Friday, January 29, 2016

PUMaC: A Princeton-run math competition for high schoolers

Almost exactly 8 years ago, I was a senior in high school and travelled to Princeton with other students from my ARML team to participate in PUMaC, which stands for Princeton University Mathematics Competition.  One memory that stuck out from the day was Adam Hesterberg, that year's director, up on stage wearing a sarong (a part of Mathcamp culture).  I didn't win anything, but had a really good experience, both in terms of the math and learning more about Princeton, which I had just applied to. 

Little did I know that PUMaC would end up playing a huge role in my college experience.  It is entirely run by Princeton undergraduate students, and during my sophomore and junior year at Princeton, I coordinated the 100+ volunteers and managed a lot of the logistical aspects of the competition.  

How does the competition work? 

Teams of 8 students compete in a power round (which is proof based and done in the week leading up to the competition), 2 individual rounds (students choose from algebra, geometry, combinatorics, and number theory), and a team round (which usually has some fun gimmick to tie it together).  There are two divisions which teams can compete in - A and B - meant to give less experienced teams a chance of success.  Individuals who excel in one of the individual tests are invited to take the individual finals, an extremely challenging proof based round, which puts them in the running for the individual awards (but doesn't count for team awards).  In the afternoon when the individual finals and grading happens, there are lots of mini-events to keep the students occupied, like a puzzle hunt, board games, a math bowl competition, and a lecture by a prominent professor.  The competition is aimed at high school students, but advanced middle schoolers occasionally attend. 

One of my favorite aspects of the competition is that each team is assigned their own proctor and room.  The proctor is a Princeton student, and often ends up talking with their team and answering questions about Princeton and college in general.  To my knowledge, PUMaC is the only competition that does this - others have students in large lecture halls, or use parents and coaches as proctors.  PUMaC's choice to do this makes a lot of extra work for the organizers and limits the size of the competition, but also makes for a better and more personal student experience.  

While I was lucky to live in easy driving distance of Princeton, teams do come from all over the country (and even all over the world), to participate.  They have also started offering a power competition which only involves the power round and does not require traveling to campus - teams submit their solutions by email.  

In addition to the math itself, on-site competitions like PUMaC are valuable because (like math circles and camps) they bring students into an environment where being excited about math is the norm, give them opportunities to develop friendships with others who share their interests, expose them to aspects of math culture, and help to cement their identities as "math kids".  

Thursday, January 28, 2016

Prime Climb: A beautiful, colorful, mathematical board game

I haven't gotten a chance to play Prime Climb (amazon affiliate link here) yet, but it looks so good that I can't help writing a post about it now!  It's meant for ages 10 and up, but could also be enjoyed  by some younger kids.  From the amazon description:
Prime Climb is a beautiful, colorful, mathematical board game designed for 2 to 4 players. 
Roll the dice and add, subtract, multiply and divide your way to the center of the board, picking up Prime cards and bumping your opponents back to start as you go. The first to land both pawns on 101 wins the game! 
Everyone can learn to multiply and divide using Prime Climb's unique color coding. Inspire deeper mathematical understanding while mastering arithmetic! Prime Climb is a perfect game for families & schools, kids & adults. Awaken your love of math, with Prime Climb!
After looking at a few reviews (such as here and here), a few things stick out to me as being really awesome:

  • Prime Climb uses color to represent prime factorization (if 2 is orange and 3 is green, then 6 is half orange and half green).  For visual learners, this game could really help them understand prime factorization.  Plus it makes it more intuitive for younger kids who aren't yet fluent at multiplication and division. 
  • To move around the board, players role two 10-sided dice, and each of the results can be added, subtracted, multiplied or divided against the number either of their pawns currently occupies.  So there are lots of possibilities to consider on every turn... like the game 24 (amazon affiliate link here).
  • If you land on a prime, you get a special prime card. 
  • The game is very adaptable... you could limit the allowable operations to make it accessible for younger kids, play as solitaire, use only one pawn per person, etc.   
Prime Climb was made by Math For Love, and their website is definitely worth a visit (especially if you live near Seattle - they have weekend classes, host an annual Julia Robinson Mathematics Festival, and do professional development workshops).  The game came out of a successful Kickstarter campaign, and has since won many awards.

