Math Problems

In theory, the Insight method of teaching math can use any problems. In practice, you can't use problems your students remember how to solve. Also, I have developed problems to suit my techniques and to teach concepts I want to teach. Finally, because I teach concepts, the problems can be different.

I regularly add problems to this list.

Reverse Arithmetic

Reverse Arithmetic I start with this type of problem. For example, find 3 numbers which add to 100. This teaches number facility, including numbers and operations as tools. It also segues nicely into the problem of expressing a pattern.

Expressing Patterns

Expressing Patterns Teaches the concept of a variable and develops facility with their use. Also, what-were-you-thinking problems.

Base 10 Operations

Natural Operations Nothing to do with base-10. Investigating the natural, real-world operations that correspond to addition, subtraction, multiplication, and division.
Packaging Problems Story problems analogous to the base-10 operations of addition, subtraction, multiplication, and division. Also used to teach other bases, or mixed bases such as yards-feet-inches.
Exploring the Representation of Number. An exploration of different methods of representing number.

Algebra problems

These are all story problems. They can be solved without algebra, but they have algebraic equivalence.

Loox problems have strange animals in a box. For example, A box contains 3 cats, 4 dogs, and one loox. There are 17 eyes in the box. How many eyes does a loox have?
Stick problems express an algebra equation using sticks. For example x + 3 = 7 would be two sticks, one unknown length and one length 3, that together are as long as a stick of length 7.
magic problems. I am thinking of a number....
Silver & Gold

Computer Programming

Computer Programming Someone has built a machine to follow instructions. Be the machine! Be the machinemaster!

Measurement

Length Constructing their own measure of length.
Communicating Communicating a unit of length to aliens.
Comparison Some basic measurement theory.
Measuring Areas

Proportions, Probabilities, and Fractions

Basic concepts
combination problems
problems that segue from combination to probability
Defining "half" (bright six-year olds)
Fractions

Geometry Foundations

Other

Frodo Problems Example: 3 is a blodo. Adding 4 to a blodo makes a new blodo. Make as many blodos as you can. These primarily teach number facility, but they also address recursion and set a foundation for understanding the axiomatization of numbers.
Numbers. Understanding the different types of numbers.
Tinkering. The process of improving on a first answer or idea.
Categories Pattern recognition, but moreso describing/defining categories.
Rules Teaching students to build mathematical rules (being a mathematician)
A Mathematical "Magic" Trick Probably teaches some basic concept about being careful in calculations

Essays:

The Inner Curriculum I define the outer curriculum as the information students learn. The inner curriculum is the skills for understanding, retaining, using, and appreciating the outer curriculum.
Understanding how to calculate the area and circumference of a circle.
Of Names and Numbers Which numbers have names? And related thoughts.
Basic Number Concepts Some problems. For the toddler, the concept of zero; for the mathphobic adult, the real numbers.
Proofs Proof as explanation, understanding, and some rigor.
Definitions How to define a concept, such as "wall". This is actually a thinking skill, but it is similar to the mathematical problem of defining a circle.

Dad: "Son, did you do all of your homework tonight?"

Seven-year-old son (admonishingly): "Dad, you know I don't have homework on Sunday night"

Dad (with a twinkle in his eye): "But did you do all of your homework tonight?"

son: (big smile) "Yes, dad."

Mathematician interacting with his son. (Logically, if there was no homework, then the son did all of it.)