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VEDIC ALGORITHMS REVISITED BY MATHEMATICS TEACHERS AND
COMPUTER SCIENCE TEACHERS: A COMPARATIVE STUDY
Yifat Ben-David Kolikant
Department of Science Teaching
Weizmann Institute of Science, Rehovot
Israel 76100
Tel: 972-8-934-2493 Fax: 972-8-934-4115
Sarah Pollack
Department of Science Teaching
Weizmann Institute of Science, Rehovot
Israel 76100
Tel: 972-8-934-2493 Fax: 972-8-934-4115
Nurit Zehavi*
Department of Science Teaching
Weizmann Institute of Science, Rehovot
Israel 76100
Tel: 972-8-934-3156 Fax: 972-8-934-4115
VEDIC ALGORITHMS REVISITED BY MATHEMATICS TEACHERS AND
COMPUTER SCIENCE TEACHERS: A COMPARATIVE STUDY
ABSTRACT
Using examples from ancient Vedic Mathematics, we describe the interaction
between graduate students in Mathematics teaching and Computer Science teaching
as they explored Vedic Mathematics. This subsequently led to a comparative study of
Vedic Mathematics in workshops for a group of computer science teachers and a
group of mathematics teachers. Each group of teachers was exposed to the Vedic
algorithms and discussed (a) the knowledge required to perform the methods, (b) the
relationship between the mathematical justification and the computer program, and (c)
the didactical value. We found that the two groups had a different point of view that
influenced their interests, the knowledge they pursued, and the procedures they
employed as they studied. However, each group gained from the encounter with the
other discipline. Based on the findings, we discuss the epistemological and
pedagogical issues that these two groups of teachers can learn from each other in
regard to algorithmic thinking.
KEYWORDS
Interdisciplinary projects, pedagogical issues, improving classroom teaching
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1. INTRODUCTION
This paper describes a comparative study conducted in order to formulate guidelines
for cross-fertilization among computer science (CS) teachers and mathematics
teachers. We begin with a survey of the literature regarding the relationship between
mathematics and computer sciences, and of previous work done on the benefits of
cooperation between mathematics education and computer science education.
Knuth (1974; 1985) specified the differences between mathematics and computer
science and the different approaches in these two disciplines. Mathematics deals with
theorems, infinite processes, and static relationships, whereas computer science deals
with algorithms, finite constructions, and dynamic relationships. Nevertheless,
mathematical thinking and algorithmic thinking overlap in thinking modes, such as
formula manipulation, representation, reduction to a simpler problem, abstract
reasoning, and generalization. Knuth claimed that computer science and mathematics
have a mutual impact. Computer science has been influencing mathematics in many
ways, such as facilitating complex computations, providing algorithms as new
techniques for mathematical proofs and most importantly, opening a new horizon of
interesting problems. Mathematics has been affecting computer science since
computer science borrowed concepts, theorems, and techniques from mathematics.
Knuth (1974) defined an algorithm as a precisely defined sequence of rules telling
how to produce specified output information from given input information in a finite
number of steps. He claimed that the ability to deal with algorithms is a powerful
general-purpose mental tool, because the attempt to formulate issues such as
algorithm leads to a deepening of the understanding of knowledge.
The possible pedagogical contribution of each subject matter in relation to the other
has been discussed in previous works. First we describe some of the work done about
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the potential contribution of CS to mathematics education. Johnson (2001) reviewed
attempts in using programming in school mathematics, within the past three decades.
According to him, the integration of programming in the curriculum of mathematics
education can provide (1) a new way to view or express mathematical concepts and
relationships, and (2) a new means for solving problems, in other words, becoming
competent in programming interwoven with the development of mathematical
thinking. He analyzed the task of assessing an algorithm’s efficiency according to
mathematical perspective, in order to demonstrate the contribution of studying
programming to enhance the development of mathematical thinking.
Papert (1991) also advocated the use of programming to enhance learning
mathematics. He coined the notion of constructionism, emphasizing that learning is
more effective when students construct a product that is meaningful to them or to their
classmates. Within this framework, Papert suggested that the students could produce
software products on the basis of what they learned in mathematics, such as software
that calculated expressions with fractions. Wilkensky (1995) reported a similar
experience, in which students who were engaged in a programming project while
studying probability developed a better understanding of the basic concepts of
probability.
Wells (1995) discussed another possible contribution of CS to mathematics: the
implementation of ideas and theories of CS in mathematics education such as the use
of the term of black-boxes to simplify the concepts of functions. Black-boxing means
enabling the manipulations of an object only through it’s interface, similarly to the use
of a mathematical function, whereas the implementation of the object is transparent to
the manipulator. Sfard & Leron (1996) drew attention to the potential cultural impact
of integrating programming into mathematics education on regarding socio-
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mathematical norms. They claimed that programming norms could help in improving
the socio-mathematical norms to be more reality-adapted, for example, to establish
the perception that difficult questions are not solved in one shot, but rather by a chain
of guessing and conjecturing interlaced with meticulous testing and debugging.
