# Download e-book A Century of Mathematics: Through the Eyes of the Monthly (MAA Spectrum Series)

Hernandez: A Pioneer. Using Humor to Combat Inequities. Jacqueline M. Dewar, Sarah J. Sylvia T. Back Matter Pages About this book Introduction This collection of refereed papers celebrates the contributions, achievements, and progress of female mathematicians, mostly in the 20 th and 21 st centuries. In inspiring and informative chapters, the authors featured in this volume reflect on the accomplishments of women in mathematics, showcasing the changes in mathematical culture that resulted as more women obtained tenure-track and tenured academic positions, received prestigious awards and honors, served in leadership roles in professional societies, and became more visibly active in the mathematical community.

Readers will find discussions of mathematical excellence at Girton College, Cambridge, in the late 19 th and early 20 th centuries; of perseverance by Polish women in mathematics during and after World War II and by Black women in mathematics in the United States from the s onward; and of the impact of outreach programs ranging from EDGE's promotion of graduate education to the Daughters of Hypatia dance performances. These essays provide compelling reading for a wide audience, including mathematicians, historians of science, teachers of mathematics, and students at the high school, college, and graduate levels.

Editors and affiliations. Janet L. Beery 1 Sarah J. Greenwald 2 Jacqueline A. Jensen-Vallin 3 Maura B. Mast 4 1. Buy options. I played the role of moderator, asking questions like: "All right, what choices do we have? Are there any others? Which of these seem promising? So, it's between b and c. Which one do you want to work on? What if we began the construction with side a? L We knew that we could construct the locus of points that made a fixed angle opposite a one locus for the vertex A.

We knew we had one locus for the point P. We needed a third piece of information about the triangle. Determine the locus of the center of the inscribed circle, given the side a and the variable vertex A that makes an angle a opposite a. Which one should we pursue? Suggestion ; We're on very shaky ground, and have no foundation for making a good judgment. It may be time for making some rough sketches.

The result of some empirical work may suggest that we choose one alternative over the other. It may even suggest an hypothesis. We try choices I and II respectively, in figures 4. If we were unsure about the conjecture, we could do a more accurate sketch. Let us not demean empirical exploration. Subproblem : Prove that the locus of P, given fixed a and variable A, is a circle that has a as a chord. Question : How do we prove such things? What do we know about circles and chords?

In this context, we know that the set of points that make a fixed angle opposite a given line segment chord is a circle. We knew that the point P lies on the intersection of che three angle bisectors I ' ' - of T, and this led to the argument in figure 4. Once we had the inscribed circle, we could finish the construction as suggested in figure 4.

I think so, although I am certainly not about to recommend that we solve every problem this way. There are times when we simply need to present information, when students need to master routine procedures, and when for any of a number of good reasons , we must ask students 'to learn and discover by themselves. Indeed, our most important function as teachers is to train our students to learn and think by themselves.

The discovery was prompted by need. It- was suggested by some empirical work. It was surprising, and it would prove useful in other constructions. The experience they had, in discovering that minor result, is similar to the experience that we have when we are engaging in reaj mathematics. It allows them to see mathematics as a living, breathing discipline in which discovery is both. What about the false starts, reversals, blind alleys, etc.?

The fact is that doing mathematics involves all of them.

Doing mathematics successfully. Discussions like the one above provide them with a means of seeing how they- can do so in a sensible and efficient way. We owe it to those who will be the mathematicians of the future, to those who will use mathematics, and to those who would like a "feeling" for mathematics, to introduce them to the problem solving experience. We hope and believe that the problem sol ving' approach to mathematicf,througnout tne curriculum a through a variety of problem courses, will convey to our students the excitement and beauty of mathematics.

To the degree that we train our students to think independently and to use the knowledge at their disposal, we will have succeeded as teachers. We solicited extensive contributions from experts in each of the types of problem solving courses represented in our survey, and received in addition a large number of suggestTons from those teachers of. Whatever facet of the subject interests you, you will find some useful resources listed below.

Of course, our listing a source in this bibliography does not constitute, an endorsement in any sense. Our familiarity with and enthusiasm for each reference can best be seen in the annotations. Similarly, exclusion should not be taken as' a negative comnent: in any sampling, some valuable sources will be overlooked.

