With a painterly eye, eight-year-old Jenny Engelke scrutinizes the pair of plastic blocks stacked on her desk and tries to sketch how a box capable of holding the blocks might look if she could unfold it and lay it flat.
Using graph paper to match the size of the cubes, she first draws a pattern with two squares in the base attached to a row of four squares. She cuts it out and tries to fold it around the blocks, but it won't go all the way around.
"Maybe you should modify your design a little bit," urges her teacher, Carmen Curtis.
Jenny returns to her pencil and scissors, adding first one, then two squares to the pattern. Delighted, she runs to show Ms. Curtis that the cubes fit.
While many of her third-grade peers around the country are spending their math classes toiling over columns of addition and subtraction problems, Jenny and her classmates at Country View School in Verona, Wis., are part of a three-year experiment to make math something children can see and feel, what Curtis calls "math with understanding."
Building on children's natural tendency to explore things visually, elementary school teachers in this district south of Madison are working with researchers at the University of Wisconsin-Madison to develop students' spatial skills at the same time they're learning about numbers.
Students encounter problems in measurement, area, volume, and way-finding - the fundamentals of geometry - then come up with ways of defining the problems and explaining the solutions. This differs from classes that use "manipulatives" - hands-on tools designed to aid children's understanding of math - which tend to focus exclusively on numbers, on counting.
Researchers have known for years of the link between spatial reasoning and success in math. Most recently, researchers at Boston College found that better spatial skills among boys had more to do with why they outscored girls on the math portion of the SAT college-entrance exam than did girls' lower self-confidence in math. "The best indicator of doing well in math isn't ability to compute; it's ability to visualize," Curtis says.
Why spatial reasoning?
By encouraging children to come up with their own ways of solving complex visual problems, students learn the value of making conjectures and then finding ways to support them through math, says Richard Lehrer, who developed the curriculum at the UW-Madison's Department of Educational Psychology.
Besides mirroring the way mathematicians and scientists think, such an approach forces students to incorporate many important concepts in math, such as geometry, probability, and proof, which are often not taught until high school, he says.
Opposition to this "layer cake" approach to teaching math is what's driving many states to adopt academic standards aimed at introducing complex reasoning skills at an earlier age.
Mr. Lehrer's work in the primary grades has led to a collaboration with fellow-researcher Leona Schauble and about 40 teachers here to also change the way science is taught. They are finding that students with a firm grounding in spatial skills were better able to create and revise models, the principal way that scientists explain the world.
In a unit last year, where kids created models of the elbow using rubber bands and dowel rods, students not in the program tended to view a good model as one that looked like an elbow, whether it acted like one or not.
But children using the new curriculum revised their models, keeping those features that matched the elbow's function and rejecting those that didn't.
Working on spatial problems also provides parents and teachers a window into how children tackle problems and where they get stymied, Lehrer says, something that isn't always evident if all you know about a child's math ability is that he can't multiply 8 times 9. Teachers can then use those insights to build a road map for teaching more effectively, what researchers refer to as cognitively guided instruction (CGI).
Even more important, educators say, visual exploration in math helps children discover their own route to solving other kinds of problems. Traditional math, for example, teaches kids to multiply fractions by multiplying the numerators together and the denominators together, but with perhaps little understanding of why this works. That is, 1/2 x 1/2 x 1/2 becomes 1 x 1 x 1 over 2 x 2 x 2, or 1/8.
But as children learn to carve up space, they also learn to visualize problems in ways that work best for them. Thus, a child might see the same problem as successive halvings of a rectangle.
Parents raised in an era in which all math problems had only one right answer are understandably skeptical. They fear another "new math," the sometimes confusing reforms schools tried a generation ago. Some critics have likened CGI to the whole-language approach to reading. Teachers are spending too much time on concepts and not enough on multiplication drills, they say.
But parents are quickly won over when they see what their children are doing - and realize they can't solve the problems themselves, Curtis says. Last year, a group of second-graders in the program performed as well or better than college honors students on an exercise to create two-dimensional representations of three-dimensional objects.
"We teach them multiplication tables, too," Curtis says. "But the kinds of kids we're teaching don't give up; they don't say 'It's too hard for me.' ... That stick-to-itiveness, that confidence is something I never saw before."
Parents: Try These With Your Children
Parents can help their children develop an understanding of space and visual mathematics through everyday experiences, beginning by playing with blocks, Legos, and other geometric shapes.
Challenge your children to use the shapes in different ways. Given a certain number of blocks, for example, ask how they might stack them so that they build something tall yet stable.
Here are some other activities you can try:
* Unfold cereal boxes and see how folding the two-dimensional form back up results in a three-dimensional structure.
* Look at buildings or other objects and have children draw the various forms that make them. Ask them to draw the forms from different perspectives (from the side, from above, etc.).
* Mapmaking gives excellent practice in position, direction, and scale, and can provide the basis for understanding coordinate systems. Ask your child to draw a map of the neighborhood, or the route from her bedroom to the kitchen. Have her describe how to get from one point to another that includes some unit of measure (for example, "two steps straight ahead, one step to the right"). Then have someone else try to follow the map.