This book is for high school and college teachers who want to know how they can use the history of mathematics as a pedagogical tool to help their students construct their own knowledge of mathematics. Often, a historical development of a particular topic is the best way to present a mathematical topic, but teachers may not have the time to do the research needed to present the material. This book provides its readers with historical ideas and insights which can be immediately applied in the classroom. The book is divided into two sections: the first on the use of history in high school mathematics, and the second on its use in university mathematics. The articles are diverse, covering fields such as trigonometry, mathematical modeling, calculus, linear algebra, vector analysis, and celestial mechanics. Also included are articles of a somewhat philosophical nature, which give general ideas on why history should be used in teaching and how it can be used in various special kinds of courses. Each article contains a bibliography to guide the reader to further reading on the subject.
Part I. History In School Mathematics: 1. History of mathematics can help improve instruction and learning; 2. The role in the history of mathematics of algorithms and analogies; 3. Using problems from the history of mathematics in classroom instruction; 4. Revisiting the history of logarithms; 5. Napier's logarithms adapted for today's classroom; 6. Trigonometry comes out of the shadows; 7. Alluvial deposit, conic sections, and improper glasses, or history of mathematics applied in the classroom; 8. An historical example of mathematical modeling: the trajectory of a cannonball; Part II. History In Higher mathematics: 9. Concept of function - its history and teaching: my favorite ways of using history in teaching calculus; 10. Improved teaching of the calculus through the use of historical materials; 11. Euler and heuristic reasoning; 12. Converging concepts of series: learning from history; 13. Historical thoughts on infinite numbers; 14. Historical ideas in teaching linear algebra; 15. Wessel on vectors; 16. Who needs vectors?; 17. The teaching of abstract algebra: a historical perspective: toward the definition of an abstract ring; 18. In Hilbert's shadow: notes toward a redefinition of introductory group theory; 19. An episode in the history of celestial mechanics and its utility in the teaching of applied mathematics; 20. Mathematical thinking and history of mathematics; 21. A topics course in mathematics; 22. Niels Henrik Abel (1802-1829): a tribute.
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