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Group Theory of Knowledge

group tok TOK is a course about critical thinking and inquiring into the process of knowing, rather than about learning a specific body of knowledge. It is a core element which all Diploma Programme students undertake and to which all schools are required to devote at least 100 hours of class time. TOK and the Diploma Programme subjects should support each other in the sense that they reference each other and share some common goals. The TOK course examines how we know what we claim to know. It does this by encouraging students to analyse knowledge claims and explore knowledge questions. A knowledge claim is the assertion that “I/we know X” or “I/we know how to Y”, or a statement about knowledge; a knowledge question is an open question about knowledge. A distinction between shared knowledge and personal knowledge is made in the TOK guide. This distinction is intended as a device to help teachers construct their TOK course and to help students explore the nature of knowledge.

TOK plays a special role in the Diploma Programme by providing an opportunity for students to reflect on the nature of knowledge. The task of TOK is to emphasize connections between areas of knowledge and link them to the knower in such a way that the knower can become aware of his or her own perspectives and those of the various groups whose knowledge he or she shares. TOK, therefore, explores both the personal and shared aspects of knowledge and investigates the relationships between them.

The raw material of TOK is knowledge itself. Students think about how knowledge is arrived at in the various disciplines, what the disciplines have in common and the differences between them. The fundamental question of TOK is “how do we know that?” The answer might depend on the discipline and the purpose to which the knowledge is put. TOK explores methods of inquiry and tries to establish what it is about these methods that makes them effective as knowledge tools. In this sense TOK is concerned with knowing about knowing.

The individual knower has to try to make sense of the world and understand his or her relationship to it. He or she has at his or her disposal the resources of the areas of knowledge, for example, the academic disciplines studied in the Diploma Programme. He or she also has access to ways of knowing such as memory, intuition, reason and sense perception that help us navigate our way in a complex world.

It is easy to be bewildered by the sheer diversity of the knowledge on offer. For example:

In physics, experiment and observation seem to be the basis for knowledge. The physicist might want to construct a hypothesis to explain observations that do not fit current thinking and devises and performs experiments to test this hypothesis. Results are then collected and analysed and, if necessary, the hypothesis modified to accommodate them.
In history there is no experimentation. Instead, documentary evidence provides the historian with the raw material for interpreting and understanding the recorded past of humanity. By studying these sources carefully a picture of a past event can be built up along with ideas about what factors might have caused it.
In a literature class students set about understanding and interpreting a text. No observation of the outside world is necessary, but there is a hope that the text can shed some light upon deep questions about what it is to be human in a variety of world situations or can act as a critique of the way in which we organize our societies.
Economics, by contrast, considers the question of how human societies allocate scarce resources. This is done by building elaborate mathematical models based upon a mixture of reasoning and empirical observation of relevant economic factors.
In the islands of Micronesia, a steersman successfully navigates between two islands 1,600 km apart without a map or a compass.

In each case above there is clearly knowledge at work, although the collection as a whole illustrates a wide variety of different types of knowledge. The task of TOK is to examine different areas of knowledge and find out what makes them different and what they have in common.

At the centre of the course is the idea of knowledge questions. These are questions such as:

what counts as evidence for X?
what makes a good explanation in subject Y?
how do we judge which is the best model of Z?
how can we be sure of W?
what does theory T mean in the real world?
how do we know whether it is right to do S?

While these questions could seem slightly intimidating in the abstract, they become much more accessible when dealt with in specific practical contexts within the TOK course. They arise naturally in the subject areas, the extended essay and CAS. The intention is that these contexts provide concrete examples of knowledge questions that should promote student discussion.

Discussion forms the backbone of the TOK course. Students are invited to consider knowledge questions against the backdrop of their experiences of knowledge in their other Diploma Programme subjects but also in relation to the practical experiences offered by CAS and the formal research that takes place for the extended essay. The experiences of the student outside school also have a role to play in these discussions, although TOK seeks to strike a balance between the shared and personal aspects of knowledge.

Recognizing the discursive aspect of the course, the TOK presentation assesses the ability of the student to apply TOK thinking to a real-life situation. The TOK essay gives the opportunity to assess more formal argumentation prompted by questions of a more general nature.

