Yours is not to reason why,But to invert and multiply.
This recipe for dividing fractions still haunts Charles Allan. Now a mathematics education consultant with the Michigan Department of Education, Allan remembers learning this procedure as a child, but doesn't recall being able to make sense of it. And when he asked his teacher for the logic behind the algorithm, he was told, "That's just the way it's done. If you want to pass, follow these procedures."
Eager to pass, Allan completed his sheet of fraction division problems but could not tell whether his answers were right or wrong or even close; only the teacher knew for sure. "We were not encouraged to ask, `Does this answer make sense?'" he says, admitting that neither he nor his peers really understood what they were doing.
"Our battles in middle school math today revolve around this very issue," states Allan. "The controversy is over how to engage students so that math builds understanding, so that it is meaningful."
Meaningful Mathematics
The fact that math is not meaningful to many of today's U.S. middle schoolers was reflected recently by the Third International Mathematics and Science Study (TIMSS), the most extensive and far-reaching comparative study of mathematics and science ever attempted. "The results indicate that U.S. mathematics education in the middle grades is particularly troubled," says William Schmidt, executive director of the U.S. National Research Center for TIMSS at Michigan State University. Schmidt points out that while U.S. 4th graders scored somewhat above the international average, U.S. 8th graders scored below it—a clear signal, he says, that the problem with U.S. middle school math is not the students but the system. He cites low expectations for student achievement; insufficient professional development; and shallow, repetitive curriculums as symptoms of "dysfunctional" math education systems.
But Glenda Lappan, president of the National Council of Teachers of Mathematics and a professor in the department of mathematics at Michigan State University, has responded to the situation with optimism. "Rather than looking at the horror of the horse race data," as she calls certain TIMSS findings, "educators should make sense of what TIMSS tells us so that we can make a difference in what we do in the classroom." Because TIMSS includes information about those systems in which student achievement is high, "the findings are very helpful," says Lappan.
One especially helpful TIMSS finding is that countries with higher math achievement explore fewer topics each year—but in greater depth. By contrast, middle school math curriculums in the United States "tend to include all the topics every year and just keep repeating them, compelling teachers to rush students through lessons in a superficial manner," notes Schmidt. He cites a 75 percent overlap in math content annually between 4th and 8th grades, resulting in only 25 percent of content that is new each year. "In other countries it is the reverse," he observes.
Experts agree that reducing repetition through curriculum mapping permits students to study fewer topics each year and raises achievement. "Kids are bored to tears with the same material repeated year after year," insists Lappan. She maintains that children in grades 6, 7, and 8 are going through developmental changes that allow them to take on challenges in mathematics that are "far beyond what we have expected of kids in the past." Besides that, says Lappan, "kids this age thrive on being able to do things well."
Lamenting a cultural trend that has "taken all the challenge out of what we're asking of American kids," Lappan encourages educators to beef up math curriculums by giving students time to wrestle with important mathematical concepts. Teachers should only present from two to four fundamental mathematical ideas within a six-week grading period, but study them in depth, she advises. "And if you can get parents to understand your objectives for that time span," she says, "you'll have a lot more support from home." As Lappan also insists that an effective math curriculum should require from 20 to 30 minutes of homework every day, she sees parent support as essential.
Professional Development
Beefing up math curriculums doesn't happen without professional development, say experts, and Lappan observes that "the kind of mathematics we're asking middle school teachers to teach is quite sophisticated." She indicates that an appropriate curriculum for middle grades includes the study of rational numbers, algebra, geometry, measurement, and statistics.
"But many middle school teachers don't know a lot about statistics or geometry," she says, noting that many states do not require content area certification for middle school teachers. Because many middle school teachers received their initial training for the elementary level, it is not uncommon to find limited mathematical expertise among them, says Lappan. "So we have to acknowledge this situation and help teachers become smarter about the subject matter and about how to engage kids with it," she says.
Increasing Student Engagement
"Middle school students get bored quickly and don't do well with rote procedures," observes Susan Birnie, chair of the math department at George Washington Middle School in Alexandria, Va. To help her Math Eight students learn and retain geometric concepts such as surface area, Birnie engages them in a parachute contest. "The idea is to make a parachute that has the longest hang time and the least surface area," she explains.
