“When are we ever going to use this?” Every math teacher (and every parent who has helped his child learn math) has heard this question more times than he cares to remember. In a new book out from Encounter in July, J. Jacob Tawney says we have been giving the wrong answer.
Teachers often defend the mental strain required of math students by pointing to the ubiquity of math in our lives. They say things like, “You won’t always have a calculator and you’ll need to know how to do this,” or describe the high pay of careers in STEM fields. Not only are these answers completely unsatisfying to most students, Tawney says they undermine mathematics itself.
In Another Sort of Mathematics: Selected Proofs Necessary to Acquire a True Education in Mathematics, Tawney says we need to reshape how we think about math and how we teach it. Math is something to be experienced, wondered at, pondered, and enjoyed. It is not a tool to be mastered that will make you rich or a box to check for college admissions.
The first few chapters give a deeply philosophical defense of mathematics as worth studying for its own sake. But, like the rest of the book, it is written with a clarity and simplicity that makes the concepts approachable.
Tawney, the chief academic officer of the classical charter schools network Great Hearts Academies, asserts that mathematics is essential and uniquely human. That “there is something unique in the human soul that can only be satisfied by wondering about mathematics.” To fully live as a human being, then, requires us to study mathematics.
If he is right— and I think he is — this is reason enough to learn math, but he doesn’t stop there. He says we should study mathematics because it is good, true, and beautiful, giving a defense for each of these three transcendentals. He goes on to situate mathematics in its rightful place in the truly liberal arts, the historic Western course of education that was “about freeing the mind to know and to know in a particular way.”
These arguments hold much more weight than the idea that a student might need to know how to count out change to a customer. Tawney correctly understands that mathematics inherently ennobles the human spirit, that it forms the habits of the mind to be truer and more precise.
This not only means it is worth studying, but also that it cannot be replaced with a calculator, smartphone, or the newest development in artificial intelligence. If to study or to do mathematics is about making the doer more fully human, we ought to avoid these shortcuts because we should not want the calculation machine to take the wonder and joy of the mathematical proof away from us.
Tawney also briefly explains the implications of this understanding for math curricula. Our current system for curricular development prioritizes volume of knowledge because the developers believe mathematics is only a tool, is only utilitarian. If it is a tool, the more facts the student can recite, the more powerful the tool.
If, on the other hand, math is about leading the student to know what is true, good, and beautiful, the teacher should not be in a rush to “get through” more material any more than he should be primarily concerned about “getting through” all of William Shakespeare. Instead, he should be more interested in spending time investigating, proving, and marveling at the truths discovered.
This is not to say that facts should not be memorized or material covered, but that these should not be the first priority of a curriculum. Teachers should prefer instilling a love of real mathematics in their students over burdening them with endless topics and obsessing over standardized test scores.
This line of reasoning may lead quite a bit farther than Tawney takes it. For centuries, young men began their study of arithmetic in their early teens, not when they were five or six years old. A glance through one of the standard medieval textbooks makes evident why. Mathematics, from arithmetic on up, was concerned with understanding quantity and magnitude, with number theory and philosophy, not just memorizing the standard algorithms. A six-year-old is simply incapable of such abstract thinking.
In the early 20th century, optimistic progressives thought the most we should expect from students by the end of sixth grade was to master the four basic operations using integers and a few common fractions. Even this was done primarily for utility, not for the purposes Tawney describes. Take a look in a fourth-grade math text, and you’ll see we have crammed a lot more down into those books. Tawney suggests cutting the number of topics studied in half. Maybe that is just the start.
The majority of the book is dedicated to 30 proofs, theorems, and unsolved problems that are worth knowing. Because of the state of mathematical education, many readers probably just thought nothing could be more tedious, but that couldn’t be more wrong. The proofs and theorems are delightful, surprising, elegant, and clearly explained. The math is, in fact, interesting in itself. As Tawney reminds us often, it is up to us whether we recognize that fact: “We must choose to find mathematics interesting.”
To tell the end of any of the chapters would spoil the discovery for the reader, so you will have to go read them yourself (the book is available for preorder now). If you do so and choose to find it interesting, your efforts will be rewarded. Simply contemplating the truth, beauty, and goodness of mathematics is well worth the time and mental effort.
“Mathematics deserves to be wondered about, and that is precisely because it is good,” Tawney writes. “In fact, you can only understand the inherent goodness of mathematics insofar as you choose to stand in awe of it and wonder about it. And that means pausing to ask not first about its usefulness or its relevance but instead about the thing itself.”
