Reflections on Self-Studying Math and Physics for 3 Years

In 2020, I embarked on a journey in self-education, Inspired by Scott Young’s MIT challenge (https://www.scotthyoung.com/blog/myprojects/mit-challenge-2/), but this time with a twist: diving into the world of Mathematics and Physics.

Utilizing an array of resources, I tailored my unique educational path. The backbone of my challenge was MIT’s Open Courseware curriculum for Math and Physics. Complementing these resources, I also incorporated a few insightful ideas harvested from various online platforms.

For the curious minds among you, a comprehensive list of the courses that I undertook during this challenge is available on my website (https://www.diegovera.org/). Furthermore, you can find the following interview I had with Scott Young (https://www.scotthyoung.com/blog/2023/02/21/diego-vera-mit-challenge-math-physics/), where we delve deeper into my self-study project.

I’ll primarily focus on discussing the most beneficial activities that have significantly aided my learning journey. These strategies, some of which I unearthed during my challenge and others in its aftermath, have proven to be high-yield in terms of knowledge acquisition and retention.

Sparking Connections and Organizing Our Toolshed

The premise of relational learning is the idea that understanding the connections between ideas can be a powerful way to deepen your understanding of a subject, enhance memory retention, and accelerate the overall learning process. Think of it like this: instead of defining what a dog is by its features (four legs, barks, etc.), you define a dog by its relationships (friend of humans, chaser of cats, etc.). This way of definition is very powerful and general and applies across many areas, more importantly, it links back to previous knowledge that you already have (this will be particularly important for one of the ensuing sections)

On the other hand, organizing your brain is more like thinking of your brain as a toolshed, and the information you learn therein as the tools. If you randomly throw tools into the shed without organizing them in any way, it would be challenging to find what you need later on (you might forget you even have certain tools or struggle to remember where you placed them). Let’s say that instead of throwing them haphazardly, you organize your tools based on their use — gardening tools in one corner, woodworking tools in another, and so forth. Now, when you need a tool for a particular task, you know exactly where to look, and when you get a new tool, you know where it fits within your existing organization system. When learning new information, you’re not just cramming facts (tools) into your brain (shed) without any structure. You’re organizing these facts based on how they relate to what you already know (the use of the tools), and how they fit within the larger context of the subject (the sections in the shed). For instance, when learning about financial models, you might connect new knowledge with what you already understand about economics, mathematics, or data analysis. Or in particle physics, you group particles based on certain traits, like their charge or the forces they interact with. By doing so, you’re creating a structured framework for your knowledge, which helps you remember and understand things better. New information has a “place” in your mind, making it easier to retrieve when you need it. Learning, then, becomes less like memorizing a jumble of disconnected facts and more like building a well-organized, interconnected network of ideas. It’s a more natural way of understanding the world, as our brains are inherently wired to seek patterns and connections.

Implementation:

A useful way in which I have implemented both ideas involves the following:

- Comparing and contrasting examples
- Asking relational questions like how is so and so related to that
- Trying to spot patterns by understanding the purpose behind concepts, helps you think about the underlying principles or reasons, promoting deeper understanding and long-term retention.
- Saying the sentence “It’s kind of like” then from there try to let your brain fill in the rest.

Insights from the rearview mirror

Just as a game player might lose a life or face a bottleneck until a particular problem is resolved, a learner may stumble on a concept or falter in a calculation. This instantaneous response to missteps serves not as an end but a means to pivot, rethink, and reconsider strategies — an encouraging nudge towards experiential learning and the development of new problem-solving skills. I wish to discuss two broad aspects: the reflective approach toward methods employed, and the quality of learning attained.

Reflective Approach to Methods

This aspect is analogous to the strategic choices you make within a game. Do you opt for an aggressive, direct strategy or a more subtle, stealthy approach? My current approach to reflecting on methods involves establishing a ‘control group’ for your learning process. Subsequently, after each distinct ‘learning experiment’, you alter a single variable, retaining those strategies that yield positive results and discarding those that do not. Once this is done you could take things one step further and consider combinations of variables, seeing which provide the most effective results, and finally, you implement your strategies in different contexts to figure out which approach should be used according to the scenario at hand. Take language learning as an instance. Perhaps you opt to learn vocabulary via flashcards while simultaneously engaging in conversations. You could adjust the balance of flashcard learning and conversational practice depending on your learning stage. Furthermore, within the conversational practice, certain discussions might prove more beneficial than others, warranting increased focus. Alternatively, you might discover that concentrating on different variations of the same conversation offers broader insights. The succeeding iterations of your ‘learning experiment’ could involve subtle tweaks depending on which aspects seemed more effective, continuously refining until you identify the most optimal approach.

