I will consider my career a success when my students no longer tell me that they are bad at (or good at) word problems.
The direction of K-12 mathematics curriculum in the United States in recent years has been towards giving students a sense of how people actually use mathematics. There are many people who use abstract mathematical ideas on a regular basis outside the mathematics classroom-physicists, statisticians, mathematicians and the like. There are many more people who regularly use mathematics to solve practical problems that matter. (See for instance, this recent New York Times article on airline bumping). But no one solves word problems such as the following unless they are in math class:
The apartments in Vincent’s apartment house are numbered consecutively on each floor. The sum of his number and his next-door neighbor’s number is 2409. What are the two numbers? (from Introductory and intermediate algebra: full citation below)
This is a classic word problem of the sort that I hope to eradicate from mathematics instruction (at least from my own).
Why eradicate them? I have already presented my first argument-that they do not represent the ways anybody actually uses mathematics. At their very very best, word problems are intriguing puzzles and perhaps like a clever lyric or a piece of a melody they provide a bit of satisfaction to the aesthetic soul. But few word problems meet this standard. Instead, they represent to many students roughly half of what mathematics is (most the other half is abstract symbolic manipulation). Students who find these problems silly come to believe that mathematics has nothing to offer them.
I believe students are right in asking When am I ever going to use this? I can honestly tell students that they will not ever be in a situation where they know the sum of two addresses, and where they know that the addresses are consecutive, but where they do not know either address.
Occasionally I let myself believe that American math teaching has made substantial strides since the release of the 1989 NCTM Standards for School Mathematics and the revised Principles and Standards 2000.
I recently had an experience that set me straight. I was in my College’s Math Center, where the developmental math courses are housed (i.e. those courses not bearing college credit because they cover remedial topics). The apartments problem cited above comes from the textbook we use for these developmental courses.
Things were slow so I browsed a small selection of books on a shelf. At least one had been culled from the college library recently. It was titled, How to solve word problems in algebra: A solved problem approach by Mildred Johnson (full citation below). I became intrigued by the Table of Contents, which closely matches the kinds of problems in our text: Numbers; Time, Rate and Distance; Mixtures; Coins; etc. Of course the problems in this book match those in our text as well. Consider:
The sum of three consecutive integers is 54. Find the integers. (p. 13)
This problem, at least, is more honestly posed as a puzzle. The apartment problem puts the same idea into an unrealistic setting. But How to solve word problems commits the same crime. Consider:
Mrs. Mahoney went shopping for some canned goods which were on sale. She bought three times as many cans of tomatoes as cans of peaches. The number of cans of tuna was twice the number of cans of peaches. If Mrs. Mahoney purchased a total of 24 cans, how many of each did she buy? (p. 14)
Seeing the similarities between the perspectives of our textbook and this supplementary book, I noticed the Preface.
There is no area in algebra which causes students as much trouble as word problems…Emphasis [in this book] is on the mechanics of word-problem solving because it has been my experience that students having difficulty can learn basic procedures even if they are unable to reason out a problem.
And here is the crux of the matter. I have already argued that the very nature of word problems is such that people’s actual experience has no bearing on solving them. But in this preface is the rarely stated truism that we can train students to work these problems even when we cannot teach them to think mathematically. Entire sections of textbooks are devoted to the translation of word problems into algebraic symbols and Ms. Johnson has written the book on it.
While I appreciate Ms. Johnson’s efforts to help students through the arcane world of word problems, I am saddened by the uncritical approach. There is no discussion of why students should be forced to learn to solve artificial word problems, nor a questioning of whether there might be better uses of their (and their teachers’) time.
I began to wonder how recently the book had been written, given the similar perspectives of our text with this supplement. The copyright is 1976. In thirty-four years, we have truly made no progress.
Back to work.
Bittinger, M. & Beecher, J. (2007). Introductory and intermediate algebra: Third edition. Boston: Pearson.
Johnson, M. (1976). How to solve word problems in algebra: A solved problem approach: Updated first edition. New York: MacGraw Hill.