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Studying the history of science is particularly important for learning mathematics. In scribe schools of Ancient Mesopotamia we find a teaching model that can be adapted perfectly to the first level of education in this discipline, because it was in this context where the first elements of arithmetic, geometry and algebra were set. In this paper we examine how the curriculum of Mesopotamian schools can help our students learn mathematics.
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Procedia Social and Behavioral Sciences 2 (2010) 1137–1141
A
vailable online at www.sciencedirect.com
WCES-2010
Learning mathematics through its history
Piedad Yuste
a
*
a
Philosophy Department, UNED, Paseo Senda del Rey 7, 28040 Madrid. Spain
Received October 9, 2009; revised December 18, 2009; accepted January 6, 2010
Abstract
Studying the history of science is particularly important for learning mathematics. In scribe schools of Ancient Mesopotamia we
find a teaching model that can be adapted perfectly to the first level of education in this discipline, because it was in this context
where the first elements of arithmetic, geometry and algebra were set. In this paper we examine how the curriculum of
Mesopotamian schools can help our students learn mathematics.
Keywords: Mathematical education; scribe schools; mesopotamian mathematics.
1. Introduction
When we teach mathematics in schools, we do it from a current perspective, by using abstract formulas and
algorithms, seldom explaining where they come from. This strategy is often applied to teaching algebra, because all
the relations established in the statement of a problem are moved to this language. Similarly in geometry we do not
stop to inquire about the origin of the most common and essential procedures. Students absorb all this information
without actually knowing what its source and foundation were. Some approaches are treated as unexplained recipes.
We have not considered whether it would be easier to learn the scientific formalizations by ascending from their
lowest conceptual level, by going back to their initial, experimental and intuitive stages, when both physical contact
with things and observation of the environment inspired early scientists. In my opinion, learning scientific subjects
would be more interesting and successful if we repeated the historical stages of the science that we try to teach in the
classroom, as a preamble or prologue to it.
The origin of mathematics is found in schools for scribes in Ancient Mesopotamia and even further back in the
practices carried out by surveyors, calculators and book-keepers. Numbers and writing emerged at the same time in
the territory controlled by the Sumerians in the late fourth millennium BC. First, as pieces of clay with different
values: tokens which meant both the quantity and the thing to be reckoned. Later, abstract numerals were invented;
their shape and size depended on the class or group that owned the quantified object, which is represented by means
* Piedad Yuste. Tel.: 00 34 91 398 6944
E-mail address: pyuste@fsof.uned.es.
1877-0428 © 2010 Published by Elsevier Ltd.
doi:10.1016/j.sbspro.2010.03.161
© 2010 Elsevier Ltd.
Open access under CC BY-NC-ND license.
Open access under CC BY-NC-ND license.
1138 Piedad Yuste / Procedia Social and Behavioral Sciences 2 (2010) 1137–1141
of a pictogram. Sumerian children learned up to thirteen different numbering systems, divided into classes and
orders of magnitude, in school. There were specific objects to symbolize different cardinals:
1 10 60 600 3600 36000
Figure 1
To count, the tabs of the same amount are grouped, and when there are enough of them, they are replaced by
another higher order. Numerous clay tablets state this procedure and this is how basic arithmetic operations,
additions and subtractions, are performed. Over time, this procedure was simplified and resulted in a sexagesimal
numeration, where only two signs were utilized to express all quantities. The use of the abacus led to the invention
of the positional system. Learning mathematics through its history means manipulating objects, displaying quantities
and forms, sorting and classifying, composing figures. And then extracting a concept: the number from the quantity,
the geometric shape from the form, the abstract relation from the geometric composition. Thus we can gradually
penetrate the disciplines that support these ideas: arithmetic, geometry, and algebra.
2. Procedure and contextualization
If we decide to teach mathematics through its history we must follow this science’s evolutionary sequence.
Children will be happy to know that centuries ago there were schools similar to theirs where students learned these
same things.
- We will start by learning the fundamentals of arithmetic: numbers and numeration. Later, we will go on to
teach how to count and the way the decimal system works. After that, the basic arithmetical operations:
addition and subtraction. Next we will explain the concept of product, as a repetition of the same amount,
and the idea of division and partition of objects (division). These notions are taught successively at lower
grade levels. We will use pieces of different size and color (or shape), to represent the units, tens, and
hundreds respectively. We also need an abacus. It is highly desirable for students to memorize addition and
multiplication tables, just as the students of schools for scribes did four thousand years ago. Fractions are
expressed by means of the distribution in parts.
