The goal should be, not to implant in students’ minds every fact the teacher knows; but rather to implant a way of thinking that enables students, in the future, to learn in one year what the teacher learned in two years. Only in that way can we continue to advance one generation to the next. Edwin Jaynes, 1993
The question: The school math curriculum includes fractions and negative numbers. Why do students who have completed it display large deficiencies in their comprehension of both?
Conjecture: They misunderstand them. They were taught: (1) "negative" and "fractional" are physical properties numbers are altered to accommodate, (2) the accommodation is reflected in number notation, with arithmetic executed by blind adherence to rules.
I have been working on a mathematical approach to school arithmetic. The idea is to develop the Integer Number System and then the Rational Number System from Natural/Whole Numbers within mathematics itself. This is not difficult, it is different though. Meaning is built into Integers and Rationals by their construction, not "discovered" from their associations. It could be said that Natural Numbers were discovered, but it must be acknowledged that Integers and Rationals were invented.
The traditional approach, carried on by Common Core Standards, is based on the "discovery" of physical entities that are intrinsically "negative" or "fractional". Initially this can be taken as convenient self-deception, but American students appear to pay a high a price when it is not corrected.
I want to solve the negative-fraction problem. It has been around forever, and should be taken as an indication of a deep fault in the traditional approach. Subsequent posting in this blog will be devoted to the effort.