Clicking image leads to an Amazon affiliate link

Wednesday, January 27, 2016

NASA Mathematics Teacher Guides: Authentic glimpses into mathematics applications

Did you know that NASA has a bunch of teacher guides with practical mathematics applications in space science? I came across their booklet on transit math:
Although planets, stars and other celestial bodies move through space in complicated ways, space is so vast that rarely do such bodies collide. However, when someone watches these movements from a distant vantage point, it sometimes looks as though collisions occur because of the perspective. The introduction of Transit Math clearly explains the apparent "collisions," eclipses, transits and occultations to middle school students. The variety of concepts in this 44-problem collection includes synodic periods, planetary conjunctions, geometry, fractions, linear equations and probability. The problems are authentic glimpses of modern science and engineering issues, often involving actual research data. Each word problem includes background information. The one-page assignments are accompanied by one-page teachers answer keys.

They also have ones focused on the mathematics of black holes, astrobiology, the moon, sun, mars, space weather, and more!  If you have a kid who's into space, or want to show your class some exciting real world applications of the math they're studying, check them out! NASA does a lot of education work!

Thanks to Tiffany Petty for sharing this resource.

Tuesday, January 26, 2016

What is the Name of this Book?: The Riddle of Dracula and Other Logical Puzzles

Martin Gardner called this book "The most original, most profound, and most humorous collection of recreational logic and math problems ever written", and I agree.  I keep coming back to What is the Name of This Book?: The Riddle of Dracula and Other Logical Puzzles (here is my Amazon affiliate link) time and time again.  I mainly use it with middle school math circles, but I recommend it for upper elementary schoolers all the way through to adults!

The most well known chapter is Knights and Knaves.  On the island of Knights and knaves, everyone is either a knight, and always tells the truth, or a knave, and always lies.  What if two inhabitants come up to you, and one says to you: "We are both knaves." Can you figure out whether each of them is a knight or a knave? Another one of my favorite chapters is based off of Alice in Wonderland.  Each chapter reads like a story (which happens to ask a lot of questions of its readers).

In the future, I think I will combine this book with activities from Camp Logic whenever I want to teach logical reasoning.

Clicking this image leads to an Amazon affiliate link

Monday, January 25, 2016

Project Origami: activities for exploring mathematics

While at HCSSiM, I took a class on the mathematics of origami from Tom Hull.  It was an awesome experience, and even now I always have a few memo cubes lying around for when I get the urge to make something.  Origami is one of the topics I like to pull out for math circle sessions, but if I was relying only on my memory and notes from that summer, it would be nearly impossible.  Luckily, we have the internet, and I found out about Tom's book Project Origami: Activities for Exploring Mathematics.

It presents 30 great origami activities, complete with handouts (which are available here), solutions, and instructor's guides.  All of the material would be a great fit for a high school math circle, and some of the activities would be accessible to middle schoolers too.  Most involve concepts from college-level math and so also have a place in courses like calculus, number theory, discrete math, abstract algebra, and topology.  That's one of the things I love about origami - it draws from so many areas of math!

I highly recommend this book, as it's been of my favorite resources throughout the years.  I especially like:
  • Folding a Parabola
  • Can Origami Trisect an Angle?
  • Solving Cubic Equations
  • Making Origami Buckyballs
  • Making Origami Tori

My Amazon affiliate link to it is here.

Clicking this image leads to an Amazon affiliate link

PRIMES-USA: a remotely mentored mathematics research program for high school students

PRIMES stands for Program for Research in Mathematics, Engineering, and Science for High School Students.  We'll focus on PRIMES-USA, which is the only part of the program available to students who don't live in the Boston metro area.  High school juniors from the across the United States can apply to work on an original research project in mathematics for a year mentored remotely by an MIT researchers. Participants go through the same steps that professional mathematicians do, including writing a paper and presenting at a conference.  Many students use their work to enter the Intel Science Talent Research.  

PRIMES is a very small and competitive program (< 20 students) as coming up with and mentoring a research project takes quite a bit of work.  There are a lot of opportunities for mathematics research at the undergraduate level, but this and RSI are the main options for high school students (besides convincing an individual faculty member to work with you). 

I think that all high schoolers could benefit from doing some sustained mathematical exploration (like this), but I think only a few are ready for PRIMES/RSI style original research (and that's perfectly okay).

