Past Teacher Courses

The method of Mass Points in Geometry enables you to solve seemingly difficult problems with ease. The talk will include a justification for the method and how to apply it to problems familiar to Math Team adepts. We will progress to some less widely known extensions of the method and to more difficult problems.

Please share this invitation with your colleagues! Note that this workshop is free, but registration is required as we expect to reach capacity.

David Hankin is a highly respected mathematics teacher whose distinguished career spanned four decades at the high school and college levels. In addition, he served as Mathematics Department Chair at Hunter College High School, Chair of the AIME (American Invitational Mathematics Examination), and is a prolific author of competition questions. Mr. Hankin was instrumental in the creation of New York Math Circle, where he taught until his retirement in 2017.

There are well-known formulas for the inradius, circumradius, and exradii of a triangle. Relationships between these quantities are less familiar. In this presentation, various approaches to some of these less widely known relationships and ideas will be considered.

Please share this invitation with your colleagues! Note that this workshop is free, but registration is required as we expect to reach capacity.

David Hankin is a highly respected mathematics teacher whose distinguished career spanned four decades at the high school and college levels. In addition, he served as Mathematics Department Chair at Hunter College High School, Chair of the AIME (American Invitational Mathematics Examination), and is a prolific author of competition questions. Mr. Hankin was instrumental in the creation of New York Math Circle, where he taught until his retirement in 2017.

Join us for a short but scenic voyage slightly outside of the usual realm of high school Geometry. The journey will take us through some results familiar to many and on to surprisingly beautiful and rare geometric gems.

Please share this invitation with your colleagues! Note that this workshop is free, but registration is required as we expect to reach capacity.

David Hankin is a highly respected mathematics teacher whose distinguished career spanned four decades at the high school and college levels. In addition, he served as Mathematics Department Chair at Hunter College High School, Chair of the AIME (American Invitational Mathematics Examination), and is a prolific author of competition questions. Mr. Hankin was instrumental in the creation of New York Math Circle, where he taught until his retirement in 2017.

You can bring your own lunch or make arrangements with us for $10 per person.

Transfinite Numbers by Marty Rudolph, retired math teacher and recipient of the “Presidential Award for Excellence in Mathematics and Science Teaching” (PAEMST) in 1997.

Transfinite Numbers Handout

How large is "large?" Is infinity the largest number? Consider the following argument between two children in the 1950s:
1: I dare you.
2: I double dare you.
1: I triple dare you.
2: I dare you times a million (said with a smirk)
1: I dare you times infinity (with finality)
2: I dare you times infinity plus one!!! (uh oh)

(Of course, in the film A Christmas Story, they all lose to a “double dog dare.”)

This is a short but scenic voyage slightly outside of the usual realm of secondary school Plane Euclidean Geometry. Highlights include a wealth of sources and resources together with a complement of delightful problems, surprising results and a collection of very useful theorems. Some topics - Collinearity and Concurrency, Cyclic Polygons and Ptolemy’s Theorem, Special Points and Lines associated with Triangles, Circles, Surprising Consequences of the Pythagorean Theorem, and more.

The registration fee covers both sessions, and lunch on both days.

In this series of informal talks, teachers are invited to discuss mathematics, work on problems, and have dinner together. If you are interested in attending any talk in the series, please register (for free) using the link above. We'll send you reminders in advance. More talks may be added, so please check this page for updates.

Each session includes a mathematical presentation by an invited speaker. After that, we'll head out to a nearby restaurant to continue the conversation over a self-hosted dinner (each participant pays for their own dinner). Topics are selected for their mathematical content, which are of interest to high school, middle school or elementary school math teachers, and participants receive a handout of interesting problems and other information that is relevant to the talk. If you'd like to give a talk or suggest a speaker, please email info@nymathcircle.org.

Scheduled Talks

• Use of Cuisenaire Rods to Support the Learning of High School Level Mathematics by Dr. Arthur Powell, Associate Professor and Chair of the Department of Urban Education, Rutgers-Newark, Dr. Paula Hajar, Senior Professional Development Specialist, Bronx Charter School for Better Learning, Dr. Ted Swartz, Director of Professional Development, Bronx Charter School for Better Learning. Monday, March 23.

  Session Notes

  Visible and Tangible Mathematics, an approach developed by Caleb Gattegno, rests on an awareness of learners as agents active in creating their understanding of everything they know about or how to do. While mathematics curricula in early elementary grades often rely heavily on “hands-on” manipulatives to promote such understanding, junior high and high school courses generally do not. Visible and Tangible Mathematics, on the contrary, incorporates Cuisenaire Rods to support learners’ mastery of mathematical concepts at all levels, including, for example, quadratic equations. Our workshop will involve participants in creating structures with the rods, designed to promote insight into the algebraic relationships that underpin and produce quadratic equations. The tangible and visible evidence of those relationships fosters deep, satisfying and lasting insight into a subject that can otherwise be dull and/or not easily accessible to many high school students.

