Finding Poetry in Mathematics
In her poem “Poetry”, Marianne Moore describes poetry as “imaginary gardens with real toads in them”. Since first reading this poem, I have that thought that this was also a wonderful description of mathematics.
Mathematics has its imaginary gardens and its real toads, although they are more commonly called pure mathematics and applied mathematics. This duality of mathematics, between theory and practice, between form and function, between the imaginary garden and its real toads, gives mathematics its power to inspire us with its beauty and elegance, and to serve us with its utility and remarkable effectiveness.
High school students can experience these two connected but distinct aspects of mathematics through their experiences with mathematical proof and mathematical modeling. Both proof and modeling call on the students to creatively use essential concepts and mathematical tools developed in class to generate their own knowledge and understanding.
The creative component of pure mathematics is found in conjecture and proof, which are central to the process of mathematical discovery. This process plays out under the strict constraints of mathematical abstraction in the idealized world of the imaginary garden. It requires precision of thought and calculation. In the garden, 𝝿 is never confused with 3.1459. The imaginary garden places no value of utility or application. There, logic and rigor, reason and verification, truth and beauty, reign supreme.
The creative component of applied mathematics is in mathematical modeling. Mathematical modeling also has constraints, but those are imposed by the real world. In the world of the toads, approximated reality is the master. But the approximation gives students flexibility in their use of mathematics that doesn’t exist in the imaginary garden. In the garden, if a theorem’s conditions are not all exactly met, the theorem doesn’t apply. It is of no use. Among the toads, if the conditions of a theorem are almost met, then we might expect the results of the theorem may be approximately realized (the entirely of Statistics rests on this notion). We constantly use approximations and even incorrect calculations (replace sin(x) with x, or assume events are independent when we know they are not) because the goal isn’t getting the correct answer (there is no “the correct answer”); the goal is gaining some imperfect, but useful, understanding of that component of the world being modeled. Practices that are essential in modeling are forbidden in theory, and this gives the modeler some enjoyable creative freedom which allows them to be imaginative modelers with far less experience than is required for being creative proof writers.
Both mathematical proof and mathematical modeling give students opportunities and permission to do it their own way. In both, students are expected to be creative, generating a conjecture and proof or a model from their own experience and insight. Ownership of the mathematics occurs when students have the flexibility to make decisions about what to solve and how to solve it themselves. This means they are thinking their way through problems rather than just remembering what they were told to do or repeating the teacher’s approach. Students gain confidence in their mathematical identity when they solve problems by themselves and in their own way. In both mathematical proof and in mathematical modeling, students can move from doing mathematics by themselves to doing mathematics for themselves.
Mathematics, like Robert Frost’s poetry, “begins in delight and ends in wisdom”.
Dan Teague, NCSSM Department of Mathematics