Last week, I was looking at mathematical coloring pages to use for math circle's holiday party, and I found this awesome free PDF by the mathematician Marshall Hampton. The patterns are beautiful and fun to color, but also involve some deep mathematics!

BTW, if you're looking for a mathematical coloring book to give as a gift, I highly recommend buying either Patterns of the Universe (Amazon affiliate link here) or Visions of the Universe (Amazon affiliate link here) by Alex Bellos and Edmund Harriss.

A collection of games, toys, books, programs, and websites for K-12 students, parents, and teachers.

## Tuesday, December 13, 2016

## Monday, December 12, 2016

### A better way to teach geometry

Today, I saw this blog post by Christopher Danielson entitled "How I Learned to Love Middle School Geometry". I thought it was super useful for parents, math circle instructors, and math teachers. A few highlights:

- Students spend a lot of time learning how to classify quadrilaterals throughout the years (see example below). But what if you asked them to do the same thing for hexagons? Suddenly, there's creativity, invention, and discovery! This seems like an awesome idea for a math circle or a school math class.

- His book entitled

- Students spend a lot of time learning how to classify quadrilaterals throughout the years (see example below). But what if you asked them to do the same thing for hexagons? Suddenly, there's creativity, invention, and discovery! This seems like an awesome idea for a math circle or a school math class.

- His book entitled

*Which one doesn't belong?*(Amazon affiliate link here) also looks awesome. On each page there are four shapes, and any one of them could be the one that doesn't belong - so instead of finding the correct answer, it's all about explaining and justifying your reasoning. For example, in the cover image below, the bottom right could be the odd one out because it has 5 sides and the rest have 6. Or the top right shape, because it's the only one that's not convex. Can you think of criteria that make each of the two leftmost shapes not belong?Clicking the image leads to an Amazon affiliate link |

## Saturday, October 15, 2016

### MouseMatics: Workbooks for Preschoolers

In reading this blog post from Musings of a Mathematical Mom, I found out about an awesome series of math workbooks (or maybe I should call them playbooks?) called MouseMatics for young children (ages 4 - 7) by Jane Kats. She also has a book of math games called Math for Dessert, and a Christmas Coloring Book.

These books aim to break the standard conventions:

They know that a picture with dots drawn in the corners of a square is called “four”, and an identical picture with an extra dot in the middle is called “five”. But can they recognize the same number five if it looks different?Always representing the number of five with the same configuration of dots allows students to memorize the picture instead of counting. This book makes sure to not let that happen:

In some problems, we have two correct solutions (a pair of friends can split a chocolate bar in different ways); some problems lack an answer (an odd number of objects cannot be divided in two). After all, in mathematics no solution is also a solution. A matching pair of shapes can be found in the same column. We build geometric figures using not only the usual squares, but also diamonds, trapezoids, and triangles. When the child is asked to find a number value, we use a variety of shapes in addition to dots: diamonds, crosses, anything. Also, their arrangements in the boxes are random, unlike the standard dice configurations.However, the thing I like most about these books is this:

The main purpose of the Mousematics series is not to teach preschoolers to count (this is something every child will inevitably learn sooner or later), but to spark the children's interest, to invite them along on an exciting journey into the world of logics and mathematics.

### Resources for advanced high schoolers

David Zureick-Brown is a math professor at Emory, and as I'm writing this he is in the middle of giving a talk to Atlanta-area undergraduate students. At the beginning of the talk, he pointed to a page on his website which lists fun math links for undergraduates. While this blog is not aimed at undergraduate students, many of the resources would also be appropriate for advanced high schoolers. Check it out!

## Wednesday, August 24, 2016

### YouCubed: Lots of great resources for students, parents, and math teachers

This morning, while scrolling through my Facebook feed, I came across an article entitled "Not a Math Person: How to Remove Obstacles to Learning Math". Being a mathematician, the people I meet are always telling me how impressed they are that I study math, how smart I must be, how they were always bad at math in school, how they just aren't a math person, etc. It's incredibly frustrating, and I still haven't worked out the perfect response - I usually acknowledge that a lot of people have had bad experiences with math at school, but try to impart that it doesn't have to be that way, that pretty much everyone is capable of learning math.

Through reading this article, I came across YouCubed and the work of Stanford math education researcher Jo Boaler. If you haven't already, take some time to read the article and browse YouCubed. Seriously. So much good stuff there for parents, students, and math teachers at all levels. After an hour browsing YouCubed, I've already used its Week of iMath to plan my first few middle school math circles. There are books, online courses, articles, lesson plans, and more.

Through reading this article, I came across YouCubed and the work of Stanford math education researcher Jo Boaler. If you haven't already, take some time to read the article and browse YouCubed. Seriously. So much good stuff there for parents, students, and math teachers at all levels. After an hour browsing YouCubed, I've already used its Week of iMath to plan my first few middle school math circles. There are books, online courses, articles, lesson plans, and more.

## Monday, August 22, 2016

### STEM GEMS: Awesome roll models for girls in STEM

This past Friday, I met Stephanie Espy, the author of the book STEM GEMS. The book tries to combat the lack of women in STEM fields by profiling 44 women who are doing amazing things in science, technology, engineering, and mathematics.

The mission of the book is "to help girls and young women to see their future selves as scientists, technologists, engineers and mathematicians, and to show them the many diverse options that exist in STEM". The book also contains a few chapters at the end giving actionable advice for how to set yourself up for success in a STEM career. The women in the book put a special emphasis on the challenges they faced along the way, helping to debunk myths common in math and other STEM subject like "The Genius Myth" and the "It-Should-Be-Easy Myth".

I was very surprised to find that 12 of the 44 women fall into the broad category of mathematician, and there was even a number theorist (Melanie Matchett Wood) featured!! In many STEM initiatives, the M is not emphasized beyond it's usefulness in S, T, and E.

Stephanie Espy was trained in chemical engineering, and so has experienced first hand what it's like to be one of the only women in the room (and oftentimes, THE ONLY). Luckily, she grew up in a family filled with STEM professionals, and so she had the role models that many young people - especially women and people of color - lack.

On browsing it in person, I found this book to be very visually appealing and not at all intimidating, with big color photos to draw you into the stories of these women. While there is value to learning about historical women in STEM, like Marie Curie, I think that seeing women in the modern world succeeding at fields like animation, global health, data science, and electrical engineering can have an even bigger impact.

The mission of the book is "to help girls and young women to see their future selves as scientists, technologists, engineers and mathematicians, and to show them the many diverse options that exist in STEM". The book also contains a few chapters at the end giving actionable advice for how to set yourself up for success in a STEM career. The women in the book put a special emphasis on the challenges they faced along the way, helping to debunk myths common in math and other STEM subject like "The Genius Myth" and the "It-Should-Be-Easy Myth".

I was very surprised to find that 12 of the 44 women fall into the broad category of mathematician, and there was even a number theorist (Melanie Matchett Wood) featured!! In many STEM initiatives, the M is not emphasized beyond it's usefulness in S, T, and E.

Stephanie Espy was trained in chemical engineering, and so has experienced first hand what it's like to be one of the only women in the room (and oftentimes, THE ONLY). Luckily, she grew up in a family filled with STEM professionals, and so she had the role models that many young people - especially women and people of color - lack.

On browsing it in person, I found this book to be very visually appealing and not at all intimidating, with big color photos to draw you into the stories of these women. While there is value to learning about historical women in STEM, like Marie Curie, I think that seeing women in the modern world succeeding at fields like animation, global health, data science, and electrical engineering can have an even bigger impact.

## Thursday, May 26, 2016

### Dice games: Quick and fun mental math practice

I was looking through ThinkFun's games today (because they have so many awesome ones!), and found one called Math Dice (Amazon affiliate link here). The idea is that you roll 2 twelve-sided dice and multiply them to get your target number, and then roll 3 six-sided dice. You want to use addition, subtraction, multiplication, and division to get as close as you can to the target number. They also have a version for younger kids (Amazon affiliate link here) that only uses addition and subtraction.

