"The human mind has never invented a labor-saving machine equal to algebra." ~ Author Unknown
We, as 'mathematicians' (I use this term loosely here), take a situation, a set of numbers, a relationship, and find a pattern. Then, we express that pattern as an expression or an equation. We do this to make predictions and represent these scenarios in a concise manner. When we see a graph or a table, we translate these into a simple equation that sums up what is there. We save ourselves more work later by making generalizations...which is a big part of algebra.
On a slightly different note, as 'mathematicians', we even develop patterns in how we, as individuals, solve problems. For example, the problem 1/3(x + 5) = 5x + 12, some people choose to distribute the 1/3 while others choose to multiply both sides of the equal sign by 3. Both are correct, and, assuming one does his/her math correctly for the rest of the steps, both will receive the same answer. However, different people choose different approaches based on their preference and how their minds work.
Any thoughts/comments on the quote?
Are there any unique approaches you have developed for problem solving over your teaching years?