Learning as a process of re-labeling

Disclaimer: This article is highly speculative, and based on my own experiences and a couple of articles I might have come across. I will be happy to remove it when I come across scientific evidence that contradicts it.

One defining feature of smart people is that they learn things fast. You tell them a concept or idea, and they’re able to understand and implement it much faster than the average person. Stupid people take much longer to understand an idea, assuming that they’re ever able to completely understand the idea. As someone who has been stupid for most of his life, who has only recently begun to be slightly “smarter”, I feel that I can shed some light on what might be causing this discrepancy.

Complexity

On a naive level, it is not difficult to believe that “simple” things are easy to understand, and that understanding “complex” things takes much more time. For example, it might be easy for us to understand that a car can travel faster than a human, but difficult for us to understand what Maxwell’s laws are saying.

What makes things “simple” or “complex”? Although there exist several ways of classifying things as simple or complex, my proposed definitions for both concepts are the following: things that we can refer to from sensory or emotional experience are “simple”, while things that are abstract and cannot be referred to from such experience are “complex”. For instance, an “apple” is a simple concept, because as soon as you say the word, you visualize a red, almost spherical object. You can remember what it smells like, what it tastes like, etc. “Moving fast” is a simple concept. You remember running on the ground, as well as traveling in a car. You implicitly know that cars travel faster than humans, because you have seen your car overtake humans on the sidewalk all the time. Maxwell’s laws, on the other hand, are a “complex” entity. You have no actual sensory or emotional experience that result from Maxwell’s laws. You may remember them as mathematical formulae written in your textbook, or perhaps some diagrams involving charges, magnets, field lines, etc. Hence, Maxwell’s laws will always be a much more complicated concept to internalize than the concept of an “apple”.

What is “simple” for humans may not be “simple” for computers. For instance, you can easily program a computer to solve Physics questions based on Maxwell’s laws. However, it is much more difficult for you to tell a computer what an apple is. What an apple might look like, smell like, or perhaps taste like. Hence, this definition of “simple” only attests to the ease with which a person may learn a concept, and not to any inherent, universal simplicity.

Anecdotal experience

So where am I going with this? We remember and learn concepts better when we can perceive them through our senses or emotions, and re-label those concepts as the corresponding sensory or emotional experience. Let me try and explain this with a couple of anecdotes.

I have tried to learn Algebraic Topology and Algebraic Geometry for a very long time. I own multiple books, all filled with all the correct formulae, and I have pored over them many a time. But after months and years of staring at those formulae, memorizing them and also occasionally repeating them in front of confused strangers, I often had to contend with the fact that I had no idea what the heck they were saying. Despite overwhelming evidence, I refused to accept the obvious inference- that I was stupid.

On the other hand, I came across strangers on the internet who had written lengthy lecture notes on these topics in high school! They had published research papers in prestigious journals, and were glorified everywhere. Sure. I could accept that they were smarter than me. Maybe even much smarter. But that much smarter? That they could learn things in middle school that I could not as a PhD student at a decent grad school? I also recently had the opportunity to talk to a PhD student, who had recently completed his thesis on Algebraic Geometry. He said that it was only after he had defended his thesis that he had begun to understand some basic concepts from his field. He felt that he should re-do all the problems from the very basic textbooks in order to really understand what was happening. I got the same reaction from multiple faculty members at a prestigious Indian research institute. Sure, I could be stupid. But all these very smart people, who had completed their degrees from some of the best institutes in the world, could not be stupid.

Now let us talk about music. I’ve always had a good ear from music. I picked up the guitar in class 7, and within a couple of months of picking it up, I could play along with most Hindi songs that played on the radio. If you played me a chord, I could play it back to you in seconds. I thought that this was how most people learned music, and did not know how rare this was until I went to college. I was one of four people in my batch of 800 who was selected for music club, and was often told that I had the best “ear” for music that they’d seen in years. Why was I good in music, whilst terrible in mathematics? Why wasn’t I uniformly stupid, or uniformly smart?

This was because of the following: when I listened to a chord, I felt an emotional experience. I felt either happy and “straight”, or mysterious and romantic, or about to launch into a speech, or some other complex emotion, and I would know right away that the chords that were being played were C major, A minor of F major respectively.

With Mathematics, I would have no such sensory or emotional feeling. I would see formulae on the paper that I would memorize, understand, and then soon forget. A particular example from Topology comes to mind: I have read and re-read about homology at least 50 times in my life. Perhaps more. I understand the formulae. The definitions. Where they come from. Calculating homology is the mathematical culmination of multiple mathematical concepts that come together beautifully. However, despite verifying and re-verifying this edifice very many times in my life, I had never actually “understood” what is happening. This was until someone told me that homology calculates the number of “holes” in an object. Since re-labeling the abstract concept of homology as the visual picture of the number of holes in an object, life has become much easier for me. Even if I can’t always calculate homology on my first go, I know that I am just calculating the number of holes in something, and then intuition takes over to guide me to the right answer.

Synesthesia

I have never been particularly good at mental math. On multiple occasions, I’ve been asked “You study mathematics right? Calculate 34\times 47“. And on many occasions, someone else would have calculated it faster than me. Of course I could rebuff it by saying something like “mathematicians are not calculators. We study ideas”. But I never did. This is because I have tried to be better at mental calculations for a long time, and have always failed. Hence, unless I join an Abacus class or something, I will probably never be much better than the layperson in multiplying two digit numbers in my head.

And then of course come the people with synesthesia. On seeing numbers, some people with synesthesia have reported seeing colors or shapes. If you ask them to multiply 34\times 47 for instance, they would see something like a yellow disc swallowing a giraffe, and then turning into a hippopotamus. And as soon as they see the hippopotamus, they’d know that the answer is 1598. You don’t believe me? See this.

Why this discrepancy? This is because I don’t have a sensory or emotional response when I see a number. I just see it as something written on a piece of paper. However, a person with synesthesia has re-labeled various numbers as emotions. And those emotions help them calculate much faster than the usual multiplication algorithm would.

So how can we use any of this stuff?

Over the past year, since I decided to fire up this blog again, I have tried to learn very many different concepts from different fields. At the beginning, I was understanding things only at a superficial level. At the same time of course, I was also having trouble understanding my own mathematical field, that Penn State might soon proclaim I am an expert at because I’ve done a PhD in it. What a load of drivel.

However, after failing to understand a particular mathematical paper despite reading it multiple times over a month, I started drawing things up on my iPad in multiple colors. In other words, I began relabeling those mathematical concepts as sketches on my iPad. Now on, whenever I would read a concept, my brain would visualize only that sketch that I’d made for it. They didn’t even need to be accurate drawings of the concept. I just needed some visual representation. This was tremendously helpful. I soon began doing the same for concepts that I tried to learn from other papers, and I think that I now understand many concepts much better than before.

Of course, if I am able to relabel those mathematical concepts as some emotions inside my head, I would have an even easier time understanding, recalling and manipulating those concepts. However, until I figure out how to do so, a sensory (visual) representation would suffice.

So if you’re trying to learn something, understand it using reason and logic. But remember it by relabeling it as something that evokes a sensory or emotional response. If you’re anything like me (let’s hope for your sake you’re not), it might greatly boost your ability to learn new things much faster.

Published by ayushkhaitan3437

Hello! My name is Ayush Khaitan, and I'm a graduate student in Mathematics. I am always excited about talking to people about their research. Please please set up a meeting with me if you feel that I might have an interesting perspective to offer- https://calendly.com/ayushkhaitan/meeting-with-ayush

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