# [Question] When do you visualize (or not) while doing math?

I’ve been doing a lot more math lately, and, of course, also thinking about how I can do it faster and with deeper understanding. When I was in college, I remember that I found visualizing immensely helpful, to the point where it’s almost what “doing math” is for me.

The most obviously useful place to visualize was in multivariable and vector calculus. Especially because so many things were set in three dimensions, visualizing expressions made it so much easier to see why they were true.

I also visualize all the way down to doing algebra. When I symbolically manipulate an equation, I visualize things like bringing the letters across, or them merging into each other.

But it has its downsides. One is that, since I’m visualizing, I’m visualizing something specific; a specific function, a specific type of set. It’s not necessarily high-resolution, but it don’t usually cover the full range of what is represented by the symbols on the paper. And so by visualizing something specific, it’s likely that I’m not keeping in mind all the types of cases.

The most obvious downside is that visualization is computationally intensive. When I’m rendering a big scene in my mind, I can feel that it’s hard. Sometimes I have to close my eyes and block out the light. Things can slow down, and it can take several tries to get the visual just right, or to maintain it all the way through the manipulation. It seems plausible that this gets me more tired than is necessary, and causes me to stop doing the task earlier.

So, it has at least occurred to me that I could “try” doing something else. I put try it quotes because visualizing is so natural that it would be effortful for me to even think of an alternative. It’s as if someone said “you know, you don’t have to shovel that snow by hand” and I’m thinking, do they mean I could do it with my feet? What they actually mean is using a snow-blower.

Are there fundamentally different ways that you do math? Are there particular tasks you’ve tried that go faster with one method or another? Are there books that talk about this type of strategy?

• I have a PhD in economics, so I’ve taken a lot of math. I also have Aphantasia meaning I can’t visualize. When I was in school I didn’t think that anyone else could visualize either. I really wonder how much better I would be at math, and how much better I would have done in math classes, if I could visualize.

• Right, so my question to you is, how do you do math?? (This is probably silly question, but I’d love to hear your humor-me answer.)

• Last time I did math was when teaching game theory two days ago. I put a game on the blackboard. I wrote down an inequality that determined when there would be a certain equilibrium. Then I used the rules of algebra to simplify the inequality. Then I discussed why the inequality ended up being that the discount rate had to be greater than some number rather than less than some number.

• #### How I do math that starts out as a mathematical expression

I learned how to do math on paper or blackboard, except for an interlude at a Montessori school, where we used physical media. After a while, any math problem that took the form of a listing of mathematical expressions was one to solve with successive string manipulations. The initial form implied a set of transformations and a write-up that I had to perform. By looking at what I just wrote, and plotting how to create a transformation closer to my final answer, but with just a few manipulations of the earlier expression, I would record successive approaches to the final answer.

“Move the expression there, put that number there, that symbol there, combine those numbers into a new number using that operator, write that new thing underneath that old thing, line up the equals signs, line up those numbers,”. Repeat down the page as I write.

#### How I do math that starts out with a verbal description(for example, an algebra word problem)

Starting from the verbal description, get an idea of the mathematical expressions that I need to write to express what’s given in the problem. Write those down. From there, go on to solve the mathematical expression with a successive transformation approach.

#### How I do math that starts out with a graphical description (for example, a graph of a function)

Similar to starting with a verbal description, get an idea of the mathematical expressions that I need to write to express what’s given in the problem. From there, go on to solve the mathematical expression with a successive transformation approach.

NOTE: sometimes a verbal description could benefit from a picture, for example, in a physics problem. In that case, I would draw a picture of the physical system to help me identify the corresponding mathematical expressions to start with but also to help me feel like I “understand” what the verbal description depicts.

#### About internal visualizing vs using cognitive aids to do math

Cognitive aids reduce cognitive load for representing information. If you can choose between having an external picture that you can look at anytime of a graph, versus an internal picture of the same graph, then for most purposes, and to allow you the most freedom of operation in approaching a mathematics problem, I would suggest that you use the cognitive aid.

That’s how I’ve always preferred to do math:

• write out a math expression or draw out the graph or diagram of the problem

• don’t reduce write-outs of successive steps if that could lead to calculation errors

• keep it all neat on the page

For higher-level math, I think it makes sense to use a computer as much as you can to handle details and visualization, relying on its precision and memory while you concentrate on the identification and use of an algorithm that produces a useful solution to the problem.

#### Conclusions about how I do math

Basically, I do it the same way I have since I was little. I remember learning my times tables by memorization and writing them out, then paper and pencil work in class, then a bit of calculator work much later on with a TI-83, but mainly for large-number multiplication or division, or to check my work.

I took a graph theory class for my math minor, and I wish I still had my notes. I suspect that some of the answers were actually pictures, but I don’t remember much of what I did in the class, it was 30 years ago.

Mathematics involving a computer is more or less the same. You write out equations, but you might be working with cell references or variable names or varying data sets, so you are basically stopping at the point that you turn a word or graphical problem description into a mathematical expression. Then you let the calculator or computer do the work.

• What’s the most important problem you are trying to solve by visualizing?

• Heh, well, see the aforementioned

it’s almost what “doing math” is for me

It also feels like you’re asking something like, “what’s the most important problem you are trying to solve by having visual perception?” It’s kind of just how I navigate the world at all (atoms or math).

But let me take your question at face value and try to answer it.

I think the main answer is something like “semantics”. So much of my experiential knowledge is encoded in this physical, 3D physics manner, and when I can match up a symbolic expression with a physical scenario, I get a whole bunch of understanding for free. So it at least makes many of the “computations” far faster.

I also use it as a working memory scratchpad. If I’m trying to multiply two numbers, I often come up with a way to FOIL them, and then I hold all four of those terms in a visual space in my mind. As I calculate, I replace terms with their new values.

(There’s probably more, but that’s what I got from the first attempt at introspection.)

• The first math problem I created was spawned from a visualization on a rare occasion smoking marijuana. I started imagining cubes falling from the sky, and just observing this process. I got curious about the chance of one cube landing on top of another. This led to the problem:

Given a square boundary of size B, place squares of side length S one at a time completely inside of the boundary at random locations. How many squares on average will you place before a pair of squares overlaps?

• First: what’s the load-bearing function of visualizations in math?

I think it’s the same function as prototypical examples more broadly. They serve as a consistency check—i.e. if there’s any example at all which matches the math then at least the math isn’t inconsistent. They also offer direct intuition for which of the assumptions are typically “slack” vs “taut”—i.e. in the context of the example, would the claim just totally fall apart if we relax a particular assumption, or would it gracefully degrade? And they give some intuition for what kinds-of-things to bind the mathematical symbols to, in order to apply the math.

Based on that, I’d expect that non-visual prototypical examples can often serve a similar role.

Also, some people use type-tracking to get some of the same benefits, though insofar as that’s a substitute for prototypical example tracking I think it’s usually inferior.