  The March 30 presentation has been rescheduled to May 11 (see below).

• A Potpourri of Delightful Mathematical Curiosities by Joy Hsiao, Math Teacher, NYC DOE. Monday, April 27.

  Session Notes and Origami Instructions

  Join us for an assortment of delicious mathematical treats, including games, magic tricks, origami, and activities. This Math & Dinner talk is intended especially for middle school teachers and parents with elementary to middle school age kids, but of course everyone is welcome.

• Discovering Pick’s Theorem by Paul Ellis, Manhattanville College and Westchester Area Math Circle. Monday, May 4.

  Session Handout, a Flowchart for Euclid, Book I, and Leslie Lamport's How to Write a 21st Century Proof

  Pick’s Theorem allows one to compute the area of a lattice polygon, knowing just the number of lattice points on its boundary and interior. Participants will be guided through the discovery process of deducing the formula, and proving it.

• Understanding Infinity by Daniel Zaharopol, Art of Problem Solving Foundation and Summer Program for Mathematical Problem Solving. Monday, May 11.

  Have you ever wondered what it means that there are infinitely many numbers? What does it mean that numbers go on forever? Even more perplexing: is every infinity the same size? Are there more natural numbers, or even numbers, or prime numbers? What about rational numbers, or real numbers?

  It is a miracle of modern mathematics that we can ask and answer questions about infinity. By asking each other simple but provocative questions, we will as a group come up with a definition for what it means when two infinite sets are the same size, or when one is bigger than another.

In this series of informal talks, teachers are invited to discuss mathematics, work on problems, and have dinner together. If you are interested in attending any talk in the series, please register (for free) using the link above. We'll send you reminders in advance.

Each session includes a mathematical presentation by an invited speaker, after which participants continue the conversation over dinner at a nearby restaurant. Topics are selected for their mathematical content, which are of interest to high school, middle school or elementary school math teachers, each talk features a handout of interesting problems and other information that is relevant to the talk. If you'd like to give a talk or suggest a speaker, please email info@nymathcircle.org.

Scheduled Talks

• Math is more than Computation: Teach Algorithms for Deeper Understanding by Frances Stern, Mathematics Education Consultant. Monday, July 7, 4-6pm. Room 202

  Session Notes

  We know that our students are likely to use calculators to perform arithmetic, as do most adults. Yet we have a gut feeling that we should be teaching arithmetic algorithms. At the same time, when we see a student use an algorithm to multiply a number by 100, or to subtract 59 from 61, we know that something has gone terribly wrong.

  			3	7	2				6	1
  	×		1	0	0		−		5	9

  So yes, I think that we should teach algorithms, but for much more than precision and not as a replacement for mental math. We can teach algorithms to deepen understanding of place value, develop number sense, and prepare for the abstract thinking of algebra. This knowledge is vital even when we use calculators or computers to perform computation.

• An Array of Matrix Explorations by Patrick Honner, Brooklyn Tech. Monday, August 4, 5-6:30pm. Room 312

  Session Notes

  The world of matrices is rich and diverse, connecting ideas across disparate disciplines and extending familiar mathematical notions into unfamiliar territory. In this talk, participants will explore some common concepts in the high school curriculum–algebra, geometry, and trigonometry–through the mathematics of matrices, where their depth and connections can be seen in exciting ways.

• Manifolds, smooth and not by Sylvain Cappell, NYU Courant. Wednesday, August 20, 4-6pm. Room 201

  Manifolds are "nice" spaces which arise in all dimensions and have many of the familiar properties of Euclidean spaces and are thus well suited for applications in many directions in mathematics, physics, etc. An example is the unit sphere in each dimension. We will review how the challenges of classifying manifolds in various dimensions and investigating their symmetries have been central research themes in the 20th century and remain a focus of current efforts. We will introduce the basic three levels of manifolds - topological, piecewise linear and smooth - and survey some of what's known and not known about the relations between these notions in varying dimensions.

We’re planning our math teacher programming for the coming year, and we’d love your input. To that end, we’re scheduling a focus group where we can meet in person and discuss our program. Please register, so that you can be part of the conversation!

When can 4 different Fibonacci numbers be the sides of a quadrilateral? What are the two smallest integer-sided triangles that agree in 5 parts [sides, angles], yet are not congruent? Ever prove things using “Mass Points”?

Gil Kessler will conduct a 5-session course for high school and middle school math teachers interested in adding to their collection of ideas, problems, and puzzles for themselves and their students. The material can be used to enrich lessons and homeworks, as well as challenge math team members. The presentation will be at a level accessible to all, with content ranging from easy to impossible [just kidding!], but always enjoyable.

[This course is independent of any other enrichment courses.]