Our coverage of the literature in languages other than English is particularly sparse. If a refer- ence that you find especially valuable does not, appear in the bibliography, please call it to our attention. The problem solving literature is vast, and may seem overwhelming. We have tried to make "initial entry" as' easy as possible.

The three sections of thf? The categories are as follows. E,I listing given to Polya's - Mathematical Discovery, is self-explanatory. The problem section, now edited by G. Alexanderson and Dale Mugler, offers a rich variety of problems and solutions from elementary to advanced levels. For a subscription, write: c A. Washington, D. Its primary focus is on classroom suggestions for elementary school teachers. The November Vol.

The editor of the Problem Section is E. The cost varies for different categories of membership in the CMS. Write: CP. It is a good way to keep on top of current events in problem solving. Bound volumes from are available. Many of our references come from The Olympiad Corners: 3 Vol.

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Write: F. It is the official journal of the Fibonacci Association. The problem editors" are A. Hillman Elementary Section and R. Whitney Advanced Section. Write to: Mr. This is an informal publication of the Ont-ario Institlite fqr Studies in Education. It is published 8 times a year and is sold in sets of 10 copies. It is especially designed to provide continuous material for children's individual reading in mathematics and to supplement material studied at school by problems, games, and investigations in which the child can be involved on his own. For students in the grades range and also for bright children in grade 4 or 'for older children taking a general mathematics or remedial arithmetic course.

The editors are Shmuel Avital and Mary Stager. Write to: Professor B. Write to: Baywood Publishing Co. Research articles on all aspects of problem solving appear with some regularity. The oroblem editor is M. It is published 4 times a vear and is included in the membership fee. Although it does not have a formal problem section, it does have a Problem Bureau, and many problems can be extracted from the papers and notes it publishes.

Write to: Honorary Treasurer, Math. Journals, p. There are 5 issues a year. Write to: A. This slim newsletter was published 8 times a year by the National Council of Teachers of Mathematics and contained a Competition Corner edited by George Berzsenyi.

Its primary focus is on useful ideas for classroom teachers at the secondary school level. Problem solving is a frequent topic of discussion. Write to: Adm. It has a problem section edited by E. It is published twice a year. There is a Problem Corner edited by Kenneth M. Write to; Douglas W. It is the official journal of the Pi Mu Epsilon honorary mathematical fraternity. There is an extensive problem section edited by Clayton W. Conferences and publications are announced and reviewed. A broad spectrum of interdisciplinary work is covered.

Box Philadelphia. PA P. It is an interdisciplinary offering, with ideas about prob-lem solving in engineering, medicine, mathematics, etc. Write: Donald Woods Dept. The Problem Department is edited by N. Kuenzi and Bob Prieh'pp. The March 1. Write: Dale M. Box Indiana 'University of Pennsylvania Indiana. PA 66 62 Journals, p.

It is published 5 times a year. Washington, 0. A The UMAP project is concerned with disseminating information and classroommaterials dealing with applications of "lathematics. Books The literature of problem books, and of books about problem solving, is immense. The best general introduction to problem solving, at virtually any level, comes from the pen of Polya. The Contest Problem Book s.

Modeling is too diverse for us to point to a single source; for the best overview of the area, see the CUPM's Recommendations for a General Mathematical SciencesProgram. For a view of research with classroom applications, see Schoenfeld's Mathematical Problem Solving. The intersection of problem solving and remediation is recent and small, but rapidly growing; we look for a forthcoming report from an M. Washington: Mathematical Association of America, Norton, Gen,cre:.

E,I This book offers a broad discussion of creativity, with many interesting examples. Aggarwal, R. San Francisco: Holden-Day, Aichele, D. The essays can serve as focal points for discussion in a teacher training class. Aleksandrov, A. Boston: MIT, Gen,His:E,I This three volume set is a rich source of material on tfie history and the background of mathematics.

