I Could Be Wrong
As a physics student, I’m taught over and over again that science is about checking to see where we have made incorrect assumptions. The goal of science is to correct our assumptions about the world, and find better descriptions for what is going on. Of course, this is a distilled version of the goal of science (and I imagine some would disagree), but my point is that science aims to perform a consistency check on our hypotheses and ideas.
If not, we would just be giving opinions all the time.
That being said, I think there’s a pressing need to remind ourselves of this message over and over again. Whenever we think something is true, it’s worth appending our thought with the words “but I could be wrong”.
I’m not just talking about science here. Every day, we tell ourselves stories about how the world works, the motivations of other people, and why certain things happen. We aren’t mind readers though, so we definitely can’t tell what another person is thinking. Sure, we can guess, but we won’t know for certain unless we ask them. Similarly, we can observe a chain of events and ascribe some sort of causality behind it, but we often don’t have the full picture. Instead, we merely have a slice of it.
Have you ever declared something to another person, only to find out that your assumptions (which you depended on) were wrong? When this happens, we have to grapple with the fact that we don’t understand the world in its entirety. We might be correct from time to time, but on the whole, we are merely making educated guesses.
This isn’t to make you feel bad about yourself. Rather, I want to encourage you to understand how many things we paper over in an effort to give ourselves some consistency and rhythm to our lives. If we start out a conversation arguing one position, we might not want to back down and change our mind, even if internally we know it’s the right thing to do. Instead, we declare our beliefs to the world and don’t even entertain the fact that we could be wrong.
I think this is worth repeating. I could be wrong. I might be right, but I should always leave the possibility of being wrong on the table. Without it, I’m simply being closed off to change.
There’s a lot more to say about this topic. In particular, you want to get to the point where this isn’t merely lip service. In an ideal world, when you say “I might be wrong”, you actually take the time to reflect on the possibility. This process reminds you to slow down in your thinking. It’s easy to get caught up in the stories of our lives and forget that the world isn’t always transparent. There’s a lot going on that we think we understand but really don’t. A good education helps us see this.
I am now trying to walk around the world with this phrase in mind. No matter how knowledgeable I think I am for a given topic, I could be wrong.
Where Does It Belong In The Web?
If you look at a typical mathematics class, you will find that they follow a similar rhythm. First, students are presented with a definition. A few examples of how that definition applies might be given, and then the rest of the class is spent proving results based on this definition.
It’s a familiar recipe, and it works well for transmitting information. If you asked students to give a presentation on the topic, I would predict that most students would follow this layout.
There’s nothing bad with this layout. However, I think there is a key missing ingredient, and it has consequences for how students perceive mathematics. It’s the answer to the question, “Where does this concept/definition/axiom/theorem fit in the grand scheme of mathematics?”
We don’t put enough emphasis on answering this, to the detriment of students. There are two main consequences of failing to answer this question. One, it makes students unaware of their place within the “space” of mathematics. And two, it makes mathematics feel somewhat arbitrary and disconnected.
Locating yourself within the web of mathematics
In physics, we have a notion called “phase space”. In classical mechanics, phase space refers to your coordinates in both position and momentum space. Since we live in three spatial dimensions, there are three position coordinates and three corresponding momentum coordinates, for a total of six coordinates. These six numbers describe all that matters when analyzing a system. Phase space is six-dimensional (when analyzing a single particle), so it’s difficult to visualize. However, if you know the tools of mathematics, you can still analyze phase space and figure out what the particle is doing.
I think a similar structure exists in all disciplines, and this structure constitutes the “totality” of the subject. You can think of it as the space where a discipline (like mathematics) “lives”. I like to think of this space as a web, since there are many connections between various topics. The key observation is that very few mathematical facts exist in isolation. In general, whenever you learn something new about a specific topic, you’re growing the space of mathematics that you know about.