Consider this a preview of a multi-session course that is in the works. This course will bring together high school students and high school math teachers to familiarize them with all aspects of the ARML and NYSML events, including team building, proof writing, special types of problems, and so forth.

One of the many goals is to create opportunities for more people to get involved as participants, coaches, assistants, and other roles. This initial meeting will be hosted by Larry Zimmerman, and will open the floor to suggestions and comments, which may be helpful in shaping the architecture of the actual course.

Originating from 16th and 17th century questions concerning games of chance, probability theory has generated many interesting and challenging problems and has been responsible for significant applications. With appropriate reviews of basic combinatorial principles and fundamental results of probability, we will discuss famous examples in this four-part mini-course, such as the Buffon Needle Problem, the Birthday Problem, and the Monty Hall Problem, as well as many unnamed gems of discrete and continuous probabilistic reasoning.

Sample Problems:

1. If a stick of length x is broken into three pieces, what is the probability that the three pieces can be used to construct a triangle?
2. Tom and Dick tell the truth only a third of the time. Tom makes a statement and Dick tells us that Tom was speaking the truth. What is the probability that Tom was actually telling the truth?
3. Suppose you return n homework assignments to your n students. What is the probability that no student gets back his or her own assignment?
4. Suppose you are waiting at a street corner where you could take either of two different buses, both running independently at 10-minute intervals. How long would you expect to wait for a bus?

Math Team for Teachers - part II is intended as a natural follow-up to Math Team for Teachers - Part I, although it is independent and would be suitable for anyone who has a very deep enjoyment of problem solving (and problem posing). Much will drawn from the many problem sets handed out in the previous sessions (additional sets will be available, of course) as well as a lot of new and novel items.

Problems will be explored, solved, extended, developed, varied and serve as pathways to special topics and techniques. Essentially, our sessions will be Problem Solving Seminars and will not focus so much on the organizational aspects of Math Team as was the case in the previous course.

The registration fee covers both sessions, and lunch on both days.

In this series of informal talks, teachers are invited to discuss mathematics, work on problems, and have dinner together. If you are interested in attending any talk in the series, please register (for free) using the link above. We'll send you reminders in advance.

Each session includes a mathematical presentation by an invited speaker and a catered dinner (free, but we welcome contributions to cover dinner costs). Topics are selected for their mathematical content, which are of interest to high school or middle school math teachers, and participants receive a handout of interesting problems and other information that is relevant to the talk. If you'd like to give a talk or suggest a speaker, please email info@nymathcircle.org.

Scheduled Talks

• The Art of Mathematical Modeling: Developing Intuition for Real World Problem Solving by Dr. Lisa Rogers, Applied Mathematician. Monday, March 31.

  Math Modeling Handout and Math Model Algorithm

  All problems, from the everyday mundane to the most complex dynamical systems, can be solved with the use of mathematical intuition and logic. The key to problem solving is the a method that is familiar to mathematicians - reduction of a problem into a simpler form that we know how to solve, and building from there. My years of experience working with students of various disciplines on modeling problems has helped me to develop the Algorithm for Constructing a Mathematical Model, a clever reinterpretation of the scientific method specific to quantitative analysis. The key to efficient, creative problem solving is organization and logic, as well as the ability to quantify even the most minute of details. Scientists and future scientists alike require the ability to observe a problem from various perspectives and with varying levels of detail in order to develop accurate, all-encompassing solutions. In this talk, we’ll discuss the Algorithm for Constructing a Mathematical Model, and how to apply it to a wide range of problems. I will also highlight the importance of developing student ability to communicate mathematical/scientific ideas effectively to scientists and non-scientists alike.

• Transfinite Numbers by Marty Rudolph, Ret. math teacher. Monday, April 7.

  Transfinite Numbers Handout

  How large is "large?" Is infinity the largest number? Consider the following argument between two children in the 1950s:
  1: I dare you.
  2: I double dare you.
  1: I triple dare you.
  2: I dare you times a million (said with a smirk)
  1: I dare you times infinity (with finality)
  2: I dare you times infinity plus one!!! (uh oh)
  (Of course, in the film A Christmas Story, they all lose to a “double dog dare.”) This sample lesson (for teachers) is most appropriate for juniors-seniors in a precalculus class in high school. However, a wider range of students could easily enjoy and learn from it.

• The Art of Teaching Math in Grades 1–4 by Yakov Kamenyar, Math 21. Monday, April 28.

  Art of Teaching Math Handout

  The art of teaching math is all about how to teach. For expert knowledge, it is useful to compare teaching styles at the international level. While the basic characteristics of expert teaching are clearly visible; catching the nuances is a challenging task. In this talk we will highlight some fine techniques of Russian math education, explained in the context of typical math problems at the elementary level.

• The Mathematical Puzzles of Literature by Elisabeth Jaffe, Baruch College Campus High School. Monday, May 12.

