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May 1, 1996
Vol. 53
No. 8

Problem-Based Mathematics—Not Just for the College-Bound

Mathematics is far more interesting to students when they get to do some real thinking. That is one of the premises of the Interactive Mathematics Program.

Picture this: You're on a planning team, consulting to the city manager. Your task is to come up with a reasonable plan for the use of 550 acres of land recently obtained by the city. The acreage includes a recently closed army base, a 300-acre farm, and abandoned mining land.
  • a maximum of 200 acres from the army base and the mining land will be used for recreation, and
  • the amount of army land used for recreation plus the amount of farm land used for development will together total 100 acres.
Not only are you dealing with opposing factions, but with improvement costs ranging from $50 to $2,000 per acre, depending on which parcel of land is involved and how it will be used. You have to satisfy everyone while minimizing the total cost for improvements. To arrive at a reasonable allocation plan will demand careful analysis and attention to detail.
If you were 16 and in a traditional high school math class, you would be enrolled in Algebra II, perhaps doing one system of linear equations after another.
Instead, it's a whole new ball game. Your task is to solve the city's planning problem. This isn't an extra credit assignment. It's a unit called Meadows or Malls? You're going to be working on it for the next six to eight weeks, learning and using algebra, geometry, and matrix operations.
Don't expect to spend your time memorizing facts. There will be no pop quizzes, and no columns of figures to work on and turn in when the bell rings.
Do expect to be working both in a group and on your own. You'll be dealing with numbers and doing matrix operations on a sophisticated calculator. You'll also be working with words—writing and explaining to the rest of the class how your group arrived at its solution to the problem.
Welcome to Year 3 of the Interactive Mathematics Program, also known as IMP.

About the Program

The Interactive Mathematics Program is currently funded by the National Science Foundation to design a comprehensive high school mathematics curriculum. It fulfills the vision of Curriculum and Evaluation Standards for School Mathematics(National Council of Teachers of Mathematics 1989). Since 1989, IMP has been used in classrooms throughout the United States to develop and test the kinds of tasks sought by NCTM, embedding those tasks within a larger vision of a complete mathematics program. It is one of five programs funded by NSF to develop new comprehensive curriculums at the high school level.
IMP's four-year program replaces the traditional Algebra I-Geometry-Algebra II/Trigonometry-Precalculus sequence. The program integrates traditional material with additional topics recommended by the NCTM Standards, such as probability and statistics, and utilizes graphing calculator technology to enhance student understanding. This new curriculum meets college entrance requirements and prepares students to use problem-solving skills at school and on the job.
Teachers need extensive support when they adopt this curriculum, beginning with inservice workshops. The optimum introductory arrangement also includes scheduling one period per day for new IMP teachers to study and to share experiences. Other forms of support include team-teaching the first year of implementation and maintaining a network of telephone contact among teachers.
We have extensively tested the four-year program, and classroom teachers and curriculum writers have continually reviewed and revised it. The program is scheduled to be published by Key Curriculum Press beginning in fall 1996.

Thinking About Mathematics

A major premise of the Interactive Mathematics Program is that most students are capable of thinking about mathematics and understanding complex concepts. This is a change from the philosophy of many traditional programs in which students do mostly rote work. The role of the program teacher also differs from that of a traditional mathematics teacher. In IMP, the emphasis is on guiding students and helping them make connections between key mathematical ideas and concepts, while minimizing time spent lecturing to the class. While students have both the teacher and their peers as resources, each is expected to think and to create in mathematics class.
The Interactive Mathematics Program is a problem-based curriculum. For example, each unit, such as Meadows or Malls?, begins with a motivating problem, too difficult for almost any of the students to solve at first. Students examine this initial situation and then look at similar, perhaps simpler, situations in shorter problems. At every step along the way, students must pose questions, look for patterns, and make connections between the current problem and the mathematics they have learned in previous units. By solving a variety of problems, students deepen their understanding, and they begin to abstract the concepts and refine the techniques needed to apply to the complex original problem.
In order for a problem to build mathematical power in the student, the student needs the opportunity to do genuine thinking about it. In IMP, this means giving students a chance to explore, conjecture, experiment, and reflect on their results. If students fail initially, they return to the problem for more exploration, new conjectures, and more experimentation. The real thinking in problem solving takes place in examining an unfamiliar situation and finding the underlying mathematical ideas. This will occur only if problems are presented without a fixed procedure or solution.

