Math furnishes a universal grammar for solving real-world problems with a man-made system of signs based on patterns, puzzles, and proofs—a system designed to increase our capacity for precision. Ironically, that synthetic system (math) trains the mind to see empirical patterns previously overlooked—but only after we inhabit the theoretical world with its symbolic order.
In this issue’s interview, Dr. James Tanton supplies us with more than enough reasons for studying mathematics with enthusiasm. According to Tanton, “We say mathematics has much practical utility, that it’s all about applications to real world problems—which is quite motivating! But, the other side of the coin is that mathematics is philosophical and joyous and artistic and all the rest, which is what I value, but people don’t tend to see that side. I want to share that side of the coin and how both sides work together. Practical application often motivates such beautiful abstract mathematics; while, at the same time, abstract thinking often motivates practical mathematics.”
As this issue goes to press, we are all living with the new reality of a national quarantine—which has me thinking about how our subject matter, mathematics, pertains to our current concerns.
Let me begin with a few examples: predictive models of the infectious virus; actuarial tables of relative risk; and economic forecasts of GDP. Each of these domains relies upon the language of mathematics to achieve its applied ends: anticipating the growth and hoped-for containment of a fateful pathogen; estimating the cost of specific behaviors undertaken by different demographics; and predicting the future productivity of a society—perhaps even the world! These and many other areas of modern life depend upon math to abstract and evaluate variables from lived experience.
But, how does mathematics accomplish such feats? It does so by dint of human reasoning and imagination. As the great 20th century mathematician George Polya explains, “The use of signs [both linguistic and mathematical] appears to be indispensable to the use of reason.” In particular, math provides us theories of number and space that improve our reasoning about the universe, by use of an imaginative world of symbols and relationships. Math furnishes a universal grammar for solving real-world problems with a man-made system of precise signs drawn from patterns, puzzles, and proofs. Ironically, that synthetic system trains the mind to see patterns previously overlooked—but only after we inhabit that theoretical world with its symbolic order. As Polya points out, “Teaching to solve problems is education of the will.”
In the modern era, mathematics has arisen to become the international lingua franca of science, enabling empirical researchers and scientists to symbolically represent all types of patterns—biological, chemical, geological, astronomical, etc. Thus, math enables science to explore physical phenomena from various vantages.
So, it seems quite fitting to look more closely at the theoretical system (mathematics) that underlies so many modern advances—in science, technology, engineering, etc. To begin, we discover the fundamental impulse behind our study of geometric propositions, as Jake Tawney takes us on a tour of “Proof,” revealing the dramatic narrative behind the mathematician’s search for truth—a genuinely poetic quest.
Then, one of our colleagues from New York City, unveils the mystery behind pure mathematics: Williams explains that the “interplay between applied math and pure math” arises from the ability “to think in a certain way about a certain sort of thing (mathematics).” While we all await the next great application, be it microprocessor or skyscraper, the truly great achievements of mathematics are bestowed on those who “first think abstractly about charming little problem[s].”
We are also pleased to have a teacher trainer and curricular author, Tom Clark, join us with his essay on the “guaranteed” path to mastery: a Socratic conversation that welcomes errors as the first steps to understanding. Learning how to retrace one’s missteps—and chart a path forward—forms the high road to a positive personal experience with math.
Finally, we have a delightful interview with Dr. James Tanton, of Exploding Dots fame. (If you haven’t seen the 10-minute YouTube video, take a peek.) For Tanton, math can never be a dull, lifeless experience, if you understand its essential nature: “It’s really a mulling process, giving students the permission to be human: to not know what to do, to flail and flummox. But let that mulling and the subconscious work on things. And then you’ll experience those epiphanies…”
Epidemiological modeling of COVID-19, risk assessment, and economic planning are representative of the myriad fields dependent upon mathematics. But, those fields and the continued advancement of modern research will only flourish if we are able to transmit the ordered, imaginative world of mathematics to the next generation. From this issue of VIRTUE, we discover the proof, charm, conversation, and epiphanies at the heart of math—which should be more than enough reason to pursue its excellence in K-12 classical education.
Dr. Robert Jackson is the chief academic officer of Great Hearts America and the founding director of the Institute for Classical Education.
In the modern era, mathematics has arisen to become the international lingua franca of science, enabling empirical researchers and scientists to symbolically represent all types of patterns.
Edna St. Vincent Millay (1892–1950) was one of America’s most talented 20th century poets, whose brilliance as a sonneteer has secured her place in numerous collections of children’s poetry.
About Virtue Magazine
Virtue is the flagship publication of the Institute for Classical Education. It disseminates stories, ideas, research and experiences in classical education to readers across the nation, helping them pursue the classical ideals of truth, goodness, and beauty.