Commerce, Colonialism, and Culture in 9^th Century Puerto Rican Arithmetic Word Problems

Jose Antonio Segarra Harvard University

Mathematics is considered by many to be a neutral and value-free discipline, and the word problems used to codify numerical relationships are considered as well to be abstract and free of cultural, social, and economic relationships. However, contrary to these assumptions, research has revealed that mathematics word problems have historically represented very concrete and complex cultural, social, and economic relationships (Smith, 1918, 1925; Cohen, 1982; Swetz, 1987, 1992). In this work I will focus on three late nineteenth century colonial Puerto Rican primary school arithmetic textbooks to illustrate some of these relationships.

Arithmetic for centuries has been associated as the foundation for all mathematics. From the 14th to the 19th century, the evolution of "arithmetic" came to be closely associated with commerce.

In Patricia Cline Cohen's book A Calculating People, she gives a history of the uses of arithmetic in England and early "America" (the United States). She says:

... in America as in England, Arithmetic would be identified with commerce.... Arithmetic was a commercial subject through and through and was therefore burdened with the denominations of commerce. Addition was not merely simple addition with abstract numbers, it was the art of summing up compound numbers in many denominations- pounds, shillings, pence; gallons, quarts, and pints (differing in volume depending on the substance being measured)., acres and rods, pounds and ounces (both troy and avoirdupois), firkins and barrels, and so on. Eighteenth century copybooks show that students had to memorise all these table of equivalencies before embarking on the basic rules and operations.... Here is a typical word problem, typical in its complexity and its use of current events to suggest the utility of arithmetic. The mathematical text and the word problems contained within them reflected a very complex array of relationships dealing with everything from monetary exchanges for goods and products, to questions attempting to establish the dietary needs for General George Washington's revolutionary troops during the war for independence against British colonialism in the 18th century:
"Suppose General Washington had 800 men and was supplied with provision for but two months, how many of his men must leave him that this provision may serve the remaining five months ? (1982, pp. 18- 21)" Historically, the discussions surrounding arithmetic texts of the last 600 years has centred around mathematical textbooks used predominantly by pupils or students being trained in the art of reckoning. A perfect example of this can be found in the 15th century Italian arithmetic text II Treviso Aritmetico. It is considered one of the first widely published and used arithmetic books for the study and teaching of the art and business of numeration. In the book Capitalism and Arithmetic, Frank Swetz, a historian of mathematics, translates the treviso from its original Italian into English and places the text within the socio-historical, economic, and educational context of 15th century Europe.

An aspect of arithmetic texts that has remained constant for centuries has been the use of the word problem. The word problem came to be associated as the means through which the gospel of arithmetic was conveyed. Word problems have come to represent one or a series of codified problematised mathematical postulate or postulates. The numbers within arithmetic word problems are imbedded in one or a series of matrices which have linguistic, historical, cultural, and economic moorings that exist in national, international, communal, and individual contexts. It was through this medium, the arithmetic word problem that a variety of economic, social, and cultural realities became codified.

A fundamental question which fuels this work is determining what economic, social, and cultural relationships were expressed in the word problems in the arithmetic books of primary and elementary schools in late 19th century Puerto Rico when it was colony of the Spanish Empire. Further, if one views these textbooks as cultural products of a colonial relationship with Spain, then, the word problem is the place where these economic, social, and cultural relationships become crystallised.

In the field of the history of mathematics, research on non-European people and thier mathematical work (i.e., arithmetical texts) is extremely scarce. For example, in terms of the history of mathematics and education in Puerto Rico, Francisco Garriga's three volume dissertation entitled, The Teaching of Mathematics in Puerto Rico During the XIX Century, completed in 1962 at the Universidad Central de Madrid, stands as the only work done on the history of mathematics of Puerto Rico. Garriga asserts that the arithmetic textbook of 19th century Puerto Rico was essential for the learning and teaching mathematics. He says:

Textbooks were almost indispensable, in the teaching of Mathematics in the XIX Century in general, and in Puerto Rico in particular. For the primary schools, the authorities would prepare a list from which to select texts to use from.... No other books than those designated by the Inspection of Studies with the approval of the Sovereign Government."

(Translated from the original Spanish.)

(Garriga, 1963, pp. 158-159).

