Teaching the Derivative through an Algebraic–Geometric Tangent Model and Dynamic Visualization

 Paper Title: 

Teaching the Derivative through an Algebraic–Geometric Tangent Model and Dynamic Visualization


Authors: 

Davit Kapanadze, Georgian National University SEU, Georgia


Abstract:

The derivative is often introduced through the limit of a difference quotient. This definition is mathematically rigorous; however, at the initial stage of learning it may be difficult for students, since it brings together several new ideas at once: increment, secant line, limit, tangent line, and instantaneous rate of change. As a result, students may learn differentiation rules mechanically without fully understanding the geometric and functional meaning of the derivative. This article presents an algebraic– geometric tangent model for the initial teaching of the derivative in the case of polynomial functions. Instead of beginning with the formal limit definition, the model first defines the tangent line by means of a double-root condition: the difference between a polynomial and its candidate tangent line has a double zero at the point of tangency. This condition uniquely determines the slope of the tangent line, and the resulting correspondence from each point to the corresponding slope is introduced as the derivative function. The model is based on concepts already familiar to students: function, graph, line, slope, divisibility of polynomials, and double root. The limit definition is not rejected in this approach; rather, it is obtained later as an analytic formalization of the already constructed tangent model. Dynamic visualization helps students connect the algebraic condition, the geometric tangent, and the transition to the classical limit formula. The article also discusses classroom and university teaching experience accumulated between 1986 and 1997, which indicates that this approach may strengthen students’ conceptual understanding of the derivative as a means of describing the local behavior of a function.


Keywords

derivative; tangent line; double root; polynomial function; limit-free approach; algebraic–geometric model; dynamic visualization; mathematics education; conceptual understanding


Volume URL: https://airccse.org/journal/ijite/vol15.html


Abstract URL: https://aircconline.com/abstract/ijite/v15n2/15226ijite02.html


Pdf URL: https://aircconline.com/ijite/V15N2/15226ijite02.pdf


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