Computations and Computing Devices in Mathematics Education Before the Advent of Electronic Calculators.

By: Freiman, ViktorContributor(s): Volkov, AlexeïMaterial type: TextTextSeries: eBooks on DemandMathematics Education in the Digital Era Ser: Publisher: Cham : Springer, 2019Copyright date: ©2018Description: 1 online resource (464 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9783319733968Subject(s): Computers-HistoryGenre/Form: Electronic books.Additional physical formats: Print version:: Computations and Computing Devices in Mathematics Education Before the Advent of Electronic CalculatorsDDC classification: 004.09 LOC classification: L1-991Online resources: Click here to view this ebook.
Contents:
Intro -- Contents -- Editors and Contributors -- Introduction -- 1 History of Computations, Computing Devices, and Mathematics Education from the Teaching and Learning Perspective: Looking for New Paths of Investigation -- Abstract -- References -- Middle East, India, Greece, and Byzantine Empire in Antiquity and Middle Ages -- 2 Computation in Early Mesopotamia -- Abstract -- Introduction -- Some Background -- Sargonic Mathematics (2340-2200 BCE) -- Mathematical Tables from the Early Dynastic III Period (2600-2340 BCE) -- Mathematical Problem Texts from Šuruppak (ca. 2500 BCE) -- Archaic Mathematics (ca. 3000 BCE) -- Calculation -- Appendix: Periodization -- References -- 3 Computational Techniques and Computational Aids in Ancient Mesopotamia -- Abstract -- Introduction: The Familiar -- Why? -- Addition and Abacus -- The Centesimal System and the Decadic Notation -- Third Millennium Difficult Division -- Long-Time Developments-Summary and Conclusion -- References -- 4 Interpreting Tables of the Arithmetical Introduction of Nicomachus Through Pachymeres' Treatment of Arithmetic: Preliminary Observations -- Abstract -- Introduction: Studying Greek Mathematics in a Late Antique Context -- Preliminary Remarks -- Basic Cultural Notions from Late Antiquity in Relation to the Study of Arithmetic -- What We Know about the Teaching and Learning of Arithmetic in Late Antiquity and Some Added Remarks -- Tables as Conveyors of Mathematical Knowledge in Nicomachus' Arithmetical Introduction -- Numbers in Relation and Their Classification into Kinds -- The Text Introducing the Generic Table Showing the Numbers in Relation -- Discussion of the Text -- The Function of Tables in Nichomachus -- Arithmetical Problem-Solving: Contextualizing the Heuristic Tool Function of the Nicomachean Generic Table -- Conclusion -- Acknowledgements.
Appendix: Tables and the ancient readers of Nicomachus -- References -- Sources -- Modern Studies -- 5 The So-Called "Dust Computations" in the Līlāvatī -- Abstract -- Introduction -- Līlāvatī: Structure of the Work -- Setting Down of Numbers for Calculation -- Multiplication -- Example of Multiplication -- Division -- Example of Division -- Square -- Example of Squaring -- Square Root -- Cube -- Example of Cube -- Cube Root -- Eight Operations of Fractions -- Eight Operations with Zero -- Concluding Remarks -- Acknowledgements -- References -- 6 Reading Algorithms in Sanskrit: How to Relate Rule of Three, Choice of Unknown, and Linear Equation? -- Abstract -- Introduction -- Description of Nārāyaṇa's Algorithms -- Choice of Unknown -- Tabular Settings -- The Rule of Three -- Relation Between Rule of Three and Unknowns -- The Rule of Three -- Choosing the Unknown for a Linear Equation -- Conclusion -- Appendix -- References -- Primary Sources -- Secondary Sources -- Far East: China, Korea, Japan -- 7 Chinese Counting Rods: Their History, Arithmetic Operations, and Didactic Repercussions -- Abstract -- Introduction -- Counting Rods: Design and History -- The Material and the Shape -- The Time of Appearance of the Counting Rods -- The Counting Board: Myth or Reality? -- Time of the Extinction of the Counting Rods in China -- Two Comments on the Computations with the Counting Rods in Japan -- Operations with Counting Rods -- Representation of Numbers with Counting Rods -- The Reason for Alternation of Orientation of the Counting Rods -- A Remark on the Range of Numbers that Could Be Represented with the Counting Rods -- Negative Numbers -- Arithmetical Operations with the Counting Rods -- Multiplication -- Multiplication in the Sun zi suan jing -- An Example of Multiplication Provided in the Sun Zi Suan Jing.
