Conceptual Curriculum for a Second Grader
Introduction
These are the things I would try to teach a second grader. Noteworthy for not being on the list is the concept of a variable. That is a basic concept, but I suspect just too difficult for a secondgrader.
I don't think this list is very long. But, I think that if a child had mastered these, the child would be a whiz at math.
Our educational system doesn't know how to teach concepts, so  as products of our educational system  most people would be deficient in almost all of these concepts. I teach these to everyone. It is just that with older students, I would try to teach other concepts too (such as the concept of a variable). In other words, do not think the skills below are easy or common.
Finally, let me note that this list includes things that I would try to teach with the hope of being just partially successful. You are just starting them on the journey.
Numbers/Algebra
 Adding and Subtracting. Able to add and subtract using his/her fingers, and understand what he/she is doing. Solve loox and stick problems (with small numbers/number size commensurate with skill), Realize that subtraction undoes addition (and vice versa).
 Solving Story Problems. Solves story problems by visualizing/imaging the situation of the story. (Not by keywords!) Can use conceptual skills such as adding and subtracting.
 Zero. As a number: realizes that zero is a number (you can have zero giraffes in a box) and that it isn't a number (you can't draw a picture of zero giraffes). Realize that zero is the limiting case as a postive number gets smaller and smaller. Uses it as a tool to add to a number without changing the number. Should understand that 237 + 0 equals 0.
 One. Should realize that adding one to a number produces the next counting number.
 Concept of Division. Should be able to divide 12 things evenly between 4 people. (I think at this age most children are not taught the numerical procedures of division.)
 Number dexterity. This is an openended skill. Here I am imagining number dexterity in terms of addition, subtraction, and moderately size numbers. I already have on this list using 0 and 1 as tools. Another sign of dexterity is recognizing patterns  seeing 5 > 8, 1 > 4, 7> 10, 31 > 34, a child should immediately perceive +3, without any thought, any calculation, or even knowing that there is a pattern. (One student, on a black box problem, had difficulty with 0 > 100, 1 > 102, 2> 104, 3 > 106, etc. He could see the pattern in the answers, but he could not find the answer, which was multiply by two and then add 100. His problem, as near as I could tell, was that to get from 1 to 102, he could imagine only multiplying by a large number. Obviously, multiplication is not on this list. I think a child of this age will just use repeated addition. I am just trying to make a point about number dexterity.)
 The Nature of Numbers. Start on this concept, learn about some of the numbers. I do not know if children this age can grasp that there are numbers between the counting numbers. *Do we define or invent them? How can they be specified?)
 The operations of base10 arithmetic. Packing and unpacking. Applies to other bases and to mixedbase calculations (such as feet and inches).
Noteworthy for not being on this list: the concept of a variable, ways of representing nonintegers, negative numbers.
Geometry & Measurement
 Understand basic shapes, such as triangle and rectangle. Could wrestle unsuccessfully with circle.
 Think about more basic concepts, such as angle and line.
 Shape dexterity. E.g., dangles, twotriangle problem. An openended skill.
 Understand the measurement of length and area.
 Understand that measurement involves at least one completely arbitrary choice.
 Understand that some of the choices are based on convenience, and ways that some choices are more convenient than others.
Problem Solving
 Aha. Having the aha experience. Understanding that it occurs. Realize that you have to work at a problem, wait for answers, and that solving problems this way can be enjoyable.
 The process of problem solving. Being able to stay calm and keep working.
 Tinkering. Able to keep improving on an answer, and appreciating that process.
 Learning through trying to solve. (too hard?) Realize that trying to solve a problem often requires and produces learning.
 Finding a quiet space to solve problems.
Mathematical Thinking
 Lack of noise in mathematics
 Any general statement applies to everything (not just the things it makes sense for). This includes taking things to their logical extreme and using the extremes to test propositions.
 Desire to be rigorous. For example, to get a definition exactly right. I do not know if this is possible for a secondgrader.
 Start on proof: Show me!
 Probably not: recursion, following instructions.
Introduction to Curriculum
