Abacus School
Encyclopedia
Abacus or abaco refers to calculations, especially the subject of direct calculations, using Hindu
numerals without the help of the abacus
(an instrument for calculating). Abacus school is a term applied to any Italian school or tutorial after the 13th century, whose commerce-directed curriculum placed special emphasis on mathematics, such as algebra
, among other subjects. These schools sprang after the publication of Fibonacci
’s Book of the Abacus and his introduction of the Hindu-Arabic numeral system
. In Fibonacci’s viewpoint, this system, originating in India
around 400 BCE. and later adopted by the Arabs, was simpler and more practical than using the existing Roman numeric tradition. Italian merchants and traders quickly adopted the structure as a means of producing accountants, clerks, and so on, and subsequently abacus schools for students were established. These were done in many ways: communes could appeal to patrons to support the institution
and find masters; religious institutions could finance and oversee the curriculum
; independent masters could teach pupils. Unless they were selected for teaching occupations that were salaried, most masters taught students who could pay as this was their main source of income.
Firstly, because mathematics
was associated with many professions, including trade
, there was an increasing need to do away with the old Roman numeral system which produced too many errors. The number of Roman characters a merchant
needed to memorize to carry out financial transactions as opposed to Hindu-numerals made the switch practical. Commercialists were first introduced to this new system through Leonardo Fibonacci, who came from a business
family and had studied Arabic math. Being convinced of its uses, abacus schools were therefore created and dominated by wealthy merchants, with some exceptions. Sons could now be trained by the best and brightest teachers to take over their family business and the fortunate poor had more access to a variety of vocations. Morality also played a role in determining the school attendance of commoners.
Secondly, reading, writing, and some elementary math as job requirements for general occupations meant that literacy
levels rose with the number of ordinary students attending institutions or being tutored at home. Sailors, for example, who wished to climb the social ladder had to present literacy and arithmetic skills on their résumé
. Aspiring abbaco masters themselves need have studied only elementary, or secondary abbaco in order to teach others.
abacus school systems differed more in their establishment than in their curriculum during the Middle Ages
. For example, institutions and appointed educators were set up in a number of ways, either through commune patronage or independent masters’ personal funds. Some abbaco teachers tutored privately in homes. All instructors, however, were contractually bound to their agreement which usually meant that they could supplement their salary
with tuition
fees or other rates. Curriculum for Abbaco masters was also universal, in that lessons were directed towards solving commercial problems. Still, these primary and secondary math schools were not to be confused with university
level math subjects.
governments and their desire to educate their sons in commercial mathematics, initiated constructions of schools with the support of other parents. Communal governments then proceeded to attract abbaco masters from elsewhere, so as to avoid partiality and inevitable dispute, and to select the best according to qualification, age and salary request. They drew up contracts that specified the number of years an appointed master must work, the number of students he was allowed to teach, and the certain percentage of fees he was to return to the commune. Contracts usually ranged from twenty to thirty ducats or florins, depending on the currency
used, for one to three years. In return, communes, would grant tax
exemptions (full or half), rights to collect fees (tuition fees, textbook and school supply sales), and a house
for personal use over and above the masters annual salary.
Independent teachers could also be hired by the commune, but for lower wages. Most times, free-lance masters were contracted by a group of parents in a similar fashion to that of communal agreements, thus establishing their own school if the number of students being tutored was significant in size. Abbaco apprentices training to become masters could also tutor household children and pay for their studies simultaneously. Upon graduation, however, apprentices were required to teach elsewhere for fear of stealing the master’s students and income.
, bookkeeping, reading and writing in the vernacular were the basic elementary and secondary subjects in the abbaco syllabus for most institutions, which began in the fall, Mondays through Saturdays. Although Grendler states that the hours students spent in school could be very long and lasting half a year to eight months, Wiesner-Hanks takes the opposite view: boys and girls only spent half a day for up to four months. Here, Grendler may be considering the wealthy commoners and higher social classes, whereas Wiesner-Hanks may be looking at the working classes.
