Maths class can be boring - or it could be the most fascinating subject. It's all up to you, the teacher to make maths engaging. In this article, we're going to be exploring how to use maths and philosophy to do just that!

This works at home, when teaching to small groups and in the classroom. It works because it's interesting, and *interest* and *curiosity* are the best reasons for learning anything.

The following is written by Anthony Barr, a homeschool graduate who grew up using the Math-U-See program and has studied history and literature at Eastern University in America. We recently wrote an article about bringing maths to life using the lens of history as well.

Here's Anthony's story:

#### Maths is About Big Ideas

Maths really clicked for me as a junior in college when I took a course called 'Mathematics in The Western Tradition' which doubled as a philosophy and maths history 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 realisation 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 realised that maths is about *Big Ideas*: beyond dry word problems and rote memorisation, the field of mathematics opens up into an ongoing conversation centred 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.

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.

The great thing about philosophy is that it is something we can all participate in these debates, at the level that we are able. You can make maths engaging for a group of first graders by helping them wonder why the sky is blue. In doing so, 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 maths, 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.

#### Conclusion on Philosophy and Maths

For most of us, our daily experience of mathematical reasoning is tied to the practical – like doubling a recipe while cooking. I hope this post opened up possibilities for engaging with mathematics on a more theoretical level - it's a very exciting journey!

#### Discussion Prompts

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 primary 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.

*Have you tried talking to your students about the link between philosophy and Maths? How did it go? Let us know in the comments below. *

We're interested to know your thoughts. We'll have more articles to help you make maths engaging coming out so keep an eye on the blog.

The Team at Maths Australia