  Math Puzzles in Literature and Parent Handout

  Mathematics possesses an inherent beauty that our students often don’t see. In this talk we will explore the poetry of mathematics introduced in literature. We will find ways to show our students that mathematics is an art. It is not simply about finding a correct answer, but it is about exploring our own creative minds. We will specifically address the mathematical puzzle introduced in The Lottery by Shirley Jackson and the numerical surprises introduced in The Housekeeper and the Professor by Yoko Ogawa. We will look at books around which we can create lessons and/or projects to inspire our students and ourselves to look beyond the one answer.

• Challenging Geometry Problems for Lower Middle School Students by Mila Martynovsky, New York Math Circle. Monday, May 19.

  Geometry Problems Handout

  This session is for anyone with an interest in mathematical problem-solving for lower middle school students. Participants will enjoy a set of non-routine problems involving complex geometrical figures, designed to stimulate scholars to explore familiar concepts from an unusual angle or point of view. Useful as a supplement to a basic mathematics curriculum, these problems lead to further explorations on specific topics and are intended to develop problem-solving techniques. Detailed solutions, hints, and specific answers for all problems will be provided.

Can you find y if 3[y]² - 7[y] + 2 = 0 and 4y is a prime (brackets represent “the greatest integer function”)? What is the sum of the first 212 terms of 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, ... (the sequence consists of consecutive terms of Pascal’s Triangle)?

Gil Kessler will conduct a 3-session course for high school and middle school math teachers interested in adding to their collection of ideas, problems, and puzzles for themselves and their students. We will discuss some series, problems that each involve multiple concepts (including Morley’s Theorem), and a variety of odd facts and puzzles. The presentation will be at a level accessible to all, with content ranging from easy to amazing, and always enjoyable.

[This course is independent of any other enrichment courses.]

In this series of informal talks, teachers are invited to discuss mathematics, work on problems, and have dinner together. If you are interested in attending any talk in the series, please register (for free) using the link above. We'll send you reminders in advance.

Each session includes a mathematical presentation by an invited speaker and a catered dinner (free, but we welcome contributions to cover dinner costs). Topics are selected for their mathematical content, which are of interest to high school or middle school math teachers, and participants receive a handout of interesting problems and other information that is relevant to the talk. If you'd like to give a talk or suggest a speaker, please email info@nymathcircle.org.

Scheduled Talks

• Jim and Nim by Japheth Wood, New York Math Circle. Monday, October 21.

  Jim and Nim Handout

  Nim is an impartial combinatorial game with a long history and a rich mathematical theory. Jim (short for Japheth's Nim) is also an impartial combinatorial game that was invented by the speaker in 2011 while jogging around Prospect Park! Participants will learn to play both Nim and Jim, and develop strategies that lead to a full understanding of the mathematical theory of both games.

• Geometric Probability: An Appetizer by Henry Ricardo, New York Math Circle. Monday, October 28.

  Geometric Probability Slides and Geometric Probability Notes

  An introduction to the idea of Geometric Probability with discussion focused on some carefully-chosen examples, both ordinary and famous. This Teachers' Math Circle session also serves as a preview of Henry's mini-course Interesting Problems of Choice and Chance.

• Euler's Formula by Kovan Pillai, Taprobane LLC. Monday, November 4.

  Euler's Formula Handout

  Of the many formulas Euler discovered, one of the most admired is

  e^iθ = cos⁡θ + i sin⁡θ. The formula provides an elegant method for evaluating complicated integrals and summing series; it also provides an elegant proof of De Moivre’s Theorem and can be used to derive many trigonometric formulas. Distinguished theoretical physicist Richard Feynman described it as “one of the most remarkable, almost astonishing, formulas in all of mathematics”. Euler's formula will be derived, and participants will solve challenging problems.

• Investigating the Mathematics of Sona: Sand Drawings from Angola by Paul Ellis, Manhattanville College and Westchester Area Math Circle. Monday, November 18.

  The Mathematics of Sona Handout

  The Chokwe people of central Africa make a curious sand drawing called Sona. At first glance, these seem like simple drawings, but the more we look, the more we see some beautiful mathematical properties.

• The Mathematics of Friday the 13th, by Sheila Krilov, Hunter College HS. Monday, December 9.

  Friday the Thirteenth Slides and Friday the Thirteenth Handout

  Does every year contain a Friday the thirteenth?, What is the maximum number of Friday the thirteenths in a year? These questions and more will be answered, just in time for you to deliver the answers to your students, on Friday the 13th! This talk is appropriate for both middle school and high school classrooms. [No need for Triskaidekaphobia!]

• Shapes and Numbers, by Alex Belopolsky, Enlightenment Research. Monday, December 16.

  Shapes and Numbers Handout

  A series of problems that appear to be algebraic or arithmetic, but benefit from a reformulation in geometrical terms.

