Guided discovery of the nine-point circle theorem and its proof

The nine-point circle theorem is one of the most beautiful and surprising theorems in Euclidean geometry. It establishes an existence of a circle passing through nine points, all of which are related to a single triangle. This paper describes a set of instructional activities that can help students discover the nine-point circle theorem through investigation in a dynamic geometry environment, and consequently prove it using a method of guided discovery. The paper concludes with a variety of suggestions for the ways in which the whole set of activities can be implemented in geometry classrooms.     

Dynamic geometry as a context for exploring conjectures

The purpose of this paper is to provide examples of ‘non-traditional’ proof-related activities that can explored in a dynamic geometry environment by university and high school students of mathematics. These propositions were encountered in the dynamic geometry environment. The author believes that teachers can ask their students to construct proofs for these propositions.     

Prompting Teacher Geometric Reasoning through Coaching in a Dynamic Geometry Software Context

This study investigated the ways in which four middle grades teachers developed mathematical knowledge for teaching (MKT) geometry as they implemented dynamic geometry software in their classrooms with the assistance of a coach. Teachers developed various components of MKT by observing coaches teach, by dynamic discourse with students, which is discourse with respect to dynamic geometry software images, and by discussions with coaches. The dynamic geometry software environment proved productive as coaches guided teachers’ growth in explanations, examples, and definitions. The environment also helped teachers discover unnoticed abilities among their low achievers. Moreover, teachers developed confidence to teach with more expert uses of dynamic geometry software.     

An epistemic model of task design in dynamic geometry environment

Dynamic geometry environment (DGE) has been a catalytic agent driving a paradigm shift in the teaching and learning of school geometry in the past two decades. It opens up a pedagogical space for teachers and students to engage in mathematical explorations that niche across the experimental and the theoretical. In particular, the drag-mode in DGE has been a unique pedagogical tool that can facilitate and empower students to experiment with dynamic geometrical objects which can lead to generation of mathematical conjectures. Furthermore, the drag-mode seems to open up a new methodology and even a new discourse to acquire geometrical knowledge alternative to the traditional Euclidean deductive reasoning paradigm. This discussion paper proposes an epistemic model of techno-pedagogic mathematic task design which serves as a theoretical combined-lens to view mathematics knowledge acquisition. Three epistemic modes for techno-pedagogic mathematical task design are proposed. They are used to conceptualize design of dynamic geometry tasks capitalizing the unique drag-mode nature in DGE that opens up an explorative space for learners to acquire mathematical knowledge.     

A geometry teacher's use of a metaphor in relation to a prototypical image to help students remember a set of theorems

This article asks the following: How does a teacher use a metaphor in relation to a prototypical image to help students remember a set of theorems? This question is analyzed through the case of a geometry teacher. The analysis uses Duval's work on the apprehension of diagrams to investigate how the teacher used a metaphor to remind students about the heuristics involved when applying a set of theorems during a problem-based lesson. The findings show that the teacher used the metaphor to help students recall the apprehensions of diagrams when applying several theorems. The metaphor was instrumental for mediating students’ work on a problem and the proof of a new theorem. The findings suggest that teachers’ use of metaphors in relation to prototypical images may facilitate how they organize students’ knowledge for later retrieval.     

A class of dual fuzzy dynamic programs

In this paper we propose a large class of fuzzy dynamic programs. By use of the notion of dual binary relation we define a dual fuzzy dynamic program in the class. We establish two duality theorems between primal and dual fuzzy dynamic programs. One is for the two-parametric recursive equations. The other is for the nonparametric. We specify maximum–minimum process and minimum–minimum process in fuzzy environment and multiplicative–multiplicative process in quasi-stochastic environment. It is shown that the duality theorems hold between primal and dual programs.     