The contribution of mathematics to CS education and later to the quality of software
engineers has also been intensively discussed. For example, Hoare (1989) claimed
that mathematical knowledge is a crucial component of the professional
programmer’s knowledge. In fact he perceived programmers as the “engineers of
mathematics.” Recently, a group of CS scholars has warned repeatedly from what
they refer to as mathematics phobia in the CS curriculum, which damages the quality
of future programmers. For a more detailed review of the relationship between
mathematical knowledge and CS studies, see Henderson et al. (2001) and the CUPM
website.
Ben-David Kolikant and Pollack (2002) described an instructional approach to
enhance the mathematical orientation in programming skills among beginning
Computer Science students, and demonstrated its application. They contend that
mathematics has three roles in CS education: (a) a knowledge domain of problems:
the understanding and solving of many problems require an adequate background in
mathematics, (b) tools and skills that facilitate the process of solution; these tools can
be provided to the student when viewing a problem from a mathematical perspective,
and (c) an anchor to concepts in CS whose association with mathematics can
contribute to the clarification of their meaning. A different approach for enabling the
cross-fertilization between mathematics education and CS education was proposed by
Gal-Ezer & Lichtenstein (1997). They discussed the possibilities of removing the
artificial boundaries between the two disciplines, which they claim exist in formal
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education, by using topics that fit both CS curricula and mathematics curricula, such
as set theory, as well as encouraging an algorithmic approach.
The motivation for the comparative study reported in this paper stems from the
experience gained in exploring ancient Vedic algorithms in a graduate course
“Integration of topics in mathematics through a Computer Algebra System (CAS)”.
The course, given by N. Zehavi (the last author), was held at the Weizmann Institute
of Science. The participants were graduate students in mathematics teaching or CS
teaching. One of the topics in the course was concerned with investigating several
methods of Vedic mathematics (see Section 2 for a historical perspective of the Vedic
Mathematics and Section 4 for a description of two Vedic mathematics methods). To
this end, the students used a computer-algebra programming environment (Section 3)
to develop and implement algorithms that simulate the Vedic methods. We noticed
that students from these two groups approached the problems differently and each
group favored different features and topics. However, these differences did not
prevent them from sharing and exchanging ideas, but rather had the effect of
stimulating them in a fertile vein of interesting new problems and solution techniques
(Section 5).
This experience motivated us to explore the perspectives that teachers of both
disciplines possess. The significance of exploring the teachers’ communities is clear
due to the their pedagogical role as formulators of socio-professional norms. We
conducted similar workshops for each of the groups. Specifically, we introduced the
teachers to two methods of Vedic Mathematics and their implementation using CAS
(see Section 6). The teachers were engaged in using these methods and in proving
their correctness, namely, they dealt with mathematical problems of an explicit
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algorithmic type. In each group we initiated a discussion about the methods and their
pedagogical value in order to explore the perspectives of each group.
The findings clearly indicate that the two groups have a different point of view, which
was demonstrated by different answers to the same questions, and different
approaches to the same assignments. The findings are described in Section 7. Finally,
Section 8 discusses the possible cooperation between the groups as well as the
benefits of sharing experiences.
2. WHAT IS VEDIC MATHEMATICS?
To obtain a historical perspective on Vedic Mathematics, briefly we discuss the
mathematical developments in India (Boyer, 1968; Katz, 1992). Archeological
excavations documented an old and highly cultured civilization in India during the
third millenium B.C.E., but no Indian mathematical documents where found from that
period. India like Egypt had its geometrical measurement in the form of a body of
knowledge known as the Sulvasutras (“rules of the cords”). The word sutra (“thread
of knowledge”) means rules expressed by aphorisms relating to rituals or science.
This primitive account, dating perhaps before the time of Pythagoras (6th century
B.C.E.), contained rules for the construction of right angles by means of triple cords.
The period of the Sulvasutras was followed by the age of the Siddhantas (“systems of
astronomy”) starting around 500 C.E., which contributed to trigonometry the notion
of the sine, namely the ratio of half a chord of a circle and half the corresponding
central angle. The trigonometry of the sine function is one of the noted contributions
of India to modern mathematics. Another marked development is our system of
numeration for integers, called the Hindu-Arabic system (about 700 C. E.). The use of
the zero symbol in India existed from the ninth century. Other developments from the
same period were indeterminate analysis (Brahmagupta) and algebraic techniques. In
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the first centuries of the second millenium spherical trigonometry and Pell equations
(Bhaskara) were developed. The discovery of power series for trigonometric functions
took place around 1500. Hindu mathematians were inclined to further develop topics
in number theory and indeterminate analysis in particular. However, these aspects did
not contribute to later developments in modern mathematics such as analytic
geometry, calculus, and algebra.
The term Vedic Mathematics refers to a set of sixteen mathematical sutras and their
corollaries derived from the Vedas. The Vedas are ancient texts written in Sanskrit;
the word Veda means “knowledge” – knowledge both within and among the senses. It
is conjectured that the vedic mathematical system was part of the Sulvasutras. This is
a system of calculations based on easy-to-follow rules and principles that can be used
effectively to solve problems in arithmetic, algebra, geometry, and trigonometry. The
Vedic system was rediscovered between 1911 and 1918 by Sri Swami Bharati Krisna
Tirthaji and has been re-structured for use in schools. It is being taught in some
schools in several countries as well as being used for scientific applications (see
references to Web sites).