This is crucial to understand, because it means that there’s utility in thinking about what you’re learning at a higher level. When you learn about logic and implications, it’s easy to dismiss this as a niche corner of mathematics that only has applications to riddles about islanders who either tell the truth or lie. However, a broader understanding of mathematics makes you realize that logic permeates all the we do in mathematics. It lets us fashion theorems, helps us prove them with various techniques, and gives us different ways to think about the same problem (through equivalent truth tables).
The unfortunate part is that a student learning the subject isn’t always privy to this “extra” information. Instead, they’re taught the minimum so that everything is self-contained. This might be good for focusing in class and not worrying about more material than is necessary, but it means that students often aren’t thinking about the “big picture”. While this won’t affect their performance in class, it will affect how they see mathematics.
To address this, we need to become more serious about adding a “motivation” section to classes. This is sometimes done by professors, but frankly I think the motivation is missed in most classes (at least, it was to me). In an ideal scenario, I’d like to exit a class after I learn a new idea and be capable of saying, “Here is what I learned, and this is why it’s important.”
The latter doesn’t “give” you anything in the sense of mathematical skill, but it does let you situate yourself in the web of mathematics, and I think this is a valuable thing in and of itself.
Just a bunch of rules
Hearing the above heading as a description of mathematics is always disheartening for me. I know that mathematics isn’t just a bunch of arbitrary rules. In fact, only a small part of mathematics is arbitrary1, and that’s often to establish a convention when ambiguity arises. Apart from that, mathematics is a connected collection of islands that are linked through logic. Unless you go very deep, you can find explanations for most mathematical facts. It’s definitely not just a bunch of rules.
At first though, this isn’t clear at all. However, the cool thing about mathematics is that as you get deeper into the subject and learn more, you find that it gets easier. What once looked like a bunch of rules actually stems from a single explanation.
This is why it’s so important to teach the context around mathematics. When we know where a definition comes from, we start to see how it’s not fully arbitrary. This turns mathematics from something that looks disconnected to a coherent framework.
A good example of this is when you study results that look like they were “reverse-engineered”. When I was studying abstract algebra, I learned about a result called Eisenstein’s criterion. It was a way of determining if a particular polynomial was irreducible. The details don’t matter, but the important bit was that the requirements for this result seemed quite ridiculous. Another example is that a number is divisible by three if the sum of its digits is divisible by three. At first glance, there’s no reason why this should work. It seems kind of random. However, once you understand how this result is a consequence of modular arithmetic, everything becomes clear.
We can’t let definitions and results float around as isolated items that only belong in one specific class. We need to make sure that the connections are brought to light. There is a lot of room for improvement within the classroom, but I also think there’s a huge opportunity for those who communicate online through blogs and videos.
When I learn new mathematical ideas now, I try to always ask myself where this idea fits within the mathematical web. It’s not that I don’t see any value in learning ideas in isolation. Rather, I think the best part of mathematics is seeing how everything is interconnected. If you’re willing to put in the effort, the web becomes clear.
It just takes a choice to move past the bare minimum of what is asked in school.
Though, we have to be careful by exactly what we mean here. Do humans make up the rules of mathematics? Sure. Therefore, in some sense, mathematics is arbitrary. ↩
Being Good At Mathematics
“How did you get so good?”
This is a question I’m asked from time to time with respect to mathematics and physics. People see the kinds of grades I get, and they want to know what my secret is. I presume they think I have a method, or at least some explanation as to why I get great grades at school. When I hear this question, I often have to hold myself back from ranting about what it means to be “good” at mathematics, and how I’m actually good at school. However, since this is my blog and we have the time to explore the nuances that are present, I want to lay out exactly what I think it takes to be “good” at mathematics.
But first, I want to get something out of the way. If I ask you to give me five words you think of when I say “mathematics”, I’m betting that a sizeable portion of people would respond with “school” as one of their choices. This is important to realize, because it shows how we tie together two different concepts. On the one hand, we have school (and more broadly, the education system). This comprises a huge chunk of students’ lives, and involves a variety of disciplines. On the other hand, we have mathematics, which is a subject that’s taught in school, but only forms a small part of the experience. In this sense, many might draw a Venn diagram of school and mathematics like this.
