I am happy to accept this as a working definition for myself:
Mathematical thinking is a whole way of looking at things, of stripping them down to their numerical, structural, or logical essentials, and of analyzing the underlying patterns."'
This is not to say that I think novices (students) are always thinking like experts (mathematicians). I don't. School math and professional math are not the same things. But I do think that students can adopt the identities of (young) mathematicians in the same way that they can adopt the identities of (young) scientists: by using some of the tools of the disciplines as they work at their own levels. Students of all ages are encouraged to construct scientific hypotheses about the world, without controversy, whereas I think it is far too easy for us to present all school math as axiomatic, or even universal, without giving students a chance to see into the mathematical world themselves.
Keeping the expert/novice divide in mind, then, here are some things that students will generally NOT be doing:
-constructing proofs by contradiction
-proceeding from rigorous logical structures like "there exists a..."
-using set theory
Some things that students will, and I believe should, be doing:
-developing their own reasoning, and "proving" things as true as they can, for any given task
-exploring truths about arithmetic, like commutativity
-systematically exploring why things like the formula for area of a triangle work for all triangles
-making generalizations using algebra, for example about a given linear pattern
-choosing the best "tool" for the job, whether it be adding, subtracting, multiplying, dividing, or higher powered tools, as they get older
-constructing models for how things "might" work, in the context of a given problem
Here is a nice piece that argues that mathematical thinking doesn't look anything like mathematics. If you don't believe that, I give your Andrew Wiles' proof for Fermat's Last Theorem: "Modular elliptic curves and Fermat's Last Theorem". Probably only a few hundred people in the world can read that.
For those keeping track, I am absolutely not arguing that any of these things are done in the absence of content.
I do believe that the content itself is the vehicle for thinking mathematically. I am also unabashedly on the side that says students aren't given enough chances to develop their own reasoning. Mathematical tools that just sit in the mental toolbox are no better than dusty old tools in my basement. They should be used.
I also think mathematical content should be presented for what it is, in discipline-based language. Even the viral Common Core problem on "5x3" was an ideal opportunity to teach kids of any age about commutativity. You might say kids can't handle that- I myself have seen grade 2s in @MrSoClassroom use the phrase "distributive property". A lot of our coaches are also exploring arithmetic in interesting ways with number talks. And yes, they are teaching standard algorithms as well, because that's part of the journey. But number talks might be an example of what Mr. Devlin is talking about-stripping down simple arithmetic into structural elements, then putting the elements together.
Students must be given the chance to work with lots of problems, in order be able to strip them down to their mathematical elements. As they learn more mathematical content, as the grades progress, they will be more able to do so. Sometimes we do that in a more structured way, such as giving a few problems that are obviously "about" Pythagorean Theorem. Other times, we might let students play with the structure of problems (I used to like having students write their own Pythagoras problems, to see what they could do with the problem structure).
Why not? Students come up with things much more interesting than the old "ladder against a wall".
Even better is using that structure to explain something in the world, in this case, how televisions are sold:
A few pictures I still have and like that I hope illustrate some of these points about student mathematical thinking:
This one represents an attempt to systematize the thousand locker problem. The deep structure of this problem is that all square numbered lockers are the only ones touched an odd number of times. For any student to understand that, they have to first come up with a tool that will help them find the structure.
This problem is something about numbers on horse stable stalls. I forget. They made it all the way through a systematic organized list before realizing they could start to generalize about their results.
This one represents a students' understanding of a dot pattern. She saw the structure as adding a group of three dots to the side each time. It was a big "a-ha" moment, and led her toward the algebraic expression that models this pattern.
This picture represents a students' choice of tool for a problem involving using a certain amount of fencing to enclose as much area as possible. Students were generalizing about the meaning of "squareness" with regard to area and perimeter.
This picture is one of my favourites, It represents a kid exploring the class of polygon called a "trapezoid", in order to attempt to figure out how you might get its area. It's exciting because it led to...
...the beginning of a generalization about trapezoids. In my experience, making statements about all cases, beyond ones that they have found themselves, is tricky. A nice article about how generalization develops in early algebraic reasoning is here.
This one represents a generalization with algebra about a version of the tables and chair problem. Gradually students realize that the "+2" stands in for the two chairs on the end, that are always there, even as more tables are added.
In summary, I believe that students are very much capable of mathematical thinking, even if they don't always "think like mathematicians". They need lots and lots of chances to break down problems into their basic structural elements, to explore their own reasoning, and to start to make interesting generalizations on their own.