Monday, May 26, 2014

Playing With Math - Rave Reviews from Early Readers

Mathematics is a creative activity, like music. It requires some technique, and the technique has to be taught, but the main point is elsewhere - it is all about creativity, a sense of enjoyment, and higher purpose. This book goes a long way in that direction.
Ivar Ekeland, author of The Cat in Numberland

Sue VanHattum has assembled a marvelously useful and inspiring book. It is filled with stories by people who don't just love math, they share that love with others through innovative math activities. Playing With Math is perfect for anyone eager to make math absorbing, entertaining, and fun.
Laura Grace Weldon, author of Free Range Learning

The Internet is presently bursting with vibrant writing about mathematics learning; yet it can be difficult to navigate this wealth of resources. Sue VanHattum has carefully collected and arranged some of the best of this writing. Imagine having a cheerful, knowledgeable, caring, and patient native interpreter accompany you on a tour of a foreign land. That's Sue in the land of math. She and the authors collected here care deeply about welcoming everyone to the world of mathematics. Whether you play with math every day, or are struggling to believe that one can play with math, Playing With Math will provide inspiration, ideas and joy.
Christopher Danielson, author of Talking Math with Your Kids and

As a homeschool mom who grew up hating math, I didn’t want to pass that attitude on to my children. I thought if I bought a textbook and relearned it, I would somehow learn to enjoy it. That didn’t seem to help. Then I read Playing With Math and discovered that math isn’t what you find in a textbook at all. It’s all around us, it’s beautiful, and most of all, it’s exciting! This book is a gem that I turn to again and again for fun and inspiration.
Shalynn Wilson

[These reviews will eventually show up at, which I'm still getting ready for prime time.]

Playing With Math - Table of Contents

Preface 9
Introduction 13

Math Circles and More: Celebrating Math

Section Introduction 19

The Art of Inquiry: A Very Young Math Circle, Julia Brodsky 23
     Puzzle: Imbalance Abundance, Paul Salomon 29

Rejoicing in Confusion, Maria Droujkova 31
     Game: Parent Bingo, Maria Droujkova 35

Parents and Kids Together, Sue VanHattum 37
     Puzzle: Foxes and Rabbits, Sue VanHattum 45

On Noticing and Fairness: A Mindful Math Circle, Rodi Steinig 47
     Puzzle: Is this for real? Avery Pickford 52

Bionic Algebra Adventures, Colleen King 53
     Story: Alexandria Jones in Egypt, Denise Gaskins 58

The Oakland Math Circle: A First Iteration, Jamylle Carter 63
     Game: Fantastic Four, Exploratorium Staff 69

A Culture of Enthusiasm for Math, Amanda Serenevy 71
     Activity: Vertices, Edges, and Faces, Amanda Serenevy 74

Seized by a Good Idea, Stephen Kennedy 75
     Puzzle: Math Without Words #1, James Tanton 81

A Prison Math Circle, Bob and Ellen Kaplan 83
     Puzzle: Math Without Words #2, James Tanton 86

Agents of Math Circles, Mary O’Keeffe 89
     Puzzle: Food for Thought, Jan Nordgreen 94

The Julia Robinson Mathematics Festival, Nancy Blachman 95
     Saint Mary’s Math Contest Sampler, Br. Alfred Brousseau 97
     Exploration: Candy Conundrum, Joshua Zucker 97

A Young Voice: Consider the Circle, Elisa Vanett 99

Homeschoolers Do Math

Section Introduction 103

Tying It All Together, Julie Brennan 105
     Game: Place Value Risk, Sue VanHattum 112

Advice from Living Math Forum, Julie Brennan 113
     Puzzles: Deep Arithmetic, Sue VanHattum 119

Transitioning to Living Math, Jimmie Lanley 121
     Game: Math Card War, Denise Gaskin 126

At the Eye of the Hurricane, Melanie Hayes 129
     Puzzle: Self-Referential Number Square, Jack Webster 142

One and a Quarter Pizzas, Holly Graff 143
     Game: Function Machine, Sue VanHattum 148

The Math Haters Come Around, Tiffani Bearup 149
     Puzzle: Magic Hexagon, Michael Hartley 155

Mapping the Familiar, Malke Rosenfeld 157
     Game: Racetrack, Sue VanHattum 159

Radically Sensible Ideas, Pam Sorooshian 161
     Game: Dotsy, Leonard Pitt, Cinda Heeren, Tom Magliery 168

