Learning, Learning to Learn, and Teaching
I’ve taught for several semesters in Berkeley now, and I think it’s high time I wrote some thoughts down before my knowledge on teaching fades away.^{1} So here’s a collection of thoughts: we begin with a light theory of how people acquire knowledge through inquiry, followed by a quick metaanalysis of learning to learn, ending with teaching.
I like to view teaching as a way of approximating the learning processes that people would normally undergo. The key difference is usually convenience and speed; learning can be quite arduous, and teaching often eases the process.
But many educators forget what it’s like to be a student. They lean on prescriptive methods of teaching, leading students to be walking reflection of their image. Through this post, I hope that students and educators alike will be able to rethink learning.
Learning
How do people learn? It all begins when a person asks “why?” and starts seeking answers for themselves.^{2}
It’s important to understand why this is successful. First, there’s the question of motivation. When a person begins questioning the world, it’s their internal motivation driving them to answer a question. This motivation allows the individual to dig deeply into their question of choice, possibly uncovering the truth they initially set out to find, possibly stumbling upon another question. Regardless of the outcome, however, it’s imperative to note that the individual’s motives do not stem from external factors. Rather, they’re internal, allowing them to try, retry, and retry again should they run into a wall.
At the end of the journey to discovery, when all the why’s are answered, all they’re left with is knowledge and satisfaction. It’s incredibly gratifying to be able to ask a question, start with only a toolbox of inquisition, and to arrive at some tangible knowledge.
Importantly, this process highlights a key component of learning: through the grueling process of hunting down hints and clues that eventually yield the answers, we implicitly learn about learning. We learn what to do, what not to do, and how to go about answering similar questions in the future. In doing so, we add new tools to our toolbox of inquisition.
Learning to Learn
Let’s examine this a bit more. If we want to be good teachers, it’s imperative to understand the mechanism through which we obtain knowledge. Formalizing the previous section on learning, we have a model that looks like the following:
 Student has initial set of facts, as well as a toolbox of methods they understand to experiment on the world.^{3} They ask some motivated question about the world around them.
 To answer this question, they initially have a broad (correct or incorrect) hypothesis of how the world should behave according to their set of facts.
 They then conduct experiments to verify or challenge their beliefs.
 Eventually, they will obtain a new set of facts that updates their understanding of the world. In the process, they might also obtain a new method of experimentation to add to their toolbox.
 Repeat from step 1, with the extra knowledge and tools from step 4.
Example: “Discovering Probability”
Let’s use this framework to imagine what it might be like to rediscover the memoryless property of coin flips.
A reasonable (even if wrong) initial set of beliefs about the world might be something like this:
 Beliefs:
 Coins have some fixed chance of landing Heads or Tails. This chance does not change between flips.
 Toolbox:
 We can flip coins to test the chance of outcomes. (physical)
 If we believe strongly that the chance of flipping Heads is fixed, we can also simulate this. (computational)
 Similarly, if we believe strongly that the chance of flipping Heads is fixed, we can mathematically prove this. (proof) [^proof]
 Initial hypothesis (belief that we want to test):
 Coins are fair, so if we get many Heads in a row, it should start giving us Tails.
Assuming that we only have the physical method available, we can easily test the memoryless property of coin flips. If our initial hypothesis is true, then we should expect to see more Heads than Tails after a Tail. However, recording large number of flips will show us that this isn’t true.
As a result, we can update our internal beliefs about coin flips.
A couple of remarks are in order.
 It’s interesting to note that if one holds the belief of coins being consistent through flips, the memoryless property falls out immediately. (Formally, conditional independence implies the memoryless property.)
 However, students will sometimes hold the simultaneous belief that coins are (1) unchanging through rolls, and yet (2) believe in some kind of “mean reversion,” or “bias toward the mean on subsequent rolls”. When confronted with this discrepancy, they’ll often doubt (1) rather than (2), making the computational and proofbased methods unavailable to them.
 Typically, as we go through this process of testing, we also run into roadblocks and contradictions. These can be frustrating, but they’re actually quite critical to learning! Since part of the process is formulating a hypothesis, it’s important to understand the actual space of viable hypotheses. Finding counterexamples allow us to understand the boundaries of the hypothesis space, which in turn allow us to ask better, moreinformed questions.
 Also, as we go through this process, we often will come up with better ways of doing things. (As the proverb goes, “necessity is the mother of invention!”) These discoveries add to our toolbox, adding to step 1 and accelerating step 3.
Teaching
With this model of learning in mind, we turn to teaching. Before we answer the question of how we should teach, there’s a more fundamental question that needs to be answered. Assuming a student is sufficiently selfmotivated and capable, why does teaching even exist? Why isn’t it always better for students to ask questions, stumble around, and get a solid, fundamental grasp of everything?
The answer is largely a question of efficiency. While it’s possible a student can answer their own questions and discover new things about the world around them, it simultaneously remains true that there are an infinite number of “wrong paths” to go down.
The Role of Teaching
So how should we teach? In its purest form, teaching should approximate the process outlined in Learning to Learn, with one important modification: the teacher is given the role of guiding a student through making insightful hypotheses and insightful wrong decisions.
In particular, this means teaching should never be prescriptive, where a method is given without rationale and told to be adhered to. All too common, we see this in public education. In calculus, for example, students are told to “just differentiate, set to zero, and solve,” and to “check the second derivative for validity.” According to our framework above, this method is wrong because it does not approximate learning! Students cannot be given a result first and then a hypothesis after, since it’s mentally unclear where this result is supposed to fit into their framework of reality.
To fix this, we should rethink teaching by starting from a foundational motivation. In the Calculus example above, instead of starting from a result, form a motivating question first: “Suppose a ball were thrown in the air. How do we calculate its highest point off the ground?” (This example can also be helped with a graph of the ball at various points describing a parabola.) With this motivating question, students can then form their own hypotheses and understand where each possible solution falls short. In the end, it’s ideal for the instructor to manually describe approaches that break, and then describe the “valid solution” that students should come to.
Shortfalls of Teaching
As with any approximation, there are shortfalls. With teaching, it’s that the process of error is sometimes completely removed since the process of constructing bad paths has become artificial.
This is a necessary sacrifice we take in favor of accelerated learning, within bounds. Many teachers forget the process of motivated learning and become prescriptive (and they wonder why concepts don’t stick in students’ heads). As educators, we can do a lot better: put yourself in your students’ shoes, rewind your brains to match theirs, and relearn the concepts with them from a clean slate.
Credits
Thanks to Andi Gu and Sean O’Brien for reading a draft of this.
Footnotes

It’s a pipe dream, but someday I’d love to be a teaching professor! School seems quite tedious and a PhD seems out of grasp, but who knows what a few years in industry will do… ⤴

I mean, have you ever talked to a toddler who just learned the word “why”? Perhaps this is why they acquire knowledge so quickly… ⤴

Actually, we also have to make the assumption that the universe is consistent and understandable. This seems obvious in our age, but it’s not a priori clear that the universe should behave the same tomorrow as it does today. Suffice it to say however that we all take this for granted, so we don’t have much need to explicitly say this. ⤴