MULTIPLE PATHS TO MATHEMATICS PRACTICE IN
AL-KASHI’S KEY TO ARITHMETIC:
A PRELIMINARY REPORT
Osama Taani
Department of Mathematical Sciences
New Mexico State University
In this paper, I discuss one of the most distinguishing features of Jamshid alKashi’s pedagogy from his Key to Arithmetic, a well-known Arabic mathematics
textbook from the fifteenth century. This feature is the multiple paths that he includes
to find a desired result. I show four different examples of his versatility in presenting
a topic from multiple perspectives. These examples are multiple definitions, multiple
algorithms, multiple formulas, and multiple methods for solving word problems. I
discuss possible implications for modern curricula.
Key Words: Jamshid al-Kashi, Key to Arithmetic, Islamic Mathematics, multiple
solutions.
“You can learn more from solving one problem in many different ways than you can from
solving many different problems, each in only one way.” (Silver, Ghousseini, Gosen,
Charalambous, & Strawhun 2005, p. 288)
HISTORICAL BACKGROUND
Between the eighth and the fifteenth centuries, Islamic civilization produced a
series of remarkable mathematicians. Among them was Ghiyath al-Din Jamshid
Mas’ud al-Kashi. Kashan, a desert town in Persia, witnessed al-Kashi’s birth in 1380.
He was entirely self-taught and he lived in severe poverty, until he moved in 1421 to
Samarqand upon the sultan Ulugh Beg’s invitation (Bagheri, 1997) to lead the
scientific movement under his patronage. Al-Kashi was the leader of the observatory
and the madrasa in Samarqand [Uzbekistan], the scientific capital of Tamerlane’s
empire. He wrote several books on astronomy, but in 1427, two years before his
death, he finished his magnum opus, the Key to Arithmetic (Miftah al-Hisab), which
is an encyclopaedic mathematics textbook that was dedicated to Ulugh Beg. This
magisterial compendium consists of five treatises in 276 pages: arithmetic of whole
numbers, arithmetic of fractions, arithmetic of sexagesimals, geometry, and algebra.
The Key to Arithmetic is written in Arabic 1 and has never been translated to other
languages except Russian (Rosenfeld, Yushkevish, & Segal, 1956). The Key to
Arithmetic was used for centuries by astronomers, architects, artisans, surveyors, and
merchants as a textbook in the Islamic world (Freely, 2009). Its influence even
extended to reach the mathematics teaching climate in Europe (Riahi, 1995). I am
working from two sources; the first is the Leiden manuscript photographically
reproduced in Rosenfeld et al. (1956). The second manuscript is from Al-Assad
1
National Library in Damascus. It is typeset in Arabic with commentary published by
al-Nabulsi (1977).
ILLUMINATING AL-KASHI’S PEDAGOGY
Islamic civilization in the middle ages, like all of Europe, had a dichotomy
between theoretical and practical mathematics. Practical mathematics was the
common subject, “whereas theoretical and argumentative mathematics were reserved
for specialists” (Abedljaouad, 2006, p. 629). Following this dichotomy, al-Kashi
designed his book for use by students who were looking to apply mathematics in their
professions. The book does not contain any theoretical proof for any problem, but it
does contain methods for solution and correctness verification, such as performing
the opposite operation to check a result, and the method of casting out nines to check
whether the product, quotient, or root is correct.
With this distancing from the theoretical climate, al-Kashi is intent on giving
multiple paths to reach a desired practical result. As I will discuss in this article, he
defines a concept from different perspectives and looks at the idea from several
points of view. He gives more than one algorithm for the same operation, and shows
different representations for formulas, and different approaches for solving word
problems. Multiple paths were a pervasive feature of al-Kashi’s pedagogy. I will give
a range of examples, and then discuss possible applications to modern teaching.
MULTIPLE DEFINITIONS
Understanding a concept from different points of view is one of the main
perspectives for al-Kashi’s teaching methodology. He adopts this as an educational
tool to anchor the meaning of the concept. Let us investigate some definitions to
make this idea clear. In Treatise I, Chapter IV, he first defines division as:
For the whole numbers, it is to partition the dividend into equal parts as many as the
number of units in the divisor, to determine the allowance of each unit, and this
allowance is called the quotient.