Operations of Multiplication in the Mathematical Compendium of Zhou Shuxue -- Division -- Division in the Sun zi suan jing: the General Algorithm -- The Division in the Sun Zi Suan Jing: An Example -- An Example from the Treatise of Zhou Shuxue -- Conclusions -- Appendix: On the picture of a counting instrument from the Suan fa tong zong 算法統宗 -- Bibliography -- Part A: Primary Sources -- Part B: Translations of Chinese Texts -- Part C: Secondary Works -- 8 Interpreting Algorithms Written in Chinese and Attempting the Reconstitution of Tabular Setting: Some Elements of Comparative History -- Abstract -- Introduction -- Description of the Algebraic Procedure -- Description of Tabular Setting -- Representation of Polynomials on the Counting Surface -- Division and Equation -- Comparative Perspective for Conclusion -- Appendix -- Translation -- References -- Primary Sources -- Secondary Sources -- 9 Same Rods, Same Calculation? Contextualizing Computations in Early Eighteenth-Century Korea -- Abstract -- Introduction -- Computation of Mathematical Officials in Writings of Nine and One -- Computation of Literati in Summary of Nine Numbers -- Conclusion -- References -- Primary Sources -- Secondary Sources -- 10 The Education of Abacus Addition in China and Japan Prior to the Early 20th Century -- Abstract -- Introduction -- The Abacus and Its Structure -- The Era of the Use of Verse -- Era of the Elaboration of Finger Techniques-Motion and Fingering -- Conclusions -- References -- 11 Teaching Computation in 19th-Century Japan: The Transition from Individual Coaching on Traditional Devices at the End of the Edo Period (1600-1868) to Lectures on Western Mathematics During the Meiji Period (1868-1912) -- Abstract -- Computation Teaching Before the Meiji Era.
Teaching Elementary Operations with the Abacus: The Jinkōki 塵劫 (Inexhaustible Treatise) by Yoshida Mitsuyoshi 吉田光由 (1598-1672), Published for the First Time in 1627 -- Computation with Counting Rods: A High-Level Algebraic Tool for Problem Solving -- Revolutionizing the Teaching of Computation During the Meiji Era -- The First Textbooks of Arithmetic According to Western Methods: The Life and Death of the Traditional Devices -- The Contributions of Fujisawa -- Conclusion -- Bibliography -- Early Modern Europe and Russia -- 12 Counting Devices in Russia -- Abstract -- Introduction -- The Counting Instruments Used in Russia Prior to the Early Eighteenth Century -- The Early History of the Russian Shyoty: The Extant Evidence -- "Russian Abacus" of John Tradescant -- Peder Von Haven's Description of Operations with the Schyoty -- The Origin of the Russian Schyoty: Spasskiĭ's and Simonov's Hypothesis -- Russian Abacus of the Seventeenth Century -- Conversion of Fractions -- Identities in the Book of sokha script of the year 7137 [=1629] -- Conclusions -- References -- A. Russian Mathematical Textbooks of the 17th Century -- B. Primary Works in Other Languages -- C. Russian and Soviet School Manuals Related to Schyoty -- D. Secondary Works -- 13 A Short History of Computing Devices from Schickard to de Colmar: Emergence and Evolution of Ingenious Ideas and Technologies as Precursors of Modern Computer Technology -- Abstract -- Introduction -- Initial Steps Toward Automatization of Calculations in the 17th Century -- Calculators of Morland and Leibniz: When Practical Needs, Philosophy, Mathematics, and Technological Creativity Push the Boundaries of Automatic Computing -- Getting Closer to Having Computations Done Automatically by Machines: The 19th-Century Breakthrough.
