I’m not good at math, I struggled with calculus and physics in college.
However some forms of math I can visually understand. Basic math like arithmetic or division I can create images in my head of what is happening (5*7=35, I can see an image of that my head and understand what the math means). Same with some aspects of Newtonian physics, I can visualize what the math means in the real world with certain acceleration problems.
But for advanced math, are people who are talented at math able to look at an equation and actually understand on an intuitive level what all that math means and how it relates to the workings of the world, or are they just following the abstract representations of the math?
In calculus class, I could perform the equations but I had no idea what they meant even if I tried to make sense of it. For people who are more talented, can they look at an advanced math equation and visualize what it means the same way I can visualize basic math (addition, subtraction, multiplication, etc)? Or do they more understand the math as an abstract concept, rather than something they can relate to the real world or visualize the workings thereof?
Sometimes I could, and still to this day, even though my professor marked me wrong in asking to derive the area of a circle from it’s circumference formal. And though he had this elaborate answer, mine was simple: the sum of all the total circumferences from zero length to the actual length being measured is the area of the circle, thus (the sum of) Pie times (d) (from length zero to current length) is Pie* r^2.
Sometimes teachers just don’t get what the OP is talking about and they penalize those who do.
Probably unrelated to the OP, but when I would work a word problem, it was visualized in reverse for me, i.e., the last step before the answer would pop into my head and I’d have to work it backwards to get full credit. Then plug in der zolution.
In my experience, it depends on the math. Some things are easier to visualize than others. I find a lot of calculus pretty easy to picture, for example, but differential equations are stubbornly abstract to me.
Note that being able to visualize it doesn’t necessarily mean you can do much with it; I’ve forgotten most of the shortcuts in calculus, so it’d be a bit of a struggle to actually use it now.
I was pretty good at Math and Physics. But eventually you hit a wall.
The first one in my case was in Group Theory. Once I got past the “easy” stuff it became a nightmare. Simple seeming formulas became a nightmare.
Another area that’s just baffling are the equations used in Quantum Theory. I really wanted to understand those because of Quantum Computing. Just impossible for me to parse.
But I’m still fairly good at a lot of stuff. E.g., the equations on the chalk board at the beginning of Good Will Hunting were quite simple to parse.
differential equations applied to geometry, where I could tell the solution by geometrical means and therefore yo momma was going to actually solve the equation system (my brain just went on strike),
a chrystallography teacher with whom the grades I got in the two first partial exams were respectively bad and worse, but the conversation when I went to see the second one (we had to “go see the exam” with the teacher every time) finally gave me the key I needed. I got a 100% on the final, having finally been able to translate the lines and planes I saw so clearly into those matrices he used. The final didn’t average with the partials, so I officially have a 100% rather than a fuckmesideways%.
multiple instances of material which made zero sense in math class, but perfect sense once we’d applied it in a different course.
and several of material we never applied, but which made sense once someone finally explained a detail I’d never been able to extract from the math teachers themselves (generally, the utility of the material in question… so, yeah, its meaning). I am eternally grateful to the poster who explained group theory to me, although to my chagrin I have to admit I don’t remember who it was.
I was ok at math, but I think I was mainly just following the rules on anything higher than basic stuff.
I am a database analyst, and I would not say I “picture” what code does to the data as much as just understanding what some piece of code is attempting to do.
For example, if I have been given a query or a bit of code written by someone else, then I can usually pretty quickly have an idea of what they were attempting to do. It takes a bit longer to think through the logic and determine if they will be successful, but I can usually write out what they were wanting to do and be accurate.
I would think a real life Sheldon would be able to do the same with math equations that fit within his wheel house. The they could look at a problem and just “understand” what the equation writer was attempting to do/solve/work on.
The only possible answer is, it depends. One of the beauties of math is that sometimes you understand the beginning and end of a long sequence of operations, but have no need to understand the intermediate steps, so long as you understand how to go from one to the next. That is, they might both be incomprehensible, but you do understand how to go from one to the next.
In the branch of math I do, we make heavy use of diagrams, what are actually called commutative diagrams. I can not illustrate them here. But I remember looking at a formula in a book I had written many years earlier and not understanding where it came from. But I turned it into a diagram (as it should have been from the beginning) and understood it immediately. I immediately revised my copy and the online copy of the book to replace the equations by a diagram. Now the young Turks in my field are using a new kind of diagram (called “string diagram”) that I don’t follow. But people who understand them feel they are even easier to process than my commutative diagrams.
A lot of math advances by introducing notation and modes of reasoning that make hard things easy. Or easier anyway.
Visualizing math. I can’t look at an equation and say that’s a hyperbola. So my formula to visual processor is weak. What I do is work on my understanding of a model and try to understand the model. Doing that with equations involved eventually leads me to associate a particular form of an equation with a particular model.
I’ve always been very good at math (and physics, which is basically just math), to the point that I could usually see where my teacher was headed when he/she started introducing a new concept.
But visualizing equations never happened for me. Show me an equation that I’m not familiar with, and nothing happens for me until you begin explaining things.
I’m not very good at math, as I was always limited to what I could be insightful about. I never got very good at manipulating things beyond the point I could intuit their meanings.
That said, I can often grasp the meaning, the message, of equations. That is true today for differential equations, as I use software to state them for field variables and then solve them using the finite element method. The software automates meshing and solving, about which I don’t have to think, but the equations themselves certainly look meaningful. I haven’t solved differential equations myself, analytically, in over 35 years.