Unlike most other ThinkFun games, I don't think it makes sense to buy these, as all you really need are the dice. I instead encourage you to buy an assortment of dice (here's an example via Amazon affiliate link) which you can use for many different games!

I wanted to note a few things about dice games:

Unlike most other ThinkFun games, I don't think it makes sense to buy these, as all you really need are the dice. I instead encourage you to buy an assortment of dice (here's an example via Amazon affiliate link) which you can use for many different games!

I wanted to note a few things about dice games:

- Most can also be played with a deck of playing cards (like this), but there are subtle differences when playing with dice... previous roles don't effect your chances, you don't need a flat surface (rolling them in a tupperware container works well), etc.
- Look for games that have some strategy involved in addition to practicing mental math.
- Customize games to your child's skill level by modifying the allowed operations, number of dice, and type of dice. You can even invent a new game!
- Dice games are great for when you have just a few minutes to fill - a round goes by very quickly, and you can easily bring them with you anywhere.
- Sets of dice made for D&D and other role-playing games work really well for math games too!
- Spend some time looking at the shapes of the various dice. Why did they chose those numbers? Could you make a fair seven-sided die? Note that most of the ones in sets are Platonic Solids.

## Wednesday, May 25, 2016

### Programming Resources for Elementary School (and Pre-K)

Today I've been looking at resources to teach elementary school students to code. There are so many amazing options! The general idea is that kids drag and drop predefined blocks of code, usually to make an onscreen character do something, like jump or turn left. This blog post by Vicki Davis on Edutopia, which was updated in 2015, has a pretty comprehensive list. I don't yet have any strong opinions on which ones are best, so try out some of the free ones that fit with your kid's age and platform of choice.

A few ones that I want to highlight:

A few ones that I want to highlight:

- Code Studio's free online computer science fundamentals series is a full introductory curriculum for ages 4+. It could be used at home or in a classroom. It mostly consists of individually paced in browser videos and activities, but there are a few lesson plans for "unplugged lessons" along the way.
- Scratch is the most well known introductory programming language for kids. It allows them to program their own interactive stories, games, and animations, and share their creations with the online community. Scratch is free, works in a computer browser, and is aimed at ages 8 and up. Hopscotch is similar to Scratch, but made for iPad (and iPhone). For the slightly younger crowd, there's ScratchJr, an iPad and Android tablet app aimed at ages 5 - 7.
- There are a lot of board games that teach programming basics without any technology. For ages 4+, there's Robot Turtles (Amazon affiliate link here) and Littlecodr (Amazon affiliate link here), and for ages 8/9+, there's Code Monkey Island (Amazon affiliate link here) and Code Master (Amazon affiliate link here).

## Tuesday, May 24, 2016

### Math in Context: Building and designing a house on a budget

This is a cool math project given to a 3rd grade class in Atlanta. The challenge is to design a home with specific constraints that stays under budget. The constraints are things like

There's so much going on in this project! There's a lot of routine calculations, but in a manner that is obviously relevant, self-motivating, creative, and requires higher-level thinking skills. This project could be adapted to a wide range of grade levels, and kids could proudly present their finished creations to the class!

Anyway, here's the project sheet. I didn't make it - a parent showed it to me.

- "Your kitchen must be twice as big as your dining room"
- "Two of the hallways must be parallel"
- "You must have a garage which is 10% of your total square footage"

There's so much going on in this project! There's a lot of routine calculations, but in a manner that is obviously relevant, self-motivating, creative, and requires higher-level thinking skills. This project could be adapted to a wide range of grade levels, and kids could proudly present their finished creations to the class!

Anyway, here's the project sheet. I didn't make it - a parent showed it to me.

## Sunday, May 22, 2016

### Long Division: Why do we care?

Long division is a contentious subject in math education. Students struggle with it, and most people fail to see its relevance in an age when you can just ask Siri. I wanted to give some reasons why long division is worth teaching and mastering. What you'll hopefully see is that it is absolutely crucial for us to have a general procedure which takes a division problem and gives us a quotient and remainder. You might have seen this written as \(19÷3 = 6 \; R \;1\), but remember that \(19÷3\) is the same as \( \frac{19}{3}\) and \(6 \;R\; 1\) really means \(6 + \frac{1}{3}\) in this situation. Another way to write it is \(19 = 3*6 + 1\), where we multiplied both sides of the equation by 3 to get rid of the fractions.

If I manage to convince you, and you're wondering how to teach it, take a look at this blog post. Hopefully, you can use what I wrote here to help motivate everyone involved in the learning process.

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If I manage to convince you, and you're wondering how to teach it, take a look at this blog post. Hopefully, you can use what I wrote here to help motivate everyone involved in the learning process.

#### Algorithms

Long division is the first big example of an algorithm. An algorithm is a step-by-step description of how to do something. You can conveniently reuse algorithms again and again in similar situations, share them with others, and teach them to a computer. Once we have an algorithm for how to complete a particular type of task, we no longer have to think about it so much. Even if we don't always use it, it's there in our back pocket, and gives us confidence that we know how to solve any problem that comes our way. Algorithms are fundamental to the closely related fields of math and computer science, and when kids learn some coding in elementary school (here are some resources for that), they are more prepared for understanding long division as an algorithm.#### Decimal Expansions

Long division connects fractions, division, and decimals. We can use long division to write 5/7 as a decimal... we just have to keep going until it terminates or repeats. In other words, it gives us an*algorithm*for finding the decimal representations of fractions. Moreover, long division explains why all rational numbers have decimal representations that terminate or repeat. Becoming comfortable with rational numbers and all of their forms is one of the most challenging and important tasks of middle school mathematics.#### Other Bases

We write numbers in base 10. So 1068 means \(1*10x^3 + 0*10^2 + 6*10 + 8\). But we could also use powers of another number, like 7. How would we write 1068 in base 7? Well, we use the division algorithm to find that \(\frac{1068}{7} = 152 + \frac{4}{7}\), or equivalently that \(1068 = 7*152 + 4\). Next, we divide the quotient 152 by 7 to get a new quotient 21 and a new remainder 5, yielding \(152 = 7*21 + 5\). Continuing in this way, we find that \(21 = 7*3 + 0\) and \(3 = 7*0 + 3\). So

$$1068 = 7*152 + 4 = 7(7*21 + 5) + 4 = 7(7(7*3) + 5) + 4 = 3*7^3 + 5*7 + 4.$$

This tells us that \((1068)_{10} = (3054)_7\), where the subscripts tell us the base the number is written in.

