Crossroads in Mathematics

Chapter 1 - Introduction

Introductory College Mathematics

One day, about one-third of the way through the spring semester, an experienced mathematics faculty member walked into her department chair's office with a worried look. She had volunteered to teach an experimental section of Intermediate Algebra using a textbook that emphasized real problem solving, group projects, and technology. Algebra techniques were introduced as they were needed to solve problems. For example, while there was considerable work that involved linear, quadratic, exponential, and logarithmic functions, as well as systems of linear equations, there was little, if any, work on such traditional algebra topics as factoring, theory of equations, and the solution of rational and radical equations. She explained to the department chair that it was really refreshing to teach the new material using innovative instructional methods. Furthermore, her students, who were hesitant at the beginning of the semester to get involved in group projects, were also beginning to enjoy the new approach to studying mathematics. So, why the worried look?

When she had volunteered to use what she considered an exciting approach to learning algebra, she thought that she understood the underlying principles of the much discussed reform in mathematics education. She wanted to get involved. Now she wasn't sure that she was doing the right thing. Her course did not include many of the topics that had been the mainstay of intermediate algebra. She was worried that she was not adequately preparing her students for the study of higher levels of mathematics either at her college or at transfer institutions.

If you were the department chair, how would you have responded to your faculty member's concerns?

The Need for Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus

". . . declining educational standards eliminate some applicants from consideration. We estimate that 30 to 40% have limited mathematics and science fundamentals, and that concerns us."

Jim Palarski, Marriott Hotels. In Mathematics Education, Wellspring of U.S. Industrial Strength, 1989, p. 2.

Higher education is situated at the intersection of two major crossroads: A growing societal need exists for a well-educated citizenry and for a workforce adequately prepared in the areas of mathematics, science, engineering, and technology while, at the same time, increasing numbers of academically underprepared students are seeking entrance to postsecondary education.

Mathematics is a vibrant and growing discipline. New mathematics is continually being developed and is being used in more ways by more people than ever before. In fact, the rate of growth in mathematically based occupations is about twice that for all other occupations (NRC, 1990). Yet, an alarming situation now exists in postsecondary mathematics education. More students are entering the mathematics "pipeline" at a point below the level of calculus, but there has been no significant gain in the percentages of college students studying calculus (Albers et al., 1992). The purpose of Crossroads in Mathematics is to address the special circumstances of, establish standards for, and make recommendations about introductory college mathematics. The ultimate goals of this document are to improve mathematics education and encourage more students to study mathematics.

The students addressed in this document are seeking Associate of Arts (AA), Associate of Science (AS), Associate of Applied Science (AAS), and bachelor's degrees. Some are traditional full-time students who are recent high school graduates. Others, particularly those at two-year colleges, are from widely diverse populations and fall into one or more of the following categories. They

"If there is danger to the status of mathematics, it does not arise from overemphasis of its 'reasonableness.' It comes from the deadly overemphasis on routine 'techniques,' and the unwholesome neglect of its 'reasonableness' and of its 'relevance' to the real world."

Moses Richardson, American Mathematical Monthly, 1942, p. 505.

All of these characteristics dramatically affect introductory college mathematics instruction.

Basic Principles

The following principles form the philosophical underpinnings of this document:
"We must equip all of our students-- regardless of age, sex, ethnic background, educational goal, occupational goal, personal history or capabilities--to think for themselves, and to solve their own problems and those of society to the very best of their individual abilities. That's what the right mathematics does."

Michael Davidson, Cabrillo College

This document makes no attempt to define "college-level mathematics," nor does it address the issue of whether courses at the introductory level should be credit bearing (to meet graduation requirements).


Introductory college mathematics must serve well all college students who are not prepared to study at the calculus level or beyond. This document offers a new paradigm for this level of mathematics education. The standards that follow in Chapter 2 are not a "quick fix" for what is wrong. Rather, they provide a strong and flexible framework for the complete rebuilding of introductory college mathematics.