3. WHY CAS?
Johnson (2001) suggested that one of the major obstacles in integrating programming
into the mathematics curriculum is the choice of the programming environment, since
learning to program is not a trivial task. However, he claimed that recently developed
computerized environments, such as the Spreadsheet and the Computer Algebra
System (CAS) are easy to manipulate and master.
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From an algorithmic point of view, the Vedic rules are sets of instructions that solve
problems, that is, algorithms. Therefore, Computer Algebra Systems, having both
mathematical and programming power, are suitable for revisiting the ancient Vedic
mathematics. The CAS technology can perform symbolic mathematics calculations.
We used Derive, which is based on the declarative paradigm of programming
(functional programming), a set of pre-defined functions, and programming tools.
There are several pedagogical advantages of using such an environment with both
mathematics and CS students:
- It provides the students with a convenient environment for inquiry of both
mathematical and algorithmic problems, mainly due to the variety of
representations as well as the built-in functions.
- Using the symbolic mechanism, it is easy to verify the correctness because it
provides immediate feedback and powerful tools for simplifying mathematical
expressions.
- The environment enables introducing students to programming. For example, the
pre-defined functions can reinforce explanations of re-use and encapsulation
(black-boxes), which are fundamental concepts of programming.
- Numbers are represented as a list of digits by the software, and therefore big
numbers can be manipulated easily, using many pre-defined functions such as
DIM(S), which counts S’ digits, ELEMENT(S, N), which returns the Nth digit of
S, and INSERT (X, S, N), which inserts the digit X in position N into number S.
Thus, we concluded that CAS software provides a suitable environment for both
communities of CS and mathematics practitioners.
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4. TWO VEDIC METHODS
Several Vedic methods were studied during the course. Here we describe two
methods, each of which is used for multiplication of two natural numbers. The
workshop described later included the use of these methods. For each method we
describe its goal, rules of use, correctness justification, the matching algorithmic
problem, and its solution. The description of both methods uses the symbol ‘|’ to
separate the prefix and the suffix of a number. The term length of a number indicates
the number of its digits.
Method 1
Multiplication of two numbers x (the multiplicand) and y (the multiplier), where the
multiplier’s digits consist entirely of nines; using the Ekanyunena Purvena sutra,
which means: “By One Less Than the One Before”.
According to this method, the result of x*y comprises two parts: a prefix and a suffix.
Case I: if the number of digits of x (x’s length) is greater or equal to the number of
digits of y (y’s length), then:
The prefix of x*y is x - 1 and the suffix is y - (x - 1).
Examples:
1. 7*9 7 – 1 | 9 – (7 – 1) 6 | 3 (meaning 63)
2. 777*999 777 – 1 | 999 – (777 – 1) 776 | 223
Case II: If x’s length is smaller than y’s length, then perform the following
procedure:
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1. Divide x with ‘|’ into two parts, where:
- The right part (x2) contains the same number of digits as y
- The left part (x1) contains the remaining digits of x.
2. Calculate the prefix as the difference x – (x1 + 1).
3. Compute the suffix simply by the difference y – (x2 – 1).
Example:
3. 15639*999 15 | 639*999 15639 – (15 + 1) | 999 – (639 –1)
15623 | 361
In both cases the number of digits of the suffix should equal the number of digits
of y, by adding leading zeroes (see example 4 for case I and example 5 for case
II).
4. 98*99 98 – 1 | 99 – (98 – 1) => 97 | 2 97 | 02
5. 5499*99 54 | 99*99 5499 – (54 + 1) | 99 – (99 – 1) 5444 | 1
5444 | 01
The matching algorithmic problem for this method is as follows: write a function that
implements the Vedic method and multiplies two natural numbers x and y, whereas
y’s digits consist entirely of nines.
For a suitable algorithm we used the pre-defined Derive function ‘FLOOR’, in order
to translate the Vedic instruction: “divide the number with ‘|’ to prefix and suffix”.
The function FLOOR(a,b) return the quotient of the operation a/b, for example,
FLOOR(758,100) = 7. The distinction between the two cases in the method, based on
the length of the two multipliers, was translated to the equivalent two cases, x y and
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1. If x y
1.1 then compute:
Prefix = x – 1
Suffix = y – ( x – 1)
1.2 Else { if x>y} compute:
x1 = FLOOR(x, y + 1)
x2 = x – FLOOR(x , y + 1) * (y + 1)
Prefix = x – (x1 + 1)
Suffix = y – (x2 – 1)
2. If the number of digits of the suffix is smaller than the number of digits of y, add leading
zeroes.
3. Compose the two parts (prefix and suffix) into one number i.e. compute:
Prefix * (y + 1) + Suffix
x > y. The cases are equivalent because y’s digits consist entirely of nines, and
therefore any number whose number of digits is equal, smaller, or greater than y
would be smaller, equal or greater than y, respectively. Figure 1 presents the
algorithm for this method.
---Insert figure 1 here---
In order to prove the correctness of the method, we show that in both cases, x y and
x > y, the application of the Vedic method yields x*y. The scheme of the proof for
case I and case II are illustrated in Figure 2 and 3, receptively.
Case I: x y. The first line in Figure 2 presents the result of applying the Vedic
method on x and y, according to case I. The result is a concatenation of the prefix x -
1, and the suffix y - (x-1). The composition of the number is determined by
multiplying the prefix, with the adequate power of ten, y + 1, and adding the suffix, as
in the second line. The third line is a simplification of the expression obtained in the
second line, which finally yields x*y.
