### 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 suggesting otherwise misleads the student.

The meaning of a number in a number system has long since been abstracted out from physical properties. A number is neither hot nor cold, long nor short, big nor small, ...Properties of numbers and their arithmetic reside in the system to which they belong. Numbers are/were attached to physical entities by a person using the numbers; this includes number lines.

Students can discover the Natural Number system; and then construct the Integer and Rational number systems from it. Construction is specified in a drawing, showing the meaningful positions of slots for Natural Numbers. The specification is a “particularization” of the “generalization” in the abstraction of “difference” for two Natural numbers.

• An Integer is the signed additive difference of two Natural Numbers.
• A Rational Number is the signed multiplicative difference of two Natural Numbers.
Additive and multiplicative differences are explained in the 9/9/17 post of this blog. This is all discussed in depth in the link at the bottom of this post.

The meaning of numbers and  arithmetic is at their conceptual level. Meaning determines how properties and relations are expressed in problem formulation. For the most part computers and estimation can take it from there.

At some point today’s arithmetic instruction must turn toward conception as the basis for understanding and using mathematics. Now this point is generally delayed until well after the student has been lead far down a dead-end path.

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