From a Student's Question to the Half-Derivative: A Constructivist Teaching Model - Part I
Abstract
This paper presents a conceptual, cognitive-constructivist model for teaching the half-derivative. The model starts from a student's question and frames it as the problem of generalizing the familiar operation of differentiation. On a suitably chosen space of monomials, under the assumptions of linearity and of lowering the exponent by one half, the requirement that the operator applied twice give the ordinary derivative produces a recurrence for the coefficients. The freedom of normalization becomes visible, and an additional convention yields a formula compatible with the monomial rule of the Riemann-Liouville half-derivative. The domain of the operator and its relation to the Riemann-Liouville and Caputo operators are treated separately. The mathematical results rest on the author's earlier preprints; the contribution of this paper is their transformation into an instructional pathway organized in phases, with diagnostic questions, an analysis of typical errors, and comparative problems. The model is a design proposal: its effectiveness has not yet been established by systematic empirical research. The interactive Socratic implementation and the detailed diagnostics are the subject of Part II.
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BibTeX
@misc{kapanadze2026from,
title = {{From a Student's Question to the Half-Derivative: A Constructivist Teaching Model - Part I}},
author = {Davit Kapanadze},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18932},
eprint = {2610.18932},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18932},
}
RIS
TY - UNPB
TI - From a Student's Question to the Half-Derivative: A Constructivist Teaching Model - Part I
AU - Davit Kapanadze
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18932
UR - https://arxiv.org/abs/2610.18932
AB - This paper presents a conceptual, cognitive-constructivist model for teaching the half-derivative. The model starts from a student's question and frames it as the problem of generalizing the familiar operation of differentiation. On a suitably chosen space of monomials, under the assumptions of linearity and of lowering the exponent by one half, the requirement that the operator applied twice give the ordinary derivative produces a recurrence for the coefficients. The freedom of normalization becomes visible, and an additional convention yields a formula compatible with the monomial rule of the Riemann-Liouville half-derivative. The domain of the operator and its relation to the Riemann-Liouville and Caputo operators are treated separately. The mathematical results rest on the author's earlier preprints; the contribution of this paper is their transformation into an instructional pathway organized in phases, with diagnostic questions, an analysis of typical errors, and comparative problems. The model is a design proposal: its effectiveness has not yet been established by systematic empirical research. The interactive Socratic implementation and the detailed diagnostics are the subject of Part II.
ER -