Mathematical problems dealt with the everyday exchange of different types of goods or monies of differing values, whether it was in demand or in good quality, and how much of it was being traded. Other problems dealt with distribution of profits, where each member invested a certain sum and may have later withdrawn a portion of that amount. Labor contracts too, where the employer agreed to a certain wage over the course of a certain term for a certain type of work that produced a specific amount of goods, but the employee decided to leave after a while, were brought up. Bookkeeping taught students to note weight
, length
, size
and other quantitative and qualitative information of goods.
----
Hindu
Hindu refers to an identity associated with the philosophical, religious and cultural systems that are indigenous to the Indian subcontinent. As used in the Constitution of India, the word "Hindu" is also attributed to all persons professing any Indian religion...
numerals without the help of the abacus
Abacus
The abacus, also called a counting frame, is a calculating tool used primarily in parts of Asia for performing arithmetic processes. Today, abaci are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of...
(an instrument for calculating). Abacus school is a term applied to any Italian school or tutorial after the 13th century, whose commerce-directed curriculum placed special emphasis on mathematics, such as algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, among other subjects. These schools sprang after the publication of Fibonacci
Fibonacci
Leonardo Pisano Bigollo also known as Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italian mathematician, considered by some "the most talented western mathematician of the Middle Ages."Fibonacci is best known to the modern...
Hindu-Arabic numeral system
The Hindu–Arabic numeral system or Hindu numeral system is a positional decimal numeral system developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arab mathematicians , and spread to the western world...
. In Fibonacci’s viewpoint, this system, originating in India
India
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
around 400 BCE. and later adopted by the Arabs, was simpler and more practical than using the existing Roman numeric tradition. Italian merchants and traders quickly adopted the structure as a means of producing accountants, clerks, and so on, and subsequently abacus schools for students were established. These were done in many ways: communes could appeal to patrons to support the institution
Institution
An institution is any structure or mechanism of social order and cooperation governing the behavior of a set of individuals within a given human community...
and find masters; religious institutions could finance and oversee the curriculum
Curriculum
See also Syllabus.In formal education, a curriculum is the set of courses, and their content, offered at a school or university. As an idea, curriculum stems from the Latin word for race course, referring to the course of deeds and experiences through which children grow to become mature adults...
; independent masters could teach pupils. Unless they were selected for teaching occupations that were salaried, most masters taught students who could pay as this was their main source of income.
Significance
Abacus schools were significant for a couple of reasons:Firstly, because mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
was associated with many professions, including trade
Trade
Trade is the transfer of ownership of goods and services from one person or entity to another. Trade is sometimes loosely called commerce or financial transaction or barter. A network that allows trade is called a market. The original form of trade was barter, the direct exchange of goods and...
, there was an increasing need to do away with the old Roman numeral system which produced too many errors. The number of Roman characters a merchant
Merchant
A merchant is a businessperson who trades in commodities that were produced by others, in order to earn a profit.Merchants can be one of two types:# A wholesale merchant operates in the chain between producer and retail merchant...
needed to memorize to carry out financial transactions as opposed to Hindu-numerals made the switch practical. Commercialists were first introduced to this new system through Leonardo Fibonacci, who came from a business
Business
A business is an organization engaged in the trade of goods, services, or both to consumers. Businesses are predominant in capitalist economies, where most of them are privately owned and administered to earn profit to increase the wealth of their owners. Businesses may also be not-for-profit...
family and had studied Arabic math. Being convinced of its uses, abacus schools were therefore created and dominated by wealthy merchants, with some exceptions. Sons could now be trained by the best and brightest teachers to take over their family business and the fortunate poor had more access to a variety of vocations. Morality also played a role in determining the school attendance of commoners.
Secondly, reading, writing, and some elementary math as job requirements for general occupations meant that literacy
Literacy
Literacy has traditionally been described as the ability to read for knowledge, write coherently and think critically about printed material.Literacy represents the lifelong, intellectual process of gaining meaning from print...
levels rose with the number of ordinary students attending institutions or being tutored at home. Sailors, for example, who wished to climb the social ladder had to present literacy and arithmetic skills on their résumé
Résumé
A résumé is a document used by individuals to present their background and skillsets. Résumés can be used for a variety of reasons but most often to secure new employment. A typical résumé contains a summary of relevant job experience and education...