A Young Voice: An Unschooler at College, Lavinia Karl 169

Passionate Teachers: In the Classroom

Section Introduction 173

Teach Less, Learn More, Sue VanHattum 177
     Game: Modular Skirmish, John Golden 181

Trust, Montessori Style, Pilar Bewley 185
     Puzzle: Measuring With Paper, by B. Zolkower, D. Abrahamson 190

Math In Your Feet, Malke Rosenfeld 191
     Game: Fizz Buzz, by Michael Hartley 197

Dinosaur Math, Michelle Martin 199
     Puzzle: Alien Math, Amanda Serenevy 201

Better Teaching Through Blogging, Kate Nowak 203
     Activity: Candy Launcher, Sean Sweeney 208
     Background:Using Math to Describe Gravity, Sue VanHattum 213

Putting Myself in My Students’ Shoes, Allison Cuttler 217
     Puzzle: What Number Am I? Jonathan Halabi 220

An Argument Against the Real World, Friedrich Knauss 221
     Puzzle: Octopus Logic, Tanya Khovanova 223

Area of a Circle, Fawn Nguyen 225
     Exploration: Coloring Cubes, Joshua Zucker 227

Textbook Free: Kicking the Habit, Chris Shore 229
     Activity: Guess My Dice, Kaleb Allinson 231

Math Is Not Linear, Alison Forster 233
     Puzzle: A Little Math Magic, Jonathan Halabi 239

A young voice: Geometric Delights, Luyi Zhang 241


Introduction and Internet Resources 247

Math and the Electronic Commons, Maria Droujkova 252

Creating Math Teachers at Play, Denise Gaskins 255

Math Playground: Designing Games for Real Learning, Colleen King 260

Supporting Girls, Sue VanHattum 266

How to Become Invisible, Bob Kaplan 280

Starting a Math Club, Maria Droujkova, Sue VanHattum 282

Conclusion 285

Sue’s Book Picks 297

Hints for Puzzles 306

Meet the Authors 311

Acknowledgements 320

Index 322

Playing With Math - the Authors and Artists

The book is almost here! We are preparing for a publicity and fund-raising campaign, and now seemed like a good time to introduce the wonderful people who came together to create this book. So, without further ado...

 Meet the Authors and Artists!

Alison Forster is currently a high school math teacher at the Doane Stuart School in Albany, NY where she teaches algebra, geometry, precalculus, and an elective of her own design called "Foundations of Mathematics." She got her start having fun with math with homeschoolers at the age of sixteen and never looked back.

Allison Cuttler is originally from New Jersey and fell in love with math at Haverford College, a small liberal arts school outside of Philadelphia. After completing her Masters in Applied Mathematics at the University of California, San Diego, she discovered her true passion for teaching math and programming at High Tech High Chula Vista, a project-based charter school serving San Diego’s South Bay community. In the fall of 2011, she moved back to NJ and has been teaching at North Star Academy College Preparatory High School in Newark, New Jersey ever since. In her free time she enjoys ultimate frisbee and running, exploring local cuisine, and blogging at

Amanda Serenevy is the executive director of the Riverbend Community Math Center, an organization that promotes access to high-quality math education for people of all ages in north-central Indiana. In that capacity, Amanda presents hands-on math activities, leads workshops for teachers, and mentors elementary, high school, and undergraduate students. After teaching in Bob and Ellen Kaplan's Math Circle program in Boston, Amanda became active in the Math Circle movement, connecting mathematicians with young students interested in mathematics. In 2007, Amanda earned a Ph.D. from Boston University with a dissertation on the dynamics of networks of inhibitory neurons. She has published research on mathematical neuroscience and iterated matrix maps, and has additional research interests in geometric topology and mathematical origami.

Avery Pickford is currently a fifth and sixth grade math teacher in the San Francisco Bay Area, and the way he teaches math does not equal the way he was taught math. In his fifteen years of teaching he has had the pleasure of teaching math and science to students from third grade to graduate school. He is always eager to discuss progressive teaching, and is especially interested in student-posed problems. In addition to his love of math, he is also an amateur photographer.

Betina Zolkower is an associate professor at Brooklyn College (City University of New York) where she teaches methods and research courses for pre- and in-service middle and high school mathematics teachers and conducts research on the functional grammar of whole-group conversations in mathematics classrooms. Betina is the founding co-director of the Grupo Patagónico de Didáctica de la Matemática (, a lesson study/instructional design collective of teachers and teacher educators in Southern Argentina whose work is inspired by Hans Freudenthal's realistic mathematics education. Betina is also a photographe.

