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Introduction and Implementation Strategies for the Interactive Mathematics Program: A Guide for Teacher-Leaders and Administrators
How the IMP® Curriculum Is Different
The IMP curriculum looks and feels dramatically different from the programs that have existed in most schools for many years.
The IMP curriculum is problem-centered.
Units of the IMP curriculum generally begin with a central problem or theme. Students explore and solve that problem over the course of the unit. How long does it take for a 30-foot pendulum to complete twelve periods? How can you predict the length of a shadow? What is the best design for a honeycomb? What is the probability that the baseball team currently in the lead will win the championship, given the current record of its closest rival? What's the best way to resolve the conflicts of a particular land-use situation within the constraints of competing political and social forces? When should a diver be released from a rotating Ferris wheel in order to land safely in a moving tub of water?
These are a sampling of IMP's central unit problems. Some are based in real-world situations; others, in more fanciful notions. These problems are generally too complex for students to solve initially. As teachers guide them through a variety of smaller problems, students develop the mathematical ideas and techniques they need in order to solve the central problem. It is common for challenges in later units to build on earlier units, requiring students to apply what they have previously learned in ever more sophisticated and complex ways.
Because the IMP curriculum is problem-based, students get a rich experience of the way mathematics is actually used. Teachers relate how gratifying it is that they never hear IMP students ask, "When are we ever going to use this?" Appendix A: A Unit-by-Unit Summary of the IMP Curriculum provides a brief description of each unit and shows the organization of the curriculum units by year.
Appendix B: Concepts and Skills for the IMP Curriculum lists the major topics covered in the IMP curriculum and correlates them to the NCTM Standards.
The IMP curriculum is integrated.
Solving a particular unit problem often requires concepts from several branches of mathematics. IMP's problem-based approach helps students to see how important ideas are related to each other. Mathematical concepts are integrated throughout all four years of the curriculum, instead of being isolated from one another. Therefore, it is inaccurate to label any of the IMP courses with such familiar titles as Algebra, Geometry, or Trigonometry. The integrated character of the IMP curriculum also includes the use of other subject areas - such as history, physics, and literature - as settings for the mathematical content.
Concepts and Skills for the IMP Curriculum gives an outline of mathematical topics in the IMP curriculum, including both the new and the traditional content, and shows where each topic can be found. It is Appendix B.
The IMP curriculum expands the content scope of high school mathematics.
The IMP curriculum follows the recommendations of NCTM's Standards to include new topics in the high school curriculum, such as probability, statistical reasoning, and discrete mathematics. These topics are essential in the mathematics education of students, and reflect the demands of present-day society on its citizens and its workforce. The curriculum also makes recommended adjustments within the traditional areas of mathematics, by increasing attention to certain topics and decreasing attention to others.
The IMP curriculum focuses on developing understanding.
The IMP curriculum is designed to help students develop an in-depth understanding of mathematical concepts and techniques and of the ways to apply them. The curriculum challenges students to explore open-ended situations actively, in a way that resembles the inquiry method used by mathematicians and scientists in their work. Students routinely experiment with examples, look for and articulate patterns, make, test and prove conjectures, and make connections among mathematical ideas.
The IMP Year 1 unit The Pit and the Pendulum provides a concrete example of what this can mean. In this unit, students are presented with the problem of whether the prisoner in Edgar Allan Poe's classic story would have enough time to escape the blade on a 30-foot pendulum that will reach him in only 12 more swings. To resolve this question, students construct pendulums and conduct experiments to find out what variables determine the length of the period of a pendulum and what the relationship is between the period and these variables.
Students are introduced through experiments and examples to the related concepts of the normal distribution and the standard deviation. Though college students usually learn about these statistics concepts as formulas to be memorized, for IMP students these concepts are tools to help them determine whether a change in a given variable really does affect a pendulum's period. Thus, they have a compelling reason to really understand the concepts, because they are using them in the context of a meaningful problem.
Once they discover what determines the period, they analyze data and use graphing calculators to find a function that fits their data closely. Finally, after deriving a theoretical answer to the problem, students build a 30-foot pendulum to test their theory. The first time teachers use this unit, they are amazed at the accuracy of their students' predictions.
The IMP curriculum can serve students of varied mathematical backgrounds in heterogeneous classrooms.
A curriculum built around complex, open-ended problems can be explored at many levels of sophistication. The central problems in IMP units have a richness that will challenge the brightest student and a concreteness that allows all students to do meaningful mathematical work.
The curriculum also includes a varied collection of supplemental problems for each unit. These supplemental problems give teachers the flexibility to meet individual student needs. Extensions can be used for students who quickly understand a concept and want more challenge. These activities require students to take ideas from the IMP curriculum farther than the basic unit does. Reinforcements can help students who need additional experience in order to better understand and synthesize what they have encountered in the unit.
Because each unit of the IMP curriculum allows for many points of entry and many levels of achievement, this curriculum is ideal for schools that want to create heterogeneous classrooms.
The IMP curriculum includes long-term, open-ended investigations.
Each unit contains several Problems of the Week (POWs). These are open-ended problems, often mathematical classics, that cannot be solved easily in a short period of time. Though POWs are embedded within the units, the mathematics of these problems is usually independent of the unit problem.
POWs help students develop thoughtfulness and perseverance, and force them to focus on their own thinking processes. Students must explain and illustrate their strategies and solutions, and must justify their reasoning in clearly written reports. Over the course of the school year, each student will make at least one oral presentation to the class on a POW.