Anderson, B. Rec:E Anderson, R. Woburn, MA: Butterworth, Antonov, N. Moscow: Mir Publishers, Gen,con:E Many problems from arithmetic, algebra, geometry and trigonometry. AHs, R. Belmont, CA: Pitman, Arnold, B. Huntington, NY: Krieger, Averbach, Bonnie and Chefn, Orin. San Francisco: W.

Freeman, Gen,Tch,Lit:E A non-threatening and well-written introductory text on recreational mathematics, using a "problem solving" format as a means of introducing the subject. The book is designed to serve as the text for an introductory level college course. Ic should be on everyone's bookshelf. Barbeau, E. Barnard, Douglas St. London: Pan -Books, Barnard, S. Martin's Press, Alg,' con:E,I Recommended reading by M.

Klamkin for a Mathematical Olympiad Program. Barr, S. New York: Thomas Crowel 1 , 1 Wl Barr, Stephen. New York: Crowel 1 , Rec:E Barr, Stephen. New York: MacMillan, Rec:E Bartholomew, D. Bartlett, M. New York: Methuen, ERIC " 67 Books, p. Birmingham: The University of Alabama, Worth, New York, Tch,Lit:E Beckenbach, E. Berlin: Springer-Verlag, Con:E,l Recommended reading by M. Beqle, E. Tch,Res:E,I A brief "state of the art" summary of research in mathematics education.

Beiler, Albert, H. New York: Simon and Schuster , His,Num,Rec Bender, E. New York: Wiley, Benson, R. Nt York: McGraw-Hill, , pages.

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Gen,con,geo:I Recommended by M. Berlekamp, E. ERIC 68 4 Books, p. Clarendon Press, Oxford, 1 Tch:E A text for prospective elementary school teachers with a great and conscious debt to Polya. Black, M. Block, James H. Bloom, B. Chicago: The University of Chicago Press, Bloom was one of the first researchers to focus on what students actually do when they work on such problems - a far cry from the logical analysis that we expect them to perform. A look at what actually goes on in the students' heads is enlightening. Boden, M. New York: Basic Books, AI,Lit Research on computer simulations of intelligent performance sheds light on thinking processes.

Bogen, 0. Gen,psy Two interesting papers naving to do with how the left side of vour cerebral cortex differs from your right side. Crudely: One" half does algebra - the other half does geometry. Bottema, 0. Groningen, Netherlands: Wolters-Noordhoff , ERIC 69 Books, p. Translated by J. Res,Psy Brams, S. The Free Press, Brooke, Maxay. Rec,E,I Don't let the title fool you, these puzzles are not just for kids.

Brooke, Maxey. New York: , Dover publications, Some are difficult. If you like these kinds of puzzles, look here first. The first few pages give a few hints on how to solve this type of problem. Bittinger, Marvin L. Reading, MA: Addison -Wesley, Braswel 1 , J. Brousseau, Brother Alfred. San Jose: The Fibonacci Association, Many nice problems involving Fibonacci numbers.

Useful for high school students or college freshmen. Con:E,I Some good contest problems are here, grouped as "elementary" and "advanced". Many of the problems here are quite clever and unusual. Bruner, O. Cambridge: Harvard University Press, This is one of his most important books. New York: McGraw-Hill, Gen,Lit:E,I The book contains problems ostensibly accessible to 10th and 11th graders. The more thought- provoking problems would keep college freshmen and sophomores busy.

Burki,ll, O-. Cambridge: Cambridge University Press, Con:E,I A collection of practice problems for the Cambridge University scholarship examination in mathematics. The problems vary from the routine to the unusual , covering algebra, geometry, trigonometry, calculus, mechanics and "misc. Bosto"nl Little, Brown and Company, Bushaw, Donald et al. Glenview: Scott, Foresman, T5en,Tch,Lit:E Very good little book.

It's out of print now, but worth looking for.

Buzen, T. London: BBC Publications, Gen:E Ideas and suggestions for organizing material, improving your memory and learning. Carrier, G. MAA summer seminar lecture notes-. Mathematical Association of America, Carroll, L. Dover New York: Lit,Rec:E Carroll's recreations are just as charming as you would expect - and there is interesting mathematics behind them. New York: Dover, Charosh, Mannis. Top Churchill, E. Richard and Linda. New York: Scholastic Book Services, 1 Clark, C. New York: Wilev. Modi me Lum ftoaeling panel recommends this.