And then he says:
The general definition is to find a number its ratio to one as the ratio of the dividend
to the divisor. (al-Nabulsi, 1977, p. 62)
In other words, for al-Kashi the first process of division is to partition the
dividend into equal parts in order to determine the allowance of each unit of the
divisor. This allowance is a number (the quotient). Second, the ratio of the quotient to
1 equals the ratio of the dividend to the divisor. So the proportion is quotient : one ::
dividend : divisor. In today’s curricula these two definitions exist at different grade
levels.
Let us consider another example from the Key to Arithmetic, Treatise IV,
Chapter VI, Section I. First he writes:
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The circular cylinder is a solid surrounded by: two equal circles, the bases, and a
plane. Its width is circular, and its length is straight, such that a plane connects the
two bases, and if a line that connects the two bases moves parallel to the line that
connects the two centres of bases, then the line will touch the surface.
And then he says:
Another definition of right cylinder: a revolution of a right quadrilateral around one
of its sides. (al-Nabulsi, 1977, p. 279)
Al-Kashi has a similar dual approach to a cone:
The circular cone is a solid surrounded by a circle, which is its base, and by a curved
surface which tapers to a point called the vertex, such that if the straight line on its
surface that connects the vertex and the base, turns around the surface, it does
touch the whole surface. The line connecting the vertex and the base center is called
the axis. (al-Nabulsi, 1977, p.279)
And then he says:
Another definition of a circular cone: a revolution of a right triangle around a side
adjacent to the right angle. (al-Nabulsi, 1977, pp. 279-280)
Al-Kashi has a consistent method in defining 3-dimensional figures. He always
gives an alternative definition. In the first definition he describes the solid by listing
its characteristic features. In the second definition he describes how to obtain the
geometric object by revolution, which aids in forming a mental image. Using modern
theoretical terminology, these two types of definition are very useful to distinguish
between the concept definition, which is the formal definition that can be accepted by
mathematicians, and the concept image, which describes the mental picture of a
concept. For more details see Moore (1994).
MULTIPLE ALGORITHMS
Five different multiplication algorithms are given in Treatise I, Chapter III. I
will describe three of them briefly without all the details given by al-Kashi. The first
one using a lattice contains as many rows as the number of multiplicand digits, and as
many columns as the number of multiplier digits; then we divide each cell diagonally
into two triangles. We place the digits of the multiplicand above the columns and the
digits of the multiplier beside the rows. Then we perform the multiplication and we
put the ones of each product in the lower triangle and the tens in the upper triangle.
After that we add diagonally. For example, the multiplication of 7806 by 175 is
represented by
Figure 1: 𝟕𝟖𝟎𝟔 × 𝟏𝟕𝟓 multiplication process represented by lattice
from al-Nabulsi (1977, p.57).
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Another way: we start by multiplying the first digit from the multiplier by every digit
from the multiplicand, and we place the ones of the second product under the tens of
the first product and the ones of the third under the tens of the second, and so on.
Then we multiply the second digit from the multiplier by every digit from the
multiplicand, and we place the ones of the first product above the tens of the first
product of the first digit, then we put the ones of the second product above the tens of
the second product of the first digit, and so on. We do that for all digits. The
multiplication of 358 by 624 is represented by the following:
1 2
0 6 2 0
1 8 1 0 3 2
3 0 1 6
4 8
-------------------2 2 3 3 9 2
Figure 2: 𝟑𝟓𝟖 × 𝟔𝟐𝟒 multiplication process representation
from al-Nabulsi (1977, p. 59).
Another way: If the number of digits in the multiplicand is large, then we duplicate
the multiplicand eight or nine times and we set these multiples in a table beside their
order; then we list the multiples, that besides each digit from the multiplicand, such
that tens of the second lies under the ones of the first, and the tens of the third under
the ones of the second, and so on. Then we add to get the desired product. The
multiplication of 2783 by 456 is represented as follows:
Figure 3: 𝟐𝟕𝟖𝟑 × 𝟒𝟓𝟔 multiplication process representation
from al-Nabulsi (1977, p. 61).
