# Precalculus Reform at the University of Michigan: Traditional UM Precalculus Before Fall, 1993

## Topics Covered in the Traditional Course

• Review of fundamental skills and concepts.
• Real numbers, basic rules of algebra, radicals and exponents, polynomials and special products, factoring, rational expressions.
• Equations and inequalities.
• Functions and their graphs.
• Linear, quadratic, polynomial, rational, inverse, exponential, logarithmic, combinations thereof.
• Trigonometry: right triangle and unit circle.
• The six trig functions and their graphs, combinations, identities, equations, solving the general triangle.

## Characteristics of the Course

• Strong emphasis on manual algebraic manipulation.
• Calculators discouraged (and forbidden on common midterm and final examinations).
• No coursewide emphasis on cooperative learning.
• Instructors generally spent most of their time lecturing.

## Homework Assignments

• Almost exclusively individual.
• Typically 10-25 exercises daily, typically of the following nature.
• Evaluate the following expressions ...
• Simplify these expressions as much as possible ...
• Find all solutions to these equations ...
• Graph these functions ...
• Prove these trigonometric identities ...
• Emphasis on algebraic skills, not concepts.
• Perhaps 2-3 "word problems," perhaps not.

## Word Problems

• Here are three typical ones for the traditional course.
• Weight of Copper Tubing. One foot of a certain type of copper tubing weighs 3.5 ounces. What is the weight of 7/8 mile of this tubing?
• Height of a Flagpole. A person who is 5.5 feet tall walks away from a flagpole toward the tip of the shadow of the pole. When the person is 25 feet from the pole, the tip of the person's shadow and the shadow cast by the pole coincide at a point 7 feet in front of the person. Find the height of the pole.
• Maximum Height of a Dive. The height of a dive in terms of the horizontal distance of the dive is given by (a certain quadratic relationship). What is the maximum height of the dive?
• Possible student (and instructor) reactions to these problems.
• Weight of Copper Tubing. This at least seems like a realistic exercise that someone might really wish to do, though the problem is actually just a very short exercise in unit conversion.
• Height of a Flagpole. Would anyone really measure the height of a flagpole this way? How about just lowering the flag, then raising it slowly while someone uses a yardstick to measure the amount of rope between the ground and the top of the flag when it is raised to its full height? Wouldn't that be much easier and perhaps more accurate?
• Maximum Height of a Dive. Where did this formula come from? Do you mean that someone has gathered enough information to be able to come up with this quadratic relationship, but they don't know the maximum height of the dive? Oh, come on!

• Students bore easily; they have seen much of this material before.
• Reinforces student belief that the essence of mathematics is algebraic manipulation, not understanding of concepts.
• Applications ("word problems") are uninteresting, implausible, and unrealistically short.
• Gives the impression that mathematics is exclusively a solitary activity.
• This is not to say that there is no solitary component to learning mathematics. As it has been put by Doug Shaw of the Geometry Center at the University of Minnesota, "There is an aspect to mathematics that consists of sitting in a quiet room, with a blank sheet of paper in front of you, and staring at that paper until your forehead bleeds." However, the extreme emphasis on solo problem-solving that often is part of the traditional approach can leave the student believing that it is somehow against the "rules of mathematics" to sit down with others and brainstorm.
• De-emphasizes writing and other communication skills.
• Does not take advantage of modern technology that can help automate routine computations (and help convey concepts).
• Corollary.The numbers used in exercises must be unrealistically simple, preferably integers.

• Many students really do need their algebraic manipulative skills enhanced.
• Weak manipulative skills may have been the reason they were placed in the course.
• It is easy to teach and test large numbers of students per class.
• The only equipment required is paper and pencils.
• Historically, the calculus course that followed had exactly the same emphasis. That has changed.

## The UM Calculus Reform Project

• Basic principles.
• The rule of four. Every topic should be studied geometrically, numerically, and algebraically, and communicated back to the instructor in a literate fashion.
• The way of Archimedes. Formal definitions and procedures evolve from the investigation of practical problems.
• Reformed text used: Calculus, by Hughes-Hallett et al., Wiley, 1994.
• Emphasis on cooperative learning, in-class and with homework groups; de-emphasis on lecturing.
• Fewer but more involved homework exercises; realistic applications; emphasis on writing skills.
• High-end programmable graphing calculators (TI-81) used.
• Large-scale testing began in the fall of 1992 in Calculus I, full implementation in Calculus I and Calculus II (except for some special-purpose sections) occurred in the fall of 1994.
• It quickly became clear that the traditional precalculus course was not the right course to precede our reformed calculus courses. This was the primary stimulus for changing the precalculus course.