When I am in the middle of a lesson, cooking along explaining things, working examples, perhaps rejoicing in the beauty of the subject matter (or my own perception of cleverness in thinking up a new analogy), there is always one question a student can ask that is guaranteed to throw me off my groove. That dreaded question is, “Why do we have to learn this?” As we get more seasoned and experienced as teachers, we perhaps learn ways to set things up in the beginning so that students are not tempted to ask this question. But even though I have been teaching for 16 years now, I still get it from time to time.
We may as well extend the question to all of mathematics. Why should students learn math at all beyond the simple skills needed to count change and pay bills? The need for learning more advanced mathematics may be obvious for students who will grow up to be scientists, engineers or financial officers. Naturally, one cannot do what an engineer has to do without a substantial background in advanced mathematics. But are algebra and geometry necessary for everybody? It is very easy to think, “Of course everyone has to take algebra! Everyone always takes algebra!” But our task is to see if this response can be justified.
For starters, we can probably all think of examples that illustrate just how challenging this question is. My own youngest daughter, now a senior in high school, struggles mightily with math and longs for the day when she can be done with it. She does feel bad about this, since I am her dad. Still, she wishes she lived in Jane Austen’s world, needing only to develop the “accomplishments” of a young lady, which happen to be the very things she loves— music, needlework, drawing, literature, and French. I, too, sometimes wish she could live in that world. That would be a nice life.
A completely different example is found in one of the great literary lights of the mid-twentieth century, Thomas Merton, author of The Seven Storey Mountain. Many of Merton’s formative years were spent in Europe, and as a youth Merton set his sights on studying at Cambridge. However, the very demanding exams he would have to take included mathematics that he had no talent for. He almost despaired of realizing his dream but then learned that he could avoid the math exams by even higher level achievement in the humanities, namely, studying his literature in the original languages and being examined accordingly. So, in addition to the classical languages he mastered and read Italian, French, and Spanish, passed the exams, and went to Cambridge.
It would be difficult indeed in contemporary times to design a school that can give prodigies like Merton what they need, and still be suitable for ordinary kids, as most of our students will be. I think I would be happy for any prodigy like Merton or Mozart to focus mainly on where his gift lies. I’m not going to worry about whether Mozart or Merton ever take algebra. But such prodigies are rare, and we must develop a rationale for our schools that applies to the other 99.999% of our students. The example of my daughter is probably a better example to challenge us as we address this question of why students should take mathematics. What about the ordinary kids? Why can’t girls be taught the way girls in Jane Austin novels were taught?
Before we develop a justification for including math in the curriculum, let’s pause for a moment to define the subject. To do this, I would like to make some observations about how the human mind works. Classical and Christian Education (CCE) schools typically emphasize the Trivium—grammar, logic, and rhetoric—and as a result students tend to exhibit above average performance in written and verbal expression through language. This is laudable, but interacting with the world and other people in the world through the written word represents only one part of human capability. The human mind is also wonderfully adept at imagining and discovering patterns, and communicating these symbolically. Moreover, as we have discovered during the past 400 years with the rise of contemporary science, our response to God’s creation is sometimes better facilitated by words, as in poetry or prose, and sometimes in the form of symbols, as in music and architecture. When the subject matter at hand deals with patterns, and with communicating ideas about specific patterns, communicating through the use of symbols is much more efficient than communicating through words.
This brings me to my definition of mathematics, a definition that is not original with me. I define mathematics as the study of patterns, a study that includes manipulating and expressing ideas about patterns symbolically and quantitatively. And though I will not be specifically addressing the Quadrivium in this essay, it seems to me that the key characteristic of the subjects in the Quadrivium, and the key thing to be preserved in education from the Quadrivium, is the centrality of searching for, identifying and describing patterns.
And now to our justification for including mathematics in the classical curriculum. Although it may sound strange to those espousing classical education, the first reason for teaching mathematics is the sheer practicality of well-developed mathematical skills. Please do not howl and stop up your ears; I am neither a modernist nor a utilitarian. But I ask, as I once was asked, “Should classical education be an ideal thing, or a realized thing?” Since we are here trying very hard to realize it at our schools, we must answer, of course, that it is to be a realized thing. Realizing any educational paradigm in any culture must involve the practical cultural question of who gets educated and why. In our culture it is not only the elite who get educated; it is everyone. This is a plain fact of democracy. We have no formal class system, we promote the freedom of the individual, and we have an educational system that has as its fundamental goal the broad education of the entire population so that every child has the opportunity to make his or her way in the world according to his or her own abilities and industry. In this country, in this century, education is for everyone and must serve the need for everyone to function in contemporary society. To do this, education must be practical. This means it will include living foreign languages, chemistry, and, of course, mathematics.
