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Would you be able to elaborate on "algebraical vs geometrical" thinking? I've noticed myself that there are two types of "math" that people tend to be good at:

Would you be able to elaborate on "algebraical vs geometrical" thinking? I've noticed myself that there are two types of "math" that people tend to be good at: 1) Algorithmic - Following a set of steps to achieve a particular result. Algorithms, discrete math, "compsci" math, and procedural and OO programming. Coders tend to be good at this kind. 2) "Abstract" - "Pure Math", what I would label as the harder kind of math, like calculating the intersection of planes, calculus, linear algebra, etc. Pure math majors and theoretical computer science folks are good at this kind. I have met folks that are good at 1 or 2, or both. Forgive me for my poor use of terms, I'm one of those folks that is not particularly great at math (mostly due to lack of practice) but I was always curious if my observation was backed up with any "real" terms or if there has been any research into this kind of thing. What you describe seems to be relatively close to what I've observed myself.

bcbrown 13y ago
I know that I personally am much stronger with algebraic math than geometrical math. For me, that means that integrals, linear algebra, etc are something that I'm strong at, but I'm poor at anything that requires spatial reasoning. That's different from your two types. For example, take your "calculating the intersection of planes." I would approach that by first writing the algebraic equations that define the two planes, then trying to find the manipulation that will let me solve for the intersection. Someone who is a geometric thinker might start by plotting the two planes, and reasoning spatially. So, that distinction is wholly separate from your algorithmic/abstract distinction, and to me, is a distinction within the abstract realm.
- darkmighty 13y ago
  Yea, this is a good illustration. For example, for an intersection of two planes going through origin, the plane equation coefficients are (a multiple of) the of the normal vector coefficients. Now, the intersection vector must be perpendicular to both planes normals, so if you take the cross product of the two vectors, you get the intersection vector. This in this case I would just look at it and instantly do a cross product of the coefficients, whereas you would eventually do precisely the same, but reasoning algebraically, perhaps just as fast. In this case, thinking algebraically doesn't seem to yield the intimate relationship with your object of study required for creativity, just because of the geometric nature of it. But this reasoning wouldn't be much use in -- for instance -- fiddling with discrete mathematics, where you may get more intimate precisely by reasoning algebraically!
jclos 13y ago
There was an essay from an extremely famous mathematician that I read not a long time ago on analytical vs geometrical oriented thinking in mathematics and I can't recall where it is and it is driving me crazy. If anyone knows what it is, please link it.

bcbrown 13y ago

I know that I personally am much stronger with algebraic math than geometrical math. For me, that means that integrals, linear algebra, etc are something that I'm strong at, but I'm poor at anything that requires spatial reasoning. That's different from your two types. For example, take your "calculating the intersection of planes." I would approach that by first writing the algebraic equations that define the two planes, then trying to find the manipulation that will let me solve for the intersection. Someone who is a geometric thinker might start by plotting the two planes, and reasoning spatially. So, that distinction is wholly separate from your algorithmic/abstract distinction, and to me, is a distinction within the abstract realm.

darkmighty 13y ago
Yea, this is a good illustration. For example, for an intersection of two planes going through origin, the plane equation coefficients are (a multiple of) the of the normal vector coefficients. Now, the intersection vector must be perpendicular to both planes normals, so if you take the cross product of the two vectors, you get the intersection vector. This in this case I would just look at it and instantly do a cross product of the coefficients, whereas you would eventually do precisely the same, but reasoning algebraically, perhaps just as fast. In this case, thinking algebraically doesn't seem to yield the intimate relationship with your object of study required for creativity, just because of the geometric nature of it. But this reasoning wouldn't be much use in -- for instance -- fiddling with discrete mathematics, where you may get more intimate precisely by reasoning algebraically!

jclos 13y ago

There was an essay from an extremely famous mathematician that I read not a long time ago on analytical vs geometrical oriented thinking in mathematics and I can't recall where it is and it is driving me crazy. If anyone knows what it is, please link it.