Humanistic Mathematics - Rediscovering Joy in Learning
This paper summarizes my personal reflections on the present state of mathematics education following my attendance at the NSF short course: Humanistic Mathematics: Fact or Folly. The course was conducted on June 4-6, 1998, by Dr. Alvin M. White, emeritus professor of mathematics at Harvey Mudd College in Claremont, CA. About 20 years ago Dr. White began a campaign for "humanistic mathematics" and he has seen his personal vision grow into an international movement which is having a signi ficant impact on mathematics education. What is "humanistic mathematics" and why has this movement gained much support but also generated controversy among mathematics educators? To answer this question I will begin with a brief history of the development of mathematics to provide a context for readers who are not mathematicians. Next I will examine the position of mathematics in present day U.S. culture and attempt an analysis of how this situation developed. Finally I will describe what is meant by "hum anistic mathematics" and why many in the mathematics community feel it is an idea whose time has come.
I. History and Background
The exact origins of mathematics will forever be shrouded in mystery since "it is certain that mankind made calculations and considered geometric figures before the invention of writing" (Freudenthal 6). However the earliest writings available, as well as physical artifacts such as Stonehenge, indicate that mathematics was developed to solve practical problems such as counting, measuring, creating calendars, surveying land and keeping financial records. At that time mathematics could be defined as arithmetic, and before 500 BC Pythagoras expressed this when he said, "Number rules the universe" (qtd. in Reid 1). Modern mathematics had its beginnings about this time when the Greeks transformed mathematics into a logical system. They began with so me basic postulates, which they considered to be obvious truths about reality, and then used deductive reasoning to prove theorems. Ironically the application of this method led to two great discoveries which precipitated the first crisis in mathematics. The first discovery was a proof of the familiar Pythagorean Theorem: the square on the hypotenuse of a right triangle equals the sum of the squares on the other two sides. The second discovery was the proof of a related theorem: If the two shorter sides o f the right triangle are equal, it is impossible to find a whole number or ratio of whole numbers which exactly measures the hypotenuse. In other words number could not rule the universe, so when Plato (about 350 BC) was asked about the acti vities of God, he said, "God eternally geometrizes" (qtd. in Reid 1). Incidentally Plato's philosophy that ideas, eternal, unchangeable, and outside the realms of space and time, were the true source of reality and beauty will become important in later di scussions of the nature of mathematics.
The proof of these two theorems by the Greeks "determined the decisive choice of form over number and set Western mathematics on the path it would follow for twenty centuries." [Reid 1] Greek mathematics was formalized and preserved in the thirteen volumes of Euclid's Elements, the second most widely circulated book of all time. The Greeks were also interested in astronomy and the earth-centered model of the universe created by Aristotle and Ptolemy remained virtually unchanged until Coperni cus published his book in 1543.
After the Greek era there were no major changes in the nature of mathematics until the mid seventeenth century when Newton and Leibniz invented the calculus. With this new tool mathematics expanded from the study of shape and number to include the study of motion, change and space. This provided the tools for scientists and engineers to develop all the technological advances that followed. By the end of the seventeenth century most of the mathematics presently taught at the secondary and college le vel had been developed. At this time mathematics was considered to be an area of absolute truth. For example, when Descartes was searching for a way to arrive at truth through philosophy, he concluded "that a sound body of philosophy could be deduced only by the methods of the geometers, for only they had been able to reason clearly and unimpeachably and to arrive at indubitable truths" (Kline 161). In one sense mathematics is certain since the solution to the equation 3x + 1 = 16 was 5 in the time of Pyt hagoras and will always be 5 in the future. However in the mid nineteenth century, another crisis in mathematics occurred when its very foundations began to be questioned. It had been thought that the basic postulates used in developing mathematics were unquestionable truths about the nature of the universe, and therefore the derived theorems were also absolute truths. However Lobatchevsky, Bolyai and others showed that it was possible to start with a different set of assumptions and produce a cons istent system which could also describe some aspects of reality. Since the initial postulates were seen to be mere assumptions and not absolute truths, people gradually came to understand that mathematics could only provide tentative models of real ity - models that may have to be changed or even abandoned as new discoveries are made.
