Teaching Mathematics: Methodology · Chapter 1

Meaning and History of Mathematics: Great Contributors

What to remember

Mathematics is the science of number, quantity, shape and pattern. It is built by logical reasoning from definitions and axioms, and it trains the mind in precise thinking.
Indian contributors: Aryabhata (Aryabhatiya; place value, sine table, value of pi), Bhaskaracharya (Lilavati, Bijaganita, Siddhanta Shiromani) and Ramanujan (number theory, infinite series, partitions).
World contributors: Euclid (Elements; axiomatic geometry), Pythagoras (right-triangle theorem) and Cantor (set theory; sizes of infinity).

Meaning and nature of mathematics

The word "mathematics" comes from a Greek word that means "learning" or "knowledge". Mathematics is often called the language of science and the queen of the sciences.

Main features:

Exact: answers are precise; a statement is either proved or not proved.
Logical and abstract: it moves from simple ideas to complex ones using reasoning.
Pattern-based: it studies relations in numbers, shapes, change and chance.
Cumulative: each topic stands on earlier topics, so a gap in the early classes hurts later learning.
Useful and creative: it is used in daily life, trade, science and technology, and also has beauty and play.

Aims and values of teaching mathematics

Value	Meaning
Utilitarian (practical)	Skills for shopping, measurement, banking, time and daily life
Disciplinary	Trains reasoning, concentration, accuracy and logical thinking
Cultural	Shows how human civilisation developed mathematical ideas
Social	Needed for trade, planning, statistics and social life
Aesthetic	Beauty in patterns, symmetry and neat proofs
Moral	Honesty and accuracy built through the habit of checking

The National Curriculum Framework stresses that mathematics learning should be child-centred and related to the child's life, with problem solving, reasoning and communication as the main goals. A teacher should use history as a tool: stories of mathematicians make the subject human and motivating.

Brief history of ideas

Numbers and counting: early people counted using fingers, pebbles and tally marks.
Indian place-value system: the decimal system with a symbol for zero grew in India. It travelled to the Arab world and then to Europe, which is why the digits 0 to 9 are called the Hindu-Arabic numerals.
Early Indian geometry: the Sulba Sutras (Baudhayana, Apastamba, Katyayana) gave rules for building altars and stated the rule now called the Pythagoras theorem.
Greek mathematics: Greeks made geometry a proof-based science. Thales, Pythagoras and Euclid are leading names.
Classical Indian period: Aryabhata, Brahmagupta and Bhaskaracharya developed arithmetic, algebra and astronomy.
Modern period: coordinate geometry, calculus, set theory and computing grew in Europe and across the world. Ramanujan, from India, added deep results in number theory.

Aryabhata

Indian mathematician and astronomer of the classical period, born c. 476 CE. He is linked with Kusumapura (near present Patna).
His work Aryabhatiya (c. 499 CE) is in verse and has four parts: Gitikapada, Ganitapada (mathematics), Kalakriyapada (time and its reckoning) and Golapada (the sphere and astronomy).
Contributions:
Gave a clear place-value method and a way to extract square and cube roots.
Gave an approximate value of pi as 3.1416 (62832/20000) and said it is "approximate".
Gave a sine table (called jya) and the idea of versine.
Gave formulas for the sum of series such as the sum of the first n natural numbers, their squares and cubes.
Gave the area of a triangle as half the base times the height.
Studied linear indeterminate equations (the "kuttaka", the pulverizer method).
In astronomy, said that the earth rotates on its axis and that day and night are caused by this rotation. He explained that eclipses are caused by shadows of the earth and the moon, not by demons.
India's first satellite, launched in 1975, was named Aryabhata in his honour.

Bhaskaracharya (Bhaskara II)

Indian mathematician and astronomer, born in 1114 CE in a place in the Karnataka region. He worked at the astronomical centre of Ujjain.
Main work: Siddhanta Shiromani (c. 1150 CE) in four parts:

1. Lilavati: arithmetic and mensuration.

2. Bijaganita: algebra.

3. Grahaganita: mathematics of planets.

4. Goladhyaya: the sphere and astronomy.

Contributions:
Treated zero and infinity in operations. He said a number divided by zero gives the "khahara" (infinite quantity) which does not change when numbers are added or taken from it.
Gave solutions to quadratic and indeterminate equations, including the Chakravala (cyclic) method for equations of the Pell type.
Worked on early ideas that lead to calculus, such as instantaneous motion of a planet.
Described rules for permutations and combinations and for surds.
Tradition links the name Lilavati with his daughter. This is a story, not a proved fact.