Bob Kaplan has worked on mathematics with people from four up, most recently at Harvard University. In 1994, with his wife Ellen, he founded The Math Circle, a program open to all comers, for the enjoyment of pure mathematics. He has also taught philosophy, Greek, German, Sanskrit, and “Inspired Guessing.” He is the author (as Robert Kaplan) of The Nothing That Is: A Natural History of Zero (Oxford 2000), and with his wife, The Art of the Infinite: The Pleasures of Mathematics (Oxford 2003), Out of the Labyrinth: Setting Mathematics Free (Oxford 2007), and Hidden Harmonies: The Lives and Times of the Pythagorean Theorem (Bloomsbury Press 2010). In the past year the Kaplans have opened over a thousand Math Circles in Brazil, each aimed at the poorest sections of the country. The program is planned to expand over the next five years. Bob lives with his wife in Massachusetts, but plays cricket for the Grange Club in Scotland, where he first became acquainted with naught.

Chris Shore teaches high school algebra, geometry, and an International Baccalaureate math course, effectively engaging adolescents in the mathematics classroom. Chris is the editor and publisher of The Math Projects Journal, a professional newsletter offering innovative math lessons, most of which are published as the book, MPJ’s Ultimate Math Lessons. As a leader in implementing instructional change, Chris has made presentations nationwide to teachers and administrators on improving math instruction and raising standardized test scores. He is the department chair at his high school and has led his team to being the highest-performing school in the county. Chris is the 2001 California recipient of the Presidential Award for Excellence in Mathematics and Science Teaching.

Colleen King is the co-founder of a mathematics learning center where she teaches K-12 students the art of problem solving. Colleen's unique approach to math instruction includes computer programming, robotics, science projects, and role-playing games. Each class is an adventure and students enjoy the unpredictable learning experiences. Colleen is probably best known for her work on, a popular educational site for elementary and middle school students. Colleen's goal is to one day design a game that captures the collaborative problem solving and "hard fun" that takes place at her math center.

Denise Gaskins says, "Math is not just rules and rote memory. It's like ice cream, with more flavors than you can imagine. And if all your children ever do is textbook math, that’s like feeding them broccoli-flavored ice cream.” As a veteran homeschooling mother of five who loves math, she wants to help other homeschoolers see the variety and richness of the subject. Denise writes the Let's Play Math! blog and started the Math Teachers at Play blog carnival to share creative ideas for learning, teaching, and understanding math. She’s also taught physics, which was just one story problem after another. What fun!

Dor Abrahamson is a professor of education at the University of California, Berkeley. He does research on how students learn mathematical concepts and invents systems for learning mathematics. These two strands of Dor's work come together in an approach called "design-based research", by which researchers can make contributions to both practice and theory. Dor is particularly interested in finding ways of helping children build on their intuition when they learn mathematics. When kids seem to get things "wrong," Dor looks for the grain of "right" in their intuition, and he creates systems that help kids connect these intuitions with formal mathematical ways of describing the world. Most of Dor's work has been on the concepts of proportionality and probability. Recently he has put out a free iPad app called the Mathematics Imagery Trainer for Proportion.

Elisa R. Vanett is currently a senior at John Adams high school. She plans on attending Indiana University South Bend and enrolling in the nursing program, pursuing research as an undergraduate, and minoring in creative writing.

Ellen Kaplan was a classical archaeologist through graduate school at Harvard and in Germany, and has also taught biology, Greek, Latin, and the history of many places and times. She began teaching mathematics to integrate an all-male department, but was so delighted by the breadth and depth of the field that she ended up co-founding the Math Circle with her husband, illustrating his book, The Nothing That Is (Oxford 2000), and writing The Art of the Infinite (Oxford 2003), Out of the Labyrinth: Setting Mathematics Free (Oxfod 2007), and Hidden Harmonies: The Lives and Times of the Pythagorean Theorem (Bloomsbury Press 2010) with him. With their son, Michael, she has written Chances Are ... Adventures in Probability (Viking 2006), and Bozo Sapiens: Why to Err Is Human (Bloomsbury). They are at work on their third book. In the past year Ellen and Bob Kaplan have opened over a thousand Math Circles in Brazil, each aimed at the poorest sections of the country. The program is planned to expand over the next five years.