Quality of Method = (Insights — Confusions) / Time

The aim is to use methods that maximize this quantity, measured based on ‘Feelings of Good Learning’ or ‘aha’ moments. Such moments signal a genuine understanding and connection with the subject matter, similar to the experience of resolving a complex in-game puzzle and suddenly comprehending the path to victory. These moments foster profound insights and knowledge. Conversely, we should seek methods that help identify where our understanding falters to reduce confusion. Lastly, we strive to minimize the time spent learning — methods that require less time are more effective.

Reflecting on Learning

This aspect pertains to an overview of your progress. Have you enhanced your skills in the areas you desired? Are there topics or skills you’re still struggling with? This reflection offers a comprehensive perspective of your learning journey and aids in fine-tuning your strategies. To gauge the effectiveness of your learning, focus on enhancing retention and the fluency of understanding.

Quality of Learning = Retention x Fluency of Understanding

In what follows we will look into methods that perform quite well on both metrics

You could also reflect on the nature of the material, figuring out what percentage is procedural and what percentage is conceptual.

Mental Alchemy

This will now consist in methods that I’ve found useful from personal experience as well as a certain amount of research I've done, in the hopes to optimize the equations we looked into apriori.

Thinking outside the box

This involves seeking new perspectives and introducing creativity into problem-solving by altering the underlying variables and seeing the second-order consequences. It’s like trying to solve a jigsaw puzzle but considering multiple configurations, not just the obvious one. In a physics problem, you might decide to for instance solve the problem by venturing into different frames of reference and not just using the most obvious one. Another instance might be that when handed an example of a particular problem, say a problem about friction you switch things up a bit, like for instance instead of having one coefficient of friction you use two coefficients, and perhaps you also switch the linear dependence the coefficient of friction has on the normal force to a quadradic dependence just to test out the second order consequences. Playing around with things in this manner gives you a more nuanced understanding of the issue at hand

Mastery Through Inquiry

The value of questions in learning is akin to using a map when exploring new territory. Good questions provide a guide, helping us understand our current location, the landscape, and where we might want to go next. If a topic doesn’t spark your curiosity, questioning its importance and relevance, its historical context, or its relation to other concepts can give you new insights, just like examining a familiar object in a different light. Some questions to follow things up with might include, Why is this useful (from a first principles perspective) What did the discoverers have in mind when figuring this out (what was their thought process)? Could there have been possible alternatives? If So how do they compare? How is this connected to the other important concepts?

Another way to deal with this is to go through examples along with practice questions (by reading them and not necessarily solving them) to get a feel for the underlying context and be able to spot meaningful patterns, even if you don’t fully understand what’s going on, that’s okay because the purpose here is just to develop general insights regarding the subject matter.

Webs of Wisdom

Gathering insights from a wide array of resources is like watching a play from different seats in the theater. Each viewpoint offers a different perspective, highlighting certain aspects while downplaying others. To understand the entire performance, you need to take in the show from multiple angles. Resources of varying quality and nature provide these different angles, helping you develop a more holistic understanding. I often found myself spending quite a while trying to understand a certain concept from one resource, then being able to understand it immediately as soon as I turned to another resource. This leads me to think that seeing things from different angles has benefits for patching up bits that the resources individually didn’t patch up. A useful way to think about this comes from David Hume’s (a philosopher) Dialectic which essentially involves merging arguments from both people in a debate into one holistic viewpoint because in doing so one viewpoint patches up the issues of the counterargument and vice versa. By sheer coincidence, I once remember reading an article studying experts largely in the fields of physics and maths, and they found this to be a common trait (they had a wide variety of resources).