Practicality is defined by the age in which one lives. Practicality used to be about computing quantities of seed for planting, figuring sizes of parcels of land, or calculating exchange rates and unit quantities for commodities. Our high-tech age brings different requirements for the citizens. Nearly every job in the professional world, and many jobs in the trades as well, involve fairly sophisticated math. One doesn’t have to be an engineer to get into budget forecasting, statistical analysis of surveys, setting up spreadsheets, pre-tax paycheck deductions, network download rates, amortization, interest and tax calculations, cost vs. benefit analysis, the storage capacity of a back-up hard drive, and on and on.
Now, if practicality is one of the reasons for teaching math to everyone, it is also one of the criteria for determining what mathematics everyone should learn. When math is taught to everybody, practicality is a primary issue. This is why it is wise to require math studies to continue at least through Algebra 2 for all students possessing average or above average mathematical ability. Just as we expect everyone to gain a serviceable level of English proficiency for reading and writing, but do not expect everyone to be a writer, so in math we set the goal of a serviceable level of math proficiency suitable for life in the contemporary world, but do not expect everyone to be an expert in calculus. For many students this goal means that studies in math continue through Algebra 2, with perhaps some introductory statistics.
A student might reasonably argue that learning exponential decay functions or rules for powers and roots goes far beyond what is practical for most people. This is a reasonable point to make, and my response to it is two-fold. First, learning these more advanced skills in Algebra 2 is analogous to athletic training. Athletes train with arduous exercises, but this does not mean they will repeat these same actions in the game. The drills are demanding and are designed to get the athletes in shape so they can handle the actual game effectively. Similarly, we will expect that some mathematical topics and problems will be taught for their training value, and not because a particular type of relationship or function will be specifically needed in later life.
Second, contemporary issues constantly require citizens to think in quantitative terms, particularly in terms of a functional relationship between two or more variables. Mathematical relationships are now ubiquitous in modern society in every discussion of medicine, climate change, computer technologies, energy efficiencies, taxes, investments, survey results, profitability, trade, and so on. Exponential and power/root functions do come up all the time in particular fields of endeavor. But more generally, learning to handle them trains the mind to think in quantitative terms, with legitimate mathematical reasoning.
We are Americans living in America, and for 200 years Americans have been world-famous for their interest in practicality. If you want to get anyone’s attention in our culture today, including the professionals who are the parents of our students, you had better have a firm grip on the practical side of your discipline. Nowhere is this more true than in math and science. The competitive, high-tech, corporate-driven world we live in is unforgiving of weaknesses in math and science. If you can’t handle the math or the physics, there are plenty of students in developing countries who can, and they will take your place at the table and leave you to work your way up to an assistance manager’s job at Best Buy. Solid skill in math and science is very practical.
This brings me to one final point I wish to make on the practicality issue: without mastery (one of my favorite topics), no practical skill has been gained, and your efforts in the classroom have been in vain. Schools cranking out graduates that cannot do math are a dime a dozen. Our challenge in the CCE movement is to find a way to break through these decades of low performance into a new realm of proficiency and competence. Is this possible in a democracy? Ultimately, I do not know. But I think if we are wise in our efforts we will have our school families on our side as we do the hard work of developing a mastery-based curriculum.
So much for the practical value of teaching and learning mathematics. But while the modern world may be driven almost exclusively by the practical, for teachers in schools espousing a classical philosophy, the practical is not nearly enough. The reason for this is that as important as all the practical things are, they do not even come close to exhausting what being human is all about, and the core of the classical model of education is the goal of developing good human beings, not merely equipping people with practical trades.
Once we crack open the door on classical considerations for why we should study math we find that the reasons are just as extensive, if not more, as those on the practical side of the question. We could, for example, consider further my earlier point about the way the human mind works, and its capacity for expression in words as well as in mathematical symbols (as well as in the forms, colors and harmonies that are the raw materials of the arts). Or we could consider the matter from Plato’s point of view. In the Republic Plato taught that the proper subject for the education of a free man is that of being, the realm of the transcendent and permanent, as opposed to becoming, the realm of the temporal and transient. This was because he recognized in humans some kind of eternal, transcendent soul, and he viewed the proper task of education as feeding that transcendent soul. He saw mathematics as deeply connected to permanent, unchanging, transcendent truth, and thus a fitting subject for human beings to study. A third direction we could go would be to consider the Christian doctrine of the cultural mandate, and our understanding that Scripture charges God’s people with using Creation and all art, science and technology to improve the lives of fellow human beings, which is part of carrying out the Second Greatest Commandment. Finally, we could consider classical education from the point of view of pursuing truth, goodness and beauty as a means toward the development of wisdom and virtue.