The problem here is that mathematics is often fashioned as something that lives within the school system. As such, when we leave school (as many of us inevitably do), mathematics becomes a long-forgotten memory. The biggest drop-off occurs after secondary school. Once students start specializing in their degree for different careers, mathematics is often dropped. Therefore, the experience of doing mathematics remains as something that happens in school, and nowhere else.
The reason I bring this up is that this perception is an illusion. Mathematics is present in the entire world around us if we are willing to take a closer look. However, when we think of mathematics as something that is only done in school, we start to merge the two concepts together. As such, when people talk about me being good at mathematics, I think what they are really getting at is that I get good grades in mathematics in school. This seems like a subtle difference, but as I’ll describe below, there’s a huge chasm between being good at mathematics in general and doing well in school.
Of course, keep in mind that these are points I think are indicators of being “good” at mathematics. Different people might have slightly (or very) different ideas. I’ve listed these traits kind of implicitly, in the form of questions.
How resourceful are you?
To be good at mathematics, you need to be capable of using what you’re given to derive consequences. This is just a fancy way of saying you need to be able to use the axioms in such a way that you produce new results.
As time goes on, you’ll see that there are basic techniques that can be used to great effect. These include the standard techniques of proof (contradiction, induction, and so on), different tools from a variety of disciplines, and general “trends” that certain problems take. For example, if I’m dealing with a problem in the field of graph theory, I know that most of the results will have to use induction in some sense (though not always). Being able to reach back to what you learned previously can be a huge help.
This brings me to something which I’m sure will be controversial: memorization is often a good thing. I’m not talking about memorizing every mathematical fact you come across, but being able to remember a core set of facts can be of enormous help. I know, everyone likes to say that mathematics is all about being able to work out knowledge from first principles, but I think it’s evident that recalling facts can speed up the process. This is just as true in a test as it is when you’re working out a problem on your own. I guarantee you that basic memorization does make people think you’re “good” at mathematics (though it’s not critical).
Also under the umbrella of resourcefulness is the need for creativity. I think someone is “good” at mathematics when they look for creative solutions to problems. Creativity is a cornerstone for coming up with new proofs and finding new insights to old concepts. If mathematics is about understanding structures of some kind, creativity allows you to wander around these structures and view them from different angles.
Do you sit with an idea that you’re struggling with, or do you move on?
The “best” people at mathematics are those who refuse to skip over their ignorance. If they get to a problem that they don’t know how to solve or are confused about a concept, they don’t just commit it to memory and move on. Instead, they reflect about the idea and try to figure out where their misunderstanding lies. This is a hugely important point that I don’t think gets made enough. Willing to be stuck is key if you want to understand something new.
Learning mathematics is in large part about building a strong foundation. Once you have a strong foundation, you can increase the abstraction and complexity. Learning new ideas becomes easier. On the other hand, if you just skip over an idea that you don’t understand and commit it to memory in order to pass, it will come back to haunt you later on. It’s almost an inevitability, since mathematics is about accumulating knowledge on top of itself. To get to the “next” level, you need to understand what you’ve looked at previously. If you aren’t willing to sit with an idea you’re struggling with, you’ll find yourself on a shaky foundation later on.
The truth is that you will find yourself stuck later on, so it’s a good idea to build up your resilience from the start. If you aren’t willing to sit with uncertainty and figure out your confusion, the result will be a lot of frustration and not many problems solved. Mathematics is a process of being wrong over and over again until you’re finally right. As such, failure isn’t something to be avoided. It’s part of the deal.
How much persevering will you do?