TOK is a course in critical thinking but it is one that is specifically geared to an approach to knowledge that is mindful of the interconnectedness of the modern world. “Critical” in this context implies an analytical approach prepared to test the support for knowledge claims, aware of its own weaknesses, conscious of its perspectives and open to alternative ways of answering knowledge questions. It is a demanding course but one that is an essential component not only of the Diploma Programme but of lifelong learning.

The ways of knowing

While there are arguably many ways of knowing, the TOK course identifies eight specific ways of knowing (WOKs). They are language, sense perception, emotion, reason, imagination, faith, intuition, and memory. Students must explore a range of ways of knowing, and it is suggested that studying four of these eight in depth would be appropriate.

The WOKs have two roles in TOK:

they underlie the methodology of the areas of knowledge
they provide a basis for personal knowledge.

Discussion of WOKs will naturally occur in a TOK course when exploring how areas of knowledge operate. Since they rarely function in isolation, the TOK course should explore how WOKs work, and how they work together, both in the context of different areas of knowledge and in relation to the individual knower. This might be reflected in the way the TOK course is constructed. Teachers should consider the possibility of teaching WOKs in combination or as a natural result of considering the methods of areas of knowledge, rather than as separate units.

The areas of knowledge

Areas of knowledge are specific branches of knowledge, each of which can be seen to have a distinct nature and different methods of gaining knowledge. TOK distinguishes between eight areas of knowledge.

They are mathematics, the natural sciences, the human sciences, the arts, history, ethics, religious knowledge systems, and indigenous knowledge systems. Students must explore a range of areas of knowledge, and it is suggested that studying six of these eight would be appropriate.

The knowledge framework is a device for exploring the areas of knowledge. It identifies the key characteristics of each area of knowledge by depicting each area as a complex system of five interacting components. This enables students to effectively compare and contrast different areas of knowledge and allows the possibility of a deeper exploration of the relationship between areas of knowledge and ways of knowing.

The Aims

The overall aim of TOK is to encourage students to formulate answers to the question “how do you know?” in a variety of contexts, and to see the value of that question. This allows students to develop an enduring fascination with the richness of knowledge.

Specifically, the aims of the TOK course are for students to:

make connections between a critical approach to the construction of knowledge, the academic disciplines and the wider world
develop an awareness of how individuals and communities construct knowledge and how this is critically examined
develop an interest in the diversity and richness of cultural perspectives and an awareness of personal and ideological assumptions
critically reflect on their own beliefs and assumptions, leading to more thoughtful, responsible and purposeful lives
understand that knowledge brings responsibility which leads to commitment and action.

TOK and International-Mindedness

“Teachers open the door, but you must enter by yourself.”

Chinese proverb

Knowledge can be seen as the shared legacy of mankind, a legacy which has been shaped and influenced by a wide range of cultures. This era of increased global interconnectedness promises unprecedented possibilities for interaction and enhancement of mutual understanding arising from the nurturing of international-mindedness.

The Chinese anticipated a period of “Tai”, a time when communication between individuals and the world at large is totally open and people are receptive to new ideas. The TOK course provides an ideal vehicle for such global exchange and beneficial action through its examination of shared and personal knowledge in an atmosphere of critical and reflective inquiry.

We have inherited rich traditions from indigenous knowledge systems, stretching back to the origins of our societies and cultures. Africa, where the human adventure began, has transmitted a treasure trove of wisdom. The Swahili proverb akili ni mali (“intelligence is wealth”) and the Gikuyu saying, “wisdom is ahead of might”, represent the clear call for the primacy of good thinking for humans to survive and flourish. Early African cultures celebrated diversity, a model for our times. The Asante proverb from West Africa tenabea nyinaa nse reminds us that all dwelling places are not alike and the Swahili kila ndege huruka na mbawa zake encourages every bird to fly with its own wings.

Responsible action underpins this respect for diversity. This is also seen in the Australian aboriginal idea of “Dreamtime”, which promotes a sophisticated ecological perspective, including a celebration of nature’s bounty in multiple art forms and careful stewardship of the earth’s resources.

