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Graphic Calculators and Micro-Straightness: Analysis of a Didactic Engineering

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Abstract

This paper concerns the analysis of a didactic engineering, the aim of which is to introduce Calculus, at secondary-school level, through the relationship between global and local points of view. It was designed for a graphic–symbolic calculator environment and structured in accordance with a learning trajectory from identifying the graphical phenomenon of local linearity to its mathematical formulation. This learning trajectory involves the reconstruction of the relationship with the tangent line to a curve at a chosen point. The analysis shows the use of different semiotic systems in order to grasp this phenomenon and construct its mathematical meaning.

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Notes

  1. Didactic engineering comprises three phases:

    1. Preliminary analysis, in which the theoretical framework and aims of the research are established.

    2. Construction and a-priori analysis.

    3. Experiments, a-posteriori analysis, and conclusions. The validation of hypothesis is based on comparison of a-priori analysis and a-posteriori analysis.

  2. ZoomIn or ZoomBox.

  3. \( {\text{y}}3\left( {\text{x}} \right) = - x^{3} - 2\left| x \right| + 4 \) at x = 0

  4. \( y4\left( x \right) = 4 + x\sin \frac{1}{x} \) at x ≠ 0, = 4 at x = 0

  5. Teacher.

  6. Italian mountain.

  7. Italian word.

  8. Italian word.

  9. Italian word.

  10. Steps:

    1. Set the system with the equation of the curve and that of the line passing through the given point (depending on a parameter).

    2. Reduce the system to a second-degree equation with a parameter.

    3. Compute the parameter imposing the condition for a square root.

  11. In the experiments following the first one, the function was changed from a polynomial of degree two to a polynomial of degree three, to make a possible algebraic solution difficult.

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  1. Department of Mathematics, University of Modena-Reggio Emilia, Via Campi 213/b, 41100, Modena, Italy

    Michela Maschietto

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  1. Michela Maschietto
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Maschietto, M. Graphic Calculators and Micro-Straightness: Analysis of a Didactic Engineering. Int J Comput Math Learning 13, 207–230 (2008). https://doi.org/10.1007/s10758-008-9141-7

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