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:
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
(Translated from the original Spanish.)
(Garriga, 1963, pp. 158-159).
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.
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
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.
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:
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.
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.
Swetz, F. (1992). ‘Fifteenth and Sixteenth Century Arithmetic Texts: What can we learn from them ?’ Science and Education, 1, pps 365-378.