USING THE HISTORY OF MATHEMATICS IN TEACHING

George Gheverghese Joseph

WHY INCLUDE HISTORY OF MATHEMATICS IN TEACHING?

“The teacher who knows little of the history of mathematics is apt to teach techniques in isolation, unrelated either to the problems and ideas which generated them or to the further developments which grew out of them”.

  From a UK Ministry of Education report of 1958

WHY INCLUDE HISTORY OF MATHEMATICS IN TEACHING?

“Students should learn to study at an early stage the great [historical] works …. instead of making their minds sterile through everlasting exercises …. which are of no use whatsoever ….. where indolence is veiled under the form of useless activity”.

(Eugenio Beltrami, 1873)

Where did Mathematics begin? The Ishango Bone: Its Location

◙

Ishango Bone

The Ishango Bone (dated 25000 – 20000 BC)

WHAT IS THE MATHS [HIS]STORY BEHIND THIS 1835 PAINTING BY TURNER?

THE 1835 PAINTING BY TURNER DEPICTS THE HOUSES OF PARLIAMENT BURNING IN 1834

BACKGROUND Some resistance to the adoption of new arithmetic

afforded by the Indo-Arabic numerals. Tally sticks were in use until the 19th century. The fire was caused by tally sticks kept in the houses.

Charles Dickens commented at the time: In 1834 ... there was a considerable accumulation of them [tally sticks]. ... The sticks were housed in Westminster…… and so the order went out that they should be privately and confidentially burned. It came to pass that they were burned in a stove in the House of Lords. The stove, over gorged with these preposterous sticks, set fire to the paneling; the paneling set fire to the House of Commons.”

A MATHEMATICAL DIFFERENCE BETWEEN TWO PAINTINGS

MELCHIOR BROEDERLAM (c. 1394) PIETRO PERUGINO FRESCO AT THE SISTINE CHAPEL (1481)

PERSPECTIVE – HOW GEOMETRY INFLUENCED ART

Florentine architect Brunelleschi (1377 –1446 ): First in Europe to carry out a series of experiments leading to a geometrical theory of perspective.

Essentially parallel lines on a horizontal plane depicted in the vertical plane meet – at the vanishing point

After his discovery artists adopted perspective and since then paintings depicting real life scenes have been more realistic. Perugino’s fresco at the Sistine Chapel (1481) clearly shows perspective while Broederlam’s 1394 painting does not (Brunelleschi had not yet discovered the rules of perspective).

Cuboid with 1 vanishing point

For ExampleMELCHIOR BROEDERLAM (1394) PIETRO PERUGINO FRESCO AT THE SISTINE CHAPEL (1481)

WHAT HAS BONE SETTING GOT TO DO WITH ALGEBRA?

WHAT HAS BONE SETTING GOT TO DO WITH ALGEBRA?

Al-Khwarizmi wrote the first treatise on algebra: Hisab al-jabr w’al-muqabala in 820 AD. The word algebra is a corruption of al-jabr which means restoration of bones.

In Spain, where the Moors from North Africa held sway for a long period, there arose a profession of ‘algebristas’ who dealt in bone setting.

álgebra. 1. f. Parte de las matemáticas en la cual las operaciones aritméticas son generalizadas empleando números, letras y signos. 2. f. desus. Arte de restituir a su lugar los huesos dislocados (translation: the art of restoring broken bones to their correct positions)

Some ways to convince students that the mathematics they study has the trace of history:-

Trace of History

Writing in English proceeds from the Left to the Right:

Just as you are reading this sentence?

THE TRACE OF HISTORY Roman numbers can be read

from the Left to the Right: C X V 100 10 5

THE TRACE OF HISTORY But Our Place value number structure proceeds from the Right to the Left:

So to interpret a whole number 72 | 611 | 134 | 942 | 342 |835

you naturally/normally proceed [in blocks of 3 places] from the Right to the Left to finally identify place value of the numeral 7.

THE TRACE OF HISTORY

Early Indian systems were both Left to the Right and Right to the Left systems.