#### Polynomial Division

The algorithm of long division can be applied to polynomials as well as positive integers. Just as long division allows us to rewrite 17/5 as 3 + 2/5, with 3 as the quotient and 2 as the remainder, it allows us to rewrite $$\frac{x^4 - 5x^2 + 6x^2 - 18}{x^3 - 3x^2} \; \; \text{as } \; \; (x - 2) - \frac{18}{x^3 - 3x^2},$$ with \(x - 2\) as the quotient and -18 as the remainder. This becomes useful in calculus, as its the first step to integrating a rational function where the degree of the numerator is at least as big as the degree of the denominator.#### Modular Arithmetic

Modular arithmetic is born out of the idea that we can create a new number system by only caring out the remainder when we divide by*n*. So, if we're looking at remainders after dividing by 7, for example, 2*4 = 1, which means that 4 = 1/2. Crazy, right? Modular arithmetic is fundamental to many areas of mathematics and computer science, including cryptography! And it's a great topic to explore in math circles for upper elementary and above, as it's interesting, useful, and reinforces many of the main concepts of elementary and middle school mathematics.#### Continued Fractions

Expressing fractions in terms of quotient and remainder allow one to make continued fractions: $$\frac{7}{10} = \frac{1}{\frac{10}{7}} = \frac{1}{1 + \frac{3}{7}} = \frac{1}{1 + \frac{1}{\frac{7}{3}}} = \frac{1}{1 + \frac{1}{2 + \frac{1}{3}}}.$$ You can even write irrational numbers as (infinite) continued fractions. Take, for example, the golden ratio \(\phi = \frac{1 + \sqrt{5}}{2}\). $$\phi = \frac{1}{1 + \phi} = \frac{1}{1 + \frac{1}{1 + \phi}} = \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \phi}}} = \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \ddots}}}}.$$ The reason that \(\phi\) has such a nice continued fraction representation is that \(1 + \frac{1}{\phi} = \phi\). Can you figure out why that's true? This is another great math circle topic idea, and is probably most appropriate for high school students.#### Generalizations of the Integers

Long division is fundamental to number theory, the area of math that I work in. We've already seen a little of this in modular arithmetic and continued fractions, but it goes even further. One of the main things we study in number theory is generalizations of the integers and rational numbers. In order to do this, we must look closely at the properties that the integers and rational numbers satisfy. One important property is unique prime factorization, and another is the existence of a division algorithm that produces a quotient and remainder. This is what is meant when the first problem set of PROMYS and ROSS state that "long division has unexpectedly deep consequences and can be generalized in far-reaching ways".## Thursday, May 19, 2016

### Math You Can Play: Math games using everyday materials

This week, I took at look at the first two books in the

*Math You Can Play*series by Denise Gaskins. The first book (Amazon Affiliate link here) focuses on counting and number bonds, and is aimed at pre-K through 2nd grade, and the second book (Amazon Affiliate link here) focuses on addition and subtraction, and is aimed at Kindergarten through 4th grade.Why play these games? From the introduction:

Math games push students to develop a creatively logical approach to solving problems. When children play games, they build reasoning skills that will help them throughout their lives. In the stress-free struggle of a game, players learn to think things through. They must consider their options, change their plans in reaction to new situations, and look for the less obvious solutions in order to outwit their opponents.

Even more important, games help children learn to enjoy the challenge of thinking hard. In the context of a game, children willingly practice far more arithmetic than they would suffer through on a workbook page, and their vocabulary grows as they discuss options and strategies with their fellow players. Because their attention is focused on their next move, they don’t notice how much they are learning.

And games are good medicine for math anxiety. Everyone knows it takes time to master the fine points of a game, so children feel free to make mistakes or “get stuck” without losing face.

If your child feels discouraged or has an “I can’t do it” attitude toward math, take him off the textbooks for a while and feed him a strict diet of games. It will not be long before his eyes regain their sparkle. Beating a parent at a math game will give any child confidence. And if you’re like me, your kids will beat you more often than you might want to admit.A good math game is still fun and challenging when the math concepts involved have been mastered, and so older siblings and parents can still enjoy joining in. For example, how can you make 24 using each of the numbers 1, 8, 4, and 3 exactly once, combining them using addition, subtraction, multiplication, and division? Even for me, there's a moment of excitement on figuring out the answer. (You can buy the 24 Game, or make your own version with a deck of cards.)

In addition to the games, there are chapters about "Workbook Syndrome", mastering the math facts, and lots of resources ranging from books to board games (including more than a few that I regularly play with friends).

If you want to try out a few games without going all in and buying the books, you can find many of them on her blog. If, like me, you're cost conscious, you can also buy the combined kindle edition for a significant discount (Amazon Affiliate link here). She also has a book entitled

*Let's Play Math: How Families Can Learn Math Together and Enjoy It*(Amazon Affiliate link here). I haven't read it, but it looks good!

## Friday, May 6, 2016

### Tiny Polka Dot: A deck of math cards for ages 3 - 8

I already posted about the first Math For Love game, Prime Climb. They recently started a new Kickstarter campaign for a specialized deck of math cards, called Tiny Polka Dot, which is aimed at kids in pre-K to 2nd grade. There are lots of mathematical games and activities that use a standard deck of cards (here are some examples), but these have 6 different suits each of which represent the numbers from 0 to 10 in a different way - ten frames, big and little dots, circles, doubles plus one, numerals, and dice patterns.

They include 12 different games which teach conservation of number, one-to-one correspondence, subitizing, addition, and subtraction. They also encourage parents and kids to make up their own games or modify standard card games for this deck. This is a great way to make mathematics playful and fun! I just backed this project and highly recommend that you do too!

Photo of Tiny Polka Cards from their Kickstarter page |

They include 12 different games which teach conservation of number, one-to-one correspondence, subitizing, addition, and subtraction. They also encourage parents and kids to make up their own games or modify standard card games for this deck. This is a great way to make mathematics playful and fun! I just backed this project and highly recommend that you do too!

## Thursday, May 5, 2016

### Blokus: A strategic game of spatial thinking

Last night, I played Blokus (Amazon affiliate link here) with a few friends (none of which consider themselves to be math people). In Blokus, four players take turns placing one of their colored tiles on the board, starting at their corner. Each piece played must touch another piece of the same color, but only on the corners. As the board fills up, players must block their opponents, protect their territory, and plan out a strategy in order to fit as many of their remaining pieces as possible. The game requires a ton of spatial thinking, and can be enjoyed by everyone from early elementary schoolers to adults.

My roommate asked an awesome question after we had finished our game: If we worked together, could we fit all the pieces on the board and still follow the rules of starting from a corner and having same-color pieces touch by corners? I immediately started unpacking the pieces so that we could give it a try. Before getting started, we did a few things:

1. We counted how many squares there were on the board and on all of our tiles, so that we could get a sense of how tightly we would need to pack them in. The board was 20 by 20, and hence had 400 squares. Each color had 1*1 + 2*1 + 3*2 + 4*5 + 5*12 = 89 squares, so there were 89*4 = 90*4 - 4 = 360 - 4 = 356 squares on the tiles. This means that we can leave 44 squares uncovered. Note that all of this was talked through out loud without needing paper or a calculator.

2. We decided that we would allow ourselves to move around pieces after placing them. We just wanted our final product to be a valid Blokus arrangement, and so didn't care about following the usual rules during the placement process.

3. My roommate suggested that we start by mirroring each other, as we had often done that for the first few moves while playing the game.

My neighbor took the lead most of the time in placing pieces, and we quickly realized that copying someone else's placement really challenged our spatial thinking skills, especially as the colors began to mingle. We liked the symmetric arrangement, and decided to try and keep that going the entire time. We also realized that it was easier to fill in the corners as much as possible with two colors, and then worry about connecting each color later:

Here was our final result, which we were all very happy with:

Although there were no children involved, this is a great example of how to encourage more mathematical thinking at home! If my roommate's question had been shot down, we would have just put away the game and been done for the night, but we followed the principal of "Yes, and..." and created something we were really proud of (as evidenced by the fact that a picture was immediately posted to Facebook). Mathematics is about asking and exploring interesting questions, and can happen naturally in the right sort of environment!

Although this didn't come up last night, I wanted to mention another mathematical aspect of Blokus:

A

Since Blokus pieces can be rotated and flipped, it makes sense in our situation to consider all of the ones pictured above to be the same. It might be fun to take out a piece of paper and try to draw and count all of the polyominos with one, two, three, four, and five cells. You will find 21 shapes which correspond exactly to the Blokus pieces!

Counting polyonimos quickly becomes a very hard problem - the number of polyonimos will n cells is only currently known up to n = 28. They are great ways to explore area, perimeter, symmetry, and counting in the elementary school classroom and beyond!