The discovery that mathematicians were free to choose their initial postulates added a new dimension of creativity to mathematics. The discovery also made it possible to separate the issues of "valid proof" from "absolute truth." But if mathematici ans were going to restrict themselves to proving theorems from a system of axioms, two important questions remained: (1) is the axiom system complete so all questions can be answered, and (2) is the system consistent so no contradictory theorems will aris e? In 1931 our view of mathematics was forever changed when a young Austrian mathematician, Kurt Gödel, proved that both of these goals are impossible to achieve. "In the axiomatization game, the best you can do is to assume the consistency of the axioms and hope that your axioms are rich enough to enable you to solve the problems of highest concern to you." [Devlin 64]
Despite these difficulties there has been a virtual explosion of mathematical knowledge in the past century and perhaps fifty new branches of mathematics have evolved. One estimate states that all the mathematics that was known in 1900 would fit in to about eighty books whereas today it would be in the neighborhood of one hundred thousand volumes (Devlin 3). The modern view of mathematics is summed up nicely by Keith Devlin.
Mathematics, the science of patterns, is a way of looking at the world, both the physical, biological, and sociological world we inhabit, and the inner world of our minds and thoughts. Mathematics' greatest success has undoubtedly been in the physical domain, where the subject is rightly referred to as both the queen and servant of the (natural) sciences. Yet, as an entirely human creation, the study of mathematics is ultimately a study of humanity itself. For none of the entities that form the substra te of mathematics exist in the physical world; the numbers, the points, the lines and planes, the surfaces, the geometric figures, the functions, and so forth are pure abstractions that exist only in humanity's collective mind. The absolute certainty of a mathematical proof and the indefinitely enduring nature of mathematical truth are reflections of the deep and fundamental status of the mathematician's patterns in both the human mind and the physical world. (6)
In any case, an examination of history would certainly seem to support Morris Kline's thesis that "mathematics has played a predominant role in the formation of modern culture" (453).
II. Mathematics and U.S. Culture
So what does all this mean to the average U.S. citizen at the end of the twentieth century? Most of the evidence seems to indicate that the answer to the question is "Very little". Anecdotal evidence to support this conclusion often appears in social situations. When a new acquaintance discovers that I teach mathematics, the predictable response is "I never was any good at math" or "I don't like math, I never could understand what it was good for." Some actually seem proud that they lack the ab ility to do mathematics. Perhaps they feel that this puts them in the "in group" socially since their comments often imply that you have to be a little "different" to enjoy mathematics. Occasionally these negative attitudes even slip into the mainstream m edia. Recent examples are the light beer commercial that contains the statement, "If life were perfect ... algebra would actually come in handy" and the talking Barbie doll that says, "Math is hard." Usually these items are withdrawn after some mathematic al organization issues a protest. In addition to negative attitudes about mathematics, most people also have misconceptions about the work a mathematician does. Because of his profession, a mathematician often hears comments such as the following: "You'd make a great treasurer for our organization" or "You figure out everyone's share of the restaurant bill" or "You keep score so we can sure it will be correct." Wayne Roberts comments on this in his book, Faces of Mathematics:
Mathematicians think of their work as that of finding an effective, preferably elegant, way to solve a problem that at first looks intractable, of finding patterns in data, of guessing and perhaps proving that a collection of accepted statements have c ertain implications that are not at all evident, and of generalizing work they have done in ways that might make it applicable to new situations. They do not see themselves as wizards who can do arithmetic computations in their heads faster or more accura tely than other people. They are reminded on a daily basis that their self-perception is not the popular perception. (205)
Not only is it common to find negative attitudes toward mathematics, studies and experience also show that many people lack the ability to do even basic mathematics. Comparative assessments such as NAEP and TIMMS consistently show U.S . high school students scoring lower than many other industrialized countries. Businesses invest huge amounts of time and money in remediation programs to teach basic math skills to their employees. In response to this, various organizations and governmen tal units have begun programs to promote higher standards and increased accountability. The National Council of Teachers of Mathematics (NCTM) has published standards for mathematics education in grades K-12. Minnesota high school graduation requirements now include passing the Minnesota Basic Standards Test (MBST). The U.S. Department of Labor has published the report, What Work Requires of Schools: A SCANS Report for America 2000.