The Secret Scholar

Sometimes, our best ideas and solutions appear when we aren’t actively thinking about the problem. It’s like planting a seed in your mind and allowing it to grow over time. Grothendieck’s nut-cracking analogy captures this beautifully. “Grothendieck (A well-known mathematician) once described two styles in mathematics. If you think of a theorem to be proved as a nut to be cracked, then one approach is to put the cutting edge of the chisel against the shell and strike hard. If needed, begin again at many different points until the shell cracks — and you are satisfied. For the second approach, think of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise, you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!”. This is an approach I very much follow now more than ever and have found to be particularly useful when it comes to patching up confusions of particular concepts, often times letting things sit for a while and learning things recursively can make a great difference in the insight you gain, having let some time pass and seeing it with fresh eyes yet again. We also see this in an observation made by Kant (a famous philosopher) remarking that in reading Hume (another philosopher) a second time (after having let it sit for a few years) awoke him from a dogmatic slumber. Isaak Azimov a writer has an interesting essay called “Eureka’s Phenomenon”, where he speaks about the importance of unifying both reason and revelation as opposed to solely relying on reason (they are complementary).

Throughout my learning journey, I enrolled in numerous courses at the same time in an attempt to ensure that I could dive into various problems or concepts, and pivot between subjects when need be, advancing my understanding in some areas while allowing my subconscious to process more complex topics. Often, I find myself revisiting challenging concepts after weeks, discovering that my subconscious mind has been silently at work, integrating the learning, resulting in newfound clarity.

Practice-Based Learning

One of the most effective strategies I’ve discovered is immediate practice, applying knowledge as soon as it’s attained (with an important caveat as we will see). This approach underpins the power of project-based learning, which promotes learning through problem-solving. Unlike traditional methods where learning precedes application, here, the immediate relevance of necessary techniques becomes tangible.

Typically, the Traditional Approach to learning follows this sequence:

Learn (Acquire Knowledge) → Practice (Apply Knowledge) → Learn (Acquire More Knowledge) → Practice (Apply More Knowledge)

However, the Practice-Based Approach follows a slightly different, and potentially more effective sequence:

Practice (Apply Knowledge) → Learn (Reflect and Acquire Knowledge from Application) → Practice (Apply New Knowledge) → Learn (Reflect and Acquire More Knowledge from Application)

A central advantage of the Practice-Based Learning approach is that it fosters a beginner’s mindset, encouraging the learner to take an exploratory, problem-solving approach rather than passively receiving information. This mindset enables the learner to construct a larger, more interconnected picture of the subject matter, facilitating a deeper understanding. Now an important caveat is that the learner need not completely try to solve the problem, the point is for them to see how ideas are being used and given the little they know apriori make a guess at the solution or what kinds of things might be useful- even if it’s incorrect- this is known as the “Hypercorrection effect”, and then once you have done so, only briefly overview the answer to know which concepts you need to know in order to solve the problem(only spot out the tools used not the solution as a whole) then from there try to review them, improve your answer and then check your answers again, now in depth.

An important aspect to consider still is the question of what the highest leverage questions are. By doing so we will be able to focus on solving those ones as opposed to the ones that don’t provide much insight. The following are a few things to look out for when it comes to identifying ‘high leverage’ questions.

- Questions that incorporate various concepts, particularly in abstract ways
- Questions that lead you to consider a problem in different scenarios (thereby requiring you to look at it from different angles)
- The question should test first principles knowledge that can be implemented to solve other problems
- After solving the question try to reflect and generalize the principles implemented to solve the problem so that you can use them later on
- By answering a lot of questions at the same time not necessarily fully but just a decent idea of how you could go about solving it, and then solving all of them recursively as opposed to one by one we find that questions often aid each other and insights from one question could be used to identify solutions to the other questions for instance
- Implement the Hypercorrection effect in the sense of answering the problem then only briefly overview the answer to know which concepts you need to know to solve the problem then from there try to review them, improve your answer, and then look at it again in depth.

Unravel and Illuminate

Another useful approach to learning is a more elaborative approach to the learning technique known as the Feynman Technique-This approach will be particularly useful in identifying weak points

1. Learn and Understand the Concept:
The first step is to learn and understand the topic you’re trying to explain. This requires immersion in the material and often involves reading, listening to lectures, conducting experiments, or practical application. For difficult concepts, you might need to dive deeper into foundational knowledge to truly understand.

2. Reflect and Digest the Information:
After you’ve initially learned the topic, take some time to reflect on what you’ve learned and let the information settle in your mind. You can do this by taking a break, sleeping on it, or even practicing something completely unrelated. The goal is to let your subconscious mind process the information.

3. Begin Drafting an Explanation:
After letting the information settle, you can begin drafting an explanation. Make sure the explanation is:

Condensed: Focus on the most essential information and try to omit any unnecessary details.

Insightful: Highlight the most significant aspects of the topic, and provide interesting and meaningful interpretations or conclusions.