Learning the Basics

  1. Algebra. Solving systems of linear equations for unknowns is an important skill in traditional algebra classes. In IMP, this topic is presented both in Meadows or Malls? and in a second-year unit called Cookies, which deals with maximizing profits from a bakery. Students don't just learn one method, but develop their own approaches in groups and then share ideas with one another. In Meadows or Malls, they also see how to use matrices and the technology of graphing calculators to solve such systems.
  2. Geometry. Basic concepts in traditional geometry include similar triangles and the Pythagorean theorem. In Shadows, a first-year IMP unit, students learn about similar triangles, develop proportion equations to solve similar triangles, and apply the concept of similarity to predict the lengths of shadows. They also extend their knowledge to see how similarity is used as the foundation of trigonometry.In Do Bees Build It Best? students develop the Pythagorean theorem experimentally, prove it algebraically or geometrically, and apply it to see why the hexagonal prism of the bees' honeycomb design is the most efficient regular prism possible. Several units later, students apply the Pythagorean theorem to develop other principles, such as the distance formula in coordinate geometry.
  3. Trigonometry. In traditional programs, sine, cosine, and tangent are introduced in the 11th or 12th grade. In IMP, students begin working with these functions in 9th grade (in Shadows), and learn their value and application over the years. Right triangle trigonometry is used in several units in the second and third year.In a fourth-year unit, High Dive, students extend trigonometry from right triangles to circular functions, defining the trigonometric functions for angles of more than 90 degrees. The problem-solving context in this unit is a circus act in which a performer jumps off a Ferris wheel into a moving tub of water. Not only are students developing and applying general ideas from trigonometry, but they are also learning principles of physics, developing laws for falling objects, and using vectors to find vertical and horizontal elements of velocity.

Word Problems with a Difference

At first glance, a problem like that in Meadows or Malls? resembles the word problems of traditional algebra courses. Both take students out of the realm of pure mathematics, requiring them to see the mathematical structure in a real-life situation. Let's ignore the fact that traditional word problems often strike students as contrived and artificial (two trains going in opposite directions from the station, for example). What's more important is the way they are presented.
Generally, in a traditional class, teachers give students a step-by-step procedure for one such problem and then ask them to practice the procedure on problems of exactly the same type. Each category of problem is narrowly designed to rehearse students on a specific skill. The result is that students do not need to solve a problem in the sense of thinking through a new situation. They merely follow a prescription.
By contrast, the central problems in the Interactive Mathematics Program units are complex, innovative, and challenging. Solving them involves a blend of different concepts and techniques. For instance, as students discover, the land-use problem in Meadows or Malls? involves solving systems of equations, understanding the geometry of how two-dimensional planes intersect in three-dimensional space, and developing and working with abstract notions such as identity element and inverse.

What About Drill and Practice?

Parents and teachers looking at a problem-based curriculum often wonder about skill development. Where, they ask, are the endless lists of problems that we all went through? How will students learn without such repetition?
Although IMP students first encounter an idea or algebraic technique through imaginative and challenging problems, they still need to practice it so that they don't have to rediscover it every time they need to use it. In this program, students are able to do this almost entirely in situations where they also need to think about mathematics. And because they construct ideas in context, instead of just memorizing definitions, mathematical concepts and methods have real meaning to them. They can, therefore, attain an appropriate level of fluency without the amount of practice needed by students for whom a technique is based just on memorization.
For example, one of the fundamental ideas in algebra is the distributive property, which is used to simplify algebraic expressions. IMP students develop this principle in several ways, including area diagrams, numerical examples, and in the context of problem situations. After acquiring a basic idea of the principle, students have some routine practice with it, but they also use it repeatedly in the context of more complex situations. This approach makes the practice of procedures more interesting and more productive.