--insert figure 2 here ---
Case II: x > y The first and the second line in Figure 3 present the calculation of x 1,
and x2, according to the Vedic method. The third line demonstrates the composition of
the number by multiplying the prefix with the suitable power of ten, y + 1, then
adding the suffix. The rest of the lines represent the process of simplifying the
expression of the number, finally yielding x*y.
---insert figure 3 here ---
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The program is presented in Figure 4.
---insert figure 4 here ---
Method 2
Multiplication of two numbers x (the multiplicand) and y (the multiplier), using
the Urdhva-Tiryagbhyam Sutra, which means: “Vertically and Cross-Wise”.
The Vedic instructions are represented visually. The method works for numbers of
any length. Here we demonstrate two cases: the multiplication of two-digit numbers
illustrated in Figure 4 and the multiplication of three-digit numbers illustrated in
Figure 5.
Case I: The multiplication of two-digit numbers ac*bd:
---insert figure 5 here ---
1. Compute the intermediate products: a*c | a*d + c*d | b*d.
2. Deal with carry over if necessary.
Examples:
1. 12*32 =384
In the first number (the multiplicand) a = 1, b = 2 and in the second number (the
multiplier) c = 3, d = 2.
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The intermediate products: 1*3 | 1*2 + 2*3 | 2*2 3 | 8 | 4 and since there is no carry
over this is the final result.
Examples:
2. 82*57 = 4674
In the first number (the multiplicand) a = 8, b = 2 and in the second number (the
multiplier) c = 5, d = 7.
The computation of the intermediate products: 5*8 | 8*7 + 2*5 | 2*7 40 | 66 |14.
Taking care of the carry over, we get the final answer: 40 | 66 + 1 | 4 40 +6 | 7 | 4
46 | 7 | 4.
Case II: The multiplication of three-digit numbers abc*def:
---insert figure 6 here ---
1. Compute the intermediate products:
a*d | a*e + b*d | a*f + b*e + c*d | b*f + c*e | c*f.
2. Deal with carry over if necessary.
Examples:
3. 194*285 = 55290
1*2| 1*8 + 9*2 | 1*5 + 9*8 + 4*2 | 9*5 + 8*4 | 4*5 →
2 | 26 | 85 | 77 |20 → 2 | 26 | 85 | 77 +2 | 0 → 2 | 26 | 85 + 7 | 9 | 0 →
2 | 26 + 9 | 2 | 9 | 0 → 2 + 3 | 5 | 2 | 9 | 0 → 5 | 5 | 2 | 9 | 0
In case that the lengths of the two multiplication numbers are not the same, we
should add zeroes at the beginning of the shortest number.
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The development of a suitable algorithm: The main difficulty is in forming a
general rule for any two numbers, based on the cases for two-digit numbers and
three-digit numbers, as presented above. The mathematical justification for the
Vedic rules using polynomial representation of the numbers, where x = 10, can be
a big help:
(ax + b)*(cx + d) = acx2 + (ad + bc)x + bd
(ax2 + bx + c) *(dx2 + ex + f) =
adx4 + (ae + bd)x3 + (af + be + cd) x2 + (bf + ce)x + cf
Figure 7 presents the intermediate products (obtained in the first step of the Vedic
method Vertically and Cross-Wise) according to the digits of the resulting number
formed by these products.
---insert figure 7 here ---
The relation between the digits’ position and the power values in the polynomial
representation led to a general representation of the multiplication: every digit Z, in
position k of the resulting number, is calculated by summing the multiplications of
pairs of digits Xi and Yj, whose positions i and j are summed up to k, as in the
following formula: Zk = Xi*Yj i+j = k. In the following algorithm (Figure 8), m and
n are the length of x and y, respectively. The number of digits in the result is the sum
of the lengths of the two numbers m + n. However, since digits were referred
according to their degree in the polynomial representation, the less significant digit is
in degree 0, whereas the most significant digit in the result number is in degree m+n-
1. The loop boundaries were set accordingly.
---insert figure 8 here ---
The program is presented in Figure 9.
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---insert figure 9 here ---
Note: The two Derive programs can be downloaded from
http://stwww.weizmann.ac.il/g-math/mathcomp/vedic
4. THE PILOT OBSERVATION
We noticed fundamental differences between the approaches to the algorithms of
Vedic mathematics among CS teaching students and mathematics-teaching students,
as they worked on the Vedic methods during the course. We will highlight the main
differences:
The students’ immediate curiosity when they were introduced to the Vedic
Mathematics was different. The mathematics teaching students’ first questions were
about the mathematical history context—they asked about the mathematical
knowledge the ancient Indians had: did they base the algorithms on the decimal place
value system? Did they have the zero number? They were interested in the
development of mathematics in India, and in particular they were curious to know
how the algorithms were invented. At the same time, the CS teaching students were
very much interested in the future. These students discussed the possible contribution
of the implementation of these methods to the solutions of other problems, such as the
multiplication of big numbers.
We also noticed that the two groups of students approached the activity of
constructing the computer program differently. For example, CS students
continuously tested numerical examples, in order to verify the correctness of the
program. In contrast, the mathematics teaching students deduced correctness from the
symbolic manipulation of (x - 1)*(y+1) + (y - (x -1)), which simplifies to xy.
