Alternative Forms of Knowing (in) Mathematics -- CONTENTS -- PREFACE -- CONTRIBUTORS -- CELEBRATING DIVERSITY, REALIZING1 ALTERNATIVES: An Introduction -- NOTES -- REFERENCES -- PART I: MATHEMATICS AND POLITICS OF KNOWLEDGE -- NOTES -- REFERENCES -- 1. MATHEMATICS AND ACCOUNTING IN THE ANDES BEFORE AND AFTER THE SPANISH CONQUEST -- ACCOUNTING AND NUMERATION IN EUROPEAN DOUBLE-ENTRY BOOKKEEPING -- THE KHIPU AND ITS METHODS OF INFORMATION REGISTRY -- CONQUEST, COLONIZATION, AND THE CONFRONTATION BETWEEN KNOTAND SCRIPT-BASED TEXTS -- CONCLUSIONS -- NOTES -- REFERENCES -- 2. CONTEMPORARY INDIGENOUS EDUCATION1: Thoughts for American Indian Education in a 21st-Century World -- THE CONTEMPORARY DILEMMA OF AMERICAN INDIAN EDUCATION -- AMERICAN EDUCATION FROM A TRIBAL PERSPECTIVE -- INDIGENOUS APPROACHES TO EDUCATION ARE VIABLE ALTERNATIVES -- INDIGENOUS EDUCATION AND ITS ROLE IN INDIVIDUAL TRANSFORMATION -- SOCIAL CONSCIOUSNESS AND INDIGENOUS EDUCATION -- NOTES -- REFERENCES -- 3. CRISIS AS A DISCURSIVE FRAME IN MATHEMATICS EDUCATION RESEARCH AND REFORM: Implications for Educating Black Children -- THE CRISIS NARRATIVE IN MATHEMATICS EDUCATION -- FRAMES AND FRAMING IN RESEARCH PARADIGMS -- THE IMPLICATIONS OF THE CRISIS FRAME IN MATHEMATICS EDUCATION -- CRISIS RHETORIC IN CRITICAL PERSPECTIVES -- ALTERNATIVE FRAMINGS IN MATHEMATICS EDUCATION RESEARCH: IMPLICATIONS FOR BLACK LEARNERS -- CONCLUSION -- REFERENCES -- 4. WHOSE LANGUAGE IS IT?: Reflections on Mathematics Education and Language Diversity from Two Contexts -- OUR CONTEXTS OF RESEARCH: ARIZONA AND CATALONIA -- SOCIO-POLITICAL ISSUES IN MULTILINGUAL MATHEMATICS CLASSROOMS -- DYNAMICS OF POWER IN THE BILINGUAL MATHEMATICS CLASSROOM -- Data from Catalonia -- Data from Arizona -- BILINGUAL STUDENTS' VIEWS OF ACCESS TO MATHEMATICS -- Data from Catalonia -- Data from Arizona -- FINAL REMARKS -- NOTES.

REFERENCES -- PART II: ETHNOMATHEMATICS -- NOTES -- REFERENCES -- 5. CONSULTING THE DIVINE: The (Ethno)mathematics of Divination -- SYNCHRONICITY AND DIVINATION -- THE I CHING -- Yin, Yang and Change -- The Trigrams and the Hexagrams -- The Coin Method of Using the I Ching -- The Yarrow Stalk Method -- The Probabilities -- Complexity -- MO:TIBETAN DIVINATION -- Complexity -- RUNES -- The Elder Futhark -- Rune Casting -- TAROT -- The Major Arcana -- The Minor Arcana -- The Readings -- Complexity -- REFERENCES -- 6. MAP-MAKING IN SÃO PAULO, SOUTHERN BRAZIL: History, Social Diversity, and Indigenous Peoples' Rights -- MATHEMATICS IS IMPORTANT FOR THE AUTONOMY OF INDIGENOUS PEOPLES -- First Week - Identity, Space, and Time: Who Are We and Where Are We Located? -- INDIGENOUS PEOPLES IN SÃO PAULO: FROM CULTURAL TRADITIONS TO THE QUESTION OF AUTONOMY -- Final Thoughts, Week 1: Autonomy, Self-determination, and Indigenous Peoples'Rights -- Guarani Reciprocity and the Economy of Gift-Exchange -- Is There Only One Mathematics? Or, Are There Different Mathematics? -- Xavante Mathematics: "The Two of Us Together" -- Palikur Mathematics: Geometry is Everywhere -- The Palikur numerical system -- Discoveries from the Discussion on Indigenous Mathematics -- Presentation of Terena knowledge by Alício Terena (Kopenoti Indigenous Area) -- Presentation of Pankararu Knowledge, by Dimas Nascimento -- Presentation of Krenak Knowledge by Lidiane Oliveira (Vanuire Indigenous Land) -- Presentation of Kaingang knowledge by Ilson Iaiati (Icatu Indigenous Land) -- Mathematics, Justice, and Respect for Human Rights -- ACKNOWLEDGEMENTS -- REFERENCES -- 7. DEVELOPING AN ALTERNATIVE LEARNING TRAJECTORY FOR RATIONAL NUMBER REASONING, GEOMETRY, AND MEASURING BASED ON INDIGENOUS KNOW -- BACKGROUND AND CONTEXT -- BRIEF THEORETICAL OVERVIEW.

