5-Year-Olds Can Learn Calculus

The familiar, hierarchical sequence of math instruction starts with counting, followed by addition and subtraction, then multiplication and division. The computational set expands to include bigger and bigger numbers, and at some point, fractions enter the picture, too. Then in early adolescence, students are introduced to patterns of numbers and letters, in the entirely new subject of algebra. A minority of students then wend their way through geometry, trigonometry and, finally, calculus, which is considered the pinnacle of high-school-level math.

But this progression actually “has nothing to do with how people think, how children grow and learn, or how mathematics is built,” says pioneering math educator and curriculum designer Maria Droujkova. She echoes a number of voices from around the world that want to revolutionize the way math is taught, bringing it more in line with these principles.

The current sequence is merely an entrenched historical accident that strips much of the fun out of what she describes as the “playful universe” of mathematics, with its more than 60 top-level disciplines, and its manifestations in everything from weaving to building, nature, music and art. Worse, the standard curriculum starts with arithmetic, which Droujkova says is much harder for young children than playful activities based on supposedly more advanced fields of mathematics.

“Calculations kids are forced to do are often so developmentally inappropriate, the experience amounts to torture,” she says. They also miss the essential point—that mathematics is fundamentally about patterns and structures, rather than “little manipulations of numbers,” as she puts it. It’s akin to budding filmmakers learning first about costumes, lighting and other technical aspects, rather than about crafting meaningful stories.

This turns many children off to math from an early age. It also prevents many others from learning math as efficiently or deeply as they might otherwise. Droujkova and her colleagues have noticed that most of the adults they meet have “math grief stories,” as she describes them. They recall how a single course—or even a single topic, such as fractions—derailed them from the sequential track. She herself has watched more than a few grown-ups “burst out crying during interviews, reliving the anxieties and lost hopes of their young selves.”

Droujkova, who earned her PhD in math education in the United States after immigrating here from Ukraine, advocates a more holistic approach she calls “natural math,” which she teaches to children as young as toddlers, and their parents. This approach, covered in the book she co-authored with Yelena McManaman, “Moebius Noodles: Adventurous math for the playground crowd,” hinges on harnessing students’ powerful and surprisingly productive instincts for playful exploration to guide them on a personal journey through the subject. Says Droujkova: “Studies [e.g.,  this one, and many others referenced in this symposium] have shown that games or free play are efficient ways for children to learn, and they enjoy them. They also lead the way into the more structured and even more creative work of noticing, remixing and building mathematical patterns.”

Finding an appropriate path hinges on appreciating an often-overlooked fact—that “the complexity of the idea and the difficulty of doing it are separate, independent dimensions,” she says. “Unfortunately a lot of what little children are offered is simple but hard—primitive ideas that are hard for humans to implement,” because they readily tax the limits of working memory, attention, precision and other cognitive functions. Examples of activities that fall into the “simple but hard” quadrant: Building a trench with a spoon (a military punishment that involves many small, repetitive tasks, akin to doing 100 two-digit addition problems on a typical worksheet, as Droujkova points out), or memorizing multiplication tables as individual facts rather than patterns.

Far better, she says, to start by creating rich and social mathematical experiences that are complex (allowing them to be taken in many different directions) yet easy (making them conducive to immediate play). Activities that fall into this quadrant: building a house with LEGO blocks, doing origami or snowflake cut-outs, or using a pretend “function box” that transforms objects (and can also be used in combination with a second machine to compose functions, or backwards to invert a function, and so on).

“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.”

Children need to be exposed to a variety of math styles to find the one that suits them best. But they also need to see meaningful (to them) people doing meaningful things with math and enjoying the experience. Math circles, where people help one another, are growing fast and are one way to achieve this. Math know-how (activities and examples) “must come with communities of practice that help newbies make sense of it,” Droujkova says. “One does not work without the other.”

Regardless, if learning is to be as efficient and deep as possible, it’s essential that it be done freely. That means giving children a voice in which activities to participate, for how long, and also the level of mastery they want to achieve. (“This is the biggest clash with traditional curriculum development,” Droujkova notes.)

Adults must be prepared for those times when a child would rather be doing something other than the planned activity. Says Droujkova: “The role of adults is to inspire, by saying things like, ‘Ooh, what a complex shape—have you noticed the curve is made out of straight lines?’ Provide math connections with whatever kids are doing. This is hard to do—it requires both pedagogical and math concept knowledge, but it can be learned. And everyone can easily give general support: ‘How very interesting, I will investigate more.’ You can then look online, or ask on a math circle forum, to find out what it means mathematically.”

It’s also helpful to have a variety of interesting materials on hand and to be okay with the idea of kids taking breaks as needed. Droujkova has noticed that in most groups, there are one or two kids do something else, while the rest do the main activity. (The non-participants still absorb a surprising amount, she adds.)

Pushback has come primarily from two very different (and usually opposing) camps. One is the “let kids be kids” cohort, which worries that legitimizing the idea of involving toddlers with algebra and calculus will tempt Tiger Mom types to push their kids into formal abstractions in these subjects at ever younger ages, even though that would completely miss the point. Other critics fall into the “back to basics” camp, which contends that all this play will prevent kids from becoming fluid in traditional calculation skills.

Droujkova views these criticisms as indicative of something much bigger: “They reflect rather deep chasms between different philosophies of education, or more broadly, differences in the futures we pave for kids. When we assign a lot of similar exercises, we picture kids in situations that require industrial precision.” Giving children logic puzzles or open projects, on the other hand, indicates aspirations of them growing up to become explorers or designers. “It does not work that directly,” she concedes, “but these beliefs dictate what mathematics education the grown-ups select or make for the kids.”

There are also some who worry about whether this approach is practical for disenfranchised populations. Droujkova says that it can be led by any “somewhat literate” adult; the key is to have the right support network in place. She and her colleagues are striving to empower local networks and enhance accessibility on all fronts: mathematical, cultural and financial. They have made their materials and courses open under Creative Commons, and designed activities that require only readily available materials.

“The know-how about making community-centered, open learning available to disenfranchised populations is growing,” Droujkova notes, citing experiments by Sugata Mitra and Dave Eggers. Online hubs can connect like-minded community members, and online courses and support are available to parents, teachers and teenagers who want to lead local groups.

Droujkova says one of the biggest challenges has been the mindsets of the grown-ups. Parents are tempted to replay their "bad old days" of math instruction with their kids, she says. With these calculus and algebra games, though, “parents say they get a fresh start. … They can experience the joy of mathematical play anew, like babies in a new world.”