Coffman, C. New York: Academic Press, Coleman, J. Press Press, Collea, F. Gen, Rem, Lit: E Materials to translate Piaget's ideas about concrete and formal thinking into the classroom. Conference Board of the Mathematical Sciences. Ginn, Gen,Tch: A beautiful collection of essays by distinguished mathematicians and educators - on axiomatics. Jones, Lax, Pollak, Rosenbloom and others. Conrad, S.

Con:E,I A good source of challenging problems. Academic Press: New York, Num,Rec:E ,. Oxford: Oxford University Press, His, Lit Cooney, T. New York: Barnes and Noble, Gen,con,geo:E,I Recommended by M. A good problem source. Nathan A. New York: Dial Press, Lit,rec:E,I Mainly essays on mathematical topics, but there are many cute problems included. Court, Nathan A.

New York: Chelsea, Klamkin for a Mathematical O'ympiad Program. Courant, Richard andRobbins, Herbert. Oxford: University Press, Gen,Rec:E,I A classic introduction to the spirit of the discipline. Joe, Higgins, 0. Worthington, Ohio: C. Jones Publishing Co. New York: Free Press, Boston: Birkhauser, Gen:I A delightful, broad introduction to the notion of what doing mathematics is all about.

There are sections on major mathematical results, on the "mathematical spirit," on philosophical controversies about the nature of mathematics, and much, much more. New York: 'Harper and Row, Cre:I de Bono, E. New York: Pelican Books, 1 CreTEl Many interesting examples of creative thinking. OeGrazia, Joseph. Rec:E This is a collection of elementary problems in recreational mathematics, ranging from "logic problems" to cryptarithmetic , etc. Oinesman, Howard P. Rec:E OiPrima, R. Dombrowski, 0. Palo Alto: Creative Publ ications, Tch,Lit:E This is a good book which gives problems and a fair discussion of them.

The authors list problem solving techniques. It seems to have been written for high school mathematics teachers. Oomoryad, Aleksandr Petrovich; translated by Halina Moss. Dover, New York, This is a book that deserves to be much better known than it seems to be. It is eclectic, it is spread over years of history, and it ranges in difficulty from elementary arithmetic to material that is frequently the subject of graduate courses.

If "a cows graze fa fields bare in c days, a' cows graze fa' fields bare in days, a" cows graze fa" fields bare in c" days, what relation exists between the nine magnitudes a to c"? It is assumed that all fields provide the same amount of grass, that the daily growth of the fields remains constant, and that all the cows eat the same amount each day. The style and the attitude ace old- fashioned, but many of the problems are of the eternally interebting kind; this is an excellent book to browse in.

ReciEl This book has puzzles. Like Dudeney' s other collections, it has a variety of puzzles of varying levels of difficulty. Dudeney, H. Rec:E7T] The puzzles are of every degree of difficulty and varied in character. Read the introduction to this book; it will give you some thoughts of a professional problemist. Beverly Hills, California: Litton, Rec:E Dym, C. Dynkin, E. New York: Gordon and Breach, Boston: D. Heath, Num:E,I Dynkin, E. Translated by Norman Whaland and Olga.

Boston: Heath, Pro:I Recommended by M. Klamkin for a Mathematice. Emmet, Eric Revel 1. New York: Barnes and Noble Books, Rec:E,I "Not only for experts" the subtitles says. Well, they're not only for beginners either. The difficulty of the problems is given in the table of contents. I Emmet, Eric Revel 1. Engel , Arthur ed. Stuttgart: Ernst Klett, Erdos, P. Num:A Ernst, G. New Yorkl Academic Press, This book describes its evolution. Though technical, the detailed level of discussion is quite interesting. Eves, Howard. His,Lit:E Eves. Howard and Starke, E.

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Monthly" Vol! Nothing more than that need be said. Boston: Al lyn and Bacon, 1 Geo:E,I Recommended by M. Faddeev, D. ERIC 77 Books, p. New York: Macmillan, Geo: I Reconmended by M. Garden City, NY: Doubleday. Fixx, James F. Frauenthal, J. UMAP Monograph, Friedland, Aaron J.