Sunday, January 24, 2016

Camp Logic: a book filled with a week's worth of logic games and activities

Camp Logic is another book from Natural Math. From its amazon description:
This book offers a deeper insight into what mathematics is, tapping every child's intuitive ideas of logic and natural enjoyment of games. Simple-looking games and puzzles quickly lead to deeper insights, which will eventually connect with significant formal mathematical ideas as the child grows. This book is addressed to leaders of math circles or enrichment programs, but its activities can fit into regular math classes, homeschooling venues, or situations in which students are learning mathematics on their own. The mathematics contained in the activities can be enjoyed on many levels.
The material is broken into 5 days, but I could also see it spanning a semester or so of a math circle (there are 18 activities), or used as warmup material for the entire year. There's a lot of helpful commentary for leaders throughout the book. I think it would be perfect for upper elementary or middle school students, and can't wait to try it out with my middle school math circle group!

Update: I've used a few activities from here with my middle school math circle group now.   They loved the Cryptarithms, and I was surprised that many of them figured out the Cryptogram Puzzle (I put it at the bottom of the sheet as a challenge problem).  The Game of Giotto didn't work so well... I think introducing it via the puzzles instead of as a full class game might have been better. 

As with the other Natural Math books, it has an open license under Creative Commons, and you can choose from paperback, ePub + Kindle + PDF, or name your price PDF. Here's my amazon affiliate link.

Clicking this image leads to an Amazon affiliate link

Al-Jabar: a board game based on abstract algebra

Robert Schneider, who is one of my fellow graduate students at Emory, created a mathematical board game with Cyrus Hettle while at University of Kentucky. According to the game's website:
The game of Al-Jabar is based on concepts of color-mixing familiar to most of us from childhood, and on ideas from abstract algebra, a branch of higher mathematics. Once you are familiar with the rules of the game, your intuitive notions of color lead to interesting and often counter-intuitive color combinations created in gameplay. 
Al-Jabar contains pieces in eight different colors. You are dealt a hand of thirteen of these pieces, and your goal is to get rid of your hand before your opponents empty theirs. By combining and exchanging pieces in your hand, you can reduce your hand, and even give your opponents additional pieces to slow them down. 
In addition to being a challenging strategy game, Al-Jabar is ideal for the classroom, either to teach color mixing theory in elementary or middle school or group theory in an undergraduate abstract algebra course. While Al-Jabar is based on highly abstract and profound mathematical ideas, gameplay consists only of making creative color combinations. Anyone can understand and play the game, even if they have no mathematical background.
On the website, you can find the rules, suggestions for how to make the game yourself, and a link to buy it.

Al-Jabar Games

Project Euler: a series of challenging math problems which require programming

According to their website:
Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems. 
The motivation for starting Project Euler, and its continuation, is to provide a platform for the inquiring mind to delve into unfamiliar areas and learn new concepts in a fun and recreational context. 
The intended audience include students for whom the basic curriculum is not feeding their hunger to learn, adults whose background was not primarily mathematics but had an interest in things mathematical, and professionals who want to keep their problem solving and mathematics on the cutting edge. 
The problems range in difficulty and for many the experience is inductive chain learning. That is, by solving one problem it will expose you to a new concept that allows you to undertake a previously inaccessible problem. So the determined participant will slowly but surely work his/her way through every problem.
I highly recommend this for students who want to learn more math and programming through solving challenging problems.  There are currently 544 on the website!

Saturday, January 23, 2016

Math Girls: a Japanese book series combining mathematical rigor with light romance

Math Girls is a book series written by Hiroshi Yuki and translated into english (from the original Japanese) by Tony Gonzalez.  I originally found out about while attending the Joint Math Meetings in January 2014.  From the publisher:
Combining mathematical rigor with light romance, Math Girls is a unique introduction to advanced mathematics, delivered through the eyes of three students as they learn to deal with problems seldom found in textbooks. Math Girls has something for everyone, from advanced high school students to math majors and educators 

In addition to the main series of novels, there's also Manga, and a Math Girls Talk About... series, which focuses on high school level concepts and is meant to be more accessible.

Apparently these books are really popular in Japan, and hopefully more will keep being translated into English!

