Teaching Philosophy

Do you agree with the following sentiment? It's really okay if you do not...

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." — Bertrand Russell, A History of Western Philosophy

Thank you for visiting! I want to give you a clear explanation of what I hope to offer, and how. Firstly, my goal is to help you feel more at ease and confident about mathematical encounters. That may sound like a tall order, because given our many experiences with math, how can we get to a place where its not so scary and difficult. And, how can I, or anyone, transmit to you this "right view" of mathematics, and do you need such a thing to succeed in the first place? Moreover, how can anyone convince us of the austere beauty that Russell wrote about so pretentiously?

Well, although I agree that mathematics is beautiful, beauty is in the eye of its beholder; and math is certainly expansive, and challenging, but I don't think there is a right view, and the seriousness around the subject does more harm than good...

We are all natural mathematicians: we make sense of the world through patterns, whether using mathematical symbols or not, and the experiences we have in school (or at home) are not remotely representative of what is possible with mathematical thinking. Culturally, we have yet to supplant the stuffy and torturous methods of the past, where rote memorization, drilling, and dreadful examinations are the norm. And although I think this comes from a place of seeking excellence as individuals and as a society, there are some widely held misconceptions about what success in mathematics looks like. For example, consider the matter around learning one's times tables. Personally, I remember being given no clear reasoning and provided no method for understanding these relationships between numbers; I was only given the objective of the task in the most abstract sense, and I only succeded, in a half-hearted way, because truly my parents and I were motivated by fear. Students are expected to provide correct answers as quickly as possible, in a needlessly competitive setting, and they are taught to parrot disparate and practically irrelevant mathematical facts. This is all very unfortunate and unnatural! Mathematical thinking, when it is extended by an academic education (in an ideal sense), should enable the student to interrogate the world with ever more wonder and creativity, but that is neither the aim nor the outcome that persists in most educational environments.

The idea central to my approach to teaching is that with a growth mindset mathematics is as tractable as anything else. Sure, this may require some creative approaches, but the science is progressing and there are promising results regarding novel strategies for teaching mathematics. A proper reorientation about our true capacities, together with confidence in the effectiveness of methods that nurture a growth mindset in students, success (in its truest sense) is tangible.

In a nutshell, tutoring with me involves collaborative problem solving, in an environment where there's a willingess to make mistakes! Math requires our full attention, but this should not be painful! I want to make it all less scary and approachable. That's how we learn! I will be an active listener and a steadfast companion, there to model and establish skills for long-term success. It all comes down to experiment, problem solving, and practice, keeping in mind that with a little hard work and dedication we can improve our abilities. I think developing enthusiasm for practicing mathematics would be an ideal byproduct — but you don't need to fall in love with mathematics to succeed. "Practice" can sound like boring repetition, but in reality it means thinking about something from different perspectives, in different contexts, thinking with one's hands, in conversation, with drawings and games, and so on!

With a small spark of curiosity, I think you have everything you need to go as far with mathematics as you wish to go. Practically speaking, the problem is that learning mathematics is largely cumulative: what you understand well enough at one step is necessary for understanding the step that follows, and so on to infinity. This can be very frustrating, for students of all experience levels! But imagine a colorful tower of wooden blocks. Three-year-olds love building these. With their own sense of taste and logic — picking the blocks up, inspecting them, setting them down — they slowly configure a veritable work of art in space. Then, knocking it down is even more fascinating, it would seem. To build a taller tower, we need to learn a few simple rules about the way our blocks stack up.

Depending on your experience, I would like to offer you opportunities to stretch a small set of mental habits to fill any gaps between all the "steps" in math. This doesn't mean rushing to cover years of content that seem critical. Instead, to prepare for the steps that will come later on in your mathematical journey, it is better to learn how to learn something well in the first place. Through open dialogue and simple exercises, I want you to develop familiarity with and confidence in handling mathematical objects, just as you would wooden blocks. But one of the coolest things about math is that if you master a single of those wooden blocks, you become a master of many things similar to it as well. Rest assured, something true in math will always be true, no matter where you go. Even though math is cumulative, it's not about memorization, but rather about deep understanding!

Ultimately, math is difficult, but we can rise to the occassion! Each of us has a personal relationship with the subject, and each of us should feel at liberty to ask questions and cultivate our own way of engaging with the world.

Yes, on a mathematics exam our answers are either right or wrong — but the way we arrive at them is a messy, emotional, human process. And it's okay to be emotional! In our studies, above all, we should guard and celebrate our humanity. Historically, from before and after the ancient Greeks, mathematics was taught as a liberal art — a way to build a bridge between us and nature. But the experience of crossing that bridge is meaningful in itself, and I don't think we ought to suffer as much as everyone pretends we should in order to find answers, whether they turn out to be right or wrong. Even Dr. Tao, perhaps the most brilliant mathematician of our time, struggles with mathematics — but precisely because he enjoys it and knows that it can lead to human flourishing.

Note on AI

Dr. Tao and many others are excited about artificial intelligence in mathematics because it has opened the door to more and more interesting and fruitful problems. Artificial Intelligence can be a useful tool, but it is important to delegate our cognitive tasks resposibly: it is not best put to use, for instance, believing it provides perfect answers; instead, at least if it is used and scaled responsibly, it actually reveals our misteps more and more quickly. And yet, it is ever tempting to follow expedient solutions, rather than meaningful ones. Consider what Gandalf said to Saruman: "He that breaks a thing to find out what it is has left the path of wisdom." The thing in question might be the nature of thinking itself! Something to think about...

If this resonates with you, I'd be glad to hear from you — reach out here to contact me about tutoring.