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SYMMETRY. 180° rotation INSPIRATION. At the request of Kai Krause STORY. While some musicians are happy to write songs and tour, Peter Gabriel is one of a handful of musicians to work across many media and use his position in the music world to influence much larger changes. His music video Sledgehammer remains one of the most visually inventive of all time and still makes me smile. His Explora CD-ROM was one of the savviest early uses of that medium. His record label and recording studio Real World has had a major effect in bring the musics of the world together. I was hardly surprised when I saw him as host of a documentary on world musics. To explore Peter’s world, see petergabriel.com.

SYMMETRY. 180° rotation. INSPIRATION. Commissioned by Daniel Rhod for his magic magazine Imagik. STORY. In February 2000 I attended the fourth Gathering for Gardner, an event for mathematicians, magicians, puzzle people and other kindred spirits of renowned science writer Martin Gardner. Magician Mark Setteducati introduced me to Daniel Rhod, founder and editor of the French magazine Imagik. Daniel's interview with me appears in the July 2000 issue. Daniel also asked me to create three new inversions to accompany the article: French magicians Robert-Houdin and De Kolta, and the name of the magazine, Imagik. Robert-Houdin (1805-71) transformed magic in the mid 1800s from a circus side show act to an elegant evening stage performance, complete with coat and tails. Among other accomplishments, he was the first to use electricity as part of a stage show. His performances so mesmerized audiences that American magician Eric Weiss took his stage name from Robert-Houdin, calling himself Houdini. Robert-Houdin was originally a clockmaker, and some of his acts involved ingenious lifelike automatons. I had the great pleasure of seeing two of his original mechanisms presented at the International Puzzle Party in Los Angeles last August by one of the leading manufacturers of stage magic equipment for other magicians. In one illusion an orange bush blossoms before your eyes and oranges grow. The performer tosses the oranges into the audience...they're real! Inside are borrowed objects from the audience. The climax: the top orange springs open, and butterflies fly out of it carrying a borrowed hankerchief. Truly magical. The other mechanism was a mechanical trapeze artist who did an elaborate act on a swinging trapeze. Through most of the act the human figure held on to be the bar with his hands — that was clearly how motion was communicated between the bar and the figure — but then at the climax the figure let go of the hands and held on only by his knees. Mechanical work of this precision is simply beyond present-day artisans. I used the same lettering style as in my recent Scott Kim inversion, because it deals well with the problem of turning the second half of an O into an I. I was pleased that the capital H lands in the right place to act as a pivot. Buatier de Kolta (1847-1903) was an ingenious inventor of stage illusions. Many of his illusions, like the Multiplying Billiard Balls and Vanishing Bird Cage, are still performed today. Some of his illusions we still don't understand. Daniel Rhod described the amazing Expanding Die, in which De Kolta places a an 8" cubical die on the stage. The die suddenly expands 20 times its original size. Lifing the enlarged die, he uncovers a seated woman. De Kolta's work is chronicled in the book Buatier de Kolta: genius of illusion, by Peter Warlock, currently out of print. Finally I created an invertible version of the Imagik logo for the cover. I tried to preserve the original lettering style as closely as possible. I dislike mixing upper and lowercase letters, but here it was unavoidable. Of course this design is not exactly the same upside down, because of the overall curve of the lettering.

SYMMETRY. Reflection about a vertical axis. Looks the same in a mirror. INSPIRATION. Commissioned by Key Curriculum Press, publisher of my book Inversions, for use at the 1999 annual meeting of the National Council of Teachers of Mathematics. STORY. Key Curriculum Press is a leading publisher of innovative high school mathematics books, videos, software and other materials. Outstanding titles include the textbook Discovering Geometry, the software Geometer's Sketchpad, and the kit Exploring Math Through Puzzles, which includes enough pieces to make 54 puzzles. Every year they dress up their product catalog with a different wacky theme. This year the theme is "Be a Math Action Hero". Scattered throughout the catalog are such mathematical superheroes as Captain Chaos, Deductiva and Calculadora, drawn in comic book style. For their trade show booth at the annual conference of the National Council of Teachers of Mathematics, they asked me to create a mirror symmetrical version of the word "Superteacher" as a card to give away to conference attendees. Only the left half of the word is printed on the folding card; the right half is a mirror that lets you view the other half of the word, and see the real math action hero: yourself. I usually prefer rotational over reflective symmetry, but this word works well, with the split nicely placed in the middle of the capital T. If you were to wear a shirt with this word emblazoned across the chest, it would read the same in a mirror.