Clicking this image leads to an Amazon affiliate link |

1. We counted how many squares there were on the board and on all of our tiles, so that we could get a sense of how tightly we would need to pack them in. The board was 20 by 20, and hence had 400 squares. Each color had 1*1 + 2*1 + 3*2 + 4*5 + 5*12 = 89 squares, so there were 89*4 = 90*4 - 4 = 360 - 4 = 356 squares on the tiles. This means that we can leave 44 squares uncovered. Note that all of this was talked through out loud without needing paper or a calculator.

2. We decided that we would allow ourselves to move around pieces after placing them. We just wanted our final product to be a valid Blokus arrangement, and so didn't care about following the usual rules during the placement process.

3. My roommate suggested that we start by mirroring each other, as we had often done that for the first few moves while playing the game.

My neighbor took the lead most of the time in placing pieces, and we quickly realized that copying someone else's placement really challenged our spatial thinking skills, especially as the colors began to mingle. We liked the symmetric arrangement, and decided to try and keep that going the entire time. We also realized that it was easier to fill in the corners as much as possible with two colors, and then worry about connecting each color later:

A recreation of our board after filling in the corners |

Here was our final result, which we were all very happy with:

Success! |

Although there were no children involved, this is a great example of how to encourage more mathematical thinking at home! If my roommate's question had been shot down, we would have just put away the game and been done for the night, but we followed the principal of "Yes, and..." and created something we were really proud of (as evidenced by the fact that a picture was immediately posted to Facebook). Mathematics is about asking and exploring interesting questions, and can happen naturally in the right sort of environment!

Although this didn't come up last night, I wanted to mention another mathematical aspect of Blokus:

A

*polyomino*is a plane geometric figure formed by joining one or more equal squares edge to edge. They are classified by how many square cells they contain. A*monomino*has one cell, a*domino*has two cells, a*tromino*has three cells, a*tetromino*has four cells, etc. How many different tetrominos are there? Well, it depends on how you define different.How many different tetrominos are pictured above? |

21 Blokus pieces in red |

## Wednesday, April 27, 2016

### The Logical Journey of the Zoombinis: A computer game teaching logical thinking skills

When I was a kid, one of my favorite computer games was The Logical Journey of the Zoombinis, which came out in 1996. Players solve puzzles to help guide Zoombinis to safety after they escape imprisonment by the evil Bloats, who have taken over their homeland. The puzzles teach logical and computational thinking skills which are fundamental to mathematics and computer programming. There were two sequels (which I never played) -

*Zoombinis Mountain Rescue*in 2001 and*Zombinis Island Odyssey*in 2002.
Although the games were made for now outdated operating systems, the original game was recently updated for iOS, Android, Windows, Mac, and Kindle Fire via a Kickstarter campaign (which I excitedly participated in). The National Science Foundation has even awarded a large grant to the creators to study the effects the game has on students' computational thinking skills and how the game could be leveraged in the classroom.

While new educational materials that compliment the game are still in development as part of the grant, you can find the 1996 guides for parents and teachers here and an awesome article written by the creators entitled

*Zoombinis and the Art of Mathematical Play*here. The article starts by describing how "play is nature's greatest educational device", but most computer math games are "essentially drill-and-practice programs that focus on a narrow set of skills" and "treat the playful elements as something distinct from the mathematics".
The game is aimed at elementary and middle school students, although I think many adults will still find it enjoyable and challenging. I would also recommend that teenagers and adults check out Portal.

iPad Screenshot |

## Friday, April 15, 2016

### Curve Stitching and Folding: A Hands on Activity

This weekend, the AMS will be hosting a curve stitching activity at the 2016 USA Science & Engineering Festival held in Washington, DC. The idea is that you can create curves by drawing, folding, or stitching a bunch of straight lines. Below is a quick example I folded (folds are drawn over for visibility), using a Project Origami activity.

You can prove that the lines I folded are all tangent to a parabola whose focus is the marked point and whose directrix is the bottom edge of the paper. This means that the parabola is the envelope of the family of lines I folded. Yarn also works well to create the lines (hence the name curve stitching), as when you thread yarn through two holes and pull it taut, you get a straight line connecting those points. The AMS website has pictures, pattern sheets with instructions, and further resources to check out. This would make a great math circle activity. Even elementary school students can do the stitching and marvel at the cool pattern they formed!

## Wednesday, April 6, 2016

### The Man Who Knew Infinity: Conversation starters about the upcoming film

Two summers ago, my advisor Ken Ono suddenly jetted off to England to help with the filming of the movie

*The Man Who Knew Infinity*(Click here for trailer). The film, featuring Dev Patel and Jeremy Irons, dramatizes the life of the Indian mathematical genius Ramanujan, focusing on his time at Cambridge working with British mathematician G. H. Hardy in the 1910s. It is based off of the Robert Kanigel's biography*The Man Who Knew Infinity: A Life of the Genius Ramanujan*(Amazon affiliate link here). Through Ken, whose role with the film has expanded from math consultant to associate producer, I've gotten to see the movie multiple times already, including at the world premiere in Toronto last September!
Since the movie is finally coming soon to a theater near you (May 13 is the date I've heard for Atlanta, but other cities may be earlier), the time has come for me to write a post about it. I obviously think it's a great film, but instead of writing a review (I assume you can find many using google), I'm going to give some conversation starters for discussing the film. Watch it with your kids/students (I'd recommend middle school age and up)!

1. One of the major themes is the relative importance of proof versus intuition in mathematics. Why are each of them valuable?

2. Another theme is the importance of reaching out to others and asking for help when you need it. Ramanujan did this successfully in the beginning of the movie, but struggled once he got to England. Why?

3. The film focuses on Ramanujan's work on partitions. Do you remember the definition? If you need a reminder, the partitions of 4 are 4, 3 + 1, 2 + 2, 2 + 1 + 1, and 1 + 1 + 1 + 1. So p(4) = 5. Try to compute a few partition numbers by yourself and look for patterns. Partitions makes for a great math circle topic, and I think that

*Integer Partitions*by George Andrews and Kimmo Eriksson (Amazon affiliate link here) is a particularly accessible introduction.
4. Ramanujan was self-taught. With only a single book, he was able to rediscover much of modern mathematics and push the field forward. Mark Zuckerberg asked (see the end of this clip): "What would have happened if he had had access to the internet?" Of course, the internet didn't exist when he was alive, but what implications does that have today, when more than half of the world still doesn't have access?

5. Mathematicians are often inundated by letters and emails from amateurs claiming to prove one of the big unsolved problems in mathematics, many of which are challenging to read because they do not follow the conventions that mathematicians are used to. How much time and effort should we devote to sifting through all of this for diamonds in the rough? Do we have a duty to look for mathematical talent in unlikely places?

I also wanted to mention The Spirit of Ramanujan Talent Search. The first stage of it is powered by Expii, and anyone can take part by solving some math problems on their website.

## Monday, March 28, 2016

### Spirographs: A classical toy that makes mathematical curves

This past Saturday, I helped out with a booth at the Atlanta Science Festival Exploration Expo about spirographs. The spirograph is a mathematical toy that has been around for quite some time, since at least the early 1900s. It consists of two plastic rings, with teeth on both the inside and outside, and a number of gearwheels. The idea is to put your pen in one of the holes of a gearwheel, and spin it around the ring to make patterns on the piece of paper.

Why is this so mathematical?

On a high level, the drawings produced are special types of mathematical curves, called hypotrochoids and epitrochoids. However, without getting any equations involved, you can ask some interesting questions. How can you predict the kind of pattern you will get based on the number of teeth on the ring and gearwheel and the distance between the pen hole and the center of the gearwheel. Conversely, if you want to make a particular type of pattern, which ring, gearwheel, and pen hole should you use?

After interacting with a bunch of students, I found that a few observations helped to push them along:

1. What would happen if you put your pen in the exact center of the gearwheel? You'd just make a circle because the distance from the pen to the ring would not change.