Theorem Justification and Acquisition in Dynamic Geometry: A Case of Proof by Contradiction

Theorem acquisition and deductive proof have always been core elements in the study and teaching of Euclidean geometry. The introduction of dynamic geometry environments,DGE (e.g., Cabri-Géomètre, Geometer's Sketchpad), into classrooms in the past decade has posed a challenge to this praxis. Student scan experiment through different dragging modalities on geometrical objects that they construct, and consequently infer properties(generalities, theorems) about the geometrical artefacts. Because of the inductive nature of the DGE, the experimental-theoretical gap that exists in the acquisition and justification of geometrical knowledge becomes an important pedagogical and epistemological concern. In this paper, we will describe and study a ‘Cabri proof by contradiction’ of a theorem on cyclic quadrilaterals given by a pair of 16 year-old students in a Hong Kong secondary school. We will discuss how their construction motivates a visual-cognitive scheme on `seeing' proof in DGE, and how this scheme could fit into the theoretical construct of cognitive unity of theorems proposed by Boero, Garuti and Mariotti(1996). The issue of a cognitive duality and its relation to visualization will be raised and discussed. Finally, we propose a possible perspective to bridge the experimental-theoretical gap in DGE by introducing the idea of a dynamic template as a visualizer to geometrical theorem justification and acquisition. This revised version was published online in July 2006 with corrections to the Cover Date.     

Making sense of qualitative geometry: The case of Amanda

This article presents a case study of a seven-year-old girl named Amanda who participated in an eighteen-week teaching experiment I conducted in order to model the development of her intuitive and informal topological ideas. I designed a new dynamic geometry environment that I used in each of the episodes of the teaching experiment to elicit these conceptions and further support their development. As the study progressed, I found that Amanda developed significant and authentic forms of geometric reasoning. It is these newly identified forms of reasoning, which I refer to as “qualitative geometry,” that have implications for the teaching and learning of geometry and for research into students’ mathematical reasoning.     

App-based scaffolds for writing two-column proofs

This paper presents apps designed to assist students in understanding and developing proofs in geometric theorems. These technologies focus on triangle congruence, triangle similarity and properties of parallelograms. Focus group discussions and initial testing of the apps revealed that the apps offered a more engaging medium for learning proving and were capable of facilitating proof-writing skills in geometry.     

An interesting property of hexagons

These notes discuss several related problems in geometry that can be explored in a dynamic geometry environment. The problems involve an interesting property of hexagons.     

An unexpected property of quadrilaterals*

These notes discuss several related propositions in geometry that can be explored in a dynamic geometry environment. The propositions involve an unexpected property of quadrilaterals.     

Pre-constructed dynamic geometry materials in the classroom – how do they facilitate the learning of ‘Similar Triangles’?

The use of dynamic geometry software (DGS) is becoming increasingly familiar among teachers, but letting students conduct inquiries using computers is still not a welcome idea. In addition to logistics and discipline concerns, many teachers believe that mathematics at the lower secondary level can be learned efficiently through practice alone. Thus, the application of DGS remains limited to demonstration and explanation. This article discusses how a set of pre-constructed dynamic geometry (DG) materials was designed to teach the ‘similar triangles’ concept. The reactions and behaviour of students with relatively low levels of mathematic achievement are also analysed. Finally, the potential value of pre-constructed DG materials, with lab sheets and teacher intervention, in inquiry activities for junior-level students is discussed.     

The effectiveness of ‘what if not’ strategy coupled with dynamic geometry software in an inquiry-based geometry classroom

In this paper we present the results of a study which was carried out in an inquiry-based teaching and learning environment with the use of ‘what if not’ methodology coupled with the integration of dynamic geometry software. The vast majority of the students reported that they perceived themselves as participants rather than spectators. Most of the prospective teachers came to the conclusion that the implementation of the findings of this study in their future teachings was a good idea and that it will raise the students’ motivation and enhance and deepen the knowledge pool of the learners.     

