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Our university has an Honors section of our "liberal arts mathematics" course. Typically 10-20 students enroll each Fall, with most of them extremely bright, but lacking the interest and/or mathematics background of many of the students we usually see in calculus.

I've taught this section twice already: once using the really good book on voting and apportionment methods by Jonathan Hodge and Richard Klima, and another topology course centered around Jeff Weeks' The Shape of Space.

Next Fall, however, I'd like to run more of a reading seminar course, in which students read and discuss several shorter books and papers aimed at a general audience. I'm having trouble however coming up with a good list of titles. So far I'm thinking of Flatland and Innumeracy. Not bad choices, but I was hoping for some more "mathematical" readings.

Any suggestions of books and/or papers? Maybe some specific expository articles in the MAA's Monthly?


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A nice book is Körner's "The pleasures of counting". –  Andres Caicedo Feb 20 '12 at 0:33
The recent book "Magical Mathematics" of Diaconis and Graham might appeal to some students. It describes some math related to cards and juggling. –  Dan Fox Feb 20 '12 at 10:21
Yau's "The Shape of Inner Space" is really fascinating, it's a little tricky at times with some advanced geometry, but I think they do a pretty good job making it digestible, and it basically explains the mathematics behind string theory. I've been reading it recently. Most people are interested in knowing about what is going on in string theory. –  Jon Beardsley Feb 20 '12 at 20:33
What about adapting ideas from ocw.mit.edu/courses/mathematics/… "street-fighting mathematics" ? –  Kjetil B Halvorsen Mar 4 '12 at 0:30
Related MO question: mathoverflow.net/questions/28695/… –  Timothy Chow Mar 5 '12 at 15:52
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17 Answers

In my opinion, one of the most important concepts to discuss in a liberal arts math course is the notion of mathematical proof—what it is, why mathematicians put so much emphasis on it, whether it is overrated, and how the concept has evolved over time. There are several ways to approach this subject.

  1. Give examples of proofs. This MO question as well as this one provide some nice examples. I'd also recommend the MAA series of books on Proofs Without Words.

  2. Discuss the role of computers and experiment in mathematics. Jonathan Borwein has co-authored several books on experimental mathematics, e.g,. The Computer as Crucible. Though much of the mathematical content may be too advanced, the introductions to these books are extremely lucid and valuable.

  3. Discuss the foundations of mathematics. My top recommendation in this category would be Torkel Franzen's Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. Again, some sections of the book may be too technical, but there are plenty of extremely well-written and valuable non-technical sections.

Even among the educated public, one frequently encounters people who have no concept of how unique mathematical proof is, who think that computers have put mathematicians out of business, and who have heard just enough about Gödel's Theorem to be dangerous. The above readings should go a long way towards dispelling these common misconceptions.

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really great comment. –  Kerry Mar 3 '12 at 3:30
There is a relevant letter to the EMS Newsletter by David Wells, ems-ph.org/journals/newsletter/pdf/2012-03-83.pdf pointing out that, contrary to some answers here, the notion of logical proof is not so unique to mathematics, as is shown by the popularity of many games and puzles; one should mention also Sudoku. I also argue that in mathematics teaching the notion of proof should be introduced partly as explanation and also in situations where the result needs a proof to convince of its truth (and beauty!). Euclidean geometry is great for this; think of the nine point circle. –  Ronnie Brown Jun 3 '12 at 10:57
@Ronnie: I think Wells's point is not that logical proof is not unique to mathematics; rather, it is that mathematics (in the form of games and puzzles) is not so foreign to the public consciousness. "Sudoku is math" is a truism for most mathematicians, but the public doesn't think of Sudoku as having anything to do with math. But never mind that; I basically agree with you that under my #1 suggestion above ("Give examples of proofs"), games and logic puzzles are a good source of examples. –  Timothy Chow Jun 10 '12 at 21:40
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You could perhaps base it on articles by Martin Gardner and others. I taught a non-honors course where the first half was group theory and then the second half was evenly divided between Flatland and probability. The group theory section used Gardner's "Ambidextrous Universe" for readings, the second section used Flatland, of course, and the third section used Keith Devlin's book "The unfinished game". All three sections were augmented with articles by Gardner. You can find my lecture notes and other course materials on my website. http://www.colby.edu/personal/s/sataylor/teaching/S10/MA111/

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I should also say that the first time I taught the course I had them also read Weyl's "Symmetry". That wasn't such a good idea-- it was way too hard for them. –  Scott Taylor Feb 19 '12 at 21:46
Another idea I hope to be using sometimes: Using design of experiments ($2^p$, $2^{p-q}$ factorial designs, see the book Box, Hunter & Hunter: Statistics for experimenters) to better the design of paper airplanes. –  Kjetil B Halvorsen Mar 4 '12 at 0:34
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Years ago in the UK system, Ronnie Brown and I developed a course that we called Mathematics in Context. We used Davis and Hersh the Mathematical Experience, as one of the background texts, but the real novelty and success of the course was that we often risked letting the students determine the topic for discussion for the next session. It worked well. We invited external speakers to talk about mathematics in various different areas, (e.g. a quality control engineer gave a training session of the use of statistics in quality control... and the problems of communicating the ideas to a non-mathematically literate workforce.)

