Proof Positive

After figuring out why many kids hate math, longtime educators Bob and Ellen Kaplan created an after-school program in which children as young as 5 formulate equations of their own—and actually enjoy doing it.

The unadorned fourth-floor classroom in Harvard University’s Science Center was an appropriate foil for the theories being discussed within it. On a chilly November evening as gray as the room itself, a rounded older man with a bushy white mustache conducted a math lesson on the concept of infinity.

“What we’re trying to figure out is if there’s a one-to-one correspondence between points on a line and points on a plane,” he told the 17 students, dragging chalk along the blackboard to create a square with dotted lines.

The students had already learned that a line consists of an infinite number of points; pick any two numbers (0 and 1, for example), and there’s always another between them (say, 0.5). The same is true for a plane, except each point is represented by two numbers, commonly called x and y coordinates, that relate to two perpendicular...