2. What happens if you use the ring has 96 teeth and the gearwheel has 24 teeth? You make a square, and 96/24 = 4. This leads to thinking about the number of teeth in each ring and gearwheel and keeping track of the results of different combinations.

Kits like this one (amazon affiliate link here) come with spiro-putty to hold down the rings. Since we didn't have spiro-putty with us at the festival, it was a challenge for the kids to keep the rings still (the parents often ended up helping with this). It was also challenging to keep the gearwheel pressed against the ring. This requires slow and controlled movement.

A lot of the parents fondly remembered playing with spirographs as kids, but it seemed to be a novel concept to most of the kids at the festival, even though a few excitedly realized after trying it out that they already had a kit lying around at home unused. I would encourage parents to not just buy one for their kids, but to open it up and start playing around with it themselves. Once you have some artwork to marvel over, offer to show your kids how to do it and stick around until they get the hang of it. Ask them questions like the ones included above, and try to spark their curiosity. Lots of toys and games become mathematical when you start asking questions, making observations and predictions, etc.

Why is this so mathematical?

On a high level, the drawings produced are special types of mathematical curves, called hypotrochoids and epitrochoids. However, without getting any equations involved, you can ask some interesting questions. How can you predict the kind of pattern you will get based on the number of teeth on the ring and gearwheel and the distance between the pen hole and the center of the gearwheel. Conversely, if you want to make a particular type of pattern, which ring, gearwheel, and pen hole should you use?

After interacting with a bunch of students, I found that a few observations helped to push them along:

1. What would happen if you put your pen in the exact center of the gearwheel? You'd just make a circle because the distance from the pen to the ring would not change.

2. What happens if you use the ring has 96 teeth and the gearwheel has 24 teeth? You make a square, and 96/24 = 4. This leads to thinking about the number of teeth in each ring and gearwheel and keeping track of the results of different combinations.

Kits like this one (amazon affiliate link here) come with spiro-putty to hold down the rings. Since we didn't have spiro-putty with us at the festival, it was a challenge for the kids to keep the rings still (the parents often ended up helping with this). It was also challenging to keep the gearwheel pressed against the ring. This requires slow and controlled movement.

A lot of the parents fondly remembered playing with spirographs as kids, but it seemed to be a novel concept to most of the kids at the festival, even though a few excitedly realized after trying it out that they already had a kit lying around at home unused. I would encourage parents to not just buy one for their kids, but to open it up and start playing around with it themselves. Once you have some artwork to marvel over, offer to show your kids how to do it and stick around until they get the hang of it. Ask them questions like the ones included above, and try to spark their curiosity. Lots of toys and games become mathematical when you start asking questions, making observations and predictions, etc.

Clicking this image leads to an Amazon affiliate link |

## Sunday, March 20, 2016

### Teaching teenagers that math is more than what they learn in school

I found slides for a presentation by Ben Brubaker titled "Exploring great mysteries about prime numbers".

On his first slide, he lists some big questions:

On his first slide, he lists some big questions:

- What is mathematics?
- What kinds of questions do mathematicians try to solve?
- What do mathematicians do all day?
- How does mathematical proof resemble a poem or a painting?
- Why should we learn mathematics?
- How can we become better at mathematics?

These are great questions, and I think that all math outreach should aim to help students answer them.

He also talks about two mistaken impressions about math that high school perpetuates:

1. Mathematics is a single path leading to an endpoint: calculus.

2. All mathematical proofs are like the two column proofs you do in geometry.

Here's a wikipedia article which talks about all the different areas of math - many of which do not use ANY calculus. It would be possible to design a high school mathematics curriculum which is leading towards some other pinnacle - like elementary number theory or probability and statistics! A mathematical proof is an argument that shows why your statement is true. It often takes the form of a paragraph, and the details you need to include include depends on the audience that you're trying to convince.

I've been thinking a lot about having high school students explore the prime numbers by writing programs to get lots of data, looking for patterns, and trying to explain them. I think that it's a great topic - prime numbers are extremely concrete and fundamental, but students can quickly make super deep conjectures and interact with the forefront of mathematical knowledge. It's important to me to give students the experience of exploring, creating, and doing mathematics - looking at examples, asking questions, making conjectures, and trying to understand why they're true. While there's not a lot that high school students can prove about primes (very quickly, the proofs get complicated and require a lot of background), they can get a good sense of what's going on, and I think that this is an extremely important but untaught skill.

## Saturday, March 19, 2016

### Lux: a new and exciting mathematical construction kit

With Lux's patent-pending snap and lock hinge, builders are now freer than ever before to make structures which curve, bend, and move. Not only do our squares make circles and spheres, but they model machines, biological organisms, mathematical relationships, and enable a user to construct whatever architecture they want. Put the creative power of nature in your hands.I found out about Lux through a facebook post by its creator Mike Acerra:

Not to get too hyperbolic about this , but yes, Lux has been proven to be exactly that. Oklahoma State Math professor...

Posted by Mike Acerra on Friday, March 18, 2016

Here's the rest of the text of the post, as you can't seem to click and ``see more'':

Henry Segerman, assistant professor in the Department of Mathematics at Oklahoma State University, does research in 3-dimensional geometry and topology, particularly involving ideal triangulations. His interest is in mathematical and typographical art of various kinds and dimensions.

In December 2015, Segerman discovered that because Lux builds using the rule of five squares around every vertex, (normally a square tile would fit four around a corner and make a “flat plane”) it can model hyperbolic space.

The significance of this is that when we tile the hyperbolic plane, we have vastly more freedom than we do in the limited Euclidean world. We can tile it with equilateral triangles, quadrilaterals, pentagons, hexagons, and so on. Not only that, we can do it infinitely many ways with each of those shapes. Angles are much more restricted in Euclidean space than in hyperbolic space.

Hyperbolic geometry has been used to better understand Einstein’s special theory of relativity because Einstein used the geodesic or elliptical geometry of Bernhard Riemann, which was also used in the invention of Buckminster Fuller’s famed “Geodesic Domes”.

And that's totally groovy.I don't own any of these blocks (yet), but they look super interesting! Let me know what you think if you've gotten a chance to play with them!

Clicking this image leads to an Amazon affiliate link |

## Monday, March 7, 2016

### Mathematical Dominoes: A fun and incentivizing classroom problem solving game

I've been thinking lately about fun and incentivizing ways to structure small group problem solving in a math circle. Last week, I posted about an activity where students solve math problems to find a secret code and unlock a password protected document. Here's something else I found: Mathematical Dominoes. Each team of 1 - 3 players works on problems at their own pace, selecting them one-by-one from a pool of available problems.

The problems are written on the back of two-sided cards, and the front of the cards look like dominos. The numbers on the domino indicate the difficulty level of the problem and the number of points to be won or lost by working on it. Students choose based on the domino side and can only flip it over afterwards.

How does scoring work for an [x:y] domino?

1. If you correctly solve the problem on the first try, you get x + y points. So [3:4] gains 7 points.

2. If you correctly solve the problem on the second try, you get max(x, y) points. So [3:4] gains 4 points.

3. If the second answer is incorrect, you lose min(x, y) points. So [3: 4] loses 3 points.

4. A [0:0] domino is special - you only have one try. A correct answers gets 10 points, and an incorrect answer has no penalty.

When the students finish with a problem, they return it to the pool and pick a new one. There are judges who check answers and keep track of points, but they don't need to know a lot of math. The competition ends after a set amount of time, and the team with the highest score wins.

There's a lot of strategy towards choosing your domino - [1:6], [3:4], [1:4], [3:6] all have slightly different reward/penalty structures, and should therefore indicate different levels of difficulty and riskiness. Because of this, choosing problems to put on the dominos seems like it would be very challenging. Luckily, Prime Factor Math Circle has some up on their website to get you started.