Assessment was by essays or other material, e.g. some student wanted to prepare some exhibition boards on self-similarity and fractals. (They were excellent.) You can find various documents relating to this on Ronnie Brown's webpages. The main thing to do is to get the students thinking and discussing mathematics, not just its history but the problems that it relates to even its philosophy (but avoid the mathematical logic view of Mathl. Philosophy)

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I should say that allowing students the opportunity to discuss professional matters in mathematics was like opening the flood gates! It was really relished, and writing about mathematics helped, as some expressed, them to come to terms with their own attitudes to mathematics, and improved their work in other courses. See the article on "The methodology of mathematics" on my web page. –  Ronnie Brown Feb 22 '12 at 10:29
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Rather than suggest specific topics, which must be your personal choice, let me recommend three collections as possible sources:

(1) Brian Hayes' collection, Group Theory in the Bedroom and Other Mathematical Diversions. Every chapter is quite good, and the title chapter is a gem.

(2) The Best Writing on Mathematics 2010 and 2011. The Forwards (by William Thurston and Freeman Dyson respectively) are already worth the price of admission. Thurston: "I have decided that daydreaming is not a bug but a feature." :-)

(3) The AMS series, What's Happening in the Mathematical Sciences, with many articles by Barry Cipra and Dana Mackenzie (some reprinted in The Best Writing).

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Thanks,Joseph. The second text here is especially useful for me. –  Brendan Foreman Feb 22 '12 at 12:51
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This may seem a bit offbeat, but I'd like to recommend a children's book, The Cat in Numberland, by Ivar Ekeland. It's a story about the Hilbert Hotel told from the point of view of a perplexed, indolent housecat, written for young children who are still learning to count.

The book has a great many virtues, not the least of which is that it's wonderfully well written. It's also very short (it's a children's bedtime book, after all). It's an easy read, yet it deals with deep mathematical ideas. I think it would lend itself to a good classroom discussion both on the nature of mathematical infinity and on the nature of mathematical exposition -- in particular, whether deep mathematical ideas are, as many people believe, inherently incomprehensible.

My real recommendation would be to invite Ekeland to visit your class to read the story (and lead the discussion) himself. Ekeland is, of course, a distinguished mathematician, but more to point here, he has a wonderful reading voice. Be sure to have the students bring their blankies and teddy bears....

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I like "Introduction to Graph Theory" by Trudeau. It's a short (a few hours of reading) introduction to some nice and accessible parts of graph theory.

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Perhaps one of the following will be suitable:

  1. Davis, Martin. Engines of Logic: Mathematicians and the Origin of the Computer. New York: Norton, 2000.

  2. Gowers, Timothy. Mathematics: A Very Short Introduction. New York: Oxford University Press, 2002.

  3. Kaplan, Robert, and Ellen Kaplan. Out of the Labyrinth: Setting Mathematics Free. New York: Oxford University Press, 2007.

  4. Lakatos, Imre. Proofs and Refutations: The Logic of Mathematical Discovery. New York: Cambridge University Press, 1976.

  5. Poincaré, Henri. Science and Hypothesis. 1905. New York: Dover, 1952.

  6. Whitehead, Alfred North. An Introduction to Mathematics. 1911. New York: Oxford University Press, 1958.

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I second recommending the book by Lakatos. I've used it for similar courses, and the students found it exciting. It's relatively short, but the students will have to read it twice, probably (and don't overlook the appendix on the history of the discovery of uniform continuity, which is fascinating story), to get the most out of it. You will also have to help them near the end when they get to Poincaré's argument (essentially mod 2 cohomology), because I don't think Lakatos is very clear about that. However, that's a small price to pay for an exciting read. –  Robert Bryant Feb 20 '12 at 16:05
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I've used Underwood Dudley's "Readings for Calculus" with such honors students, and I think it was successful. Don't let the title fool you, there are very few "math problems" in the book, and lots and lots of discussion questions about the nature of mathematics, its history, and its place in the world.