Join us this summer for our 5th annual workshop for middle school math teachers. The mathematical theme is Volume and Surface Area, which we will approach as learners, educators, and mathematicians. We offer collaborative problem sessions each morning, followed in the afternoons by enriching extension talks and seminars, all led by NY Math Circle instructors and Bard math professors. Participants will enjoy the backdrop of Bard College’s Annandale-on-Hudson campus, complete with its summer arts offerings.

We will address the following questions, and more:

How are surface area and volume defined, and what are the connections between them?

How are volume and surface area formulas (such as for prisms, cones, and spheres) established?

What is the history of the subject, from Euclid and Archimedes to the modern day?

What are some challenging (yet accessible) problems, as well hands-on math activities that will inspire our students and enrich our math classrooms?

Accommodations are provided. Graduate credit is available.

Polynomials are the familiar workhorses of algebra, precalculus, and calculus courses and they show their utility in more advanced areas of algebra and analysis as well. In this course we will explore the interesting and appealing world of polynomials and their connections to number theory, algebra, analysis, geometry, and probability. There will be a mix of theory and applications, using basic algebra as well as some tools from elementary calculus. Here are some samples from Henry's extensive cache of examples and problems:

1. The polynomial n² + n + 41 yields distinct primes for n = 1, 2, …, 39. [Euler, 1772]
2. Let f be a polynomial of degree n such that f(k) = k/(k+1) for k = 1, 2, …, n Find f(n+1).
3. Consider the equation x² + Ax + B = 0. Let constants A and B be chosen independently with the uniform distribution on the interval [0,1]. Calculate the probability that the equation has real roots.
4. Find all real polynomials f(x) satisfying the equation f(x²) = f(x) f(x-1).

This mini course is designed for teachers who intend to coach/advise a Math Team, a Math Club, or a related extracurricular activity. It can also be taken by those who simply enjoy doing mathematics and solving Math Team type questions. Participants will learn how to conduct a Math Team, become acquainted with many different types of Math competitions, and get to know many of the sources and resources that are available.

Some mathematical topics essential for Math Team will be presented, accompanied by a wealth of examples. We will examine ways of creating original problems to aid in the development of appropriate, effective practice materials. Techniques for training students, motivational devices, and philosophy of the Math Team experience will be addressed. Appreciation of problem solving is, of course, a key component. Special emphasis is placed upon potential for student growth as a consequence of Math Team.

The registration fee covers both sessions, and lunch on both days.

Not a course! Just informal gatherings for teachers, to discuss some mathematics, work on some problems, and have dinner together. If you are interested in any talk in the series, please register (for free) using the link above, so that we know the level of interest.

• May 23, 5pm. "A Taste of Celestial Navigation" by Dmitry Sagalovskiy, New York Math Circle.

  Celestial Navigation Handout

  It is fascinating that by looking at the sky, you can figure out your latitude and longitude on Earth. Celestial navigation relies on spherical trigonometry, a beautiful branch of mathematics, which was important throughout history until the advent of GPS hid it from public view. In this brief tour, we will try using a sextant, and discuss the mathematics of converting its measurement into a location on Earth. We will discuss measurements and coordinates on a sphere, and delve into some spherical geometry and trigonometry. Fun problems and puzzles related to the mathematics of the Earth will be provided.

• June 6, 5pm. "The Stretched Chords Problem" by Ben Blum-Smith, Graduate Student at NYU-Courant Institute, and Japheth Wood, New York Math Circle

  Stretched Chords Handout

  In July 2010, teacher participants at the Park City Math Institute (PCMI) encountered the following haunting problem: Put n evenly spaced points on the unit circle, with one point at (1,0). Then draw chords from this point to all the other points. Then multiply the chords’ lengths. What do you get? We will discuss this classic problem and the even-more-shocking extension, which we call the Stretched Chords Problem: Scale the circle vertically by a factor of √5. Scale all the chords too. What is the product of the lengths now?

When can 4 different Fibonacci numbers be the sides of a quadrilateral? What are the two smallest integer-sided triangles that agree in 5 parts [sides, angles], yet are not congruent? Ever prove things using “Mass Points”?

Gil Kessler will conduct a 5-session course for high school and middle school math teachers interested in adding to their collection of ideas, problems, and puzzles for themselves and their students. The material can be used to enrich lessons and homeworks, as well as challenge math team members. The presentation will be at a level accessible to all, with content ranging from easy to impossible [just kidding!], but always enjoyable.

[Courses in our enrichment series don't overlap material. We recommend you take them all (in any order) for greatest enjoyment.]

A minimum of 10 participants are required to run this course.

Can you find y if 3[y]² - 7[y] + 2 = 0 and 4y is a prime (brackets represent “the greatest integer function”)? What is the sum of the first 212 terms of 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, ... (the sequence consists of consecutive terms of Pascal’s Triangle)?

