As the number of children who are home-schooled grows — an estimated 1.1 million nationwide — some parents like Ms. Walter are opting for what is perhaps the most extreme application of the movement’s ideas. They are “unschooling” their children, a philosophy that is broadly defined by its rejection of the basic foundations of conventional education, including not only the schoolhouse but also classes, curriculums and textbooks.

In some ways it is as ancient a pedagogy as time itself, and in its modern American incarnation, is among the oldest home-schooling methods. But it is also the most elusive, a cause of growing concern among some education officials and social scientists.

“It is not clear to me how they will transition to a structured world and meet the most basic requirements for reading, writing and math,” said Luis Huerta, a professor of public policy and education at Teachers College of Columbia University, whose national research includes a focus on home schooling.

There is scant data on the educational results of unschooling, and little knowledge about whether the thousands of unschooled children fare better or worse than regularly schooled students. There is not even reliable data on how many people are unschooling, though many experts suggest the number is growing.

Here in Chicago, a group called the Northside Unschoolers has 100 families registered on its online list. There are similar organizations coast to coast, including the San Francisco Bay Unschooling Network, Unschoolers Unlimited in Guilford, Conn., and the Unschoolers of the Ozarks, serving Missouri, Oklahoma and Arkansas, although accurate figures for the number of families they serve are hard to come by. Adherents say the rigidity of school-type settings and teacher-led instruction tend to stifle children’s natural curiosity, setting them up for life without a true love of learning.