I usually think about math nonverbally. I am not usually doing such thinking to come up with proofs. My background is in engineering, so I got a different sort of approach to math in my education about math than the people who were in the math faculty at the university I attended.
Sometimes I do go through a problem step by step, but usually not verbally. I sometimes make notes to help me remember things as I go along. Constraints, assumptions, design goals, etc. Explicitly stating these, which I usually do by writing them on paper, not speaking them aloud, if I’m working by myself on a problem, can help. But sometimes I am not working by myself and would say them out loud to discuss them with other people.
Also, there is often more than one way to visualize or approach a problem, and I will do all of them that come to mind.
I would suggest, to spend more time thinking about math, find something that you find really beautiful about math and start there, and learn more about it. Appreciate it, and be playful with it. Also, find a community where you can bounce ideas around and get other people’s thoughts and ideas about the math you are thinking about. Some of this stuff can be tough to learn alone. I’m not sure how well this advice might work, your mileage may vary.
When I am really understanding the math, it seems like it goes directly from equations on the paper right into my brain as images and feelings and relations between concepts. No verbal part of it. I dream about math that way too.
I only got to a nonverbal level of understanding of advanced math fairly recently, and the first time I experienced it I think it might have permanently changed my life. But if you dream about math...well, that means I still have a long way to go and deeper levels of understanding to discover. Yay!
Follow-up question (just because I’m curious): how do you approach math problems differently when working on them from the angle of engineering, as opposed to pure math?
It seemed to me that the people I knew who were studying pure math spent a lot of time on proofs, and that math was taught to them with very little context for how the math might be used in the real world, and without a view as to which parts were more important than others.
In engineering classes we proved things too, but that was usually only a first step to using the concepts to work on some other problem. There was more time spent on some types of math than on others. Some things were considered to be more useful and important than others. Usually some sort of approximations or assumptions would be used, in order to make a problem simpler and able to be solved, and techniques from different branches of math were combined together whenever useful, often making for some overlap in the notation that had to be dealt with.
There was also the idea that any kind of math is only an approximate model of the true situation. Any model is going to fail at some point. Every bridge that has been built has been built using approximations and assumptions, and yet most bridges stay up. Learning when one can trust the approximations and assumptions is vital. People can die if you get it wrong. Learning the habit of writing down explicitly what the assumptions and approximations are, and to have a sense for where they are valid and where they are not, is a skill that I value, and have carried over into other aspects of my life.
Another thing is that math is usually in service of some other goal. There are design constraints and criteria, and whatever math you can bring in to get it done is welcome, and other math is extraneous. The beauty of math can be admired, but a kludgy theory that is accurate to real world conditions gets more respect than a pretty theory that is less accurate. In fact, sometimes engineers end up making kludgy theory that solves engineering problems into some sophisticated mathematics that looks more formal and has some interesting properties, and then it has a beauty of its own, although some of the beauty comes from knowing how it fits into a real world phenomenon.
Also, engineers tend to work in teams, not alone. So communicating with each other, and making sure that all the people on the team have a similar understanding of a situation, is a non-trivial part of the work. You don’t want a situation where one person has one type of abstraction in their head, and another person has a different one, and they don’t realize it, and when they go off to do their separate work, it doesn’t match up. This can lead to all sorts of problems, not limited to cost overruns, design flaws, delays, and even deaths. So, if you hear engineers discussing nitpicky details and going over technical concepts more than once, that is one major reason why. You really need people to be on the same page.
Teamwork is so important to engineering that when taking classes, we were encouraged to talk to each other and work together on problems, before submitting answers. Whereas the people over in math were forbidden to talk to each other about their work before handing it in. That policy might be different at different schools. But I think it shows an important difference in culture.
Math is certainly something that can be enjoyed and practiced solo. But especially on some of the most tricky concepts of math that I have learned, I benefitted a lot from being able to discuss it with people, and get new insights and understanding from their perspectives. Sometimes I didn’t even realize that I didn’t properly understand a concept until I attempted to use it, and got a completely different answer from someone else who was attempting to use it.
I said it can get kludgy, and that the focus is on real world problems, but there are times when it does feel clean and pure, especially when people make real world objects that correspond pretty well to ideal mathematical objects. For example, using 4th-order differential equations to calculate the bending moments for I-beams felt peaceful and pretty, once I got the hang of it, and I think not it is not unlike something you might find in a pure math course.
I’m pretty enthusiastic about math, it’s one of my favourite things to think about and do.
