Nature. Objcct~vesand
Approaclies to Teaching
of Mathematics
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Notes : a) Write your answers in the space given below.
b) Compare your answers with those given at the end of the unit.
10. Give two reasons for the need to change the school curriculum in mathematics.
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11. Mention three important principles that should be borne in mind before deciding the
syllabus for a particular class.
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1.6 CONTRIBUTION OF INDIAN MATHEMATICIANS
The Hindu civilisation had climbed to lofty heights in the knowledge of mathematics but the
path of ascent is not traceable. Much of Hindu mathematics remained merely a servant to
astronomy and it remained chiefly in the hands of the priests who put in verses all the
mathematical results, which therefore became unintelligible to the common man. While the
Greek mind was pre-eminently geometrical, the Hindu mind was arithmetical. Numerical
symbolism, the science of numbers and algebra attained far greater heights in India than they
had previously reached in Greece. Hindu geometry was merely mensuration, unaccompanied
by demonstration.
Since ancient times, Indians have made very significant contributions to mathematics.-The
system of numeration and the concept of zero are a gift of Indians to the world of mathematics.
Mathematicians like Arya Bhatta, Bhaskaracharya and Shridharacharya gave many important
concepts like the value of .rr (pai), the functions, solutions of quadratie equations and many
others to the world. During recent times, mathematicians like Ramanujan and P.L. Bhatnagar
have been shining stars on the horizon of mathematics.
He clearly stated the relationship between the sides of the right angled triangle in one of his
verses. This statement was recognised as Pythagoras Theorem. Mahavira belonged to 9th century
and wrote Ganit Sar-Sangrah. He gave the method of finding LCM, which is used even now.
He also gave the formula for finding the area of a quadrilateral :
The area of a quadrilateral = JS(S - a)(S - b ) ( -~C)(S - d ) where a,b,c,d, are the sides and S is
the semi-perimeter of the quadrilateral.
'This formula is true only for the cyclic quadrilaterals. Mahavira has also given the fo.mula for
calculat~ngthe area of an ellipse. Sridhar Acharya wrote three books - Trisanthika, Patiganit,
and Algebra. He gave a method of solving quadratic equations. He also gave the geometrical
interpretation of algebraic formula. Sripati wrote two books -Siddhanta-Shekhar and GanitTilak. After Sripati, Bhaskaracharya's contribution is notable. A work entitled SiddhantaSiromani was written by Bhaskaracharya (A.D. 1 150). The two important chapters in this bock
are -the Lilavathi (the beautiful) and Vija-ganita (rod-extraction) devoted to arithmetic and
algebra. Lilavathi was the lone daughter of Bhaskara and it is said that Bhaskara decided to
teach his daughter, who became a widow at a very young age, to keep her mentally alert and
occupied. The text is believed to be devoted to her. It contains problems stated in the form of
verses in Devnagari script, on arithmetic, algebra and geometry and are posed to the intelligent.
An example is stated below for the benefit of the reader.
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Nature, Need and Place o f
klathemrtics in
School Curriculum
Out of a swarm of Wees, a number equal to the square root of half'their number went to the
Malati Flowers;
-89 ths of the total number also went to the same place. A male bee enticed by
the fragrance of the lotus got into it. But when it was inside it, night fell, the lotus closed, and
the bee was caught inside. To its buzz, its consort was replying from outside.
What is the number of bees?
For the solution of the problem, we have the equation
He has corrected the concept of zero as laid down by Mahavira and Brahmagupta.
Pbst-Medieval Period (A.D. 1200 to A.D. 1800)
In this period, very little original work was done. There were foreign invasions in North India;
therefore mathematical development was checked in this period. At this time, South India was
considered as the centre of learning science and mathematics. During this period, Kamlakar
(A.D. 1600) wrote Siddhaata-Tatva.Vivek Narain wrote Ganit-Kaumudyand Neel Kantha (in
A.D. 1587) wrote Tajik Neel Kanthia. Pandit Jagannath (A.D. 1731) translated the works of
Ptblemy and Euclid into Sanskrit. This book is known as Samrat-Siddhanta. The present
terminology of geometry is mostly taken from this book.
Those who wish to know more about the Indian contibutions to mathematics in early periods
may consult the following source books :
'
i)
ii)
The History of Hindu Mathematics by A.D. Singh and V.P. Dutta.
The History ofAncient Indian Mathematics by C.N. Srinivasaiyengar (World Press Pvt.
Ltd., Calcutta, 1967). .
Ancient Period (3000 B.C. To 500 B.C.)