Rec:E,I A number of unusual and interesting puzzles. Friedman, B. San Francisco: Hoi den-Day, Friedrichs, K. Tch,res:E A delineation of the behaviorist position of how people learn. It's important to know, because these ideas have shaped the curriculum. New York: The Viking Press, Rec:E A clever set pf problems for the layman; some are old classics and some jre less familiar.

Gardner, Martin. Freeman and Co. I Gardner, M. New York: Knopf, Rec:El Gardner, Martin.

Rec:E Gardner, Martin. New York: Penguin, T Rec:E Gardner, M. New York: Simon and Schuster, New YorFi Simon and Schuster, Freeman and Company, Portland, ME: J. Weston Walch, Gelbaum, B.

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Ana,gen:I,A Each of these counterexamples is the solution to a good problem, for example: Does there exist. There are some exceptionally nice examples here. Glaoser, Georges. Glaeser, Georges. Glazman, I. The book is, in effect, a new kind of textbook of finite-dimensional linear algebra and linear analysis. The chapters follow one another in logical dependence, just as they do in textbooks of the conventional kind: Linear operators. Bilinear functional s, Normed spaces, etc.

It gives definitions and related explanatory background material with some care. The main body of the book consists of problems; they are all formulated as assertions, and the problem is to prove them. The proofs are not in the book. There are references, but the reader is told that he will not need to consult them. ERLC 84 80 Books, p.

The ingenious idea of the authors is to present to a beginning student the easy case, the transparent case, the motivating case, the finite-dimensional case, the purely algebraic case of some of the deepest analytic facts that functional analysts have discovered. The subjects discussed include spectral theory, the Toeplitz- Hausdorff theorem, the Hahn-Banach theorem, partially ordered vector spaces, moment problems, dissipative operators, and many other such analytic sounding results. A beautiful course could be given from this book I would love to give it , and a student brought up in such a course could become an infant prodigy functional analyst in no time.

Example: the canonical prbjection from a vector space to a quotient space is called a "contraction", and what most people call a contraction is called a "compression. One can only hope that a sequel will bring us up to date annual updates for the competitions can be found in the Monthly ; see the "articles" section. Gold, H. Goldin, G. Res:E,I This research volume classifies problem solving variables into four categories dealing with 1 syntax, 2 content and context, 3 structure, and 4 heuristic behaviors.

Goldberg, S. Graham, L. New York: Dover7n New York: Dover Publishers, Palo Alto: Creative Publications, Gregory, John, and Seymour, Dale. Greitzer, Samuel L. Con:E,I A good source of challenging problems, discussed by an able and dedicated problemist. Grosche, Gunter. Grosswald, E. Washington: National Council of Teachers of Mathematics, New York: Springer- Verlag, Num:I,A Haberman, R. Hadaraard, Jacques. His,Psy:E A detailed "gesta. This book is of substantial historical interest, though of questionable practical or theoretical value.

Hadwiger, Hugo and DeBrunner, Hans. Translated by Victor Klee with a new chapter and other materials supplied by the translator. New York: Reinhold Pub! Cre:E'] Wider view on creativity than discussed by most texts. Hall and Knight. London: Macmillan and Co. Alg:E,I Recommended reading for M. Klamkin's Mathematical Olympiad Program. A wonderful collection of rather old-fashioned but amusing problems, many from old Tripos exams.

Hardy, G. Littlewood and G. Cambridge: The University Press, Ana: I, A A classic. Recommended by M. Hardy, Godfrey H. Num:I,A Another classic. Harnadek, Anita. Harvey, John G. Madison, WI : University of Wisconsin, These studies, supplemented by a review of thirty- one parallel studies, give a good sense of the ithematics education literature of the ' s. Hatfield, L. Tch,Res:E,I Five papers discussing research and instruction in problem solving. Hayes, J. Homewood, IL: Dorsey Press, Psy,cre:E,I An introduction to the area.

