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Meta-Representational Fluency: Math Majors’ Visualization of the l’Hospital’s Rule in a Dynamic Geometry Environment

Gunhan Caglayan
p. 63-101


The present qualitative study on mathematics majors’ visualization of the l’Hospital’s Rule in a dynamic geometry environment highlights student-generated representations in a theoretical framework drawn from visual thinking in calculus and meta-representational competences perspectives. Students developed their own personal meanings of the l’Hospital’s formalism; their expression of meta-representational fluency occurred in different, yet not necessarily hierarchical modes of visualizations. Whereas global tangential visualization appeared in the form of static, successive, or simultaneous visualizations of the tangency points on both function graphs; functional visualization (local-tangential functional, static derivative functional, dynamic derivative functional) appeared as an additional key construct in students' treatment of the l’Hospital’s Rule by primarily focusing on the function graphs, without taking tangent lines into account.

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Background and Rationale for the Study
Limit Conceptions
Derivative Conceptions
Fostering Visual Thinking in Calculus: Meta-Representational Competence
Context and Methodology
Results: Computing limits using L’Hospital’s Rule
A-Priori Analysis: Tangential vs. Local-Tangential Visualizations
Global Tangential Visualization
Functional Visualization
Forms that are equivalent to the 0 / 0 indeterminate form
The 0 ∙ ∞ indeterminate form
The ∞ − ∞ indeterminate form
Discussion and Conclusions
A Summative Look at Math Majors’ Knowledge of the L’Hospital’s Rule
Reconciling Multiple Visualizations: Meta-Representational Fluency

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In 1696, the French mathematician Guillaume François Marquis de l’Hospital published the differential calculus text Analyse des Infiniments Petits, a compilation of the brilliant Swiss physician and mathematician Johann Bernoulli’s material, which also included a section on calculus of variations (Eves, 2006; Struik, 1963). “L’Hospital’s Analyse is probably most famous as the source of l’Hospital’s Rule—which should probably be renamed Bernoulli’s rule—for calculating limits of quotients in the case where the limits of both numerator and denominator are zero” (Katz, 2009, p. 576). The origin of l’Hospital’s Rule is based on the proposition: “Let AMD be a curve (AP = x, PM = y, AB a) such that the value of the ordinate y is expressed by a fraction, of which the numerator and denominator each become 0 when a, that is to say, when the point P corresponds to the given point B. It is ...

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Gunhan Caglayan, « Meta-Representational Fluency: Math Majors’ Visualization of the l’Hospital’s Rule in a Dynamic Geometry Environment », Éducation et didactique, 11-2 | 2017, 63-101.

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Gunhan Caglayan, « Meta-Representational Fluency: Math Majors’ Visualization of the l’Hospital’s Rule in a Dynamic Geometry Environment », Éducation et didactique [En ligne], 11-2 | 2017, mis en ligne le 06 décembre 2019, consulté le 21 février 2018. URL : ; DOI : 10.4000/educationdidactique.2789

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Gunhan Caglayan

New Jersey City University Mathematics Department

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