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Ancient period may be further sub-divided into three main periods :
i)
Vedic period (3000 B.C. to 1000 B.C.) : In this period, the number system and the
decimal system were invented. During this period, various interpretations were derived
from Vedas and put.into the form of Samhitas (dhl).
At one place in the thirteenth
Samhita (6/2,4,5) they have given the following relationship :
39' = 36' + 25'
This is a particular case of Pythagoras theorem.
ii)
Sulva period (I000 B.C. to 500 B.C.) : The knowledge of geometry was one of the
important features of this period. It was the belief of Aryans that yajna is the only means
for the attainment of supreme life. Yajna was performed at different geometrical altars.
The yajna was useful only when performed during the specified period at the sp~cified
altar. This was not possible without the knowledge of mathe1na:ics. The altar requires
the knowledge of geometry. Different geometrical theorc~nsarc ..t: :ed in Shlva Sutra.
~ u l v was
a considered as the rule of cord used in the construction of ~ltars.The knowledge
ofPythagoras theorem was also prevalent during this period. T h ~ IS
s stated by Baudhayan
(1000 B.C.).
iii)
Vedanta period (1000 B.C. to 500 B.C.) :This period is famous for astronomical work.
Astronomy, as a branch of mathematics, studies the positions and movements of st :rs
and planets.
Early Period (500 B.C. to A. D. 500)
Only Jain religion Grantha Bookand some pages of Bokl~shaliGunit of this period are available
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Nature, Objectives and
Approaches to Teaching
of Mathematics
at
a)
b)
c)
d)
e)
resent. The following" inventions were made in this ~ e r i o d:
Place value system of writing numbers.
Invention of zero.
Introduction of algebra.
~ e v k l o ~ m eof
n t arithmetic (Bakhshali Ganit).
Development of astronomy and Surya Siddhanta.
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In the sixth century, Varaha Mihira Acharya wrote his Pancha Siddhaktika which gives a
summary of Surya Siddhanta and four other astronomical works, then in use. The concepts of
Interest and Per cent were also known in this period.
Medieval Period (A.D. 500 to A.D. 1200)
This period starts from Aryabhata and continues up to Bhaskar 11. Aryabhata has consolidated
the rules of mathematics into 33 verses in his book Aryabhatiya.
62832
Aryabhata is considered a pioneer in the field of algebra. He gave the value of x as 20,000 which
is correct upto fourth decimal. He stated that the earth is moving while the stars are stationary.
This is described in "Aryabhatiya Goal-pad".
In the 5th century an anonymous Hindu astroaomical work was known as Surya Siddhanta'
&hich is regarded as a standard work. Brahmagupta wrote Brahma-Szrtra-Siddhanta - the
revised system of Brahma. It is believed that it is Brahmagupta who gave zero its status. He has
given a colicept of infinity in this book. According to him, if a positive or a negative number is
fT;rm *)
He also gave
divided by zero the quotient will be infinity. He wrote (mi
the formulae for finding out the volumes of the prism, cone and pyramid.
Volume of pyramid =
H
7
[A + A'
I?&+
Where H i s the height, A is the area of base and A' is the &ea of other surfaces.
I
He also gave the formula to find the sum of geometric series. If a is the first term, r is the
common ratio and n is the number of terms, then
a(rn-1)
....n terms'. = r-1
He also stated the relationship between the two intersecting chords of a circle. ifAB and CD are
two chords intersecting at E, the mid-point of CD, then
4 AE x BE = ( C D ~
This reiatibnship i s also described in the Euclidean Geometry. Only the chapter of Kuttaka
(Indeterminate equation) of Brahma-Sutra Siddhanta is available and due to this method algebra
was known as Kuttaka Ganit.
(Reproduced from 'Content-Cum-Methodology of Teaching Mathemothics', published by N.C.E.R.T.)
1.7 LET US SUM UP
Any consideration of the nature of mathematics leads to finding out answers to two questions:
i) What is the nature of mathematical proof; and ii) What is the nature of mathematical truth?
The first question is for psychologists to answer and the second for philosophers. This unit
discusses a few elementary ideas about these two questions. The nature of arguments, types of
proofs, symbolism and logic are ideas related to the first question. The utility of mathematics,
axiomatic structure, undefined and defined notians, and the nature of mathematical laws are
ideas related tb the second question:
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The unit also provides an insight into the value and importance of mathematics. The tr?&tional
approach of classifying values as social, disciplinarian, cultura1,'voc~ionaland pleasure has
been modified to reflect a broader perspective. The elements of courses of study (curriculum)
has been mentioned. These include modern views on determinants of the course content in
20
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