. Aspiring abbaco masters themselves need have studied only elementary, or secondary abbaco in order to teach others.
School system
ItalianItaly
Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...
abacus school systems differed more in their establishment than in their curriculum during the Middle Ages
Middle Ages
The Middle Ages is a periodization of European history from the 5th century to the 15th century. The Middle Ages follows the fall of the Western Roman Empire in 476 and precedes the Early Modern Era. It is the middle period of a three-period division of Western history: Classic, Medieval and Modern...
. For example, institutions and appointed educators were set up in a number of ways, either through commune patronage or independent masters’ personal funds. Some abbaco teachers tutored privately in homes. All instructors, however, were contractually bound to their agreement which usually meant that they could supplement their salary
Salary
A salary is a form of periodic payment from an employer to an employee, which may be specified in an employment contract. It is contrasted with piece wages, where each job, hour or other unit is paid separately, rather than on a periodic basis....
with tuition
Tuition
Tuition payments, known primarily as tuition in American English and as tuition fees in British English, Canadian English, Australian English, New Zealand English and Indian English, refers to a fee charged for educational instruction during higher education.Tuition payments are charged by...
fees or other rates. Curriculum for Abbaco masters was also universal, in that lessons were directed towards solving commercial problems. Still, these primary and secondary math schools were not to be confused with university
University
A university is an institution of higher education and research, which grants academic degrees in a variety of subjects. A university is an organisation that provides both undergraduate education and postgraduate education...
level math subjects.
Establishment
Wealthy merchants, because of their substantial influence on publicPublic
In public relations and communication science, publics are groups of individuals, and the public is the totality of such groupings. This is a different concept to the sociological concept of the Öffentlichkeit or public sphere. The concept of a public has also been defined in political science,...
governments and their desire to educate their sons in commercial mathematics, initiated constructions of schools with the support of other parents. Communal governments then proceeded to attract abbaco masters from elsewhere, so as to avoid partiality and inevitable dispute, and to select the best according to qualification, age and salary request. They drew up contracts that specified the number of years an appointed master must work, the number of students he was allowed to teach, and the certain percentage of fees he was to return to the commune. Contracts usually ranged from twenty to thirty ducats or florins, depending on the currency
Currency
In economics, currency refers to a generally accepted medium of exchange. These are usually the coins and banknotes of a particular government, which comprise the physical aspects of a nation's money supply...
used, for one to three years. In return, communes, would grant tax
Tax
To tax is to impose a financial charge or other levy upon a taxpayer by a state or the functional equivalent of a state such that failure to pay is punishable by law. Taxes are also imposed by many subnational entities...
exemptions (full or half), rights to collect fees (tuition fees, textbook and school supply sales), and a house
House
A house is a building or structure that has the ability to be occupied for dwelling by human beings or other creatures. The term house includes many kinds of different dwellings ranging from rudimentary huts of nomadic tribes to free standing individual structures...
for personal use over and above the masters annual salary.
Independent teachers could also be hired by the commune, but for lower wages. Most times, free-lance masters were contracted by a group of parents in a similar fashion to that of communal agreements, thus establishing their own school if the number of students being tutored was significant in size. Abbaco apprentices training to become masters could also tutor household children and pay for their studies simultaneously. Upon graduation, however, apprentices were required to teach elsewhere for fear of stealing the master’s students and income.
Curriculum
Arithmetic, geometryGeometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, bookkeeping, reading and writing in the vernacular were the basic elementary and secondary subjects in the abbaco syllabus for most institutions, which began in the fall, Mondays through Saturdays. Although Grendler states that the hours students spent in school could be very long and lasting half a year to eight months, Wiesner-Hanks takes the opposite view: boys and girls only spent half a day for up to four months. Here, Grendler may be considering the wealthy commoners and higher social classes, whereas Wiesner-Hanks may be looking at the working classes.
