Changing how kids learn math

5-year-olds and calculus. [Link]

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.”