Splitting, Covariation, and Their Role in the Development of Exponential Functions

Exponential and logarithmic functions are typically presented as formulas with which students learn to associate the rules for exponents/logarithms, a particular algebraic form, and routine algorithms. We present a theoretical argument for an approach to exponentials more closely related to students' constructions. This approach is based on a primitive multiplicative operation labeled “splitting” that is not repeated addition. Whereas educators traditionally rely on counting structures to build a number system, we suggest that students need the opportunity to build a number system from splitting structures and their geometric forms. We advocate a “covariation” approach to functions that supports a construction of the exponential function based on an isomorphism between splitting and counting structures.