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However, we noticed a mutual appreciation between the CS-teaching students and the
mathematics-teaching students. The CS-teaching students found the mathematical
inquiry to be productive since it was used to improve the efficiency of the algorithms.
The mathematics-teaching students appreciated the ability of the CS-teaching students
to “talk” with the computer at a higher level than merely using manipulations as a
way to provide answers. Eventually, they participated in a programming tutorial
session that the CS-teaching students held.
5. A COMPARATIVE STUDY
The aim of the comparative study was to identify veins for cross-fertilization among
CS teachers and mathematics teachers. To this end we conducted two identical
workshops, one to a group of 30 secondary mathematics teachers, and the second to a
group of 18 CS teachers. Each workshop consisted of the presentation of each of the
two Vedic methods previously described. First, we introduced the teachers to the
method and discussed the Vedic method in comparison with the traditional method
regarding the criteria of the knowledge required to perform the method. Second, we
presented the implementation of the Vedic method as a computer program, explaining
the development of the algorithm, and analyzed the correctness of the method and the
program. We then discussed the importance of the mathematical proof and its
relationship to the algorithm. Finally, we discussed the didactical opportunities in
using these methods.
The discussions were documented. We analyzed the responses of the teachers of the
two groups in order to characterize the differences between the groups.
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6. FINDINGS
For each group we will start by describing the teachers’ general attitude during the
workshop. Then we will present the teachers’ responses to our three questions as
reflected in the discussions: (a) the knowledge required to apply the methods in
comparison with the traditional method, (b) the relationship between the mathematical
correctness proof and the algorithms that implement the methods, and (c) the
didactical potential of the methods and their correctness proof.
The Mathematics teachers
General attitude - The teachers were impressed from both methods and appreciated
the beauty and the simplicity of the application “in the head”.
Knowledge required (for method 1) – Most of the teachers, when asked to solve
243*999 in the traditional method, transformed the task to 243*1000 - 243. For them
this is ‘tradition’! In this method one needs to know that multiplying 1000 (or any
multiple of ten) is the result of adding three zeros to the multiplicand, and then one
must master subtraction from zeros. They admitted that the Vedic method requires
less: one only has to master subtraction from 9-digit numbers, which is easier, with no
need to master multiplication in its entirety. The difficulty in using the Vedic method
is the need to remember the rule.
Knowledge required (for method 2) – Multiplication of two three-digit numbers by
pencil and paper requires nine multiplications of one-digit numbers, locating the inter-
mediate results correctly according to the two multiplicands, dealing with the carry
over and then summing up these results, and again dealing with the carry over. In the
Vedic method we still have the same nine multiplications of one-digit numbers, but
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the locations of the results are clearer and carry over is being dealt with only once,
after all the multiplications are over. Thus, the usefulness of the Vedic method stems
from the option of performing the intermediate multiplications freely and the simpler
location of the intermediate results.
The relationship between the justification and the algorithm (in method 1) –
The teachers argued that justifying the method requires algebraic knowledge. Most of
the teachers discussed the algebraic structures of the mathematical justification. Some
of the teachers spontaneously concentrated on justifying each of the methods using
arithmetic tools. For example, the following justification to method 1 simplified the
Vedic expression into the traditional multiplication:
243 * 999 = 243 – 1 | 999 - (243 –1) =
(243 - 1) * 1000 + 999 – 243 + 1=
= 243000 – 1000 + 1000 – 243 = 243 * (1000 - 1) = 243 * 999
A few of the teachers insisted on going in the opposite direction, from the exercise to
the Vedic rule, in an attempt to “explain how the Vedic came to invent the rule”,
where x y:
243 * 999 = 243 * (1000 - 1) = 243 * 1000 – 243 =
243 * 1000 + 1000 - 1000 – 243= (243 - 1) *1000 + 999 + 1 - 243 =
243 – 1 | 999 - (243 - 1)
Similarly, they backtracked an example for case 2, where x > y:
5489 * 99 = 5489 * (100 – 1) = 548900 – 5489 =
548900 – 5400 – 100 + 99 – 89 + 1 =
(5489 – (54 + 1)) * 100 + 99 – (89 – 1) =
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5489 – (54 + 1) | 99 – (89 –1)
The algorithm and the program did not help these teachers in their justification of
the method. They claimed that the execution of the program could reflect its
correctness but still a solid proof needs to be constructed.
The relationship between the justification and the algorithm (in method 2) –
The algorithm that implements method 2 was based on the polynomial
representation of the multiplication of the numbers. This representation was a
satisfactory proof for the mathematics teachers.
The didactical value – In general, most of the mathematics teachers preferred the first
method because it was piquant. They commented on a painful familiar obstacle: what
would the students say regarding the benefit of these methods compared to the simple
use of calculators. The benefit of working with big numbers in a rather convenient
representation of the number (rather than with using the exponent, which is built-in in
calculators) sounded like a good argument to them. Most of the teachers, however,
considered these methods as intellectual amusement, for enrichment and variety. Only
two teachers believed that these methods could be used as curricular activities: one
suggested using the methods as a type of verbal problems for which the students need
to translate the verbal information into algebraic patterns. The other suggested
preparing exercises of symbolic manipulations on patterns. In general the
mathematics teachers agreed that the instruction of these methods fit junior high
students better than high school students, because of their arithmetical characteristics.
