### Why do American Students Tend to Wiff on Fractions and Negative Numbers?

The short answer is they do not understand them. There is no reason to believe they do not understand what they have been taught about them.

School mathematics instruction starts by laying out number facts and arithmetic rules. Both the facts and rules are anchored in physical objects. This can work because the numbers and the arithmetic are based on counting discrete objects, thereby providing a connection of Natural Number arithmetic to a physical activity available to the youngest student.

Fractions are in the Rational Number system, negative numbers are in the system of Integer Numbers. Both Integers and Rationals are constructed with Natural Numbers; they are not extensions of them. A negative number is not the “opposite” of a Natural Number. A Rational number is not a “part” of a Natural number. Natural Numbers are in one number system, Integers in a second, and Rationals in a third; A number in one of these number systems is not embedded in another number system. Instruction …

School mathematics instruction starts by laying out number facts and arithmetic rules. Both the facts and rules are anchored in physical objects. This can work because the numbers and the arithmetic are based on counting discrete objects, thereby providing a connection of Natural Number arithmetic to a physical activity available to the youngest student.

Fractions are in the Rational Number system, negative numbers are in the system of Integer Numbers. Both Integers and Rationals are constructed with Natural Numbers; they are not extensions of them. A negative number is not the “opposite” of a Natural Number. A Rational number is not a “part” of a Natural number. Natural Numbers are in one number system, Integers in a second, and Rationals in a third; A number in one of these number systems is not embedded in another number system. Instruction …