%0 Journal Article %1 Confrey94 %A Confrey, Jere %A Smith, Erick %D 1994 %J Educational Studies in Mathematics %K bibtex-import covariance covariation ijtme2006 mythesis rate sequences unit voice %P 135--164 %T Exponential Functions, Rate of Change, and the Multiplicative Unit %U http://www.springerlink.com/link.asp?id=pr357700113173q5 %V 26 %X Describes a covariational, rather than correspondence, approach to functions that emphasizes rate of change. Proposes three ways of understanding rate of change in relation to exponential functions.

@article{Confrey94, abstract = {Describes a covariational, rather than correspondence, approach to functions that emphasizes rate of change. Proposes three ways of understanding rate of change in relation to exponential functions.}, added-at = {2006-03-07T17:56:12.000+0100}, author = {Confrey, Jere and Smith, Erick}, biburl = {https://www.bibsonomy.org/bibtex/21e348ddb62cf24b484188cf1695ca63c/yish}, citeulike-article-id = {532424}, comment = {Unit, Rate, Covariation Propose covariation as an alternative view of functions. Whereas the traditional approach sees a function as a relationship between x and y values, described as y = f(x), a covariation approach focuses on the advancement from one term to the next in parallel sequences of x's and y's. Typically, the x's increase by 1. The object of study is the pattern that links one y value and the next, what Confrey and Smith call a unit. They define unit as "the invariant relationship between a successor and its predecessor". Such an approach emphasizes rate of change, and leads the authors to extend the epistemological definition of rate. In their framework, a rate is a "unit per unit comparison". In other words, the relationship between the units of x and those of y. The concept of rate evolves through elaboration of this relationship. The most primitive notion of rate is that of equality. This notion extends to additive rates (linear functions), and in turn to multiplicative rates (exponentials). More complex relationships can be built upon these primal notions by combining them, and finally by applying fractions to the units on both columns, extended to rational numbers. - Claim unit / rate view is "natural" and provides a good pathway into calculus. - Multiplicative rates are less common, but not foreign to children. - Fail to distinguish between general phenomena and details which are a derivative of their tools. Use a system called "function probe" which facilitates particular types of explorations. - Ignore the fact that as long as the x unit is 1, it is redundant. Voice and perspective}, interhash = {80b0ca7c5d4a30cabc9b004b994bd181}, intrahash = {1e348ddb62cf24b484188cf1695ca63c}, journal = {Educational Studies in Mathematics}, keywords = {bibtex-import covariance covariation ijtme2006 mythesis rate sequences unit voice}, pages = {135--164}, priority = {2}, timestamp = {2008-04-27T13:33:18.000+0200}, title = {Exponential Functions, Rate of Change, and the Multiplicative Unit}, url = {http://www.springerlink.com/link.asp?id=pr357700113173q5}, volume = 26, year = 1994 }