However, despite all the rhetoric about the importance of mathematics from educators, politicians and business people, in the final analysis it seems even they are satisfied if the mathematical performance of high school and college graduates remains a t the elementary school level. This was confirmed in a recent study completed by Lynn Steen for SciMathMN * , What Employers and Educators Test: The Mathematics That Really Counts. "Except for the SAT-II test, none of the tests used for employment or entrance into postsecondary education place any significant emphasis on topics that are not included a strong eighth and ninth grade curriculum (15). Both employers and educators signal through these tests that hig h school mathematics is really optional (17). Indeed they appear to suggest a widespread hypocrisy about public commitment to standards (1). Under these circumstances it is no wonder that many students see high school mathematics as superfluous (5). The f act that assessment is not aligned with expectations and the fact that many teachers, especially at the K-6 level, lack adequate training in mathematics makes it seem likely that these new programs will be only marginally successful in raising the level of mathematical literacy among U.S. citizens.
III. How Did We Get Into This Situation?
The history of mathematics and how educators have interpreted that history can help us understand why there are so many people with high math anxiety and low math ability. First, as mentioned earlier, the Greeks transformed mathematics from intuitive discoveries into a formalized system. Second, most of the mathematics we presently teach had been developed by the end of the seventeenth century. Since this mathematics was deemed to be important it became part of school curriculums. Gradu ally the history and context and excitement of discovery were lost and mathematics was taught as a formal system of facts, formulas, techniques and algorithms to be mastered and memorized. Instead of being shown as a methodology of problem solving with br oad applications, "many students perceive a mathematics course as haphazard collection of definitions, theorems and problems that are beyond intuitive understanding" (White, Teaching 40). The lecture/demonstration method became the preferred method of teaching because it was thought to be the fastest way to transfer knowledge from teacher to student. The lecture was generally followed by an assignment where students worked 20-30 exercises to practice the techniques demonstrated by their teachers. S tudents often saw little relation between these classroom experiences and the things they judged to be relevant to their lives. M. Polanyi details the typical transition from the excitement of the discoverer to the dull routine present in many classrooms:
As we pursue scientific discoveries through their consecutive publications... we observe that the intellectual passions aroused by them appear gradually toned down to a faint echo of their discoverer's first excitement at the moment of illumination. A transition takes place here from a heuristic act to the routine teaching and learning of results. The impulse which in the original heuristic act was a violent irreversible self-conversion of the investigator and may have been followed by an almost equall y tempestuous process of converting others, is first repeated as a milder version of itself...and will thus assume finally a form in which all dynamic quality is lost. (172)
With a curriculum devoid of historical context, lacking in unity, and appearing to have little practical application, teachers of mathematics are all familiar with the question, "What is this good for?" Often the only answer students receive is, "Y ou'll need this in the next math course." Romberg and Carpenter note that in many classrooms, the textbook is "the authority on knowledge and the guide to learning [and that] ... many teachers see their job as 'covering the text'. Further it was noted tha t mathematics and science were seldom taught as scientific inquiry - all subjects were presented as what experts had found to be true. Ownership of mathematics rests with the textbook authors" (25). It is interesting to compare student reaction to this la ck of ownership with the reaction of someone who had a deep personal involvement in solving a problem. Tradition tells us that Pythagoras was so thrilled at finding a proof for his famous theorem "that he sacrificed to the gods a hecatomb (100) of oxen, c ausing the theorem to be known during the Middle Ages as inventum hecatomb dignum" (Reid 6).
Alan Schoenfeld of the University of California believes that traditional methods of teaching mathematics simultaneously teach students some unintended lessons. Beliefs acquired by students include:
Morris Kline sums up the situation very well in his book, Mathematics and Western Culture. After detailing the ways in which mathematics has been a major influence in the development of western culture, Kline says:
Despite these by no means modest contributions to our life and thought, educated people almost universally reject mathematics as an intellectual interest. This attitude toward the subject is, in a sense, justified. School courses and books have present ed 'mathematics' as a series of apparently meaningless technical procedures. Such material is as representative of the subject as an account of the name, position, and function of every bone in the human skeleton is representative of the living, thinking, and emotional being called man. Just as a phrase either loses meaning or acquires an unintended meaning when removed from its context, so mathematics detached from its rich intellectual setting in the culture of our civilization and reduced to a series o f techniques has been grossly distorted. Since the layman makes very little use of technical mathematics, he has objected to the naked and dry material usually presented. Consequently, a subject that is basic, vital, and elevating is neglected and even sc orned by otherwise highly educated people. Indeed, ignorance of mathematics has attained the status of a social grace. (vii)
To be fair, two things must be said at this point. First, mathematics is not the only subject whose teaching is subject to these problems. For example, history can be taught as merely a collection of names, dates and events, or it can be taught as a way of understanding our present culture and hopefully avoiding the repetition of past mistakes. Second, there are mathematics teachers who by their dynamic personalities, love of the subject or inventive teaching methods manage to overcome the aforemen tioned problems and instill in their students an understanding of and love for mathematics. But then we are faced with the question, "If it takes extraordinary teaching to overcome the problems associated with the present system, why not change the system ? I believe "humanistic mathematics" is a step in the right direction.