One attribute of the algorithm that was appreciated by the mathematics teachers was
the calculation of the length of the result of multiplication according to the length of
the multiplicands. For example, the multiplication of an m-digit number by an n-digit
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numbers will result in a number with m + n digits, or m + n -1. They offered using
this calculation as a verification method of products calculations and suggested
integrating this method into the class activities. However, several teachers claimed
that junior high students might find the polynomial representation, used in method 2,
difficult.
Finally, most of the teachers agreed that relating the history background would
interest the students and motivate them to explore these methods. It will also serve as
an opportunity to reflect on the nature of mathematical development, which most of
mathematics students are totally alien to.
The CS teachers
General attitude - The CS teachers enjoyed the workshop and described it as
stimulating. They appreciated the gimmick of performing different algorithms to
obtain familiar goals.
Knowledge required (for method 1) – The first questions of the CS teachers concerned
the domain of numbers to which this method can be applied, or in their words, the
possible input. These questions were raised immediately after the first example was
given: 7 * 9 = (7 - 1) | (9 - (7 - 1)). The following questions are examples of the
questions of the CS teachers: dose it work for all the one-digit numbers? Must the first
multiplicand be a one-digit number? Does it work for two-digit numbers or is there
another rule? Simultaneously, they started testing the method rules on various
numbers.
Next, the CS teachers dissected the process into the basic operations of addition and
subtraction. They stated that the only difficulty in performing method 1 was
remembering its rules. They found the traditional method, in this case long and
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difficult, because it demanded mastering multiplication and partition as well as
addition and subtraction, and dealing with the carry over. These activities require a
mastery of the multiplication board, a calculation of the position of mid results, and
the mastery of adding numbers including the transmission of the left digits of the mid
results.
Most of the CS teachers used calculators when asked to solve 243*999 traditionally.
After the tutor presented the rule for case II, where x > y, one teacher pointed out that
this rule is suitable for both cases as the numbers can be padded in zeros from their
left. The other teachers approved.
Knowledge required (for method 2) – The CS teachers similarly to the mathematics
teachers recognized that the knowledge required for mastering method 2 included
multiplication and addition operations. However, they also mentioned operations that
are unique to their profession, such as assignment and comparison, and two algebraic
operations, DIV and MOD, that were not discussed by the mathematics group. They
pointed out that the difficulties of using this method are due to the operations of
composition and decomposition of numbers, as well as remembering the method,
although its name “vertically and crosswise” is a big help. They also stated that the
method is inconvenient for applying on large numbers. The traditional method also
required multiplication and addition operations. In addition, it required dealing with
the carry over, and the positions of mid results and shifting. The major difficulty of
using the traditional method stems from the multitude operations that have to be
performed, each relying on another: intermediate multiplication, intermediate
addition, shifting, transmitting, and handling positions of numbers, as well as from the
need to master the multiplication board. When they examined the algorithm, they
22
were concerned about the representation, especially in transforming the result from a
string to number, an issue that naturally was not discussed in the mathematics group.
The relationship between the correctness proof and the algorithm - Six teachers
argued that understanding the mathematical correctness justification contributes to
understanding the dynamics of the algorithm but claimed it was not essential. One
teacher referred to the process of implementing an algorithm and claimed that one
need not understand why it is correct mathematically in order to implement it,
whereas another teacher claimed that an algorithm and a mathematical proof are
basically the same. Most of the teachers naturally assumed the Vedic methods are
correct and showed no propensity to verify them.
All the teachers agreed that developing the algorithms requires an understanding of
the rationale and underlying methods, but not all of them agreed that the rationale is
the mathematical proof, but rather classified mathematical patterns and proofs as one
resource for ideas a developer can use.
The didactical value - Most of the CS teachers preferred the second method because it
was more general than the first method, and declared the first method as nice but
useless. This attitude can be explained by the fact that finding the regularities, and
aspiring for a general solution is a basic activity in the CS class.
All the CS teachers except for one agreed that these methods could be used as
curricular activities for teaching and practicing fundamentals in CS. They suggested
exercises to practice the DIV and MOD operations, to exemplify and practice
representative input examples, and to demonstrate the concepts of complexity and
efficiency of algorithms. Only one teacher suggested teaching these methods as they
are, for enrichment. In addition many CS teachers indicated that Method 2 has the
23
potential to serve as a different but efficient solution to the practical, known problem
of multiplying two long numbers.
7. DISCUSSION
First we will characterize the differences between the approaches of the two groups to
the two methods. Then we will discuss the cross-reference and the potential cross-
fertilization in order to formulate our recommendation for mutual mathematics and
computer science workshops.
Procedural versus declarative point of view - The difference between the groups was
shown even in their responses to the allegedly objective question: “what is the
knowledge required to perform the methods”. The mathematics teachers described the
required knowledge for both methods in terms of other algebraic methods, such as
subtracting from zero, for method 1, and dealing with the carry over for method 2.
They did not specify the basic operations that comprise these methods. In contrast, the
CS teachers specified each of the very basic operations that comprise the process. For
example they detailed the operations that constitute the method of dealing with the
carry over.
