Conceptual Curriculum for a Second Grader
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 second-grader.
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.
Noteworthy for not being on this list: the concept of a variable, ways of representing nonintegers, negative numbers.
- 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 open-ended 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 base-10 arithmetic. Packing and unpacking. Applies to other bases and to mixed-base calculations (such as feet and inches).
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, two-triangle problem. An open-ended 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.
- 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.
- 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 second-grader.
- Start on proof: Show me!
- Probably not: recursion, following instructions.
Introduction to Curriculum