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.”