(Mathematical Association of America Notes and Reports, No. 34)
Joby Milo Anthony (Editor)
Mathematical Association of America (MAA) | 1994 | 209 páginas | DjVu (3.6 mb)
link
Howard Eves celebrated his eightieth birthday in 1991. To honor that occasion, the University of Central Florida sponsored a conference that focused on the life-long interests of this prominent American mathematician namely, the history of mathematics, the teaching of mathematics, and geometry. Howard is well-known for his contributions to all three areas. He is author of one of the best known books that has ever been writen on the history of mathematics, has a long association with the Mathematical Association of America, and has published extensive research in geometry.
The conference was unique. Conference participants included pre-college mathematics teachers, community college and university teachers, and research mathematicians. Papers were delivered in sessions devoted to the classroom teacher, to the history of mathematics,and to pedagogical and research interests in geometry. Many lectures combined these subjects. This book presents some of those lectures. anyone involved with teaching or producing mathematics can find something in this volume that will be interesting to them.
Some of these papers are speciically for the classroom teacher. They discuss a use of technology, or the organization of a class for some specific purpose. Other articles will provide teachers with examples of mathematical problems or historical episodes that can be used as part of a mathematics class. Still other papers deal with the philosophy of mathematics education. There are articles in this volume that present new insights into some of the history of mathematics, and there are other articles that deal with some new results in geometry. This is really the legacy of Howard Eves. He has been both a mentor for teachers at every level, and a colleague of research mathematicians. And so some part of this volume wil be appropriate for anyone interested in mathematics.
The conference was unique. Conference participants included pre-college mathematics teachers, community college and university teachers, and research mathematicians. Papers were delivered in sessions devoted to the classroom teacher, to the history of mathematics,and to pedagogical and research interests in geometry. Many lectures combined these subjects. This book presents some of those lectures. anyone involved with teaching or producing mathematics can find something in this volume that will be interesting to them.
Some of these papers are speciically for the classroom teacher. They discuss a use of technology, or the organization of a class for some specific purpose. Other articles will provide teachers with examples of mathematical problems or historical episodes that can be used as part of a mathematics class. Still other papers deal with the philosophy of mathematics education. There are articles in this volume that present new insights into some of the history of mathematics, and there are other articles that deal with some new results in geometry. This is really the legacy of Howard Eves. He has been both a mentor for teachers at every level, and a colleague of research mathematicians. And so some part of this volume wil be appropriate for anyone interested in mathematics.
Contents
Howard Eves by Joby Anthony vii
Howard Eves—The High Elf by Donald J. Albers xv
Geometry, History, and Pedagogy
1. Geometry—an Innate Part of Mathematics Page 1
by Bruce Meserve
Professor Meserve begins this article with a short survey of the thirteen books of Euclid's Elements. He then uses Euclid to discuss the geometric representation of numbers and of algebraic concepts leading to the use of geometry to model the physical universe. He concludes with the roles of geometries in mathematics and science, and some objectives for the inclusion of geometry in the undergraduate mathematics curriculum.
2. Connecting Geometry and Number Theory Page 17
by Peter Hilton and Jean Pedersen
In this paper Professors Hilton and Pedersen show how an approximation with any desired degree of accuracy to any regular convex Af-gon (for example a regular pentagon) may be constructed from folding a straight strip of paper in a specific, repetitive pattern. In addition they show how the folding process for the regular convex Af-gons may be modified to construct approximations to all of the regular star polygons. The paper-folding process leads to some surprising results in number theory.
History and Pedagogy
3. The Meanings of Mathematical Proof: On Relations Between History and Mathematical Education Page 41
by Evelyne Barbin
Professor Barbin has been deeply involved with the development of mathematics education programs in France for some time. In this paper she examines carefully the historical notion of "proof" in mathematics, and draws some conclusions about the role of proof for mathematics teachers. She addresses the question "Is proving a means to convince or to enlighten?" She concludes that when teaching "proof" the question "How do you prove this result?" should be replaced with "How have you obtained this result?"
