[com2kid]

Example: when I see a math expression x times y, I mentally see a rectangle with side length labeled "x" and perpendicular labeled "y". So understanding (x+h)(x+h) = x^2 + 2xh + h^2 was totally natural for me. It's not abstract symbols to me, it's pictures in my head. I see a tiny square in the upper right labeled "h^2", and a big square in the lower left labeled "x^2", and two rectangles along the edges labeled "xh".*

For me, it is the exact opposite. Teachers would spend ages trying to explain this to me, eventually I just had to accept the equation as being true, it never has really made intuitive sense to me. I dealt with it by coming to the conclusion that it is just the way math is agreed upon being done in the particular syntax we have all agreed upon using.

A good # of math courses later (enough for a minor in mathematics) and that is still basically my understanding. Physical diagrams don't really mean much. For some things they are useful, mostly for intro calculus concepts (which is a subject I find to be amazingly visualizable in general), but I just had to basically take all of Algebra on faith as being "agree upon syntax".

[nitrogen]

Does "FOIL" (first outer inner last) or the distributive property make more sense for your understanding of the expansion of (x+h) * (x+h) in algebraic terms? One thing I love about math is how there are different ways of representing the same concepts -- geometric, algebraic, etc. I wonder if this is because different mathematicians in history had different thinking styles, and paved the way for us modern folk to learn math in the format that works for our own unique minds.

[com2kid]

Neither make sense. Memorizing it as a rule made sense.

Eventually learning about different algebraic systems made it all click for me.

A lot of math is intuitive for me, the distributive property never has been though. I just had to memorize it and move on to things that contained meaning for me.