Ancient Asian civilizations have bequeathed profound insights which continue to guide our thinking. The Chinese were among the first cultures to recognize knowledge (“Shi”), its power, and the deep respect for learning and the wise sage figure permeates educational systems in that part of the world. The understanding of the self is seen as the essential foundation to effective membership and action in ever expanding spheres of community. The Indian concept of “Brahman” links the individual knower to a boldly conceived “universal spirit”, a sense of human and cosmic unity.

The Chinese sage, Confucius, inspired a tradition of inclusive and merit-based education allied to critical thinking: “A gentleman can see a question from all sides without bias”. Inheriting the inquiring spirit of Indian Vedanta, the Buddha boldly linked human suffering and dissatisfaction not only to a craving for physical and worldly pleasures but also to an attachment to ideas, opinions, and beliefs, to be replaced by a more dynamic and open-minded approach to knowledge construction. Greek thinkers introduced the notion of political democracy and the important foundations of modern science and mathematics, while their dramatists confronted audiences with complex characters and multiple perspectives. The deep understandings of these traditions were preserved and enriched in the golden age of Islamic civilization in the 10th to 12th centuries CE, a renaissance of learning and artistic flowering that continues to inspire our knowledge quest.

Students and teachers today are the inheritors of this grand journey. The path ahead, as usual, presents us with both opportunities and challenges. The TOK classroom invites a unique partnership of learning, for global controversies often rest on significant knowledge questions that can provide useful starting points for TOK explorations and TOK, in turn, can contribute significantly to the understanding of these large questions. The IB vision of internationally minded individuals implies a global engagement, embodying a commitment to address these 21st century challenges. TOK exists at the very core of the quest, as we strive toward an enlightened and fulfilled humanity.

Assessment objectives

It is expected that by the end of the TOK course, students will be able to:

identify and analyse the various kinds of justifications used to support knowledge claims
formulate, evaluate and attempt to answer knowledge questions
examine how academic disciplines/areas of knowledge generate and shape knowledge
understand the roles played by ways of knowing in the construction of shared and personal knowledge
explore links between knowledge claims, knowledge questions, ways of knowing and areas of knowledge
demonstrate an awareness and understanding of different perspectives and be able to relate these to one’s own perspective
explore a real-life/contemporary situation from a TOK perspective in the presentation.

From Diploma Programme Theory of knowledge guide, International Baccalaureate, Cardiff, Wales, 2013

Group 6 - The Arts

Group 6 subjects have currently been substituted with Chemistry HL from Group 4.

The Laurier IB Programme offerings are currently under review.

Group 5 - Mathematics

Mathematics – Standard Level

This [Math SL] course caters for students who already possess knowledge of basic mathematical concepts, and who are equipped with the skills needed to apply simple mathematical techniques correctly. The majority of these students will expect to need a sound mathematical background as they prepare for future studies in subjects such as chemistry, economics, psychology and business administration.

The course focuses on introducing important mathematical concepts through the development of mathematical techniques. The intention is to introduce students to these concepts in a comprehensible and coherent way, rather than insisting on the mathematical rigour required for mathematics HL. Students should, wherever possible, apply the mathematical knowledge they have acquired to solve realistic problems set in an appropriate context.

The internally assessed component, the exploration, offers students the opportunity for developing independence in their mathematical learning. Students are encouraged to take a considered approach to various mathematical activities and to explore different mathematical ideas. The exploration also allows students to work without the time constraints of a written examination and to develop the skills they need for communicating mathematical ideas.

The aims are to enable students to

enjoy mathematics, and develop an appreciation of the elegance and power of mathematics
develop an understanding of the principles and nature of mathematics
communicate clearly and confidently in a variety of contexts
develop logical, critical and creative thinking, and patience and persistence in problem-solving
employ and refine their powers of abstraction and generalization
apply and transfer skills to alternative situations, to other areas of knowledge and to future developments
appreciate how developments in technology and mathematics have influenced each other
appreciate the moral, social and ethical implications arising from the work of mathematicians and the applications of mathematics
appreciate the international dimension in mathematics through an awareness of the universality of mathematics and its multicultural and historical perspectives
appreciate the contribution of mathematics to other disciplines, and as a particular “area of knowledge” in the TOK course.