I usually think about math nonverbally. I am not usually doing such thinking to come up with proofs. My background is in engineering, so I got a different sort of approach to math in my education about math than the people who were in the math faculty at the university I attended.
Sometimes I do go through a problem step by step, but usually not verbally. I sometimes make notes to help me remember things as I go along. Constraints, assumptions, design goals, etc. Explicitly stating these, which I usually do by writing them on paper, not speaking them aloud, if I’m working by myself on a problem, can help. But sometimes I am not working by myself and would say them out loud to discuss them with other people.
Also, there is often more than one way to visualize or approach a problem, and I will do all of them that come to mind.
I would suggest, to spend more time thinking about math, find something that you find really beautiful about math and start there, and learn more about it. Appreciate it, and be playful with it. Also, find a community where you can bounce ideas around and get other people’s thoughts and ideas about the math you are thinking about. Some of this stuff can be tough to learn alone. I’m not sure how well this advice might work, your mileage may vary.
When I am really understanding the math, it seems like it goes directly from equations on the paper right into my brain as images and feelings and relations between concepts. No verbal part of it. I dream about math that way too.
I only got to a nonverbal level of understanding of advanced math fairly recently, and the first time I experienced it I think it might have permanently changed my life. But if you dream about math...well, that means I still have a long way to go and deeper levels of understanding to discover. Yay!
Follow-up question (just because I’m curious): how do you approach math problems differently when working on them from the angle of engineering, as opposed to pure math?
It seemed to me that the people I knew who were studying pure math spent a lot of time on proofs, and that math was taught to them with very little context for how the math might be used in the real world, and without a view as to which parts were more important than others.
In engineering classes we proved things too, but that was usually only a first step to using the concepts to work on some other problem. There was more time spent on some types of math than on others. Some things were considered to be more useful and important than others. Usually some sort of approximations or assumptions would be used, in order to make a problem simpler and able to be solved, and techniques from different branches of math were combined together whenever useful, often making for some overlap in the notation that had to be dealt with.
There was also the idea that any kind of math is only an approximate model of the true situation. Any model is going to fail at some point. Every bridge that has been built has been built using approximations and assumptions, and yet most bridges stay up. Learning when one can trust the approximations and assumptions is vital. People can die if you get it wrong. Learning the habit of writing down explicitly what the assumptions and approximations are, and to have a sense for where they are valid and where they are not, is a skill that I value, and have carried over into other aspects of my life.
Another thing is that math is usually in service of some other goal. There are design constraints and criteria, and whatever math you can bring in to get it done is welcome, and other math is extraneous. The beauty of math can be admired, but a kludgy theory that is accurate to real world conditions gets more respect than a pretty theory that is less accurate. In fact, sometimes engineers end up making kludgy theory that solves engineering problems into some sophisticated mathematics that looks more formal and has some interesting properties, and then it has a beauty of its own, although some of the beauty comes from knowing how it fits into a real world phenomenon.
Also, engineers tend to work in teams, not alone. So communicating with each other, and making sure that all the people on the team have a similar understanding of a situation, is a non-trivial part of the work. You don’t want a situation where one person has one type of abstraction in their head, and another person has a different one, and they don’t realize it, and when they go off to do their separate work, it doesn’t match up. This can lead to all sorts of problems, not limited to cost overruns, design flaws, delays, and even deaths. So, if you hear engineers discussing nitpicky details and going over technical concepts more than once, that is one major reason why. You really need people to be on the same page.
Teamwork is so important to engineering that when taking classes, we were encouraged to talk to each other and work together on problems, before submitting answers. Whereas the people over in math were forbidden to talk to each other about their work before handing it in. That policy might be different at different schools. But I think it shows an important difference in culture.
Math is certainly something that can be enjoyed and practiced solo. But especially on some of the most tricky concepts of math that I have learned, I benefitted a lot from being able to discuss it with people, and get new insights and understanding from their perspectives. Sometimes I didn’t even realize that I didn’t properly understand a concept until I attempted to use it, and got a completely different answer from someone else who was attempting to use it.
I said it can get kludgy, and that the focus is on real world problems, but there are times when it does feel clean and pure, especially when people make real world objects that correspond pretty well to ideal mathematical objects. For example, using 4th-order differential equations to calculate the bending moments for I-beams felt peaceful and pretty, once I got the hang of it, and I think not it is not unlike something you might find in a pure math course.
I’m pretty enthusiastic about math, it’s one of my favourite things to think about and do.