USING THE HISTORY OF MATHEMATICS IN TEACHING

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USING THE HISTORY OF MATHEMATICS IN TEACHING
George Gheverghese Joseph
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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

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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)

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Where did Mathematics begin?
The Ishango Bone Its Location
?

Ishango Bone
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The Ishango Bone (dated 25000 20000
BC)
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What is the maths hisstory behind this 1835
painting by Turner?
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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.

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A MATHEMATICAL DIFFERENCE BETWEEN TWO
PAINTINGSMELCHIOR BROEDERLAM (c. 1394)
PIETRO PERUGINO FRESCO AT THE

SISTINE
CHAPEL (1481)
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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. Peruginos
  fresco at the Sistine Chapel (1481) clearly shows
  perspective while Broederlams 1394 painting does
  not (Brunelleschi had not yet discovered the
  rules of perspective).

Cuboid with 1 vanishing point
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For ExampleMELCHIOR BROEDERLAM (1394)
PIETRO PERUGINO FRESCO AT THE
SISTINE CHAPEL (1481)
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WHAT HAS BONE SETTING GOT TO DO WITH ALGEBRA?
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WHAT HAS BONE SETTING GOT TO DO WITH ALGEBRA?

• Al-Khwarizmi wrote the first treatise on algebra
  Hisab al-jabr wal-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)

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?

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

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Trace of History

• Writing in English proceeds from the Left to the
  Right
• Just as you are reading this sentence?

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THE TRACE OF HISTORY

• Roman numbers can be read from the Left to the
  Right
• C X V
• 100 10 5

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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.

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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.

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SIMILARLY OPERATIONS WITH THE
INDO-ARABIC NUMERALS

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The Spread of the Indian Numerals
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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

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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

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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

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LOOKING FOR CALCULATOR MAGIC IN THE HISTORY OF
MATHS

• Heros (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 an1 ½ (an P/
  an)

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CALCULATOR MAGIC FROM THE HISTORY OF MATHS
HEROS 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 an1 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...
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CALCULATOR MAGIC FROM THE HISTORY OF MATHS
HEROS ITERATION (Continued)

• Heros iteration is an1 ½ (an P/ an)
• Suppose an ? L as n ? 8
• Then iteration an1 ½ (an P/ an) ? L ½ (L
  P/ L) as n ? 8
• Solving L ½ (L P/ L) gives L2 P.
• That is the limit L vP

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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

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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
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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?
Bhaskaras Explanation Behold!
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MAKE MATHEMATICS MORE VISUAL
FROM A BLAND STATEMENT a2 - b2 ( a b)(a
b)

• TO AN INTERESTING STATEMENT/ACTIVITY FROM THE
  HISTORY OF MATHS
• - b
  
• a
• 
  a
• 
  a-b
• 
  b
  a
• 
  b
• 
  
  a-b

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MAKING BLAND STATEMENTS VISUAL (Contd)

• - b
  a-b a
• a
  b
• 
  b
• 
  a-b
• 
  
• 
  
  ab
• 
  
• 
  a-b

Area at start a2 b2 Area at end (a b)(a
b)
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MAKE QUADRATIC EQUATIONS MORE VISUAL

• From the fifth rule of al-Khwarizmis 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(3?x)
Green square side 3
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MAKING ADVANCED SCHOOL MATHS INTERESTINGJAMSHID
AL-KASHIS 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 xn1 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

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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 context
• Ishango Bone
• .... all this should increase interest in
  learning mathematics in our multi-ethnic world

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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

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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.

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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
  teachers own interest and excitement
• 15. Provides opportunities for cross-curricular
  work with other teachers or subjects