4. Some Mathematical Applications from the History of Cryptology for the Mathematics Classroom Page 53
by Helen Skala
This paper contains a brief history of some simple enciphering and deciphering techniques along with a collection of intriguing little "coding" problems. These problems could be used in a typical high school classroom as "enrichment" exercises that will absorb the attention of bright and inquisitive students.
Problem Solving
5. Reflections of a Problems Editor Page 63
by Clayton Dodge
For many years Professor Dodge was a colleague of Howard Eves at the University of Maine where he assisted Howard with the Elementary Problem Department of the American Mathematical Monthly. Presently he is the editor of the problem section of the Pi Mu Epsilon Journal. This paper contains a collection of problems, mostly in geometry, that have challenged Professor Dodge over the years.
6. Mathematical Creativity in Problem Solving and Problem Proposing II Page 73
by Murray Klamkin
Professor Klamkin is the former coach of the American Mathematical Olympiad team. He presents a very well organized list of problem-solving strategies, and then applies them to a collection of fascinating problems that have appeared in various contests, books, and papers.
Geometry
7. Bi-centric Quadrilaterals and the Pedal n-gon Page 97
by Gordon Bennett
A "pedal triangle" is formed in the following way. From a point P in the interior of a triangle, drop perpendicular line segments to the three sides of the triangle. The triangle whose vertices are the points of intersection of these line segments with the sides of the original triangle is called a "pedal" triangle. (A generalization of this concept produces the definition of a "pedal" n-gon.) A "bi-centric quadrilateral" is a quadrilateral with both a circumscribed and inscribed circle. Both bi-centric quadrilaterals and pedal n-gons have an extensive history. This paper uses advanced euclidean geometry to give some surprising results in both subject areas along with an interesting pedal n-gon conjecture and a new euclidean construction for the bi-centric quadrilateral.
The Fundamental Group as a Beautiful Functor Page 107
by Peter Hilton
Professor Peter Hilton is one of the best known of contemporary American mathematicians. This paper shows how techniques in algebraic topology can establish deep and beautiful links between geometry and algebra. Here are some comments by one of the referees. "It is not very difficult to learn a certain theory so that we are familiar with basic definitions and results and possibly aware of some applications. It is much more difficult to acquire a sufficient understanding of a theory so that we can effectively use its tools in new situations. The third stage is understanding how the studied theory fits within mathematics as a whole. This usually takes years of serious and devoted research and possibly some special qualities of mind. Most of us never really get there. Fortunately, it is possible to get glimpses of that world from mathematicians who have achieved that stage. This paper is in that category."
Calendars, Pacemakers, and Quasicrystals Page 117
by Marjorie Senechal
What is a "blue moon?" The answer may surprise you, and the mathematical connections between "blue" moons, pacemakers, and the study of crystals that don't quite have the symmetry that crystals are supposed to have (quasicrystals) will certainly surprise you. Professor Senechal shows how "circle" maps can be used to unify some aspects of the study of these apparantly disparate phenomena.
Two Families of Cubics Associated with a Triangle Page 127
by Peter Yff
A cubic curve is the locus of points in the plane that satisfy a third-degree polynomial equation. This paper in advanced euclidean geometry establishes some remarkable connections between ordinary triangles and families of cubic curves that are associated with that triangle.
Howard Eves by Joby Anthony vii
Howard Eves—The High Elf by Donald J. Albers xv
Geometry, History, and Pedagogy
1. Geometry—an Innate Part of Mathematics Page 1
by Bruce Meserve
Professor Meserve begins this article with a short survey of the thirteen books of Euclid's Elements. He then uses Euclid to discuss the geometric representation of numbers and of algebraic concepts leading to the use of geometry to model the physical universe. He concludes with the roles of geometries in mathematics and science, and some objectives for the inclusion of geometry in the undergraduate mathematics curriculum.
