A Publication of the
Applied Math and Science Education Repository

The AMSER Science Reader Monthly aims to provide educators with a useful package of information about a particular topic related to applied math and science by combining freely available articles from popular journals with curriculum, learning objects, and web sites from the AMSER portal. The AMSER Science Reader Monthly is free to use in the classroom and educators are encouraged to contact AMSER with suggestions for upcoming issues or comments and concerns at [email protected].

This month's AMSER Science Reader Monthly topic is applied mathematics, in an unexpected context.

The Mathematics of ... Juggling
Article by Bill Donahue from Discover Magazine
Synopsis and resource annotations by Max Grinnell

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Can reciting a number sequence such as "6-6-1-5-1-5" help a person juggle? According to Allen Knutson, a mathematics professor at the University of California at Berkeley, these sequences might be the best technique to use while attempting to learn and perform this rather challenging activity. Knutson makes juggling look easy, and for him, it's equal parts dexterity and algebraic combinatorics.

In this profile from Discover Magazine, Bill Donohue describes how Knutson deploys his nuanced knowledge of algebraic combinatorics to further his juggling prowess and mastery. Knutson also uses his juggling skills to demonstrate the basic premise of discrete mathematics in his classroom. Briefly stated, the premise of discrete mathematics is that one input (or throw) will yield one output (or falling ball).

The aforementioned juggling sequences are part of siteswap, which is a mathematical language that describes juggling routines. Essentially, siteswap assigns a number to each juggling motion, so a 3 is a throw that goes about chin high and stays in the air for approximately three beats of time. Odd-number throws are passed from one hand to another, and so on. The rather amazing thing about all of this is that siteswap allows jugglers to codify routines and share information with other aficionados of this craft. Knutson comments that juggling is much like a solid mathematical theorem: "It holds together. It makes sense, and it also delivers pleasant surprises."

Found below is a list of useful resources that will illuminate and enhance understanding of several of the topics explored within this piece. Broadly, the first three links lead to resources for college level mathematics instructors and students with a curious streak. The next two offer interactive mathematics resources, and the last link provides information on Knutson's own academic specialty, combinatorics.

The first entry leads to a set of tutorials on college algebra created by Kim Steward at West Texas A&M University. The second entry will take visitors to the homepage of the Macalester College Problem of the Week, which features math questions with titles that include "A Tale of Two Sons" and "Charge Your Batteries". The third entry leads to a series of fine online tutorials, from Harvey Mudd College, on topics such as derivatives, linear algebra, and differential equations. The fourth entry will take visitors to the Maths Online Gallery, which includes some interactive multimedia units on variables, equations, and sets. The fifth entry leads to the Fun Mathematics Lessons website. Here visitors can take in well-illustrated lessons on the mathematics of cartography and fractal geometry. The final entry leads to a page that provides some informal insights into various aspects of combinatorics including Dedekind's problem, solving magic squares, and much more. Overall, these resources should provide greater scope and help contextualize the ideas and concepts found within in the featured work. The list provides links to resource records in the Applied Math and Science Education Repository (

The introduction to this site remarks, "If you need help in college algebra, you have come to the right place." Their statement is accurate, as the staff members at the West Texas A&M University's Virtual Math Lab have done a fine job creating a series of online algebra tutorials for students and anyone else who might be returning to the world of algebra. First-time visitors should look at their online guide to the tutorials to learn how their tutorials are organized. After that, they should feel free to browse through any of the 59 tutorials offered here. Each tutorial contains information about learning objectives, full explanations, and numerous examples of how to correctly solve problems.

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Back in 1968, Professor Joe Konhauser at Macalester College started a tradition by creating a math problem for his students every week. Since that time, this long-standing tradition has migrated to the web. Along with the current problem of the week, visitors can view previous editions, dating back to the fall of 1995. Currently, Professor Stan Wagon oversees the problem of the week, and visitors can browse through these problems as they see fit. The problems are meant to be accessible to first-year college students, so they can be used in a host of instructional settings, or potentially as extra-credit. Visitors can sign up to receive the problem each week via email, and it is worth noting that the solution to each problem will be posted the following week.

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Created as an addition to undergraduate mathematics courses at Harvey Mudd College, this impressive collection of tutorials is useful for anyone needing a study aid or refresher for various topics. The list includes over 40 tutorials that address methods and theorems in pre-calculus, single and multi variable calculus, linear algebra, and differential equations. Quizzes at the end of each tutorial are only accessible to registered students, but the rest of the material is open to anyone. For offline viewing, the lessons can be downloaded in Adobe Acrobat (.pdf) format.

Provided by the University of Vienna’s futureMedia initiative, the Maths Online Gallery consists of a large collection of extremely useful interactive learning units that demonstrate mathematical concepts. A large number of interactive modules exist in such areas as analytic geometry, trigonometric functions, probability and statistics, integration, Fourier series, as well as model-building and simulation. The gallery was started in 1998 and new learning units are added regularly. This is an especially good resource for college and university teachers looking for in-class interactive illustrations of a large array of basic and advanced mathematical concepts.

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To the unconverted, the words “fun” and “mathematics” might not seem to be words that should be that close together in any one sentence or phrase. Educator and mathematics guru Cynthia Lanius proves any potential naysayers wrong as she offers up over two-dozen interesting and engaging math exercises for educators to use in their classrooms on this site. She’s served as a consultant for the Math Forum@Drexel and other educational websites, and her work here includes some great examples of sound educational pedagogy. While some of the activities are geared towards young children, some of the activities (such as “Mathematics of Cartography” and “Online Geometry”) will work well with older students who might be enrolled in a developmental math course at the high school or community college level. One other nice feature of this site is that some of the activities are also available in Spanish.

This site includes about 40 "informal notes" by Kevin Brown on combinatorics including: Dedekind's problem, cumulative permutation sequences, the factorial number system, Hardy and Ramanujan on partitions, solving magic squares, collecting k complete sets of coupons, and many more.

AMSER Science Reader Monthly is published by Internet Scout at the University of Wisconsin-Madison in conjunction with the National Science Digital Library with funding from the National Science Foundation. If you have questions or suggestions please e-mail us at [email protected].