teaching

Here, “teaching” does not refer to listing facts or data. It is straightforward to tell someone a fact. It is harder to explain a complicated idea that builds on pre-existing knowledge. Everyone has slightly different ways of understanding the world, and so they will digest new ideas in slightly different ways.

A reasonable starting point is to understand their current worldview. That sounds obvious, but it is easy to imagine yourself as “the knowledgeable one” and the student as a sponge. In other words, be weary of giving long answers to short questions right away.

If you’re speaking 1-on-1, asking questions is great. Your initial goal is to discover their current worldview as quickly and accurately as possible. But be careful. Your end goal is to build the missing stuff on top of the stuff they already know. If you’re only aware of the pieces they are missing, what concepts will you plan to build upon? You need to have at least a guess of what your audience knows. That’s why we group students into years and define prerequisites for classes. That’s why we refer to a “tree” of knowledge and why textbooks go in linear order.

Preferably, you would have more than just a guess. So if a student asks for help understanding progressive tax rates (or orbital mechanics) and you spend a couple minutes chatting, and all you learn is that “the student does not understand progressive taxes at all” (or orbital mechanics), are you ready to start explaining? I would say no. I think that you should continue to back up until you find a good starting point. The alternative is jumping into an explanation and hoping that the student knows everything right up to but not including progressive taxes (or orbital mechanics).

It seems like some people take a different approach. They will start explaining at the highest level of detail, and back up if the student doesn’t understand. In my experience, this tends to go poorly for a couple of reasons:

1. A lot of time, words, energy, and trust get wasted trying to find a good starting place.
2. Saying “I’m not familiar with that concept” is easier than saying “I’m completely lost. Your last three sentences did nothing for me.”
3. The student is expected to differentiate “confusion because this is a new concept, and the explanation is not done” from “confusion because I am missing prerequisite knowledge, and I will become more confused the longer this goes on.” The student is unlikely to know the difference.

Instead, I prefer to let students start explaining themselves and see where their knowledge starts breaking down. It gives me a clear understanding of what they know and gets them into the habit of speaking their thoughts aloud.

Encouraging an open, two-way dialog is one of the most important elements of teaching.

Lecturing at a student is like building a house with your eyes closed. You know which piece of the puzzle comes next, but without seeing how the previous piece landed, you’re just hoping that it will all fit together nicely. Ideally, as you spend more time with a student, you are encouraging bi-directional communication. There are two huge reasons why this is helpful:

1. You have feedback on how the student is understanding your words.
2. The student feels excited to grapple with your explanations.

I find that the second point is often neglected because it is more “feelings”-based and less “information”-based. However, there are a lot of reasonable concerns bouncing around in a student’s brain during a conversation:

• Was my question the right question?
• Does this person think how I think?
• Am I wasting my time?
• If I tell them that this conversation is not helpful, will they get offended? Will they double down?
• Does my lack of understanding mean that I do not belong here?
• Are they hinting that I am taking up too much of their time?
• If I ask for clarification, will I sound dumb?
• If I sound dumb, will that close doors in the future?
• Should I just nod along?

Each one of these thoughts adds to a student’s cognitive load, which prevents them from engaging with the contents of your words. Despite humanity’s lofty achievements, most people are bad at controlling their attention second-to-second. To compensate, explicitly address these questions and avoid unnecessary confusion. Recognizing these concerns hints at why “jumping in and backing up” is so costly. It confuses students and sets off a chain of unhelpful thoughts.

Encouraging an open dialog is easy or hard, depending on the student. Regardless, I consider it a prerequisite, so I personally won’t proceed until I have some level of back-and-forth.

Some things that I like to say:

• What I heard you say was…
• It sounds like you’re asking about…
• If I say “X” does that mean anything to you?
• How much do you know about…
• Interrupt me as soon as I say something that you don’t understand.
• You are in charge here.
• Am I explaining myself well?
• Stop me immediately if you already know this.

