I have a question for anyone who spends a fair amount of their time thinking about math: how exactly do you do it, and why?
To specify, I’ve tried thinking about math in two rather distinct ways. One is verbal and involves stating terms, definitions, and the logical steps of inference I’m making in my head or out loud, as I frequently talk to myself during this process. This type of thinking is slow, but it tends to work better for actually writing proofs and when I don’t yet have an intuitive understanding of the concepts involved.
The other is nonverbal and based on understanding terms, definitions, theorems, and the ways they connect to each other on an intuitive level (note: this takes a while to achieve, and I haven’t always managed it) and letting my mind think it out, making logical steps of inference in my head, somewhat less consciously. This type of thinking is much faster, though it has a tendency to get derailed or stuck and produces good results less reliably.
Which of those, if any, sounds closer to the way you think about math?
(Note: most of the people I’ve talked to about this don’t polarize it quite so much and tend to do a bit of both, i.e. thinking through a proof consciously but solving potential problems that come up while writing it more intuitively. Do you also divide different types of thinking into separate processes, or use them together?)
The reason I’m asking is that I’m trying to transition to spending more of my time thinking about math not in a classroom setting and I need to figure out how I should go about it. The fast kind of thinking would be much more convenient, but it appears to have downsides that I haven’t been able to study properly due to insufficient data.
I’m only a not-very-studious undergraduate (in physics), and don’t spend an awful lot of time thinking about maths ourside of that, but I pretty much only think about maths in the nonverbal way—I can understand an idea when verbally explained to me, but I have to “translate it” into nonverbal maths to get use out of it.
I don’t tend to do a lot of proofs anymore. When I think of math, I find it most important to be able to flip back and forth between symbol and referent freely—look at an equation and visualize the solutions, or (to take one example of the reverse) see a curve and think of ways of representing it as an equation. Since when visualizing numbers will often not be available, I tend to think of properties of a Taylor or Fourier series for that graph. I do a visual derivative and integral.
That way, the visual part tells me where to go with the symbolic part. Things grind to a halt when I have trouble piecing that visualization together.
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.
As someone employed doing mid-level math (structural design), I’m much like most others you’ve talked to. The entirely non-verbal intuitive method is fast, and it tends to be highly correct if not accurate. The verbal method is a lot slower, but it lends itself nicely to being put to paper and great for getting highly accurate if not correct answers. So everything that matters gets done twice, for accurate correct results. Of course, because it is fast the intuitive method is prefered for brainstorming, then the verbal method verifies any promising brainstorms.
Correct and Precise may have been better terms. By correct I mean a result that I have very high confidence in, but that is not precise enough to be useable. By accurate I mean a result that is very precise but with far less confidence that it is correct.
As an example, consider a damped oscillation word problem from first year. You are very confident that as time approaches infinity that the displacement will approach a value just by looking at it, but you don’t know that value. Now when you crunch the numbers (the verbal process in the extreme) you get a very specific value that the function approaches, but have less confidence that that value is correct, you could have made any of a number of mistakes. In this example the classic wrong result is the displacement is in the opposite direction as the applied force.
This is a very simple example so it may be hard to separate the non-verbal process from the verbal, but there are many cases where you know what the result should look like but deriving the equations and relations can turn into a black box.
One of my current problems is that I don’t understand my brain well enough for nonverbal thinking not to turn into a black box. I think this might be a matter of inexperience, as I only recently managed intuitive, nonverbal understanding of math concepts, so I’m not always entirely sure what my brain is doing. (Anecdotally, my intuitive understanding of a problem produces good results more often than not, but any time my evidence is anecdotal there’s this voice in my head that yells “don’t update on that, it’s not statistically relevant!”)
Does experience in nonverbal reasoning on math lend actually itself to better understanding of said reasoning, or is that just a cached thought of mine?
Doing everything both ways, nonverbal and verbal, has lent itself to better understanding of the reasoning. Which touches on the anecdote problem, if you test every nonverbal result; you get something statistically relevant. If your odds are more often than not with nonverbal, testing every result and digging for the mistakes will increase your understanding (disclaimer: this is hard work).