Simple: Use plain language that can be easily understood. Avoid jargon and complex vocabulary unless it’s absolutely necessary.

4. Make Connections:
Always aim to connect the new concept to knowledge that your audience already possesses. This helps learners relate the new concept to something familiar, aiding their understanding and memory of the topic.

5. Provide Clear Motivation:
This can be achieved by explaining the real-world implications of the topic, its historical significance, or its usefulness in a particular context. This helps the learner understand why it’s important and worth learning.

6. Practice the Explanation:
Try explaining the topic to others or to yourself. Notice which parts are difficult to explain or where your audience gets confused, and revise those parts for clarity and simplicity.

7. Review and Revise:
After you’ve given the explanation, revisit it after some time, dive deeper into the resources, and try to improve upon it. As you learn more about the topic, you’ll be able to explain it in more depth and with better clarity.

8. Rinse and Repeat:
This process is iterative. Each time you explain the concept, you’ll understand it more deeply and be able to explain it better.

This learning and teaching approach is recursive and continuous. It requires tolerance for ambiguity and gradual improvement and values simplicity, intuition, and condensation. It embodies the core principles of several learning theories and has proven to be very effective in promoting deep, enduring understanding. This approach essentially attempts to combine the following teaching techniques into one.

“Kaleidoscope Concept”: This technique presents a single concept from multiple perspectives, much like looking at an object through a kaleidoscope. This can enhance understanding and retention.

“Crystal Clear Clarifier”: This method involves explaining complex concepts in the simplest possible terms, striving for clarity and simplicity.

“Time Travel Teaching”: This method explains concepts by moving back and forth in time, showing the historical development of ideas and their future implications.

The Unseen Growth Engine

There is a quote I often like to remark on, “The broader the base, the taller the building”, and when it comes to learning that’s very true.

Let’s take a brief tangent. For those of you who have some acquaintance with differential equations, hear me out and consider the following.

dy/dt = ky — -> y = e^kt

In simpler terms, what this equation is telling us is that whenever the growth of a certain quantity depends on how much of it you have apriori, it results in exponential growth. Another way to see this is “The more you learn the easier it is to make connections and learn new things”(And the results are quite significant). This is of course to a certain limit since it’s impossible to learn things within nanoseconds, but the essence is still valid. Einstein once said, “Compound interest is the eighth wonder of the world. He who understands it, earns it … he who doesn’t … pays it”. This also follows the idea often spoken of in Artificial Intelligence of “To Learn is to Generalize”(i.e link to previous schemas and keep building them up)

Examples:

Word of Mouth:

Compound interest:

You'll notice that in both cases there is exponential growth underlying

Lollapalooza Effect

Charlie Munger, vice chairman of Berkshire Hathaway and partner of Warren Buffett, often refers to the “Lollapalooza Effect” in his discussions about decision-making and human behavior. The Lollapalooza Effect is the confluence of multiple cognitive biases or tendencies acting in the same direction, which often leads to extreme outcomes. It’s not simply additive, but rather a multiplicative or exponential function, resulting in an outcome that is significantly larger or different than what you might expect from the sum of the individual parts.

For example, consider the phenomenon of auction fever, where people at an auction tend to bid higher than the intrinsic value of the item being auctioned. This could be due to several cognitive biases acting together: social proof (everyone else is bidding, so it must be the right thing to do), commitment and consistency (I made a bid, so I should keep bidding), scarcity bias (this is the only item of its kind), and contrast effect (compared to other high bids, my increase doesn’t seem that big). Think about productivity, too. It’s not just about mastering one strategy but combining several that work well together. For instance, typing faster, meditating, working out, and taking cold showers — these diverse activities can synergize to boost your overall productivity. But this doesn’t just apply to these activities, even some of the greatest basketball players like Michael Jordan who was a great defense and offense, or boxers like Mike Tyson who were quick and could also hit hard, these were anomalies because before them people were often either one or the other, you can see how these skills complemented one another and resulted in something greater than the sum of their parts.

Now, let’s apply this concept to learning theory.

We often talk about strategies in isolation, but to be a truly optimal learner I have found that being able to implement these strategies at the right moments ensures that they complement one another when need be, hence creating an effect where the system as a whole is greater than the sum of its parts.

Although this is the end of one chapter, the Journey is far from over, after having finished the challenge I spent a few weeks compiling everything and putting it up on my website, and embarked on another one the day after, which means that you’ll be hearing much more from me in the coming months and years.