Gil Kessler will conduct a 3-session course for high school and middle school math teachers interested in adding to their collection of ideas, problems, and puzzles for themselves and their students. We will discuss Mathematical Induction and Series, problems that each involve multiple concepts (including Morley’s Theorem), and a variety of odd facts and puzzles. The presentation will be at a level accessible to all, with content ranging from easy to amazing, and always enjoyable.

[This course is independent of any other enrichment courses.]

Explore a potpourri of sequences, a gallimaufry of series in Henry Ricardo’s minicourse. Among the topics to be discussed are Farey sequences, recursive sequences, Fibonacci numbers, sequences and series related to e, the harmonic series, telescoping series and products, and Riemann sums. Examples and problems will come from algebra, geometry, set theory, calculus, and number theory. The emphasis will be on precalculus methods, although Calculus Lite will be invoked from time to time.

Join us this summer for another week-long residential program of stimulating workshops and activities led by New York Math Circle instructors and Bard Math professors. This summer features math enrichment for middle school math teachers at all levels.

Share in the rich environment of creative and insightful mathematical problem solving with other interested colleagues. All that is required is a strong interest in Mathematics. Accommodations are provided. Graduate credit is available.

Not a course! Just informal gatherings for teachers, to discuss some mathematics, work on some problems, and have dinner together. If you are interested in any talk in the series, please register (for free) using the link above, so that we know the level of interest. More talks will be added shortly.

• March 27, 5pm. "Exploring coin flipping and probabilities" by Larry Guth, Professor at NYU's Courant Institute.

  Coin Flipping Exploration Handout

  Suppose that one student flips a coin a thousand times and writes down the sequence of heads and tails, let's say an H for each head and a T for each tail. The sequence might start THHT... Now suppose that another student just writes down a sequence of a thousand Hs and Ts. The second student doesn't flip any coin, but they try their best to imitate a coin. We are shown these two sequences of Hs and Ts. Do we have any hope of distinguishing the real coin from the human imitation? Starting from here, we'll discuss various issues about coin-flipping and probabilities.

• April 3, 5pm. "The Infinitude of the Primes - Proofs from THE BOOK" by Shaun Ault, Fordham University.

  This talk highlights five branches of Mathematics: Geometry, Group Theory, Number Theory, Analysis, and Topology, by using techniques of each to prove that there are infinitely many prime numbers. The proofs are based on five of the six proofs found in Aigner and Ziegler's Proofs from THE BOOK. This talk is intended for an audience familiar with lower-level undergraduate mathematics. It is independent of, though related to, the upcoming Museum of Math talk on "Proofs from THE BOOK." In particular, proofs will be discussed in detail. Questions are welcomed!

• May 15, 5pm. "A Question of Definition" by Ted Diament.

  Mathematics contains many fascinating definitions that capture core insights, some of which took centuries to clarify. We will explore some of my favorite definitions, and I will try to make the case that exploring definitions in the classroom can help build confidence and be, in itself, highly thought-provoking.

• May 29, 5pm. "Exploring Projectile Motion - with TI-Nspire CAS Graphing Calculator" by Lyubomir Detchkov, Evanthia Basias & Sheila Krilov, Hunter College High School.

  Location: Hunter College High School

  Come explore the TI-Inspire as we delve into projectile motion! This session will be held at Hunter College High School and is an exciting opportunity to hear a wonderful presentation and try out some new classroom technology!

• CANCELLED June 12, 5pm. "The Stretched Chords Problem" by Ben Blum-Smith and Japheth Wood.

  Japheth Wood and Ben Blum-Smith will present their talk during the fall semester instead, so stay tuned for information about our fall teacher programming!

When can 4 different Fibonacci numbers be the sides of a quadrilateral? What are the two smallest integer-sided triangles that agree in 5 parts [sides, angles], yet are not congruent? Ever prove things using “Mass Points”?

Gil Kessler will conduct a 4-session course extracted from the one he gave in February 2011. It is for high school and middle school math teachers interested in adding to their collection of ideas, problems, and puzzles for themselves and their students. The material can be used to enrich lessons and homeworks, as well as challenge math team members. The presentation will be at a level accessible to all, with content ranging from easy to impossible [just kidding!], but always enjoyable.

[This course is independent of any other enrichment courses.]

Join Henry Ricardo on a 4-session excursion through the marvelous world of mathematical inequalities.

Despite being a rich and important subject, the topic of inequalities doesn’t fit neatly into any of the conventional areas of mathematics and so the systematic study of inequalities has been neglected in traditional high school, college, and graduate curricula.

Come explore the classical inequalities, as well as other useful results that appear frequently in mathematical competitions and in the problem sections of mathematical journals. We will discuss history, proofs, and problem solving strategies and demonstrate connections to geometry, linear algebra, and calculus, among other areas. Participants will share in Professor Ricardo's treasure-trove of examples and have opportunities for hands-on problem solving.