This brings us to the idea of perseverance. If you want to be “good” at mathematics, you can’t give up when you’re learning. There will be setbacks and challenges, but they are all surmountable as long as you make a commitment to learning just a little bit more every day. If you go in with the mindset that everything should make sense on the first read through, you’re deluding yourself. It’s much more likely that you won’t understand a thing at first. Then, with a lot of work, you can slowly improve and wrap your mind around a subject.
When people tell me that I’m “good” at mathematics, what they’re commenting on is my ability at this particular part of mathematics. There are many more concepts within the subject that I haven’t wrapped my head around yet, but other people don’t see that. When we say someone is “good” at an activity, we probably aren’t thinking about it in an absolute sense. Instead, we look at their accomplishments and achievements in this specific instance, and generalize from that.
Being “good” at mathematics is not about solving problems quickly
This is an unfortunate result of our school system, and it’s something I want to talk about until everyone is sick of hearing me say it. You don’t have to solve problems quickly in order to be good at mathematics. If you think you need to be quick, you’ve been taught a bad lesson from the society around you.
Now, am I saying that there’s no utility to doing things quickly? Of course not. On a practical level, being quick at mathematics gives you more opportunities later on. After all, if you can work through problems quickly, you will have an easier time finishing tests and doing well. This will lead to external indicators that you are “good” at mathematics, which will in turn reinforce the idea that mathematics is your “thing”. As such, I would definitely say that being quick can help you, but I would argue against the idea that you need to be quick to be good.
If I think back to the moments where I most enjoyed learning about mathematics, they weren’t where I blazed through some reading and absorbed a bunch of ideas in a small amount of time. Instead, they were when I sat down and worked slowly through some difficult problems. After having a flash of insight and understanding the problem, that’s when I enjoyed mathematics the most. Trying to have this experience over and over again is what I would argue has made me “good” at mathematics.
Do I have some kind of genetic predisposition for mathematics? Maybe, I don’t know. What I do know is that any predisposition pales in comparison with the actual work required to go beyond talented and become good.
These were just a few cores ideas I’ve been thinking about in relation to being “good” at mathematics. I personally don’t like the discussion around being “good”, because I think it’s (mostly) a function of the effort and work you’re willing to put in. Sure, you may not be good in an “absolute” sense, but I think you can always improve with respect to your past self. If you learn about a new idea, you’re advancing your understanding of mathematics. It doesn’t have to be a groundbreaking new idea, just something that interests you.
This implies that improving in mathematics is an inevitability, if you’re willing to take the time to understand. Really, it’s a question of how committed you are to learning. I realize that there are extraordinary circumstances which don’t conform to this idea, but I think there’s still a broad applicability. That’s why I don’t like to use the word “good”. I much rather talk about the time a person has invested, since that indicates how serious they are and how much experience they have. I think these are much better indicators of ability in mathematics than just being “good”.
Hopefully, this changes the way you think about how people are skilled in mathematics. Remember, there’s “good” in the sense of school, and “good” in the sense that I explored here (invested time and effort). Overall, I think the latter better encapsulates the spirit of mathematics than the narrow, school-focused one.
Finally, I just want to make it clear that I am by no means “good” in the sense of professional mathematicians or even compared to high-achieving students. As always, when someone takes a judgement about being “good”, they are implicitly referring to a relative scale. Therefore, when I talk about being “good” at mathematics, I’m not trying to imply I am good with respect to those who are the best in their field. Rather, I’m comparing myself more to the average person.
Clarity and Words
A cornerstone of physics and mathematics is the process of labeling. If you want to learn any subject, you need to get familiar with the jargon of the field. In physics, terms like “work”, “resistance”, “capacitance”, “energy”, “potential”, and many more have precise meanings. Likewise, the terms “function”, “continuous”, and “one-to-one” have precise meanings in mathematics. When you first start learning the subject, it’s easy to be overwhelmed by the sheer number of terms. By the time you remember one term, several more have been introduced. Frankly, it can be frustrating and can ultimately turn you off from the subject.
Alternatively, I’ve also seen people just accept the words and end up using them without really knowing what they mean. We learn when these words apply, but we don’t have a mental picture of what the term means.
