Students in classrooms where mathematical thinking is encouraged from a very young age learn how to estimate and predict.  (“How many pencils do you think there are in the whole school?  Is there a way we could know for sure without counting?”)  They acquire basic skills in the process of solving meaningful problems -- often with their peers.  They may use calculators, as adults often do, so that they can tackle more challenging and engaging problems than would be possible if they had to direct their energy to computation.  In contrast to a classroom whose main activities are listening to the teacher and filling out worksheets, such a learning environment is distinguished by students “sitting in groups, discussing ideas, doing experiments, making diagrams, using concrete objects to test their conjectures, following blind alleys, and now and then experiencing the satisfaction of discovering something they did not know before.”[17]

When traditionalists insist that it’s most important for kids to “know their math facts,” we might respond not only by challenging those priorities but by asking what is meant by know.  The key question is whether understanding is passively absorbed or actively constructed.  In the latter case, math actually becomes a creative activity.

By thinking through the possibilities, students come up with their own ways of finding solutions.  They have to invent their own procedures.  What that means in practice is as straightforward as it is counterintuitive:  teachers generally refrain from showing their classes how to do problems.