MULTIPLE FORMULAS
Al-Kashi gives three different formulas to find the area of a triangle in general
and four extra formulas for equilateral triangle area. The first formula is half the base
multiplied by the altitude. The second is the radius of the inscribed circle multiplied
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by half the sum of all sides. In modern symbols and figures this becomes
���� ×
𝐴𝑟𝑒𝑎 = 𝑑𝑏
𝑎+𝑏+𝑐
2
Figure 4: These two figures show how the triangulation of a triangle can be
used to construct almond shapes
In fact, this formula is noticeably related to Islamic architecture and the use of
the almond shape (kite) 2 (one can see the three kites in Figure 4), which is very
important for decoration in mosaics, buildings, and other wooden and metallic
decorations (Ozdoral, 1995). Al-Kashi explains how to find the area of an almond
and explains extensively how almonds can be used in finding the surface area and
volume of muqarnas, one of the most characteristic designs for Islamic buildings. See
Figure 5.
Figure 5: at left, muqarnas from al-Nabulsi (1977, 380-381). At right, composition of
interleaved almonds from Ozdural (1995, p. 59).
The third formula is Heron’s formula: the area of a triangle with sides 𝑎, 𝑏,
and 𝑐 is given by �𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑑), where 𝑠 = (𝑎 + 𝑏 + 𝑐)/2.
This formula allows one to find the area without finding the altitude. Because
of this it was important for practitioners, who, for example, could measure the area of
a field without entering the field to prevent crops from damage.
Al-Kashi gives multiple formulas, and each formula depends on different
known quantities. This gives more flexibility in finding the required result using the
quantities that are known.
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MULTIPLE SOLUTIONS
Chapter IV in Treatise V is devoted to examples. Section I contains 25 general
word problems, Section II discusses seven word problems in inheritance solved
algebraically with extensive fraction arithmetic, and Section III is devoted to solving
word problems geometrically.
The sixth problem in Section I is:
A piece of jewellery is made from gold and pearl. Its weight is three methqals 3 and its
price is twenty-four dinars. The price of one methqal of gold is five dinars, and of
pearl is fifteen dinars. We want to know the weight of each kind.
The Key to Arithmetic introduces four different methods to solve this problem.
In modern symbols these become
By algebra: let 𝑥 be the gold weight. Then the pearl weight becomes 3 − 𝑥.
The pearl price = 15(3 − 𝑥 ) = 45 − 15𝑥
The jewellery price = 45 − 15𝑥 + 5𝑥 = 45 − 10𝑥 = 24
We get 𝑥 = 2.1 methqals, the gold weight. Then the pearl weight is 0.9 methqal.
In this first approach, an algebraic equation is formulated with the gold weight as
unknown, and then it is solved by a standard use of algebra.
Chapter III in Treatise V consists of fifty rules for ratios, geometry, algebra,
number theory, and other topics. One of al-Kashi’s methods to solve word problems
is using these rules; he refers to this as ‘maftoohat’ (‫)ﻣﻔﺘﻮﺣﺎﺕ‬.
By maftoohat: the gold weight is (translating the original Arabic words to
mathematical symbols)
3 × 15 − 24 21
=
= 2.1
10
15 − 5
In this second approach al-Kashi considers the jewellery as made entirely from pearl;
then the price is increased by 21 dinars, and then he divides this increase by the
difference between the two prices, to get the gold weight.
Another way: the pearl weight is
9
24 − 3 × 5
=
= 0.9
10
15 − 5
Then the gold weight is 2.1 methqals.
Similarly, in this third approach, al-Kashi supposes that the jewellery is made entirely
from gold, and he proceeds as in the second way.
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By geometry 4: We represent the problem as in the following picture.
Figure 6: Geometric illustration from al-Nabulsi (1977, p. 500).
Let ���
𝑎𝑏 = 3 be the sum of the gold weight 𝑎𝑐
��� and the pearl weight ���
𝑐𝑏 . We want to
���
find the length of 𝑎𝑐
��� and 𝑐𝑏, such that
��� × 5 dinars + ���
𝑎𝑐
𝑐𝑏 × 15 dinars = 24 dinars.
Let ���
𝑐𝑑 = 15 and 𝑐𝑔
��� = 5, then 𝑓𝑔 = 𝑎𝑒 − (𝑐𝑒 + 𝑎𝑔), but 𝑎𝑒 = ���
𝑎𝑏 × ���
𝑐𝑑 = 3 × 15 =
���
45. Furthermore, 𝑐𝑒 = 𝑐𝑏 × 15 and 𝑎𝑔 = 𝑎𝑐
��� × 5 , then 𝑐𝑒 + 𝑎𝑔 = 24 dinars.