Mathematical problems dealt with the everyday exchange of different types of goods or monies of differing values, whether it was in demand or in good quality, and how much of it was being traded. Other problems dealt with distribution of profits, where each member invested a certain sum and may have later withdrawn a portion of that amount. Labor contracts too, where the employer agreed to a certain wage over the course of a certain term for a certain type of work that produced a specific amount of goods, but the employee decided to leave after a while, were brought up. Bookkeeping taught students to note weight
Weight
In science and engineering, the weight of an object is the force on the object due to gravity. Its magnitude , often denoted by an italic letter W, is the product of the mass m of the object and the magnitude of the local gravitational acceleration g; thus:...
, length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...
, size
Size
The word size may refer to how big something is. In particular:* Measurement, the process or the result of determining the magnitude of a quantity, such as length or mass, relative to a unit of measurement, such as a meter or a kilogram...
and other quantitative and qualitative information of goods.
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See also
- Italian RenaissanceItalian RenaissanceThe Italian Renaissance began the opening phase of the Renaissance, a period of great cultural change and achievement in Europe that spanned the period from the end of the 13th century to about 1600, marking the transition between Medieval and Early Modern Europe...
- Italy in the Middle AgesItaly in the Middle AgesThis is the history of Italy during the Middle Ages.- Transition from Late Antiquity :Italy was invaded by the Visigoths in the 5th century, and Rome was sacked by Alaric in 410. The last Western Roman Emperor, Romulus Augustus, was deposed in 476 by an Eastern Germanic general, Odoacer...
- MathematicsMathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
- Islamic mathematicsIslamic mathematicsIn the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics or Arabic mathematics, covers the body of mathematics preserved and developed under the Islamic civilization between circa 622 and 1600...
Further reading
- Black, Robert. “Italian Renaissance Education: Changing Perspectives and Continuing Controversies”. Journal of the History of Ideas. Vol. 52. No. 2. University of Pennsylvania Press. April – June 1991.
- Carruthers, Bruce G. and Espeland, Wendy Nelson. “Double Entry Bookkeeping and the Rhetoric of Economic Rationality”. The American Journal of Sociology. Vol. 97. No. 1. The University of Chicago Press. July, 1991.
- Hanlon, Gregory. “Early Modern Italy: A comprehensive bibliography of works in English and French”. 9th ed. Baroque History Enterprises. Canada. December 2005.
- Høyrup, Jens. “Practitioners – school teachers – “mathematicians”: The divisions of pre-Modern mathematics and its actors”. Contribution to the conference Writing and Rewriting the History of Science 1900–2000. 5–11 September 2003.
- Jones, Phillip S. “The History of Mathematical Education”. The American Mathematical Monthly. Vol. 74, No. 1. Part 2. Fiftieth Anniversary Issue. Mathematical Association of America. January, 1967.
- Kraemer, Joel L. “Humanism in the Renaissance of Islam: A Preliminary Study”. Journal of the American Oriental Society. Vol. 104. No. 1. Studies in Islam and the Ancient Near East Dedicated to Franz Rosenthal. American Oriental Society. January – March 1984.
- Little, Charles E. “The Italians and Their Schools”. Peabody Journal of Education. Vol. 10. No. 4. Lawrence Erlbaum Associates (Taylor & Francis Group). January 1933.
External links
- Carson, Stephen W. "Fibonacci and Mises". Online Posting. Mises Economics Blog. Ludwig von Mises InstituteLudwig von Mises InstituteThe Ludwig von Mises Institute , based in Auburn, Alabama, is a libertarian academic organization engaged in research and scholarship in the fields of economics, philosophy and political economy. Its scholarship is inspired by the work of Austrian School economist Ludwig von Mises...
. May 25, 2006. Sunday, March 23, 2008. - "Mathematical decimal system introduced in 4BC". Did You Know?. Sunday, March 23, 2008.
- Trabia, Carlo. Arabic Numerals In Sicily. Best of Sicily Magazine. 2006. Sunday, March 23, 2008.