So, essentially, there isn’t actually any way of getting around the hard work. (I think I already knew that and just decided to go on not acting on it for a while longer.) Oh well, the hard work part is also fun.
Which of those, if any, sounds closer to the way you think about math?
Each serves its own purpose. It is like the technical and artistic sides of musical performance: the technique serves the artistry. In a sense the former is subordinate to the latter, but only in the sense that the foundation of a building is subordinate to its superstructure. To perform well enough that someone else would want to listen, you need both.
This may be useful reading, and the essays here (from which the former is linked).
As someone with a Ph.D. in math, I tend to think verbally in as much as I have words attached to the concepts I’m thinking about, but I never go so far as to internally vocalize the steps of the logic I’m following until I’m at the point of actually writing something down.
I think there is another much stronger distinction in mathematical thinking, which is formal vs. informal. This isn’t the same distinction as verbal vs. nonverbal, for instance, formal thinking can involve manipulation of symbols and equations in addition to definitions and theorems, and I often do informal thinking by coming up with pretty explicitly verbal stories for what a theorem or definition means (though pictures are helpful too).
I personally lean heavily towards informal thinking, and I’d say that trying to come up with a story or picture for what each theorem or definition means as you are reading will help you a lot. This can be very hard sometimes. If you open a book or paper and aren’t able to get anywhere when you try do this to the first chapter, it’s a good sign that you are reading something too difficult for your current understanding of that particular field. At a high level of mastery of a particular subject, you can turn informal thinking into proofs and theorems, but the first step is to be able to create stories and pictures out of the theorems, proofs, and definitions you are reading.
I’m a math undergrad, and I definitely spend more time in the second sort of style. I find that my intuition is rather reliable, so maybe that’s why I’m so successful at math. This might be hitting into the “two cultures of mathematics”, where I am definitely on the theory builder/algebraist side. I study category theory and other abstract nonsense, and I am rather bad (relative to my peers) at Putnam style problems.
I don’t see a clear verbal vs. non-verbal dichotomy—or at least the non-verbal side has lots of variants.
To gain an intuitive non-verbal understanding can involve
visual aids (from precise to vague): graphs, diagrams, patterns (esp. repetitions), pictures, vivid imagination (esp. for memorizing)
acoustic aids: rhythms (works with muscle memory too), patterns in the spoken form, creating sounds for elements
abstract thinking (from precise to vague): logical inference, semantic relationships (is-a, exists, always), vague relationships (discovering that the more of this seems to imply the more of that)
Note: Logical inference seems to be the verbal part you mean, but I don’t think symbolic thinking is always verbal. Its conscious derivation may be though.
And I hear that the verbal side despite lending itself to more symbolic thinking can nonetheless work its grammar magic on an intuitive level too (though not for me).
Personally if I really want to solve a mathematical problem I immerse myself in it. I try lots of attack angles from the list above (not systematically but as it seems fit). I’m an abstract thinker and don’t rely on verbal, acoustic or motor cues a lot. Even visual aids don’t play a large role though I do a lot of sketching, listing/enumerating combinations, drawing relations/trees, tabulating values/items. If I suspect a repeating pattern I may tap to it to sound it out. If there is lengthy logical inference involved that I haven’t internalized I speak the rule repeatedly to use the acoustic loop as memory aid. I play around with it during the day visualizing relationships or following steps, sometimes until in the evening everyting blurs and I fall asleep.
Personally, the nonverbal thing is the proper content of math—drawing (possibly mental) pictures to represent objects and their interactions. If I get stuck, I try doing simpler examples. If I’m still stuck, then I start writing things down verbally, mainly as a way to track down where I’m confused or where exactly I need to figure something out.
I don’t really draw that distinction. I’d say that my thinking about mathematics is just as verbal as any other thinking. In fact, a good indication that I’m picking up a field is when I start thinking in the language of the field (i.e. I will actually think “homology group” and that will be a term that means something, rather than “the group formed by these actions...”)
I’d say that my thinking about mathematics is just as verbal as any other thinking.
Just to clarify, because this will help me categorize information: do you not do the nonverbal kind of thinking at all, or is it all just mixed together?