For the present we will consider only the last of these possibilities, the pursuit of truth, goodness and beauty.
A good definition for classical education is the development of wisdom and virtue through the pursuit of truth, goodness and beauty. This ancient trilogy, reflected so vividly in Scripture in passages such as Philippians 4:8, focuses our attention on the deepest aspects of our humanity. G. K. Chesterton once wrote, “Art is the signature of man.” Creating or studying art requires the appreciation of truth, goodness, and beauty. Interestingly, so does making progress in fundamental scientific research. Let’s briefly consider truth, goodness and beauty and their relationship to mathematics each in turn.
The nature of truth has become clearer since the mid-twentieth century, for now we recognize that science and math are not so much concerned with discovering “truths” about the universe as they are modeling the universe. Students do not generally appreciate this until we lead them into discussion about it. Instead, they tend to take the findings of math and science as givens, as unchanging, eternal verities. But then we lead them to consider that science is not about discovering truth; it is about modeling the apparently infinite complexity of the natural world in an unending attempt to understand it better. And we lead them to understand that a similar principle applies to mathematics. The most glorious discoveries have
been realized through learning the language of nature, mathematics, beautiful structures that can only be described mathematically, such as Maxwell’s Equations describing electromagnetism or Einstein’s General Theory of Relativity describing gravity.
But we also know that the connection between mathematics and truth is elusive. Kurt Gödel’s 1931 theorem demonstrated that mathematics can be consistent or comprehensive but not both. And before that the nineteenth century realization that Euclidean geometry was merely one convenient geometry among many geometries, and did not carry truth about the structure of the universe the way people had thought it did since the days of Euclid himself, brought many a philosopher to tears. If Euclidean geometry was not true, what was it? A great question; one we continue to explore. As I said, students do not appreciate these things unless we lead them into the discussion. However, once we distinguish these studies from truth itself and begin to use the arena of mathematics and science as a field for the continuing pursuit of truth, a deep and fruitful discussion begins.
Goodness is all around us in math and science for the simple reason that God declared his creation “good.” Thus, an element of our interaction with nature through math and science is the recognition that it is good that the apparent diameters of the sun and the moon as viewed from earth are nearly the same. It is good that the constant of proportionality in the relationship between mass and energy is simply the speed of light squared. It is good that the number of ancestors in each generation back from a given honeybee is given by the Fibonacci sequence. It is good that the planets’ orbits may be characterized accurately (though not exactly!) by Kepler’s Third Law of Planetary Motion. The double helix of our DNA with its multiple layers of instructional encoding and its capability for self-replication is inexpressibly good. So are the navigational abilities of migrating birds, the Doppler- shift detection capabilities of bat sonar, and the hexagonal shape of ice crystals. It is very good that all of nature displays a magnificent, sublime mathematical order that even non-Christian scientists have described as essentially miraculous. And it is very, very good that we humans have the cognitive ability to perceive and describe this order— these patterns—with mathematics. When students learn mathematics, the doors to see these things open before them. What could possibly be better than learning the language in which nature speaks to us, a language that enables us to behold the very goodness of God?
The third object of our pursuit as we develop wisdom and virtue is beauty. The relationship between mathematics and beauty is nothing short of mystical. It has been written about for ages, and illustrated in countless ways by countless writers, and yet we never tire of the subject. For many decades now scientists have recognized that the most valuable physical theories are those that are expressible in beautiful equations. Beauty has become a research tool, enabling us to probe the mathematical structure of the creation further and further. As with truth, leading students to see and appreciate the deep relationship between beauty and mathematics takes no small amount of effort. One has to begin by defining beauty. Then we have
to establish the criteria we all use, usually subconsciously, when we make aesthetic judgments of all kinds. Finally, we have to demonstrate how these same aesthetic criteria apply in the domain of mathematics. As I said in the beginning, mathematics is the study of patterns, and patterns amaze and enthrall us with their beauty, from the patterns in a carbon nanostructure to those in the endlessly fascinating Mandelbrot Set. It is worth the effort to lead students to the point where they can consider and ponder beauty through the lens of mathematics.
Why should students study mathematics? We have seen that the study of mathematics is eminently practical, as practical as knowing how to read and write. And we have seen how mathematics provides a forum and a framework for the exploration of truth, goodness and beauty, a pursuit at the heart of our humanity and at the heart of the classical understanding of how humans should be educated. So on the question of students studying math, I think at this point it is safe to ask, why shouldn’t they?