SYMMETRY. 180 degree rotation. Turn this design upside down and each name reads the same both ways. INSPIRATION. Created during a week-long dance residency in Thousand Oaks, California, April 21-25. STORY. I improvised these two designs on the first names of the daughter and son of the organizer of our residency. Since the two names are related, I tried to make the lettering styles related. These two inversions illustrate two of the most common liberties I take with letterforms. If there is more than one copy of a letter in the same word, I prefer to make the shapes as similar as possible. In BRIANNA, however, the two A's are fundamentally different. With names, I prefer correct capitalization: either all capitals, initial capital and the rest lowercase, or if necessary. In JORDAN, however the J is uppercase and ODAN are lowercase, but R is uppercase.

SYMMETRY. 180° rotation. INSPIRATION. Created for a talk on Inversions at the Exploratorium. STORY. The Exploratorium is a trend-setting interactive science museum in San Francisco. Founded in 1969 by physicist and educator Frank Oppenheimer, it features highly involving exhibits that invite visitors to participate in exploring ideas in science and perception.

SYMMETRY. 180 degree rotation. Turn this design upside down and it reads the same both ways. INSPIRATION. Created for a performance by the Dr. Schaffer and Mr. Stern Dance Ensemble, February 28, at the Performing Arts Center of the California Polytechnic State University in San Luis Obispo. STORY. Drawing is not something you usually think of doing on stage, but I have always enjoyed the drama of creating inversions for people while they watch. So when Karl Schaffer, Erik Stern and I choreographed the show The Secret Life of Squares, we decided to turn inversions into one of the acts. I start by holding up signs with common invertible words: E turns into M, MOM turns upside down into WOW and NOON turns into itself. Then I show a few original inversions, usually including the name of the city in which we are performing. Finally I take a name from the audience and draw it as an inversion on the spot. At the end of February we performed at Cal Poly in San Luis Obispo. "San Luis Obispo" proved to be particularly challenging. I considered opting out and inverting some other name like "California" or "Cal Poly". In the end I decided to persist. and eventually found this solution. S into O worked well enough, but running the N into the L was quite a desperate trick. A into P is a familiar combination, but I usually avoid it because it isn't very strong. And UI into B runs the risk of "San Luis Ohispo".

SYMMETRY. 180° rotation INSPIRATION. I first saw Cirque du Soleil in 1988 when their traveling big top first touched San Francisco. I immediately fell in love with their new take on what a circus can be, and have been immensely pleased to see them grow into a world-wide phenomenon. I set pencil to paper, and found that "Cirque du Soleil" works well. Fourteen years later I set mouse to pad and rendered what I had sketched. STORY. Started in 1984 as part of a government sponsored celebration in Quebec, Cirque reinvented the circus as a musical visual experience with great beauty and mystery, minus the animals and huckster vibe. I am reminded a bit of the great French magician Robert-Houdin, who reinvented magic as a sophisticated evening entertainment, rather than a carnival side show. At once traditional and avant garde, Cirque has struck a chord with audiences all over the world, and has become an institution with multiple shows and performing groups. "Circus" comes from the same root as "circle", referring to the circular stage in which traditional circuses are staged. "Soleil" means "sun", and the sun is the symbol of Cirque du Soleil. I have used the circle as the setting for this design, and incorporated elements that suggest moon, sun and stars. Explore the world of Cirque du Soleil at http://cirquedusoleil.com.

SYMMETRY. 180 degree rotational symmetry, bent into a circle, with three repetitions of the word INSPIRATION. One of several variations on the word "infinity" created for my book Inversions. STORY. I like to write words so the form expresses the meaning; this one works particlarly well. Notice that you can read the words at both the top and bottom of the circle reading clockwise and counter clockwise. For instance, the FI turns into the Y. I programmed this design originally in JaM, the predecessor to PostScript. The mathematical transformation from straight to circular lettering was programmed by John Warnock, the co-creator with Martin Newell of JaM, and co-founder with Charles Geschke of Adobe Systems.

