The Interactive Mathematics Program (IMP, 1994) is a project designed
to implement change in the high school mathematics curriculum. The paper
will describe how the project began, what it intends to do, and what pedagogical
problems it addresses. A specific unit will be analyzed as far as its theme,
structure, and mathematical content. The advantages the IMP offers teachers
are discussed throughout the first several pages. The paper will conclude
by discussing some of the disadvantages of the project as well as some of
the results of evaluations on the project.
Since the early 1980s, it has become evident that the standard, American, high school, mathematics curriculum (Algebra I, Geometry, Algebra II, and Trigonometry) is in need of change. In 1989, NCTM published its Curriculum and Evaluation Standards to provide a guideline for the changes. Four curriculum reformers from California; Diane Resek, Dan Fendel, Lynne Alper, and Sherry Fraser, set out to design a curriculum that would "embody the vision of the Standards," and replace the standard curriculum with a four-year, integrated, problem-based curriculum. The curriculum would contain standard high school mathematical topics as well as new topics such as statistics, probability and discrete mathematics. The reformers believed the curriculum should enhance problem solving, reasoning and communication skills in high school students, so that they may be prepared for an active role in society. The curriculum should also alleviate many of the pedagogical problems teachers encounter. The Interactive Mathematics Program was the result of the reformer's efforts.
The pedagogical problems and curricular concerns that this project addresses are numerous and closely tied to the project's intentions. Many educational resources may not consider pedagogical problems, and so the IMP (1994) offers several attributes that other curriculum sources may not. One pedagogical concern to be addressed is the lack of group work in mathematics classes. One of the underlying beliefs of the project is cooperative environments promote student learning and develop necessary social skills. The IMP uses in-class activities intended for groups of two to four students and requires the groups to be switched regularly so that all students may to work together. Another problem is undeveloped mathematical communication skills. IMP has assignments and in-class group work designed to promote such skills. For example, the homework exercises often ask for a student to explain an answer to the class or to write an extended answer to a question. The teacher's manual also stresses that students be allowed to do most of the discussing, while the teacher does most of the listening. In essence, the role of the teacher needs to change from a giver of information to a facilitator, and the well-developed IMP activities should be used since they promote learning through discovery and experimentation. The IMP also addresses the lack of opportunity for all students in the standard four year curriculum. The IMP classes are a mix of race, gender, socio-economic status and ability, so that all students are exposed to the same opportunities. Another problem dealt with by the IMP is the lack of attention paid to problem solving and mathematical reasoning. The IMP curriculum is all based on problem solving and reasoning and not topics. This is clearly reflected in the structure of the project's units. Each unit is based on a central problem that requires several mathematical topics to solve it. Students develop the necessary skills and concepts required to solve the unit problem by working various smaller problems. Students are not asked to learn a skill if it is not needed for the unit problem, and practice exercises for skills are scarce. In fact, the IMP curriculum has almost no skill or manipulation problems or exercises. Another pedagogical problem is a lack of appropriate assessment techniques. The IMP believes students should be assessed in a number of different ways other than pencil-and-paper tests. Open ended questions, group discussions, student portfolios, oral presentations, and self assessment are all used to determine student learning. Also, the IMP considers assessment to be a crucial way to help students understand what aspects of mathematics are important. The lack of use and misuse of teaching materials and technology in the mathematics classrooms is another concern addressed by the IMP. Manipulatives and technology, such as graphing calculators and computers, should be available at all times and should be used to elucidate abstract concepts. Finally, the IMP is concerned with the lack of teacher preparation time, which it tries to alleviate in two ways. First, it demands that teachers using the project be given additional planning periods. Second, each unit is written as a structured series of daily lesson plans to cut down on the preparation work needed to be done by teachers.
A feeling for the nature of the project may be received by analyzing a specific unit. A first-year unit, called "The Pit and the Pendulum," (IMP, 1994) will serve as a representative unit, and it is assumed that all units contain similar structure and activities. Only the content may be different. The unit "The Pit and the Pendulum," is based on the Edgar Allen Poe short story, in which a prisoner is chained to a table underneath a large pendulum designed to kill him. With each swing, the pendulum drops, bringing its razor-sharp bob closer to the prisoner. After much thought, the prisoner develops a plan for escape, and realizes that he only has about twelve more swings of the thirty-foot pendulum in which to execute it. The plan succeeds and the prisoner escapes with his life. The students are asked to use information from the story to decide if the prisoner actually had enough time to escape. Thirty days will be needed to develop the skills and concepts necessary to solve this problem. Each day in the unit is related to the previous day and is designed to bring the students one step closer to solving the unit problem. The lessons typically begin by discussing the previous night's homework. Key aspects and trouble spots of the assignment are pointed out in the teacher's manual, so that they are sure to be discussed. Following, the homework are the in-class activities. These typically include experiments, group work, reports, and class discussions. The lesson concludes with the next homework assignment, which may include data collection, experiments, problem solving, or writing assignments.
