Real Math for School Math

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.

"The mathematics of the elementary and middle school curriculum is not trivial, and the underlying concepts and structures are worthy of serious, sustained study by teachers."  National Research Council. (2001).  Adding It Up : National Academy Press.       Access to the linear presentation of Integer and Rational number construction is provided through       a link at the end of this post. The Difference Concept I am going to get ahead of myself here to take a crack at meaning and interpretation. At this stage you can take it as conjecture based on the feeling that whatever turns out will make a sense we might have anticipated. Active learning is not just following an argument, it is more like provoking one, even with yourself. Let's not worry about number systems yet and assume "numbers" can do what we ask of them. We know two numbers can be either added or multiplied.  And we know that one number can be different from another. We are talking about a m

Overview of Number Construction The goal is sequential construction of number systems in School arithmetic, Natural to Integer to Rational , in a reasoned process involving only numbers themselves, and then understanding  the architecture of that process. A large fraction of American students emerges from School arithmetic not understanding negative numbers and fractions. Generally they have been taught to look for “real world” objects that are intrinsically “negative” or “partial”, and then to understand numbers in terms of such objects. I assume this approach is taken over from understanding natural numbers as enumerators of discrete objects. This works because we all agree on what discreteness/oneness/individual is, and we can base our understanding of natural numbers on that. This is not the case for “negative” or “fraction”. To use negative numbers or fractions we must understand the numbers themselves. Only in accommodating academic environments can we

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 diffi