Mathematics

Using the knowledge framework to explore Mathematics

An effective way to explore the different areas of knowledge is to use what the IB is referring to as a ‘knowledge framework’. This gives you five different consideration points, so you can think further about each AOK, understand the way they all link together, and reflect on your own relationship with knowledge.

1. Scope and applications

What is the social function of mathematics?	Usage in our daily lives (change calculation, cooking, financial calculations (taxes/discounts), statistics, etc.)
How many different forms does it encompass (eg calculus, algebra, applied maths, etc.)?	Geometry, Trigonometry, Calculus, Number Theory, Algebra
What are their separate aims?	Geometry à Study of shape, areas, positions, etc. Has Physics applications as well Trigonometry à Study of the lengths, angles, and relations of triangles Calculus à The study of change (derivatives and integrals) Number Theory à The study of number systems like complex and imaginary numbers Algebra à The study of mathematical operations and equations
To what extent is mathematics influenced by the society and culture in which it is pursued?	Whether or not culture and society are accepting of intellectual development Need arises for specific areas of maths to be pursued (like the statistical t-Test for scientific experiments) Shared information, not personal information
How important is mathematics?	Mathematics is probably the most important academic pursuit of all. Almost all our scientific knowledge derives from math; math is pure logic, and it is the framework from which all other knowledge comes. The statement 1+1 = 2 leads to 2-1=1…we eventually get to E=mc2.

2. Concepts and language

How do we use language to express the knowledge found within mathematics?	Wide variety of concepts, can be expressed differently in different languages Symbols contain the essence of mathematic language
To what extent does this differ according to different forms of mathematics?	Different languages focus on different areas of mathematics
Are there any central concepts for which we need specific language before approaching mathematics?	Things like addition and other operations need to be understand and have language available for them so we can move on to more advanced concepts Specific mathematical things like shapes, angles, rays, etc. terms that are used to further the general advancement of areas of math like geometry

3. Methodology

Which ways of knowing do we use in order to connect with, and understand, mathematics?	We can use language and precise definitions to connect to mathematics Reason and logic to understand mathematics Emotion to discover new mathematic concepts
Which ways of knowing do mathematicians themselves use as they research and work?	Memory, language, reasoning Always objective and stable, so intuition doesn’t work Remember theorems and formulas to solve more advanced problems Reasoning used in mathematics to convert pure to applied Language used to communicate in maths

 4. Historical development

How has our understanding and perception of mathematics changed over time?	Math is unchanging, but builds upon itself We can develop abstract concepts using mathematics
How has the role of mathematics within society developed?	Initially a system for simple bartering and calculations like the number of cows required to sustain a beef farm Slowly advanced through the discoveries of scholars and philosophers like Galileo and Newton who found applications in Astronomy and Physics Currently used as the basis of all sciences in a wide variety of fields, all technology derives from the application of mathematics
To what extent has the nature of mathematics changed?
What relationship does today’s mathematics have with that of the past? (to paraphrase Newton, does it ‘stand on the shoulders of giants’?)	Current math is an adaptation of the past Math trumps over all Adapting it is “standing on the shoulder of giants”, you’re seeing far ahead only by building on the work of past people

5. Links to personal knowledge

To what extent are you involved with mathematics?	I am quite involved with Mathematics since I am a student who studies higher level mathematics almost every day I’m in school and spend a lot of personal time studying and reviewing mathematic theories and practicing mathematical problems.
How is your perception of the world, and your position it in, affected by mathematics?	I have a greater awareness for the critical role plays in mathematics