AREAS OF KNOWLEDGE:
MATHEMATICS
“Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.”
Google Doodle by Bene Rohlmann celebrating the mathematician Gauß who developed the Theorema Egregium, a method of calculating the curvature of a surface using angles and distances, as well as the famous bell curve in statistics. At the age of 24, he wrote Disquisitiones Arithmeticae which laid the foundation for modern number theory and is widely regarded as one of the most influential mathematics texts of all time. One of the highest honors in mathematics, the Gauß Prize, bears his name
CLASS ACTIVITIES
What sets pure mathematics apart from other areas of knowledge? Is mathematics invented or discovered?
Here are some class activities that will help students to explore the scope of mathematics. Students will reflect on their own relationship to mathematics as a revered academic discipline, and if there is room for mathematicians to bring their own perspectives to the ever growing edifice of mathematical knowledge. They will encounter the distinct methods and tools of mathematics, especially the nature of mathematical proof. Finally, they will encounter some of the ethical conundrums confronted by mathematicians.
Proof
Solve a quadratic
Sum of the angles in a triangle
The Monty Hall problem
Thinking about proof and intuition
Ideal gas law compared to Euler’s relation
Pure and applied mathematics
The path from metaphor to algorithm
Mathematical induction
Revisit Pascal's triangle
Build a house of cards
The special case of proof by mathematical induction
House of cards resolved
This Statement is False
The liar's paradox
The barber's paradox
Non-Euclidean geometry
Infinities
Beguiling with statistics
In progress
Platonists and Formalists
Written assignment
KNOWLEDGE QUESTIONS
The new Theory of Knowledge Guide (2020) provides 385 Knowledge Questions for student exploration. Here are my personal favorites from the mathematics section.
SCOPE
How have technological innovations, such as developments in computing, affected the scope and nature of mathematics as an area of knowledge?
Is absolute certainty attainable in mathematics?
Does mathematics only yield knowledge about the real world when it is combined with other areas of knowledge?|
PERSPECTIVE
How significant have notable individuals been in shaping the nature and development of mathematics as an area of knowledge?
What is the role of the mathematical community in determining the validity of a mathematical proof?
METHODS AND TOOLS
What is meant by the term “proof” in mathematics, and how is this similar to, or different from what is meant by this term in other areas of knowledge?
What does it mean to say that mathematics is an axiomatic system?
ETHICS
How are unethical practices, such as “data dredging,” used by statisticians to deliberately manipulate and mislead people?
CONNECTING TO THE CORE THEME
Why do you think mathematics enjoys a privileged status in many education systems?
What steps can we take to help ourselves avoid being misled by statistics used in unclear or disingenuous ways in the media?
Indian postage stamp depicting Indian mathematician Srinivasa Ramanujan (1887 - 1920)