What exactly is Mathematical Thinking?

One of the comments I receive is, "Mathematical Thinking, what is that?"  I am going to steal part of a blog at Delvin's Anlge.  

"What is mathematical thinking, is it the same as doing mathematics, if it is not, is it important, and if it is different from doing math and important, then why is it important? The answers are, in order, (1) I’ll tell you, (2) no, (3) yes, and (4) I’ll give you an example that concerns the safety of the nation. If you had any difficulty following that first paragraph (only two sentences, each of pretty average length), then you are not a good mathematical thinker. If you had absolutely no difficulty understanding the paragraph, then either you are already a good mathematical thinker or you could acquire that ability pretty quickly. "

Mathematical Thinking is not doing traditional math problems or doing higher arithmetic. It is logic, it is sequencing, it is patterns, it is being able to think your way out of a box. Mathematical Thinking is a skill that is so highly sought that employers will pay decent money to anyone with a math major or minor. It is also something that is becoming so rare that those majors/minors are 14 of the top 15 majors on a recent ThinkAdvisor article declared as "Hot Majors." It is problem solving at its core. Not book problem solving but life problem solving.  

But how do we do that in the classroom. We engage students in activities that entice them into thinking they are doing one thing when in reality they are developing a way of thinking. Many would correlate these to logic puzzles, Ken Kens, Zupelz... But they are so much more. Here is a very simple example that a 4th grader should be able to do. A beginners version of Mathematical Thinking for this prompt would be counting 6 books for each box and adding them. A better Mathematical Thinker would setup a multiplication array. In this example, students are asked to play a game. A student who shows basic Mathematical Thinking would realize they need to leave their opponent with 4 coins. A student showing strong Mathematical Thinking realizes they already won when there are 16 coins left, or better, declares winning after the first move (It happened to me, note - not by me!).

As a teacher, what can you do to get students to think this way? That is actually the easy part.

1. Expose students to this style of problem.
2. Don't scaffold problems beyond the learning. Typically, we want to help every student. So much that we have a tendency to take the learning steps out of the problem entirely.
3. Allow students to struggle. Note - struggle, not get frustrated.
4. Allow students to work in groups. Talking among others helps connect their thoughts together.
5. Help students when struggling by giving them one prompt. Then walking away to let them think some more.

This is hard to do because, like stated earlier we like to help students. Seeing them struggle and sometimes fail ultimately makes us feel like we haven't done our job. On the contrary, it probably means we just need to find the right prompt to get the student to move forward.