Based on Garriga's assertion as to the central role of texts in general and in mathematics in particular, I will be looking at three arithmetic textbooks which were published in the late 19th century Puerto Rico and used in the island. These arithmetic texts illustrate through the word problem a fascinating constellation of economic, social, and cultural relationships.

The arithmetic textbooks used in this work

The first arithmetic text in this work is the Elementary Textbook of Arithmetic written by Don Ernesto Ollero, a Commander of Artillery on the island of Puerto Rico. The book was printed by the "Boletin Mercantil" or the "Mercantile Bulletin" in San Juan, Puerto Rico in 1883. This work has 200 pages of text and is composed of seven books or libros, and contains 71 word problems dealing with a wide array of arithmetic operations and procedures.

Written in 1888, by Don Ramon Martinet Garcia, Professor and Gentleman (Caballero) of Isabel la Catolica, his textbook titled Arithmetic for Elementary and Advanced Schools, 3rd Edition was declared an official text by the Governor General of the island. He was also the ex-Director of the Escuela Superior of San Juan, the capital of Puerto Rico. Printed by the "Mercantile Bulletin" his work has a total of 166 pages of text and is made up of two parts. The first part contains seven chapters, while the second contains two chapters. The second part of the book is composed mostly of word problems dealing with arithmetic mercantile operations, whereas the first part deals mainly with fundamental arithmetic operations such as addition, subtraction, multiplication and division. The Martinez-Garcia text contains 50 word problems dealing with a variety of arithmetic operations and procedures.

Don Emiliano J. Diaz, a Professor of Primary Instruction, in 1898 published the 10th edition of his work, Elemental Arithmetic: A Theorefical-PracticaI Work by the Manuel Lopez Printing in Pence, Puerto Rico containing 102 pages of text. The Diaz text contains 144 word problems, and like the Ollero and Martinez-Diaz texts, it covers an impressive array of arithmetic operations and procedures. This work (like the other two texts in the study) was obtained at the University of Puerto Rico in the Puerto Rican Collection.

There are two important reasons for looking at these textbooks. These books were as accepted as the formal texts on the island for primary instruction in arithmetic by the sovereign government of Spain. Secondly these textbooks were printed by the Merchants Bulletin of Puerto Rico, the state sanctioned association for publishing these arithmetic texts, which also published informational periodicals regarding the state of commercial and economic affairs on the island during the period under study.

Historians of Mathematics Look at Word Problems

The idea of the use of word problems in arithmetic education and in textbooks has been studied by a number of scholars. Some mathematical historians that have dealt with the history of arithmetic books have focused on the explicit connections present between commercial exchanges and word problems. Frank Swetz makes explicit the connections between the emerging mercantile capitalist system in Western Europe (fifteenth and sixteenth century) and the commercial word problems related to trade, commerce, and financial institutions.

In the article, Fifteenth and Sixteenth Century Arithmetic Texts: What can we learn from them ? Frank Swetz discusses the word problems played in late fifteenth and throughout the sixteenth century. He says

While commercial arithmetics emerged as a mathematical genre in the late fifteenth century and throughout the sixteenth century, their influence prevailed in the teaching of arithmetic up until the beginning of the twentieth century. Their content, format and instructional style, with a great reliance on the use of problems, set the standards for arithmetic teaching for centuries. The impact of the commercial texts had raised the popular understanding of arithmetic to a new level; before the fifteenth century, arithmetic was viewed as an abstract science reserved for the intellectual diversion of the academic elite but with its new commercial outlook and emphasis on real world problem solving, it became the basis for accessible, lucrative careers. Quite simply, arithmetic became noticeably useful .... These problems covered a variety of applications including: the reckoning of accounts, exchange, transportation, profit and loss, percentage, discount, salaries, rents, taxes, interest, and partnership(l992,p. 377). Patricia Cline Cohen's work discusses the historical evolution of commercial word problems from early Britain to post-revolutionary seventeenth century United States arithmetic books. David Eugene Smith's work makes the links between commerce and arithmetic, (i.e., The First Great Commercial Arithmetic (1925)ISIS, 25, V111, i, 41-49., and Mathematical Problems in Relation to the History of Economics and Commerce American Mathematical Monthly ,(1918) 24, 221- 223) and uses word problems to bring his point across. Within their work, each author, has defined, analyzed, and made connections between the historical nature and role of commercial word problems and arithmetic.

