Concrete activities have enormous emotional benefits:
I certainly encourage the use of concrete manipulatives, but drawings, or notes that might include symbols are just as important for students that are concrete as the iconic symbols connect to physical objects and interaction on them.
Is money concrete?
It doesn't represent the value in a concrete manner. In fact this is supported by the fact that children have a difficult time learning the value of pennies, nickels, dimes, and quarters.
Is touch math?
Maybe. Is counting hands-on? If it is, then touch math is hands-on. If counting isn't, then touch math isn't.
The worst thing about touch math is once students begin to use counting as a strategy for solving problems, they will continue to use it. It becomes very difficult to change to something better and they won't until an alternative is found more useful, easier, and comfortable. Touch math is usually used when teachers don't understand better ways to facilitate student's learning. See development of number sense, place value, and operations.
Concrete, semi concrete or iconic, and formal operational
See the example of proving that ODD + ODD = EVEN in three ways: concrete, semi concrete or iconic, and formal operational thinking.
If a middle grade class was challenged to prove that when: an odd number was added to any odd number the answer is always even.
There could be examples from each of the three levels, but most would be more concrete than formal operational. however, most students would not need to actually manipulate physical objects, but would choose to represent numbers with drawings and physical actions. Concrete representations of odd and even in setting up the problem and throughout the procedure used to achieve a solution.
Concrete and visual, meaning a picture or drawing that illustrates concretely the property of odd and the property of even as groupings of physical objects or as numbers with physical manipulation of moving the value of one from odd to even or even to odd.
Most solutions will probably be argued with specific examples that are true. However, since they didn’t check every number, it isn't a mathematical proof. however, using objects, or drawings of objects, or using symbols can all be used to provide examples that could be used to represent it is true for every number as well as an algebraic proof.
Manipulatives for upper grades are most often paper to draw and write notes, paper grids, photo copies of different shapes and objects, and drawings that fit with an activity or problem.
Concrete manipulatives that are appropriate for middle grades are Cuisenaire rods, multi-colored disks, and geometric manipulatives. However, a ruler and colored pencils can be used for all problems also.