Dynamic Geometry Reconstruction

AbstractDigital transformation has made possible the implementation of environments in which mathematics can be experienced in interplay with the computer. Examples are dynamic geometry environments or interactive computational environments, for example GeoGebra or Jupyter Notebook, respectively. We argue that a new possibility to construct and experience proofs arises alongside this development, as it enables the construction of environments capable of not only showing predefined animations, but actually allowing user interaction with mathematical objects and in this way supporting the construction of proofs. We precisely define such environments and call them “mathematical simulations.” Following a theoretical dissection of possible user interaction with these mathematical simulations, we categorize them in relation to other environments supporting the construction of mathematical proofs along the dimensions of “interactivity” and “formality.” Furthermore, we give an analysis of the functions of proofs that can be satisfied by simulation-based proofs. Finally, we provide examples of simulation-based proofs in Ariadne, a mathematical simulation for topology. The results of the analysis show that simulation-based proofs can in theory yield most functions of traditional symbolic proofs, showing promise for the consideration of simulation-based proofs as an alternative form of proof, as well as their use in this regard in education as well as in research. While a theoretical analysis can provide arguments for the possible functions of proof, they can fulfil their actual use and, in particular, their acceptance is of course subject to the sociomathematical norms of the respective communities and will be decided in the future.

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Creativity as a function of problem-solving expertise: posing new problems through investigations

ZDM ◽

10.1007/s11858-021-01228-3 ◽

2021 ◽

Author(s):

Haim Elgrably ◽

Roza Leikin

Keyword(s):

Problem Solving ◽

Dynamic Geometry ◽

Problem Posing ◽

Dynamic Geometry Environment ◽

Mathematics Majors ◽

Individual Interviews ◽

University Mathematics ◽

Domain Specific ◽

Mathematics Test ◽

Different Types

AbstractThis study was inspired by the following question: how is mathematical creativity connected to different kinds of expertise in mathematics? Basing our work on arguments about the domain-specific nature of expertise and creativity, we looked at how participants from two groups with two different types of expertise performed in problem-posing-through-investigations (PPI) in a dynamic geometry environment (DGE). The first type of expertise—MO—involved being a candidate or a member of the Israeli International Mathematical Olympiad team. The second type—MM—was comprised of mathematics majors who excelled in university mathematics. We conducted individual interviews with eight MO participants who were asked to perform PPI in geometry, without previous experience in performing a task of this kind. Eleven MMs tackled the same PPI task during a mathematics test at the end of a 52-h course that integrated PPI. To characterize connections between creativity and expertise, we analyzed participants’ performance on the PPI tasks according to proof skills (i.e., auxiliary constructions, the complexity of posed tasks, and correctness of their proofs) and creativity components (i.e., fluency, flexibility and originality of the discovered properties). Our findings demonstrate significant differences between PPI by MO participants and by MM participants as reflected in the more creative performance and more successful proving processes demonstrated by MO participants. We argue that problem posing and problem solving are inseparable when MO experts are engaged in PPI.

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Drawing dynamic geometry figures online with natural language for junior high school geometry

The International Review of Research in Open and Distributed Learning ◽

10.19173/irrodl.v13i5.1217 ◽

2012 ◽

Vol 13 (5) ◽

pp. 126 ◽

Cited By ~ 1

Author(s):

Wing-Kwong Wong ◽

Sheng-Kai Yin ◽

Chang-Zhe Yang

Keyword(s):

Distance Learning ◽

Natural Language ◽

Knowledge Base ◽

Junior High School ◽

Dynamic Geometry ◽

Preliminary Evaluation ◽

Web Page ◽

High School Geometry ◽

Geometric Figures ◽

Teachers And Students

<p>This paper presents a tool for drawing dynamic geometric figures by understanding the texts of geometry problems. With the tool, teachers and students can construct dynamic geometric figures on a web page by inputting a geometry problem in natural language. First we need to build the knowledge base for understanding geometry problems. With the help of the knowledge base engine InfoMap, geometric concepts are extracted from an input text. The concepts are then used to output a multistep JavaSketchpad script, which constructs the dynamic geometry figure on a web page. Finally, the system outputs the script as an HTML document that can be visualized and read with an internet browser. Furthermore, a preliminary evaluation of the tool showed that it produced correct dynamic geometric figures for over 90% of problems from textbooks. With such high accuracy, the system produced by this study can support distance learning for geometry students as well as distance learning in producing geometry content for instructors.<br /><br /></p>

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The Interpretative Flexibility, Instrumental Evolution, and Institutional Adoption of Mathematical Software in Educational Practice: The Examples of Computer Algebra and Dynamic Geometry

Journal of Educational Computing Research ◽

10.2190/ec.39.4.d ◽

2008 ◽

Vol 39 (4) ◽

pp. 379-394 ◽

Cited By ~ 2

Author(s):

Kenneth Ruthven

Keyword(s):

Computer Algebra ◽

Mathematical Knowledge ◽

New Technologies ◽

Early Stage ◽

Dynamic Geometry ◽

Educational Practice ◽

Explicit Recognition ◽

Mathematical Techniques ◽

Institutional Adoption ◽

Interpretative Flexibility

This article examines three important facets of the incorporation of new technologies into educational practice, focusing on emergent usages of the mathematical tools of computer algebra and dynamic geometry. First, it illustrates the interpretative flexibility of these tools, highlighting important differences in ways of conceptualizing and employing them that reflect their appropriation to contrasting practices of mathematics teaching. Second, it examines the cultural process of instrumental evolution in which mathematical frameworks and teaching practices are adapted in response to new possibilities created by these tools, showing that such evolution remains at a relatively early stage. Third, it points to crucial prerequisites, at both classroom and systemic levels, for effective institutional adoption of such tools: explicit recognition of the interplay between the development of instrumental and mathematical knowledge, including the establishment of a recognized repertoire of tool-mediated mathematical techniques supported by appropriate discourses of explanation and justification.

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