They also list some of the reasons why this game has worked so well:

The problems are written on the back of two-sided cards, and the front of the cards look like dominos. The numbers on the domino indicate the difficulty level of the problem and the number of points to be won or lost by working on it. Students choose based on the domino side and can only flip it over afterwards.

How does scoring work for an [x:y] domino?

1. If you correctly solve the problem on the first try, you get x + y points. So [3:4] gains 7 points.

2. If you correctly solve the problem on the second try, you get max(x, y) points. So [3:4] gains 4 points.

3. If the second answer is incorrect, you lose min(x, y) points. So [3: 4] loses 3 points.

4. A [0:0] domino is special - you only have one try. A correct answers gets 10 points, and an incorrect answer has no penalty.

When the students finish with a problem, they return it to the pool and pick a new one. There are judges who check answers and keep track of points, but they don't need to know a lot of math. The competition ends after a set amount of time, and the team with the highest score wins.

There's a lot of strategy towards choosing your domino - [1:6], [3:4], [1:4], [3:6] all have slightly different reward/penalty structures, and should therefore indicate different levels of difficulty and riskiness. Because of this, choosing problems to put on the dominos seems like it would be very challenging. Luckily, Prime Factor Math Circle has some up on their website to get you started.

They also list some of the reasons why this game has worked so well:

• It’s rules and format are as interesting and exciting for a 3rd-grader as for a 9th-grader

• It engages each and every student in active problem solving

• It can be played for as long as it is needed and can be stopped at any moment

• It requires very few helpers

• It allows all students to work at their pace and their level

## Monday, February 29, 2016

### A Brilliant Young Mind: A film about love and math

I've seen the movie A Brilliant Young Mind (released in England as X + Y) twice in the past year - I randomly came across it on a flight back from England, and then saw a screening of it about 6 months later at the Joint Math Meetings. From wikipedia:

The film, inspired by Beautiful Young Minds, focuses on a teenage English mathematics prodigy named Nathan (Asa Butterfield) who has difficulty understanding people, but finds comfort in numbers. When he is chosen to represent Great Britain at the International Mathematical Olympiad, Nathan embarks on a journey in which he faces unexpected challenges, such as understanding the nature of love.The film is very emotional - for example, early on in the movie, Nathan is diagnosed with autism and soon afterwards his father dies tragically in a car accident while Nathan is sitting in the passenger seat. It's rated PG-13, and I highly recommend it for teenagers and adults!

## Sunday, February 28, 2016

### Decrypt the Secret Document: A mathematical classroom challenge

I found this activity made by Anne Ho of Coastal Carolina University by way of Facebook. It's written for a calculus class, but could be adapted for other levels. The idea is that some secret document is encrypted with a 6 digit code, and the students solve math problems to find the code. There are 6 sets of 6 problems, and in each set the answers sum to a digit of the code. The students have two tries to unlock the document, and so must work together to ensure that all of the answers are correct.

Obviously, the format limits the types of problems that can be asked, and requires some work to force the sum of 6 answers to be a one-digit number. But it sounds like it would be a lot of fun, and would encourage team work and careful checking of answers.

In a graded class, the document could be solutions to next week's quiz, the statement of one of the problems that will be on the upcoming exam, etc. I'm not sure what the content of the document would be for a non-graded program like a math circle... maybe you bring in some sort of special treat and put its location in the document (or maybe they'll find out that the cake is a lie)? Or the document contains a funny math joke? Or a silly picture of the instructor? Be creative!

I want to try this with my middle school math circle group... it could be a fun way to incorporate some "school math" (which we usually shy away from)... maybe with each group of 6 at a different level (some requiring algebra 1, some only using 5th grade math) so that everyone can participate.

Obviously, the format limits the types of problems that can be asked, and requires some work to force the sum of 6 answers to be a one-digit number. But it sounds like it would be a lot of fun, and would encourage team work and careful checking of answers.

In a graded class, the document could be solutions to next week's quiz, the statement of one of the problems that will be on the upcoming exam, etc. I'm not sure what the content of the document would be for a non-graded program like a math circle... maybe you bring in some sort of special treat and put its location in the document (or maybe they'll find out that the cake is a lie)? Or the document contains a funny math joke? Or a silly picture of the instructor? Be creative!

I want to try this with my middle school math circle group... it could be a fun way to incorporate some "school math" (which we usually shy away from)... maybe with each group of 6 at a different level (some requiring algebra 1, some only using 5th grade math) so that everyone can participate.

## Friday, February 26, 2016

### Towers of Hanoi: A Mathematical Puzzle

The Tower of Hanoi is a mathematical puzzle. As you can see below, there are three rods and a number of disks of difference sizes.

The puzzle starts with the disks stacked on one rod in ascending order of size (with the smallest on the top, and the largest on the bottom). The objective is to move the entire stack to another rod. The hard part is that you can only move one disk at a time, and you can never put a larger disk on top of a small disk.

If you have n disks (in the picture above, n = 10), what's the minimum number of moves this can be accomplished in? It turns out that the answer is 2^n - 1. So 3 disks would take 7 moves, but the 10 disk example above would take 1023 moves! This is a good illustration of how quickly exponential functions grow.

The Tower of Hanoi makes for a great lecture, math circle activity, and toy to play with! Have your students make a table of examples, building up from n = 1, try to figure out a pattern, and prove it!

There's also a great legend behind the puzzle. According to Wikipedia:

Clicking this image leads to an Amazon affiliate link |

The puzzle starts with the disks stacked on one rod in ascending order of size (with the smallest on the top, and the largest on the bottom). The objective is to move the entire stack to another rod. The hard part is that you can only move one disk at a time, and you can never put a larger disk on top of a small disk.

If you have n disks (in the picture above, n = 10), what's the minimum number of moves this can be accomplished in? It turns out that the answer is 2^n - 1. So 3 disks would take 7 moves, but the 10 disk example above would take 1023 moves! This is a good illustration of how quickly exponential functions grow.

The Tower of Hanoi makes for a great lecture, math circle activity, and toy to play with! Have your students make a table of examples, building up from n = 1, try to figure out a pattern, and prove it!

There's also a great legend behind the puzzle. According to Wikipedia:

The puzzle was invented by the French mathematician Ã‰douard Lucas in 1883. There is a story about an Indian temple in Kashi Vishwanath which contains a large room with three time-worn posts in it surrounded by 64 golden disks. Brahmin priests, acting out the command of an ancient prophecy, have been moving these disks, in accordance with the immutable rules of the Brahma, since that time. The puzzle is therefore also known as the Tower of Brahma puzzle. According to the legend, when the last move of the puzzle will be completed, the world will end. It is not clear whether Lucas invented this legend or was inspired by it.

If the legend were true, and if the priests were able to move disks at a rate of one per second, using the smallest number of moves, it would take them 264−1 seconds or roughly 585 billion years or 18,446,744,073,709,551,615 turns to finish, or about 127 times the current age of the sun.

## Thursday, February 18, 2016

### Hanabi: A card game where everyone but you can see your cards

Troy Retter, another graduate student in my department, wrote a paper on the card game Hanabi (Amazon affiliate link here) as part of the Rocky Mountain-Great Plains Graduate Research Workshop in Combinatorics (GRWC):

One of many traditional social festivities at GRWC is game night, where we played a cooperative card game known as Hanabi. A few games and strategic conversations later, Hanabi became its own research project. In Hanabi, a player can not see the cards in her hand, and must rely on the actions of the other players to gain information about her cards. Based on ideas used in hat guessing games, we developed two strategies for Hanabi which performed well in computer simulations.Hanabi works with 2 - 5 players and is aimed at ages 8 and up. I own it and have played it a few times. Troy is planning on doing a math circle based on the game, and when that happens I'll include a link to his handout so that you can get a better sense of the mathematics. Until then, play the game, talk about strategies afterwards, and use it as in introduction to hat guessing games.