Justification techniques – The mathematics teachers justified the correctness of
method 1 by moving backward and forward regarding specific numerical examples.
This justification served two goals: to justify correctness and to satisfy their eagerness
to understand the historic origins. CS teachers worked in the opposite direction. They
mapped the (input) domain into categories of numbers; for each they chose a few
numerical examples and applied the method, only to verify that it ‘works’.
Appreciation of problems – The mathematics teachers characterized these methods as
piquant. They looked for unique cases that they described as ‘beautiful’, and therefore
24
they preferred method 1 over method 2. The CS teachers appreciated the possible
contribution for solving real-world problems, such as multiplication of big numbers
under the constraints of the space limitation of the digital computer. They also
appreciate generality. Therefore, the CS teachers preferred method 2 over method 1,
claiming that there is no point in teaching method 1 because it is useless whereas
method 2 is general and useful.
Integration of the methods in the instruction – Most of the mathematics teachers
group perceived the instruction of the Vedic methods as a valuable enrichment
activity, in addition to the regular syllabus. The CS teachers, in contrast, pointed out
on several issues that can be practiced by using the Vedic methods, based on their
own analysis of the required knowledge.
Mutual impact – The mathematics teachers group was enthusiastic about calculating
the number’s length as a result of arithmetic operations. Some of them decided to
integrate it into their instruction, because they thought that these calculations might
encourage the students to perceive numbers as objects rather than just performing
arithmetic manipulations. The CS-teachers group was excited about the power of
mathematics, as reflected in using method 2 to solve realistic problems. However,
some of them argued that a programmer does not necessarily have to understand the
idea underlying an algorithm, as long as he knows that the algorithm is correct. We
concluded that teachers of both disciplines borrowed ideas from the other to improve
teaching their subject matter.
Recommendations
In this work we learned that exposing two similar yet different methods to CS
teachers and mathematics teachers groups resulted in different responses. We showed
25
that the CS teachers and mathematics teachers groups had different preferences and
areas of interest, and consequently, their pedagogical decisions were essentially
different. Yet the exposure of the teachers to both mathematical and CS knowledge
regarding the Vedic methods, prompted each group to appreciate and acquire bits of
knowledge usually possessed by the other group. We therefore recommend
conducting a mutual workshop in which the teachers from both disciplines will
together investigate and develop algorithms. We believe that there will be enough
interest and that the disagreement that may arise between these two groups because of
their different points of view would contribute to a fertile discourse and to a mutual
enrichment.
Mathematics teachers will benefit from the mental tool of building algorithms (Knuth,
1974). In addition, the encounter with CS teachers can provide them with new images
of familiar objects as had occurred in this workshop (the partition of numbers to
prefix and suffix, calculation of the number length, and the use of a polynomial to
develop an algorithm). This encounter can also result in acquiring more pre-defined
functions to their teaching repertoire, such as FLOOR and DIM.
CS teachers will also benefit from this encounter. An encounter with a community
that tends to perform investigations might influence the CS teachers group to inquire
about the problem, rather than just focusing on the implementation. Consequently,
their solution would become more elegant and more efficient. Finally, the meeting
between practitioners of the old science of mathematics and the young science of CS
can teach the CS teachers to appreciate the historical perspective of mathematical
developments and later to project them toward the perspective of CS development.
An additional advantage of a meeting between CS teachers and mathematics teachers
is the reinforcement of mutual activities in school. For example, teaching topics that
26
fit both curricula. Moreover, a cross-reference attitude may enrich the students with a
more holistic attitude toward problem solving.
Finally, the CAS technology offers a suitable environment for activities of that type,
since it can address the needs of both mathematical inquiry and algorithm
development, and therefore is convenient for both groups.
ACKNOLEDGEMENT
The authors are grateful to Dr. Pushpa Agashe for her remarks regarding the Vedic
Mathematics.
27
REFERENCES
Agrawala, V.S. (1971), Vedic Mathematics, New Delhi: Motilal Banarsidass.
Ben-David Kolikant, Y. & Pollack, S. (2002). Improving mathematically oriented
programming skills. The Conference of Frontiers In Education, Boston 2002, session
T1G, 3-8.
Boyer, H. F. (1968). A History of Mathematics. New York: Wiley.
Gal-Ezer, J. & Lichtenstein, O. (1997). A mathematical-algorithmic approach to sets:
a case study. Mathematics and Computer Education, 31(1), 33-42.
Kats, V. J. (1993). A History of Mathematics. New York: Harper Collins.
Henderson, P., B., Baldwin, D. et al. ( 2001). Striving for mathematical thinking.
ACM SIGCSE Bulletin Inroads, 33(4), 114-124.
Hoare, C. A. R. (1989). Computer science. In C. A. R. Hoare & C.B Jones, Essays in
Computing Science (pp. 89-101). Englewood Cliff, NJ: Prantice-Hall.
Integrating Mathematical Reasoning into Computer Science Curricula, CUPM
Discussion Papers and Interim Reports, (2001). http://www.maa.org/news/cupm.html.
Johnson, D. (2000). Algorithms and programming in the school mathematics
curriculum: support is waning - is there still a case to be made? Education and
Information Technologies, 5 (3), 201 – 214.