Ramanujan (Srinivasa Ramanujan)

Born on 22 December 1887 at Erode, Tamil Nadu. He grew up in Kumbakonam. He died on 26 April 1920, at the age of 32.
He had little formal training in advanced mathematics but made large discoveries by his own work. He wrote his results in notebooks.
He wrote to G. H. Hardy of Trinity College, Cambridge, who invited him to England. Together they worked on number theory.
He became a Fellow of the Royal Society in 1918 and was also elected a Fellow of Trinity College.
Contributions:
Partition of integers and the Hardy-Ramanujan asymptotic formula.
Infinite series for pi that are the basis of modern fast computation.
Continued fractions, magic squares, highly composite numbers, mock theta functions.
The Ramanujan number 1729, the smallest number that is the sum of two cubes in two ways: 1729 = 1³ + 12³ = 9³ + 10³. It is also called the taxicab number (Hardy's story of a taxi).
National Mathematics Day is celebrated in India on 22 December, his birthday. It was declared in 2012, the 125th birth anniversary year.
A film "The Man Who Knew Infinity" is based on his life.

Euclid

Greek mathematician of Alexandria, c. 300 BCE. He is called the Father of Geometry.
His work Elements has 13 books (chapters). It starts from a few definitions, axioms (common notions) and postulates, and builds geometry by proofs.
Five postulates (first group):

1. A straight line may be drawn from any point to any other point.

2. A terminated line can be produced indefinitely.

3. A circle can be drawn with any centre and any radius.

4. All right angles are equal to one another.

5. If a line meets two lines so that the interior angles on one side are together less than two right angles, the two lines meet on that side when produced (the parallel postulate).

Axioms (common notions) include: things equal to the same thing are equal to one another; if equals are added to equals the wholes are equal; the whole is greater than the part.
Euclid's division lemma: for positive integers a and b there are unique q and r with a = bq + r, 0 ≤ r < b. The Euclid's division algorithm finds the HCF. Example: HCF(18, 12): 18 = 12 × 1 + 6; 12 = 6 × 2 + 0, so HCF = 6.
He proved that there are infinitely many prime numbers.
The fifth postulate led later to non-Euclidean geometries.

Pythagoras

Greek philosopher and mathematician, c. 570 to 495 BCE, born on the island of Samos. He later set up a school in southern Italy.
Pythagoras theorem: in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: c² = a² + b². Sets such as (3, 4, 5) and (5, 12, 13) are Pythagorean triples.
His followers (Pythagoreans) held that "all is number" and studied numbers, music and geometry. They also discovered that √2 cannot be written as a ratio of whole numbers (an irrational number).
The rule was known in India earlier through the Sulba Sutras. This does not reduce the value of the Greek proof.

Georg Cantor

German mathematician, born in St Petersburg, Russia (1845), died 1918. He worked at the University of Halle.
He is the founder of set theory.
Contributions:
Defined a set as a well-defined collection and developed one-to-one correspondence to compare sizes of sets.
Showed that some infinite sets are larger than others: the set of natural numbers is countable, but the set of real numbers is uncountable (the diagonal argument).
Introduced cardinal numbers and the symbol ℵ₀ (aleph-null) for the size of the natural numbers.
The Cantor set is a famous example of a set built by repeated removal of middle thirds.
His ideas met strong opposition at first but later became the base of modern mathematics.

Comparison table

Mathematician	Place / period	Famous work or idea
Aryabhata	India, c. 5th century CE	Aryabhatiya; pi = 3.1416; rotation of earth
Bhaskaracharya	India, 12th century CE	Siddhanta Shiromani; Lilavati; Bijaganita
Ramanujan	India, 19th to 20th century	1729; partitions; infinite series for pi
Euclid	Alexandria, c. 300 BCE	Elements; postulates; division lemma
Pythagoras	Greece, 6th century BCE	Right-triangle theorem
Cantor	Germany, 19th century	Set theory; countable and uncountable sets

Exam traps

Aryabhata wrote the Aryabhatiya; Bhaskaracharya wrote Lilavati and Bijaganita. Do not swap them.
Ramanujan was born in Erode but lived in Kumbakonam; his number 1729 is a sum of two cubes, not two squares.
National Mathematics Day is 22 December (Ramanujan), not 5 September (Teachers' Day).
The Pythagoras theorem applies to right-angled triangles only.
Euclid's fifth postulate is the parallel postulate, not "all right angles are equal" (that is the fourth).
Euclid's division algorithm finds the HCF, not the LCM directly.
Cantor is linked to set theory, not to geometry or algebra.
Aryabhata's pi is 3.1416, not 22/7 (22/7 is an approximation used in school).