I’m not really conscious of the distinction, unless you’re talking about outright auditory things like rehearsing a speech in my head. The overwhelming majority of my thinking is in a format where I’m thinking in terms of concepts that I have a word for, but probably not consciously using the word until I start thinking about what I’m thinking about. Do you have a precise definition of “verbal”? But whether you call it verbal or not, it feels like it’s all the same thing.
I don’t really have good definitions at this point, but in my head the distinction between verbal and nonverbal thinking is a matter of order. When I’m thinking nonverbally, my brain addresses the concepts I’m thinking about and the way they relate to each other, then puts them to words. When I’m thinking verbally, my brain comes up with the relevant word first, then pulls up the concept. It’s not binary; I tend to put it on a spectrum, but one that has a definite tipping point. Kinda like a number line: it’s ordered and continuous, but at some point you cross zero and switch from positive to negative. Does that even make sense?
Alright, that works too. We’re allowed to think differently. Now I’m curious, could you define your way of thinking more precisely? I’m not quite sure I grok it.
So, I’d say there are three modes of thinking I can identify:
Normal thinking, what I’m doing the vast majority of the time. I’m thinking by manipulating concepts, which are just, well, things.
Introspective thinking, where I’m doing the first kind of thinking, and thinking about it. Because the map can’t be the territory, when I’m thinking about thinking the concepts I’m thinking about are represented by something simpler than themselves—if you’re thinking about thinking about sheep then the sheep you’re thinking about thinking about can’t be as complex as the sheep you’re thinking about. In fact they’re represented either by words, or by something isomorphic to words—labels for concepts. So when I’m thinking about thinking, the thinking-about-thinking is verbal—but the thinking isn’t (although there’s a light-in-the-fridge effect that might make one think it was).
Auditory thinking, where I’m thinking in words in my head, planning a speech (or more likely a piece of writing—and most of the time I never actually write or say it). This is the only kind of thinking I’m conscious of doing that really feels verbal, but it feels sensory rather than thinking in words; I’m hearing a voice in my cartesian theater.
I have a question for anyone who spends a fair amount of their time thinking about math: how exactly do you do it, and why?
To specify, I’ve tried thinking about math in two rather distinct ways. One is verbal and involves stating terms, definitions, and the logical steps of inference I’m making in my head or out loud, as I frequently talk to myself during this process. This type of thinking is slow, but it tends to work better for actually writing proofs and when I don’t yet have an intuitive understanding of the concepts involved.
The other is nonverbal and based on understanding terms, definitions, theorems, and the ways they connect to each other on an intuitive level (note: this takes a while to achieve, and I haven’t always managed it) and letting my mind think it out, making logical steps of inference in my head, somewhat less consciously. This type of thinking is much faster, though it has a tendency to get derailed or stuck and produces good results less reliably.
Which of those, if any, sounds closer to the way you think about math? (Note: most of the people I’ve talked to about this don’t polarize it quite so much and tend to do a bit of both, i.e. thinking through a proof consciously but solving potential problems that come up while writing it more intuitively. Do you also divide different types of thinking into separate processes, or use them together?)
The reason I’m asking is that I’m trying to transition to spending more of my time thinking about math not in a classroom setting and I need to figure out how I should go about it. The fast kind of thinking would be much more convenient, but it appears to have downsides that I haven’t been able to study properly due to insufficient data.
I’m only a not-very-studious undergraduate (in physics), and don’t spend an awful lot of time thinking about maths ourside of that, but I pretty much only think about maths in the nonverbal way—I can understand an idea when verbally explained to me, but I have to “translate it” into nonverbal maths to get use out of it.
I don’t tend to do a lot of proofs anymore. When I think of math, I find it most important to be able to flip back and forth between symbol and referent freely—look at an equation and visualize the solutions, or (to take one example of the reverse) see a curve and think of ways of representing it as an equation. Since when visualizing numbers will often not be available, I tend to think of properties of a Taylor or Fourier series for that graph. I do a visual derivative and integral.
That way, the visual part tells me where to go with the symbolic part. Things grind to a halt when I have trouble piecing that visualization together.
This appears to be a useful skill that I haven’t practiced enough, especially for non-proof-related thinking. I’ll get right on that.
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.
