Proof validation and modification in secondary school geometry

Abstract

Proof validation is important in school mathematics because it can provide a basis upon which to critique mathematical arguments. While there has been some previous research on proof validation, the need for studies with school students is pressing. For this paper, we focus on proof validation and modification during secondary school geometry. For that purpose, we employ Lakatos' notion of local counterexample that rejects a specific step in a proof. By using Toulmin's framework to analyze data from a task-based questionnaire completed by 32 ninth-grade students in a class in Japan, we identify what attempts the students made in producing local counterexamples to their proofs and modifying their proofs to deal with local counterexamples. We found that student difficulties related to producing diagrams that satisfied the condition of the set proof problem and to generating acceptable warrants for claims. The classroom use of tasks that entail student discovery of local counterexamples may help to improve students' learning of proof and proving. first category encompasses studies in which different types of mathematical argument (visual, inductive, generic, or deductive) are presented to students and the students are asked to evaluate whether the arguments are personally convincing and whether the arguments can be considered to constitute proofs (e.g., Healy & Hoyles, 2000; Martin & Harel, 1989; Segal, 1999). A second category relates to students' understanding of given correct proofs. Based on models of proof comprehension (Mejia-Ramos, Fuller, Weber, Rhoads, & Samkoff, 2012; Yang & Lin, 2008), studies in this category seek to identify effective proof comprehension strategies and examine the effectiveness of specific training to improve students' proof comprehension (Hodds, Alcock, & Inglis, 2014; Samkoff & Weber, 2015). A third category involves proof validation; here, researchers show students purported deductive proofs and ask them to determine whether the proofs are valid or invalid (

Task Design Principles for Heuristic Refutation in Dynamic Geometry Environments

“…Hence, in the same way as in Lakatos's research, it is important to develop tasks whose conditions are purposefully implicit so that students can avail themselves of, and consider, particular proofs and counterexamples. This is supported by our previous studies demonstrating that specific tasks, in the form of proof problems with diagrams that include hidden conditions, are helpful for students to explore proofs and counterexamples in secondary school geometry (Komatsu, 2017;Komatsu et al, 2017).…”

“…This task condition led the students to proof construction for the given diagram and enabled them to produce the six types of diagrams, which had the potential to refute their conjecture, with the help of the dragging function of the DGE in Task 1-2. In our previous study (Komatsu et al, 2017), conversely, many secondary school students were found to encounter difficulties drawing diagrams that refuted their proofs in paper-and-pencil environments. The tasks used in that study were more difficult than those in case 1; nevertheless, without the use of dragging it would have been challenging for the three students in case 1 to produce various diagrams different from the one given in the original problem and thus to tackle their task also.…”

“…Lampert, 1990;Larsen & Zandieh, 2008), but these studies do not deal with task design. Given the importance of task design, we have worked on this issue by addressing task design in the context of geometry education at the secondary school level (Komatsu, 2017;Komatsu, Jones, Ikeda & Narazaki, 2017;Komatsu, Tsujiyama, Sakamaki & Koike, 2014). Our previous studies were conducted in paper-and-pencil environments, and we found that the developed tasks were generally helpful for students' work on heuristic refutation.…”

“…Second, a number of studies have reported that students encounter difficulties in producing appropriate counterexamples, irrespective of whether the abovementioned implicitness of task conditions is involved (e.g. Hoyles & Küchemann, 2002;Ko & Knuth, 2013;Komatsu et al, 2017). Thus, it is important to prepare tools that help students produce counterexamples.…”

El Contraejemplo en la Elaboración de la Definición de Función Convexa por Estudiantes Universitarios

“…En el campo de la enseñanza-aprendizaje de la matemática se han desarrollado trabajos relativos a la formulación de conjeturas y al empleo del contraejemplo para el tratamiento de conceptos, propiedades y relaciones matemáticas. En particular, en las investigaciones reportadas por Weber (2009), Komatsu (2010), Knuth y Ko(2013), Giannakoulias, Mastorides, Potari y Zachariades, (2010), Komatsu, Jones, Ikeda y Narazaki (2017), García y Morales (2013), Klymchuk, (2010), Zazkis y Chernoff (2008), Huang (2014) se han identificado que las mismas van dirigidas a profesores y estudiantes, en cada una de ellas se pone de manifiesto que la elaboración y utilización del contraejemplo es una herramienta didáctica para favorecer los procesos de validación y comprensión del conocimiento matemático.…”