Knuth, D. (1974). Computer science and its relation to mathematics. American
Mathematical Monthly, 81(4), 323-343.
Knuth, D. (1985). Algorithmic thinking and mathematical thinking. American
Mathematical Monthly, 92(3), 170-181.
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Papert, S. (1991). Situating Constructionism. In I. Harel & S. Papert (Eds.)
Constructionism (pp. 1-12), Norwood, NJ: Ablex.
Pollack, S., Yeshno, Z., Levi-Rashti, D., & Zehavi, N. (2001). Ancient mathematics
and 20th century technology: programming of Vedic Mathematics using Derive. The
Derive Newsletter, 43, 8-13.
Sfard, A. & Leron, U. (1996). Just give me a computer and I will move the earth:
programming as a catalyst of a cultural revolution in the mathematics classroom.
International Journal of Computers for Mathematical Learning, 1(2), 189-195.
Vedicmaths.org. Retrieved April, 1, 2003 from http://www.vedicmaths.org.
Vedic Mathematics. Retrieved April, 1, 2003 from http://vedmaths.tripod.com/
Wells Charles (1995). Communication mathematics: useful ideas from computer
science, American Mathematical Monthly, 102(5), 397-408.
Wilensky, U. (1995). Paradox, programming and learning probability: a case study in
a connected mathematics framework. Journal of Mathematical Behavior, 14(2), 253-
280.
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VITAE
Yifat Ben-David Kolikant has submitted her doctoral thesis at the Department of
Science Teaching at the Weizmann Institute of Science where she is a member of the
computer science group. She has developed materials for computer-science teachers
and conducted in-service teacher training. Her doctoral research explored the
dynamics of knowledge evolution of high school students who studied concurrent and
distributed computation. Her research interests involve the characterization of
dynamic learning processes both from the cognitive and social aspects in order to
design effective instructional methods.
Sarah Pollack has recently received her Ms.c degree form the Department of
Science Teaching at the Weizmann Institute of Science where she is a member at
the computer science group. She has developed materials for computer-science
teachers and conducted in-service teacher training. In her research she designed
learning materials for project-based learning in computer science and evaluated
these materials.
Dr. Nurit Zehavi has been working on curriculum development and research
in the Science Teaching Department at the Weizmann Institute of Science
since 1972. She has coordinated mathematical software development and
professional training courses for teachers on using computers in the
classroom. Her current research interest is in using computer algebra for
teaching mathematics. She is the head of the MathComp project, which was
initiated in 1996 with the aim of integrating computer algebra systems
into the mathematics curriculum.
30
LIST OF FIGURES
Figure 1: The algorithm for method 1
Figure 2:the correctness proof of the Vedic method “By One Less than the One
Before” for case I
Figure 3: the correctness proof of the Vedic method “By One Less than the One
Before” for case II
Figure 4: The program that implements method 1.
Figure 5: The multiplication of two-digit numbers
Figure 6: The multiplication of three-digit numbers
Figure 7: The components of each digit in the multiplicand x, multiplier y, and the
result x*y
Figure 8: The algorithm for calculating the intermediate products for method 2
Figure 9: The program that implements method 2
31
Figure 1: The algorithm for method 1
32
If x y
then compute:
Prefix = x – 1
Suffix = y – ( x – 1)
Else { if x>y} compute:
x1 = FLOOR(x, y + 1)
x2 = x – FLOOR(x , y + 1) * (y + 1)
Prefix = x – (x1 + 1)
Suffix = y – (x2 – 1)
If the number of digits of the suffix is smaller than the number of digits of y, add leading
zeroes.
Compose the two parts (prefix and suffix) into one number i.e. compute:
Prefix * (y + 1) + Suffix
Figure 2:the correctness proof of the Vedic method “By One Less than the One Before” for case I
33
[x-1, y – (x-1)]
(x – 1) * (y + 1) + y – (x – 1)
=x*y + x – y – 1 + y – x + 1
= x*y
Figure 3: the correctness proof of the Vedic method “By One Less than the One Before”
for case II
34
x1 = FLOOR( x / (y + 1))
x2 = x – x1* (y + 1)
(x – (x1 + 1)) * (y +1)+ y – (x2 – 1) =
(x – (x1 + 1)) * (y + 1) + y – (x – x1* (y + 1) – 1) =
x* (y + 1) – x1* (y+1) – (y+1) + y – x + x1* (y + 1) + 1 =
x * y + x - x1* y - x1 – y – 1 + y – x + x1* y + x1 + 1 = x * y
Figure 4: The program that implements method 1.
35
Figure 5: The multiplication of two-digit numbers
36
a b
c d
Figure 6: The multiplication of three-digit numbers
37
a b c
d e f
4 3 2 1 0Digit
position (k)
a b c x (i)
d e f y (j)
a*d a*e + b*da*f + b*e
+ c*db*f + c*e c*f x*y
Figure 7: The components of each digit in the multiplicand x, multiplier y, and the result x*y
38
for each k between 0 and m+n-1 do
Z[k] 0
i 0
for each i < m do
j 0
for each j < n do
if k = j + i then Z[k] Z[k] + x[i] * y[j]
Figure 8: The algorithm for calculating the intermediate products for method 2
39
Figure 9: The program that implements method 2
40