One-liners

1. Mathematics is called the language of science and the queen of the sciences.
2. The digits 0 to 9 are called Hindu-Arabic numerals.
3. Aryabhata wrote the Aryabhatiya, which has four parts.
4. India's first satellite was named after Aryabhata.
5. Aryabhata gave pi as 62832/20000 = 3.1416.
6. Siddhanta Shiromani has four parts, including Lilavati and Bijaganita.
7. Bhaskaracharya is linked with the Chakravala method.
8. Ramanujan was born at Erode on 22 December 1887.
9. 1729 is the Hardy-Ramanujan number: 1³ + 12³ = 9³ + 10³.
10. Euclid's Elements has 13 books.
11. Euclid is called the Father of Geometry.
12. Georg Cantor founded set theory.

Practice questions

The famous work of Aryabhata is
1. Lilavati
2. Siddhanta Shiromani
3. Aryabhatiya
4. Bijaganita
Answer
C. Aryabhatiya
Aryabhata wrote the Aryabhatiya in verse.
India's first satellite was named after
1. Ramanujan
2. Brahmagupta
3. Bhaskaracharya
4. Aryabhata
Answer
D. Aryabhata
The satellite honoured the classical astronomer-mathematician Aryabhata.
Aryabhata gave the approximate value of pi as
1. 3.14159265
2. 3.1416
3. 3.16
4. 22/7 exactly
Answer
B. 3.1416
He gave 62832/20000 = 3.1416.
Which idea about the earth is credited to Aryabhata?
1. The earth rotates on its own axis
2. The sun goes round the earth daily
3. The earth does not move
4. The earth is flat
Answer
A. The earth rotates on its own axis
He explained day and night through the earth's rotation.
Bhaskaracharya's major work is
1. Aryabhatiya
2. Siddhanta Shiromani
3. Elements
4. Surya Siddhanta
Answer
B. Siddhanta Shiromani
It has four parts including Lilavati and Bijaganita.
Lilavati, a part of Siddhanta Shiromani, deals mainly with
1. arithmetic and mensuration
2. the sphere
3. algebra
4. motion of planets
Answer
A. arithmetic and mensuration
Lilavati is the arithmetic part; Bijaganita is algebra.
Bijaganita deals with
1. arithmetic
2. trigonometry tables
3. set theory
4. algebra
Answer
D. algebra
Bijaganita means seed-arithmetic, that is, algebra.
The Chakravala (cyclic) method for solving indeterminate equations is associated with
1. Pythagoras
2. Euclid
3. Bhaskaracharya
4. Cantor
Answer
C. Bhaskaracharya
Bhaskaracharya gave the cyclic method for equations of Pell type.
Srinivasa Ramanujan was born at
1. Tirupati
2. Chennai
3. Erode
4. Kumbakonam
Answer
C. Erode
He was born at Erode and grew up in Kumbakonam.
National Mathematics Day in India is observed on
1. 11 October
2. 22 December
3. 14 November
4. 5 September
Answer
B. 22 December
It marks the birthday of Srinivasa Ramanujan.
Ramanujan worked with which Cambridge mathematician?
1. Georg Cantor
2. Bertrand Russell
3. Isaac Newton
4. G. H. Hardy
Answer
D. G. H. Hardy
Hardy invited Ramanujan to Trinity College, Cambridge.
1729 is famous as the smallest number that
1. is the sum of two cubes in two different ways
2. is a perfect square
3. is a prime number
4. is the sum of two squares in two ways
Answer
A. is the sum of two cubes in two different ways
1729 = 1³ + 12³ = 9³ + 10³.
Euclid's famous book on geometry is
1. Principia
2. Almagest
3. Elements
4. Arithmetica
Answer
C. Elements
The Elements builds geometry from axioms and postulates.
Euclid is known as the
1. Father of Geometry
2. Father of Calculus
3. Father of Algebra
4. Father of Set Theory
Answer
A. Father of Geometry
His Elements organised geometry as a logical system.
The Elements of Euclid is divided into how many books?
1. 10
2. 23
3. 5
4. 13
Answer
D. 13
The Elements has 13 books.
Euclid's fifth postulate is also known as the
1. line postulate
2. parallel postulate
3. right angle postulate
4. circle postulate
Answer
B. parallel postulate
It deals with lines meeting when interior angles sum to less than two right angles.
The Pythagoras theorem holds for
1. right-angled triangles
2. all triangles
3. equilateral triangles only
4. obtuse triangles
Answer
A. right-angled triangles
It relates the hypotenuse to the other two sides in a right-angled triangle.
Pythagoras was born on the island of Samos, which is in
1. Italy
2. Egypt
3. India
4. Greece
Answer
D. Greece
Samos is a Greek island.
Georg Cantor is regarded as the founder of
1. trigonometry
2. coordinate geometry
3. set theory
4. probability theory
Answer
C. set theory
Cantor developed set theory and the study of infinite sets.
The ancient Indian texts that gave rules for geometric altar construction are the
1. Elements
2. Sulba Sutras
3. Aryabhatiya
4. Siddhanta Shiromani
Answer
B. Sulba Sutras
Baudhayana, Apastamba and Katyayana composed the Sulba Sutras.
The digits 0 to 9 used worldwide are called
1. Greek numerals
2. Roman numerals
3. Babylonian numerals
4. Hindu-Arabic numerals
Answer
D. Hindu-Arabic numerals
The decimal place-value system with zero began in India and passed through the Arab world.
The missing side of a right triangle with hypotenuse 13 and one leg 5 is