Educational Implications of the Arithmometer from the Late 19th Century to the First Half of the 20th Century -- Concluding Remarks -- References -- 14 Computation Devices in Nineteenth-Century Mathematics Instruction in Europe -- Abstract -- Introduction -- The Logarithmic Tables -- France -- England -- Italy -- The Netherlands -- Germany -- The slide rule -- France -- The Instruments -- Use in Schools -- Other European Countries -- Conclusion -- Acknowledgements -- References -- 15 Toward a History of the Teaching of Calculation in Russia -- Abstract -- Introduction -- A Brief History of the Teaching of Arithmetic in Russia -- The Symbolism of Calculation: Number Alphabets -- The Technology of Calculation: The Russian Abacus -- Certain Examples of the Teaching of Calculation -- Addition and Subtraction of Integers in Kurganov's "Number Book" -- The Addition of Integers in F. Busse's Textbook -- The Addition of Integers in V. Agloblin's Textbook -- The Addition of Integers in the Textbook by A. F. Malinin and K. P. Burenin -- The Addition of Integers in N. S. Popova's Textbook -- Toward a Methodological Analysis of Changes in School Textbooks -- Mental Calculation in Russian Schools -- Conclusion -- References -- Conclusion -- 16 The Unsettling Playfulness of Computing -- Abstract -- A Frolicsome Start -- Computing: An Art in Decay? -- Playful Computations -- Unsettling -- Some Educational Suggestions, Because Why Not? -- References -- 17 Calculating Aids in Mathematics Education Before the Advent of Electronic Calculators: Didactical and Technological Prospects -- Abstract -- Didactical Prospects -- Influence of Psychological Theories -- Influence of Behaviorists -- Influence of Gestalt Psychologists -- Influence of Mathematicians and Cognitive Psychologists -- Technological Prospects: Aspects of Design.
Teaching Machines: Aspects of Design and Use in Schools.
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Intro -- Contents -- Editors and Contributors -- Introduction -- 1 History of Computations, Computing Devices, and Mathematics Education from the Teaching and Learning Perspective: Looking for New Paths of Investigation -- Abstract -- References -- Middle East, India, Greece, and Byzantine Empire in Antiquity and Middle Ages -- 2 Computation in Early Mesopotamia -- Abstract -- Introduction -- Some Background -- Sargonic Mathematics (2340-2200 BCE) -- Mathematical Tables from the Early Dynastic III Period (2600-2340 BCE) -- Mathematical Problem Texts from Šuruppak (ca. 2500 BCE) -- Archaic Mathematics (ca. 3000 BCE) -- Calculation -- Appendix: Periodization -- References -- 3 Computational Techniques and Computational Aids in Ancient Mesopotamia -- Abstract -- Introduction: The Familiar -- Why? -- Addition and Abacus -- The Centesimal System and the Decadic Notation -- Third Millennium Difficult Division -- Long-Time Developments-Summary and Conclusion -- References -- 4 Interpreting Tables of the Arithmetical Introduction of Nicomachus Through Pachymeres' Treatment of Arithmetic: Preliminary Observations -- Abstract -- Introduction: Studying Greek Mathematics in a Late Antique Context -- Preliminary Remarks -- Basic Cultural Notions from Late Antiquity in Relation to the Study of Arithmetic -- What We Know about the Teaching and Learning of Arithmetic in Late Antiquity and Some Added Remarks -- Tables as Conveyors of Mathematical Knowledge in Nicomachus' Arithmetical Introduction -- Numbers in Relation and Their Classification into Kinds -- The Text Introducing the Generic Table Showing the Numbers in Relation -- Discussion of the Text -- The Function of Tables in Nichomachus -- Arithmetical Problem-Solving: Contextualizing the Heuristic Tool Function of the Nicomachean Generic Table -- Conclusion -- Acknowledgements.