Here are my amazon affiliate links for the books:

Clicking this image leads to an amazon affiliate link

Who Wants to be a Mathematician? A mathematical game show for high school students

From their website:
Who Wants to Be a Mathematician is a game in which high school students compete for cash and prizes by answering multiple choice mathematics questions.
There are two versions - a national competition, held annually at the Joint Math Meetings, and regional competitions, which are held in six or so locations around the country every year, and are often part of larger outreach events. For both versions, qualification comes by taking one or two rounds of school administered tests. Not a lot of schools know about it - so if you want to participate, you're most likely going to have to ask.  It is possible for younger students to qualify, but the questions pull from all of pre-calculus math.

RSI: a summer research program for high school students

According to their website:
Each summer, 80 of the world's most accomplished high school students gather at the Massachusetts Institute of Technology (MIT) for the Research Science Institute (RSI). RSI is the first cost-free to students, summer science & engineering program to combine on-campus course work in scientific theory with off-campus work in science and technology research. 
Participants experience the entire research cycle from start to finish. They read the most current literature in their field, draft and execute a detailed research plan, and deliver conference-style oral and written reports on their findings. 
RSI scholars first participate in a week of intensive STEM classes with accomplished professors. The heart of RSI is the five week research internship where students conduct individual projects under the tutelage of mentors who are experienced scientists and researchers. During the final week of RSI, students prepare written and oral presentations on their research projects.
This is one of the most competitive programs to get into as a STEM-focused high school junior.  Why? It's free, and unlike other summer programs for high schoolers, it's about doing research and not taking classes.  The students who attend in mathematics tend to already have excelled in other summer math programs (PROMYS, HCSSiM, MathCamp, etc).  Many use their projects to enter the Intel Science Talent Search, the nation's most prestigious science research competitions for high school seniors.

Friday, January 22, 2016

Expii: a free platform for crowdsourced explanations and practice problems

According to the website, Expii is
a free, open, and interactive platform for crowdsourced explanations and practice problems in math and science. Learn, discover, explain, and explore. Together, we can ensure that everyone has access to a personalized education.
One thing I really like about it is the focus on multiple ways of explaining the same concept - if a bunch of people all explain something in the way that makes most sense to them, every learner can find an approach that clicks.  They also release occasional problem sets, which require creative thinking, vary in difficulty, and are written by Po-Shen Loh (a math professor who coaches the US IMO team).

This website is still in its early stages, and there are a lot of topics which don't yet have any explanations. But I think that teaching is one of the best ways to learn, so I highly encourage students to contribute! Eventually, I think it will be a great resource not only for students who want to learn on their own or are struggling to understand a concept in school, but also for future and current teachers who or are looking for alternate ways to explain something.

Thursday, January 21, 2016

Musings of a Mathematical Mom: a blog filled with ideas for injecting math into your young child's life

I know A. O. Fradkin from when I was an undergrad at Princeton, and she has an awesome blog called Musings of a Mathematical Mom, about her educational adventures with her two daughters (currently ages 3 and 6, I believe) as well as the enrichment math classes she teaches for ages 4 - 10. You'll also find a list of kid-oriented math reads on her website. If you want some ideas for how to inject mathematics into your everyday interactions with your child or plan enrichment activities for them and their friends, this is a great place to look.

I've even used ideas from her blog for the middle school group of Emory Math Circle - for example this post about the game Nim.

Wednesday, January 20, 2016

Spot It: a matching game leading to interesting mathematical questions

Spot It! (amazon affiliate link here) is a matching game. On each card, there are 8 symbols (with varying size and orientation) out of a possible 57 (I think? I haven't counted and the internet does not have a consensus), and each pair of cards has exactly 1 symbol in common.  You're goal is to find that common symbol as quickly as possible.  The interesting thing about this game is understanding how they were able to design the game so that each pair has exactly one symbol in common.

There are a lot of blog posts explaining the mathematics behind Spot It! and how it could be used as a math circle topic, for example this, this, and this. There are also a few StackExchange questions, such as this and this. I've asked students: If you have S total symbols with C on each card, what's the largest deck of cards you can make so that each pair has exactly 1 symbol in common? With smaller values of S and C, this problem is more manageable, and students can even use their solutions to design their own (easier) versions of the game!

Also, it's interesting to note that with C = 8 and S = 57, you can make 57 cards, while the Spot It! game only has 55.  Can you figure out which two cards are missing?

