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.     

Origami,geometry and art

The purpose of this paper is to describe the mathematics that emanates from the construction of an origami box. We first construct a simple origami box from a rectangular sheet and then discuss some of the mathematical questions that arise in the context of geometry and algebra. The activity can be used as a context for illustrating how algebra and geometry, like other branches of mathematics, are interrelated.     

Standard identities for skew-symmetric matrices

Let n be a positive, even integer and let K_n(F) denote the subspace of skew-symmetric matrices of Mn(F), the full matrix algebra with coefficients in a field F. A theorem of Kostant states that K_n(F) satisfies the (2n-2)-fold standard identity s_2n-2. In this paper we refine this result by showing that s_2n-2 may be written non-trivially as the sum of two polynomial identities of K_n(F).     

Hjelmslev's geometry of reality

During the first half of the 20th century the Danish geometer Johannes Hjelmslev developed what he called a geometry of reality. It was presented as an alternative to the idealized Euclidean paradigm that had recently been completed by Hilbert. Hjelmslev argued that his geometry of reality was superior to the Euclidean geometry both didactically, scientifically and in practice: Didactically, because it was closer to experience and intuition, in practice because it was in accordance with the real geometrical drawing practice of the engineer, and scientifically because it was based on a smaller axiomatic basis than Hilbertian Euclidean geometry but still included the important theorems of ordinary geometry. In this paper, I shall primarily analyze the scientific aspect of Hjelmslev's new approach to geometry that gave rise to the so-called Hjelmslev (incidence) geometry or ring geometry.     

Self-dual lattice identities

Given a finitely based self-dual variety of lattices, is it definable, modulo lattice theory, by a single self-dual lattice identity? There are infinitely many examples with “yes” as the answer and infinitely many with “no.”. Received December 20, 1999; accepted in final form July 11, 2002.     

A design and a geometry for the group Fi 22

The Fischer group Fi ₂₂ acts as a rank 3 group of automorphisms of a symmetric 2-(14080,1444,148) design. This design does not have a doubly transitive automorphism group, since there is a partial linear space with lines of size 4 defined combinatorially from the design and preserved by its automorphism group. We investigate this geometry and determine the structure of various subspaces of it.      

High school students’ understandings of geometric transformations in the context of a technological environment

This study investigated the nature of students’ understandings of geometric transformations, which included translations, reflections, rotations, and dilations, in the context of the technological tool, The Geometer’s Sketchpad. The researcher implemented a seven-week instructional unit on geometric transformations within an Honors Geometry class. Students’ conceptions of transformations as functions were analyzed using the APOS theory and were informed by an analysis of students’ interpretations and uses of representations of geometrical objects using the constructs of drawing and figure. The analysis suggests students’ understandings of key concepts including domain, variables and parameters, and relationships and properties of transformations were critical for supporting the development of deeper understandings of transformations as functions.     

An interesting family of identities

This paper investigates a general identity which expresses an apparently complicated and intriguing sum of fractions as an elegant and straightforward sum of simple terms.     

Constructive struggle in geometry classrooms

The purpose of these notes is to provide examples of large-scale geometry problems that have the potential to promote constructive struggling in high-school geometry classrooms. These problems are suitable for being explored in a dynamic geometry environment. The notes also contain brief sketches of solutions to the two problems.     

Symbolic computations in applied differential geometry

The main aim of this paper is to contribute to the automatic calculations in differential geometry and its applications, with emphasis on the prolongation theory of Estabrook and Wahlquist, and the calculation of invariance groups of exterior differential systems. A large number of worked examples have been included in the text to demonstrate the concrete manipulations in practice. In the appendix, a list of programs discussed in the paper is added.     

Diameter preserving surjections in the geometry of matrices

We consider a class of graphs subject to certain restrictions, including the finiteness of diameters. Any surjective mapping φ:Γ→Γ^′ between graphs from this class is shown to be an isomorphism provided that the following holds: Any two points of Γ are at a distance equal to the diameter of Γ if, and only if, their images are at a distance equal to the diameter of Γ^′. This result is then applied to the graphs arising from the adjacency relations of spaces of rectangular matrices, spaces of Hermitian matrices, and Grassmann spaces (projective spaces of rectangular matrices).     

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.     

An example of the geometry of a 5th-order ODE: The metric on the space of conics in CP2

As an application of the method of [4], we find the metric and connection on the space of conics in ${\mathbb{CP}}^2$ determined as the solution space of the ODE (1). These calculations underpin the twistor construction of the Radon transform on conics in ${\mathbb{CP}}^2$ described in [5]. Two further examples of the method are provided.     

Differentiators and the geometry of polynomials

In 1959, Davis introduced the concept of a differentiator of an operator on a finite-dimensional Hilbert space. We prove that every such operator possesses a differentiator. We also use the theory of differentiators to solve several problems in the geometry of polynomials. For instance, we answer in the affirmative a twenty year old unsolved conjecture of Schoenberg, a related conjecture of Katsoprinakis and a fifty year old unsolved conjecture of De Bruijn and Springer.     

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.     

How students structure their investigations and learn mathematics: insights from a long-term study

This paper reports on the mathematical thinking of participants of a long-term study, now in its 17th year, who did mathematics together through their public school and early university years. In particular, it describes how fundamental ideas and images of a cohort group of students are elaborated and presented in symbolic expressions of generalized mathematical ideas while exploring problems in grades 10 and 11. From high school and university interview data, we learn from participants how they viewed their mathematical activity in structuring their investigations and justifying their solutions.     

On some geometry and equivalence classes of normal form games

Equivalence classes of normal form games are defined using the discontinuities of correspondences of standard equilibrium concepts like correlated, Nash, and robust equilibrium, or risk dominance and rationalizability. Resulting equivalence classes are fully characterized and compared across different equilibrium concepts for 2 ×  2 games; larger games are also studied. It is argued that the procedure leads to broad and game-theoretically meaningful distinctions of games as well as to alternative ways of representing, comparing and testing equilibrium concepts.     

The mathematical beliefs and behavior of high school students: Insights from a longitudinal study

There is a documented need for more research on the mathematical beliefs of students below college. In particular, there is a need for more studies on how the mathematical beliefs of these students impact their mathematical behavior in challenging mathematical tasks. This study examines the beliefs on mathematical learning of five high school students and the students’ mathematical behavior in a challenging probability task. The students were participants in an after-school, classroom-based, longitudinal study on students’ development of mathematical ideas funded by the United States National Science Foundation. The results show that particular educational experiences can alter results from previous studies on the mathematical beliefs and behavior of students below college, some of which have been used to justify non-reform pedagogical approaches in mathematics classrooms. Implications for classroom practice and ideas for future research are discussed.     

Authority and whole-class proving in high school geometry: The case of Ms. Finley

Students’ experiences with proving in schools often lead them to see proof as a static product rather than a negotiated process that can help students justify and make sense of mathematical ideas. We investigated how authority manifested in whole-class proving episodes within Ms. Finley’s high school geometry classroom. We designed a coding scheme that helped us identify the proving actions and interactions that occurred during whole-class proving and how Ms. Finley and her students contributed to those processes. By considering the authority over proof initiation, proof construction, and proof validation, the episodes illustrate how whole-class proving interactions might relate to students’ potential development (or maintenance) of authoritative proof schemes. In particular, the authority of the teacher and textbook limited students’ opportunities to engage collectively in proving and sometimes allowed invalid arguments to be accepted in the public discourse. We offer suggestions for research and practice with respect to authority and proof instruction.