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"The world of mathematics: a Four-Volume Set" was mentioned in earlier comments. It has a lot of nice readings as well. –  Todd Eisworth Feb 19 '12 at 19:05
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I recommend Journey Through Genius. It runs through many different topics, and is focused on proving theorems that were major (but whose proofs are still elementary). For example, it starts with the infinitude of primes, the quadrature of the lune, eventually discussing uncountable sets. I like it because not only does it cover many time periods, but it also conveys well the idea of proof in mathematics - which is a part of math which often is foreign to non-math majors.

The chapters make it quite suitable to skip around as well.

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Strongly seconded. –  Steven Landsburg Mar 2 '12 at 18:22
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Few books to consider:

1)Godel, Escher, Bach: An Eternal Golden Braid, By Douglas R Hofstadter : This Pulitzer prize winning book will be good for your needs. Has short and to great extent independent chapters and goes into the the meaning of mathematics, art, music and computing. Though the book looks voluminous it gives a possibility for selected reading.

2)What is the name of this book?: The riddle of Dracula and other logical puzzles, By Raymond M Smullyan ("The most original, most profound, and most humorous collection of recreational logic and math problems ever written." — Martin Gardner.)

3)Letter to young mathematician: the art of mentoring, By Ian Stewart : Subjects ranging from the philosophical to the practical--what mathematics is and why it's worth doing, the relationship between logic and proof, the role of beauty in mathematical thinking, the future of mathematics, how to deal with the peculiarities of the mathematical community are all discussed here.

4)An Imaginary Tale: The Story of i [the square root of minus one], By Paul J. Nahin : Discusses imaginary numbers from a historical perspective.

5) (added later)The Creative Process: Reflections on the Invention in the Arts and Sciences : Collection of essays written by some of the best minds like Poincare, Einstein, Mozart on how they got original ideas. Especially Poincare's essay is very insightful.

6) (added later)I Want to Be a Mathematician. Springer-Verlag. By Paul Halmos : It is good to introduce students to Halmos writing. I am surprised nobody mentioned this.

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James Newman edited a 4 volume compendium of mathematical articles. (The title escapes me at the moment. Edit: but it did not escape Joseph O'Rourke. Thanks to him for providing "A World of Mathematics: A Four-Volume Set". End Edit.) I would be surprised if you could not find a couple of suitable articles in there for your class.

Gerhard "It Had Mathematics In It" Paseman, 2012.02.19

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Further, if some of them are interested in science writing, Barry Cipra may have some good suggestions. Gerhard "Ask Me About System Design" Paseman, 2012.02.19 –  Gerhard Paseman Feb 19 '12 at 18:52
"The World of Mathematics: A Four-Volume Set" –  Joseph O'Rourke Feb 19 '12 at 18:53
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  1. Make sure to show the Geometry Center movie "Not Knot".

  2. Someone suggested "Gödel, Escher, Bach" which is a fun book but way too long. Some reading from Torkel Franzén's book "Gödel's Theorem: An Incomplete Guide to its Use and Abuse" might be a better choice. Discussing Gödel's theorem does seem like a good idea.

  3. (Added later): I guess someone should mention Logicomix. It keeps getting recommended in other places. It looks interesting but I haven't read it myself yet.

  4. I liked James Gleick's popular book "Chaos". It describes the discovery of the Lorenz attractor and all that good stuff.

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The following books by American Mathematical Society are very good.

1) Fixed points, by Yu.A.Shashkin

2) Stories about Maxima and Minima, By V.M.Tikhomirov

3) Intuitive Topology, By V.V.Prasolov

These books translated from Russian are very insightful and are not voluminous.

You can also look at the following combinatorial game theory book:

Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. By Donald Knuth.

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I think it would help to achieve a synthesis of mathematics and history. I do not mean how the development of mathematics influenced history(or, on the other hand how various historical phenomenon influenced mathematics), but how mathematicans acted as historical figures. It would be interesting to know how mathematicans communicated with each other, how cooperation helps to engender beautiful mathematics and how inevitable disagreements follow from different personality and world-view. Controversies in mathematics, though rare, helps to clarify issues that made further development possible.

It is not possible to make all students get used to mathematical thinking, for many of them must come from a varied background having little to do with science. Instead one should focus on the human side of the story to let them believe mathematics is a subject that focused on original(not magic or mechnical) contributions, sharp individual insights, and the mathematican community is no different from other professional groups of natural science. Instead of showing 'how spectacular math is!' and drill the students with introductory level texts that marvel them, one should help them realize how mathematical research is being produced in real life. For example, it is often being misunderstood that mathematicans do not make experiments; that sitting all day in front of a computer doing huge amount of calculations is the typical way of making progress. One should make the student understand the process of constructing a theory that might help us understand some underlying structure.