Ever Salazar has been teaching math and physics in Ciudad Guayana, Venezuela for four years. He has always been interested in math, and loves to solve puzzles from math competitions, which were his only source of real math in high school. When he turned nineteen, he discovered Martin Gardner's books, and since then his passion for math has been entangled with the need to show this awesomeness to other people. And when he discovered ViHart, MinutePhysics, CGPGrey, Veritasium and other educational channels on Youtube, he knew that was his place. He is currently teaching Calculus at Universidad Católica Andrés Bello and illustrating for the Youtube channel MinuteEarth.

Fawn Nguyen has been teaching geometry, algebra 1, and sixth-grade math for the last ten years at Mesa Union Junior High in Somis, CA. Prior to teaching math, she was a middle school science teacher for fourteen years. Inspired by her father, who taught math for over thirty years, Fawn has always had a deep love for mathematics, especially problem solving. She is passionate about making math accessible, relevant, and fun for students. She blogs about teaching at Fawn currently is a presenter of the University of California, Santa Barbara’s Mathematics Project and is also helping to lead the Thousand Oaks Math Teachers' Circle.

Friedrich Knauss worked for close to two decades as a software engineer, and then decided to reboot his career, switching tracks to the teaching profession. Like most new teachers, he assumed that teaching was mostly a matter of presenting information in a clear and logical fashion, and the eager and hungry young minds would eat it up. It took one year at an inner city Los Angeles school to realize that subject knowledge was the least part of teaching; he has been using his skills as an engineer and scientist to improve his craft ever since. He blogs at

Holly Rebekah Graff is an unschooling mom and former public school science teacher. She believes that every child deserves the freedom, time, and support necessary to pursue her passions and construct her own rich understanding of the world. She has worked with a diverse group of students, urban and rural, pre-kindergarten through high school, in a variety of settings from crowded urban classrooms to intimate groups of homeschoolers. She currently teaches science classes for homeschoolers at her home in the Catskills of New York. She blogs at Unschool Days about the school-free lifestyle. Her interests include writing, theater, creating collage, swing dancing, snowboarding, baking, and gardening.

Jack Webster did his undergraduate degree in mathematics at Cambridge University, and is particularly interested in set theory and formal logic. He works in radio communications now as a programmer and mathematician. Jack blogs at, where he has posted a number of other puzzles he has created.

James Tanton has been doing puzzles all his life. He's created Math Without Words, a lovely book of puzzles, along with a number of other books and videos taking a playful approach to math. You can find links to all of this and more at

Jamylle Carter is a mathematician and a musician. In 2009 she joined the full-time mathematics faculty at Diablo Valley College in Pleasant Hill, California. Before then, she trekked all over the country for mathematics: bachelor’s degree from Harvard University; Ph.D. from University of California, Los Angeles; and postdoctoral positions at a large public research university, a science museum, and two National Science Foundation mathematics institutes. She has published research on applied mathematics for image processing. Jamylle has also played piano since the age of five. A finalist in a Los Angeles songwriting competition, she has been a director, arranger, and pianist for choirs nationwide. Jamylle is currently a pianist and choir director for the East Bay Church of Religious Science in Oakland, California.

Jan Nordgreen started writing his blog, think again, in Santa Cruz, Bolivia. Through moves to France, the Cayman Islands, and back to Bolivia, and through hurricane Ivan, France Télécom, and satellite-only connections, he kept the blog going. During his two-year stay in Thailand he renamed the blog ”thnik again” and it shot to the number one Google result for ”thnik.” He currently resides in Lanzarote, Spain.

Jimmie Lanley is the mom of one creative daughter. After seven years of teaching in public schools, she became a stay-at-home mom when Mel was three years old and the whole family moved to China. She has taught Mel at home ever since. Her research into curricula and homeschooling philosophies led her to a Charlotte Mason style, which she finds very satisfying. Jimmie likes sewing, writing, traveling, and cooking from scratch. She blogs at and

John Golden is a math teacher educator, elementary and secondary, at Grand Valley State University in West Michigan. He is interested in how people learn math and how to support teachers in the classroom, with particular interest in learning-math-game connections and dynamic geometry. He blogs at, tumbls at and tweets from @mathhombre.