There are a bunch of different versions of Spot It! aimed at various demographics, as you can see below (images lead to Amazon affiliate links).


Set: a fun game leading to lots of mathematics

Since the 1990's, the game SET has been a staple at math summer camps and other gatherings of the young and mathematically inclined.  Here's my amazon affiliate link. The goal of the game is to find as many sets as possible - a set consists of three cards which satisfy all of the following conditions:
  • They all have the same number, or they all have different numbers
  • They all have the same symbol, or they all have different symbols
  • They all have the same shading, or they all have different shadings
  • They all have the same color, or they all have different colors
Playing the game involves pattern recognition, but there is a lot of mathematics lurking in the background.  For example, some people like to deal the last card facedown, as it's possible to figure out what it is using the other cards on the table.  You can ask a lot of combinatorial questions: What's the probability of producing a set from three randomly drawn cards? What's the largest group of cards you can put together without creating a set? How many unique sets are there in a deck? What are the odds that there will be no set after 12 cards are dealt? You can model the game using four-dimensional vectors over the finite field with 3 elements.  You can create other versions, like Projective Set, based on other mathematical models.

Clicking this image leads to an Amazon affiliate link

One of my favorite games based off of set is Swish (amazon affiliate link here), which strengthens spatial-recognition skills.

Clicking this image leads to an Amazon affiliate link

Monday, January 18, 2016

Canada/USA Mathcamp: a mathematics summer camp for high school students

From their website: 
Canada/USA Mathcamp is an intensive 5-week-long summer program for mathematically talented high school students, designed to expose these students to the beauty of advanced mathematical ideas and to new ways of thinking. More than just a summer camp, Mathcamp is a vibrant community, made up of a wide variety of people who share a common love of learning and passion for mathematics. At Mathcamp, students can explore undergraduate and even graduate-level topics while building problem-solving skills that will help them in any field they choose to study.
At Mathcamp, the schedule is jam packed, but everyone on it is optional: students can choose what they want to do and when.   This makes Mathcamp unique and particularly well-suited for repeat campers.  There are classes at a wide variety of levels, and so this program is the obvious choice for the most advanced students (i.e, the ones who already know a lot of undergraduate mathematics).

The culture of Mathcamp is hard to describe (I've never been, but have known enough Mathcampers to get a pretty good sense), but is definitely on the fun and silly side.  For example, see this invented language and this a cappella video.

CTY: identifying and nurturing talented youth

John Hopkins Center for Talented Youth is all about identifying and nurturing young people of great academic promise. They run an annual talent search for grades 2 - 8, which qualifies students for their other programs.  I took the SAT in 8th grade through them.  Although I didn't participate in any of the programs it qualified me more, it was still helpful in convincing my high school to let me take more advanced math classes.  After you become part of their network, you can take online classes through them (including self paced math courses, which are quite good for acceleration), attend their summer programs, and even participate in family programs. 

Although they do have math courses in their summer program, I wouldn't quite consider it to be in the category of math camps like PROMYS, HCSSiM, MathPath, Canada/USA Mathcamp, etc.  For example, students are admitted based on standardized test scores instead of their work on challenging untimed proof based problem sets. But it is certainly a nerd camp, and I know many people who had a great experience attending! 

Sunday, January 17, 2016

HCSSiM: a summer program for high school students who appreciate yellow pigs and the number 17

Another program in the category of residential math camps for high school students is HCSSiM, which stands for Hampshire College Summer Studies in Mathematics.  From their website:
HCSSiM is an intensive six-week encounter with college-level mathematics for talented and highly motivated high school students. It is demanding and expanding. Participants spend a major portion of each day actively engaged in doing mathematics (not simply learning the results of mathematics).
 The daily schedule includes 4 hours of morning classes (Mon-Sat), the pre-supper Prime Time Theorem, and evening problem sessions. Afternoons are devoted to reading, rest, recreation, occasional trips to town, and informal study. Participants have unparalleled access to faculty members in classrooms, at meals, and in the program dorm. Productive collaborations continue long after the program, and many lifelong friendships are forged.
I was a student here in 2007, and for me it definitely was a transformative experience.  I spent 8 hours a day doing math, and I loved it! This was when I first realized that mathematics was creative, and was given the opportunity to discover mathematics for myself instead of being lectured about it.  Looking back, I really appreciate that we were encouraged to stop thinking about our problem sets once the evening session was over, and that I got a solid foundation in proof writing that helped me succeed as a math major at Princeton.

