Jane Smith was in disarray. Her son had enrolled in a math program where educators promised to teach in a new, more effective way. It wasn’t long before she realized that all was not well. Instead of learning rules and formulas, her son and his classmates spent most of their time trying to invent their own methods to solve problems. “He was very frustrated,” says Smith. “I’d say, ‘Look in the book, it will explain.’ He’d say, ‘Mom, there is no book!” In some schools, students spend much time engaged in games and activities that are designed to help them discover math concepts themselves.  They often work in groups using hands-on materials. They propose different schemes to find solutions, discuss the merits of alternative approaches, and test their methods through trial and error. The idea behind such “discovery learning” is that if you arrive at knowledge for yourself, it will mean more to you and will last longer in your brain.  The ardor for this approach is fueled by a deep conviction that learning flourishes best when children are allowed to make their own sense out of the world, unhampered by the straitjacket of externally dictated rules and formulas imposed...

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