In an ideal world -- at least, a world that most mathematics educators would probably see as ideal -- almost everyone would see the need to have some mathematical skills, and would make the effort to acquire them. Moreover, in that same ideal world, everyone would, at some stage in their education, get a good overall sense of mathematics and its importance, either by attending a "broad brush stroke" survey course on mathematics or by reading one of a small but growing number of excellent expository books on mathematics.
Unfortunately, the world we live in is far from being that mathematician's ideal. Rather, the real world is one in which there is math phobia among a sizeable minority, some level of math anxiety among many more, a general antipathy toward mathematics in the majority, and ignorance about the true nature of mathematics on the part of practically everyone but the professional mathematician.
Surely, none of these observations comes as a surprise to anyone. But to judge by much of the current national debate about "math standards", you would think that my remarks constitute the discovery of the century.
At the risk of spoiling an enjoyable tussle, let me try to relocate the Great Math Education Debate from the ideal world of the students we would like to have -- i.e., copies of our younger selves as we remember that long lost golden age of our youth -- to the real world of the students who actually populate our classrooms.
I'll start with two observations concerning that ideal world I described in my opening paragraph:
Fact 1: Acquiring mathematical skills involves dedication and hard work. As such, it requires motivation. That is already problematic, since for most people the payoff comes later in life.
Fact 2: Getting a general sense of mathematics requires nothing more than interest.
At present, we put immense effort into trying to develop mathematical skills in our students, and we wring our hands endlessly when, for the majority of our students, we fail. At the same time, we rarely try to provide our students with a good, overall picture of the mathematical enterprise.
The paradox in this state of affairs is that Fact 2 probably provides the key to overcoming the obstacles stated in Fact 1. By providing our students with a good overall sense of mathematics, including the many major roles it plays in all our lives, we might well be able to provide the motivation the students need to spend some time acquiring basic skills.
By persisting with a largely unmotivated attempt to force feed the population with a set of perceived essential mathematical skills, we simply turn (what I think is) the majority of people off mathematics altogether and produce significant math anxiety in far too large a minority. As a result, even when people do subsequently find themselves in need of some mathematics, they are often too math phobic to acquire that knowledge.
I believe we need to reduce drastically the time we spend teaching basic skills in middle and high school mathematics classes. I do not see this as a great loss. The plain fact is, few citizens in modern society need or make real use of any appreciable knowledge of, or skill in, mathematics. What mathematics they need and use they have probably already met by the time they are twelve years old.
On the other hand, the continuance of modern society requires a steady supply of a relatively small number of individuals having considerable training in mathematics. In order that the critical future supply of mathematicians does not dry up, we must ensure that all high school and university students are made aware of the nature and importance of mathematics, so that those who find they have an interest in and aptitude for the subject can choose to study it in depth.
For the middle and high school grades, the main goal in the math class should be to create an awareness of the nature of mathematics and the role it plays in contemporary society. To do this, mathematics should be taught in much the same way as history or geography or English literature -- not as a utilitarian toolbox but as a part of human culture.
In my view, an educated citizen should be able to answer the two questions:
Existing methods turn off students in droves and produce math anxiety in many, and this is counterproductive. Teach mathematics as a part of our culture and the result will be many more students who are motivated to want to learn mathematics. Surely, the aim of a mathematics education should be to produce an educated citizen, not a poor imitation of a $30 calculator.
I should stress that I am not saying that basic numerical skills are not important. On the contrary, I would put quantitative literacy on the same crucial footing as ordinary literacy: both are so fundamental in today's society that they are everybody's responsibility. The development of basic quantitative skills are as much the responsibility of, say, the social studies or the English teacher as language and presentation skills are the responsibility of the math and science teachers. To leave the development of quantitative skills to the mathematics teacher sends quite the wrong message to the student.
By changing our present education system radically so that, for the vast majority of students, the primary goal in the mathematics class is to create an awareness of the what, the how, and the why of mathematics, rather than the development of skills that, apart from a tiny majority, none of them will ever make use of, we will achieve two important goals:
Turning to goal 2, any university mathematics instructor will tell you that the present high school mathematics curriculum does not prepare students well for university level mathematics. Nor is success at high school mathematics a good predictor of later success in mathematics. The reason is simple. School mathematics is largely algorithmic: To succeed, the student needs only to learn various rules and procedures and know when and how to apply them. In contrast, university level mathematics is highly creative, requiring original thought and the ability to see things in novel ways. Since the creative mathematician does often need to apply rules and use algorithmic thinking, many successful mathematicians did indeed excel in the high school mathematics class. But many university mathematics students who shone in high school find they struggle with and eventually give up the subject at university when they discover that algorithmic ability on its own is not enough. And the fact that some of the very best professional mathematicians did poorly at high school mathematics, but by some fluke were drawn to the discipline later in life, suggests that our present system of school mathematics education probably turns off a significant number of students who have the talent for later mathematical greatness.
So there you have it. Reduce skills teaching and concentrate on the big picture. And, please, don't pay so much attention to those international comparisons of math skills attainment. Parents and educators have been berating declining educational standards since the time of Euclid. There is surely something vaguely comical about the nation that leads the world in science and technology, and which virtually dominates the world in the development of computer hardware and software, constantly lamenting the poor math skills of its population. Sure the USA has to import a great deal of mathematical talent. That is because there are plenty of Americans with the talent, mathematical ability, and drive to generate a large demand for such people, a far greater demand than in any other country in the world. The time to worry would be when there is a major outflux of American mathematicians to one of those competitor countries we keep worrying about. Frankly, I don't see that happening any time soon.
That's not a "complacent, self-satisfied American" talking, by the way. Hey, I'm a Green Card carrying immigrant with the stamp on my entry visa barely a decade old. Now, if you want to know about the poor state of math education in my native Britain . . . but that's another story.
See also: "Is Mathematics Necessary?" by Underwood Dudley, College Mathematics Journal, vol. 28, November 1997, pp. 360--364.
Dr. Keith Devlin (email@example.com) is Dean of Science at Saint Mary's College of California and a Senior Researcher at Stanford University's Center for the Study of Language and Information. A shorter version of the above article first appeared in an editorial by Devlin in the December 1997 issue of the MAA newsletter FOCUS, which Devlin edited from 1991 to 1997.