Designing a Web-based Learning Support System for Flow-chart Proving in School Geometry

Abstract

As international research confirms, many secondary school students can find it difficult to construct mathematical proofs. In this article, we explain the pedagogical and technological underpinnings of a web-based learning support system for students who are just starting to tackle deductive proving in geometry. We show how the system was designed to enable students to access the study of proofs in geometry by tackling proof problems where they can 'drag' sides, angles and triangles from the figural diagram of the problem to on-screen cells within a flow-chart proof format. When doing so, the system automatically converts the figural elements to their symbolic form and identifies any of four kinds of errors in the learners' proof attempts, providing relevant feedback on-screen. We use empirical examples from our pilot studies to illustrate how this combination of technological features and systematic feedback can support student understanding of the structure of proof and how to plan one. Finally, we point out some limitations to mathematical expression and the usage of the flow-chart format, and indicate the prospect of minimizing such limitations by adopting a learning progression for the introductory lessons concerning deductive proofs.

Students’Ability in Analyzing by Using a Flow Proof to Solve Problems in Real-Analysis Lecture

“…The conclusion of the study by [10] also showed that the use of Flow Proof can overcome students' difficulties in performing proof. Similarly to research by [11], the use of Flow Proof with open problems encouraged student's development toward understanding the structure of proof by providing assumptions with conclusions.…”

Learners’ use of domain-specific computer-based feedback to overcome logical circularity in deductive proving in geometry

“…2). As we showed in an earlier study, adopting a flow-chart format and closed/open problems can enrich the learning experience of the use of universal/singular propositions and hypothetical syllogism (see Miyazaki et al 2017b).…”

From Argumentation to Proof in Geometry Within a Collaborative Computer-Based Environment

“…In this sense, on the one hand, open problems (Arsac et al, 1991;Boero et al, 1999;Hadas et al, 2000;Pedemonte, 2008;Miyazaki et al, 2017) that require the formulation of a conjecture for their resolution are extremely effective in introducing the concept of proof. On the other, the production of arguments is certainly favoured by collaborative activities, in which students have to explain their own ideas, take into account those of others and compare them to reach a shared conclusion.…”

“…For example, many researchers have focused on the formalization of proofs, with the aim of designing environments able to support students in the creation of step-by-step proofs, for example by using flowcharts (Anderson et al, 1986(Anderson et al, , 1995. Other researchers, with particular reference to geometry, have designed web environments that allow secondary school students to construct different steps of a proof by dragging single elements (side/angle/ triangle) of a diagram within a flowchart (Miyazaki et al, 2017).…”

“…Many scholars have investigated the potential offered by dynamic geometry for supporting design activities (Sinclair & Robutti, 2012) to promote the production of arguments and proof by students. However, the combination of dynamic geometry with other digital environments could also support students in the formalization of proofs (Anderson et al, 1986(Anderson et al, , 1995Miyazaki et al, 2017).…”