Mathematics, not surprisingly, means different things to different people.

- To the public, mathematics means "problems on a page" or "scores on
a test." One participant characterized this as an "emaciated" view of
mathematics.

- To many teachers, mathematics means "mathematical power" as
expressed in the NCTM Standards--a robust mixture of problem solving,
communication, and reasoning.

- To employers, mathematics is just part of the teamwork required for
employees to do their jobs well. "In today's workplace, communicating
ideas is far more important than applying technical skills."

This widespread fear poses significant challenges for the mathematical community. Mathematics educators need to help students learn to communicate about mathematical ideas with those for whom quantitative thinking is not a welcome mode of discourse. Students' future jobs will likely depend on their skill in this endeavor.

At the same time, mathematicians and mathematics educators themselves need to communicate to the public that mathematics is not really as they imagine it in their terror-filled dreams. The world is changing, and so is mathematics--both at work and in school.

- To develop a plan to sterilize laboratory equipment in an autoclave, a
worker must gather and organize sufficient information to convince various
supervisors and review boards that the
plan is defensible. This requires analysis of data to prove that the plan
will work, discussion of alternatives to be sure that nothing less costly
would do the job, consideration of extreme cases that may not fit the
typical scenario, and responsiveness to relevant safety regulations.

- To reconcile inconsistencies in monthly inventories, a worker needs to
think backwards to determine all the possible ways in which records might
be wrong or items misplaced. This requires the ability to imagine the
workings of a complex system, to identify plausible sources of error or
failure, and to consult with various individuals in order to gather ideas and
confirm (or refute) hunches.

- To design an airplane in a modern competitive economy, one needs
more than the traditional airfoil analysis that produces optimal lift for the
wings. Now one seeks to minimize not only fuel consumption, but also
manufacturing costs: for a given lift, which design of an airfoil is cheapest
to manufacture? To answer that question requires the combined skills of
construction workers, design engineers, financial experts, and senior
management.

These problems illustrate several features of modern problem solving that are rarely addressed or developed in the schools For example, individuals are expected to recognize weaknesses in their own analysis of a problem, and then to seek assistance from their team (or their supervisors) in addressing these weaknesses. The habit of volunteering possible flaws in an argument does not come naturally to students who have become habituated by the pressure for grades to living in a world of bluff and desperation.

Whether in retail or manufacturing, finance or health, students who enter the world of work must be prepared to put forth their ideas and then to work with others to improve on these initial thoughts. School only prepares students to answer questions posed by others. Instead, students need to learn to say: "Here's my idea. Where are the holes?"

Yet the public, and many employers, still emphasize acquisition of basic skills and scores on standardized tests. Many believe that mastery of basic skills is a prerequisite to higher order problem solving, and that calculators provide young children with an excuse for not mastering these basic skills. "You must first master fundamentals, or little else can be accomplished."

Because test scores are the public surrogate for mathematical
achievement, the basic skills monitored by these tests become a bellwether
for public support of mathematics education. "If we don't satisfy the
public that students' basic skills are OK, then they will not trust us
with the changes we want to make in other parts of the curriculum." The
issue is not *whether* students should learn basic skills, but *
when* and *how*. Are they best learned in context, in parallel
with higher order problems, or in specific units to be mastered before
moving on?

Although nearly everyone agrees on the fundamental skills of elementary school mathematics, there is not much agreement among teachers, parents, employers, and others concerning the "basic skills" of high school mathematics. How important is it, really, that every citizen be able to use the quadratic formula, or factor algebraic expressions? Many believe that what really matters about school mathematics are not specific facts or procedures but experience in working with patterns, in thinking logically, in recognizing that the order of operations matters.

If employees are to solve the kinds of problems they face every day at work, they need to recognize that all input is valuable. Asking questions and challenging orthodoxy is often an individual's most important contribution to the group. Those who can only see their own viewpoints, or who put others down, tend to destroy creativity and cancel the benefits of the group. Unfortunately, this strong commitment to individual performance is all too often a correlate of high achievement in school mathematics.

There are several compelling reasons that the study of mathematics highlights individual effort over teamwork. First, major parts of school mathematics (routine problems) do not require or benefit much from group work. Other parts (complex projects) that are more suited to teamwork account for only a small portion of the curriculum. Second, students and their parents know that college admission is based on individual effort, including SAT scores, course grades, and writing samples. Parents want high schools to focus on preparation for college, and are thus very nervous about highlighting teamwork as a priority. Third, and ironically, assessing an applicant's potential as a team member is often not a priority in business hiring practices. "People who may do a good job as part of a team may have difficulty getting in the front door."

Nonetheless, employers stressed that teamwork produces a synergy not possible from individual effort. "I expect team solutions. People have different skills, and contribute differently to a team. Educators too often force all students in to the same mold. The era of the lone gunman is over."