A Decade of the Berkeley Math Circle: The American Experience (MSRI Mathematical Circles Library)

by Zvezdelina Stankova, Tom Rike (Editor)

Many mathematicians have been drawn to mathematics through their experience with math extracurricular programs exposing teenage students to advanced mathematical topics and a myriad of problem-solving techniques and inspiring in them a lifelong love for mathematics. Founded in 1998, the Berkeley Math Circle (BMC) is a pioneering model of a U.S. math circle, aspiring to prepare our best young minds for their future roles as mathematics leaders. Over the last decade, 50 instructors—from university professors to high school teachers to business tycoons—have shared their passion for mathematics by delivering more than 320 BMC sessions full of mathematical challenges and wonders. Based on a dozen of these sessions, this book encompasses a wide variety of enticing mathematical from inversion in the plane to circle geometry; from combinatorics to Rubik's cube and abstract algebra; from number theory to mass point theory; from complex numbers to game theory via invariants and monovariants. The treatments of these subjects encompass every significant method of proof and emphasize ways of thinking and reasoning via 100 problem-solving techniques. Also featured are 300 problems, ranging from beginner to intermediate level, with occasional peaks of advanced problems and even some open questions. The book presents possible paths to studying mathematics and inevitably falling in love with it, via teaching two important thinking creatively while still obeying the rules and making connections between problems, ideas, and theories. The book encourages you to apply the newly acquired knowledge to problems and guides you along the way, but rarely gives youready answers. Learning from our own mistakes" often occurs through discussions of non-proofs and common problem-solving pitfalls. The reader has to commit to mastering the new theories and techniques by "getting your hands dirty" with the problems going back and reviewing necessary problem-solving techniques and theory and persistently moving forward in the book. The mathematical world is you'll never know everything but you'll learn where to find things how to connect and use them. The rewards will be substantial. "

Genres: Mathematics

326 pages, Paperback

First published December 12, 2008

Reviews

[Murilo Andrade · Rating 4 out of 5]

This very written book acts as a vulgarisation tool for Olympiad Maths and Math Circles. I highly recommend it to Math Circle teachers. Olympiad beginners can also profit from it, although it is not the most efficient book for that.

The book is divided into 12 chapters, each of them related to a topic exposed in Berkeley Math Circles sessions. Excellent writers like Stankova, Tom Rike, Zeitz ( from Art and Craft of Problem Solving), Gabriel Carroll expose topics in a simple, accessible way to all (olympiad) grades. In general no prior knowledge is necessary for this book besides basic Math.

I quite liked PST's ( problem solving technique) spread all over the book. Once a new technique is used in a problem, they state it as a PST. Through the book we have:

1. Inversion , Stankova
Inversion transformation is defined, and Stankova shows how to pass from a normal problem to its "dual", "inversed" problem, which might be easier to solve ( e.g. Ptolomy's Theorem). In general, it is used when several circles intersect, and will take some of the intersection points as inversion point, whereas the radius doesn't usually matter. She also describes how inversion affects distances through the formula A'B' = AB*r2/(OA*OB), which is obtained through similar triangles. She concludes by explaining how we can hide well known geometry facts with Inversion into nice , interesting problems.

2. Combinatorics, Zeitz
First class in combinatorics, addition principle, multiplication principle, subtraction principle, ball in urns problem, etc

3. Rubik's Cube, Tom Davis
Not very interesting for olympiads, but still nice. Davis describes relation between rubik cube and group algebra. He introduces basic group definitions (e.g. identity, orders, cycle) and its operations. This chapter has two nice results: Every "macro" (i.e. set of moves) has an order , and if the order is a multiple of two or 3 it is often interesting to apply only half ( or 1/3) of the order, to get interesting results ( e.g. to swap two mini cubes)

4. Number Theory , Stankova
First class in number theory ( prime number, fundamental theorem of arithmetic, bezout, congruencies,divisibility etc )

5. A few words about proofs, Mira Bernstein
First class about proofs ( what does a proof look like ? What are different types of proofs? ). Proof by Contradiction,

6. Induction , Quan Lam
Induction, no big novelties here

7. Mass Point Geometry , Tom Rike
This was one of the best chapters IMO. He introduces Mass Point Geometry, where you associate masses to points ( in the vertices of a triangle usually), to solve problems. The main idea here is the one of mass center, when used with associativity of the sum of mass-points. After calculating in two different ways the mass center, you often get a nice result ! Menelaus, Ceva, Problems of concurrency of cevians, bisector angle, areas in a triangle ( routh theorem)), can all be calculated easily using mass point geometry. One of the nice things in this chapter is the way to prove <=>, where you can prove the second part by contradiction + use first part.

8. More on Proofs ( part 2 ), Mira Bernstein, Stankova
Induction again, Pigeonhole principle.

9. Complex Numbers, Tatiana Shubin
first class on complex numbers. no olympiad here

10. Invariants, Vandervelde
Invariants, coloring. The game of escape of clones has a very elegant solution ! A must read!

11 Favorite Problems at BMC, Ivan Matic
Problems in circle geometry. In general the circle is a "phantom", you have to artificially insert it in the picture. Be careful with a picture! You shouldn't rely on it and should analyse all possible cases (i.e. of intersection) instead.

12. Monovariants, Carroll
Excellent introduction to monovariants. Several variations of the mansion problem, each with a slightly change in the way we should think about monovariants.