Slide 1:
THE PHILOSOPHY OF MATHEMATICS
ED466 Workshop 2
7/8/08 Slide 2:
Like the crest of a peacock, like the gem on the head of a snake, so is mathematics at the head of all knowledge
(Vedanga Jyotisa, 500BC) Slide 3:
What is philosophy?
What is the philosophy of education?
What is the philosophy of mathematics?
What are the main philosophical viewpoints regarding mathematics? Two views of mathematical knowledge :
Two views of mathematical knowledge Absolutist View
Critique of Absolutist View Slide 5:
2. Fallibilist view The development of mathematical ideas :
The development of mathematical ideas The mathematics we know today is the result of an accumulation of content and processes developed over a period of 4000 years;
Real people in early civilisations developed ideas and ways to solve problems they faced in agriculture, and needs for weights and measures, religious worship, astronomy etc
They invented number systems to help them make calculations; geometry ideas to meet construction needs etc. Early maths civilizations :
Early maths civilizations Spread of mathematical ideas :
Spread of mathematical ideas Slide 9:
Mathematical Traditions:
Babylonian mathematics – large (base 60) number system used to construct tables for calculations. Strong in Algebra. Development of calendar.
Egyptian mathematics – used maths as practical tool in agriculture (harvest & storage, control flooding. Geometry strong. Devised & used Calendar. Fractions.
Indian mathematics – invention of the Number system (base 10 system). Some geometry & trigonometry.
4. Greek mathematics – many famous names e.g. Plato, Aristotle, Pythagoras, Thales. Greatest developments in math ideas. Strongest in astronomy & geometry.
5. Arabic mathematics – Centred on Iran & Iraq. Algebra was greatest contribution. Also astronomy & geometry. Islamic religious rituals impacted development of geometry & astronomy (moon). Egyptian Mathematics :
Egyptian Mathematics Used mathematics as practical tool to solve problems e.g. construction, trade. Used calculations.
Calculations based on addition & 2x tables
Preferred unit fractions (⅓, ⅛, ½, ¼ etc) – greater accuracy
Trade: no currency but traded using goods such as bread & beer
Much construction work e.g. canals, pyramids
6. Notable achievements – development of Lunar Calendar; Pyramids Indian Mathematics :
Indian Mathematics Ancient Hindu society regarded as sacred & pure discipline. Concealed math development & ideas.
2. Hindu Brahmins (highest caste) saw maths as related to stars & planets, and the gods/heaven & hell.
Math knowledge & study accessible only to high-caste children, so math was knowledge of the elite.
Greatest early mathematicians were Brahmins – Brahmagupta, Mahavira, Bhaskara etc
3. Indian mathematicians known outside India – Ramanujan & Shakuntala Devi – worked in number patterns & theorems Greek Mathematics :
Greek Mathematics Inherited legacy of accumulated knowledge from Babylon & Egypt. Improved & perfected ideas into a formal discipline of study – practical to abstract maths.
Greek were renowned astronomers
Important areas of discovery – Algorithms (how & why they worked), Theorems & mathematical Proofs, geometry, astronomy
Elements of Euclidean geometry ##
4. Schools of Plato, Aristotle & Pythagoras opened study to women.
5. Greek mathematicians: Thales, Pythagoras, Aristotle, Pascal Math ideas from Europe :
Math ideas from Europe Bernoulii – brilliant math family from Netherland
2. Euler – son of Protestant minister. Was minister but studied mathematics. Renowned for Algebra,Calculus
Laplace
Lagrange
Fourier
Gauss
Hardy
Polya – Problem solving
Von Neumann
etc Famous Women Mathematicians :
Famous Women Mathematicians 1. HYPATIA (Greek) - First woman mathematician. Daughter of renowned Prof of Math. Well educated, trained in arts, literature, science & philosophy. University lecturer, author, philosopher, mathematician. Executed by the Christian church.
MARIA AGNESI (Italian) – daughter of Prof of Math. Private tutoring, highly educated. University position.
SONYA KOVALESKY (Russian) – Tutored in language & arts. Father allowed her to be tutored in math, became exceptional & highly respected.
EMILY NOETHER (French) – First female doctoral degree. Prof of Math, strongly opposed by men
More!! Some mathematical history ideas :
Some mathematical history ideas 1. Number systems in India & Babylon, and now
2. Multiplication methods in Egypt & India, & now
3. Fractions in Egypt.
4. A mathematics problem from India
5. Tangrams (puzzles) & Abacus from China
6. Fractions as 2 numbers, num / den from China
7. Algebra in Arab countries
8. The Calendar in Babylon, Egypt, and now
9. Geometry (Pythagoras, Euclid) in Greece
10. Finger counting in Africa
Measurement by body parts e.g PNG, Fiji etc.
Counting systems e.g. Fijian society How can “history” ideas help mathematics learning? :
How can “history” ideas help mathematics learning? Gives mathematics a human face
Make mathematics less frightening
Showing pupils how concepts developed will help their understanding.
Changes pupils’ ideas of mathematics.
Increases motivation for learning.
Provides opportunities for investigation.
Past obstacles help to explain what today’s pupil find hard. Cont. :
Cont. Pupils derive comfort from realising that they are not the only one with problems.
Encourages fast learners to look further.
Helps develop a multicultural approach.
Helps maths teacher to maintain excitement & interest in maths.
Explains the role of maths in society.
Indicates link between maths & other subjects
Very good way of introducing a lesson Ways of using ”history” in class. :
Ways of using ”history” in class. As introduction to new concepts
As anecdotes (short, amusing or personal accounts) during the lesson.
Link concepts they’re learning as answers to historical problems faced in the past.
As a drama to demonstrate the mathematical interaction. cont. :
cont. As classroom or homework exercises
Poster displays and projects with a historical theme.
Use examples to illustrate particular techniques or methods in the past e.g. multiplication.
Use errors and alternative views from the past to help resolve difficulties today A word of caution. :
A word of caution. The history of mathematics is not taught as a subject on its own, but used only as a means to help teaching & understanding of maths.
Pick only those portions which are of real assistance to the learner.
The teacher must collect information about “history” and teach them in context.