting, addition and subtraction facts, place value, as well as algorithms and procedures for complex addition and subtraction. Emphasis is placed on the dissemination of formal definitions, easily understood by the students who are viewed as "blank slates" onto which information is etched by the teacher, in a didactic manner (Goodrow 1998). The character of that information is mainly procedural knowledge about standard algorithms and methods for arithmetic problem solving. With this "authoritative" pedagogy of arithmetic, children ought to start thinking in a certain way in order to solve numerical problems. Therefore, students might be forced to change their own way of thinking. We think that this approach restricts the variety of useful representations for mathematical thinking. We also hypothesize that the procedural knowledge, taught at traditional schools in the form of "ready to use" methods and algorithms, narrows down children's mental strategies for problem solving.
Constructivism approach
There is strong evidence that the early teaching of standard procedures for arithmetic problem solving "thoroughly distorts in children’s mind the fact that mathematics is primarily reasoning." (Kamii et al.1993). In order to address the above problem, new mathematics curricula have been introduced, based on the Piaget theory of Constructivism. This approach suggests that logico-mathematical knowledge, apart from empirical or social knowledge (Novick 1996), is a kind of knowledge that each child must create from within, in interaction with the environment, rather than acquire it directly (almost "being donated") from the environment. Students are viewed as thinkers with emerging theories about the world while teachers generally refrain from teaching procedures and algorithms but instead, behave in an interactive way with the students, encouraging them to invent their own methodologies for all four arithmetical operations (Goodrow 1998).
Examples and evaluation of the two approaches
Experimental research showed that students in classes using the constructivist approach developed better number sense and at the same time came up with several different representations of arithmetic values and expressions, leading to significantly better performance on numerical operations, compared to their age-mates from "traditional" classes. For example, constructivist students were able to represent the same number with several terms and more than one operation (i.e. 150 = 50x3 = 70+60+20 = 500-400+50) while traditional (non-constructivist) students used only two terms and one operation (12 = 6+6 = 5+7 = 4+8). Performance on 2-digit addition was more or less the same between students of the two groups, however, on 2-digit subtraction and especially in cases with place values, the performance of constructivist students was much better. They could use several different ways to decompose and regroup the numbers, having an excellent number sense, while their non-constructivist age-mates were using the traditional, column-wise from right to left algorithm, making, in a lot of cases, several mistakes with the place value, indicating a poor sense of numbers.
For example, for the simple calculation 28-9, a non-constructivist second-grade student replied:

8-9 is impossible so you cross out 2 and make it 1 and now, take that one and put it with 8, and then 8 has two numbers, and then 9 plus 9 equals 18, so the answer is 19.

One constructivist second-grade student approach was:

28 minus 8 is 20, so the answer is 19 as I have 28-9 instead of 28-8.

Another constructivist approach was:

8-9 is minus 1 so 20 minus 1 is 19.

The last approach is amazing since it involves calculations with negative numbers, which are a difficult notion for second-grade elementary school students. Traditional students would not be able to perform such reasoning, probably because they are not familiar yet with algebra of negative numbers.
Generally, children perform much better using their own ways of thinking rather than standard, taught procedures. The majority of errors among the non-constructivist students are mainly caused because of a lack of number sense and a poor generalization of the standard procedures on unknown problems. That is very natural and can be nicely explained by Minsky’s theory: "Formal definitions lead to meaning networks as sparse and thin as possible." The useful "meanings" are not the flimsy chains of definition but the much harder-to-express networks of ways to remember, compare, and change things. "A logic chain can break easily, but you get stuck less often when you use a web of cross-connected meaning-network."
It should be noted that we are talking about the constructivist approach applied to early classes of elementary school, mainly first and second grade. At this level, the main goal is for children to accumulate a good number sense and practice many different kinds of reasoning (Papert’s Principle). Nevertheless, we believe that a more balanced approach between conceptual knowledge and procedural knowledge should be followed at the next elementary school levels. New algorithms and numerical techniques taught in later classes of elementary school can be built on top of well established, early developed skills like strong number sense. Most likely such a combination of educational approaches would make children expand their numerical and mathematical abilities, without them being reluctant to tamper with their old, tested problem solving strategies (Investment’s Principle).

To answer our initial question, whether number sense is innate or learned: It should be clear by now that both elements, nature as well as nurture, influence a persons early arithmetic reasoning skills. Mathematical reasoning is neither innate nor learned, but most likely a combination of both.
Finally, it should be emphasized that education and educational models are coupled with every society’s culture. Unfortunately, if a versatile and multidimensional arithmetical and mathematical education model stands orthogonal to the prominent cultural ideal, for example, that children’s idols are basketball or football stars, then any educational model proposed, be it constructivist or not, is doomed to fail.