So, 𝑓𝑔 = 𝑎𝑒 − (𝑐𝑒 + 𝑎𝑔) = 45 − 24 = 21, then
𝑓𝑔 21
�� =
��� = ��
𝑎𝑐
𝑓𝑑
=
= 2.1 methqals
����
𝑑𝑔 10
This last approach presents a way to think geometrically about this problem, by
considering weight and price magnitudes represented by sides of a rectangle, and the
product of these magnitudes is the rectangle area.
DISCUSSION
In March 2010, I used some excerpts from al-Kashi in my Pre-Calculus class of 39
students. The goal was to prove the Law of Sines, following al-Kashi’s explanation of
different ideas for finding the area of a triangle, altitude of a triangle using three
different ways (one geometric and two algebraic), and finding one side or one angle
of a triangle given certain data. The purpose of this pilot experiment was to get data
from students about using historical sources in the classroom. However, I did not pay
attention to the multiple ways of solving a problem using multiple methods. One
question on the questionnaire was “What do you think was most interesting in the
given activity?” Almost one half of students’ answers emphasized proving the Law of
Sines by reading from a historical source. Although the goal of this activity was the
Law of Sines, some students answered “the most helpful thing was the different
way[s] to find the height [of a triangle]”, “different ways to solve the same problem”,
“how al Kashi invented all these different problems and find [found] ways to solve
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the same problem”. These data inspired me to investigate al-Kashi’s pedagogy of
multiple paths.
The materials presented in this paper can be implemented in a classroom in two ways.
The first way, as an ongoing part of my PhD research, is to present al-Kashi’s
pedagogy using his original text. This will allow students to benefit from reading
primary sources and live the experience of discovering how mathematics was
developed in the 15th century. This will “hone students’ verbal and deductive skills
through reading,..., provide practice moving from verbal descriptions of problems to
precise mathematical formulations [see multiple definitions and multiple methods],...,
offer diverse approaches to material which can serve to benefit students with different
learning styles through exposure to multiple approaches... [and] engender cognitive
dissonance (dé paysement) when comparing a historical source with a modern
textbook approach, which to resolve requires an understanding of both the underlying
concepts and use of present-day notation.” (Barnett, Lodder, Pengelley, Pivkina, &
Ranjan, in press)
The second way is to develop or adapt certain topics from modern curricula based on
al-Kashi’s teaching philosophy. The teacher has to be careful when he or she chooses
alternative methods or concepts. This depends on the class level; since at a given
level some ideas might be more clear conceptually or procedurally than others. For
example, introducing the three mentioned different formulae for the area of a triangle
in 4th or 5th grade is not suitable, since students at this level are typically not familiar
with the concept of square root or inscribed circle. In the classroom, the teacher can
promote students’ thinking and draw attention to why different methods work, by
asking, which method would you like to choose? Why did you choose this? What are
the good and bad things about this method? Which method is clear? And then
students will move to a method that can be understood and explained. Providing
multiple paths also meets the needs of diverse learners in a classroom.
I will discuss here some benefits that may be gained by using multiple paths to
mathematics practice as an instructional technique.
Providing flexibility of thinking: While learning one standard way limits one’s
flexibility of thinking and even narrows the extension of the concept, introducing
more than one definition or method encourages students to think and compare these
definitions or methods, check their validity, and explain the various facets and
approaches. For instance, thinking of division as a partition and also as
proportionality of ratios increases the intellectual dimension of the whole concept.
Furthermore, when multiple solutions are presented, students look to the similarities
and differences, and this “may be a critical and fundamental pathway to flexible,
transferable knowledge.”(Rittle-Johnson & Star, 2007, p. 562).
Providing opportunities for comparison: Sharing and comparing solution methods
and multiple formulae is an effective way of enhancing the competency of solving
other problems. In an empirical study, Rittle-Johnson and Star (2007) concluded that
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comparing multiple methods to the same problem facilitates learning, particularly
procedural knowledge. Perhaps, this opportunity of comparison will also increase the
confidence of students.
Improving creativity: Exploring more than one solution to the same problem should
stimulate students to rely on their own understandings rather than on rules; this will
provide students with access to a range of solution strategies that can be useful in
future problem solving encounters. This also may challenge students’ misconception
that there is only one way to solve a certain problem. (Gurung, Chick, & Hayni.