Humanistic mathematics is not a new invention but dates back to Plato and many prominent mathematicians who have written about the beauty, creativity and imagination they saw in mathematical ideas. The English mathematician, G. H. Hardy, wrote that what distinguishes significant from trivial mathematics is beauty: "The mathematician's patterns, like the painter's or the poet's must be beautiful, the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test" (85). My first calculus professor, Dr. Sigurd Mundhjeld, was fond of saying, "Mathematics is a very creative subject but it gets embalmed in textbooks." The major premise of humanistic mathematics is that lack of motivation is the root cause of the literacy and attitude problems in the learning of mathematics. The movement "seeks to return to the educational process the excitement and wonderment of the moments of discovery and creation" (White Process 130). Mathematics educators with a human istic philosophy are people who "consider mathematics to be one of the humanities; who understand mathematics to have much in common with literature, philosophy, art, music, and criticism; who eschew dogmatic teaching styles that expect students to parrot the lecturer" (White 1994 128). They agree with the conclusions of Csikszentmihalyi who wrote:
The chief impediments to learning are not cognitive. It is not that students cannot learn; it is that they wish not to. If educators invested a fraction of the energy they now spend trying to transmit information in trying to stimulate the students' en joyment of learning, we could achieve much better results. (115)
The following example shows the startling results that can occur when this advice is applied to a classroom.
California children were recently (1994) assessed using standardized tests. The results were discouraging. Most children were seriously deficient ...in mathematics, although there were some exceptions. ...Some called for "back to basics" as a solution again although that idea has led to failure.
There are however happier stories. A fifth-grade teacher in a small district in southern California was enrolled by others in a mathematics problem-solving workshop. On her arrival she was dismayed. Mathematics was not her interest. Her interest was li terature and art; mathematics was given very little attention. The promise of help and support persuaded her to stay. She learned about problem solving for fifth graders and was inspired by the workshop leaders. Her students were enrolled in the internati onal Mathematical Olympiads for Elementary Schools (MOES). About five years after the workshop there were 72,000 contestants in the MOES. Seventeen children had perfect scores and three of those perfect scores were from this teacher's class!
A local TV station sent a crew to interview the students and their teacher. All of the children were working on problems and talking. Very few were sitting. When a student solved a problem, the teacher checked it, and the student moved to help those wh o were still working. Before the workshop the teacher lectured or talked to the students from the front of the classroom. After the children experienced interesting mathematics in an active way, they demanded more of the same. Passivity for the teacher an d her students was rejected. (White Process 122)
The modern rediscovery of humanistic mathematics is largely due to the efforts of Dr. Alvin White of Harvey Mudd College in Claremont, CA. In June of 1986 he received a grant from the Exxon Educational Foundation to sponsor a conference to examine mathematics as a humanistic discipline. The enthusiastic response to the conference led to the creation of the Humanistic Mathematics Network Journal and Dr. White's vision has spread to thousands of mathematics educators throughout the world. In 1 994 Dr. White wrote an article for the American Behavioral Scientist which includes the following description of humanistic mathematics.
Two related themes that emerged from the conference were (a) teaching mathematics humanistically and (b) teaching humanistic mathematics. The first theme sought to place the student more centrally in the position of inquirer than is generally the case, while at the same time acknowledging the emotional climate of the activity of learning mathematics. What students could learn from each other and how they might better come to understand mathematics as a meaningful rather than an arbitrary discipline wer e among the ideas of the first theme.
The second theme was focused less on the nature of the teaching and learning environment and more on the need to reconstruct the curriculum and the discipline of mathematics itself. The reconstruction would relate mathematical discoveries to personal c ourage, discovery to verification, mathematics to science, truth to utility, and, in general, it would relate mathematics to the culture in which it is embedded.
Humanistic dimensions of mathematics discussed included the following:
In June of 1998, Dr. white convened a similar conference sponsored by the NSF that I had the privilege to attend. Educators from around the country gathered at Claremont College to exchange views and share ideas about improving mathematics education. Near the end of the conference we compiled a list of characteristics which might be true of a mathematics class taught in a humanistic way:
It was agreed that not all of these characteristics would be observed in every humanistically taught mathematics classroom. Teachers are individuals and will select the techniques and methods that are best suited to their personal interests and str engths. Someone new to this philosophy of teaching might begin by trying just one area such as forming cooperative learning groups or the addition of some interesting problems not found in the textbook. When this is working smoothly, other changes can be added incrementally.
