In this paper we report on a study of a short-term mathematics program for grade 7-8 gifted students that integrated open-ended mathematics tasks with the arts (poetry and drama) and with technology. The program was offered partially online and partially in a classroom setting. The study sought to investigate (a) students' perceptions of their school-based mathematics experience, (b) students' perceptions of the program we offered, and (c) students' mathematical thinking while engaged in the activities of our program. The study provides insights into the design of challenging mathematics experiences for gifted students, the integration of the arts with mathematics, and the use of technology in mathematics teaching and learning.Gifted students are characterized by their ability to give sustained attention to problem solving; by their propensity to question, experiment, and explore; by their inventive solution strategies; and by their ability to engage their imaginations to generate highly original ideas (Johnsen, 2004). Gifted students are motivated by choice (Gentry, Gable, & Springer, 2000) and prefer collaboration when tasks are sufficiently challenging (Diezmann & Watters, 2001). However, gifted students spend most of their time in regular classrooms with their

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Our work is inspired by the book Imagining Numbers (particularly the square root of minus fifteen), by Harvard University mathematics professor Barry Mazur (Imagining numbers (particularly the square root of minus fifteen), Farrar, Straus and Giroux, New York, 2003). The work of Mazur led us to question whether the features and steps of Mazur's re-enactment of the imaginative work of mathematicians could be appropriated pedagogically in a middle-school setting. Our research objectives were to develop the framework of teaching mathematics as a way of imagining and to explore the pedagogical implications of the framework by engaging in an application of it in middle school setting. Findings from our application of the model suggest that the framework presents a novel and important approach to developing mathematical understanding. The model demonstrates in particular the importance of shared visualizations and problemposing in learning mathematics, as well as imagination as a cognitive space for learning.

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International standardized tests are showing an underperformance of students in geometry and spatial ability relative to other content domains. In this research, we examined the relationship between a teacher's judgment of third-and fourth-grade students' geometry and spatial ability (i.e., grades), three spatial ability tasks (Water-Level-Task, the Rod-and-Frame Test, and the Mental-Rotations-Test), composite spatial ability score, and standardized test scores of geometry and spatial ability.Results showed that this teacher's judgment of a student's geometry and spatial ability was more ambitious than evidence from the other measures. While two of the spatial ability tasks and the composite spatial ability score showed improvement over time, the teacher's judgment of students' geometry, and spatial ability did not show improvement over time. Implications for learning will be discussed. K E Y W O R D Sassessment, geometry, spatial, teacher judgment