Other authors have looked at mathematical word problems as a literary and/or linguistic genre (Gerofsky, 1996; Gilbert, 1996; Morgan, 1996). Each of these authors' ideas and methods are compelling and provocative, in regards to mathematics word problems. These authors analyse word problems as language constructions and do not consider the commercial and historical nature of word problems or the cultural, economic and social connections within mathematics textbooks.

Word problems as codified expressions of arithmetic thinking and education.

Educational sociologist Pierre Bourdieu (1977,1990) has developed a theoretical framework which explores and complexities capital's inextricable relationship with education. Central to Bourdieu's argument, capital has any one of these three forms: cultural, social, or economic.
Cultural capital can exist in one of three forms: in the embodied state i.e., in the form of long-lasting dispositions of the mind and body; in the objectified state, in the form of cultural goods (pictures, books, dictionaries, instruments, machines, etc.), which are the trace or realisation of theories or critiques of these theories, problematics, etc.; and in the institutionalised state, a form of objectification which must be set apart because, as will be seen in the case of educational qualifications, it confers entirely original properties on the cultural capital which it is presumed to guarantee....Social capital, made up of social obligations ("connections"), which is convertible, in certain conditions, into economic capital and may be institutionalised in the form of a title of nobility... .Economic capital is immediately convertible into money and may be institutionalised in the form of property rights(Bourdieu in Halsey, 1997, p.46). Of these three forms Bourdieu understands cultural capital as existing in one of three forms: embodied, objectified, or institutionalised (i.e., books, dictionaries, and paintings according to Bourdieu are examples of these forms of cultural capital as they exist in an objectified state).. While in an objectified state a book, for example, can be bought, sold, exchanged or used as a resource of information.

From this theoretical construction and understanding of books, I contend that nineteenth century arithmetic books from Puerto Rico are examples of cultural capital in the objectified state. Further, the word problems within these books, represent what Bourdieu calls a 'codification's of, or a formalised mathematical problematisation of commercial relationships. An analysis of these problematisations would provide a rich variety of insights into the cultural, social, and economic relationships of nineteenth century Puerto Rico.

According to Bourdieu (1990), "Codification is an operation of symbolic ordering, which minimises ambiguity and vagueness, in particular interactions, which introduces the possibility of a logical control of coherence of a formalisation. This formalisation makes possible the establishment of an explicit normativity, that of grammar or law. ... This grammar or law become the "rules of the game.".... That being said, formalisation, understood both in the sense of logic or mathematics..., is what enables you to go from a logic which is immersed in the particular case to a logic independent of the individual case. .... Formalisation is what enables you to confer on practices, above all practices of communication and co-operation that constancy which ensures calculability and predictability over and above individual variations and temporal fluctuations "(pp. 76-86).

What is fascinating about the word problems in these texts is how the conversions or the transformations of economic capital (through commodities or exchange values of money and commodities) are codified into a form of cultural capital, in the form of a word problem. These arithmetic textbooks are forms of cultural or social capital and the word problems in these textbooks are seen as the problematisations of what Bourdieu calls "symbolic capital," is the place where the relationships of and between values are codified and manipulated so as to derive an outcome, a resolution or a solution to a given problematisation. For a tangible way to illustrate how this occurred in each of the texts one needs to look at some concrete representations.

A way of concretely exploring this is by looking at the word problems in each of the texts and finding which of them dealt with these commodities and economic exchanges. For example the word problems dealing with commodities in the Ollero text (1883) focused on: properly, ribbons, labor wages, cloth, work, coal for steamships, wheat, food for soldiers, stocks and bonds, and economic contracts. An example of a word problem from the text is:

140 pesetas have been paid for 5 pieces of cloth each of which is ?O 1/3 meters. How much does a meter cost ? The total length of the 5 pieces of cloth that is left each one has ?O 1/3 meters. How much does a meter cost ? The total length of the 5 pieces of cloth is 10 1/3 x 5 = 155/3 of a meter. while the price of a meter will be 140: 155/3 = 420/155 = 2 22/31 (1883, pp. 72-74). Within the Martinez-Garcia text (1888), the commodities expressed in the word problems dealt with: sugar, labor, cloth, silk handkerchiefs, coffee, oil, land, tabacco, silk, garbanzos, paÔ os, and books. Problematising the value of cloth in pesetas, Martinez-Garcia writes:
386 meters of cloth cost 350478 pesetas; how much will one meter of the same cloth cost ? (1888, p. 55). Of the three textbooks the Diaz (1898) book provides the most extensive array of commodities in its word problems. Some examples of these commodities were: honey, sugar, coffee, liquor, tabacco, cloth, bayetas, white linens, ship cargo, cornmeal, codfish, working instruments, letras de cambio, cacao, art work, earnings for a jornalero, house paint, cheese, canvas, gold, acres of land, wine, rice, bonds in francs, cotton, paiio, oil, meat and tisu. From the Diaz text for example came the problem:
A particular town produces 59, 786 pesos of sugar annually; 29, 960 in honies; 356, 978 in coffee; 31, 660 in rum, and 4616 in tabacco. How much is the combined value of these products? It is 473. 000 pesos (1898, p. 19). The aforementioned examples illustrate the variety of word problems and their explicit connection with commerce. Students in the public schools of late 19th century Puerto Rico were expected to memorise the word problems and be able to recite them from memory.

Discussions of this research and study in the history of mathematics.

Through this work I found that in the Ernesto Ollero text of 1883 that of 71 word problems 34, or 48%, dealt with some form of economic or commodity exchange. Of the 50 word problems in the Martinet Garcia text from 1888, 39, or 78 % of the word problems dealt with some form of economic or commodity exchange. In the Diaz text 97 out of its 145 word problems, or 67 % of the word problems dealt with some form of economic or commodity exchange. And of a total of 266 word problems contained in all three texts, 170, or 64 % of all the word problems dealt with economic or commodity value exchange(s). It seems empirically sound to conclude that the number of word problems dedicated to different forms of commodity or economic exchange is significant since at least 48% and at most 78% of the word problems from each text dealing with these issues.

The purpose of this work is to begin developing in the history of mathematics research an understanding of some of the relationships that exist within the commercial arithmetic word problems from these three colonial nineteenth century arithmetic textbooks. This research is being done so as to broaden and deepen the understanding of mathematics educational history, and the explicit connections which word problems have to cultural, social, and economic life. This preliminary study sets the stage for further investigation to explore the connections between arithmetic word problems, commerce, and the history of mathematics education in Puerto Rico in the nineteenth century.

References

Bourdieu, P. (1977). Reproduction in Education, Society and Culture. London: Sage Pubiications.

Bourdieu, P. (1990). In Other Words. Stanford: Stanford University Press.

Cohen, P. C. (1982). A Calculating People The Spread of Numeracy in Early America. Chicago: The University of Chicago Press.

Comas y Muntaner, J. (1896). Introduction to the Study of Arithmetic. San Juan: Merchants Bulletin Printers.

Garriga, F. (1965). ‘The Teaching of Mathematics in Puerto Rico During the XIX Century Vol. I, 11, III’. Unpublished doctoral thesis, Universidad Central de Madrid.

Gerofsky, S. (1996). ‘A Linguistic and Narrative View of Word Problems in Mathematics Education’. For the Learning of Mathematics, 16, 2, pps 36-45.

Gilbert, S. K. (1996). ‘Arithmetic Story Grammar: Using Literary Devices to Analyse and Categorise Story Problems’. Unpublished paper presented at American Educational Research Association.

Halsey, A.H. (1997). Education Culture Economy Society. Oxford: Oxford University Press.

Martinez Garcia, R. (1888). Arithmetic for Elementary and Advanced Schools, 3rd Edition. San Juan: Merchants Bulletin Printers.

Morgan, C. (1996). ‘"The Language of Mathematics": Towards a Critical Analysis of Mathematics Texts". For the Learining of Mathematics, 16, 3, pps 2-10.

Ollero, E. (1883). Elementary Textbook of Arithmetic. San Juan: Merchants Bulletin Printers.

Smith, D. E. (1925). The First Great Commercial Arithmetic. ISIS, 25, VIII, 1, pps 41-49.

Smith, D. E. (1918). ‘Mathematical Problems in Relation to the History of Economics and Commerce’. American Mathematical Monthly, 24, 221- 223.

Swetz, F. (1987). Capitalism and Arithmetic. Illinois: Open Court Publishing.