Theory of Arithmetic

Liping Ma and I have written two articles that describe a unified view of arithmetic in elementary grades.

The theory has two sections. The first concerns whole numbers, and the second fractions. Important features of the first section are the definitions of number, unit, like number, sum, and product. In learning arithmetic with whole numbers, students’ initial conceptions of each are established. In learning multiplication with whole numbers, students’ conceptions of unit expand to include many-as-one unit. These features are described in our open-access article here.

The second section of the theory concerns fractions. Students’ conceptions of unit expand again to include fractional unit. The underlying definition of number—a collection of like units—remains the same, but students’ conceptions of number expand to include fractions and their conceptions of unit expand to include many-as-one fractional unit. The underlying definition of sum (and addition and subtraction) remains the same. In multiplication with a fractional multiplier, students use the conception of many-as-one fractional unit. These features are described in our draft article here.

Update: The article on fractions has been published in the open-access Asian Journal for Mathematics Education.