Like all classical subjects, mathematics should be taught with the aim of fostering students’ humanity. Too often mathematics is taught using a “factory” approach: Here is a theorem (never mind where it came from or what it really means), memorize it, here’s eight steps to apply it, now do thirty practice problems. This state of affairs is unfortunate because it does not allow students to truly reason, one of the many ways that students can reflect the image of God and make his glory manifest. Consequently, classical mathematics classes should avoid the factory method and instead nurture advanced analytical reasoning. They should let students wrestle with historically significant mathematical problems before showing students how to solve them. They should require students to not just answer mathematical questions but to justify mathematical reasoning, ask good mathematical questions, exercise creativity, develop mathematical intuition, and practice presenting mathematical ideas.
Mathematics should moreover be taught out of a love of wisdom and the good, the true, and the beautiful. Mathematics isn’t just a tiresome hoop to jump through to become engineers or doctors or make lots of money. Rather, it is one of the many ways to wonder actively, to deeply contemplate simple ideas, and to delight in abstraction, certainty, order, rigor, etc. A classical mathematics class should take time to let students simply stand in awe, whether in awe of the beauty of fractals, the inventiveness of calculus, or the unfathomability of infinity. Teaching mathematics classically should involve helping students to love the great ideas that have inspired mathematicians for ages, which in turn can help students to adopt a posture of humility, joy, and wonder in the presence of God’s glory.
Lastly, mathematics should be taught holistically, as part of the larger narrative of the great ideas. Ideally, a classical mathematics course would include historical context and time to think philosophically about the presuppositions at play in whatever mathematical framework is being studied. The goal of a classical mathematics course is not to replicate the symptoms of understanding mathematics (i.e. being able to get the right answers to a specific set of prescribed, isolated problems), but rather to pass on a love and understanding of the ideas that have motivated mathematicians to inquire the way they have down through the centuries. This way mathematics doesn’t seem so inhumanely abstract, so arbitrary, so at odds with curiosity and joy.