Title lettering for a 10-videodisk anthology of computer animation, published in Japan. I met the folks behind this wonderful anthology at the annual Siggraph conference. The design appeared on the box spine and disk label: naturals since these are things that normally turn upside down. Other natural applications for inversions include backs of playing cards (check out Set and click on Triology to see the wonderful inversion John Landon did for the card game Triology), tires (Discover magazine reported several years ago that someone at a major tire company had patented a lowercase alphabet in which all letters turn upside down to become letters), and satellites (I did a logo for Ford Aerospace for GOES, the Geostationary Operational Environmental Satellite). The most interesting treatment of the design they came up with was a title animation on the videodisk. Instead of rotating the words so that "computer" becomes "graphics", which would have been the obvious choice, they let the words be stationary and instead suggested the symmetry by adding animated silhouettes of plant forms in rotational symmetry to the background. A wonderful, subtle and economical solution. Most of this inversion works rather naturally; however, I resorted to a desperate trick to make the two central P's turn upside down.

All about the art of ambigrams — words that read in more than one way. See my ambigrams with different symmetries, and watch a live demonstration of how to create an ambigram (on the interviewer's son's name). An online presentation for the MAA (Mathematical Association of America), 14 March 2020. Hosted by Tim Chartier, mathematical mime.

In his Aug 1998 article "A Quarter-Century of Recreational Mathematics" in Scientific American magazine, Martin Gardner wrote For 40 years I have done my best to convince educators that recreational math should be incorporated into the standard curriculum. It should be regularly introduced as a way to interest young students in the wonders of mathematics. So far, though, movement in this direction has been glacial. Martin Gardner was a hero to me and many other budding mathematicians. He opened the door to the joys of mathematics for more people than anyone else in history. So to hear him despair at his inability to influence educators is sobering. But not unexpected. All of us here understand the deep joy that doing mathematics can bring. And we love sharing that joy with others. But if you ever tried to bring that joy into schools, you know how resistant schools are to change. And it’s not just schools. Society at large is stubbornly mathphobic. If you tell someone at a party that you are a mathematician, you will hear stories of pain, embarrassment, and fear. As Justin Reich chronicles in his book “Failure to Disrupt,” education reformers have tried and failed to budge the school system for over a century. Every new technology, from television to VR, has been touted as the thing that will change schools forever. They’ve all failed to make progress because education is not a tech issue; it’s a social issue. So, if we want to spread joyful mathematics and change attitudes at a societal scale, we need to understand what we’re up against to figure out what new actions we need to take. In this post, I'll break down the barriers to the widespread adoption of recreational mathematics in education and discuss what it will take to overcome them. Audience exercise: in your experience, what are the biggest barriers to spreading joyful mathematics? Recreational Mathematics Regarding recreational mathematics, Gardner writes: The line between entertaining math and serious math is a blurry one. In general, math is considered recreational if it has a playful aspect that can be understood and appreciated by nonmathematicians. It encompasses mind-bending paradoxes, ingenious games, and bewildering topological curiosities such as Möbius bands and Klein bottles. Gardner gives several examples of the playful mathematics he featured in his Mathematical Games column, which ran from 1956 to 1981. Check out the original article for details. His examples include A magic trick involving a matrix of numbers — understanding why you always end up with the same sum involves a surprising moment of insight. A notorious paradox that has been dubbed the Monty Hall problem — it’s not a logical paradox but rather a crisis of intuition, where a seemingly simple probability problem sparks fierce debates, even among mathematicians. A classic 3D puzzle called the Soma Cube, invented in 1933, is in which 7 pieces must be assembled to form a 3x3x3 cube or other interesting shapes. It’s a perenially popular toy enjoyed by kids and adults alike. These experiences have four qualities that set them apart from typical classroom exercises: Entertaining. All these experiences hook the audience with a riveting premise that keeps you on the edge of your seat. In contrast, typical classroom experiences give you little reason to care other than “it will be on the test.” Exciting. These experiences evoke curiosity, awe, and excitement — emotions which open you up to learning. Typical classroom experiences evoke boredom, anxiety, and fear of failure. Participatory. Curiosity leads naturally to classroom discussion, where students are eager to understand what is happening. Conventional math education overexplains what you are supposed to learn, leaving no room for natural curiosity or divergent opinions. Approachable. Finally, these experiences never overload the audience with difficult prerequisites. The elements of the problem are familiar and easy to understand. In the case of the Soma cube, the pieces are physical, making them pleasurable to touch and handle. Together, these four qualities make magic tricks, paradoxes, and puzzles ideal ways to engage learners at the start of a lesson. Teachers are rarely trained in the art of creating captivating experiences, but as every teacher knows, their first responsibility is to engage the students’ attention. Now, let’s look at the five barriers to the widespread adoption of recreational mathematics. 