Representations in the Sixteenth-Century Arithmetic Books

The research on the History of Mathematics and Mathematics Education has on textbook a useful tool to provide diverse types of information; this fact has led to the realization of many different studies focus on them. In this context, this work analyzes eight different sixteenth-century arithmetic books to know the different types of representations included in them. For this, it has used the historical analysis supported by the content analysis of textbooks. The results obtained show diversity between the representations included in each book. Even though verbal and numerical ones are part of every one of them, graphic representations vary from one book to another.


Introduction
The importance of research about History of Mathematics and Mathematics Education is that it allows finding out and highlighting different moments, situations, institutions, people or topics which in a particular moment meant a change or an advance for both the History of Mathematics and the History of Mathematics Education [10].
Old textbooks have a relevant role in these studies, because in its pages there is different information about the habits, the society during the time when they were written, the mathematics in that period and even how that mathematics was taught.Schrubing [20, p.41] stated: "teaching practice is not so much determined by ministerial decrees and official syllabuses as by the textbooks used for teaching".This has motivated a large amount of studies which analyze historical textbooks like Glaeser [4], Sierra, González and López [21], Maz y Rico [11], Gómez [5].
This article is focused on the Sixteenth-century arithmetic books; these books were started to print in the end of the Fifteenth due to the development of the written press and were mainly a tool to help traders and businessman in their commercial trade.For that reason many of them appear in cities which were important commercial places and they contain only the mathematical knowledge that traders need, mainly in the form of problems.For example, they present exercises to calculate the value of different objects, to make monetary exchanges, share-outs, to solve problems about wages [13].
In addition, the contents of all this arithmetic books were pretty similar: Positional numeral system, algorithms for the basic operations with positive integers, basic fractions and operations with them, rule of three, progressions, some of them include also extracting square roots and cube roots and algebra [13].
In this context, the aim of this paper is to analyze the different representations which were part of eight sixteenth-century arithmetic books; all of them represent this practical trend, common during this century in Spain, of relating mathematics with business deals [15]; in order to help business people learning mathematics.
Verbal statements and visual organizations, graphic or symbolic, are the most used means in the emission, transmission and reception of mathematical knowledge, which gives them great importance in the processes of teaching and learning this subject.Particularly, representations are considered an essential part to analyze the teaching process and comprehension of mathematics [16].For this different authors have studied them in history like González [7].
Representations are the symbolic or graphic notations, specifics for each notion, through which mathematics concepts and process and its more relevant properties and characteristics are expressed.This kind of representations is external, because they have a trace or a tangible physical support.Furthermore, external representations allow expressing or communicating the concepts or ideas to the people that uses them [3].
Gómez adds about systems of representations that they can be used to represent different aspects of a concept or mathematical structure [6].
Therefore, throughout this work it pretends to know the different ways in which the sixteenth-century authors of arithmetic books explain the concepts included in their works.

Methodology
This research is exploratory and descriptive which historical-mathematical nature.
We have used primary sources and selected eight books, all of them written in Spanish and published during the sixteenth-century.These books had a large influence during this century and the followings; they were used to learn mathematics at universities, houses of trade, etc.Indeed, many of them had versions which were printed even in the XVIII-century.
The chosen books are: Conpusicion de la arte de la arismetica y Juntamente de geometría written by Juan de Ortega in 1512 [12], Sumario breve de la practica de arithmetica de todo el curso del arte mercantivol written by Juan Andres in 1515 [1], Arithmetica Practica written by Juan de Yciar in 1549 [23], Suma de Arihtmetica Practica y de todas Mercaderias con la horden de contadores written by Gaspar de Texeda in 1546 [22] , Libro primero de Arithmetica Algebratica written by Marco Aurel in 1552 [2], Arte breve y muy provechoso de cuenta castellana y Arismetica written by Juan Gutierrez de Gualda in 1564 [8], Arithmetica written by Antich Rocha in 1564 [17] and Libro de arithmetica especvlativa, y práctica, intitvlado, el Dorado Contador written by Miguel Geronimo de Santa Cruz [19] (its first edition was published in 1594, but the one used in this work was printed in1625).
The methodology used to examine these books is the historical analysis supported by the content analysis of textbooks.All the paragraphs from the books have been considered analysis units, which have been read, categorized and finally analyzed.

Results
As it was stated before, representations in this work mean the signs, notations, figures or expressions through which mathematics are expressed [3].In these books three general types of representation can be found, the classification made is based on the one suggested by Maz-Machado, López y Sierra [9]: • Verbal representation: It is the main type of representation in the books.Most concepts, methods and properties are explained using words; for example, in Figure 1 is shown a verbal representation.Tables and diagrams are used to present assembled information which helps readers to understand the different contents.For example, in Figure 3        They use geometric graphics, which are, in general, basic polygons, to explain concepts which relate to square roots, like in Figure 7.

Figure 1 .
Figure 1.Verbal representation from Juan Andres' book (Page LVI).• Numerical representation: This kind of representations is combined in general with verbal representations.The authors used numbers, lines and symbols to present different operations, exercises, examples... Figure 2 shows an example of numerical representation.
and 4 there are two examples of different multiplication tables included in the books.

Figures
Figures are used to illustrate the contents of the books, mainly geometric ones, like Figures5 and 6.
Figures are used to illustrate the contents of the books, mainly geometric ones, like Figures5 and 6.