It is equally important to let the students understand that mathematical proofs is not so different from other forms of formal logic employed in real life. From my experience mathematicans often suffer from the shallow opinion by the public that they are being too critical on details and could not see the whole picture behind the main argument. It important to let them realize that while insisting on consistency and simplicity, mathematical proofs are ultimately human-mind products like the process of using rational reasoning to reach a conclusion in other fields.

One may organize a reading course by mimicking Carl Siegel's seminar at a much lower level. There are plenty of historical manuscripts, important papers, etc that has already been translated into English and worth discussing in a class. Manuscripts I can come up with are:

Abel's proof of the impossibility of solving the quintic equation in radical;

Riemann's speech "On the hypotheses which lie at the foundation of geometry";

Siegel's book "Topics in Complex Function Theory", Vol I should suffice for the purpose.

E.T.Bell's book "Men of Mathematics".

Hao Wang's book "A Logical Journey: From Gödel to Philosophy".

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I disagree with the idea of a synthesis. I have been dismayed to find that some historical mathematicians whose work is beautiful had some unpleasant political beliefs or objectionable/criminal actions, such as those who campaigned to purge universities of Jewish mathematicians in the 1920s and 30s. I think it's great that mathematicians try to judge mathematical work on its own merits, and bringing in history may both displace mathematics and degrade it. –  Douglas Zare Mar 3 '12 at 0:48
@Douglas Zare: One should note that in every historical period there are heros and villains like now. My point of achieving a synthesis of history and mathematics is not to point out that Newton quarreled with Flamsteed, or Laplace had been dishonest, or Galois had been wronged, or Luzin was slapped by Kolmogorov, etc. Such things are everywhere to be seen and nowhere to be avoided. I mean how one's mathematical idea influenced the mathematical community as a whole. I thought I have clarified this already in the first paragraph. –  Kerry Mar 3 '12 at 3:23
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Another good book though starts at an elementary level but covers lot of fundamental topics such as geometry, topology and calculus and could be highly recommended for the honors programs is: What is Mathematics? By Richard Courant and Herbert Robbins.

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You might like to look at the article with Tim Porter

"Making a mathematical exhibition" http://pages.bangor.ac.uk/~mas010/icmi89.html

As far as content goes we agreed that the exhibition should:

  1. suggest that the making of mathematics is a natural human activity, part and parcel of the usual methods by which man has explored, discovered, and understood the world;
  2. present each item with a purpose and context, and not just because it was something that could be shown or demonstrated;
  3. convey an impression of some of the key methods by which mathematics works;
  4. show mathematics in the context of history, art, technology and other applications.

In the end it became "mathematics and knots", but knots were used to show to the general public some methods of mathematics, namely:

  • Representation
  • Classification
  • Invariants
  • Analogy
  • Decomposition into simple elements (and laws for combination)
  • Applications

To see how much of this we felt we achieved, read the article, and judge yourself by seeing the web version (done years later) as part of
http://www.popmath.org.uk together with a gallery of sculptures by John Robinson. Many of these have a mathematical flavour (examples of fibre bundles in bronze!) although he knew almost no mathematics!

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I am somehow missing what are your goals - perhaps introduction to proofs, exposition to higher mathematics, or indicating what does it involve to do research?

Nevertheless, you may be interested in the book The Enjoyment of Mathematics: Selections from Mathematics for the Amateur by H. Rademacher and O. Toeplitz if you can get hold of it (very unfortunately, it is out of print, but the table of contents can be checked on amazon). It contains short pieces not requiring anything beyond high school math, yet many of them are full of elegance and joyful to read. As the students certainly should be able to get through the chapters independently, covering few of them can provide a good start.

In a similar direction, but more advanced, is Proofs from the book.

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My goals are to expose these students to a bit of higher mathematics without getting lost in too much background material. Some good ideas I've seen have been books of essays (e.g. Mathematics in the Bedroom) that will spark conversation, books devoted to biographies of mathematicians that will encourage a context of history, and books on the philosophy of mathematics (e.g. Letter to a Young Mathematician) that will prompt a more broad discussion about the field itself. –  brian loft Mar 3 '12 at 16:04
Later in the course, I'm hoping each student will be interested in a particular theorem/mathematician/branch of mathematics, and I can guide her (in the past, 27 of my 30 students were female) to a paper or book that will be the main source for a paper to be presented to the rest of the class. –  brian loft Mar 3 '12 at 16:05
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