1. 8
2. 10
3. 12
4. 144
Answer
C. 12
√(169 - 25) = √144 = 12.
Which of the following is a Pythagorean triple?
1. 7, 24, 25
2. 7, 24, 26
3. 6, 8, 12
4. 5, 11, 13
Answer
A. 7, 24, 25
49 + 576 = 625 = 25².
By Euclid's division algorithm, HCF(18, 12) is
1. 3
2. 2
3. 12
4. 6
Answer
D. 6
18 = 12 × 1 + 6; 12 = 6 × 2 + 0; HCF = 6.
By Euclid's division algorithm, the HCF of 100 and 75 is
1. 15
2. 25
3. 50
4. 5
Answer
B. 25
100 = 75 × 1 + 25; 75 = 25 × 3 + 0; HCF = 25.
Using Euclid's division lemma, 17 = 5q + r with 0 ≤ r < 5. The values of q and r are
1. 2 and 7
2. 3 and 2
3. 4 and 3
4. 3 and 5
Answer
B. 3 and 2
5 × 3 = 15; remainder 2.
Aryabhata's value of pi, 62832/20000, when simplified decimally equals
1. 3.1623
2. 3.1428
3. 3.1416
4. 3.1400
Answer
C. 3.1416
62832 ÷ 20000 = 3.1416.
Ramanujan's number 1729 can be written as the sum of the cubes of 1 and
1. 12
2. 10
3. 11
4. 9
Answer
A. 12
1 + 1728 = 1729, since 12³ = 1728.
1729 is also the sum of the cubes of 9 and
1. 10
2. 8
3. 12
4. 11
Answer
A. 10
729 + 1000 = 1729.
Euclid's first set of postulates has how many postulates?
1. 3
2. 5
3. 7
4. 13
Answer
B. 5
Euclid gave five postulates.
A ladder 10 m long reaches a wall at a height of 8 m. The distance of its foot from the wall is
1. 18 m
2. 2 m
3. 6 m
4. 4 m
Answer
C. 6 m
√(100 - 64) = 6 by the Pythagoras theorem.
The value of 1³ + 12³ is
1. 1331
2. 1728
3. 1730
4. 1729
Answer
D. 1729
1 + 1728 = 1729.
A set with 3 elements, as studied in set theory, has how many subsets?
1. 8
2. 6
3. 9
4. 3
Answer
A. 8
2³ = 8 subsets.
The disciplinary value of teaching mathematics means that it
1. gives skills for shopping
2. trains reasoning, concentration and accuracy
3. shows the history of civilisation
4. gives pleasure from patterns
Answer
B. trains reasoning, concentration and accuracy
Disciplinary value concerns mental training.
The utilitarian value of mathematics refers to its use in
1. pure beauty of proofs
2. training logical thinking
3. knowing past culture
4. daily life such as measurement and banking
Answer
D. daily life such as measurement and banking
Utilitarian means practical use.
The cultural value of teaching mathematics is that it
1. makes students faster at shopping
2. only builds speed
3. removes need for reasoning
4. shows how human civilisation developed mathematical ideas
Answer
D. shows how human civilisation developed mathematical ideas
History of mathematics links the subject with culture.
Consider the statements. 1. Aryabhata wrote the Aryabhatiya. 2. Aryabhata wrote Lilavati. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Lilavati is by Bhaskaracharya.
Consider the statements. 1. Euclid's Elements is based on definitions, axioms and postulates. 2. Euclid's division algorithm is used to find the HCF. Which is/are correct?
1. 1 only
2. Both 1 and 2
3. 2 only
4. Neither 1 nor 2
Answer
B. Both 1 and 2
Both are correct.
Consider the statements. 1. The Pythagoras theorem is true for all triangles. 2. The Pythagoras theorem is true for right-angled triangles. Which is/are correct?
1. 1 only
2. Both 1 and 2
3. 2 only
4. Neither 1 nor 2
Answer
C. 2 only
It requires a right angle.
Consider the statements. 1. Cantor invented coordinate geometry. 2. The set of natural numbers is countable, in Cantor's theory. Which is/are correct?
1. 1 only
2. Both 1 and 2
3. Neither 1 nor 2
4. 2 only
Answer
D. 2 only
Cantor founded set theory; natural numbers are countable.
Consider the statements. 1. Euclid's fourth postulate is the parallel postulate. 2. Aryabhata's value of pi is exactly 22/7. Which is/are correct?
1. Neither 1 nor 2
2. 1 only
3. 2 only
4. Both 1 and 2
Answer
A. Neither 1 nor 2
The fifth postulate is the parallel postulate; Aryabhata gave 3.1416.
Consider the statements. 1. Aryabhata explained eclipses as effects of shadows of the earth and moon. 2. Ramanujan worked with Hardy at Cambridge. Which is/are correct?
1. 1 only
2. Both 1 and 2
3. 2 only
4. Neither 1 nor 2
Answer
B. Both 1 and 2
Both statements are correct.
Match the mathematician with the work. P. Aryabhata Q. Bhaskaracharya R. Euclid 1. Elements 2. Aryabhatiya 3. Lilavati
1. P-3, Q-2, R-1
2. P-2, Q-1, R-3
3. P-2, Q-3, R-1
4. P-1, Q-3, R-2
Answer
C. P-2, Q-3, R-1
Aryabhatiya, Lilavati and Elements.
Match the mathematician with the contribution. P. Ramanujan Q. Cantor R. Pythagoras 1. Set theory 2. The number 1729 story 3. Right-triangle theorem
1. P-2, Q-3, R-1
2. P-1, Q-2, R-3
3. P-2, Q-1, R-3
4. P-3, Q-1, R-2
Answer
C. P-2, Q-1, R-3
Ramanujan - 1729, Cantor - set theory, Pythagoras - right-triangle theorem.
Match the part of Siddhanta Shiromani with its subject. P. Lilavati Q. Bijaganita R. Grahaganita 1. Algebra 2. Arithmetic 3. Mathematics of the planets
1. P-3, Q-1, R-2
2. P-1, Q-2, R-3
3. P-2, Q-3, R-1
4. P-2, Q-1, R-3
Answer
D. P-2, Q-1, R-3
Lilavati is arithmetic, Bijaganita algebra, Grahaganita planets.

