The ‘knowledge framework’ is designed to help students explore, discuss, and form an understanding of each of the eight areas of knowledge. The knowledge framework isn’t a formal element of the TOK essay or presentation (ie, it isn’t directly assessed); instead, it is designed to present various consideration points that that can then be used to compare and contrast the different areas of knowledge, as well as tying them to the ways of knowing.
What is the social function of mathematics? How many different forms does it encompass (eg calculus, algebra, applied mathematics, etc.)? What are their separate aims? To what extent is mathematics influenced by the society and culture in which it is pursued? How important is mathematics?
How do we use language to express the knowledge found within mathematics? To what extent does this differ according to different forms of mathematics? Are there any central concepts for which we need specific language before approaching mathematics?
Which ways of knowing do we use in order to connect with, and understand, mathematics? Which ways of knowing do the mathematicians themselves use in order to study mathematics and communicate their understanding of it?
How has our understanding and perception of mathematics changed over time? How has the role of mathematics within society developed? To what extent has the nature of mathematics (for example, the different forms of mathematics) changed? What relationship does today’s mathematical understanding have with that of the past? (to paraphrase Newton, does it ‘stand on the shoulders of giants’?)
To what extent are you involved with mathematics? How is your perception of the world, and your position it in, affected by mathematics?
Is there a distinction between truth and certainty in mathematics? Is mathematics independent of culture? Is mathematics discovered or invented?
Mathematics is founded on a set of more or less universally accepted definitions and basic assumptions. It proceeds from a system of axioms using deductive reasoning to prove theorems or mathematical truths. These have a degree of certainty unmatched by any other area of knowledge, making it excellent raw material for study in TOK.
Despite, or rather because of, the strict confines of mathematical logic, mathematics is an enormously creative subject, asking of its practitioners great leaps of the imagination. Pure mathematics requires no prior sense perception at the start of inquiry but the application of mathematics to real-world situations requires techniques such as those used in the natural and human sciences. Indeed, most research in the natural and human sciences is underpinned by mathematics. There are also often close links between mathematics and the arts where formal requirements for harmony or symmetry impose mathematical structures on a work.