Jonathan Halabi lives in the Bronx, where he teaches high school math. He is the founding mathematics teacher at the High School of American Studies at Lehman College (2002) and designed and planned much of that school's curriculum. Jonathan has also taught college math, middle school enrichment, and methods and content to preservice math teachers. He is a union activist, and is interested in problem solving, numbers, and social justice. He often speaks on mathematical problem solving.

Joshua Zucker is the founding director of the Julia Robinson Mathematics Festivals, which bring deep, collaborative problem solving to a wide range of students. He discovered his love for number theory at Dr. Arnold Ross's summer program at Ohio State University over twenty years ago. Joshua taught at Stanford’s Education Program for Gifted Youth, community colleges, and public and private high schools, before becoming a freelance math teacher. In 2006, he helped begin the Math Teachers' Circle project at the American Institute of Mathematics. He currently is a part-time instructor for the Art of Problem Solving, as well as a leader at several math circles in the San Francisco area.

Julia Brodsky is a homeschooling mom with three naughty and curious kids. When she is not with her family, she works as a rocket scientist for NASA Goddard, runs a weekly Art of Inquiry math circle for elementary school students, organizes the annual Math Kangaroo Olympiad for Montgomery County kids - and still keeps some sanity. She is constantly fascinated by the way children learn and solve problems. Julia grew up in Russia, where she was a mediocre student in one of the best math magnet schools of St. Petersburg. Later, she had a lot of fun working as an International Space Station astronauts' instructor at Johnson Space Center. Julia also enjoys writing poetry, hiking, and watching somebody else working instead of her.

Julie Brennan hosts the Living Math Forum, a five thousand-member Yahoo group engaged in discussion and sharing of math education experience and resources. Her website,, is full of information on teaching and learning math in non-traditional ways. Julie’s homeschooling experience is reflected in the site’s content and articles, but many parents of schooled children and teachers also benefit from the information. Julie sells Living Math History lesson plans on the site, a fascinating approach to learning math through the study of the masters who discovered it. Julie worked professionally as a CPA and financial consultant prior to staying home with her four children. She currently teaches classes for homeschoolers.

Kaleb Allinson is a high school math teacher and the department head at Lake Stevens High School in Lake Stevens, WA. He taught middle school during his first four years of teaching and has taught at the high school for eleven years. He currently teaches Geometry, Advanced Algebra and AP Calculus. Kaleb has always enjoyed attempting to solve problems that he has never seen before. When he’s not teaching he's quite busy with his six energetic kids.

Kate Nowak teaches mathematics at Charlottesville High School in Charlottesville, Virginia. She has loved puzzles, logic, and origami from an early age. In addition to teaching for eight years, Kate has also written real-world lessons at Mathalicious, completed an engineering degree, and fixed airplanes for the U.S. Navy. She is passionate about showing kids that mathematics is fun and fascinating, and improving her craft in collaboration with colleagues around the globe. She has written the popular blog f(t) since 2007.

Lavinia Karl was unschooled. She earned a B.A. in math from Knox College. Now she's in her twenties, exploring life's possibilities.

Linda Palter is a chiropractor in West Michigan. She can also be found square dancing, crafting, and playing fetch and frisbee with a very high-energy dog.

Luyi Zhang is presently an undergraduate math major at the Massachusetts Institute of Technology and an instructor of online math courses at Art of Problem Solving. In middle and high school she participated in numerous math contests, placing in the top ten statewide in MATHCOUNTS and qualifying for the USA Math Olympiad. She has taught middle school students through Breakthrough Collaborative and has worked with gifted students at the math camps MathPath and Epsilon. She blogs about her original geometric creations such as Sierpinski triangle brownies and beaded teddy bears on her website, Geometric Delights.

Malke Rosenfeld is a percussive dance teaching artist, math explorer, curriculum designer, editor, and writer. Her interdisciplinary inquiry focuses on the intersections between percussive dance and mathematics and how to best illustrate these connections for students. In her Math in Your Feet program, percussive dance becomes the platform for a robust choreographic inquiry into mathematical thinking, practices and topics. You can find out more about Malke’s many collaborative math and making projects at

Dr. Maria Droujkova is a curriculum developer and mathematics education consultant. She organizes meetings with project and community leaders in the Math 2.0 interest group, an online collaboration of hundreds of researchers and educators interested in modeling software, computational tools, and social media in mathematics education. The group has held more than one hundred events since 2009, and has given rise to several ongoing research and development projects. Natural Math, the company Maria founded in 2001, provides a unique forum where researchers and developers join parents and teachers for discussions of family mathematics, early algebra, individualized instruction, and math clubs.