2. Connecting Geometry and Number Theory Page 17
by Peter Hilton and Jean Pedersen
In this paper Professors Hilton and Pedersen show how an approximation with any desired degree of accuracy to any regular convex Af-gon (for example a regular pentagon) may be constructed from folding a straight strip of paper in a specific, repetitive pattern. In addition they show how the folding process for the regular convex Af-gons may be modified to construct approximations to all of the regular star polygons. The paper-folding process leads to some surprising results in number theory.
History and Pedagogy
3. The Meanings of Mathematical Proof: On Relations Between History and Mathematical Education Page 41
by Evelyne Barbin
Professor Barbin has been deeply involved with the development of mathematics education programs in France for some time. In this paper she examines carefully the historical notion of "proof" in mathematics, and draws some conclusions about the role of proof for mathematics teachers. She addresses the question "Is proving a means to convince or to enlighten?" She concludes that when teaching "proof" the question "How do you prove this result?" should be replaced with "How have you obtained this result?"
4. Some Mathematical Applications from the History of Cryptology for the Mathematics Classroom Page 53
by Helen Skala
This paper contains a brief history of some simple enciphering and deciphering techniques along with a collection of intriguing little "coding" problems. These problems could be used in a typical high school classroom as "enrichment" exercises that will absorb the attention of bright and inquisitive students.
Problem Solving
5. Reflections of a Problems Editor Page 63
by Clayton Dodge
For many years Professor Dodge was a colleague of Howard Eves at the University of Maine where he assisted Howard with the Elementary Problem Department of the American Mathematical Monthly. Presently he is the editor of the problem section of the Pi Mu Epsilon Journal. This paper contains a collection of problems, mostly in geometry, that have challenged Professor Dodge over the years.
6. Mathematical Creativity in Problem Solving and Problem Proposing II Page 73
by Murray Klamkin
Professor Klamkin is the former coach of the American Mathematical Olympiad team. He presents a very well organized list of problem-solving strategies, and then applies them to a collection of fascinating problems that have appeared in various contests, books, and papers.
Geometry
7. Bi-centric Quadrilaterals and the Pedal n-gon Page 97
by Gordon Bennett
A "pedal triangle" is formed in the following way. From a point P in the interior of a triangle, drop perpendicular line segments to the three sides of the triangle. The triangle whose vertices are the points of intersection of these line segments with the sides of the original triangle is called a "pedal" triangle. (A generalization of this concept produces the definition of a "pedal" n-gon.) A "bi-centric quadrilateral" is a quadrilateral with both a circumscribed and inscribed circle. Both bi-centric quadrilaterals and pedal n-gons have an extensive history. This paper uses advanced euclidean geometry to give some surprising results in both subject areas along with an interesting pedal n-gon conjecture and a new euclidean construction for the bi-centric quadrilateral.
The Fundamental Group as a Beautiful Functor Page 107
by Peter Hilton
Professor Peter Hilton is one of the best known of contemporary American mathematicians. This paper shows how techniques in algebraic topology can establish deep and beautiful links between geometry and algebra. Here are some comments by one of the referees. "It is not very difficult to learn a certain theory so that we are familiar with basic definitions and results and possibly aware of some applications. It is much more difficult to acquire a sufficient understanding of a theory so that we can effectively use its tools in new situations. The third stage is understanding how the studied theory fits within mathematics as a whole. This usually takes years of serious and devoted research and possibly some special qualities of mind. Most of us never really get there. Fortunately, it is possible to get glimpses of that world from mathematicians who have achieved that stage. This paper is in that category."
Calendars, Pacemakers, and Quasicrystals Page 117
by Marjorie Senechal
What is a "blue moon?" The answer may surprise you, and the mathematical connections between "blue" moons, pacemakers, and the study of crystals that don't quite have the symmetry that crystals are supposed to have (quasicrystals) will certainly surprise you. Professor Senechal shows how "circle" maps can be used to unify some aspects of the study of these apparantly disparate phenomena.
