Computer-Generated Geometry Proofs in a Learning Context (original) (raw)
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The Interplay of Teacher and Student Actions in the Teaching and Learning of Geometric Proof
Educational Studies in Mathematics, 2005
Proof and reasoning are fundamental aspects of mathematics. Yet, how to help students develop the skills they need to engage in this type of higher-order thinking remains elusive. In order to contribute to the dialogue on this subject, we share results from a classroom-based interpretive study of teaching and learning proof in geometry. The goal of this research was to identify factors that may be related to the development of proof understanding. In this paper, we identify and interpret students' actions, teacher's actions, and social aspects that are evident in a classroom in which students discuss mathematical conjectures, justification processes and student-generated proofs. We conclude that pedagogical choices made by the teacher, as manifested in the teacher's actions, are key to the type of classroom environment that is established and, hence, to students' opportunities to hone their proof and reasoning skills. More specifically, the teacher's choice to pose open-ended tasks (tasks which are not limited to one specific solution or solution strategy), engage in dialogue that places responsibility for reasoning on the students, analyze student arguments, and coach students as they reason, creates an environment in which participating students make conjectures, provide justifications, and build chains of reasoning. In this environment, students who actively participate in the classroom discourse are supported as they engage in proof development activities. By examining connections between teacher and student actions within a social context, we offer a first step in linking teachers' practice to students' understanding of proof.
1 Moving Toward More Authentic Proof Practices in Geometry
2016
Abstract: Through the Standards documents, NCTM has called for changes related to Reasoning and Proof and Geometry. There is some evidence that these recommendations have been taken seriously by mathematics educators and textbook developers. However, if we are truly to realize the goals of the Standards, we must pose problems to our students that allow them to play a greater role in proving. We offer nine such problems and discuss how using multiple proof representations moves us toward more authentic proof practices in geometry.
Proofs produced by secondary school students learning geometry in a dynamic computer environment
Educational Studies in Mathematics, 2000
As a key objective, secondary school mathematics teachers seek to improve the proof skills of students. In this paper we present an analytic framework to describe and analyze students' answers to proof problems. We employ this framework to investigate ways in which dynamic geometry software can be used to improve students' understanding of the nature of mathematical proof and to improve their proof skills. We present the results of two case studies where secondary school students worked with Cabri-Géomètre to solve geometry problems structured in a teaching unit. The teaching unit had the aims of: i) Teaching geometric concepts and properties, and ii) helping students to improve their conception of the nature of mathematical proof and to improve their proof skills. By applying the framework defined here, we analyze students' answers to proof problems, observe the types of justifications produced, and verify the usefulness of learning in dynamic geometry computer environments to improve students' proof skills.
Proofs through exploration in dynamic geometry environments
International Journal of …, 2004
The recent development of powerful new technologies such as dynamic geometry softwares (DGS) with drag capability has made possible the continuous variation of geometric configurations and allows one to quickly and easily investigate whether particular conjectures are true or not. Because of the inductive nature of the DGS, the experimental-theoretical gap that exists in the acquisition and justification of geometrical knowledge becomes an important pedagogical concern. In this article we discuss the implications of the development of this new software for the teaching of proof and making proof meaningful to students. We describe how three prospective primary school teachers explored problems in geometry and how their constructions and conjectures led them "see" proofs in DGS.
The mathematical nature of reasoning-and-proving opportunities in geometry textbooks
International calls have been made for reasoning-and-proving to permeate school mathematics. It is important that efforts to heed this call are grounded in an understanding of the opportunities to reason-and-prove that already exist, especially in secondary-level geometry where reasoning-and-proving opportunities are prevalent but not thoroughly studied. This analysis of six secondary-level geometry textbooks, like studies of other textbooks, characterizes the justifications given in the exposition and the reasoning-and-proving activities expected of students in the exercises. Furthermore, this study considers whether the mathematical statements included in the reasoning-and-proving opportunities are general or particular in nature. Findings include the fact that the majority of expository mathematical statements were general, whereas reasoning-and-proving exercises tended to involve particular mathematical statements. Although reasoning-and-proving opportunities were relatively numerous, it remained rare for the reasoning-and-proving process itself to be an explicit object of reflection. Relationships between these findings and the necessity principle of pedagogy are discussed.
In this paper we classify student's proving level and design an interactive help system (IHS) corresponding with these levels in order to investigate the development of the proving process within a dynamic geometry environment. This help system was also used to provide tertiary students with a strategy for proving and to improve their proving levels. The open-ended questions and explorative tasks in the IHS make a contribution to support students' learning of proving, especially during the processes of realizing invariants, formulating conjectures, producing arguments, and writing proofs. This research wants to react on the well-known students' difficulties in writing a formal proof. The hypothesis of this work is that these difficulties are based on the lack of students' understanding the relationship between argumentation and proof. Therefore, we used Toulmin model to analyze student's argumentation structure and examine the role of abduction in writing a deductive proof. Furthermore, this paper also provides mathematics teachers with three basic conditions for understanding the development of the proving process and teaching strategies for assisting their students in constructing formal proofs.
The impact of different proof strategies on learning geometry theorem proving
2004
I thank my advisor Kurt VanLehn for his kind and considerable intellectual support. Thanks also my committee members Peter Brusilovsky, Christian Schunn, James Greeno, and Ken Koedinger. I have learned a lot from every single meeting with those great scholars. Also, I thank Sara Masters, Eri Seta, and Kwangsu Cho for their patience during early pilot-testing as well as very many constructive comments on the software.
GPT: A Tutor for Geometry Proving
2017
Geometry is a mandatory subject for secondary school students, where they learn geometric figures and their properties, and the axioms, postulates and theorems involving them. A key topic in Geometry class is proving, where learners are required to derive and prove a certain property is true based from the given properties and by using various axioms, postulates and theorems. This is where most learners encounter difficulty. In this paper, we describe Geometry Proof Tutor, a learning environment where learners can practice Geometry proving through multiple representations two-column proof and proof tree. With the use of a knowledge base to model Geometry concepts, the software validates the learner’s proof statements and provides corrective feedback accordingly. Test results showed that the learners found the availability of the proof tree to be useful in tracking their progress. The presence of complete reasons to use for proofs also helped them understand Geometric Proving better....
Proof, Explanation and Exploration: An Overview
2001
This paper explores the role of proof in mathematics education and provides justification for its importance in the curriculum. It also discusses three applications of dynamic geometry software-heuristics, exploration and visualization-as valuable tools in the teaching of proof and as potential challenges to the importance of proof. Finally, it introduces the four papers in this issue that present empirical research on the use of dynamic geometry software.