It should be noted that not everyone in the educational community agrees with the humanistic approach. The NCTM document, Curriculum and Evaluation Standards for School Mathematics, includes many aspects of the humanistic philosophy. For exa mple, the Standards list five general goals for all students: (a) to learn to value mathematics, (b) to become confident in their ability to do mathematics, (c) to become mathematical problem solvers, (d) to learn to communicate mathematically, and (e) to learn to reason mathematically (5). However, a number of states, including California, are rejecting these standards in favor of a "back to the basics" drill and practice approach to learning mathematics. Another example of the disagreement is cal culus reform, which is causing of a lot of discussion in college mathematics departments between the reformers and those who want no change in the traditional approach. In some ways this disagreement was predictable. Some of people in the field are there because they enjoyed the formal logic, abstractness, symbol manipulation and certainty they saw in the traditional mathematics classroom. To present mathematics in a more creative way or even make it "fun" may somehow seem to cheapen it or make it seem un fair: "I learned mathematics by many grueling hours of toil which I didn't always enjoy and students now should have to do the same." "If we add all this other stuff, we will have to skip some of the things that I had to learn."
In my view, even if we cover one less chapter of the text, the trade-off would be worth it. Maybe more people in the next generation could look back on their math classes as positive experiences. Hopefully they would be more confident of their abil ity to use the mathematics they learned in school and see its connections to real life situations and problems. They would see mathematics as a major influence on modern western culture. Perhaps they could even see some of the beauty and creativity in mathematics. A fringe benefit to college math departments would be attracting more majors to the field.
"Educators ignore humanistic mathematics at their peril. Without it, educators may teach students to compute and to solve, just as they can teach students to read and to write. But without it, educators can't teach students to love or even like, to appreciate or even understand, mathematics" (Tymoczko 14).
Csikszentmihalyi, M. "Literacy and Intrinsic Motivation." Daedalus. 119 (1990): 115-140.
Devlin, Keith. Mathematics: The Science of Patterns. New York: Scientific American Library,
Freudenthal, Hans. Mathematics Observed. New York: McGraw-Hill Book Company, 1967.
Hardy, G. H. A Mathematician's Apology. Cambridge, England: Cambridge University Press, 1967. (Originally published 1940)
Kline, Morris. Mathematics in Western Culture. New York: Oxford University Press, 1964.
Minnesota Basic Standards Practice Test. St. Paul, MN: Minnesota Department of Children,
Families, and Learning, 1997.
National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School
Mathematics. Reston, VA: NCTM, 1989.
Polanyi, M. Personal Knowledge. New York: Harper and Row, 1964.
Reid, Constance. A Long Way From Euclid. New York: Thomas Y. Crowell Company, 1963.
Roberts, A. Wayne. Faces of Mathematics. 3rd ed. New York: Harper Collins Publishers, 1995.
Romberg, T. A., & Carpenter, T.P. "Research on Teaching and Learning Mathematics: Two
Disciplines of Scientific Inquiry." Handbook of Research on Teaching. 3rd ed. Ed. M. C.
Wittrock. New York: Macmillan, 1986. 850-873.
Schoenfeld, Alan H. "When Good Teaching Leads to Bad results: The Disasters of 'Well-
Taught' Mathematics Courses." Educational Psychologist. 23 (1988): 145-166.
Secretary's Commission on Achieving Necessary Skills. What Work Requires of Schools: A
SCANS Report for America 2000. Washington, DC: U.S. Department of Labor, 1991.
Steen Lynn. What Employers and Educators Test: The Mathematics That Really Counts. St.
Paul, MN: SciMathMN, 1998.
Tymoczko, Thomas. "Humanistic and Utilitarian Aspects of Mathematics." Essays in
Humanistic Mathematics. Ed. Alvin White. Washington, DC: The Mathematical
Association of America, 1993. 11-14.
White, Alvin. "Teaching Mathematics as Though Students Mattered." Teaching as Though
Students Mattered. Ed. J. Katz. San Francisco: Jossey-Bass, 1983. 39-48.
White, Alvin. "The Process of Education." American Behavioral Scientist. 38 (1994): 122-