What are good questions that could be used to demonstrate the nature of mathematics study? Although I try to tell people about what we actually do in mathematics, sometimes I can't translate the activity very well. For example, when telling them that we study were do mathematical ideas come from, they usually understand it with an arbitrary self-explanation, for example: When someone learns the derivative, they are able to see that it comes from the concept of limit. But that wasn't really my point, for the concept of limits, there are some other concepts that ought to be learned in order to give the limits genesis and, who knows what other kinds of concepts could form a deeper construction that, perhaps, could be used to construct all mathematics?

Until the present date, I have only a handful of examples of mathematical phenomena and I guess that the most interesting question I've been able to produce is:


  Are there simple objects which could be used in arbitrary combinations to build all known mathematics? If yes, what is the proof for that?


I have some questions now:


What are good questions that could be used to demonstrate the nature of mathematics study?
Is that question I've formulated suitable for my purpose? I believe it is but I'd like to hear what other people with more competence than me could think about it.
Contrary to the gospel, I submit that mathematics is the physical science of the amazing variety and number of aspects and implications of quantity. I find it very disturbing that mathematics is transmogrified as abstract, unworldly, about logic and thinking and cognition and skill and philosophy and stylized as so-called "definitions" that do not define in the sense of inform, and "theorems" that give no clue as to their origin and relevance. The objects of mathematics are even more real than the objects of the other sciences. The fact that we can prove is due to the fact that the simplest variety of quantity is immediately discernible to us and many if not all other animals without language or training or mathematics. The objects of the other sciences are not directly observable so are not concrete and immediate. In this sense mathematics is the most concrete of the sciences. Mathematics is habitually disguised as the emperor in the invisible clothes of esoterica. For example, contrary to your (and everyone else's) mantra, calculus is not about limits. The limit is merely a device that enables us to obtain the derivative or integral of a particular function. It is not at all what calculus is about. Differential calculus is about a very interesting and useful point property of functions called the instantaneous rate of change that is characterized as slope or grade. Integral calculus is about the very curious antiderivative of a function which is a point property that very simply gives an interval property, a remarkable thing indeed. I'm sure that if we divest mathematics of its esoteric and procedural draperies we could grasp ten times as much in one tenth of the time and appreciate it accordingly.
Contrary to the gospel, I submit that mathematics is the physical science of the amazing variety and number of aspects and implications of quantity. I find it very disturbing that mathematics is transmogrified as abstract, unworldly, about logic and thinking and cognition and skill and philosophy and stylized as so-called "definitions" that do not define in the sense of inform, and "theorems" that give no clue as to their origin and relevance. The objects of mathematics are even more real than the objects of the other sciences. The fact that we can prove is due to the fact that the simplest variety of quantity is immediately discernible to us and many if not all other animals without language or training or mathematics. The objects of the other sciences are not directly observable so are not concrete and immediate. In this sense mathematics is the most concrete of the sciences. Mathematics is habitually disguised as the emperor in the invisible clothes of esoterica. For example, contrary to your (and everyone else's) mantra, calculus is not about limits. The limit is merely a device that enables us to obtain the derivative or integral of a particular function. It is not at all what calculus is about. Differential calculus is about a very interesting and useful point property of functions called the instantaneous rate of change that is characterized as slope or grade. Integral calculus is about the very curious antiderivative of a function which is a point property that very simply gives an interval property, a remarkable thing indeed. I'm sure that if we divest mathematics of its esoteric and procedural draperies we could grasp ten times as much in one tenth of the time and appreciate it accordingly.