Although it is necessary to discuss pedagogy and structure, it is the main purpose of the IMP to teach integrated mathematics. Since the project is a four year curriculum, it would be difficult to assimilate in one paper the entire mathematical content of the project. Therefore, only the content of one representative unit, "The Pit and the Pendulum" will be discussed. The mathematical content in this unit begins with the concept of periodic motion, for which the swing of a pendulum serves as a perfect example. Students measure various periodic events; such as the length of time between human heartbeats, or the length of a person's stride; in order to learn about controlled experiments, sampling distributions, and bar graphs. Students are then asked to collect several data sets, to be used as example distributions throughout the unit. Bar graphs of each distribution are created and used to introduce measurement error and normal distributions. Students learn about various aspects of the normal distribution such as its symmetry about the mean and its concavity. They also learn to use estimates of areas under the curve to introduce the concept of standard deviation. Students learn the geometrical relationship between standard deviation and the normal distribution as well as how to calculate it from a set of data. The difference between a statistic and a parameter is also taught. Students learn to use a calculator to find the mean and standard deviation of a distribution as well as how to use the statistics in problem situations. The situations relate back to the unit question, and students should realize that progress towards solving the unit problem has been made.
At this point, the unit moves away from statistics and begins to discuss the relationships between functions, graphs of functions and data sets. Students learn ways in which graphs of statistical data can be manipulated so that the graphs become misleading. Students learn to estimate maximum values of sketched graphs the relationship between equations and their graphs, and function notation. Lines and quadratic equations are specifically dealt with in this unit. A key concept taught in the unit is the use of curve fitting to predict the results of an experiment. After curve fitting is mastered and the students realize only length affects the period of a pendulum, all the tools necessary to solve the unit problem are available. So, students collect data on the period of pendulums with varying lengths, and use the resulting data to form a prediction equation relating period to length. A value for a thirty-foot pendulum is obtained from the equation, and the unit concludes by testing this predicted value.
It should be seen that the IMP curriculum offers many advantages to teachers and schools over a standard four year curriculum and its related materials. However, there are several disadvantages of the project which have not been mentioned. In fact, one disadvantage may be the nature of the project itself. Since the IMP curriculum is very different from the standard curriculum, this may cause problems in the acceptance of the project by students, parents and teachers. Many students may enjoy the activities created by the IMP, may learn from them better, and may develop a love of mathematics because of them. However, many students may fail to see the purpose of units, or they may believe that they are not being taught significant mathematics. many students use to individual work may not like working in groups. Parents may complain because they believe their children are not being taught appropriate skills. Parents may also be concerned that the IMP curriculum does not include calculus. The IMP suggests that parent education will be needed, but this may create added aggravation for teachers and administrators. Teachers may also have problems accepting the IMP curriculum. First, the pedagogical methods used in the IMP may be unfamiliar to many teachers. The project suggests that teachers take several courses to learn to use the IMP materials, but this could create added pressure on teacher's already limited time. Because the material is integrated and contains many non-standard topics, teachers may be uncomfortable teaching the new and unfamiliar material. Many teachers may find the structure of the units too confining. IMP units do not allow for much deviation from the unit objectives. Heterogeneous classes could also cause a problem. A common pace for all students may instill boredom in some students and confusion in others. Another problem in the IMP is the severely limited number of skill and manipulation exercises. Certainly, many teachers believe more problem solving and reasoning are needed, but students also need to be able to practice skills. The compatibility of the IMP curriculum with the standard curriculum is also in question. It is not known whether the IMP covers enough mathematics in order to prepare students for college or an active role in society. Hence, evaluation of the project is needed before it can be implemented in high schools throughout the country.
A short evaluation of the project can be found in the project report to the National Science Foundation (Schoen, 1993). The report considers the compatibility issue and states the "IMP course content has been approved by several important arbiters of mathematics curricula." The arbiters include universities, state education references and other projects. Because of the support of the arbiters, the conclusion is made that the IMP curriculum is compatible with the standard curriculum. The report uses the SAT as an ability measure for comparing IMP students with non-IMP students. The project was piloted in three California high schools, two urban and one rural, from 1989 to 1992. The student populations of the three schools represented students from different races, genders, socio-economic statuses, and ability. In general, quantitative SAT scores of IMP students were similar to non-IMP students, and in some cases better. Verbal scores were not reported. The reports also claims the Project has had an impact on student attitude. It suggests IMP students believe they are more mathematically able, and more likely to perceive mathematics as having applications than non-IMP students. The report states IMP students also have more positive attitude towards mathematics than non-IMP students. The report claims teachers also enjoy the IMP materials more than standard curriculum materials. The author of the report provides no significant evidence for most of these claims.
The Interactive Mathematics Program has definite plans for the future curriculum of the United States. The project began with just three schools in 1989 and has since grown to include more than 130 high schools across the nation. The curriculum accomplishes its goal "to embody the vision of the Standards, " but only time and further evaluation will tell if the IMP is the curriculum of the future.
National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.
Schoen, H. L. (1993). Report to the National Science Foundation on the impact of The Interactive Mathematics Project. Madison, WI: Wisconsin Center for Education Research.
Interactive Mathematics Program. (1994). The Interactive Mathematics Program. [Brochure]. Emeryville, CA. Interactive Mathematics Program.