- In subsequent courses, teaching geometry will start with the observation of flat irregular shapes:
quadrilaterals. Then we will show rectangles, and squares as special cases in which the sides of the
rectangle measure the same. Triangles are conceptualized from the rectangles. Finally, the circles. We
should explain the rule of averages to find the measure of surfaces and the possibilities offered by the
circles. It will be necessary to use geometric shapes and a blackboard.
- Finally, at a higher educational stage, we will initiate students into the knowledge of algebra and into its
peculiar strategy. For this we also need simple geometric shapes: squares and rectangles. A computer will
be very useful for our goal, as seen below. Object manipulation will enable the students to extract the
fundamental rules of algebra and express mathematical relationships in this language.
3. Learning techniques
Three areas of teaching mathematics: arithmetic, geometry, and algebra are implemented in a number of issues
that we will explain and develop using examples. We will take some exercises from the scribe schools and use them
as paradigms of education in each of these disciplines.
Piedad Yuste / Procedia Social and Behavioral Sciences 2 (2010) 1137–1141
1139
3.1. Arithmetic
Children should begin to study arithmetic by manipulating objects. Once they have assimilated the basic ideas,
they will learn routines with the help of numerical tables.
- The first contact with arithmetic begins with the idea of number. Schoolchildren will hold pieces of the
same size and shape to learn to count. Later we explain that when you have ten of these objects, we can
exchange them for another one of equivalent value. The same thing happens when we have a hundred.
- We should teach our students to count and to recognize how many objects of a certain class have, for
example, how many pencils, how many fingers, etc. They will also learn to collect objects to do additions
and subtractions.
- Subsequently, we will move this numbering system and its core operations to an abacus. This way, students
will learn the decimal positional method and they will be able to make arithmetical computations.
- A third step corresponds to multiplication: they will learn to form classes of objects with the same amounts.
Once they have captured the notion of the product as a repetition of elements "n times m," we recommend
the use of multiplication tables to expedite calculation.
- The idea of division is based on the distribution of objects. Later, students need to understand that this
operation is the inverse of the product. The pieces (calculi) are essential in this task.
3.2. Geometry
The science of geometry has its origins in the measurement of fields. Expert surveyors of Ancient Mesopotamia
and Egypt determined the sides of the land with the aid of ropes and rods. Due to the irregularity of their shapes,
they calculated the area using the "surveyor’s rule" or "average formula," but the results were only approximate:
-
22
WwLl
S

u
Figure 2
- It is important to note the way the calculation of the areas is determined by taking so many times the width
by the length.
In our schools we could follow the development this discipline underwent:
- We will begin by studying an irregular quadrilateral. The measurement of the area is calculated by applying
the surveyor’s rule. These figures should not be rigid, allowing us to experiment with many other geometric
shapes; this way, we can observe how the average formula gives a good result.
Then, we will go into the study of other geometric shapes, and their equivalences, by means of gnomons:
- Rectangles: sides, surface.
- Squares: sides, surface.
- Right triangles: sides, surface.
- Equilateral triangles: sides, surface.
- Right trapezoids: sides, surface.
W
w
l
L
1140 Piedad Yuste / Procedia Social and Behavioral Sciences 2 (2010) 1137–1141
- Circles have a peculiar approximation from the square of their diameter.
At a more advanced level of learning, we would teach the following concepts:
- Intuitive demonstration of the Pythagorean rule.
- Calculation of
2 in a square.
- Measurement of the diagonal in rectangles.
- Isosceles trapezoids: sides, height and surface.
It is necessary to underline the importance of touching and manipulating different shapes before drawing on
blackboards, paper, etc. The computer allows us to visualize all these trials in a very pleasant and imaginative
manner.
3.3. Algebra
The mathematicians of Ancient Mesopotamia transferred the more complex arithmetical relations - those in which squares
appear- to the context of geometry. We call them quadratic equations and we have a specific algorithm to find their solutions.
Usually we teach this algorithm at school without specifying where it comes from, or how it came into use. In addition, we
provide the language and rules by which students must express correspondences between elements of a problem, but all this does
not go beyond a mere catalogue of instructions.
Before initiating students into the tools of algebra, they should become familiar with the basic ideas:
- Both the square of the sum and the square of the difference of two given magnitudes.
- The difference of squares.
- The equivalence between a rectangle and the difference of two squares.