Clicking this image leads to an Amazon affiliate link |

## Tuesday, February 16, 2016

### Recipes for Pi: A math teacher's blog

I just came across a blog post about the game of Snugglenumber, which teaches place value and works well with a wide variety of ages - according to the post, it was a hit with grades 3 - 10.

Update: I just played this with my middle school math circle, and it was definitely a big hit. One parent emailed me afterwards to say:

The blog's author, Anna Weltman, who is a math teacher in Brooklyn, also wrote a mathematical art activity book. It looks awesome, and is aimed at ages 9 and up. On Amazon, there seems to be an American version called

Update: I just played this with my middle school math circle, and it was definitely a big hit. One parent emailed me afterwards to say:

When [name] and [name] got back, they told me that they loved yesterday's class. It seemed to be their favorite so far.There's some other good stuff on the blog, which is called Recipes for Pi, but it hasn't been updated in a bit.

The blog's author, Anna Weltman, who is a math teacher in Brooklyn, also wrote a mathematical art activity book. It looks awesome, and is aimed at ages 9 and up. On Amazon, there seems to be an American version called

*This is Not a Math Book*(amazon affiliate link here) and a British version called*This is Not a Maths Book*(amazon affiliate link here) but both are only available from third party sellers. Here's a blog post with more information about the book.Clicking image leads to Amazon affiliate link |

## Thursday, February 11, 2016

### Chocolate Fix: A sweet logic game of deductive reasoning

ThinkFun makes a bunch of great games, including Rush Hour (Amazon affiliate link here), Swish, and Robot Turtles (Amazon affiliate link here). But the one I want to focus on in this post is Chocolate Fix (Amazon affiliate link here). In this one-player logic game, there are 9 pieces of chocolate, in three colors and three shapes, and you have to figure out where to place them in the chocolate box using the clues on your challenge card. A few things that I really like about chocolate fix:

1. You don't need paper and pencil, as you have physical chocolate pieces to work with (and cardboard circles for partial information).

2. You have to figure out where the patterns on the challenge card go in the chocolate box. For example, on the picture below, the two diagonal patterns only have one possible spot, but the third pattern could a priori go in two different locations.

3. With four levels (and no words or numbers), it works well with a wide variety of ages... I'd say preschool/kindergarten through adult.

I bought a few copies for my 2016 Julia Robinson Math Festival, and the students seemed to really like it!

1. You don't need paper and pencil, as you have physical chocolate pieces to work with (and cardboard circles for partial information).

2. You have to figure out where the patterns on the challenge card go in the chocolate box. For example, on the picture below, the two diagonal patterns only have one possible spot, but the third pattern could a priori go in two different locations.

3. With four levels (and no words or numbers), it works well with a wide variety of ages... I'd say preschool/kindergarten through adult.

I bought a few copies for my 2016 Julia Robinson Math Festival, and the students seemed to really like it!

Clicking image leads to an Amazon affiliate link |

## Monday, February 8, 2016

### MathILy: Serious mathematics infused with levity

sarah-marie belcastro, who spent many years teaching at and later co-directing HCSSiM, founded her own math summer program, called MathILy, in 2013. MathILy is all about "serious

I know sarah-marie and Tom (the other lead instructor) from when I was a student at HCSSiM, and therefore can comfortably recommend the program without having attended myself.

There's also MathILy-Er, which is like MathILy, but "adapted for students who are slightly earlier in their chronology or mathematical development". There's a single application process for both

**math**ematics**i**nfused with**l**evit**y**". It's five weeks long, aimed at high school students, and located at Bryn Mawr College. According to the website:The weeks break down into a 2-1-2 schedule: We start with two weeks of Root Class, which consists of a gallimaufry and melange of mathematics that gives all students a base on which to grow. This is followed by Week of Chaos, in which there are many many short classes with topics suggested by students and instructors alike. The denouement of the program offers more advanced Branch Classes in the final two weeks.The classes are all taught using inquiry based learning, which means that the instructor is more of a facilitator, and the students are up at the board asking questions, making definitions and conjectures, and coming up with proofs. This is one of the things that sets MathILy apart from other programs. It also has a lot of silliness (for example, take a look at the website for the "alter ego program" MathIGy).

I know sarah-marie and Tom (the other lead instructor) from when I was a student at HCSSiM, and therefore can comfortably recommend the program without having attended myself.

There's also MathILy-Er, which is like MathILy, but "adapted for students who are slightly earlier in their chronology or mathematical development". There's a single application process for both

## Sunday, February 7, 2016

### A Decade of the Berkeley Math Circle: The American Experience, Volume I

The Berkeley Math Circle is one of the oldest in the US, and

*A Decade of the Berkeley Math Circle*(Amazon affiliate link here) takes 12 of its sessions and turns them into textbook like chapters with exposition and exercises. Unlike most math circle books, this is aimed at students directly as well as teachers looking for math circle lesson plans and topic ideas.
I think this book would be most appropriate for the type of high school students who typically attend top-tier math circles, i.e. they are substantially beyond the average high schooler in terms of mathematical experience and being good at and interested in math is already part of their identity. However, as is always the case, some of the material could be adapted to other audiences.

I really like a lot of the language in the introduction about math circles, and have quoted from it on the Emory Math Circle website. However, I haven't used any of the mathematical material yet, as I mostly teach our younger group (grades 6 - 9).

There's also a volume II, which I have not personally looked through (Amazon affiliate link here).

## Friday, February 5, 2016

### Vi Hart's YouTube videos are the best

To copy from the wikipedia article, Vi Hart is a self-described "recreational mathemusician" who is most known for creating mathematical videos on YouTube.

I occasionally use these videos in math circle, and think they're awesome! The videos are fun and silly and have a lot of great mathematical content. They can also work in other environments, and I know a bunch of students who watch them at home (so at least one of your students will likely have already seen the video you're about to show). I'll tell you about a few that I've used:

Making hexaflexagons is one of the best hands on math circle activities, and works for pretty much all ages (I've done it with middle schoolers). I used this video as an introduction and used this video later on in class (pausing at appropriate points for students to ask questions, make their own, etc).

There's a whole playlist called

*Pi and Anti-Pi*, which I used on Pi Day. Some of the videos are good for middle school, but others mention radians and trigonometry and therefore would be more appropriate for high schoolers. I think that students enjoy hearing about the Pi vs Tau argument, and it makes them think about the importance of definitions and that the way they're taught is not the only way.
Infinity is a great and mind-blowing topic for math circles, and Vi Hart has a playlist called The Infinite Series. I haven't used these videos myself, but I think it's a great topic for high schoolers.

I also really like the story of Wind and Mr. Ug, which is about living on a Mobius strip. I used it in conjunction with Math Improv: Fruit by the Foot for an exploration of what happens when you make

*n*half twists in a paper strip before taping the ends together. I used this with middle schoolers.## Thursday, February 4, 2016

### Fluxx: A card game where the rules and goals are constantly changing

Fluxx (Amazon affiliate link here) is a card game where the rules and conditions for winning are constantly changing as you play. There are a ton of different themed versions, a few of which are pictured below (images lead to Amazon affiliate links).

How does the game work? You start out with the basic rules: draw a card, then play a card. Among the cards that can be played at goals (which tell you how to win the game), new rules (which change the rules of the game), keepers (most goals involve collecting a certain combination of keeper cards), and actions (when you play the card, you do whatever it says).

What makes this game mathematical?

How does the game work? You start out with the basic rules: draw a card, then play a card. Among the cards that can be played at goals (which tell you how to win the game), new rules (which change the rules of the game), keepers (most goals involve collecting a certain combination of keeper cards), and actions (when you play the card, you do whatever it says).