Heath, Royal Vale. Rec : E Heofford, Phillip. Hill, Claire Conley. New York: Nichols Publ ishing Company, Gen,Lit:E,I This volume offers extensive annotations for more than different sources in the problem solving literature. Topics covered include -"problem solving in using associations," "problem solving in forming and testing hypotheses", "problem solving as a goal", etc. The coverage is broad and of general interest. Hill , Thomas, Ed. Gen,con:E Hill, Thomas, Ed. Gen,con:E Hillman, Abraham P. Boston: Allyn and Bacon, Combinatorial topics and inequalities are featured prominently. Hlavaty, Julius H.

Gen:E 27 enrichment topics for academically talented students in grades Each chapter provides a wealth of problems. Holt, John. New York: Pitman, TchTE Holt writes about his experience as a grade school teacher, but his descriptions of classroom exchanges raise issues at all levels of instruction.

Honsberger, Ross. Rec,lit:E,I Whether it be a discussion of gems, morsels, plums or ingenuity, Honsberger offers fascinating problems and nice discussions of them. ERIC 88 84 Books, p. Rec, Lit:E,I Beautifully written exposition of some classic problems. Rec,Lit:I Honsberger, Ross. Hoppensteadt, R. Howson, AG , Ed. Cambridge: Cambridge University Pre'is, Tch,rec:E,I Essays on mathematical education, including problem solving, originating in the 2nd International Congress on Mathematical Education. Piaget and Polya were featured speakers at the Congress.

Hughes, Barnabas, O. Gen,tch,lit:E This is a text book for teaching heuristics. As the title suggests, it is a straightforward introduction to the tools of the trade.

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## 90th Birthday - The Ross Mathematics Program

Hunter, J. Rec:El Hunter, J. Rec:E This book has some puzzles but its content is mainly dis- cussions of some of the popular puzzle types. It's well worth reading for these discussions alone. New York: Dover Publications, Inhelder, B. Psy :E The book is an absolute classic in psychology. While it's not "directly" related to mathematical problem solving at the high school or college level, it is critical for us to understand that children are not simply miniature versions of adult computers.

Instrument Society of America, Jacobs, H. San Francisco; W. Lit,rec:E A charming, entertaining introduction to some interesting mathematical ideas. Johnson, Donovan. Pasadena, CA: Webster, Geo Judson, Horace F. It's part of a large package including films on the topics and a teacher's resource book.

See the films if you can. Kalomitsines, Spyros P. Lit,rec:E,I Easy reading, well written and accessible to practically anyone. A nice introducation to elementary mathematics. Lit,rec:I Beautiful, unusual problems. Kemeny, J. Gen,lit:E,I A pioneer text with clever problems. Boston: MIT Press, Kennedy, Joe and Thomas, Diane.

Lit:E 50 imaginative and humorous story problems designed to encourage students to read. The level of the mathematics is easy enough that middle school and high school students shouldn't be intimidated. Kespohl, Ruth Carwell. Geo:E Kilnatrick, J. Tch,res:E This series of volumes presents translations of research, classroom procedures, and theoretical discussions about mathematical learning and teaching.

The range of articles provides insights into the developments in this field over the past several decades and gives an idea of current practice in the USSR. The Soviet "teaching experiments" have had an impact on mathematics education research in the U. Klahr, D. Klambauer, G. New York: Dekker, It is an excellent and exciting book. It does have some faults, of course, including some misprints and some pointless repetitions, and the absence of an index is an exasperating feature that makes the book much harder to use than it ought to be.

It is, howaver, a great source of stimulating questions, of well known and not 87 Books, p. It should be on the bookshelf of every problem lover, of every teacher of analysis from calculus on up , and, for that matter, of every serious student of the subject. Inequalities, Sequences and series, and Real functions.

Here are some examples from each that should serve to illustrate the range of the work, perhaps to communicate its flavor, and, I hope, stimulate the appetite for more. A simple but striking oddity is this statement: if m and n are distinct positive integers, then "The chapter on inequalities contains many of the famous oneS' H 51der, Minkowski, Jensen , and many others that are analytically valuable but somewhat more specialized and therefore somewhat less famous. Acta Academica Scientiarum.