The most beautiful thing we can experience is the mysterious.  It is the source of all true art and science.
                    --- Einstein, What I Believe, 1930.

The whole of science is nothing more than a refinement of everyday thinking.
                    --- Einstein, Physics and Reality, 1936. 

Certainly few things are more mysterious to a high school freshman than trying to link physical relationships and measurement to the world of mathematics and arithmetic he/she has previously experienced. Yet if science is just a "refinement" of everyday thinking, how can it be such a mystery? Probably a paradox there somewhere. 

How Do We Know? Presents a system for review of arithmetic and algebra which can remove the “mystery” of arithmetic calculations and enable students to see the beauty and mystery of mathematics and science.

How Do We Know? Contains operant conditioning in the form of Precision Teaching which is behavoristic conditioning using a conditioned response approach, and reflexive thinking in the form of justification tables reminiscent of formal geometric proofs.

There is a reflexive nature in most of the Precision Teaching exercises. This reflexive thinking is in the from of justification tables reminiscent of formal geometric proofs. These activities can serve as an introduction to a reflective, metacognitive process. One definition of metacognition is thinking about how we think.

Linking reflexive thinking and operant conditioning; there is probably a paradox there somewhere also.

Precision Teaching is a powerful tool for motivating students to learn. It has found wide acceptance in the field of Special Education. Extensive research has validated this approach to operant conditioning in educational settings. I hope I have presented a structure to help introduce this system to mainstream teachers.