What About Results?

One test of success will be what happens to graduates of the Interactive Mathematics Program after they leave high school. A major long-term study currently under way—conducted by Norman Webb of the Wisconsin Center for Education Research—will provide important data on the program's effectiveness.
Some things we already know, however. For example, IMP students are staying with mathematics longer than students in traditional programs. Although nationally only 60 percent of all students take more than two years of high school mathematics (National Center for Education Statistics 1993), a significantly higher percentage of IMP students continue beyond two years (Webb et al. 1993). Considering that many students who would have otherwise gone into remedial mathematics classes are in the Interactive Mathematics Program, this finding takes on greater relevance. Other encouraging reports about the program indicate an increase in the number of minority and female students completing three years of college-qualifying mathematics.
One of the features of the program mentioned earlier is the expansion of the curriculum to include new topics. For example, students learn about normal distribution and standard deviation, regression and curve fitting, and matrix algebra for both equation solving and geometric transformations—areas of mathematics that most high school students never see. In spite of studying these extra topics, IMP students are still having success with traditional tests. Several studies have shown that they are doing at least as well as students in traditional mathematics classes on such tests as the SAT, even though IMP students spend far less time than do traditional students on algebra and geometry skills (IMP Evaluation Update 1995).
In another study of high schools with high concentrations of low-income and lower-achieving students, Interactive Mathematics students obtained greater achievement growth over the course of a school year than students in traditional general math and college preparatory courses (White et al. 1995). Student performance was measured by comparing pre- and post-test results on a mathematics achievement test composed of test items from the National Assessment of Educational Progress (NAEP).
We have been collecting anecdotal evidence on the program as well. Juan, now age 19, graduated from an inner-city San Francisco high school and was the first person in his family to attend a university. He says the Interactive Mathematics Program helped prepare him for a college statistics class. "The topics in the textbook were all things I learned in IMP," he said.
After her first semester of a traditional college math course, Theresa, from San Antonio, said, I always ask questions. The others don't. I thought it was because they knew it already, but then after class, they would ask me questions. I realized that they are just scared to ask. They don't know what is going on. Oh yes, I got an A.
Teachers also appreciate the curriculum. Over the years, we have worked with many talented teachers across the country who are now master IMP teacher/trainers in their own communities. Reflecting on her experience, a California teacher (Bussey 1992) wrote: Too many kids are flunking out of Algebra I or getting turned off to mathematics. Algebra I acts as a sieve, keeping only a select few, while filtering out many talented kids. Isn't there more than one way to "learn" mathematics and "do" mathematics?...When I'm in my classroom and witness my students working in groups, debating mathematical principles, and developing their own ideas to solve meaningful problems, that is when I feel most successful. My students have proven to me that they all can learn, that learning can be meaningful and relevant—and fun. What more can one ask of the educational process?

Bussey, J. A. (1992). "Reflections of a Teacher Caught up in the Storm of Reform." Unpublished paper, West High School, Tracy, Calif.

IMP Evaluation Update. (Spring 1995). Emeryville, Calif.: The Interactive Mathematics Program, pp. 1-4.

National Center for Education Statistics. (May 1993). Data Compendium for the NAEP 1992 Mathematics Assessment of the Nation and the States (Report No. 23-ST04, pp. 398-399). Washington, D.C: U.S. Government Printing Office.

National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: NCTM.

Webb, N. L., H. Schoen, and S. D. Whitehurst. (April 1993). Dissemination of Nine Precollege Mathematics Instructional Materials Projects Funded by the National Science Foundation 1981-91. Madison: Wisconsin Center for Education Research, School of Education, University of Wisconsin-Madison.

White, P., A. Gamoran, and J. Smithson. (1995). Math Innovations and Student Achievement in Seven High Schools in California and New York. Madison: Consortium for Policy Research (CPRE) and Wisconsin Center for Education Research, School of Education, University of Wisconsin-Madison.

End Notes

1 Names are pseudonyms.

Lynne Alper has been a contributor to Educational Leadership.

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