Appendix: Tables and the ancient readers of Nicomachus -- References -- Sources -- Modern Studies -- 5 The So-Called "Dust Computations" in the Līlāvatī -- Abstract -- Introduction -- Līlāvatī: Structure of the Work -- Setting Down of Numbers for Calculation -- Multiplication -- Example of Multiplication -- Division -- Example of Division -- Square -- Example of Squaring -- Square Root -- Cube -- Example of Cube -- Cube Root -- Eight Operations of Fractions -- Eight Operations with Zero -- Concluding Remarks -- Acknowledgements -- References -- 6 Reading Algorithms in Sanskrit: How to Relate Rule of Three, Choice of Unknown, and Linear Equation? -- Abstract -- Introduction -- Description of Nārāyaṇa's Algorithms -- Choice of Unknown -- Tabular Settings -- The Rule of Three -- Relation Between Rule of Three and Unknowns -- The Rule of Three -- Choosing the Unknown for a Linear Equation -- Conclusion -- Appendix -- References -- Primary Sources -- Secondary Sources -- Far East: China, Korea, Japan -- 7 Chinese Counting Rods: Their History, Arithmetic Operations, and Didactic Repercussions -- Abstract -- Introduction -- Counting Rods: Design and History -- The Material and the Shape -- The Time of Appearance of the Counting Rods -- The Counting Board: Myth or Reality? -- Time of the Extinction of the Counting Rods in China -- Two Comments on the Computations with the Counting Rods in Japan -- Operations with Counting Rods -- Representation of Numbers with Counting Rods -- The Reason for Alternation of Orientation of the Counting Rods -- A Remark on the Range of Numbers that Could Be Represented with the Counting Rods -- Negative Numbers -- Arithmetical Operations with the Counting Rods -- Multiplication -- Multiplication in the Sun zi suan jing -- An Example of Multiplication Provided in the Sun Zi Suan Jing.

Operations of Multiplication in the Mathematical Compendium of Zhou Shuxue -- Division -- Division in the Sun zi suan jing: the General Algorithm -- The Division in the Sun Zi Suan Jing: An Example -- An Example from the Treatise of Zhou Shuxue -- Conclusions -- Appendix: On the picture of a counting instrument from the Suan fa tong zong 算法統宗 -- Bibliography -- Part A: Primary Sources -- Part B: Translations of Chinese Texts -- Part C: Secondary Works -- 8 Interpreting Algorithms Written in Chinese and Attempting the Reconstitution of Tabular Setting: Some Elements of Comparative History -- Abstract -- Introduction -- Description of the Algebraic Procedure -- Description of Tabular Setting -- Representation of Polynomials on the Counting Surface -- Division and Equation -- Comparative Perspective for Conclusion -- Appendix -- Translation -- References -- Primary Sources -- Secondary Sources -- 9 Same Rods, Same Calculation? Contextualizing Computations in Early Eighteenth-Century Korea -- Abstract -- Introduction -- Computation of Mathematical Officials in Writings of Nine and One -- Computation of Literati in Summary of Nine Numbers -- Conclusion -- References -- Primary Sources -- Secondary Sources -- 10 The Education of Abacus Addition in China and Japan Prior to the Early 20th Century -- Abstract -- Introduction -- The Abacus and Its Structure -- The Era of the Use of Verse -- Era of the Elaboration of Finger Techniques-Motion and Fingering -- Conclusions -- References -- 11 Teaching Computation in 19th-Century Japan: The Transition from Individual Coaching on Traditional Devices at the End of the Edo Period (1600-1868) to Lectures on Western Mathematics During the Meiji Period (1868-1912) -- Abstract -- Computation Teaching Before the Meiji Era.