Mary O'Keeffe is a founding advisor of Albany Area Math Circle, a wonderful community of problem solvers with whom she has been happily making mistakes since 2001. She is also a public policy economist, specializing in public finance and mathematical economics. She teaches economics at Union College in Schenectady, New York, where her students run a free Volunteer Income Tax Assistance (VITA) site for low-income working families, people with disabilities, and senior citizens. She is also the Associate Director of the Math Prize for Girls, which brings together hundreds of young women from across the U.S. and Canada for a celebration of extreme problem solving each year. Her latest initiative is launching the Guerrilla Math Circles movement.

Melanie Hayes has made it her life’s work to help gifted children find their niche and achieve their goals. She is passionate about creating a world where all children are encouraged to wonder, explore, and think. Melanie holds a M.Ed. with an emphasis in the intellectual, social, and emotional needs of gifted and talented children. She is also a credentialed teacher and educational consultant with experience in evaluation, assessment, intervention, professional development, teaching, and mentoring. Melanie has gifted twins, so she experiences life with gifted children on a personal level as well, both the joys and the hardships. She writes about her work and family on her blog, Life Among the Gifted. Melanie homeschools her children and enjoys watching them learn and grow through their daily activities. She loves to travel, paint, sculpt, garden, write, and hike in the wilds.

Michael Hartley, creator of Dr. Mike's Math Games for Kids, was raised in Perth, Western Australia. He's loved math since he was a child, so it was natural for him to do a Ph.D. in mathematics, in a branch of geometry. After a ten-year stint teaching in colleges and universities in Malaysia, he returned to Perth to work as a mathematician in the oil and gas industry.

Michelle Martin is a public school teacher at Prairie Creek Community School in Northfield, Minnesota, where she works with a class of fourth and fifth graders. She loves having the opportunity to weave math and other subjects together. Her favorite moments are those when students glimpse the wonder and awe of mathematics. She writes about the work of her class at The Rookery.

Nancy Blachman strives to make math cool, fun, and engaging, finding her inspiration in the work of Vi Hart and Martin Gardner. Nancy is co-founder of the Nueva Math Circle, founder of and the Julia Robinson Mathematics Festival, and is a member of the organizing committee for the Gathering for Gardner. Nancy has taught after-school math classes and math camp for four years and Mathematica classes for ten years. Nancy earned a B.Sc. in Mathematics from the University of Birmingham, UK, a Masters in Operations Research from the University of California at Berkeley, and a Masters in Computer Science from Stanford University.

Pam Sorooshian has been teaching college-level economics and statistics since 1976. She's also been a homeschooling mom to three now-grown children. Pam is a proponent of unschooling, in which there are no lessons, assignments, tests, or grades, and children learn naturally while following their own interests with the encouragement and strong support of their parents. Pam has been a speaker at many homeschooling conferences and an American Educational Research Association conference, and is on the Board of Directors of the National Home Education Network and the HomeSchool Association of California.

Paul Salomon is a math nerd.teacher.artist living in Saint Louis. He designed his imbalance problems during his time at Saint Ann’s School in Brooklyn, where he taught math to grades 5 through 12 and helped develop a mathematical art program. Paul shares his own mathematical art through Twitter (@lostinrecursion) and his blog, Lost in Recursion. Paul also coauthors Math Munch, a weekly math blog written with middle schoolers in mind, aimed at helping them dig in to the mathematical world that exists outside of math class.

Pilar Bewley holds two AMI Montessori certifications (ages 3 to 6 and 6 to 12), as well as a M.Ed. in Montessori Education. A self-professed "math hater" from childhood, she discovered the beauty of mathematics and geometry during her Montessori training courses. She lives with her math geek husband and Montessori baby in San Diego, CA.