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Chapter quiz · 10 questions

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1.Vinod, Kajal and Arpit invest ₹1,040, ₹1,390 and ₹1,890, respectively, to start a business. If the profit at the end of the year is ₹1,440, then what is the share of Arpit in the profit? 6077
2.In a quadrilateral WXYZ, ∠ W = 74° and ∠ X = 58°. The bisectors of ∠ Y and ∠ Z meet at O. What is the measure of ∠ ZOY?
3.In a quadrilateral MNST, ∠ M = 63° and ∠ N = 99°. The bisectors of ∠ S and ∠ T meet at O. What is the measure of ∠ TOS?
4.In △ ABC, BD ⟂ AC at D and ∠ DBC = 80°. E is a point on BC such that ∠ CAE = 37°. What is the measure of ∠ AEB?
5.In △ ABC, BD ⟂ AC at D and ∠ DBC = 72°. E is a point on BC such that ∠ CAE = 71°. What is the measure of ∠ AEB?
6.In △ ABC, BD ⟂ AC at D and ∠ DBC = 21°. E is a point on BC such that ∠ CAE = 27°. What is the measure of ∠ AEB?
7.In a quadrilateral IJKL, ∠ I = 17° and ∠ J = 51°. The bisectors of ∠ K and ∠ L meet at Z. What is the measure of ∠ LZK?
8.In a quadrilateral ABCD, ∠ A = 32° and ∠ B = 58°. The bisectors of ∠ C and ∠ D meet at O. What is the measure of ∠ DOC?
9.In quadrilateral PQRS, the bisectors of ∠ R and ∠ S meet at the point T (inside the quadrilateral) and ∠ STR = 33°. If the ratio of ∠ P to ∠ Q is 3: 8, then what is the difference between the measures of ∠ Q and ∠ P?
10.In quadrilateral PQRS, the bisectors of ∠ R and ∠ S meet at the point T (inside the quadrilateral) and ∠ STR = 18°. If the ratio of ∠ P to ∠ Q is 2: 7, then what is the difference between the measures of ∠ Q and ∠ P?

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