Two Families of Cubics Associated with a Triangle Page 127
by Peter Yff
A cubic curve is the locus of points in the plane that satisfy a third-degree polynomial equation. This paper in advanced euclidean geometry establishes some remarkable connections between ordinary triangles and families of cubic curves that are associated with that triangle.
History
11. Symmetry Through the Ages - Highlights from the Page 139
History of Regular Polyhedra
by Benno Artmann
There are only five "regular" solids; the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. Three of these, the tetrahedron, the octahedron, and the icosahedron, have faces that are equilateral triangles. One, the cube, has faces that are squares; and one, the dodecahedron, has faces that are regular pentagons. In this paper Professor Artmann discusses the history of these regular polyhedra from the time when they first appear in the writings of Greek mathematicians in antiquity to their modern connections with abstract algebra.
12. G. A. Miller: Mathematician, Historian of Mathematics, and Mathematics Educator Extraordinaire Page 149
by Richard Davitt
Certainly one of the most prolific and influential mathematicians of the first part of the twentieth century was Professor G. A. Miller. This paper concentrates on Professor Miller's 44-year career at the University of Illinois. Along with some anecdotes about the Professor, is a compendium of his contributions to the founding of the American Mathematical Society and the Mathematical Association of America. This is an excellent summary of the life and work of a brilliant researcher, a dedicated teacher, and an active, responsible member of the community of scholars.
13. The History of the Mathematical Representation of Electromagnetic Theory Page 159
by Domina Spencer and Uma Shama
Much of the mathematical representation of electromagnetic theory began in the last half of the nineteenth century. This paper traces the development of that theory from the early work of Oersted, Ampere, Gauss, and others through the publication of Maxwell's Treatise on Electricity and Magnetism in 1873. It is shown that current formulations of electromagnetic theory depend on the interplay of two distinct traditions. Gauss suggested the idea of developing the entire subject from a single retarded equation for the force between moving charges which depends on the relative velocity of source and receiver. Maxwell developed the mathematical formulation of the field concepts of Faraday. These mathematical representations are now a fundamental tool of contemporary physics.
14. From Howard Eves to Africa Counts to Multicultural Mathematics Education Page 171
by Claudia Zaslavsky
Before there was much attention being paid to multicultural education there was Africa Counts. This paper gives a brief description of how Professor Zaslavsky came to write one of the earliest and best known of the books now used as references for multicultural mathematics education. Pedagogy
15. An Alternative to the Conventional Wisdom of a Term Paper in History of Mathematics Survey Courses Page 175
by Joe Albree
We certainly want our mathematics students to be literate, to be able to use language to communicate accurately and correctly. And they must have something to say. How do they attain such knowledge and skill? Professor Albree lists a number of problems with conventional term papers that he has encountered. Then he states some of his goals for students in a history of mathematics course: (1) confidence with basic library research skills, (2) organizational and writing skills, and (3) succesful completion of a major piece of
writing. This paper then explains how these goals are met with a "progression of literature reports."
16. DERIVEing a New Approach to Calculus Page 183
by Steven Amgott and Jane Friedman
There is much discussion today about the use of computers in mathematics classes. This paper explains how computer laboratories and the DERIVE software can be used to teach the beginning calculus courses.
17. Alternatives to the Lecture in Teaching Page 191
by William Bonnice
Teachers usually teach the way that they were taught. Lectures have been the mainstay of mathematics teaching for many years. In this paper Professor Bonnice discusses a variety of teaching techniques that can supplement or replace lectures. Included are cooperative groups, journals, and "writing to learn."
Calculus in Context: Integrating Calculus Concepts and Finite Mathematics in a First College Course Page 201
by Stephen Turner
This paper discusses ways to introduce the concepts of finite mathematics and calculus in a mathematics course designed for business majors. The use of computer systems utilizing computer algebra software, graphics programs, and spreadsheets is a key element in the course design. A very nice example of building a mathematical model of a simple business decision situation illustrates most of the methodology. Professor Turner was a student of Howard Eves at the University of Maine. He concludes this paper with some recollections of Howard as a teacher.