Teaching Elementary Operations with the Abacus: The Jinkōki 塵劫 (Inexhaustible Treatise) by Yoshida Mitsuyoshi 吉田光由 (1598-1672), Published for the First Time in 1627 -- Computation with Counting Rods: A High-Level Algebraic Tool for Problem Solving -- Revolutionizing the Teaching of Computation During the Meiji Era -- The First Textbooks of Arithmetic According to Western Methods: The Life and Death of the Traditional Devices -- The Contributions of Fujisawa -- Conclusion -- Bibliography -- Early Modern Europe and Russia -- 12 Counting Devices in Russia -- Abstract -- Introduction -- The Counting Instruments Used in Russia Prior to the Early Eighteenth Century -- The Early History of the Russian Shyoty: The Extant Evidence -- "Russian Abacus" of John Tradescant -- Peder Von Haven's Description of Operations with the Schyoty -- The Origin of the Russian Schyoty: Spasskiĭ's and Simonov's Hypothesis -- Russian Abacus of the Seventeenth Century -- Conversion of Fractions -- Identities in the Book of sokha script of the year 7137 [=1629] -- Conclusions -- References -- A. Russian Mathematical Textbooks of the 17th Century -- B. Primary Works in Other Languages -- C. Russian and Soviet School Manuals Related to Schyoty -- D. Secondary Works -- 13 A Short History of Computing Devices from Schickard to de Colmar: Emergence and Evolution of Ingenious Ideas and Technologies as Precursors of Modern Computer Technology -- Abstract -- Introduction -- Initial Steps Toward Automatization of Calculations in the 17th Century -- Calculators of Morland and Leibniz: When Practical Needs, Philosophy, Mathematics, and Technological Creativity Push the Boundaries of Automatic Computing -- Getting Closer to Having Computations Done Automatically by Machines: The 19th-Century Breakthrough.

Educational Implications of the Arithmometer from the Late 19th Century to the First Half of the 20th Century -- Concluding Remarks -- References -- 14 Computation Devices in Nineteenth-Century Mathematics Instruction in Europe -- Abstract -- Introduction -- The Logarithmic Tables -- France -- England -- Italy -- The Netherlands -- Germany -- The slide rule -- France -- The Instruments -- Use in Schools -- Other European Countries -- Conclusion -- Acknowledgements -- References -- 15 Toward a History of the Teaching of Calculation in Russia -- Abstract -- Introduction -- A Brief History of the Teaching of Arithmetic in Russia -- The Symbolism of Calculation: Number Alphabets -- The Technology of Calculation: The Russian Abacus -- Certain Examples of the Teaching of Calculation -- Addition and Subtraction of Integers in Kurganov's "Number Book" -- The Addition of Integers in F. Busse's Textbook -- The Addition of Integers in V. Agloblin's Textbook -- The Addition of Integers in the Textbook by A. F. Malinin and K. P. Burenin -- The Addition of Integers in N. S. Popova's Textbook -- Toward a Methodological Analysis of Changes in School Textbooks -- Mental Calculation in Russian Schools -- Conclusion -- References -- Conclusion -- 16 The Unsettling Playfulness of Computing -- Abstract -- A Frolicsome Start -- Computing: An Art in Decay? -- Playful Computations -- Unsettling -- Some Educational Suggestions, Because Why Not? -- References -- 17 Calculating Aids in Mathematics Education Before the Advent of Electronic Calculators: Didactical and Technological Prospects -- Abstract -- Didactical Prospects -- Influence of Psychological Theories -- Influence of Behaviorists -- Influence of Gestalt Psychologists -- Influence of Mathematicians and Cognitive Psychologists -- Technological Prospects: Aspects of Design.

Teaching Machines: Aspects of Design and Use in Schools.

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