I have recently started to develop my mathematical intuition. In the past I saw math as a mere game of symbol manipulation, whosoever was able to see patterns and cram formulas and apply them upon those symbols won. However,I now feel that the truth is contrary to the aforementioned,this is because I see in each new chapter that I study in calculus or linear algebra, there is an inherent geometrical intuition that can be easily visualized. At other instances there may not be a geometric interpretation, however there are ways of understanding why the proved results are true. But there is still a void of intuition in the most basic algebra that I am encountering. The following link should demonstrate what algebra I am talking about:
<http://link.springer.com/chapter/10.1007/978-0-8176-4549-6_1#page-2>

Another example further in this text:
Finding the roots of the equation: (xy − 7)^2 = x^2 + y^2

Although, the aforementioned can be easily solved after looking at solved examples, however, for each step my motivation is to basically juggle the symbols until I find a pattern I am looking for. What that pattern represents and the symbol juggling itself seems like an opaque layer to me? Please, can anybody give me suggestion to how I can intuitively plan out my steps doing basic algebra. Is there a way of doing math on an intuitive basis from the bottom up? Preferably, in a visual way?