HCSSiM certainly has a unique culture which needs to be experienced to really understand - but be prepared to appreciate the number 17 and yellow pigs (for example, see here).  For me, it's on the fun and quirky side of summer math camps.

Yellow Pigs Day in 2007

Julia Robinson Math Festivals: a noncompetitive problem solving event for middle and high school students

Julia Robinson Mathematics Festivals inspire students to explore the richness and beauty of mathematics through activities that encourage collaborative, creative problem-solving. The Festival provides an alternative to the competitive culture of math contests, with tables dedicated to different problems and activities, each with a mathematician standing by to offer guidance and encouragement without giving away too much. Says the Festival's Mathematics Director, Joshua Zucker, "We offer intriguing and challenging problems, puzzles, and activities, and a supportive setting for kids who like to take their time working on a problem. Students make whatever progress they make, they get help, they work with other kids, and eventually come to some kind of stopping point and go find another activity of interest."

Most festivals are aimed at students in grades 6 - 12, although some will be more tightly focused or accommodate younger students. They can be set up for children come with their parents, or as a school field trip.  Some regions have them annually, but they are not yet super wide-spread.  Emory will be hosting our first next weekend, and I'm super excited! Some (but not all) of the problems and activities used at these festivals are available online, and would also work well for math circles.

ARML: a regional team math competition for high schoolers

I grew up in the Lehigh Valley, which in the math world is well known for having one of the best ARML teams.  The American Regions Mathematics League is an annual high school mathematics competition held concurrently at four sites (University of Iowa, Penn State, UNLV, and University of Georgia).  Students compete in teams of 15, usually representing a large geographic region like a state or a major city (although some elite schools can field their own teams).  It consists of:
  • a team round, where the team works together to solve 10 problems in 20 minutes
  • a power round, where the team works together for 1 hr to solve a multi-part problem requiring explanations and proofs in 1 hr
  • an individual round, where each member works individually to solve 5 pairs of problems, with 6 minutes given per pair
  • a relay round, where the team is split into groups of 3, and within each group the first team member solves a problem and passes the solution to the next team member, who plugs that answer into their question, and so on. There are two relay questions, and 6 minutes are given for each (although extra points are awarded for finishing in 3 minutes)
There's a lot of variation on how much a team will practice beforehand and how they will select their members (usually through some sort of preliminary test).  Many regions will have multiple teams (for example, Lehigh Valley has Fire, Ice, Lightning, Thunder, and Storm), some of which compete in the A division and others of which compete in the B division.  

I never made it onto the top Lehigh Valley team (although I was the captain of Ice my senior year, and we got 2nd place in the B division!), and I think more than the math itself I enjoyed the social aspect of ARML.  During the competition, I got to stay on a college campus, hang out with my fellow math nerds, and see friends from other parts of the northeast.  A lot of Mao and Mafia was played, and there was even a song contest.

Note: There are fees for registration and housing, but my team always had sponsors to cover them.  I'm not sure if this is the case more generally. 

Saturday, January 16, 2016

PROMYS: a summer program for high school students

From their website:
PROMYS is a challenging program designed to encourage ambitious high school students to explore the creative world of mathematics. Each summer, approximately 80 high school students from around the country gather on the campus of Boston University for six weeks of rigorous mathematical activity. Through their intensive efforts to solve an assortment of unusually challenging problems in Number Theory, participants will practice the art of mathematical discovery. The problem sets encourage students to design their own numerical experiments and to employ their own powers of analysis to discover mathematical patterns, formulate and test conjectures, and justify their ideas by devising their own mathematical proofs.
I absolutely love the PROMYS curriculum and this style of learning, but in terms of summer math camps, it's definitely on the serious side.  Students have to learn to manage their time, deciding how to spend their hours outside of class and how much time each day to devote to their problem set, which they are unlikely to be able to complete.  This feels a little bit too much like grad school to me... where there's no such thing as finishing your work, and so whenever you take some time away from it you feel a little guilty.  It also is one of the few programs to have a midterm and final exam.  However, in exchange for all this, you really do learn a huge amount, and if you love the mathematics you are discovering you'll have a great summer.

