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This book is for college and secondary school 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 the 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 secondary school mathematics, and the second on its use in university mathematics. So teachers planning a discussion of logarithms, will find here the historical background of that idea along with suggestions for incorporating that history in the development of the idea in class. Teachers of abstract algebra will benefit by reading the three articles in the book dealing with aspects of that subject and considering their ideas for presenting groups, rings, and fields.
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.
This book grew out of a conference in Norway which brought together mathematicians and mathematics educators from a dozen countries who were interested in the use of the history of mathematics as a pedagogical tool in the teaching of mathematics. .Since the conference which provided the genesis of this book took place in Norway near the home where Niels Henrik Abel spent his final days, the book's title comes from a note scribbled in one of Abel's notebooks: "It appears to me that if one wants to make progress in mathematics one should study the masters." The authors hope that readers will benefit from Abel's advice and show their students how they too can LEARN FROM THE MASTERS.
Part 1 History in School Mathematics
While many would agree that the teaching of mathematics at all levels can be enriched by historical reflection, perhaps that consensus is even stronger when directed at the secondary school level. At this level, historical enrichment can have a profound effect! For it is at the secondary or high school level that students first experience the power of mathematics and begin to realize the wide scope of its applications and possibilities. Hopefully, this cognitive impact can be stimulating, resulting in an anticipation and enthusiasm for a deepening of mathematical knowledge, but, disconcertingly, it can also be intimidating, especially for a student who has lacked obvious structure in his or her mathematical learning. It is in this latter instance, particularly, that the history of mathematics can supply a structure of understanding relating reasons with results. History can provide a logic between the definition of a mathematical concept and its application or, more historically correct, between the application and the definition-theory of a concept. The following discussions seek to identify and clarify some techniques and pedagogical approaches for using the history of mathematics in secondary teaching.
Shmuel Avital makes a broad case for the learning/teaching benefits resulting from historical awareness. Building on his teacher education experiences in Israel, Avital is a strong proponent in teachers' studies on and about the history of mathematics.
Phillip Jones examines several concepts related to the general idea of "number" and describes their evolution. He then notes how a familiarity with this evolution lends itself to understanding how mathematics and mathematical research work.
Frank Swetz contributes three papers to this section: the first advocated the use of actual historical problems in classroom instruction; the second documents the development of trigonometry as it could be revealed to students; the third considers the history of a particular topic in mathematical modeling. All of these contributions also help to illustrate the multicultural aspects of the development of mathematics, yet another dimension that can be used to make mathematics learning "interesting."
Are logarithms are dead topic? Not according to John Fauveel who shows that through historical considerations there is a lot of mathematical life still left in this topic.
Victor Katz also discusses logarithms and supports their importance as a mathematical topic. He reviews the historical evolution of the concept, concluding that a Napieran approach to teaching logarithms may be well suited to today's classrooms.
Jan van Maanen shows how a fourteenth century legal problem can supple a bounty of worthwhile mathematical learning experiences; how seventeenth century drawing instruments lead to modern problems on conic sections and how the struggles of Torricelli, Sluse and Huygens in attempting to understand the mathematical properties of infinitely extended solids can help students today. Van Maanen's discussions further emphasize the rich variety of historical materials available for classroom use.
Thus, at the school level, whether one is interested in improving calculation, geometric reasoning, analytic thinking or trigonometry applications, the history of mathematics can assist in these efforts.
Part I: History in School Mathematics
1. History of Mathematics can Help Improve Instruction and Learning by Shmuel Avital
2. The Role in the History of Mathematics of Algorithms and Analogies by Phillip S. Jones
3. Using Problems from the History of Mathematics in Classroom Instructions by Frank J. Swetz
4. Revisiting the History of Logarithms by John Fauvel
5. Napeir's Logarithms Adapted for Today's Classroom by Victor J. Katz
6. Trigonometry Comes Out of the Shadows by Frank J. Swetz
7. Alluvial Deposits, Conic Sections, and Improper Glasses, or History of Mathematics Applied in the Classroom by Jan A. can Maanen
8. An Historical Example of Mathematical Modeling: The Trajectory of a Cannonball by Frank J. Swift
Part II: History in Higher Mathematics
9. Concept of Function-Its History and Teaching by Man-Keung Siu
10. My Favorite Ways of Using History in Teaching Calculus by V. Fredrick Rickey
11. Improved Teaching of the Calculus Through the Use of Historical Materials by Michel Helfgott
12. Euler and Heuristic Reasoning by Man-Keung Siu
13. Converging Concepts of Series: Learning from History by Joel P. Lehmann
14. Historical Thoughts on Infinite Numbers by Lars Mejibo
15. Historical Ideas in Teaching Linear Algebra by Victor J. Katz
16. Wessel on Vectors by Otto B. Bekken
17. Who Needs Vectors? By Karen Reich
18. The Teaching of Abstract Algebra: An Historical Perspective by Israel Kleiner
19. Toward the Definition of an Abstract Ring by David M. Burton and Donovan H. Can Osdol
20. In Hilbert's Shadow: Notes Toward a Redefinition of Introductory Group Theory by Anthony D. Gardiner
21. An Episode in the History of Celestial Mechanics and Its Utility in the Teaching of Applied Mathematics by Eric J. Aiton
22. Mathematical Thinking and History of Mathematics by Man-Keung Siu
23. A Topics Course in Mathematics by Abe Shenitzer
Niels Henrik Abel (1802-1829): A Tribute
About the Authors