2009, p. 265).
Creating a network of ideas: When students go through different ways to think about
a solution, they will consider the variety of possible ideas, and this can facilitate
connections between the problem posed and different elements of mathematical
knowledge they are familiar with. For instance, solving the Jewellery problem with
three different approaches requires different levels of skill. The first method requires
algebra skills, the second requires ratio skills, and the third requires knowledge in
finding areas. By connecting all these skills and ideas, students will have a broader
understanding for tackling another problem. This may enhance the conceptual
network that helps students solve new problems.
CONCLUSION
Al-Kashi, as an astronomer, put mathematics in the land of applications, in
astronomy, architecture, surveying, merchandising, etc. These fields have different
contexts and different applications and different background data. It was very
important for him as a mathematics teacher to give different points of view, and
multiple methods, to increase the flexibility of the applications to practical
mathematics. I have shown in this article multiple perspectives regarding definitions,
algorithms, formulas, and problem solving techniques. These perspectives may be
able to be used in the modern classroom to motivate students to value the processes
of mathematization, and gain the predilection to look at problems through different
eyes. I propose that looking at these multiple perspectives will help students to
explore patterns, not just memorize formulas, and it will expand their thinking to use
different approaches. In other words, students will have opportunities to study
mathematics as an exploratory, connected subject, rather than as a solid mass of
formulas and laws to be memorized.
REFERENCES
Abdeljaouad, M. (2006). Issues in the history of mathematics teaching in Arab
countries. Paedagogica Historica, 42(4&5), 629-664.
Al-Nabulsi, N. (1977). Jamshid al-Kashi: Miftah al-Hisab [Jamshid al-Kashi: Key to
Arithmetic]. Damascus, Syria: Ministry of High Education.
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Bagheri, M. (1997). A newly found letter of al-Kashi on scientific life in Samarkand.
Historia Mathematica, 24, 241-256.
Barnett, J. H., Lodder, J., Pengelley, D., Pivkina, I., & Ranjan, D. (in press).
Designing Student Projects for Teaching and Learning Discrete Mathematics and
Computer Sciences Via Primary Historical Sources. In Recent Developments in
Introducing a Historical Dimension in Mathematics Education (Refereed, Eds.
Katz, V. & Tzanakis, C.). Washington, D.C.: Mathematical Association of
America.
Freely, J. (2009). Aladdin’s Lamp: How Greek Science Came to Europe Through the
Islamic World. New York: Vintage Books.
Gurung, R. A., Chick, L. N. & Haynie, A. (2009). Exploring Signature Pedagogies:
Approaches to Teaching Disciplinary Habits of Mind. Sterling: Stylus.
Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in
Mathematics, 27, 249-266.
Ozdural, A. (1995). Omar Khayyam, mathematicians, and “conversazioni” with
artisans. Journal of Society of Architectural Historians, 54(1), 54-71.
Riahi, F. (1995). An early iterative method for the determination of sin 10. The
College Mathematics Journal, 26(1), 16-21.
Rittle-Johnson, B. & Star, J. R. (2007). Does comparing solution methods facilitate
conceptual and procedural knowledge? An experimental study on learning to solve
equations. Journal of Educational Psychology, 99(3), 561-574.
Rosenfeld, B. A., Yushkevish, A. P., & Segal, V. S. (1956). Klyuch Arifmetiki;
Traktat ob Okruzhnosti. (The Key to Arithmetic; The Treatise on the
Circumference). Moscow: Gosudarstvennoe Izdatel’stvo Tekhniko-teoreticheskoi
Literatury.
Silver, E. A.,Ghousseini, H.,Gosen, D., Charalambous, C., & Strawhun, B. T.
F. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having
students consider multiple solutions for problems in the mathematics classroom.
Journal of Mathematical Behavior, 24, 287–301.
ACKNOWLEDGEMENT
I wish to thank my PhD professors Patricia Baggett and David Pengelley for their
helpful comments and their continuous support in preparing this manuscript.
NOTES
1
Translation from Arabic to English is by the author.
A kite is a quadrilateral with two distinct pairs of equal adjacent sides.
3
A methqal is a weight unit used in the Islamic world, 1 methqal = 4.61 grams.
4
This solution is in the margin, and it might be al-Kashi’s solution or from someone else.
2
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