Indefinite Kähler submanifolds with positive index of relative nullity

We deal with complex submanifolds in indefinite space forms. In particular, submanifolds with large index of relative nullity are emphasized. In that context, we prove cylinder theorems in terms of indefinite metrics. We also give a systematic way of constructing a family of new complete and closed indefinite complex submanifolds in the projective setting.In the appendix, we show that the method used for complex cases can be applied to real indefinite geometry. We prove real cylinder theorems including B-scrolls in the general signature. We also show two decomposition lemmas which clarify the relationships between the Hartman-Nirenberg cylinder theorem and slanted cylinder theorems in indefinite geometry.     

几何定理机器证明三十年

由于传统的兴趣和多种原因,几何定理的机器证明在自动推理的研究中占有重要的地位.自吴法发表至今30年,几何定理机器证明的研究和实践有了很大的进展.对无序几何命题而言,代数方法、数值方法均能有效地判定其真假,面积法(消点法)、搜索法更能生成其可读的证明.几何不等式机器证明的研究,由于多项式完全判别系统的建立,也有了突破.研究领域已由机器证明扩展为包括几何作图在内的一般几何问题的机器求解,并有了实际的应用.     

Unpacking learner’s growth in geometric understanding when solving problems in a dynamic geometry environment: Coordinating two frames

The emergence of dynamic geometry environments challenges researchers in mathematics education to develop theories that capture learner’s growth in geometric understanding in this particular environment. This study coordinated the Pirie-Kieren theory and instrumental genesis to examine learner’s growth in geometric understanding when solving problems in a dynamic geometry environment. Data analysis suggested that coordinating the two theoretical approaches provided a productive means to capture the dynamic interaction between the growth in mathematical understanding and the formation/application of utilization scheme during a learner’s mathematical exploration with dynamic geometry software. The analysis of one episode on inscribing a square in a triangle was shared to illustrate this approach. This study contributes to the continuing conversation of “networking theories” in the mathematics education research community. By networking the two theoretical approaches, this paper presents a model for studying learner's growth in mathematical understanding in a dynamic learning environment while accounting for interaction with digital tools.     

Software Tools for Geometrical Problem Solving: Potentials and Pitfalls

Dynamic geometry software provides tools for students to construct and experiment with geometrical objects and relationships. On the basis of their experimentation, students make conjectures that can be tested with the tools available. In this paper, we explore the role of software tools in geometry problem solving and how these tools, in interaction with activities that embed the goals of teachers and students, mediate the problem solving process. Through analysis of successful student responses, we show how dynamic software tools can not only scaffold the solution process but also help students move from argumentation to logical deduction. However, by reference to the work of less successful students, we illustrate how software tools that cannot be programmed to fit the goals of the students may prevent them from expressing their (correct) mathematical ideas and thus impede their problem solution.This revised version was published online in September 2005 with corrections to the Cover Date.     

The role of multiple solution tasks in developing knowledge and creativity in geometry

This paper describes changes in students’ geometrical knowledge and their creativity associated with implementation of Multiple Solution Tasks (MSTs) in school geometry courses. Three hundred and three students from 14 geometry classes participated in the study, of whom 229 students from 11 classes learned in an experimental environment that employed MSTs while the rest learned without any special intervention in the course of one school year. This longitudinal study compares the development of knowledge and creativity between the experimental and control groups as reflected in students’ written tests. Geometry knowledge was measured by the correctness and connectedness of the solutions presented. The criteria for creativity were: fluency, flexibility, and originality. The findings show that students’ connectedness as well as their fluency and flexibility benefited from implementation of MSTs. The study supports the idea that originality is a more internal characteristic than fluency and flexibility, and therefore more related with creativity and less dynamic. Nevertheless, the MSTs approach provides greater opportunity for potentially creative students to present their creative products than conventional learning environment. Cluster analysis of the experimental group identified three clusters that correspond to three levels of student performance, according to the five measured criteria in pre- and post-tests, and showed that, with the exception of originality, performance in all three clusters generally improved on the various criteria.