In Conversation with You — What Truly Sets My Math Teaching and Solutions Apart

I want to touch people with my art. I want them to say “he feels deeply, he feels tenderly.”

– Vincent van Gogh

(Click here to listen to my recording of this article.)

Hey everyone!

Time flies like an arrow over a vector! We’ve been together for a little more than a month. How’s everything going so far? Do you like the solutions and formula sheets I made? Today, I’d like to talk about the philosophy I’ve been following in my solutions and teaching style.

I don’t know if you’ve noticed, but the “Foreword” box on my teaching page is excerpted from an article I wrote: “Thoughts on What Good Math TAs Should Be Like.” I’m really serious about this article. I wrote it during the 2022 Fall Break, and over the past 15 months, I’ve revisited and refined it very consistently. The current set of 10 + 1 principles is linearly independent and spans my entire teaching style, meaning they in a way serve as a basis for our discussion space. The true essence of the article lies in the postscript:

In addition to knowledge, as a teaching assistant, what I hope the most is to convey a sense of passion for math to my students. When we love math, learning it will become fun and attractive, and we will become more creative and innovative. While some teachers like to focus on a kind of macroscopic guiding where they constantly ignore the basics of math and leave most details to their students, I don’t believe in that teaching style at all. A person will be led by their interest and enthusiasm in math to set up a dream of pushing the math field forward and truly act on it, while such interest and enthusiasm in math, to a large extent, come from the joy and confidence they gain in understanding and applying math to solving problems, and the only way to gain such joy and confidence is to be able to comprehend the detailed logic in existing math knowledge, instead of being intimidated by its mysteriousness. Therefore, if their teacher can deliver such detailed logic to help them truly understand seemingly complicated math knowledge, then they will automatically earn a sense of control in math, which will offer them the best platform for exhibiting their creativity and the help they really need in pursuing their dreams. This is what I believe in, and this is what guides my teaching style. I’ve always tried to do that as a teacher, and I always will.

Following this guideline, I’ve chosen to share all detailed solutions and discussion notes with you in my easy-to-understand style. You might think the most dominant characteristic of my solutions and formula sheets, as well as my teaching style in general, is the tasteful humor and the artistic smiley faces or cats in different colors. But, I don’t think that’s the biggest feature of my style. I believe the only thing that truly sets me and my solutions apart is the incredible clarity and ease of understanding. Many mathematicians tend to overlook this user-friendliness aspect of math teaching or solutions, assuming “all solutions are the same” or “if you know the knowledge well then you’re automatically a good teacher.” But let me be honest, in my very humble opinion, the term “user-friendly” doesn’t really apply to most, if not all, college or graduate-level math solutions or textbooks I’ve seen. My solutions are the single best solutions I’ve ever seen in my entire life, very objectively speaking, not just because I’m the author. I’ll explain why.

First, have you ever wondered why it’s so common to hear “Math is hard”? Let me tell you: it’s because most mathematicians are super used to hiding logic with words. They really are. They somehow think there’s some magic in using “complete sentences” in their work. But, you know what? These sentences are mostly a complete disguise of logic; logic is what truly makes math super easy for the human mind to grasp. Let me give you an analogy: a logically organized solution or teaching is like a roadmap, intuitively showing you how to get from Point A to Point B. Picture this: if you’ve got a roadmap, you can immediately know how to get to your destination from where you are, right? You’ll know which road to take, and which road seems the shortest, without needing any words to describe the roadmap. On the other hand, if you don’t have the roadmap, but only have someone else’s instructions in words, like “First, turn left to go to the overpass. Thus, it follows that we have to leave the overpass at a crossroads since the destination is not on the overpass. Therefore, we obtain that we can continue going straight after leaving the overpass…” This instruction sounds formal and professional, agree? But who the heck knows when and where to leave the overpass? Who the heck understands how and why “Therefore, we obtain that we can continue going straight”? Who the heck knows how to finally get to the destination? Come on, you don’t tell us, you want us to guess or what? Of course, once you’ve traveled this road or been to the destination before, everything makes sense from your description above because you already know the steps to take. — That’s why so many experienced mathematicians can’t recognize how bad their math notes or math teaching is: yes, they think their notes or their peers’ notes are clear enough and easy to understand, but that’s because before teaching they’ve understood everything, just like before traveling they’ve known all the roads. However, for beginners, such super confusing descriptive words would take an incredibly long time to figure out how to follow, because we beginners don’t already have a roadmap in our minds and everything is completely new to us. How can you expect us to figure out every omitted logical step from such ambiguous and incomplete words? What we truly want is a roadmap, which gives us an incredibly intuitive idea about which way we really should go and when we should go this way. After having this roadmap, not only can you reach the destination super easily, but you can also use words to teach and confuse others with the roadmap in mind (although I hope you don’t do it.)