Rodi Steinig is the founder, director, and leader of the Talking Stick Math Circle. Her goal is to awaken the inner mathematician and to shepherd the unfolding of abstract reasoning in every child. Her formal training is in economics and education. Rodi continues to hone her craft of math circle leadership under the gentle guidance of Bob and Ellen Kaplan. Her mathematical interests are logic, history, hydrodynamics, sacred geometry, the misapplication of statistics, and the expression of mathematical concepts via multiple modalities. Rodi blogs at

Sean Sweeney currently teaches high school Algebra and Calculus just outside of Philadelphia at Woodlynde School, a college prep school for students with learning disabilities. He constantly looks for exciting ways to engage students in math who have often had bad experiences with it in the past. More activities, ideas and songs from Sean can be found at

Stephen Kennedy first learned the pleasures of mathematics at Stonehill College in Massachusetts and ever since has wondered why nobody told him earlier. He teaches mathematics at Carleton College in Northfield, Minnesota. He co-directs the Carleton College Summer Mathematics Program for Women and served a five-year stint as co-editor of the Mathematical Association of America’s magazine for undergraduates, Math Horizons.

Sue VanHattum has been teaching math at the community college level for over twenty years, and recently branched out to teach math at her son’s freeschool. She created the Richmond Math Salon, a monthly event she hosted at her home, gathering kids and their parents together to play with math. She blogs about math and math education at Math Mama Writes. Sue appreciates both the community-building possibilities inherent in public schools, and the freedom to learn naturally available through homeschooling. Outside of math, teaching, and writing, her interests include gardening, living simply, activism for the rights of all, and children’s books. Sue is a single parent, raising one son.

Tanya Khovanova received her Ph.D. in Mathematics from Moscow State University in 1988. At that time her research interests were in representation theory, integrable systems, super-string theory, and quantum groups. Her research was interrupted by a period of employment in industry, where she became interested in algorithms, complexity theory, cryptography, and networks. Several years ago she resigned from industry to return to research. Her current interests lie in combinatorics, number theory, probability theory, and recreational mathematics. Her website is located at She also writes a highly popular math blog, and produces the Number Gossip website.

Tiffani Bearup used to be a pretty average, run-of-the-mill, standard-issue mom trying to figure out life in suburbia and her place in it. Then she started unschooling, and things got a little crazy! Crazy good! As this book went to press she and her three kids were all traveling through South America. She blogs at

Sunday, May 18, 2014

Using Math to Describe Gravity (from Playing With Math)

We are nearing completion of the book, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers. Our copy edit process was really a deeper editing process, and took over a year, with us working through a few chapters a week at first, getting everything just right. We finished it on May 9, Mother's Day. (My son was born on Mother's Day. I didn't realize until this moment how much I like it that our copy edit finished on Mother's Day too!)

One of the last pieces to go through copy edit was Sean's catapult activity that goes with Kate Nowak's chapter, Better Teaching Through Blogging. I had to build a catapult to test out the instructions Sean gave. I am not a crafts sort of person, so that had to wait until I had plenty of time to deal with it. Last fall, after I finally made my catapult, made my small adjustments to Sean's piece, and sent it to our copy editor, she  asked a lot of questions about the math. I began to realize that we needed an explanation of the math behind the catapult project.

So I wrote one last piece for the book, Using Math to Describe Gravity. I had fun writing this one. Most of what I wrote for the book took me a long time (and lots of agonizing) to write. This one was easy and quick. I finally realized that I really enjoy writing explanations. (I think I know my next book project...) I thought this might be useful to have online, so I'm including a modified version here. Enjoy!

Using Math to Describe Gravity

In the picture above, we see water shooting upward. Whenever you squirt water from a hose pointed up at an angle, it follows a similar path. Have you ever wondered why? Math is helpful whenever we want to think about how something is changing.

For example, the idea of velocity tells us how position is changing with respect to time. (Unlike speed, which is just a positive number, velocity has direction, and can be negative to indicate a downward direction.) If you are driving at 70 miles per hour, in one hour you will have driven 70 miles. In two hours, 140 miles. Mathematicians generalize this idea by writing distance = rate * time.

The path of the water in the fountain makes the shape of a parabola. The physics of gravity explains why the water follows that path. As the force of gravity pulls us toward the center of the Earth, it creates an acceleration – a change in velocity. When we are near the surface of the Earth, that acceleration is always 32 feet per second squared (downward). That’s a weird unit, isn’t it? It means that if you are headed straight down, your velocity will increase by 32 feet per second each second. So when you drop something, one second later it has already gone from a speed of 0 feet per second to a speed of 32 feet per second. Over 20 miles per hour! In metric units, that would be 9.8 meters per second squared. (We’ll mostly stick with metric from here on out.)