What makes this game mathematical?

- Playing this game develops mental flexibility, which is very important for mathematics (and life in general)! For example, you might need to switch to a different method when solving a problem, or change your research question as you investigate. You have to be willing to let go your original idea when it's not working and try something new.
- Understanding and keeping track of the current rules and goals require a good amount of mental effort - especially when a bunch of new rule cards are active at once or you have one like inflation (which adds one to all the numbers on cards). I see this process as very similar to that of carefully reading a math problem and working to understand the situation it's describing and the question it's asking. I find that a lot of students struggle with this. In most games, learning the rules is a once and done kind of thing, but in Fluxx it's something that you have to do constantly.

One frustrating thing about this game is that the amount of chance involved makes it hard to play strategically. The box says ages 8 and up, but I've only played it with middle schoolers through adults. It takes 2 - 6 players, and usually is pretty short (< 30 minutes).

## Wednesday, February 3, 2016

### AMC 8, 10, and 12: Multiple choice math competitions that are just the beginning

The Mathematical Association of America runs a number of competitions for middle and high schools students, which are usually administered in schools, and are the most well known and widely taken math competitions in the US. However, the high school competitions are actually the first step of many in the selection process for the US International Mathematical Olympiad (IMO). There are a lot of acronyms in this post, so I included a cheat sheet and a flow chart at the bottom.

**AMC 8:**

- Anyone in grades 8 and below can participate
- Students have 40 minutes to answer 25 multiple choice questions
- Takes place in November
- Students are not penalized for incorrect answers

**AMC 10/12**

- Anyone in grades 10 and below (AMC 10) or 12 and below (AMC 12) can participate
- Students have 75 minutes to answer 25 multiple choice questions
- Draws on material up through algebra and geometry (AMC 10) or precalculus (AMC 12)
- Offered twice in February (A and B version)
- Students are penalized for incorrect answers
- 150 is perfect score

**For the top scorers, the AMC 10/12 is just the beginning!**

- High scorers (usually 120+ on AMC 10 or 100+ on AMC 12) are invited to take the AIME.
- The AIME consists of 15 questions of increasing difficulty where each answer in an integer between 0 and 999 inclusive.
- A combination of AMC and AIME score is used to determine qualification for the USAMO (for students who took AMC 12) and USAJMO (for students who took AMC 10)
- The USAMO and USAJMO are proof based exams and are spread over two days. During each day, students are given four and a half hours to answer 3 questions.
- Top scorers on the USAMO and USAJMO (who are US residents) are invited to MOSP and considered for the US IMO team.
- MOSP is an intensive summer program meant to select and train the US IMO team held at Carnegie Mellon University.
- From MOSP, 6 students are invited to represent the United States in the IMO.
- The IMO is proof based and spread over two days. During each day, students have four and a half hours to answer 3 questions.

**My thoughts:**

I think that all middle and high school students who like math should take the appropriate AMC. As I've gotten older, I've appreciated the quality of the problems more and more. For students who are serious about math competitions, it's worth practicing with old exams, which you can find here.

It also should be noted that students can take more than one of these exams per year, and can chose to take one that is higher than their grade level. For example, a middle schooler can take (AMC 8) and (AMC 10A or AMC 12A) and (AMC 10B or AMC 12B). A 9th or 10th grader can take (AMC 10A or AMC 12A) and (AMC 10B or AMC 12B). An 11th or 12th grader can take (AMC 12A) and (AMC 12B). This kind of thing mainly makes sense for students who want to improve their chances of advancing to the AIME and beyond. If you want to take more than just the AMC 8 as a middle schooler or the AMC 10/12 A as a high schooler, you will likely have to be more proactive with your school (or look for an enrichment center or university that offers the exams).

**Acronym Cheat Sheet:**

- AMC = American Mathematics Competitions
- AIME = American Invitational mathematics Examination
- USAMO = United States of America Mathematical Olympiad
- USAJMO = United States of America Junior Mathematical Olympiad
- MOSP = Mathematical Olympiad Summer Program (sometimes abbreviated MOP)
- IMO = International Mathematical Olympiad

## Tuesday, February 2, 2016

### Prove It! Math Academy: A new summer math program focused on the transition from problem to proof

Prove It! Math Academy is a new summer math program held in Colorado. From their website:

Another unique thing about the program is that it's run by one family (father, mother, daughter, 2 sons, and son-in-law) who all happen to be mathematicians that care a lot about teaching and outreach. I know many of them well from Lehigh Valley ARML and the Emory REU.

It's a short program - in 2016 it runs July 24 to August 7, and so would be a great choice for students who want to do a summer math program, but don't want it to take up their entire summer.

Almost all secondary school and freshman-level undergraduate mathematics classes and many summer camps and online programs are calculation-based — where students perform some sequence of computations to arrive at a numerical answer. At the other end of the spectrum are research competitions and advanced undergraduate level mathematics courses that require the ability to read and write mathematical proofs — objectively verifiable explanations of why a certain mathematical statement must be true. Yet there are very few resources available to help students make the transition from calculation-based problem solving to mathematical proof-based activities.Prove It! Math Academy is designed to fill this gap - teaching mathematical proof, while also strengthening problem solving skills and providing camaraderie. Other summer programs, like HCSSiM and PROMYS, do put a lot of emphasis on helping students make this transition, Prove It! Math Academy is the only one I know about that makes it the main focus.

Another unique thing about the program is that it's run by one family (father, mother, daughter, 2 sons, and son-in-law) who all happen to be mathematicians that care a lot about teaching and outreach. I know many of them well from Lehigh Valley ARML and the Emory REU.

It's a short program - in 2016 it runs July 24 to August 7, and so would be a great choice for students who want to do a summer math program, but don't want it to take up their entire summer.

### SWIM: Summer workshop in mathematics for female high school students

SWIM (which stands for Summer Workshop in Mathematics) is a free 9-day workshop for female students who are rising seniors in high school and interested in mathematics. It was held in 2009 and 2010 at Princeton (I was an RA and TA in 2010), and will be held this summer (2016) at Duke. Students are housed in dorms, attend two math courses, and spend their afternoons working in groups on an exploration topic related to one of their courses.

The reason that this program has not been held every single summer is that it requires a good amount of funding. Every participant receives full support for travel and accommodation, which is extremely rare in programs for high school students.

Sadly, the application deadline just passed for this summer, but I think it's still worth posting about the program so that more people are aware of it in the future.

For many women, being in an environment that is dominated by men is challenging and discouraging. So programs like this (the IAS also runs a Women and Mathematics program for undergrads and graduate students) are incredibly helpful in encouraging more women to stick with math.

## Monday, February 1, 2016

### Splash: a weekend-long learning extravaganza

According to Learning Unlimited's website:

Splash is a weekend-long extravaganza of classes at a local college or university, where pre-college students are invited to learn about everything and anything from passionate university students.Most are aimed at high schoolers, but some will let middle schoolers attend too. Some of the longer running programs are at MIT, Stanford, UChicago, Duke, Yale, and Boston College. They are all student run and tend to have a lot of fun / silly / unusual classes. You could learn how to write parody songs or make your own play-doh (I taught that one once), but could also take a class on math or computer science. Here's last year's MIT Splash course catalogue.

Learning Unlimited was founded by MIT alumni who wanted to make it easier for other college students to run educational programs like Splash. Their website has a list of programs throughout the country. Some student groups also run other programs. For example, MIT's Educational Studies Program (which is a student group, even though it sounds so official) also has Spark, HSSP, Cascade, Junction, Delve, and ProveIt.

If you can find one near you, I highly recommend attending! There tends to be a lot of math-y people, both among the university students and the pre-college students. I helped with one at Princeton in 2013.

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