1. Lack of Exposure Barrier: most people have never been exposed to joyful mathematics. That includes both students and teachers. Their only experience of mathematics is through school, and most of that is boring, anxiety-provoking, and meaningless. When older kids and adults finally experience joyful mathematics, they are often pissed — why didn’t I know this sooner? Solution: This is the most basic barrier, and it’s the easiest to overcome. We must create more joyful mathematics experiences and get them in front of kids. That’s what the Julia Robinson Math Festival is doing with its in-school festivals and what the Museum of Mathematics in New York City is doing with its museum and many outreach programs for kids and adults. It’s what Thinkfun is doing with its wonderfully designed puzzle toys for families and books like The Number Devil, and You Can Count on Monsters are doing for young readers. Which leads us to 2. Lack of Distribution Barrier: we need to reach more people, especially those scared off by the word “math.” Solution: distribute through the most widely viewed channels, like YouTube. And use language and presentation styles that appeal to people who don’t like math. The most successful example of excellent distribution for an inventive educational experience is Sesame Street, which exploded on public television in 1969. These days, math YouTubers like Grant Sanderson of 3 Blue 1 Brown and Derek Muller of Veritasium are making a serious dent in math education by producing seriously entertaining and highly personal videos on deeply mathematical topics. YouTuber Vi Hart has done an outstanding job of reaching young women who don’t love math by leading with a voiceover about how boring math class is. But much great recreational mathematics needs better distribution. I recently hooked up with Alex Rosenthal at TEDed, which has produced a spectacular video library of over 100 mathematical riddles. With over 19 million subscribers, they have certainly reached a large audience. But like many nonprofit educational institutions, their funding is used primarily to pay their small staff, which means their beautifully produced material is less well known than it should be. I think the lack of attention to distribution and marketing stems from the fact that teachers teach to a captive audience and thus don’t have to market themselves. In contrast, the market forces on YouTube, where everyone competes for the viewer’s attention, force content creators to design videos that hook and hold the viewer’s attention — the best YouTube math videos are highly entertaining. Next, we have the active barriers — forces that actively resist change. 3. Mental Model of Math Barrier: if you believe the narrow definition of mathematics taught in school, you will reject anything playful as “not real mathematics.” Solution: widen people’s understanding of what mathematics is. When I take math games into classrooms, students and teachers have a good time and generally understand the value of what they are experiencing. But then, class goes back to “normal”. Teachers treat recreational mathematics as a brief break from real mathematics rather than an integral part of good education. The underlying reason for this disconnect is that school trains people to believe that mathematics consists only of memorizing and accurately reproducing canned formulas. Under that definition, any experience that involves ambiguity, asking questions, or creativity does not qualify as mathematics. A similar situation once existed in English class. When I went to elementary school, I learned the rules of grammar and how to write an outline before writing a paper. Completely missing from this curriculum are the much messier things that real writers do — searching for topics, drawing mind maps, and most importantly rewriting. Thank goodness school now teaches the complete writing process, through programs like the widely adopted Readers and Writers Workshop programs, published by Heinemann. We need to do the same for mathematics. And indeed, Heineman also publishes a “cognitively guided instruction” math program with the same philosophy as Readers and Writers Workshop. Doing math is more than memorizing formulas and computing correct answers. Doing math also includes noticing patterns, being curious, asking questions, trying things out, being wrong, and trying again. Under that much broader definition of mathematics, everyone is already a mathematician. 4. Curriculum Standards Problem: curriculum standards leave no room for recreational mathematics Solution: revise the standards and build recreational mathematics into the core curriculum. Teachers resist including more recreational mathematics in their teaching — even the ones who love it — because they don’t have time. Teachers are under increasing pressure to stick to state-mandated curriculum standards that prescribe precisely what to teach, and when. As a result, teachers race through a curriculum that is “a mile wide and an inch thin”, without pausing to make sure that kids understand what they are learning. California — often the leader in social change — is now embroiled in a highly contentious effort to revise state math curriculum standards. Parents and teachers are wary of new standards, and for good reason — every math reform I’ve lived through has been royally botched. For instance, the recent Common Core standards were originally drafted by state legislators who had no expertise in education (educators swooped in to triage the damage), and implemented without any funds or plans for producing revised textbooks or retraining teachers. To move things forward, we need successful examples of progressive mathematics education that meet accepted standards. I’d like to see radically different approaches to teaching math that produce twice the results in half the time, with greater engagement and retention. Finally, we come to the most stubborn barrier of all. 5. Social Norms Barrier: the widespread belief that math is a painful subject that must always be taught in the same way. Solution: start a social movement that lets people heal math abuse and reclaim mathematical empowerment. Math anxiety is a generational trauma passed on from one generation to another by parents and teachers who never learned to love math when they were kids. Math empowerment needs to become a society-wide movement. People need to heal from mathematical trauma, reclaim their right to a healthy relationship with mathematics, and know that mathematics can be joyful. This movement is already happening with STEM, but somehow, math gets left out in the excitement of promoting science, technology, and engineering. M may not be as obviously flashy as STE, but you can’t have STE without M. Starting a social movement is a grassroots effort that requires many leaders. I look to other examples of social change movements, like abolishing slavery and allowing women to vote, for inspiration. Within education, society already transformed literacy from a skill possessed by a few scribes to a skill enjoyed by the entire population. We can do the same for math. Don’t be discouraged by the scale of the problem. Instead, be energized by the scale of the opportunity. As Margaret Mead once said, “Never doubt that a small group of thoughtful, committed citizens can change the world; indeed, it's the only thing that ever has.”