I was a counselor at PROMYS the summer after I graduated college, in 2013, and found that even as someone about to start a PhD in number theory, I learned a lot from the number theory course.  I just was left with the impression that as a high schooler, this program would have been a little too much for me.

The PROMYS program is based off of another math camp called ROSS, which takes place at Ohio State.  I haven't had any personal experience with ROSS, but I definitely know people who attended and loved it.  The two programs are similar enough that geography could reasonably be a deciding vote.

PROMYS also runs a concurrent program for teachers, which I think is a fantastic idea.  It's associated with Math for America Boston, Focus on Mathematics, and the Master's Degree in Mathematics for Teaching at Boston University.  

Zometools: a construction kit for kids, artists, scientists, and educators

Zometools are pretty much my favorite mathematical toy.  Well, toy is an understatement - it's a modeling kit that be can used to build complicated and beautiful 3-D structures, and is used by kids, artists, scientists, and educators.

  • I've done enrichment activities with elementary schoolers (and my friends and family) involving dipping Zometool structures in bubbles.  See this article for more information.  Here's my amazon affiliate link to their Zometool Crazy Bubbles Kit, but any zometools kit can be used with bubbles!
  • Zometool has a pdf of lesson plans for teaching with the Zome system.  I haven't used any of them directly, but have certainly looked at them for inspiration!
  • Once in college, Manjul Bhargava used some Zometool models in our 400-level representation theory class. 
  • I've often used them when teaching platonic solids, although you'll need a kit with special green struts to make all five. 
  • I've spent lots of time playing with them myself!
The main drawback of Zometools is that they break easily, but they'll replace your broken parts for free, so it's hard to complain. 

I got started with the Creator 1 Constructor Kit (amazon affiliate link), but it does not have any of the green struts.  You can buy them separately (amazon affiliate link), or get the Kepler's Cosmo Kit (amazon affiliate link).

Me at the Children's Museum of Atlanta

Math Kangaroo: a fun competition for grades 1 through 12

Math Kangaroo is an annual math competition for students in grades 1 through 12.  It's multiple choice, and I've found the questions to be interesting, challenging, and age appropriate.  It's one of the few competitions that includes elementary school students (as long as they can read and answer a multiple choice test independently). I consider this contest to be more on the fun side (rather than the serious side) - for example, students get a tshirt, small gift, and mathematica license (for grades 6 and above) when they participate.

Sign up is in the fall, and the competition happens in March.  It's not something that most schools offer, but depending on your location you may be able to find (or start!) a testing center near you.

AOPS: books, online courses, and community for secondary school students

Art of Problem Solving was founded by Richard Rusczyk in 2003 to create interactive educational opportunities for avid math students. Here of some of the things you'll find:
  • Challenging and rigorous textbooks and online classes for prealgebra through calculus (including some topics not usually taught in school, like number theory and combinatorics).  Most mathy homeschoolers I know in grades 6 - 12 use these as their primary textbooks. 
  • Textbooks and online classes focused on problem solving and math contest preparation.
  • AoPS wiki, which includes a repository of old math contest problems, lists of math summer programs and competitions, and lots of other resources.
  • A free online learning system called Alcumus and a MATHCOUNTS style online game called For the Win!
  • A forum, which is especially good for students who don't have a lot of math-y peers nearby. 
This is definitely a website to check out for middle and high school students who are into math competitions and problem solving.  I still use it regularly! 

Math Circles: in person math enrichment during the school year

To describe what a math circle is, I'll follow Circle in a Box (one of the best resources for starting a math circle), and quote Mark Saul:

Mathematical circles are a form of outreach that bring mathematicians into direct contact with pre-college students. These students, and sometimes their teachers, meet with a mathematician or graduate student in an informal setting, after school or on weekends, to circle work on interesting problems or topics in mathematics. The goal is to get the students excited about the mathematics they are learning; to give them a setting that encourages them to become passionate about mathematics. 
Math circles can have a variety of styles. Some are very informal, with the learning proceeding through games, stories, or hands-on activities. Others are more traditional enrichment classes, but without formal examinations. Some have a strong emphasis on preparing for olympiad competitions; some avoid competition as much as possible. Models can use any combination of these techniques, depending on the audience, the mathematician, and the environment of the circle. Athletes have sports teams through which to deepen their involvement with sports; math circles can play a similar role for kids who like to think. One feature all math circles have in common is that they are composed of students who enjoy learning mathematics, and the circle gives them a social context in which to do so.

