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). 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.įar 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). “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. They also lead the way into the more structured and even more creative work of noticing, remixing and building mathematical patterns.”įinding 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. Says Droujkova: “Studies have shown that games or free play are efficient ways for children to learn, and they enjoy them. 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. 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.”ĭroujkova, 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. They recall how a single course-or even a single topic, such as fractions-derailed them from the sequential track. Droujkova and her colleagues have noticed that most of the adults they meet have “math grief stories,” as she describes them. It also prevents many others from learning math as efficiently or deeply as they might otherwise. This turns many children off to math from an early age. It’s akin to budding filmmakers learning first about costumes, lighting and other technical aspects, rather than about crafting meaningful stories. They also miss the essential point-that mathematics is fundamentally about patterns and structures, rather than “little manipulations of numbers,” as she puts it. “Calculations kids are forced to do are often so developmentally inappropriate, the experience amounts to torture,” she says. 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. 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. 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. A minority of students then wend their way through geometry, trigonometry and, finally, calculus, which is considered the pinnacle of high-school-level math.īut 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. Then in early adolescence, students are introduced to patterns of numbers and letters, in the entirely new subject of algebra. The computational set expands to include bigger and bigger numbers, and at some point, fractions enter the picture, too. The familiar, hierarchical sequence of math instruction starts with counting, followed by addition and subtraction, then multiplication and division. For example, consider the following four sequences and their different behaviors as n\to \infty (see Figure 2): Since a sequence is a function defined on the positive integers, it makes sense to discuss the limit of the terms as n\to \infty. Calculate the limit of a sequence if it existsĪ fundamental question that arises regarding infinite sequences is the behavior of the terms as n gets larger.
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