Joyful math. It’s a phrase that’s catching on in the math popularization community. The Julia Robinson Math Festival is on a mission to “spread joyful math”. The Global Math Project seeks to “cultivate a world united through the joyful learning of mathematics.” And joyful-math.com declares that math is “a beautiful, interesting, playful, creative, and human-centered sense-making endeavor.” So why the exuberant adjective? Why do we need to proclaim math to be joyful? Simple. Most people are convinced that math is inherently boring and lifeless. And I’m not just talking about people who dislike math. Even teachers who enjoy teaching math, and STEM experts who enjoy sharing the joys of Science, Technology and Engineering, are often hard pressed to muster any enthusiasm for Math itself. My colleague Dave Hendry, with whom I worked at Age of Learning, once produced a series of documentaries where he interviewed top scientists and engineers in many industries to learn how they used mathematics in their job. To his astonishment many of his subjects claimed not to use any math at all. When he pointed out the many ways they did use math, they slowly realized he was right. The reason they didn’t immediately recognize mathematics in their own work was that the math they did at work looked and felt different from the math they did in school. So in their mind, it didn’t count as mathematics. In this article I’m going to explore what joyful math is, what kills math joy, and the challenges that face those of us who want to spread math joy. What Joyful Mathematics Is Joy, of course, means happiness, but it’s a deeper and often more lasting emotion than everyday happiness, which can be fleeting and circumstantial. Think of the joy of meaningful events, like being with friends, graduating from school, or getting your first driver’s license, or listening to a favorite song for the first time. Joyful mathematics is mathematics that brings you deep and lasting happiness, inspires a deep sense of being connected to something larger than yourself, in the same way that great music, art or literature can stir your soul. It can be as simple as having your mind twisted by Möbius Strips, as mind-expanding as number bases other than 10, as ineffable as the mystery of infinity, or as world-changing as the concept of calculus. Or it can be as bluntly mundane as the inescapable fact that 2+3=5. Well that’s nice, but Houston, we have a problem. If you already experience math as joyful, then you know what I mean. But if you don’t experience math as joyful, this description may strike you as alien, hopelessly geeky, and not for you. So I’ll simply report that for me, math is a joyful activity that connects me to my friends, fires up my imagination, stuns me with beauty, and invites me to participate in a boundless tradition of creativity and invention. I’m eager to share math joy with others, and deeply sad that so many people have never experienced it. I don’t expect most people to share my level of enthusiasm, but I do want people to have tasted this sweet fruit. What Kills Math Joy What kills math joy? School. Math, as presented in school, is an unending cycle of memorizing meaningless formulas in order to pass tests, then clearing them from your head to make room for the next lesson. It’s a race to see who can get the right answer first, and an endless marathon that eventually humiliates everyone into feeling stupid. Sure, you have a vague sense that some of it might be useful some day, and you may get some pleasure out of mastering pieces of it, but that’s about it. So what’s going on here? Certainly math is a field where there ARE right answers, so we can’t pretend that precision doesn’t matter. And concepts in math, more than in most other subjects, do rest on a tower of prerequisite concepts, which must be mastered in sequence. But to make a long story short, math as taught in school focuses solely on teaching the mechanics of math. The notation, the algorithms, the formulas. Without ever getting to anything meaningful, beautiful, or awe-inspiring. That’s a very narrow slice of mathematics. And guess what? You can kill interest in any subject by just teaching the mechanics without the meaning. If all you ever learned in English class was spelling and grammar, and you never read any great books (or even knew that books existed), you’d hate language class. If you all you ever learned in Music class was learning the rules of proper notation and music theory, and you never listened to or played any music, you’d hate music class. If all you ever learned in History class was