I’ve done this before, and I hate it. I feel like I’m a fraud, using terms to blend in with the others, hoping no one asks me any questions that are too specific. To give you a simple example, for the longest time I didn’t have a good conception of what a dielectric was. I simply associated it with something that popped up in my studies of electromagnetism. I never really thought of what it was. By the time I wanted to know the answer, it felt strange to ask. Nonetheless, I asked my professor and he explained that it was simply a non-conducting material.
Upon hearing this, I was immediately able to understand what he was talking about. That’s because I have a mental picture of what a conductor is. It’s a material that has mobile electrons which can move around and produce a current. Therefore, a dielectric was the opposite of that, meaning a sort of insulator. This definition made sense to me, and it allowed me to fill this hole in my knowledge.
I think a lot of terminology we use is stupid. It often only makes sense in hindsight, but a good majority of terms have no relation to their actual meaning (think of ideas named after people). This is needlessly confusing for beginners. I understand that we sometimes have terms that just stick in the canon of a field, but it’s something I fight against whenever I can.
What I find amazing is that we carry these holes in our knowledge all the time. There are terms we don’t know, but we go through our education without figuring out what they mean. I get it, a lot of terms are stupid and a description would be better. However, we can’t easily change what the majority of physicists or mathematicians do, so we need to become comfortable with the terminology.
I would suggest to avoid using terms you don’t know. This is something I try to adhere to all the time, because it’s precisely where your ignorance gets put on display. If you start using terms because you heard them from others (but you don’t know what they mean), you’re setting yourself up for questioning. This will only create an awkward situation in which you can’t explain what the term is, even though you used it. You will look ridiculous, and it isn’t a good learning experience for you. Instead, use the words and terms that you do have a clear mental picture for, because this will make explanations so much easier. I’m not saying you can’t get anything wrong, but be honest with yourself about what terms you do and don’t know.
Mental pictures and pinpointing misunderstandings
Terms might not be fun to learn and memorize, but they provide a springboard for launching new ideas. If everyone has the same idea of what “energy” means, there won’t be any debate when we use the term. This makes conversations easier, so it’s worth building up those mental images of various terms you learn.
When I’m working with a student and they don’t understand something, my greatest wish is to peek into their minds and see what they’re imagining. If I could do this, it would be so much easier to help them out. I know that a lot of misunderstandings stem from talking about the same word, yet having different mental pictures of what’s going on.
This is why I like to draw diagrams and pictures while teaching. My hope is that, by sharing a little about what I see for the problem, the student will be able to figure out where our thinking diverges. They can then stop me and we can have a more fruitful discussion about what’s happening.
Understanding the jargon of a field is a process with a steep initial learning curve that slowly gets easier with time. There are two important points to remember when learning the terms. First, make sure you have a mental picture of what’s going on. There are so many tools available for effective visualization that there is virtually no excuse for not having a good sense of what a term means. In physics and mathematics, diagrams are super helpful for this. I could tell you in words what it means to have a four-vector that is spacelike or timelike in general relativity. Or, I could just show you a diagram that makes this distinction clear. This really is a situation where a good diagram is worth a thousand words.
Second, avoid using terms you don’t know. This sounds simple and obvious, but it’s funny how often I find myself in situations where using words I don’t know can occur. This is a bad habit to have, and it’s one I recommend you avoid as much as you can. If you feel like you’re going to use a word you don’t know, I recommend asking someone else for clarification about what it means. This then lets you use the word without falling into the trap of saying words which have no meaning to you.
Finding your footing in a subject isn’t easy, but having a good grasp of the terminology certainly helps. The best way I’ve found to boost your knowledge is to build a lot of mental pictures of the terms. Then, you can refer to them when you hear them versus having a completely blank mind.
Don’t let yourself be swept up in the sea of confusing terms. Build yourself a raft, and learn to be comfortable with the ideas. (No matter how stupid the names are!)