TOWARDS AN INDIGENOUS KNOWLEDGE BASEDMATHEMATICAL LEARNING TRAJECTORY -- Current Curriculum Developments -- The Square Is the Central Shape within IK-HLT -- ASSESSMENT DEVELOPMENT -- ASSESSMENT REQUIREMENTS -- CONCLUSION -- NOTES -- REFERENCES -- 8. IN SEEKING A HOLISTIC TOOL FOR ETHNOMATHEMATICS: Reflections on Using Ethnomodeling as a Pedagogical Action for Uncovering Ethnomathematical Practices -- ETHNOMATHEMATICS AS A HOLISTIC APPROACH TO MATHEMATICS EDUCATION -- ETHNOMODELING: A DEFINITION -- PLACING AN ETHNOMODELING PERSPECTIVE INTO THE MATHEMATICS CURRICULUM -- ETHNOMATHEMATICS AND MATHEMATICAL MODELING -- ETHNOMODELING -- EXAMPLES OF ETHNOMODELING -- DEFINITION OF ETHNOMODELS -- Measuring Land -- The Symmetrical Freedom Quilts -- Some Considerations about Ethnomodeling -- FINAL CONSIDERATIONS -- NOTES -- REFERENCES -- 9. FROM ETHNOMATHEMATICS TO ETHNOCOMPUTING: Indigenous Algorithms in Traditional Context & Contemporary Simulation -- FROM CSDTS TO PCSDTS -- OBSERVATIONS OF STUDENTS USING PCSDTS -- Observer 1 -- Observer 2 -- ANALYSIS OF PRELIMINARY RESULTS. -- CONCLUSION -- ACKNOWLEDGEMENT -- REFERENCES -- PART III: LEARNING TO SEE MATHEMATICALLY -- NOTES -- REFERENCES -- 10. THE WORK OF SEEING MATHEMATICALLY1 -- THE LIVING/LIVED WORK OF SEEING MATHEMATICALLY -- Case 1: What Makes a Cube a Cube? -- Case 2: Mathematical Seeing while Proving the Angle Sum of a Triangle -- OF PERCEPTUAL WORK AND ACCOUNTS OF PERCEPTION -- IMPLICATIONS FOR PEDAGOGY -- DIRECTIONS FOR RESEARCH -- NOTES -- REFERENCES -- 11. RUNNING THE NUMBERS: A Conversation -- RESONANCE -- NOTES -- REFERENCES -- 12. TO KNOW HOW TO SEE: The Realities of Learning and Teaching Mathematics -- SEEING THE MATHEMATICS -- SEEING THE STUDENT -- SEEING ONESELF -- CONCLUSIONS -- REFERENCES -- PART IV: MATHEMATICS EDUCATION FOR SOCIAL JUSTICE -- REFERENCES.

13. QUANTITATIVE FORM IN ARGUMENTS1 -- CHANGING THE FORM TO UNDERSTAND LARGE QUANTITIES -- QUANTITATIVE FORM IN ARGUMENTS -- QUANTITATIVE FORM IN AURAL ARGUMENTS -- QUANTITATIVE FORM IN ARTISTS' ARGUMENTS -- ADDITIONAL CONSIDERATIONS ABOUT QUANTITIES IN ARGUMENTS -- CONCLUSION -- NOTES -- REFERENCES -- 14. CONNECTING COMMUNITY, CRITICAL, AND CLASSICAL KNOWLEDGE IN TEACHING MATHEMATICS FOR SOCIAL JUSTICE1 -- INTRODUCTION -- CONNECTING COMMUNITY, CRITICAL, AND CLASSICAL KNOWLEDGE -- An Example of Connecting the Three C's in Practice -- CONCLUSION -- POSTSCRIPT -- NOTES -- REFERENCES -- EPILOGUE: Why Bother about Diversity of Mathematical Practices? -- REFERENCES.