These concepts will be explained by using geometric compositions. Later, we will try to show in this way the
relations that appear in some simple equations:
1)
2
abac: 2)
2
abac: 3)
2
acba :
b
a
a
1
2
b
Figure 6
a
b
a
Figure 7
2
b
a
b
Figure 8
Figure 3 Figure 4
Figure 5
Piedad Yuste / Procedia Social and Behavioral Sciences 2 (2010) 1137–1141
1141
Once this methodology has been assimilated (cut and paste method, by Jens Høyrup), we will exhibit some
simple examples drawn from the Arithmetic of Diophantus of Alexandria, with emphasis on the theoretical step
toward formalizing the statements previously proposed. At this stage of education, we will also mention Arabic
mathematicians’ approach, especially Al-Khwarizmi’s approach.
4. Conclusions
When we teach science, we must never neglect the study of its particular history. Learning mathematics through
its historical development is essential not only from a cultural and humanistic perspective, but also because it
facilitates understanding its contents. Currently, the history of mathematics is an introduced subject in the secondary
schools, but it has been excluded in the lowest levels of education, probably because it thinks children do not need to
be aware of developments in this discipline. Mathematical practices carried out in the scribe schools can be easily
understood by our own students and constitute the main tool for their initiation into the knowledge of this subject:
arithmetic calculation, geometric constructions, and introduction to algebra.
Manipulating objects, displaying evidence, and searching for a geometric reasoning are all the tools to put this
endeavour in practice. Training mathematics should not start at the final stage: the passive assimilation of abstract
concepts and their correlations; it should, in contrast, be the students themselves who obtain the general notion. This
is the way to go back to the initial moment of a scientific proposition. The teacher will guide students in the
discovery process; from simple and obvious things provided by experience to the formalization of real theoretical
structures. This is how students learn to think, to draw general laws from empirical data. In this way, we will
stimulate students’ creative abilities and their capacity to solve problems intuitively. Also they will understand the
issues addressed better.
This experience should be tested in small groups of students aged 3 to 12 years old, by noting the degree of
acceptance achieved among children and how this program affects the assimilation of concepts and routines. The
success of this approach will depend on the care and persistence that we put in this work.
Acknowledgments
To my friend Nancy Konvalinka for her generous assistance with the intricacies of English.
References
Etayo Miqueo, J. J.; García Hoz, V. (1995). Enseñanza de las matemáticas en la educación secundaria. Madrid: Rialp.
Guzmán, M. de (1996).
El rincón de la pizarra. Madrid: Pirámide.
(2002).
La experiencia de descubrir en geometría. Madrid: Nivola.
Hoyrup, J. (2007). “The roles of Mesopotamian Bronze Age Mathematics. Tool for state formation and administration-carrier of teacher’
professional intellectual autonomy”,
Educational Studies in Mathematics 66, pp. 119-129.
Nissen, H. J.; Damerow, P.; Englund, R. K. (1993):
Archaic Bookkeeping: Early writing and techniques of economic administration in the
Ancient Near East,
Chicago & London: University of Chicago Press.
Proust, Ch. (2008): “Quantifier et calculer: usages des nombres à Nippur ",
Revue d’Histoire des mathématiques 14, pp. 143-209.
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(2008). “Geometry in Mesopotamia and Genesis of Algorithms”,
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Enseñanza de las matemáticas en la educación secundaria
  • J J Etayo Miqueo
  • V García Hoz
Etayo Miqueo, J. J.; García Hoz, V. (1995). Enseñanza de las matemáticas en la educación secundaria. Madrid: Rialp. Guzmán, M. de (1996). El rincón de la pizarra. Madrid: Pirámide.
El rincón de la pizarra La experiencia de descubrir en geometría
  • M Guzmán
  • De
Guzmán, M. de (1996). El rincón de la pizarra. Madrid: Pirámide. (2002). La experiencia de descubrir en geometría. Madrid: Nivola.
Quantifier et calculer: usages des nombres à Nippur”, Revue d’Histoire des mathématiques 14
  • Ch Proust
Proust, Ch. (2008): "Quantifier et calculer: usages des nombres à Nippur ", Revue d'Histoire des mathématiques 14, pp. 143-209. Schmand-Besserat, D. (1992): Before writing I: From counting to cuneiform. Austin: Univ. of Texas Press.
Tools and models in the origin of algebra”
  • P Yuste
Yuste, P. (2007). "Tools and models in the origin of algebra", Patras Univ. Press, vol. III, pp. 1107-1114. (2008). "Geometry in Mesopotamia and Genesis of Algorithms", Historia Scientiarum 18-1, pp. 24-35.