As someone employed doing mid-level math (structural design), I’m much like most others you’ve talked to. The entirely non-verbal intuitive method is fast, and it tends to be highly correct if not accurate. The verbal method is a lot slower, but it lends itself nicely to being put to paper and great for getting highly accurate if not correct answers. So everything that matters gets done twice, for accurate correct results. Of course, because it is fast the intuitive method is prefered for brainstorming, then the verbal method verifies any promising brainstorms.
Could you please explain what you mean by “correct” and “accurate” in this case? I have a general idea, but I’m not quite sure I get it.
Correct and Precise may have been better terms. By correct I mean a result that I have very high confidence in, but that is not precise enough to be useable. By accurate I mean a result that is very precise but with far less confidence that it is correct.
As an example, consider a damped oscillation word problem from first year. You are very confident that as time approaches infinity that the displacement will approach a value just by looking at it, but you don’t know that value. Now when you crunch the numbers (the verbal process in the extreme) you get a very specific value that the function approaches, but have less confidence that that value is correct, you could have made any of a number of mistakes. In this example the classic wrong result is the displacement is in the opposite direction as the applied force.
This is a very simple example so it may be hard to separate the non-verbal process from the verbal, but there are many cases where you know what the result should look like but deriving the equations and relations can turn into a black box.
Right, that makes much more sense now, thanks.
One of my current problems is that I don’t understand my brain well enough for nonverbal thinking not to turn into a black box. I think this might be a matter of inexperience, as I only recently managed intuitive, nonverbal understanding of math concepts, so I’m not always entirely sure what my brain is doing. (Anecdotally, my intuitive understanding of a problem produces good results more often than not, but any time my evidence is anecdotal there’s this voice in my head that yells “don’t update on that, it’s not statistically relevant!”)
Does experience in nonverbal reasoning on math lend actually itself to better understanding of said reasoning, or is that just a cached thought of mine?
Doing everything both ways, nonverbal and verbal, has lent itself to better understanding of the reasoning. Which touches on the anecdote problem, if you test every nonverbal result; you get something statistically relevant. If your odds are more often than not with nonverbal, testing every result and digging for the mistakes will increase your understanding (disclaimer: this is hard work).
So, essentially, there isn’t actually any way of getting around the hard work. (I think I already knew that and just decided to go on not acting on it for a while longer.) Oh well, the hard work part is also fun.
Each serves its own purpose. It is like the technical and artistic sides of musical performance: the technique serves the artistry. In a sense the former is subordinate to the latter, but only in the sense that the foundation of a building is subordinate to its superstructure. To perform well enough that someone else would want to listen, you need both.
This may be useful reading, and the essays here (from which the former is linked).
reads the first essay and bookmarks the page with the rest
Thanks for that, it made for enjoyable and thought-provoking reading.
As someone with a Ph.D. in math, I tend to think verbally in as much as I have words attached to the concepts I’m thinking about, but I never go so far as to internally vocalize the steps of the logic I’m following until I’m at the point of actually writing something down.
I think there is another much stronger distinction in mathematical thinking, which is formal vs. informal. This isn’t the same distinction as verbal vs. nonverbal, for instance, formal thinking can involve manipulation of symbols and equations in addition to definitions and theorems, and I often do informal thinking by coming up with pretty explicitly verbal stories for what a theorem or definition means (though pictures are helpful too).
I personally lean heavily towards informal thinking, and I’d say that trying to come up with a story or picture for what each theorem or definition means as you are reading will help you a lot. This can be very hard sometimes. If you open a book or paper and aren’t able to get anywhere when you try do this to the first chapter, it’s a good sign that you are reading something too difficult for your current understanding of that particular field. At a high level of mastery of a particular subject, you can turn informal thinking into proofs and theorems, but the first step is to be able to create stories and pictures out of the theorems, proofs, and definitions you are reading.
I’m a math undergrad, and I definitely spend more time in the second sort of style. I find that my intuition is rather reliable, so maybe that’s why I’m so successful at math. This might be hitting into the “two cultures of mathematics”, where I am definitely on the theory builder/algebraist side. I study category theory and other abstract nonsense, and I am rather bad (relative to my peers) at Putnam style problems.