There are lots of math circles throughout the US (I run one in Atlanta, for example), and I highly recommend checking out this list and doing a little googling to find out if there's one near you!

Mastermind: a game of logic and codebreaking

Mastermind is a two-player codebreaking game, and is recommended for ages 8 and up. Click on the pictures for my Amazon affiliates links for the regular version and the slightly easier kids version.
Just playing the game develops logic and reasoning skills, but you can also use the game as a jumping off point for more mathematical thinking. For example: Can you devise a strategy which guarantees you can always break the code in 10 moves? 5 moves? See this blog post for some of the mathematics.

I've also seen Mastermind turned into a one-player puzzle, like the one below. You could even have students try to create their own!

Puzzle made by Elisa Giaccaglia for an elementary school enrichment class

MathPath: a residential math camp for middle schoolers

MathPath is a four week residential math camp for middle school students.  It's kind of like MathCamp, but is designed for younger students and therefore has way more supervision and structure.  For example, at MathPath most of the kids who fly in are met at the gate instead of baggage claim, there are certain streets that can only be crossed with a counselor, and everyone takes classes during the same time slots.

Some (but not all) of the participants are quite advanced mathematically (for example, already knowing much of the high school math curriculum), and so breakout courses at a wide variety of levels are offered.  Students are doing math for about 6 hours most days, which depending on the kid could be a dream come true or a nightmare.  I strongly believe that the decision to attend a program like this should be led by the student, and not the parent.

I was a counselor here in 2010.  Like other summer math camps, admission is largely based on performance on the qualifying test.  This is not like a regular test - it's not timed, the thought process and explanation (and not the answer) matter most, and students are not expected to get complete solutions to every problem.  If you're not sure if MathPath would be a good fit for you, try out the qualifying test - if you enjoy working on it and can make significant progress on most of the problems, than the program would likely be a good fit.

Friday, January 15, 2016

MATHCOUNTS: a mathematics competition for middle school students

MATHCOUNTS is a nationwide school-based competition for middle school students.  Not all schools offer it - if your school doesn't I encourage you to push a bit and help them find an appropriate coach (teacher, parent, older student, etc).  There are four levels: school, chapter, state, and national, and the problems get harder as you go. There are four rounds: sprint (individual written exam), target (four 6-minute mini-exams), team (group written exam), and countdown (oral fast-paced head to head round).  While some students thrive on the focus on speed and the competitive nature of the competition, it's not attractive to all students.

The MATHCOUNTS foundation offers a few other middle school mathematics program, like The National Math Club, which provides math club leaders with resources and materials needed to run other clubs.  Unlike the competition series, math circles and other tutoring / enrichment centers can register.

I was not able to participate in MATHCOUNTS myself, as my school had stopped offering it a few years before, but I know many people who have.

Moebius Noodles: adventurous math for the playground crowd

Ever wanted to introduce the concept of a function to your three year old? Moebius Noodles is one of my favorite books to help parents engage with their young children (including toddlers!) about mathematics, and is available as a 'name your price' PDF as well as a physical book.  Each chapter is focused on a different topic (like functions), and includes ideas for conversations, games, and activities appropriate for a variety of ages.  I think this book will be useful for all parents, ranging from the math phobic to the math professors, who want to enjoy playful math with their children.  This would also be a great starting point for a pre-k / elementary math circle.

Amazon affiliate link: Moebius Noodles.

Moebius Noodles is a project of Natural Math, a community for families, math circles, and other learning groups, and includes online courses, books, local events like math circles, a newsletter, blog, and forum.   It's all about giving young children a chance to observe, play with, and ultimately make their own math. I hope to feature some of their other resources in separate posts, but for now take a look at their website!

Beast Academy: a new curriculum for grades 2 - 5

Beast Academy is an awesome curriculum for students in grades 2 - 5 (as of Jan 2016, this is still in progress - but grades 3 and 4 are out) being produced by the Art of Problem Solving.  It is done in am engaging and age-appropriate comic book style, while presenting material at a deeper and more challenging level than a typical elementary curriculum.  Perfect for homeschoolers or as a supplement to school!

I haven't used these books myself, but have recommended them before and heard good things back!