1. Historical Development (Adhiraj Vallabh)
1.1. This is how mathematics has developed over the years
1.1.1. Questions such as how has our understanding of mathematics changed over time are asked
1.1.1.1. We ask such questions to analyse the thought process of humans throughout history.
1.1.1.1.1. Hence finding the difference between ours and their thoughts
1.1.1.1.2. And to see how pieces of History have been proved wrong in our time
1.1.1.1.3. And to see how they made use of maths in different fields
1.1.2. Our perception and the importance of maths has also changed over time
1.1.2.1. In the beginning Maths was used to count the number of animals killed by prehistoric men
1.1.2.1.1. Now days maths is one of the most essential parts of humanity
1.1.2.1.2. It is used in nearly every single Aspect of our lives
1.2. Mathematics is the study of topics such as quantity, measurement, etc.
1.2.1. Over the years Many new topics of maths have developed such as:
1.2.2. Graphing and Coordinate Geometry
1.2.2.1. Such topics have become more and more mainstream due to technology
1.2.2.1.1. In UWC, we use a GDC (graphing display calculator) to help us do coordinate geometry
1.2.2.2. Now Coordinate geometry is one of the essentials basics of maths, which is compulsory in IGCSE
1.2.3. Calculus is also a relatively new concept
1.2.3.1. Calculus is a subject common in higher educations
1.2.4. Conjecture in Mathmatics is when a conclusion is formed based on incomplete Information
1.2.4.1. Whereas a theorem is a conjecture if proved with a good amount of evidence and complete information
1.3. Mathematics has a long and interesting History
1.3.1. Math was first used by hunter-gatherers to identify number of animals they hunted
1.3.2. During the classical period, Egyptians made the most of amount of significant developments in Maths
1.3.2.1. They used Maths to build amazing architectural marvels
1.3.3. The Greeks further expanded this information to promote systematic study of Maths and took Maths to a great level
1.3.3.1. One of the most famous and useful Mathematical Theorems in the World was written in this era
1.3.3.2. It is called the Pythagoras Theorem and was written by Greek Philosopher, Pythagoras
1.3.3.2.1. This theorem is still widely used and recognised despite the many changes in Maths
1.3.4. The Golden age of Mathematics in India
1.3.4.1. India is one of the countries credited with the most development in Maths
1.3.4.2. Mathematicians like Aryabhata used zero as an actual number rather than making it just a place holder
1.3.4.3. India is also credited for the early development of Calculus
1.4. Some KQ's regarding Mathematics
1.4.1. Does maths need language to be understood?
1.4.2. Is emotion irrelevant to the construction of Mathematical knowledge?
1.4.3. How much statistical data should be used to determine the reliability of a result?
1.4.4. To what extent does mathematics describe the real world?
2. Scope & Application(Linda)
2.1. Sole function
2.1.1. Solve problems in society
2.1.1.1. eg. analysing surveys, patterns, creating models...
2.1.1.1.1. largely used in Human/Natural sciences
2.2. Different forms and aims
2.2.1. Calculus
2.2.1.1. model making
2.2.2. Algebra
2.2.2.1. Discover theories and relationships between factors
2.2.3. Geometry and Trig
2.2.3.1. Architecture
2.2.4. Logic
2.2.4.1. Deductive reasoning
2.2.4.2. solving real life problems
2.2.5. Many theories and concepts are USELESS
2.2.5.1. purely theoretical
2.2.5.1.1. Pure math vs. Applied math
2.3. Interpreted the same way around the world in all languages
2.3.1. no culture context
2.3.2. Almost all societies encourage math
2.3.2.1. Used as a measure of intelligence
2.3.3. Share knowledge
2.3.3.1. People are very unlikely to question math
2.3.3.1.1. A timeless knowledge/truth
2.4. Other functions
2.4.1. A measure of beauty/art
2.4.1.1. symmetry
2.4.1.2. golden ratio
2.5. Benford's Law
2.5.1. Making predictions about the future
2.5.1.1. Number 1 is more prevalent than other numbers
2.6. Views
2.6.1. Platonists
2.6.1.1. a code to understand the world around us
2.6.2. Constructivists
2.6.2.1. the nature of Mathematics is artificial and abstract
2.6.2.1.1. Mathematical Humanism
2.6.3. Formalists
2.6.3.1. an abstract game played according to invented rules
3. The main WOK used in Mathematics is Reason, it is a very binary subject where the answer is either true or false
4. Consider the WOKs in relation to your experiences, how have these affected what and how you know in this AOK?
5. What is the nature of the contribution of individuals you know personally to this area, in terms of your experience?
5.1. We don’t necessarily contribute to mathematical knowledge
6. Methodology (vivek)
6.1. Axioms
6.1.1. Building blocks of math
6.1.2. Cannot be disputed
6.1.3. e.g. the value of PI
6.2. Theorem
6.2.1. Statement that is supported by a proof
6.2.1.1. traditional mathematic proof
6.2.1.1.1. aim remove all doubt
6.2.1.1.2. there are certain ways to use grammar which differs from every day use e.g. or, if, then
6.2.1.1.3. using statements that are already proven inconclusive to find statements that you don't already know
6.2.1.1.4. in a formal proof
6.2.2. Generalisation of proofs
6.3. Logical deduction
6.3.1. creating situations which are true in any circumstance
6.3.1.1. Proof by substitution
6.3.1.1.1. all humans can think -> all sentient beings can think -> therefore all humans are sentient beings
6.3.1.1.2. characteristics of larger categories apply for groupings inside of the larger category
6.3.1.2. Proof by contradiction
6.3.1.2.1. if statements contradict each other they are false
7. Language & Concepts sam
7.1. Language
7.1.1. Numbers
7.1.1.1. an object used to count and measure
7.1.1.2. Where did numbers come from?
7.1.1.2.1. Numbers originated from India in the 6th or 7th century.
7.1.1.2.2. The Indians created a system of 10 digits 1,2,3,4,5,6,7,8,9,0
7.1.2. Symbols
7.1.2.1. This represents the numbers
7.1.3. Ethnomathematics
7.1.3.1. The study of mathematics and culture
7.1.3.1.1. Within cultures there are different types of mathematical language. In Germany they say 67 and 7,6
7.1.4. Global agreement around modern mathematic language
7.1.4.1. However some countries still have their unique languages around mathematics
7.2. Concepts are mental representations of ideas
7.2.1. Number
7.2.2. Length
7.2.3. Area
7.2.4. Volume
7.2.5. Symmetry
7.2.6. Set theory
7.2.6.1. Real numbers
7.2.6.2. Natural
7.2.6.3. Irrational
7.2.6.4. Rational
8. Personal Knowledge
8.1. What responsibilities rest upon YOU by virtue of YOUR knowledge in this area?
8.1.1. Applying the skills learnt in maths to life: Analytical skills, problem solving, critical thinking etc...
8.2. What are the implications of this area of knowledge in terms of YOUR individual perspective?
8.3. What assumptions underlie YOUR own approach to this knowledge?
8.3.1. There is a right answer
8.3.2. There are multiple ways to arrive at an answer
8.3.3. Axioms can be proved if they are assumed to be true, this is called a proof