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'Modeling' in math
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A startling effect has been achieved by using models to teach mathematics and science to elementary school students in Wisconsin: fifth graders performing at 12th grade levels.
Those fifth graders are in the Verona, Wis., public schools, and they're using models of all sorts to learn the process, not just the results, of science. They're modeling well, for they visualize and interpret data like high school seniors, as measured by questions taken from national assessment tests.
"Models give students room to think on their own two feet." |
Verona teachers in grades K-5 have worked on modeling for four years with researchers from the University of Wisconsin-Madison. Their goal: Help students invent, test and revise models in mathematics and science.
The researchers are Richard Lehrer and Leona Schauble, members of the educational psychology faculty with appointments in the Wisconsin Center for Education Research.
Among the models used by students:
- a compost column to observe rotting.
- a map to display fruit fly density based on available food.
- coin-flipping to represent birds' choices in foraging for two kinds of food.
- graphs and other visual displays to represent variability and rate of change as plants and insects grow.
- triangles to model the relationship between the height of objects and the length of the shadows they cast.
"The students are visualizing their thinking through models," says Lehrer. "Historically, mathematics started with geometry, which is very visual, but in the past two centuries, mathematics has increasingly emphasized abstract algebra and related forms of symbolization."
The Verona project runs counter to that trend. By using more visual, observation-based mathematics, students are given the tools they need to model.
Despite the potential of computers for new forms of visual mathematics, says Schauble, "Students don't typically emphasize modeling in their science study until they reach undergraduate college classes." Before that point, they're usually told what scientists have discovered, not how they have made those discoveries.
"We are not abstractly dropping math into children's heads," says Lehrer. "That approach is why a lot of people don't understand math. Instead, we help students learn math by building on their experience."
Second graders, for example, slide toy cars that they construct out of Lego building blocks down inclines of different steepness. "The students explain variations in the speed of the cars by considering things like friction and the weights of the cars, and by mathematically representing the steepness of the inclines," says Schauble.
That modeling exercise and others, says Lehrer, "encourage students to think about regularities and patterns that describe and organize what they observe around them. It's what scientists do."
The researchers and teachers are helping students see, through the peephole called mathematics, what the scientific method means in action, not just in a book. What they are aiming at was captured by Ian Stewart and Martin Golubitsky in their book "Fearful Symmetry: Is God a Geometer?"
"Scientists use mathematics to build mental universes," the authors say.
"They write down mathematical descriptions - models - that capture essential fragments of how they think the world behaves. Then they analyze their consequences. This is called 'theory.' They test their theories against observations: this is called 'experiment.' Depending on the result, they may modify the mathematical model and repeat the cycle until theory and experiment agree."
One of the pedagogical blessings of models is how they can help teachers avoid the trap of "cued elaboration," a form of regurgitation. That's when a teacher asks the class a question with an expected answer in mind and continues to shape the question until someone gives the "right" answer.
Models, in contrast, give students room to think on their own two feet. For instance, when the compost column was used for observation of rotting, students were interested in mold but had a basic misconception: The mold couldn't be alive.
Rather than simply telling the students they were wrong and moving on, the teacher designed a way of enabling students to observe more mold, this time on bread. When they examined it under a microscope, they concluded that the mold was indeed alive. This modeling allowed students to expand their web of inquiry, just as a scientist does.
That web was also expanded when groups of students were given materials from a hardware store - dowels, springs, Styrofoam balls, rubber bands - and asked to build a model of a human elbow.
Their first efforts produced models that looked like an elbow but bent a full 360 degrees. Their revisions - not the teacher's - led to more functional models, which enabled them to learn the relationship of such variables as (a) the weight of a load being lifted by the elbow's action, and (b) the distance it's lifted.
"By generalizing about the relationship of those variables," says Schauble, "students worked with equations in ways that look very much like algebra."
The model elbows allowed them to discover these mathematical ideas, instead of reading about them in a book.
"Textbooks hide models," says Lehrer. "They just list the facts of science, which are the outcomes of models."
That's why it's important that modeling transcends mere fact listing, says Schauble: "Making knowledge, which students are doing with models, is far different from just consuming it."
- Jeff Iseminger, (608) 262-8287
Posted April 7, 1999
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