This acceleration affects the relationship between distance and time because the speed, or rate, is changing. In the driving example, if your speed goes from 55 miles per hour to 60 miles per hour to 70 miles per hour, it makes it harder to calculate how far you have driven. (Calculus is great for understanding situations where your rate of change is changing, but I've written this for people who aren't familiar with the concepts of calculus, sticking to algebraic ideas.)

When considering the physics of situations like the fountain, we can analyze the vertical and horizontal motion separately. Gravity isn’t affecting the horizontal motion, so that stays constant. (If you were moving very fast, air resistance would slow you down. But at these speeds, we can ignore the effect of air resistance.) When you throw something upward at an angle, gravity pulls straight down, changing the vertical component of the velocity. Since the horizontal part of the motion is constant, this gradually changes the direction the object is headed, making the parabolic path you see above.

But how do we know that it’s in exactly the shape of a parabola? To see why it is, we’ll start with a simpler experiment, throwing a ball directly up. Now the path is no longer a parabola, because the horizontal position is not changing. But, amazingly, if we were to draw a graph of height versus time, that graph would still be a parabola. To describe parabolas algebraically, we use equations like y = at2+bt+c. In this case, y is the height and t is the time.

[Note: Depending on the flavor we want, we say the same thing in lots of different ways. rate * time = distance, R*T=D, velocity * time = height, h=v*.]

We can figure out a lot about a, b, and c by using what we know about the physical situation. When t=0, y = a*02+b*0+c = c, so c can be filled in by knowing your initial height. If we had no gravity (and no air resistance), what we threw upward would keep going up with a constant speed. So its height would be given by rate (velocity) times time plus initial height. Do you see why b is the initial velocity the ball has as it leaves your hand?

That leaves a. The value of a will always be half the value of the gravitational constant - hmm, why half? In one second, the acceleration of gravity would increase velocity from 0 to 9.8 meters per second. So the average velocity during that second is 4.9 meters per second. In t seconds, we would increase from 0 to 9.8t meters per second, with an average of 4.9t meters per second times t seconds, for a height change of 4.9t2 meters.

Now we can see the effects of initial height, initial velocity, and gravity combining to make an equation of the form y = at2+bt+c for height versus time. Gravity will actually affect an object in this way no matter which direction it’s pointed. And since the horizontal motion is constant, this same sort of relationship holds when we look at height versus horizontal position, although the values for a, b, and c will change.

When an object is launched at an angle, the value for a is determined by both gravity and the launch angle, b is determined by both the initial speed of launch and the launch angle, and c is still the initial height. To find the values for a and b, we can use the symmetry of parabolas across their vertex. If we can find the coordinates for the position of the vertex, we can use that to help use find the values for a and b.

All this thinking can help us understand the catapult activity better:
  • When launching from the floor, where beginning and ending heights are the same, the x-coordinate of the vertex is just half the distance. So the vertex will be reached halfway through the time in the air.
  • From the time the projectile reaches the vertex until the time it hits the floor, its height is decreasing at the same rate as an object that has been dropped, so you just use 4.9t2 to find how far it dropped, which tells you how high it was.
  • The vertex form for the equation of a parabola is y = a(x - h)2 + k, where (h,k) represents the vertex, which we just found. If we assume that we launched from the origin, plugging zero in for x and y allows us to find the value for a.
  • With the values for a, h, and k filled in, we have an equation in x and y. We can simplify it (change it to the form y = ax2+bx+c), and then modify it for raised launches by simply changing the value of c from 0 to the height of the launch surface. 
  • We can use the new equation to figure out where to put a target we want to hit! The target will be on the ground, where the height is 0, so we can plug in y=0, and find x. In real life, the numbers that show up are almost never simple enough for factoring to work, so we’d need the quadratic formula. Measuring the horizontal distance from the floor right below the catapult, to the positive x-value from the quadratic formula, and centering the target there, we should be able to hit it. Candy bombing, here we come! 

My thanks to John Golden for his help improving this.

Friday, May 2, 2014

Linkfest for Friday, May 2

Sometime in the past week, I added to my feedly feed. They compile posts from all the math blogs they know of, with an interesting mechanism that takes you right to the original blog, instead of just linking to it. Yikes! I was already following hundreds of math blogs, and suddenly my feed doubled or tripled. I'll have to let it go eventually, it's just too much. But I found lots of interesting posts on blogs I'd never seen before, so today's linkfest will be more diverse than usual.

Crazy, how much good stuff there is to read. How can anyone absorb all this?

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