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Mathematics seminar for “non-mathematicians”

Next term I am leading a seminar for students, who will become teachers for elementary school i.e. for kids of age 6-10. The students in the seminar will have no mathematical background beyond the "Abitur". They are supposed to give a 45 minutes talk on a (not too difficult) mathematical topic, and they have to write an exposition of a few pages. My first ideas cover geometry of triangles, basics around Fibonacci sequences ect. Did somebody on MO teach a similar class already?

In brief: I would appreciate further suggestions for suitable topics very much!

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Community Wiki? – J.C. Ottem Mar 25 2011 at 13:42
What is the "Abitur"? What mathematical content can be found in it? – Joel Reyes Noche Mar 25 2011 at 14:05
@JNR: Until somebody who knows this better comes along a tentative answer: "Arbitur" is the name used in Germany for the final exam at the end of high-school (so age 18/19 after 12/13 years of school). In other words: the background is only high-school mathematics. What this high-school maths means in detail in the place where the questioner is, is unknown to me. – quid Mar 25 2011 at 15:25
@JNR: I have not much to add to the answer of unknown(google). In fact those attending the seminar will have 13 years of school behind them. (But we will soon switch to a new system where one only has to go to school for 12 years before the so-called Abitur.) – Sebastian Petersen Mar 26 2011 at 15:42
@Ottem: To be honest, I never understood what Community Wiki is. But feel free to make it such a Communiki Wiki, if this fits better, or let me know how to do that. – Sebastian Petersen Mar 26 2011 at 15:43
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Have a look at the exhibition on knots on http://www.popmath.org.uk . I have used this and other material for talks advertised as "How mathematics gets into knots" for ages 8-80: I was then criticised for suggesting one should stop at 80!

The material I used in a recent general lecture in Kansas (see my my preprint page) on the Dirac string trick, and the related belt trick and Phillipine wine trick, and on the pentoil, can be done for any age; it is essential to involve the audience in doing the tricks, or demos. One can also do addition of knots, and analogies between knots and numbers, getting over the point that while knots and numbers are quite different, the relations between knots can be analogous to the relations between numbers. Note also that my apparatus for the Dirac String Trick was home made: two piece of board, coloured ribbon, bulldog clips to attach the ribbon (in case it gets too tangled). My pentoil was made in a science workshop from copper tubing.

In giving such a presentation to children and teachers once, a teacher came up to me afterwards and said:"That is the first time in my mathematical career that anyone has used the word "analogy" in relation to mathematics!" Thus a real concern is that there is so little discussion about mathematics, and its context.

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The main thing I (speaking of course only for myself) would want students to learn from this exercise is that it is possible to study mathematics, as opposed to using mathematics to study other things. So I'd steer clear of applications to voting or music or gambling and steer them toward writing about prime numbers (say, the infinitude of primes, or the fundamental theorm of arithmetic, with applications to the irrationality of the square root of 2) or topology (konigsberg bridges, or the euler characteristic of a sphere, or the connect-three-houses-to-three-utilities problem, except without trying to make it sound like applied math) or geometry (platonic solids) or set theory (uncountability of the real numbers) or quadratic forms (e.g. sums of two squares).

These are of course all terribly unoriginal ideas but what matters is that they're new to your students, not to us.

A student who could explain what a group is, and give a few examples, and prove that the identity element of a group is unique, would have accomplished something quite substantial for this age.

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 Obligatory link: pauli.uni-muenster.de/~munsteg/arnold.html (go down to "What is a group?") – Kevin Lin Mar 26 2011 at 19:31

I highly recommend teaching arithmetic in different bases and base conversion because it aids in the conceptual understanding of our ordinary base ten arithmetic and representations of numbers. If these students are going to become elementary school teachers, they're primarily going to be teaching kids how to add, subtract, multiply, and divide so it is really important for them to have a firm grasp on why these procedures actually work.

For a related "magic" trick, you can ask them why the telephone number trick works.

In general, a variety of discrete math topics including propositional logic and basic number theory would also seem suitable.

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The following books might provide suggestions of topics to cover and might prove useful resources for the teachers:

• Enzensberger, Hans Magnus. Der Zahlenteufel: Ein Kopfkissenbuch für alle, die Angst vor der Mathematik haben. Deutscher Taschenbuch Verlag, 1999.

• Hart, George W., and Henri Picciotto. Zome Geometry: Hands-on Learning with Zome Models. Emeryville, California: Key Curriculum Press, 2001.

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

• Petzold, Charles. Code: The Hidden Language of Computer Hardware and Software. Redmond, Washington. Microsoft Press, 1999. [This title is very good on binary numbers.]

• Schwartz, Richard Evan. You Can Count on Monsters: The First 100 Numbers and Their Characters. Natick, Massachusetts: A.K. Peters, 2010.

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I once taught a successful class to second graders, on euler's theorem for polyhedra by handing out colored cardboard models and letting them count the facets, etc... This would seem a suitable topic also for elementary school teachers.

(When reading elementary presentations of triangle geometry for children one sometimes encounters false statements, such as the claim that the rigidity of a triangle made of straws implies SSS congruence. Of course rigidity implies the moduli space is discrete rather than that it is a singleton, i.e. a similar proof shows (the false theorem) "SSA congruence".)

Another favorite topic, harder than euler, is modular arithmetic applied to computing what day of the week someone was born on.

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In the United States, this kind of courses is not uncommon, which means that there are lots of textbooks that gather material at this level (of difficulty and generality). You may want to look at the table of contents of a few of them to get a feel for what they cover. For instance, my school currently uses this book.

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I once taught a similar course. I had some success discussing mathematics and voting. Larry Bowen of Alabama has a webpage with some basic ideas and exercises:

http://www.ctl.ua.edu/math103/voting/mathemat.htm

Voting was also the topic for Mathematics Awareness Month April, 2008.

http://mathaware.org/mam/08/

We covered mathematics in music. The magical mathematics of music, by Jeffrey Rosenthal may be a good introduction for the students.

http://plus.maths.org/content/magical-mathematics-music

On this topic, I also recommend the work of Rachel Wells Hall (St. Joseph's). Here is a link to her homepage.

http://www.sju.edu/~rhall/research.htm

We discussed some probability. The students found the Monty Hall problem intriguing.

I hope this helps.

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 I agree that the connections of mathematics and social justice are interesting and easily accessible. (Most involve only algebra, some use a little basic calculus.) I recommend you also look at Fair Division: en.wikipedia.org/wiki/Fair_division. An excellent paper is found here: ams.org/notices/200611/fea-brams.pdf. – Joel Reyes Noche Mar 25 2011 at 14:02