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## There Are Integrated Programs, and There Are Integrated Programs

January 2001

There are two types of "integrated programs." There are those like the ones in Singapore, which teach some arithmetic or algebra and some geometry each year with connections among them used. This is something that mathematicians have been proposing for a very long time (Mathematical Association of America 1923). Similar programs exist in many other countries. Singapore is mentioned because textbooks in that country are good, are written in English, and are accessible in the United States. (See www.singaporemath.com.)

The other is illustrated by some of the National Science Foundation (NSF)-funded programs from the NSF call for new programs in the early 1990s. These have a different focus, trying to teach mathematics in the context of real-world problems. It is this second type of integrated program that has become the focus of controversy in the United States.

To explain some of my concerns, consider an example from Contemporary Mathematics in Context (CMIC), written by the Core Plus staff. This example appears on pages 311–13 in Course 1A, near the end of the first semester of what for most students will be ninth grade. The real-world problem is how to get an estimate for the average time when doing a task many times by doing a sample study:

Compute the average time for each task in the feasibility study project using the formula:

 Average time = best time + 4(most likely time) + worst time. 6

Nothing is written in the textbook about where this formula comes from, but the following information is given on page T313 in the Teacher's Guide.

Background information: This is a commonly used way to compute the expected value of the task time. It assumes a specific type of probability distribution (called a beta probability distribution) and is only an estimate. It is, of course, neither necessary nor desirable to pursue this point technically, but it can be used as some quick justification for students who want to know there the "4" came from. This will be followed up a bit in a later probability unit in Course 2.

Ignoring the fact that the "explanation" says nothing at all about where the 4 comes from, another serious problem is illustrated by this "explanation." The beta distribution is the integrand in the beta function integral.

Notice that there is no 4 anywhere. In fact, the beta distribution has nothing to do with this formula. The formula is just the basic building block of Simpson's rule, which allows one to exactly integrate a cubic polynomial.

Where did the idea that the beta distribution had anything to do with this come from? I had no idea, so I posted a question to one of the authors on the Math Forum discussion group math-teach. The answer that I received said, in part, "In management science, they often use the expected value of a beta distribution to estimate the 'average' task time."

The author then gave a source for the following quotation: "[The equation for the average or expected time is] based on the assumption that the uncertainty in activity times can be described by a beta probability distribution...." So one major problem with real-world applications as a basis for school mathematics education is that the developers do not know enough to get things right. Also, if they copy problems from books in many applied areas, little or no mathematical reason will be given for the use of specific parts of mathematics. A similar formula to the one given above is used in the treatment of volumes in CMIC. This is also done very poorly, with essentially no geometry being used. This appears in the second half of the first course. In particular, no geometric reasons are given for the factor of 1/3 in the formula for the volume of a pyramid.

There are other reasons to be concerned about the type of programs similar to this one. The following quotation is from the introduction to "Applications and Misapplications of Cognitive Psychology to Mathematics Education" (Anderson, Reder, and Simon).

There is a frequent misperception that the move from behaviorism to cognitivism implied an abandonment of the possibilities of decomposing knowledge into its elements for purposes of study and decontextualizing these elements for purposes of instruction.

A program that postpones a proof of the Pythagorean theorem until the third year of high school because students do not learn how to expand until then is unacceptable by world standards, and that is what we have to consider. That is the case for Contemporary Mathematics in Context.

References
Anderson, John R., Lynn M. Reder, and Herbert A. Simon, "Applications and Misapplications of Cognitive Psychology to Mathematics Education." act.psy.cmu.edu/personal/ja/misapplied.html. World Wide Web.

Coxford, Arthur, et al. Contemporary Mathematics in Context, Course 1, Part A. Chicago, Ill.: Everyday Learning, 1998.

Mathematical Association of America. The Reorganization of Mathematics in Secondary Education: A Report by the National Committee on Mathematical Requirements under the Auspices of the Mathematical Association of America, Inc. Mathematical Association of America, 1923. www.singaporemath.com. World Wide Web.

To read the complete discussion about this issue, go to forum.swarthmore.edu/discussions/epi-search/nctm.l.html. In the search, use the phrase "question for Eric Hart" and click on "that exact phrase" and "complete words only."

 Richard Askey has taught at the University of Wisconsin—Madison since 1963. His research is in special functions, which includes "higher trigonometry," that is, functions on higher-dimensional spheres, and extensions of the gamma and beta functions.

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