"computation" absolutely did arise from noticing how the merely formal could faithfully adhere and correspond to the concrete. So it is "grounded" in that sense. There continue to be arguments of how best to ground the formalities.
See, that is the gist. 1 + 1 = 2 is formal. Not merely formal, actually formal. It requires the student to understand and own a formal system. The set of integers and the additive operation. So what we are really talking about here is what actually occurs when a student takes ownership of a formal system. What difference does it make if one apple and one apple always makes two apples in the real world if 1 + 1 isn't always 2 in the student's head, all by itself? Over time, that formal system grows and becomes more involved and more self sustaining.
However, as we learned in New Math, 1 + 1 = 2 is premised on concepts of base and we need to know what set we're operating in. You say integers and it's good that you say that, as in a modulo arithmetic game, 1 + 1 + 1 might equal 1, as we're adding modulo 2. Or 1 + 1 might equal 10, because we're using base 2.
I do not regret learning the above at a young age. I learned that "addition" is not just one operation, nor is multiplication, but that a family resemblance across language games. Similarities in usage, allow us to use the "addition" as the in-common name for these ops. That's how we develop our formalisms.
Learning about bases is still highly practical, as computers work in base 2 and their set of integers is sometimes closed in the sense of modulo. In some integer type implementations, adding 1 to a maximum flips you to a maximum negative, so that you go in a circle.
In some math classes, playing with robotic devices such as the Arduino is de rigeur (a required / expected activity), so learning about a closed int type makes plenty of sense. Fat paper-wasting math textbooks that don't cover that should not be where the money goes.
School boards that buy truck loads of heavy textbooks instead of Arduinos and Raspberry Pis are likely made up of civilian know-nothings who waste money out of ignorance.
These kinds of extra-curricular centers, where more practical / relevant math is taught, are spreading, staffed by geeks who may or may not be unionized teachers (more likely not these days):
(San Francisco -- one of our OCN faculty was just visiting)
Students with this kind of background are more likely to know about projects like this one:http://vimeo.com/focusforwardfilms/semifinalists/51764445
(has an MIT flavor)
When you say "how to best ground the formalities" what you really mean is how to reach students unaware of the formalities, but the formalities themselves are grounded in one thing only, formal thinking (is that better than "reasoned thought"?). If you fail with the formalities then you have failed period and you fail with the formalities when you have failed to invoke formal thinking in the student.
I think formalities often come through with contrast. Compare formalities with one another.
In Euclidean geometry, there's the formality of the infinitely tiny point, so small that any point you make is too big to really be a point.
In non-Euclidean geometry, that concept may not survive intact. Perhaps the formality is you have "lumps" and these are defined to always have an inside and outside, so one may speak of a point's "interior" (Karl Menger, mathematician, describes such a geometry).
There may be a concept of a zero point or empty point but it's only fleetingly achieved i.e. there's a sense of passing through and never pausing at zero. That sounds like a different metaphysics, and it is.
The important thing is to realize that a line going to infinity in both directions, that is infinitely thin, and made of points of zero dimension -- that's all metaphysics too, i.e. philosophy. These concepts are not "proved" but are axiomatic. They're inherited from ancient Greek philosophers.
Younger students may not remember "board games" but older adults do. When you buy a new board game, or take one off the shelf at your assisted living facility, you read the back of the box or under the lid for directions. The directions spell out the rules of play.
Maths are a lot like that: a huge stack of board games that interconnect with one another in various logical ways (or don't -- is more of a disconnected graph with islanded connected parts, if we want to bring graph theory into it).
You said that the "mapping" between formal thinking and the real world is important. How so? I spend hours, days, and even weeks pondering some problems with nary a single thought of the real world. But I agree, when we apply mathematics to the real world, that mapping is very important. But this is irrelevant to
Maybe you should give an example.
my claim. If you have not developed formal thinking then any notion of mapping is moot, is it not? The real issue that I am concerned with here is what role does common sense, namely concrete examples, play in the development of formal thinking. After searching deep, the only role I can come up with is as an
The usual answer is special case examples have things in common that remain, which you abstract away the differences, and these commonalities are what we call the formal aspects.
For example, we look at one container after another, starting with simple balls and/or polyhedrons.
When inside, the surface tends to curve toward us and focus reflected light towards a center. There's a concentrating aspect.
When outside, the surface tends to bend ways from us and light is dispersed. We look at this over and over in different special case circumstances and from these experiences we develop the concepts of "concave" and "convex" respectively.
What we notice about concavity and convexity is they go together i.e. are two aspects of the same thing. This is fairly common in formal conceptualization: two concepts will co-define one another.
For example, the United States is understandable as having an inside and an outside. Inside and outside go together. The interface between inside and outside is the world of borders, customs inspections, screening, filtering. It's also the space of treaties, agreements, relationships with other sovereignties. Drive across the border, into the Warm Springs reservation, and you're in another nation with its own laws and practices.
aid in communicating the basic properties of the formal system you are trying to teach and as a tangible check against which you test the formal reasoning you are developing. I gave examples of things (coordinate transformations etc.) that are entirely of formal origin and I could fill page after page of similar examples. This tells me that formal reasoning, once developed, is entirely grounded in its own domain, separate of the real world.