This study investigated the ways in which the technological tool, The Geometer's Sketchpad, mediated the understandings that high school Honors Geometry students developed about geometric transformations by focusing on their uses of technological affordances and the ways in which they interpreted technological results in terms of figure and drawing. The researcher identified different purposes for which students used dragging and different purposes for which students used measures. These purposes appeared to be influenced by students' mathematical understandings that were reflected in how they reasoned about the physical representations, the types of abstractions they made, and the reactive or proactive strategies employed.

### Karen F. Hollebrands, AnnaMarie Conner and Ryan C. Smith

Prior research on students' uses of technology in the context of Euclidean geometry has suggested it can be used to support students' development of formal justifications and proofs. This study examined the ways in which students used a dynamic geometry tool, NonEuclid, as they constructed arguments about geometric objects and relationships in hyperbolic geometry. Eight students enrolled in a college geometry course participated in a task-based interview that was focused on examining properties of quadrilaterals in the Poincaré disk model. Toulmin's argumentation model was used to analyze the nature of the arguments students provided when they had access to technology while solving the problems. Three themes related to the structure of students' arguments were identified. These involved the explicitness of warrants provided, uses of technology, and types of tasks.

### Michael J. Lawson and Mohan Chinnappan

Our concern in this study was to examine the relationship between problem-solving performance and the quality of the organization of students' knowledge. We report findings on the extent to which content and connectedness indicators differentiated between groups of high-achieving (HA) and low-achieving (LA) Year 10 students undertaking geometry tasks. The HA students' performance on the indicators of knowledge connectedness showed that, compared with the LA group, they could retrieve more knowledge spontaneously and could activate more links among given knowledge schemas and related information. Connectedness indicators were more influential than content indicators in differentiating the groups on the basis of their success in problem solving. The tasks used in the study provide straightforward ways for teachers to gain information about the organizational quality of students' knowledge.

### Cathy Jacobson and Richard Lehrer

In 4 Grade 2 classrooms, children learned about transformational geometry and symmetry by designing quilts. All 4 teachers participated in professional development focused on understanding children's thinking in arithmetic. Therefore, the teachers elicited student talk as a window for understanding student thinking and adjusting instruction in mathematics to promote the development of understanding and used the same tasks and materials. Two of the 4 teachers participated in additional workshops on students' thinking about space and geometry, and they elicited more sustained and elaborate patterns of classroom conversations about transformational geometry. These differences were mirrored by students' achievement differences that were sustained over time. We attribute these differences in classroom discourse and student achievement to differences in teachers' knowledge about typical milestones and trajectories of children's reasoning about space and geometry.

### M. Katherine Gavin, Tutita M. Casa, Jill L. Adelson and Janine M. Firmender

The primary goal of Project M^{2} was to develop and field–test challenging geometry and measurement units for all K—2 students. This article reports on the achievement results for students in Grade 2 at 12 urban and suburban sites in 4 states using the Iowa Tests of Basic Skills (ITBS) mathematics concepts subtest and an open–response assessment. Hierarchical linear modeling indicated no significant differences between the experimental (n = 193) and comparison group (n = 192) on the ITBS (84% of items focused on number); thus, mathematics concepts were not negatively impacted by this 12–week study of geometry and measurement. Statistically significant differences (p < .001) with a large effect size (d = 0.89) favored the experimental group on the open–response assessment. Thus, the experimental group exhibited a deeper understanding of geometry and measurement concepts as measured by the open–response assessment while still performing as well on a traditional measure covering all mathematics content.

### Patricio G. Herbst

Two questions are asked that concern the work of teaching high school geometry with problems and engaging students in building a reasoned conjecture: What kinds of negotiation are needed in order to engage students in such activity? How do those negotiations impact the mathematical activity in which students participate? A teacher's work is analyzed in two classes with an area problem designed to bring about and prove a conjecture about the relationship between the medians and area of a triangle. The article stresses that to understand the conditions of possibility to teach geometry with problems, questions of epistemological and instructional nature need to be asked—not only whether and how certain ideas can be conceived by students as they work on a problem but also whether and how the kind of activity that will allow such conception can be summoned by customary ways of transacting work for knowledge.

### William F. Burger

The elementary school mathemalics curriculum contains no substitute for the study of informal concepts in geometry. in geometry, children organize and structure their spatial experiences. Also, geometry provides a vehicle for developing mathematical reasoning abilities about visual concepts, for example, through the study of planar shapes. In this article, I shall focus on how reasoning can be developed through the study of two-dimensional shapes, their properties, and the relationships among them. Additional topics, such as tessellations with shapes, motions and symmetry, congruence, similarity, geometric constructions, and measurement, are also highly useful in developing reasoning in geometry. The Bibliography includes many materials that teachers have recommended on these topics.

### Henry P. Manning

* These theorems are given with full details in the “Geometry of Four Dimensions” soon to be published by The Macmillan Company.

### Pauline L. Richards

Finding suitable instructional aids for my fifth-grade geometry class was a problem until I discovered the versatility of Tinkertoys.

### Carol Ann Alspaugh

Kaleidoscopic geometry is an interesting type of mirror geometry that could be utilized to introduce geometrical topics such as regular polygons, coordinates of points in a plane, reflections, and symmetry. Most children enjoy the possible explorations offered by this geometry, which would make it useful to the teacher desiring to develop interest and motivation when introducing new materials.