Confrey & Smith (1995)

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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.


Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26(1), 66–86.
author = {Confrey, Jere and Smith, Erick},
doi = {10.2307/749228},
journal = {Journal for Research in Mathematics Education},
number = {1},
pages = {66--86},
title = {{Splitting, covariation, and their role in the development of exponential functions}},
url = {},
volume = {26},
year = {1995}