This mini course is designed for teachers who intend to coach/advise a Math Team, a Math Club, or a related extracurricular activity. It can also be taken by those who simply enjoy doing mathematics and solving Math Team type questions. Participants will learn how to conduct a Math Team, become acquainted with many different types of Math competitions, and get to know many of the sources and resources that are available

Some mathematical topics essential for Math Team will be presented, accompanied by a wealth of examples. We will examine ways of creating original problems to aid in the development of appropriate, effective practice materials. Techniques for training students, motivational devices, and philosophy of the Math Team experience will be addressed. Appreciation of problem solving is, of course, a key component. Special emphasis is placed upon potential for student growth as a consequence of Math Team.

The registration fee covers both sessions, and lunch on both days.

Given 6 points, can you connect each pair with a red or blue string, yet form no “monochromatic” triangles (whose vertices are from those points)? For what acute angle x does 8 sinx cosx cos2x cos4x = 1? A (right circular) cylindrical hole is drilled directly through the center of a sphere; if the height of the cylinder is 6”, what is the volume of sphere remaining?

Gil Kessler will conduct a 3-session course for high school and middle school math teachers interested in adding to their collection of ideas, problems, and puzzles for themselves and their students. The material can be used to enrich lessons and homeworks, as well as challenge math team members. The presentation will be at a level accessible to all, with content ranging from easy to impossible (just kidding!), but always enjoyable.

This course is independent of any previous enrichment courses, all are welcome.

Join us at Bard this summer for a week-long residential program focused on the investigation of inequalities and optimization. Enjoy an environment of creative and insightful mathematical problem solving for middle school and high school math teachers who wish to deepen their mathematical understanding. No prior experience with inequalities required, just an interest in doing mathematics in a community of teachers. Workshops and activities led by NYMC instructors and Bard math professors.

Math & Dinner sessions are gatherings for teachers, to discuss some mathematics, work on some problems, and have dinner together. This pair of sessions is free, but please register using the link above, so that we know how many people to expect.

Gauss said "Mathematics is the queen of the sciences and number theory is the queen of mathematics." To take it further, quadratic forms would be the queen's crown, with quadratic reciprocity the crown jewel. A major area of research for many mathematicians, notable contributors included Fermat, Euler, Legendre, and Gauss. Gauss referred to quadratic reciprocity as the "golden theorem".

This two-session mini-course will take you on an introductory incursion into this classical area of the theory of numbers. The goal is to motivate from an historical perspective the theory of binary quadratic forms as developed by Fermat, Euler, Legendre, and Gauss. We will present the highlights of the more elementary portions of the theory, including quadratic reciprocity, in order to whet the appetites of those who wish to pursue more deeply this still lively area of mathematics.

When can 4 different Fibonacci numbers be the sides of a quadrilateral? What are the two smallest integer-sided triangles that agree in 5 parts (sides, angles), yet are not congruent? Ever prove things using “Mass Points”?

Gil Kessler will conduct a 6-session course for high school and middle school math teachers interested in adding to their collection of ideas, problems, and puzzles for themselves and their students. The material can be used to enrich lessons and homeworks, as well as challenge math team members. The presentation will be at a level accessible to all, with content ranging from easy to impossible (just kidding!), but always enjoyable.

This course is intended for math teachers, whether new or experienced. Participants will be taken on a mathematical journey through topics in number theory, algebra, and geometry, exploring fundamental notions and special topics. Problem posing and problem solving are emphasized throughout. Particular attention is given to the structural underpinnings of mathematics that are relevant to sound and effective teaching.

Although mathematical content is mostly on a high school level, motivated middle school teachers are welcome to register as well.

These all-day problem workshops will feature a wealth of specially selected problems ranging from relatively simple to very challenging. Solutions, extensions and associated topics will be included. The sessions are a nice follow-up to the Math Team for Teachers mini-course offered last year, although prior attendance is not a pre-requisite.

The mathematical content of these sessions is mostly on a high school level. The registration fee covers both sessions, and lunch on both days.

This week-long residential program is focused on creative and insightful mathematical problem solving for middle school math teachers. The high level mathematics activities and classes are led by NYMC instructors and Bard math professors.

Participants will engage in a wide variety of mathematical investigation, problem solving, and classes that explore and develop a deep appreciation and understanding of the Theorem of Pythagoras and its consequences. Specific topics include geometric dissections, quadratic number system extensions, Diophantine equations, historical development, and a study of Euclid's original proof, which dates back over 2000 years.

This three session mini-course will take participants through a variety of wonderful mathematics problems. Participants will see both classic well-known problems, and little-known gems.

There will be an additional related "Math & Dinner" session on Wednesday 3/17 (see registration below). It will be open for free to all interested math teachers. This is a perfect opportunity to bring along colleagues.

