The erosion of core curricula in the humanities and social sciences is widely discussed, but it’s also important to examine the natural sciences and mathematics, which are critical to a liberal education as well. The problem is that it’s challenging to develop substantial, engaging science and mathematics courses for students who for the most part aren’t majoring in these disciplines.
The natural sciences have done a pretty good job, and many schools have fascinating science courses aimed at non-majors. Some, particularly in the areas of astronomy and cosmology, deal with important developments in current scientific research.
Great examples of this are the astronomy courses (available via podcast) offered by Richard Pogge of Ohio State University. His Astronomy 141 course, Life in the Universe, introduces astrobiology. The course description states: “We will learn about scientists’ ongoing quest for answers to some of the most fundamental human questions: How did life originate on Earth? Is there life on other worlds? Are we alone in the universe? What is the long-term future of life in the universe?”
Now there’s a course that should appeal to almost every student.
The one area of general education that has been almost totally ignored is mathematics. Most of the required courses are dull, pedestrian, and often repetitive of what students took in high school. At many schools College Algebra is the mathematics course that students must take. However, this course is really second-year high school algebra. It asks of students no more than they were required to do in order to graduate from high school.
Some schools have bumped this requirement up a notch so to include a mixture of pre-calculus topics such as trigonometry and analytic geometry. While more advanced, such courses still consist of material that is part of the standard high school curriculum.
Another common math requirement course is statistics. This, too, is a high school-level course; indeed, many students arrive at college with AP credit for it. And statistics focuses narrowly on the analysis of data. It treats only superficially the important and interesting mathematical ideas that lie at the foundation of the statistical techniques being studied.
Students should be offered the opportunity to have an engaging and meaningful experience with important mathematical ideas. It isn’t easy to devise such a course for students whose mathematical background is limited, but it can be done. If we aspire to produce graduates who truly are liberally educated, it should be done.
In the late 1990s, I devised a course, entitled Great Ideas of Modern Mathematics, to address this need. The course begins by posing the question: “What is mathematics?”
A facetious, although not entirely useless, definition was suggested to me by a colleague some years ago. Mathematics is what mathematicians do. A more satisfying definition is the one offered by the English biologist T.H. Huxley. [Mathematics] is that [subject] which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation.
Huxley describes mathematics by telling us what it is not. Observation and experiment play no role in mathematics. Therefore, mathematics is not science. The truth claims of mathematics in no way depend on observations or experiments.
This leads us to a very provocative definition, formulated by the English mathematician J.J. Sylvester, contrasting mathematics and science: The object of pure Physics is the unfolding of the laws of the intelligible world; the object of pure Mathematics that of unfolding the laws of human intelligence.
In order words, mathematical knowledge is self-knowledge. It tells us something of how we as human beings think, how our minds work, how we process ideas, and it demonstrates our natural propensity for finding connections between seemingly unrelated concepts.
Students deal wtih numbers all the time, but unfortunately their only conception of numbers it that they are things you compute with. Our schooling, with its relentless emphasis on solving real-world problems, teaches us that we can use numbers to model all sorts of phenomena and that crunching these numbers can enable us to better understand real-world situations. It is unfortunate that, beyond computational applications, there is little appreciation of all that underlies the concept of numbers.
Yes, for the most part students understand rational numbers (i.e., fractions), but they have no understanding or even exposure to the concept of an irrational number (i.e., a number that can’t be expressed as a simple fraction, such as pi or the square root of two). In the course, we address, in a rigorous manner and with careful attention to formal argument and proof, questions such as:
- What does it mean for a number to be irrational?
- Do these numbers actually exist?
- Can we give specific examples of such numbers?
One of the weaknesses of math courses aimed at the general student is that they often morph into a math history course, thereby transforming the experience into something other than a course in mathematics per se. Another pitfall is that many such courses are survey courses organized around a multitude of topics. Each topic receives only superficial treatment and the student misses the experience of delving deeply into an important mathematical theory.
I prefer to focus on only three topics, chosen carefully to meet several criteria. Each topic must be one that students probably haven’t encountered before and represents a truly important mathematical idea. Each topic should be modern—one that originated relatively recently and is an active area of continuing research. This will help show students that, mathematics, like science, is an active and cutting-edge discipline. Finally, it must be possible to present a topic to students without loss of mathematical rigor. This last criterion is important since the whole point is to convey a meaningful sense of the mathematical experience—and math is rigorous.
Two topics with which I have had great success are group theory and the theory of infinite sets. These are relatively recent mathematical theories and are the subject of active and ongoing research. The mathematics represented by these theories is so unlike anything students have encountered before that the experience is one of a whole new world opening up before their eyes. They see for the first time that mathematics is not the dry, skills-based subject they encountered in high school but, rather that mathematics deals with important and provocative ideas that stimulate the mind and lead us in a variety of unexpected directions.
If taught properly, a course like that can convey to students some deep and fascinating mathematical ideas. It can engender in them a sense of wonder and appreciation, as well as an aesthetic experience they never would have thought possible from anything connected with mathematics. A theorem such as the irrationality of the square root of 2 is important, surprising, simple to prove, and easy to generalize. This combination of attributes imparts a genuine aesthetic quality to the theorem, which in turn leads students to see it as a thing of beauty.
I don’t claim that this is the perfect course. There are many paths one might follow to find an appropriate place for mathematics in the general education curriculum of our colleges and universities. But it is important that schools give serious thought to introducing a course that gives students an inspiring, meaningful, and engaging experience with mathematics. We should do no less if we aspire to graduate students who are broadly and liberally educated.