I don’t see a clear verbal vs. non-verbal dichotomy—or at least the non-verbal side has lots of variants. To gain an intuitive non-verbal understanding can involve
visual aids (from precise to vague): graphs, diagrams, patterns (esp. repetitions), pictures, vivid imagination (esp. for memorizing)
acoustic aids: rhythms (works with muscle memory too), patterns in the spoken form, creating sounds for elements
abstract thinking (from precise to vague): logical inference, semantic relationships (is-a, exists, always), vague relationships (discovering that the more of this seems to imply the more of that)
Note: Logical inference seems to be the verbal part you mean, but I don’t think symbolic thinking is always verbal. Its conscious derivation may be though.
And I hear that the verbal side despite lending itself to more symbolic thinking can nonetheless work its grammar magic on an intuitive level too (though not for me).
Personally if I really want to solve a mathematical problem I immerse myself in it. I try lots of attack angles from the list above (not systematically but as it seems fit). I’m an abstract thinker and don’t rely on verbal, acoustic or motor cues a lot. Even visual aids don’t play a large role though I do a lot of sketching, listing/enumerating combinations, drawing relations/trees, tabulating values/items. If I suspect a repeating pattern I may tap to it to sound it out. If there is lengthy logical inference involved that I haven’t internalized I speak the rule repeatedly to use the acoustic loop as memory aid. I play around with it during the day visualizing relationships or following steps, sometimes until in the evening everyting blurs and I fall asleep.
Personally, the nonverbal thing is the proper content of math—drawing (possibly mental) pictures to represent objects and their interactions. If I get stuck, I try doing simpler examples. If I’m still stuck, then I start writing things down verbally, mainly as a way to track down where I’m confused or where exactly I need to figure something out.
I don’t really draw that distinction. I’d say that my thinking about mathematics is just as verbal as any other thinking. In fact, a good indication that I’m picking up a field is when I start thinking in the language of the field (i.e. I will actually think “homology group” and that will be a term that means something, rather than “the group formed by these actions...”)
Just to clarify, because this will help me categorize information: do you not do the nonverbal kind of thinking at all, or is it all just mixed together?
I’m not really conscious of the distinction, unless you’re talking about outright auditory things like rehearsing a speech in my head. The overwhelming majority of my thinking is in a format where I’m thinking in terms of concepts that I have a word for, but probably not consciously using the word until I start thinking about what I’m thinking about. Do you have a precise definition of “verbal”? But whether you call it verbal or not, it feels like it’s all the same thing.
I don’t really have good definitions at this point, but in my head the distinction between verbal and nonverbal thinking is a matter of order. When I’m thinking nonverbally, my brain addresses the concepts I’m thinking about and the way they relate to each other, then puts them to words. When I’m thinking verbally, my brain comes up with the relevant word first, then pulls up the concept. It’s not binary; I tend to put it on a spectrum, but one that has a definite tipping point. Kinda like a number line: it’s ordered and continuous, but at some point you cross zero and switch from positive to negative. Does that even make sense?
It makes sense but it doesn’t match my subjective experience.
Alright, that works too. We’re allowed to think differently. Now I’m curious, could you define your way of thinking more precisely? I’m not quite sure I grok it.
So, I’d say there are three modes of thinking I can identify:
Normal thinking, what I’m doing the vast majority of the time. I’m thinking by manipulating concepts, which are just, well, things.
Introspective thinking, where I’m doing the first kind of thinking, and thinking about it. Because the map can’t be the territory, when I’m thinking about thinking the concepts I’m thinking about are represented by something simpler than themselves—if you’re thinking about thinking about sheep then the sheep you’re thinking about thinking about can’t be as complex as the sheep you’re thinking about. In fact they’re represented either by words, or by something isomorphic to words—labels for concepts. So when I’m thinking about thinking, the thinking-about-thinking is verbal—but the thinking isn’t (although there’s a light-in-the-fridge effect that might make one think it was).
Auditory thinking, where I’m thinking in words in my head, planning a speech (or more likely a piece of writing—and most of the time I never actually write or say it). This is the only kind of thinking I’m conscious of doing that really feels verbal, but it feels sensory rather than thinking in words; I’m hearing a voice in my cartesian theater.