Agree fully with lbrandy's point. Just to add a little more, people seem to get confused between the expression of mathematical thoughts and having mathematical

Agree fully with lbrandy's point. Just to add a little more, people seem to get confused between the expression of mathematical thoughts and having mathematical thoughts. Mathematical symbols are abbreviations for ideas that, if expressed in natural language, would become tedious. Mathematical thinking is something people do everyday. For example, parents have to use combinatorial optimization everyday to figure out a best strategy for running errands. Few parents are schooled to a formal expression of the mental processes they go through in determining a solution but they weigh different goals and optimize nonetheless. Oddly, a study that is all about exploration of ideas and abstraction has been narrowly defined in many people's minds. Hopefully mathematics educators can break these limited views in years to come.