11. Symmetry Through the Ages - Highlights from the Page 139
History of Regular Polyhedra
by Benno Artmann
There are only five "regular" solids; the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. Three of these, the tetrahedron, the octahedron, and the icosahedron, have faces that are equilateral triangles. One, the cube, has faces that are squares; and one, the dodecahedron, has faces that are regular pentagons. In this paper Professor Artmann discusses the history of these regular polyhedra from the time when they first appear in the writings of Greek mathematicians in antiquity to their modern connections with abstract algebra.
12. G. A. Miller: Mathematician, Historian of Mathematics, and Mathematics Educator Extraordinaire Page 149
by Richard Davitt
Certainly one of the most prolific and influential mathematicians of the first part of the twentieth century was Professor G. A. Miller. This paper concentrates on Professor Miller's 44-year career at the University of Illinois. Along with some anecdotes about the Professor, is a compendium of his contributions to the founding of the American Mathematical Society and the Mathematical Association of America. This is an excellent summary of the life and work of a brilliant researcher, a dedicated teacher, and an active, responsible member of the community of scholars.
13. The History of the Mathematical Representation of Electromagnetic Theory Page 159
by Domina Spencer and Uma Shama
Much of the mathematical representation of electromagnetic theory began in the last half of the nineteenth century. This paper traces the development of that theory from the early work of Oersted, Ampere, Gauss, and others through the publication of Maxwell's Treatise on Electricity and Magnetism in 1873. It is shown that current formulations of electromagnetic theory depend on the interplay of two distinct traditions. Gauss suggested the idea of developing the entire subject from a single retarded equation for the force between moving charges which depends on the relative velocity of source and receiver. Maxwell developed the mathematical formulation of the field concepts of Faraday. These mathematical representations are now a fundamental tool of contemporary physics.
14. From Howard Eves to Africa Counts to Multicultural Mathematics Education Page 171
by Claudia Zaslavsky
Before there was much attention being paid to multicultural education there was Africa Counts. This paper gives a brief description of how Professor Zaslavsky came to write one of the earliest and best known of the books now used as references for multicultural mathematics education. Pedagogy
15. An Alternative to the Conventional Wisdom of a Term Paper in History of Mathematics Survey Courses Page 175
by Joe Albree
We certainly want our mathematics students to be literate, to be able to use language to communicate accurately and correctly. And they must have something to say. How do they attain such knowledge and skill? Professor Albree lists a number of problems with conventional term papers that he has encountered. Then he states some of his goals for students in a history of mathematics course: (1) confidence with basic library research skills, (2) organizational and writing skills, and (3) succesful completion of a major piece of
writing. This paper then explains how these goals are met with a "progression of literature reports."
16. DERIVEing a New Approach to Calculus Page 183
by Steven Amgott and Jane Friedman
There is much discussion today about the use of computers in mathematics classes. This paper explains how computer laboratories and the DERIVE software can be used to teach the beginning calculus courses.
17. Alternatives to the Lecture in Teaching Page 191
by William Bonnice
Teachers usually teach the way that they were taught. Lectures have been the mainstay of mathematics teaching for many years. In this paper Professor Bonnice discusses a variety of teaching techniques that can supplement or replace lectures. Included are cooperative groups, journals, and "writing to learn."
Calculus in Context: Integrating Calculus Concepts and Finite Mathematics in a First College Course Page 201
by Stephen Turner
This paper discusses ways to introduce the concepts of finite mathematics and calculus in a mathematics course designed for business majors. The use of computer systems utilizing computer algebra software, graphics programs, and spreadsheets is a key element in the course design. A very nice example of building a mathematical model of a simple business decision situation illustrates most of the methodology. Professor Turner was a student of Howard Eves at the University of Maine. He concludes this paper with some recollections of Howard as a teacher.
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