If you are empathetic while teaching, then you will eventually notice that some people have very different worldviews. They might think about a specific idea through algebra when you know it through geometry, or they might focus on a system where you see a set of components. These differences are both beautiful examples of diversity, and also sources of frustrating teacher-student interactions.

In general, if you are so inclined, spending extra time to digest the student’s point of view is a rewarding endeavor. If it is a fatally flawed perspective, then you can discover why. Correctly identifying the flaw in their current understanding can motivate the student to discard this maladaptive worldview. Furthermore, it shows that you value the students thoughts. That can gain their trust and respect.

If their perspective seems fine, then you can adopt it and build your explanation inside of their existing framework. Along the way, you might discover new insights of your own. If you ever encounter this specific point of view again, you will be more prepared. I would dare to say that most important concepts have multiple worthwhile perspectives. Knowing only one paradigm is not always sufficient.

Bridging large differences is an acquired skill. As the differences get larger, the conversation looks more and more philosophical. At first, this might sound like a dangerous waste of time. However, remember that every field of study is ultimately built upon philosophy. Once a popular consensus emerges, we put the philosophical debate to bed. If you want to understand a concept well enough to reinvent it, then you should strive to understand the philosophical decisions that went into building it.

At the very least, you should be humble about your level of knowledge. Often, the question “Why do we do it this way and not that way?” has only one truthful answer: “I don’t know, but I’ve never seen that other way and my way feels natural to me now.” In all likelihood, the student did not accidentally stumble upon an excellent alternative. However, it does no good to haphazardly invent a reason that their view is wrong. A good explanation to dismantle their perspective is helpful, but a bad explanation will just cause more confusion and justified distrust.

To encourage this intellectual humility, remember that many philosophical debates took years to resolve. Some are still ongoing. These questions have material effects on how fields of study are taught or practiced. Here are some examples that come to mind:

• Bayesian vs. Frequentist
• Western Music Theory vs. Eastern Music
• Great Man Theory vs. Historical Determinism
• Copenhagen vs. Many Worlds Interpretations of Quantum Mechanics

Earlier, I said that the student should “feel compelled to grapple with your explanations.” Most people who are asking questions are interested in learning, but not always. The final office hours before a problem set is due are famous for very low ratios of learning per question asked.

There are many things that can inhibit learning. They include emotional stress, external pressure, guilt, physical discomfort, and a desire to impress. There is also a desire to get the right answer as quickly as possible.

On some level, it is hard to fault students for this mentality. They are told that hard work will result in success. Hard work means completing the problem sets assigned to them. Everything in their lives revolves around getting the right answer. And yet, a conversation around getting the right answer looks very different from understanding the material.

It is important to recognize when a student wants understanding vs. wants to be given the right answer. It is not important to recognize this because you want to deny a student easy points. It’s important because you do not want to optimize your teaching style for students looking for points.

Many teachers will habitually adopt an adversarial relationship with all students, where they answer every question in oblique and cryptic ways. This is destructive to communication and, very often, the wrong way to go about pedagogy.

Sometimes, you are obligated to withhold the answer to a problem set for the sake of fairness. If that is the case, then try to explicitly communicate this, rather than slipping into an adversarial relationship. For example:

• Say, “Maybe that would work, but I can’t answer such a specific question about the problem set” rather than “maybe.”
• Say, “Generally, yes, that is a good thing to try. It’s probably worthwhile to try it here as well, but I can’t tell you if it will work on this particular problem” rather than “maybe.”
• Say, “I don’t fully understand, it doesn’t sound very familiar; If you explain it more, I might be able to help,” rather than “maybe.”

If you are not trying to withhold information from the student, then it is often good to encourage students to drive the interaction. Some lines that I like to say:

• How can I be the most useful to you right now?
• This is your time to ask for whatever you need.
• What specifically are you looking to accomplish?
• Do you want to know how to get the answer, or why that is the right answer?
• Do you want to know how to solve this problem, or all problems like it?

In my experience, teachers and students normally have the exact same incentives. The act of “aligning incentives” is normally a communication challenge rather than a game theory challenge.

