Some of us are visual thinkers and understand things easier if there are images attached :-). Personally i think almost completely in images so i'd have to somehow visualize something to understand it :-P someone doing it for me helps immensely.
Of course i can also think more abstract but it feels as if i'm trying to send megabytes of data to another city through TCP/IP with one byte per send().
Everyone is a visual learner but everyone can also understand a simple arithmetic expression. Everyone thinks almost completely in images. The only difference is that mathematicians have had more practice learning how to transform expressions into visual images. It's a bit like looking at sheet music and being able to hum it in your head.
Learning how to read sheet music isn't something that shouldn't even be attempted because you're an "auditory learner". I also think that spending a little extra effort to learn how to read an arithmetic expression is worthwhile and a skill we should foster in society.
>Everyone is a visual learner but everyone can also understand a simple arithmetic expression. Everyone thinks almost completely in images.
I think it is hard to separate your experience from the collective here. Much of the time I am working on math or programming, I don't think in images; I am just manipulating symbols in a patterned fashion that I have already convinced myself is sound (in a logic sense).
I definitely have never thought about floating point as anything other than binary scientific notation. I am quite comfortable with the understanding of exponent, significand, and sign.
You've never drawn a linked list? A Venn diagram to think about your logical sets? A Hasse diagram for showing set inclusion? A function to see its growth? Never visualised numbers as being on a real line, with in-betweenness properties?
The visual cortex is one of the most developed part of our brains. Even blind people use it for mathematics:
Even when you're doing symbol pushing "blindly", you're still using visual skills to place symbols, different fonts, you align variables, superscripts, subscripts. There's a certain geometry to mathematical notation itself.
> You've never drawn a linked list? A Venn diagram to think about your logical sets? A Hasse diagram for showing set inclusion? A function to see its growth? Never visualised numbers as being on a real line, with in-betweenness properties?
I never said that I don't visualize things. Just that it isn't the only (or main) way that I think about things.
By all means, methods of visualization can be very powerful. Designing different ways to visualize things can make more ideas easily accessible. Many things are also illuminated by logical proofs, sound visualizations (for lack of a better word), or narrative.
I think it is hard to extrapolate from a single experience what the best way to understand is for most people. You may be a very visual learner, and not realize that many other people don't gain as much from visualizations as you do.
> Even when you're doing symbol pushing "blindly", you're still using visual skills to place symbols, different fonts, you align variables, superscripts, subscripts. There's a certain geometry to mathematical notation itself.
The particular symbols and syntax matters little. I certainly am not considering different fonts when I work things out. Often manipulations I am doing are only in my head and don't have a concrete image until I need to write it down.
I'll step in here. I've met people who are visual like me, and do math using their visual thinking. For me, algebra and such is like tetris or a legos but with symbols. To this day, when I do algebra in my head, it involves visualizing symbols as if they are on a page and manipulating that picture.
Others however are not visual, but yet they can do algebra fine using something like logical rules, and to be honest, often tend to be more accurate with it. It sounds like you're one of those people, so it's unfair for what the above poster said regarding everyone being "visual". I think it's not really a matter of ability, it's more about what methods people prefer and some people just don't prefer visualization.
Funny side story: I think I can do the logical thing but it isn't my usual modus operandi, and I had to work on sharpening my "logical" side. I remember when I was in particle physics, and my office mate was so good at the algebra in QFT class and always very accurate. I on the other hand made frequently many little errors, but I'd always be able to "see the forest from the trees." I remember him once showing me the correct way to derive some transition amplitude that was in the homework, and just seeing it for the first time, I said "ah, that reduces to 1 + blah" because I could see that immediately; while he hadn't realized that even after working on it for a couple of hours. I think both types of people (left and right brain I guess) sort of complement each other in research, and we both have to approach math in our own ways.
Of course i can also think more abstract but it feels as if i'm trying to send megabytes of data to another city through TCP/IP with one byte per send().