There has been much discussion over the years about the usefulness of mathematical studies. Everyone seems to have a different viewpoint on the issue. Some believe that mathematics has little use in the working world and so is not a subject that should be taught at higher levels in secondary school. Others argue that mathematics does serve a profound purpose, albeit one that is subtle and not obvious in the vocational world. G. H. Hardy and Underwood Dudley, two great mathematicians of the twentieth century, have differing views, and our current Secretary of Education Richard Riley has his thoughts as well. So who is right? Who has a stronger argument? Here we will take a closer look at what is the main objective of all mathematics instruction.
Hardy once wrote, "Very little of mathematics is useful practically, and that little is comparatively dull" [2]. However, in the next sentence, Hardy states that the power, the importance, the usefulness of a mathematical idea is not in its practical application, but rather in the power of the thought. Dudley agrees with this statement; his main premise is that mathematics, from the world’s viewpoint, is a way of thinking as opposed to the thought. Dudley argues that mathematics is not taught so students can someday go out on the job with confidence of knowing the derivative of sin(x), but that the subject exists in the classroom to teach students how to think [1]. Mathematics is a method of thinking, a powerful way of looking at the world, but it is not a way of making use of oneself in practical living.
In 1996 Robin Ria II and David Burghes conducted an investigation of the need for math skills of young, perspective employees in business, industry, and public service. Ria and Burghes interviewed numerous employment representatives from these various fields, and found that most employers did not even test for mathematical proficiency. Among those that did were the Army, rubber manufacturers, and the nursing profession. Ria and Burghes’ findings showed that these proficiency tests examined the most basic arithmetic skills; even basic high school mathematics were neglected [5].
Underwood Dudley noted that fully seventy percent of engineers, those persons who are renowned for actually having to implement mathematics in their daily work, do not need calculus to perform their work. Dudley’s obvious conclusion: if the majority of engineers do not need calculus, then how many of the thousands of high school and college students who are required to take calculus will actually use it someday? [1] The answer seems to be minutely few. However, many people use computer programs daily that have been written using calculus and other mathematics in their binary code, and numerous jobs require the implementation of mathematical formulas that have been derived from calculus, such as a formula for the volume of a sphere. Does this mean that the person using the computer program or the formula needs to know calculus? No. Nevertheless, we must think once more of the thought process developed by mathematics. A student who has mastered the techniques of calculus has enhanced their own logical reasoning abilities thus has gained something from the mathematics.
Michael Wood discusses an aspect of mathematics that is often overlooked or just ignored by many instructors of mathematics. Professor Wood calls this aspect the "crunchy methods" of mathematics, wherein a student relies on his own reasoning to work through complex and not-so-difficult problems. Wood notes that this technique allows students to get a feel for the reasoning behind the solution, which is a better way of learning than memorizing the techniques of famous mathematicians of past centuries. He goes on to say that the use of mathematical reasoning for practical purposes is often used by persons who are not experts in mathematical techniques, such as in the areas of management and engineering [7]. Again, it is not formal techniques such as differentiation or integration that are used in everyday problems, but rather the thought process developed by the study of subjects such as calculus.
The implementation and manipulation of formal mathematical techniques may reveal the cleverness of the user, but they may not be the most powerful method. This statement is supported by the defeat of world chess champion Gary Kasparov by the supercomputer Deep Blue. Kasparov had the supposed advantage of intuition and clever strategies, but the computer had the ability to calculate more moves in advance. The computer could also better calculate the consequences of those moves. This crunchy method, or number crunching, was thus shown to be a more effective approach than the cleverness and intuition of the most talented human being [7].
We have seen that mathematics, or at the very least mathematical reasoning, is an essential part of the education process. Our Secretary of Education Robert Riley made this conclusion in a speech he gave last year. He went so far as to say that almost ninety percent of new jobs require a greater proficiency in mathematics than is taught in high school [6]. This seemingly contradicts an earlier statement by Underwood Dudley until we focus on the words "new jobs." Mathematics does have some merit in the working world after all. In this age of technology that is creating so many new occupations, mathematics may very well be a key factor.
The debate within mathematics education that is making headlines today involves drastic reforms in the way mathematics is taught. In California the debates have escalated into what are now nationally known as the "Math Wars." One group of reformers, led by the National Council of Teachers of Mathematics (NCTM), which was established in 1986, suggests five major shifts in mathematics education. These shifts focus on students finding their own answers from logical and mathematical evidence as opposed to the teacher being the sole keeper of the right answer. The main theme of the NCTM is the teaching of mathematical reasoning and not so much memorizing procedures [3].
William G. Quirk criticizes Secretary Riley and the NCTM for ignoring mathematics altogether. He argues that the reform methods focus on "discovery learning" and independent "process skills," and that reformers deny the importance of remembering the "old facts." Quirk insists that both understanding and reasoning are heavily dependent on facts already in the brain, the things that have been memorized [4].
The best technique of mathematics instruction seems to be an emphasis on independent reasoning, basing that reasoning on the mathematical skills the student has learned previously. Yet there is no obvious solution to the debate within mathematics education. However, it is clear that mathematical reasoning of the simplest nature is an incredibly powerful tool that should not and must not be ignored by educators. Although some higher-level mathematics courses may not seem to have any practical use in the world, the reasoning developed by implementing the mathematical techniques will be useful in all walks of life. The goal of education in mathematics should be as Michael Wood explained it, "we [teachers] want to help students develop as powerful and reliable a framework as possible with as little pain as possible" [7]. Mathematics should be a framework of reasoning; it should not be hated by students because of its apparent uselessness. Mathematics must be appreciated and respected for it’s best face, that of the tool of reasoning and logic.
[1] Dudley, Underwood. "Is Mathematics Necessary?" The College Mathematics Journal. 28 (Nov 1997), 360-364.
[2] Hardy, G.H. A Mathematician’s Apology. Cambridge: The University Press, 1941.
[3] Lacampagne, Carole B. State of the Art: Transforming Ideas for Teaching and Learning Mathematics. July 1993. <http://www.ed.gov/pubs/StateArt/Math> (10/16/99).
[4] Quirk, William G. The Anti-Content Mindset: The Root Cause of the "Math Wars". <http://www.wquirk.com/content.html> (10/16/99).
[5] Ria II, Robin, and David Burghes. Mathematical Needs of Young Employees. <http://www.ex.ac.uk/cimt> (10/16/99).
[6] Riley, Richard W. "The State of Mathematics Education: Building a Strong Foundation for the 21st Century." Conference of American Mathematical Society and Mathematical Association of America. 8 Jan. 1998. <http://www.ed.gov/Speeches/01-1998/980108.html>. (10/16/99).
[7] Wood, Michael. The Case for Crunchy Numbers in Practical Mathematics. <http://www.soton.ac.uk/~gary/Wood98.htm> (10/16/99).
This paper
was peer-reviewed by: Brad Stark, Bucknell '03.