Conjecture: The Arabs following the practice of writing Arabic naturally adopted the Right to the Left and transmitted it Westwards.

SIMILARLY OPERATIONS WITH THE INDO-ARABIC NUMERALS

norientatioleftright

ArabicinarithmeticOur

−

×←×←×

880508

''444721721721

22121

The Spread of the Indian Numerals

WHY IS 1= 0.99999.......? A PROOF FROM HISTORY FOR A SCEPTIC IN A SENIOR CLASS

Derived from a 16th century mathematical manuscript (Yuktibhasa) from Kerala, South India:

1 = 1 . 10 = 1 . [9 + 1] = 1 . [1+ 1] 9 10 9 10 9 9 10 9

Use Identity 1 1 . [1+ 1]: and replace last 1 by the identity. 9 10 9 9

Hence 1 = 1 . [1+ 1 .[1 + 1 ] = 1 + 1 + 1 [1 + 1] 9 10 10 9 10 102 102 9

Keep going .......

1 = 1 + 1 + 1 + 1 + ................ 9 10 102 103 104

WHY IS 1= 0.99999.......? A PROOF FOR THE SCEPTIC (continued)

1 = 1 + 1 + 1 + 1 + ................ 9 10 102 103 104

Can generalise by replacing 1 by any non-zero numeral a:

a = a + a + a + a + ................ 9 10 102 103 104

Now a = 9 implies

9 = 9 + 9 + 9 + 9 + ................ 9 10 102 103 104

1 = 0.999999........ Q. E . D

WHY IS 1= 0.99999.......? THE PROOF GENERALISED FOR GEOMETRIC SERIES

Show: 1/(1-c) = 1 + [1/(1- c)]c  Substitution for (1/(1- c)) gives   1/(1-c) = 1 + (1 + [1/(1- c)]c) c = 1 + c + (1/(1- c))c2

  Repeated substitution gives   1/(1 – c) = 1 + c + c2 + …………….+ cn-1 + cn/(1 – c)

Rearranging gives:

(1 – cn)/(1 – c) = 1 + c + c2 + …………….+ cn-1

LOOKING FOR ‘CALCULATOR MAGIC’ IN THE HISTORY OF MATHS

Hero’s (1st century AD, Alexandria) algorithm for the square root of a number P.

Step 1: Guess an approximate square root a1 for P.

Step 2. Second guess is a2 = ½ (a1 + P/ a1)

Third guess is a3 = ½ (a2 + P/a2)

Fourth guess is a4 = ½ (a3 + P/ a3)

In general: (n + 1)st guess is an+1 = ½ (an + P/ an)

‘CALCULATOR MAGIC’ FROM THE HISTORY OF MATHS: HERO’S ITERATION

Step 1: Guess an approximate square root a1 for P.

Step 2. Second guess is a2 = ½ (a1 + P/ a1)

Third guess is a3 = ½ (a2 + P/a2)

Fourth guess is a4 = ½ (a3 + P/ a3)

In general: (n + 1)st guess is an+1 = ½ (an + P/ an)

a1 a2 a3 a4 Sq. Root P =

P = 30 5.5 5.47727.. 5.4772255. 5.4772255. 5.477225575...

P = 27 5.5 5.20454545... 5.19615919.. 5.196152423.. 5.196152423...

P = 37 5.5 6.113636364... 6.082840487.. 6.082762531.. 6.08276253...

‘CALCULATOR MAGIC’ FROM THE HISTORY OF MATHS: HERO’S ITERATION (Continued)

Hero’s iteration is an+1 = ½ (an + P/ an)

Suppose an → L as n → ∞

Then iteration an+1 = ½ (an + P/ an) → L = ½ (L + P/ L) as n → ∞

Solving L = ½ (L + P/ L) gives L2 = P.

That is the limit L = √P

ARE YOU ALREADY USING HISTORY OF MATHS WITHOUT REALISING IT?

The Indian mathematicians Bhaskara II (1114-1185) developed this proof for the theorem of the right angled triangle: a b

b c

a a c

c

b

b a

ARE YOU ALREADY USING HISTORY OF MATHS WITHOUT REALISING IT?

