Issues and Challenges in Instrumental Proof

Richard, Philippe; Venant, Fabienne; Gagnon, Michel

doi:10.1007/978-3-030-28483-1_7

Cited by 15 publications

(5 citation statements)

References 20 publications

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“…The dragging feature in dynamic geometry environments (DGE) allows its users to quickly generate infinitely many dynamically linked examples. The use of dragging also has the added benefit of maintaining the invariant properties of a set of cases that are the result of the construction process (Richard et al, 2019). Moreover, the measuring facility in DGEs makes it convenient for students to gather data and look for numerical relationships.…”

Section: Mathematical Reasoning With Dynamic Technologymentioning

confidence: 99%

“…Moreover, the measuring facility in DGEs makes it convenient for students to gather data and look for numerical relationships. From this process, students can recognize visual or numerical patterns and abstract the invariant properties of the construction, and then use that information to form generalizations and justifications (Richard et al, 2019;Sinclair & Robutti, 2012).…”

Section: Mathematical Reasoning With Dynamic Technologymentioning

confidence: 99%

“…While these concerns exist and have been raised, there have been responses defending the use of dynamic technologies in mathematics learning. Researchers have argued that dynamic technologies have to be a tool used to develop mathematical thinking rather than simply a tool that executes mathematical operations (Richard et al, 2019). By using dynamic technologies as ways to build knowledge inductively, students can be supported to focus on the more general aspects of the cases they examine and move to thinking more deductively (Baccaglini-Frank, 2019; Komatsu & Jones, 2019;Lachmy & Koichu, 2014).…”

Section: Mathematical Reasoning With Dynamic Technologymentioning

confidence: 99%

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Connections between Empirical and Structural Reasoning in Technology-Aided Generalization Activities

Yao¹,

Elia²

2021

INT ELECT J MATH ED

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Mathematical generalization can take on different forms and be built upon different types of reasoning. Having utilized data from a series of task-based interviews, this study examined connections between empirical and structural reasoning as preservice mathematics teachers solved problems designed to engage them in constructing and generalizing mathematical ideas aided by digital tools. The study revealed closer connections between naïve empiricism and result pattern generalization, between naïve empiricism and recognizing a structure in thought, between reasoning by generic example and process pattern generalization, and between reasoning by generic example and reasoning in terms of general structures. Results from this study imply that the ability to generalize based on perception and numerical pattern does not necessarily lead learners to generalize based on mathematical structure.

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Section: Mathematical Reasoning With Dynamic Technologymentioning

confidence: 99%

Section: Mathematical Reasoning With Dynamic Technologymentioning

confidence: 99%

Section: Mathematical Reasoning With Dynamic Technologymentioning

confidence: 99%

See 1 more Smart Citation

Connections between Empirical and Structural Reasoning in Technology-Aided Generalization Activities

Yao¹,

Elia²

2021

INT ELECT J MATH ED

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“…At some point, they might also use tools (ruler, compass, dynamic geometry software, etc.) to help with this process [18]. Overall, this whole process is extremely different from what a computer would do.…”

Section: Differences Between a Human's And A Computer's Reasoningmentioning

confidence: 99%

Automating the Generation of High School Geometry Proofs using Prolog in an Educational Context

Font

Cyr

Richard

et al. 2020

Electron. Proc. Theor. Comput. Sci.

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When working on intelligent tutor systems designed for mathematics education and its specificities, an interesting objective is to provide relevant help to the students by anticipating their next steps. This can only be done by knowing, beforehand, the possible ways to solve a problem. Hence the need for an automated theorem prover that provide proofs as they would be written by a student. To achieve this objective, logic programming is a natural tool due to the similarity of its reasoning with a mathematical proof by inference. In this paper, we present the core ideas we used to implement such a prover, from its encoding in Prolog to the generation of the complete set of proofs. However, when dealing with educational aspects, there are many challenges to overcome. We also present the main issues we encountered, as well as the chosen solutions.1 Context The QED-Tutrix softwareThe QED-Tutrix software [15,19] provides an environment where a high-school student can solve geometry proof problems. One of its key features is that it allows the student to provide proof elements in any order, not limiting them to forward-or backward-chaining. For instance, when solving the simple problem "prove that a quadrilateral with three right angles is a rectangle", the student can provide any element of any possible proof, such as a direct consequence of the hypotheses ("if two lines are perpendicular to a third, they are parallel"), a necessary premise for the conclusion ("a rectangle is a quadrilateral that has four right angles"), or anything in between ("the quadrilateral ABCD is a parallelogram"). A second key feature is the tutoring aspect. When the student is stuck is the resolution, the software is able to provide them with relevant messages. In the previous example, if the student entered "the quadrilateral ABCD is a parallelogram" and is stuck afterwards, the software identifies that they are working on a proof using parallelogram properties, and will provide them messages such as "what is the definition of a parallelogram?" or "is there a relation between parallelogram and rectangle?"These features, the flexibility in exploration and the tutoring, are very interesting from a mathematics education perspective, but come with a cost. Indeed, to allow this behavior, the software must know, first, all the various mathematical elements that can be used at any step of any proof for the problem at hand, and second, how these elements are used in the various possible proofs of the problem. This requires,

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“…Otra evolución importante es la información de salida que entregan estos sistemas y cómo se articulan con otros artefactos digitales, en particular con los CAA. Si los trabajos de comienzos de siglo, los CAS entregaban principalmente un objeto de salida, como las soluciones de una ecuación o un gráfico (Artigue, 2002;Hoyles et al, 2004;Kieran y Drijvers, 2006;Lagrange, 2000;Ruthven, 2002;Trouche, 2000;Yerushalmy, 1999), actualmente, los sistemas han evolucionado entregando no solo la respuesta, sino que las soluciones paso a paso de un cálculo (Barzel et al, 2019), o en entornos interactivos para trabajar el razonamiento y la prueba automatizada (Botana et al, 2015;Richard et al, 2019) o para automatizar la evaluación, validando respuestas equivalentes, diferenciando expresiones según algunas características (por ejemplo, si una expresión está factorizada o no) y dando un feedback en función del enunciado o de la respuesta del estudiante (Gaona et al, 2018;Gaona y Menares, 2021;Sangwin, 2007,0).…”

Section: Introductionunclassified

Aprendizaje de los números complejos usando diferentes sistemas de cálculo simbólico y un sistema de evaluación en línea en formación inicial de profesores

Gaona¹,

López²,

Delgadillo³

2022

Preprint

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Issues and Challenges in Instrumental Proof

Cited by 15 publications

References 20 publications

Connections between Empirical and Structural Reasoning in Technology-Aided Generalization Activities

Connections between Empirical and Structural Reasoning in Technology-Aided Generalization Activities

Automating the Generation of High School Geometry Proofs using Prolog in an Educational Context

Aprendizaje de los números complejos usando diferentes sistemas de cálculo simbólico y un sistema de evaluación en línea en formación inicial de profesores

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