In Mindstorms, Papert makes the case that we find the study of mathematics difficult only because it is foreign and formal. He argues that, if we were immersed in an environment rich in appropriate materials, then we would construct our own mathematical understanding intuitively simply by interacting with the world around us. It would be like learning French while growing up in France rather than trying to learn French through the unnatural process of foreign-language instruction in a classroom.
Papert describes how his early love of automobiles and differential gears helped him make sense of equations later in school. Equations made sense to him because he was able to integrate them into his cognitive model for gears… a model he had constructed through years of play. Learning is easier when we have models that can help us relate to and explain new concepts. Fitting new information into pre-existing cognitive models is a process that Piaget called assimilation.
“Slowly I began to formulate what I still consider the fundamental fact about learning: Anything is easy if you can assimilate it to your collection of models. If you can’t, anything can be painfully difficult. […] What an individual can learn, and how he learns it, depends on what models he has available. This raises, recursively, the question of how he learned those models.”
Papert does not suggest that children should start studying math at an earlier age. Nor does he suggest that all children should play with gears. Gears worked for him because he fell in love with them. Instead, Papert believes the key is creating a world rich in materials so that all children can construct their own models by pursuing their own interests. And not just one or two models, but hundreds of them. In a material-rich environment, learning math would be like learning to walk or talk… personalized, experiential, and without formal instruction. Most of us learn to walk and talk easily and naturally because our environments are littered with the materials we need to construct the necessary models.
So, how do we create a world rich in mathematical materials? In many ways, this is a classic chicken-and-egg dilemma. Which comes first… the materials that enable children to grow up as math natives or the math natives capable of creating the appropriate materials?
“The computer is the Proteus of machines. Its essence is its universality, its power to simulate. Because it can take on a thousand forms and can serve a thousand functions, it can appeal to a thousand tastes. This book is the result of my own attempts over the past decade to turn computers into instruments flexible enough so that many children can each create for themselves something like what the gears were for me.”
Papert suggests that the computer may make up for our material deficit by functioning as a general-purpose material that children can craft into whatever form they need. But he also recognizes that creating new materials and adding them to the landscape is not enough. The lack of mathematical materials in our environment has shaped our culture for hundreds of generations. We have long-held and self-perpetuating beliefs about who can learn math and what learning math looks like. That culture and our beliefs can contaminate and warp any materials we might create… even materials created by children themselves. To counteract this culture, the computer must also function as a carrier of cultural “germs”, insulating and protecting young minds from the current culture and instilling a new one.
Thirty-five years later, the influence of Mindstorms can be seen in three educational trends: (1) the emphasis on personalized and experiential learning; (2) the creation of more mathematical materials to add to the landscape; and (3) the concentration of mathematical materials in after-school centers and maker spaces. But will these trends succeed where Papert’s Proteus of machines has so far failed? I’m not sure. Papert’s vision of change felt more abrupt and disruptive to me. The current path we are on feels more incremental: each generation provides a slightly richer environment so that the next generation can provide an even richer one… until we reach a tipping point. Will our self-perpetuating “mathophobic” culture allow incremental progress or is the current love affair with robotics and 3D printers another fad that will simply blow over? I’ve decided to hedge my bets.