Area of large square = (a + b)2

Which is made up of inner square of area c2 and 4 triangles each of area ½ ab

So (a + b)2 = c2 + 4 ½ ab

Or a2 + b2 + 2ab = c2 + 2ab

Thus a2 + b2 = c2

HISTORY OF MATHS – ANOTHER PROOF OF ‘PYTHAGORAS’ THEOREM’

Bhaskara II developed another proof for the theorem of the

right angled triangle using this diagram:

A

B C

How does the figure help show BC2 = AC2 + AB2?

Bhaskara’s Explanation Behold!

An ‘Action Proof from the Yuktibhasa (1550) Given a Right -angled Triangle

(i) Construct the squares on the smaller sides:

(ii) Move smaller square to align with larger one:

(iii) Construct two triangles congruent to original triangle within the above configuration:

(iv) Cut, re-assemble and paste these two constructed triangles to form square on hypotenuse:

This proves in a n original but convincing way that the sum of the squares

on the smaller two sides of a right angled triangle is equal to the squa re on

the hypotenuse.

MAKE MATHEMATICS MORE VISUAL

TO AN INTERESTING STATEMENT/ACTIVITY FROM THE HISTORY OF MATHS

- b = a

a

= a-b b a b a-b

FROM A BLAND STATEMENT: a2 − b2 = ( a – b)(a + b)

MAKING BLAND STATEMENTS VISUAL (Contd)

- b = a-b a

a b

b

a-b

= a+b

a-b

Area at start = a2 – b2

Area at end = (a – b)(a + b)

MAKE QUADRATIC EQUATIONS MORE VISUAL

From the fifth rule of al-Khwarizmi’s algebra :

How to solve x2 - 6x = 40

x 3

The problem is 6x + 40 = x2 or x2 - 6x = 40.

x

So Orange area = x2 - 6x + 32 3

x2 - 6x + 32 = 40 + 32 = 49.

So (x - 3)2 = 49.

Thus x - 3 = 7 or -7. Hence x =10 or -4

Orange square = Sq. side x – 2 rectangles(3x) + Green square side 3

MAKING ADVANCED SCHOOL MATHS INTERESTINGJAMSHID AL-KASHI‘S FIXED POINT ITERATION

To solve a cubic such as c = 3x – 4x3 Al-Kashi re-arranged it as x = (c + 4x3)/3 and called it x = g(x) where g(x) = (c + 4x3)/3.

And then performed the iteration xn+1 = g(xn).

This is exactly the fixed-point iteration used in pure mathematics. Good A level books will provide some kind of rationale for this: y = x

y = g(x)

location of exact solution

x0 x1 x2

WHY INCLUDE HISTORY OF MATHEMATICS IN TEACHING?

It provides cross-curricular links Art, Spanish

It presents mathematics as a global endeavour rather than a monopoly of any single culture. Spread of Indian Numerals

It locates mathematics in a cultural contextIshango Bone .... all this should increase interest in

learning mathematics in our multi-ethnic world

BRAINSTORMING: REASONS FOR USING HISTORY IN MATHEMATICS EDUCATION

1. Increase motivation for learning

2. Humanizes mathematics

3. Helps to order the presentation of topics in the curriculum

4. Showing how concepts have developed and helps understanding

5. Changes students' perceptions of mathematics

REASONS FOR USING HISTORY IN MATHEMATICS EDUCATION (Continued)

6. Comparing ancient and modern helps in understanding the value of the latter

7. Helps develop a multicultural approach

8. Provides opportunities for investigation

9. Past difficulties a good indication of present pitfalls

10. Students derive comfort from realizing that they are not the only ones with problems.

REASONS FOR USING HISTORY IN MATHEMATICS EDUCATION (Continued)

11. Encourages quicker learners to look further

12. Helps to explain the role of mathematics in society

13. Makes mathematics less frightening

14. Exploring history helps to sustain a teacher’s own interest and excitement

15. Provides opportunities for cross-curricular work with other teachers or subjects