Not a course! Just informal gatherings for teachers, to discuss some mathematics, work on some problems, and have dinner together. Please register using the link above, so that we know how many people to expect.

March 17, 5pm, "A Pleasing Potpourri of Paradoxes and Puzzles", by Gil Kessler. This talk will be open to both Great Problems mini-course participants and non-participants as well. Come see a selection of unusual problems, then join us for dinner at a nearby restaurant.

April 21, 5pm. "Exploration of Unique Factorization, Part I", with Ben Blum-Smith. The "Fundamental Theorem of Arithmetic" is the name for the familiar fact that every natural number factors into primes in just one way. We'll begin with some provocative problems exploring the importance of this fact, and then take a historical journey to understand why a proof of it is necessary.

April 28, 5pm. "Exploration of Unique Factorization, Part II", with Ben Blum-Smith, in Room 202. This picks up directly where the April 21 session leaves off. Participants will be led to collectively create one of the classic proofs of the Fundamental Theorem of Arithmetic. The proof will reveal some surprising connections between factorization and other aspects of arithmetic.

May 12, 5pm. "Solving the Cubic", with Japheth Wood. Although solution methods for quadratic equations had been known to the ancient Babylonians, it wasn't until the 16th century that solutions for the cubic began to surface in Renaissance Italy. Come join us to learn the fascinating history of the cubic and see for yourself how it works!

May 19, 5pm. "Exploration of Unique Factorization, Part III", with Ben Blum-Smith. We will prove the unique factorization theorem and extend its reach beyond the integers (We might even get to see why it fails in Z[√-5]). There will be a recap for participants who were not at the April 21 and 28 sessions, so everyone is welcome to attend.

We are pleased to offer a math circle session for middle school teachers at Stuyvesant HS this election day. We hope you can join us for an enjoyable day of mathematics, as always in a supportive environment. Space is limited, so please write to japheth@nymathcircle.org soon if you are interested. Your RSVP is required to attend.

Topics from first year Calculus are revisited and viewed from several perspectives.

Emphasis is placed on the underlying structural principles and pervasive unifying themes. Both classic and non-routine problems are presented together with some special theorems and non-standard proofs. In addition, we will offer tips for presentation and course design.

This course would be especially helpful to those teaching or planning to teach Calculus, and will provide insight and practical suggestions based on many years of experience teaching it.

This mini course is for teachers who intend to coach a Math Team, a Math Club, or a related extracurricular activity. It can also be taken by those who simply enjoy doing mathematics and solving math team type questions. Participants will learn how to conduct a Math Team, and will become acquainted with different types of competitions, and with many available sources and resources.

Some mathematical topics essential for Math Team will be presented, accompanied by a wealth of examples. Techniques for training students, motivational devices, and philosophy of the Math Team experience will be addressed. Appreciation of problem solving is a key component. Special emphasis is placed upon potential for student growth as a consequence of Math Team.

Not a course! Just informal gatherings for teachers, to discuss some mathematics, work on some problems, and have dinner together. Two sessions are planned:

August 6th, 5pm, Interactive Math Investigation, by Jesse Johnson and Ben Blum-Smith. Jesse and Ben recently attended the Math Circle Summer Teacher Training Institute, run by Bob and Ellen Kaplan (who visited both our teachers' and students' classes this spring) and will share their favorite topic with us. As always, the style of this evening of mathematics will be interactive, with everybody welcome to participate in deriving (from scratch) a classical result from ancient Greek mathematics.

August 13th, 5pm, Pianos and Mathematics, by Jeff Suzuki, Brooklyn College. Why does Asian music sound so distinctive, and a C-D chord sound so jarring? What are flats, sharps, and what do they have to do with irrational numbers? This talk will begin with an introduction to music theory and its mathematical challenges. Along the way, we'll learn how to tune a piano (or at least, learn why a piano is tuned the way it is) and explore some of the many connections between mathematics and music.

This course is a continuation of Math Enrichment for Teachers II started in Fall 2008. See description.

We are forming a math circle for middle school math teachers, with the aim of building a community of teachers interested in developing problem solving skills, deepening content knowledge and discussing pedagogy. The Middle School Math Teachers' Circle will meet on the second Wednesday of each month, followed, at the end of July, by a three day immersion workshop at Union College.

If you'd like to register for the summer workshop but can't attend the monthly meetings, use the registration link above, and mention this in the comment box on the form.

This course is intended for math teachers, whether new or experienced, who wish to enrich their classes. Participants will continue a mathematical journey through number theory, algebra, and geometry, exploring fundamental notions and some special topics. Both new participants and those continuing the Fall course are welcome.

The course will include several guest speakers. Problem posing and problem solving will be emphasized throughout, and opportunity will be provided to work on challenging problems during class time.