Using plastic from trash bags, string, and a paperclip as the skydiver, her students experiment with shapes and dimensions, calculating surface areas of rectangles, triangles, circles, and even heart shapes. Some make double-decker parachutes, and some design their parachutes with the string off to one side so that their chute will descend at an angle, hoping this will result in a longer hang time.
With stopwatch in hand, Birnie and her 8th graders launch their parachutes, taking careful notes on each descent. Then, they learn to graph the results on a coordinate grid (x = hang time, and y = surface area). Soon they discover which portion of the grid is the winning region and have fun while learning concepts in both geometry and graphing, says Birnie.
Another way to increase student engagement is to pose math problems in authentic contexts, says Allan. For example, he recommends having students analyze the per-unit cost of breakfast cereals in a supermarket. "In the course of doing this," he says, "they're doing an awful lot of computational work, so they're learning procedures. But they're also learning to analyze data in the context of something that is meaningful to them." Such an active approach helps students take a greater interest in mathematics and helps them develop a more intuitive sense of how numbers work, he says.
A Coherent Curriculum
Although fun ideas and activities engage student interest, there's a potential downside when teachers pick and choose from a variety of resources, warns Barbara Reys, professor of mathematics education at the University of Missouri, Columbia. "When kids study isolated ideas every day that don't connect to previously learned math concepts," she says, "this is what we call an incoherent curriculum."
A coherent curriculum, explains Reys, means that the sequence of instruction relates to previously studied concepts. In a coherent curriculum, students can develop a web of related ideas and understand how they connect, she argues. "The teacher's job is to continually remind kids how the current idea connects to ideas studied earlier," she says, "because the connections aren't always obvious to the learners when they're first experiencing them."
A tool that helps students grasp the coherence of math lessons is the calculator. Nancy Berkas, a 30-year veteran of middle school math teaching and a consultant with the Math and Science Consortium of the North Central Regional Education Lab (NCREL), remarks that "technology has made some mathematics obsolete, some mathematics do-able, and some mathematics possible."
Berkas recommends using graphing calculators to help students more deeply understand math concepts such as the graph of a parabola (y = x 2). "When students graph y = x 2-1, and then -2, and then -3, they immediately see what those numbers do to the U shape of the graph," she says. "If you try to get 7th graders to do those four graphs by hand, you'll spend the whole 45-minute period. In the meantime, your students have lost track of the point of the lesson." Whether they know their times tables or not, all students can quickly enter an equation on a calculator, and as they explore mathematical ideas this way, she observes, they can more readily connect concepts.
A Functional System
To make middle school mathematics education coherent—to transform it from a dysfunctional to a functional system—requires commitments from many constituents, says Lappan. "It takes more than just a new set of textbooks. If there's one thing I wish more people would understand," she says, "it is that a classroom of kids and a teacher operate within a system." High expectations coupled with a developmental understanding of middle schoolers, professional development, coherent curriculums, parent support, effective teaching—all must be part of the reform process, argues Lappan. "The whole system has to function together," she says, "and when it does, it produces better learning for kids."
The Show-Me Project
The Show-Me Project
Middle school math educators who want to help more of their students make sense of math should explore the Show-Me Project (http://www.showmecenter.missouri.edu), advises Barbara Reys, project director. Funded by the National Science Foundation and headquartered at the University of Missouri, the Show-Me Project helps middle grades educators implement math reforms based on the standards of the National Council of Teachers of Mathematics (NCTM).
Along with providing professional development opportunities, the Show-Me Project features five middle school math curriculums that are "quite different from traditional math textbooks," says Reys. Rather than offering daily lessons with practice exercises, these curriculum modules offer math explorations that require class discussions and several days to complete, she explains.
Reys cautions against adopting the materials without an accompanying professional development plan. "These curriculums require a high degree of mathematical knowledge," admits Reys, "but the teachers who have been using them are very excited," she says. "They see that kids are learning more, and both teachers and kids find math more meaningful."
An Action Plan
An Action Plan
Every Child Mathematically Proficient: An Action Plan is a helpful, 24-page booklet that outlines strategies for achieving a coherent K–12 mathematics curriculum. Published by the Learning First Alliance, of which ASCD is a member, the action plan is available on the Web at http://www.learningfirst.org.