Mathematics and Theory of Knowledge

The Theory of knowledge guide (March 2006) identifies four ways of knowing, and it could be claimed that these all have some role in the acquisition of mathematical knowledge. While perhaps initially inspired by data from sense perception, mathematics is dominated by reason, and some mathematicians argue that their subject is a language, that it is, in some sense, universal. However, there is also no doubt that mathematicians perceive beauty in mathematics, and that emotion can be a strong driver in the search for mathematical knowledge.

As an area of knowledge, mathematics seems to supply a certainty perhaps missing in other disciplines. This may be related to the “purity” of the subject that makes it sometimes seem divorced from reality. However, mathematics has also provided important knowledge about the world, and the use of mathematics in science and technology has been one of the driving forces for scientific advances.

Despite all its undoubted power for understanding and change, mathematics is in the end a puzzling phenomenon. A fundamental question for all knowers is whether mathematical knowledge really exists independently of our thinking about it. Is it there “waiting to be discovered” or is it a human creation?

Students’ attention should be drawn to questions relating theory of knowledge (TOK) and mathematics, and they should be encouraged to raise such questions themselves, in mathematics and TOK classes. This includes questioning all the claims made above. Examples of issues relating to TOK are given in the “Links” column of the syllabus. Teachers could also discuss questions such as those raised in the “Areas of knowledge” section of the TOK guide.

Mathematics and International Dimension

Mathematics is in a sense an international language, and, apart from slightly differing notation, mathematicians from around the world can communicate within their field. Mathematics transcends politics, religion and nationality, yet throughout history great civilizations owe their success in part to their mathematicians being able to create and maintain complex social and architectural structures.

Despite recent advances in the development of information and communication technologies, the global exchange of mathematical information and ideas is not a new phenomenon and has been essential to the progress of mathematics. Indeed, many of the foundations of modern mathematics were laid many centuries ago by Arabic, Greek, Indian and Chinese civilizations, among others. Teachers could use timeline websites to show the contributions of different civilizations to mathematics, but not just for their mathematical content.

Illustrating the characters and personalities of the mathematicians concerned and the historical context in which they worked brings home the human and cultural dimension of mathematics.

The importance of science and technology in the everyday world is clear, but the vital role of mathematics is not so well recognized. It is the language of science, and underpins most developments in science and technology. A good example of this is the digital revolution, which is transforming the world, as it is all based on the binary number system in mathematics.

Many international bodies now exist to promote mathematics. Students are encouraged to access the extensive websites of international mathematical organizations to enhance their appreciation of the international dimension and to engage in the global issues surrounding the subject.

Examples of global issues relating to international-mindedness (Int) are given in the “Links” column of the syllabus.

Group 5 Assessment objectives

Problem-solving is central to learning mathematics and involves the acquisition of mathematical skills and concepts in a wide range of situations, including non-routine, open-ended and real-world problems. Having followed a DP mathematics SL course, students will be expected to demonstrate the following:

Knowledge and understanding: recall, select and use their knowledge of mathematical facts, concepts and techniques in a variety of familiar and unfamiliar contexts.
Problem-solving: recall, select and use their knowledge of mathematical skills, results and models in both real and abstract contexts to solve problems.
Communication and interpretation: transform common realistic contexts into mathematics; comment on the context; sketch or draw mathematical diagrams, graphs or constructions both on paper and using technology; record methods, solutions and conclusions using standardized notation.
Technology: use technology, accurately, appropriately and efficiently both to explore new ideas and to solve problems.
Reasoning: construct mathematical arguments through use of precise statements, logical deduction and inference, and by the manipulation of mathematical expressions.
Inquiry approaches: investigate unfamiliar situations, both abstract and real-world, involving organizing and analysing information, making conjectures, drawing conclusions and testing their validity.

From Diploma Programme Mathematics SL guide, International Baccalaureate, Cardiff, Wales, 2012