by Henry O. Pollak, Teachers College, Columbia University
Excerpted with permission from "Why Numbers Count: Quantitative Literacy for Tomorrow's America," Copyright (c) College Entrance Examination Board, 1997. All rights reserved.
Problem solving is at the heart of quantitative literacy--the use of mathematics in everyday life, on the job, and as an intelligent citizen. Real-world problem solving involves not only mathematics, but also some situation outside of mathematics or some real-world difficulty crying out for systematic understanding.
Not surprisingly, there is a large literature on problem solving. What is surprising, however, and rather disappointing, is that this literature shows very little agreement on what the phrase "problem solving" actually means. ... [T]wo poles of meaning ... are illustrated in Webster's definitions for the term "problem": "Definition 1: In mathematics, anything required to be done, or requiring the doing of something. Definition 2: A question ... that is perplexing or difficult." Thus, problem solving sometimes involves repeated routine exercises to achieve fluency with a particular technique, ... [and at other times] solving problems that are "problematic." In this sense, solving problems is not routine drill, but is at the heart of mathematics itself.
What fundamental idea do these two definitions have in common? What features are shared by a grade schooler's long multiplication, a question about centroids in a calculus course, and a professional mathematician's settlement of a twenty-year old conjecture--other than that it is common usage to call all three "solving" a problem?
They do have something in common: the reduction of something you don't already know how to do to something you do already know how to do! This definition is dynamic, not static. What you already know how to do varies enormously during your education. The grade schooler knows addition and single-digit multiplication facts, to which a previously unsolved "problem" of long multiplication can be transformed. The calculus student examines the notion of centroids, and ultimately reformulates the question in terms of certain integrals which are by now familiar. ... [And] when the twenty-year old conjecture is transformed by a series of ingenious insights into previously known mathematics, the problem is said to have been "solved."
What is common to all these examples is that once the problem has been "transformed"--the mathematician likes to say "reduced"--to a formulation in terms of previous mathematics, the "problem solving" is in fact over. What follows, the carrying out of previously learned techniques or application of earlier mathematics, is no longer problem solving.
Real-world problem solving must meet the standards both of mathematics and of the external situation to which mathematics is being applied. This need to serve two masters is the main difference between problem solving in mathematics and mathematical problem solving in the real world. The interplay between the two adds great richness over and above problem solving in mathematics alone.
In the real world situation, as in mathematics itself, the basic meaning of problem solving is the process of reducing something you don't know how to do to something familiar. The collection of the familiar, of what each of us knows how to do, keeps growing with time.
A student's mathematical education is simply not complete if that student has not experienced the usefulness of mathematics in the larger world. This experience comes through real-world problem solving. Thus success can not be measured entirely through assessment in mathematics itself, or in terms of mathematical preparation for the next level of courses. We must also look for the ability to examine in a mathematical way situations in everyday life, on the job, and as a citizen.
Henry O. Pollak is a visiting professor at Teacher's College, Columbia University. Previously he served as director of the Mathematics and Statistics Research Center of AT&T Bell Labs. He can be reached by e-mail at firstname.lastname@example.org.
Last Update: July 17, 1997