So, the way I write my solutions is to make them like roadmaps, and all other mathematicians’ solutions I’ve seen so far are like descriptive words. What I’ve tried my best to do in my solutions is to unveil all the logic behind each step, connecting the solution into a clear logical whole. I don’t usually write many words in my solutions or on the board in discussions, because they’re not necessary anymore with all the clear logic. With this logical design, I am 100% confident that reading my notes can be extremely fast; even if you initially find a step hard to follow, after pondering over it for less than 1 minute (rather than 1 year!), everything will make sense. Because in my solutions or teaching, when I say A → B by “Formula Circle i”, where “Formula Circle i” is a theorem or result learned in lecture that states exactly “Given A, we have B,” it’s really hard to imagine how anyone can’t understand it super fast! Yes, that’s how I craft each step in my solutions. So, basically, you learn how to trust your own ability to understand this writing style; once you trust it, you fly. You know what? Given the choice, I’d exchange all my wealth for an algebraic geometry textbook crafted in the way I compose our linear algebra solutions. But unfortunately, it seems like the math field is waiting for me to be the first to do that.

But why do I care so deeply about the ease of understanding of my solutions? Because, as a math student myself, I feel deeply disheartened each time I get lost in the mysteriousness of a proof that could’ve been elucidated using my methodology, in a current math textbook I’m using. I feel deeply frustrated about the norm the current math papers and textbooks follow. What should’ve taken me only 1 minute to understand using my way of teaching would take me 6 hours using the current textbook. For the most part, that’s a complete waste of time. I can’t pass that kind of sadness onto you. I just can’t. I know how depressing it is. If I write solutions, I must do it right. As a math educator, I can’t write and share junk.

So, if I’ve given you all the details of the solutions published, what am I doing each time we meet in discussion? Merely reading and copying my notes? No, I’ll tell you what I’ve always tried to do there. There’s actually still a missing piece of the puzzle in my detailed logical solutions — motivation: why and what’s guiding us to think of deducing the next step from the current step. My solutions already tell you how to deduce B from A (which is, by “Formula Circle i”), but the motivation tells you why we think of using this formula and performing this step. In other words, it tells you I didn’t just pull this step out of the hat. Unlike logic, these motivations are better conveyed verbally (in words) and understood intuitively. That’s why I choose to disclose most of them in discussion instead of including them in the solutions. Again, without those motivations, my solutions can still be easy to understand but would lack a soul. With all those motivations, you’ll understand why I choose to go this way on the roadmap, not just how to get to the destination. Thus, you’ll be able to build your own roadmap in your mind to get to other destinations, meaning to solve other math problems when solutions are not available. You’ll be able to understand which formulas you should use, when to use them, and how deeply to use them. With that kind of understanding, excelling on exams is almost a given. They’ll pave your way by informing you about what’s in the box, allowing you to truly think out of the box in your future academic journey and career.

As a dedicated and kind person, I take your experience exceptionally seriously and personally, because I just value it. When some of you are not satisfied with your performance, I feel sad too. I never judge anyone by their scores or performance, but when you care about it, I care about it as well. Because I care about you. In my teaching career thus far, I don’t think I’ve ever turned down any requests from my students, as long as they are ethical and I have the discretion to make the decision. That’s because I believe in your judgment in your unique situation, which is hard for anyone else to truly empathize with, including me. I believe your decision-making in your academic life can do the best for you because you know yourself best.

I hope the information provided offers a clearer insight into our discussions and my teaching approach: to make math incredibly easy to understand in a friendly and attractive way. No matter whether you agree I’m fulfilling this commitment well or not, I promise I’m trying my best to do well, and I will do better and better by learning from you. I aim to make your view of math a little clearer and guide you toward informed decisions down the road. I eagerly look forward to seeing your continued growth and academic success!

Warmly,

Ivan :)

12:30 PM On February 11, 2024

Last Update: Feb 25, 2024