memorizing dates and place names, and you never learned the stories of the people in history, you’d hate history class. Oh wait, that’s exactly how I learned history in school, which is why I hated history class. But I now I love watching history videos on YouTube, because they tell fascinating stories that help me make sense of the world. All great teachers know that their first job is to capture their student’s interest. And so they lead with intriguing stories, dramatic demonstrations, or awesome performances. But the sad truth is that public school teachers here in the United States, and to various degrees in other countries, are under enormous pressure to race through a curriculum that is a mile wide and an inch thin. It’s a race that guarantees that most learning will be lost in the dust. Schools get away with this form of intellectual murder because they have a captive audience. And that has created generations of adults who have been beaten into complacency, convinced them that rote formulas are all there is to mathematics. The truth is that math is something far larger and more interesting, and that everyone is born a mathematician. You can’t be alive and not think mathematically. Spreading Math Joy So what’s a math evangelist to do? The first step is simply to expose the public to joyful math. That’s why James Tanton started the Global Math Project Eoin Gill and Sheila Donegan started the country-wide global maths week in Ireland Cindy Lawrence and Glen Whitney started the Museum of Mathematics in New York City Nancy Blachman started the Julia Robinson Math Festival in California Brianna Donaldson started Math Communities through the American Institute of Mathematics. These institutions are already reaching millions of kids and adults worldwide, introducing them to joyful math art projects, games and activities where everyone can participate and succeed. I encourage everyone to check out these resources. Many are freely available online. They’re fabulous. All of these groups are doing fabulous work. It’s necessary work. But it’s not sufficient. That’s because we are fighting deeply entrenched institutionalized attitudes that are difficult to change. The second step is to anticipate and counter the inevitable backlash to joyful math. Here’s what happens when I bring joyful math into classrooms. First I beg teachers to give me time to play math games with students. I bring in games like Rush Hour, Prime Climb, or Set to class. Everyone plays and has a good time. Teachers and students alike understand that they are learning larger concepts that help them understand other aspects of mathematics. But then everything goes back to “normal”. I’ve just shown them “fun” math. But in their minds, teachers and students wall that off as an aberrant experience, like a once-a-year field trip. It is a nice supplement, but not “real” math. And when it comes down to school policy and choice of textbooks, the pressure to make conventional choices is overwhelming. Only schools that operate outside of public funding have the freedom to operate differently. And even there, pressure from parents usually direct the rivers of curriculum to flow along conventional channels. A few students, parents and teachers do have their world permanently enlarged. A few teachers do go out of their way to implement extracurricular programs like Math Kangaroo and MathCounts. But they are the exception. How do we counter the backlash? I think there are many answers, working both inside and outside of conventional institutions. And we must try them all. Some of the strategies: create replacement curriculum, create memorable joyful mathematics events that take over whole schools, provide ongoing support to teachers and parents, offer free online meetups for kids. My personal belief is that conventional math education is so badly broken that it cannot be incrementally repaired. I personally do not want to engage in curriculum standard wars. Instead, I prefer the carrot over the stick. I advocate establishing an attractive alternative that shows everyone a better way, that brings them joy AND satisfies conventional expectations of what math education can do. Sesame Street enticed a generation of inner city kids to master early learning concepts, not through mandate, but through an effective, entertaining, accessible TV show. We can do the same. Getting to the promised land will require all of us working together. That’s why I’m thrilled to see all of the joyful math groups starting collaborate and share resources. Let’s turn our individual ripples into an unstoppable tidal wave of joy.