Some 10-year-olds are now performing as well as 17-year-olds, according to researchers from the Wisconsin Centre for Education Research and the National Mathematics and Science Centre.
One of the researchers, Richard Lehrer, said that the new system of teaching had enabled elementary-school pupils to visualise and interpret data as well as high-school seniors.
"Historically, mathematics started with geometry, which is very visual, but in the past two centuries, mathematics has increasingly emphasised abstract algebra and related forms of symbolisation," he said.
Teachers in Verona, Wisconsin, have been working with researchers from the University of Wisconsin-Madison for four years. The aim has been to help children invent, test and revise models in maths and science rather than depend on textbooks.
Among the models they have used were a compost container that enabled them to observe rotting, coin-flipping to represent birds' choices in foraging for two kinds of food, and triangles to demonstrate the relationship between the height of objects and the length of the shadows they cast.
Lehrer's colleague, Leona Schauble, explained: "We are not abstractly dropping maths into children's heads. That approach is why a lot of people don't understand maths. Instead, we help students learn maths by building on their experience."
Seven-year-olds have been encouraged to build Lego cars and then slide them down inclines of different steepness. "The children explain variations in the speed of the cars by considering things like friction and the weights of cars, and by mathematically representing the steepness of the inclines," Leona Schauble said.
The compost container proved useful in a science lesson because it helped to puncture their belief that mould was an inert substance. And materials from a hardware store - dowels, springs, Styrofoam balls and rubber bands - were used in a lesson about elbows.
The pupils' first efforts produced models that looked like an elbow but bent 360 degrees. Their subsequent models enabled them to learn the relationship of such variables as the weight of a load being lifted by the elbow's action and the distance it is lifted.
"By generalising about the relationship of these variables children worked with equations in ways that look very much like algebra," said Leona Schauble. "Making knowledge, which are doing with models is far different from just consuming it."
Richard Lehrer can be contacted on 001-608-262-0836.