“You can take any branch of mathematics and find things that are both complex and easy in it,” Droujkova says. “My quest, with several colleagues around the world, is to take the treasure of mathematics and find the accessible ways into all of it.”

She started with algebra and calculus, because they’re “pattern-drafter tools, designer tools, maker tools—they support cool free play.” So “Moebius Noodles” includes activities such as making fractals (to foster an appreciation of the ideas of recursion and infinitesimals) and “mirror books” (mirrors that are taped to each other like the covers of a book and can be angled in different ways around an object to introduce the concepts of infinity and transformations). (Another book in this genre is “Calculus by and for Young People,” by Don Cohen.)

“It’s not the subject of calculus as formally taught in college,” Droujkova notes. “But before we get there, we want to have hands-on, grounded, metaphoric play. At the free play level, you are learning in a very fundamental way—you really own your concept, mentally, physically, emotionally, culturally.” This approach “gives you deep roots, so the canopy of the high abstraction does not wither. What is learned without play is qualitatively different. It helps with test taking and mundane exercises, but it does nothing for logical thinking and problem solving. These things are separate, and you can’t get here from there.”

She doesn’t expect children to be able to solve formal equations at age five, but that’s okay. “There are levels of understanding,” she says. “You don’t want to shackle people into a formal understanding too early.” After the informal level comes the level where students discuss ideas and notice patterns. Then comes the formal level, where students can use abstract words, graphs, and formulas. But ideally, a playful aspect is retained along the entire journey. “This is what mathematicians do—they play with abstract ideas, but they still play.”

Droujkova notes that natural math—whose slogan is “make math your own, to make your own math”—is essentially a “freedom movement.” She explains: “We work toward freedom at many levels—the free play of little kids, the agency of families and local groups in organizing math activities, the autonomy of artists and makers, and even liberty for us curriculum designers. … No single piece of mathematics is right for everyone. People are different, and people need to approach mathematics differently.”

For example, in a group learning about the properties of rhombuses, an artistically inclined person might prefer to draw a rhombus, a programmer might code one, a philosopher might discuss the essence of rhombi, and an origami master might fold a paper rhombus.

Nor does everyone need to learn any particular piece of mathematics, aside from what’s essential to function in his or her culture. Many people live to a ripe and happy old age without knowing calculus, for example. “At the same time, the world would be better off with a higher literacy for mathematics, and humanity as a whole needs advanced math to make it through the next 100 years, because there are pretty complex problems we’re facing.”