(The following post is Part 3 in a four-part series on studying math through the lens of other disciplines. We believe that students thrive when they can form meaningful connections across different areas of study. And we also know that many students who are disillusioned with math can find fresh inspiration in seeing how math interacts with other subject areas that they care about. With this in mind, in this series we want to provide ideas for supplementing math education to reinvigorate your student’s learning. Previous installments in the series have included History and Art. Our final entry in the series will include studying math through music.)
If your student is like me, textbook math is often…less than engaging. Order of operations, differential equations, and long division…yawn! We need to learn all of these things, but sometimes it’s more fun to get lost down a rabbit trail of ideas.
Math is About Big Ideas
Math really clicked for me as a junior in college when I took a course entitled Mathematics in The Western Tradition which doubled as a philosophy and history of math course – while also counting for my quantitative reasoning requirement. In this course, we worked our way through the development of mathematics, from the mystery cults of the Pythagoreans through Cantor’s radical realization that there can be greater and lesser sets of infinite numbers (I still only somewhat understand this one.) For the first time in my life, I realized that math is about Big Ideas: beyond dry word problems and rote memorization, the field of mathematics opens up into an ongoing conversation centered on exciting philosophical questions like, “why is there something rather than nothing?”
Your student may find the philosophy of mathematics to be an engaging way to supplement their mathematical education, and a source of inspiration for those times when the discipline feels tedious or dry. Sometimes it’s worth putting the textbook aside (it’ll be there waiting when you return), and putting on your philosopher cap.
Related related blog post: How and When to Take a “Math Break”
You and your student can start by just asking questions: What is a number, and does it exist? How do you know?, and letting the conversation unfold from there. I have some sample “Discussion Prompts” at the end of this post, but before I list those, here’s some background on the philosophy of mathematics.
Ontology: What Is a Number, and Does It Exist?
Numbers are profoundly strange. We have names for them, we talk about them, and we use them in our daily activities. But we don’t see numbers the way we can see butterflies, or feel numbers the way we can feel the wind on our skin. And while music and math are very obviously linked, particularly in terms of rhythm, we don’t hear 2 singing out good morning. So in what way can we say that a number is “something” that exists?
The subfield of philosophy that these questions fit into is ontology, the study of being, essence, and existence. Various philosophers have addressed these very questions in differing ways. The ancient Greek philosopher Plato believed that numbers exist as Forms – immaterial and invisible realities that shape the material world. Medieval theologians like Augustine often taught that numbers exist as ideas in the mind of God. The 20th century mathematician Bertrand Russell believed that all math is fundamentally a self-contained system of logic. And many postmodern thinkers maintain that math is basically a made-up language that helps us achieve practical goals but isn’t true in any absolute sense.
I have my own beliefs about what a number is, but that isn’t the point here. The great thing about philosophy is that it is something we can all participate in these debates, at the level that we are able. When a group of first graders wonder why the sky is blue, they are just as engaged in philosophy as the tenured Harvard professor.
Epistemology: How Do We Know?
Okay, so we might have strongly held beliefs on what numbers are, but how do we go about justifying those beliefs to ourselves and others? How do we know that our beliefs about numbers are true?
Epistemology is the subfield of philosophy that deals with questions about how we know what we think we know. The goal in epistemological investigations is to produce what is called a “justified true belief,” a fancy way of saying that we want to demonstrate both the right belief and the right reasons for that belief. Suppose, for example, that I was locked in a room without windows in the month of December and I said, “I believe it is snowing outside, because it is winter.” Now, it could very well be snowing outside which would make my belief true, but it doesn’t snow every day in winter, and so there’s no adequate justification for my belief. But suppose my room did have a window, and I observed it to be snowing. Most of us would readily accept that I have adequate justification for my belief. Though of course it could conceivably be the case that I’m trapped in a virtual reality simulation and am being tricked into seeing snow where there is none, the far more likely scenario is that it is indeed snowing.
When it comes to the epistemology of math, we can produce proofs (leading to justified true belief) for all sorts of things, such as demonstrating that 1 + 1 = 2 or that “a squared plus b squared equals c squared.” But a belief in the validity of something like the Law of Noncontradiction – which says for example that a shape cannot be both a square and a triangle – is something that is not provable but must be held as a self-evident first principle. Of course, the question of what is a first principle and what is a contingent (and provable) idea gets tricky fast, and that’s yet another dimension of the ongoing conversation that is mathematics.
For most of us, our daily experience of mathematical reasoning is tied to the practical – like doubling a recipe while cooking. But I hope this post opened up possibilities for engaging with mathematics on a more theoretical level. For another take on engaging with math in unconventional ways, check out this post on exploring math through its often incredibly exciting history.
– For the youngest ages, I would simply start by asking, “what is a number,” and seeing what your student comes up with. I’m often shocked by how perceptive even the youngest learners are, and how far their curiosity and wonder naturally take them.
– For middle school students, consider also asking about other immaterial things we speak of as existing, like love, and how numbers are or are not similar to these other entities.
– For high school students, consider supplementing your discussion by reading this article on Plato’s famous Meno dialogue, which will be sure to provide all sorts of interesting questions to ponder and discuss. If your student feels particularly inspired, they can even read the full Meno dialogue.
– What does mathematics have to do with virtue formation? According to Francis Su, former president of the Mathematical Association of America, studying and practicing mathematics can help cultivate virtues like perseverance, integrity, and love.
– One core belief that we have at Demme Learning is that students need to be active participants in their own learning, and part of that means being invited to see themselves as already being mathematicians participating in the tradition of mathematics.
– The understanding that mathematics is inherently philosophical fits well within a classical education framework.