How Do Different People Conceptualize Mathematics?

[muppet]

Now our exams for the year are done my housemate and I are free to wallow in esotric nerdishness without recourse to computation or the relevance of our discussion matter to.. well, anything really
One thing that's emerged from our ramblings is that we view maths in completely different ways. I like to think about things in terms of english wherever I possibly can; he has an incredible functional understanding of maths (which stands him in great stead in exams) but thinks about everything that isn't explicitly geometrical (which he also has a great capacity to picture) in computational terms.
To take a clear cut example, say you're solving simultaneous equations using matrices. I've come to understand it by regarding the matrix of coefficients as a linear map that takes an unknown point (x y z) to some definite point (a b c), and see inverting the matrix as finding the inverse of this linear map and applying it to the point (a b c). The way he sees it, when you multiply both sides of the equation by the inverse of the matrix of coefficients it's inherently obvious that you're left with (x y z) equal to some definite vector, and he figures I just make my life difficult for myself.

I was just curious as to how everyone else thought about this sort of thing?

 

[lisab]

Now our exams for the year are done my housemate and I are free to wallow in esotric nerdishness without recourse to computation or the relevance of our discussion matter to.. well, anything really
One thing that's emerged from our ramblings is that we view maths in completely different ways. I like to think about things in terms of english wherever I possibly can; he has an incredible functional understanding of maths (which stands him in great stead in exams) but thinks about everything that isn't explicitly geometrical (which he also has a great capacity to picture) in computational terms.
To take a clear cut example, say you're solving simultaneous equations using matrices. I've come to understand it by regarding the matrix of coefficients as a linear map that takes an unknown point (x y z) to some definite point (a b c), and see inverting the matrix as finding the inverse of this linear map and applying it to the point (a b c). The way he sees it, when you multiply both sides of the equation by the inverse of the matrix of coefficients it's inherently obvious that you're left with (x y z) equal to some definite vector, and he figures I just make my life difficult for myself.

I was just curious as to how everyone else thought about this sort of thing?

Hi muppet-

I've thought a lot about this!

When I was in college (a long time ago!), I was always pressed for time. I hardly had time to get my homework done.

I observed that if I "spoke" through the problem in English (in my head), it just took too long. Eventually I found a different, more efficient way to think. I learned how to turn off language, and just...well, think in math. It was a lightning fast way to solve problems!

But I found that there were problems with this. My spelling went to hell; I couldn't explain in English how I did problems. My language skills in general deteriorated.

Once I got my degree, I stopped thinking like that and my language skills came back.

 

[mordechai9]

Some times, when trying to understand a proof, I visualize things in my head. I imagine a function of two or more variables as a surface which sits above another surface, with maximum features as peaks and minimums as troughs and so on. Most of the time when I am solving problems or computing things, I don't actually think about anything visually, and I just do calculations. It seems like the hardest problems require me to visualize things in order to understand, but simpler problems and calculations I spend very little time conceptualizing, it's more like just using any inspiration and known methods to try and solve the problem.

 

[scottie_000]

It is pretty much like learning any other language: when you're learning it can be easier to translate back and forth (between maths and english/pictures), but once you get going it's possible to think in symbols without going back
Personally I'm still learning and it helps to do both; you can do things quicker in exams, but then step back and see what you're dealing with...

 

[Gwilim]

Personally I can't tell the difference between just using the mathematical notation and thinking the problem through in english, the mathematics seems to me to be simply helpful abbreviations of what would otherwise be clumsy and unwieldy sentences in english. Doing proofs however is different, as long as I understand what is being proved I can always talk an interlocutor through the problem in english and convince them that it is so, even if I can't always see how to prove it on paper.

 

[MathematicalPhysicist]

Hi muppet-

I've thought a lot about this!

When I was in college (a long time ago!), I was always pressed for time. I hardly had time to get my homework done.

I observed that if I "spoke" through the problem in English (in my head), it just took too long. Eventually I found a different, more efficient way to think. I learned how to turn off language, and just...well, think in math. It was a lightning fast way to solve problems!

But I found that there were problems with this. My spelling went to hell; I couldn't explain in English how I did problems. My language skills in general deteriorated.

Once I got my degree, I stopped thinking like that and my language skills came back.

in what topic did you graduate?
in maths?

wasn't this a problem in exams where you need to be able to express your thoughts and proofs exactly?

 

[TMM]

Interesting thread!

I haven't really thought about this much before. When I do arithmetic, algebra, or other computations I think in math, hardly translating what I'm doing into English. When my classmates ask me how to do a problem, I switch my thinking, and I have to decode what I was actually doing. I have no problem with this, but I don't usually think about it when I'm doing a problem; I just do it.

 

[rootX]

Interesting thread!

I haven't really thought about this much before. When I do arithmetic, algebra, or other computations I think in math, hardly translating what I'm doing into English. When my classmates ask me how to do a problem, I switch my thinking, and I have to decode what I was actually doing. I have no problem with this, but I don't usually think about it when I'm doing a problem; I just do it.

I just use pictures (visualizing surfaces, objects ... ). If problem in tough (application problems), then I rewrite the whole question and translate it into picture+math (it takes 70% time translating and 30% time solving on average)
And if someone asks me, I just say it goes from here to there; I don't know how (I cannot decode ) I cannot explain things mathematical proofs, but luckily I never get asked.

 

[The|M|onster]

To me, mathematics is like the wind. It's anywhere and everywhere, but it is always close to my heart.

 

[muppet]

Thanks for the responses guys, keep 'em coming! I'd really like to know Mathwonk's answer to this question if he's reading...

 

[eastside00_99]

Maybe it is because the first real math course I took was topology, but I think of math in terms abstraction and counterexamples. For instance, when learning about homology & cohomology theory, you typically don't start with the 7 axioms of homology theory, but first you have at least of some experience of calculating simplical, singular, CW homology, et cetera. Then you see how the subject goes. Once you have that doing abstract homology theory is easier because you can just fix an example such as singular homology theory on topological spaces which will give you most of the intuition you need (and you could change examples depending on what kind of problem you are trying to solve).

A more elementary example is basic abstract algebra. We all know basic algebra by the time we graduate from high school, and we can employ examples of group, rings, fields to give us intuition for how to solve problems.

But, I was forced, in topology, to come up with a lot of examples and counterexamples on my own as well as forced to reason abstractly when I didn't have expertise to do this.

 

[PowerIso]

I guess I think of terms in English (spanish) but that's because I really have no idea what it means to not do so. When working through a proof, I simply talk myself through it and ask myself what is the next logical step or what does my previous statement imply about what I am trying to prove? It works for me.

 

[DavidWhitbeck]

A more elementary example is basic abstract algebra. We all know basic algebra by the time we graduate from high school, and we can employ examples of group, rings, fields to give us intuition for how to solve problems.

This must be the royal "we" because I certainly didn't learn group theory etc in high school!

 

[PowerIso]

Sure you did! You knew the group integers Z with addition operations.