As an alumni, when visiting Cornell’s campus, I sat in the Math Support Center. A girl taking calculus was waiting for a tutor to be available. Because I still looked like a student and because I had done it many times as an undergraduate student, I impersonated a tutor.

She was solving for the maximum of a 2-variable polynomial over a rectangle. She had found the critical points on the interior of both the rectangle and the four edges. She was confused about the next steps. As I went over her understanding, two things became very clear: (1) she wanted a conceptual understanding, the why, and (2) she was doing the problem correctly, and she knew the final step (checking the corners). It seemed like she was just unwilling or unsure about following a procedure that didn’t make sense to her.

I double-checked this hypothesis—because it is awkward to find yourself trying to explain a concept to a student who just wants the answer—and then started explaining why a maximum on the edge would not be found by a critical point search in the interior. She had learned this procedure but not internalized its purpose. As soon as she understood the rationale, she drove the conversation to the rectangle’s corner.

I generally talked about 3D vs. 2D vs. 1D, mentioned recursion, and told her to focus on just the bottom edge of the rectangle. We sat in silence until her face lit up, and she started explaining the final step to me.

I would have provided the answer at any time. She did not ask because she trusted that I had given her the information that she needed, and she wanted to figure it out for herself. I am glad that I was able to assist in the discovery rather than spoil it.

To teach is to foster an understanding of the world within another’s mind, a process that demands a variety of skills and techniques. My own journey in teaching stems from a disorganized thought process and a constant search for mutual understanding, which ironically has made me adept at conveying complex ideas. Over time, I’ve honed my teaching abilities and identified several effective, yet often overlooked, strategies.

Emphasizing One-on-One Conversations

I place great importance on one-on-one interactions. Effective communication in these settings often translates into successful group communication.

Building on Pre-existing Knowledge

Teaching extends beyond reciting facts; it involves explaining complex ideas that build on a learner’s existing knowledge. Understanding a student’s current worldview is crucial. It’s easy to fall into the trap of seeing oneself as the expert and the student as merely receptive. Instead, aim to gauge their understanding through targeted questions, avoiding the pitfall of lengthy explanations to simple queries.

Strategic Questioning

When speaking one-on-one, the initial goal is to swiftly and accurately assess the student’s current knowledge base. This understanding is essential for effectively building upon what they already know. Grouping students by knowledge level and prerequisites in educational settings is a reflection of this need.

Bi-directional Communication

Fostering an open, two-way dialogue is critical. Lecturing without feedback is akin to building a house blindfolded. It’s essential to ensure that each piece fits by gauging the student’s understanding as you progress. This not only helps in adjusting the teaching approach but also makes learning an engaging process for the student.

Addressing Emotional and Cognitive Barriers

Students often harbor concerns that can hinder learning, such as doubts about their questions or the fear of appearing ignorant. Addressing these concerns directly can significantly reduce their cognitive load, allowing them to focus more on the learning material.

Navigating Different Worldviews

Encountering diverse perspectives can be challenging but rewarding. Spending time to understand and possibly integrate the student’s viewpoint can lead to richer teaching experiences and insights. This approach not only respects the student’s perspective but also enhances trust and learning effectiveness.

Philosophical Foundations

Teaching often involves bridging philosophical differences that underpin various disciplines. Acknowledging these differences and the limitations of one’s knowledge can foster a humble and inquisitive learning environment. It’s important to recognize that not all questions have clear answers and that some debates are ongoing.

Aligning Incentives

Recognizing whether a student seeks understanding or merely the right answer is crucial. This distinction influences teaching strategies, emphasizing the need for clarity and avoiding unnecessarily cryptic responses. Open communication about the learning objectives and constraints can enhance the effectiveness of the interaction.

A Personal Reflection

During a visit to Cornell’s campus, I assisted a student in a way that underscored the importance of understanding over mere answers. This experience highlighted the value of guiding students towards self-discovery rather than providing immediate solutions.

This revision aims to clarify your points, streamline the flow of information, and correct grammatical nuances, making your insights into teaching more accessible and engaging.