that she had constructed for herself, requiring her instead to engage in a teacher's path requiring her trust that it lead her to the answers she wanted.
Brother Joe recognized that her independence in thinking about the mathematical ideas was intertwined with creating her rainbow in her own way, so he pushed her to think about the ideas in the context of making her rainbow. His choice is consistent with his teaching practice, reflecting the values system of the school, rooted in the principles of Kwanza, by choosing to respect her determination to solve the problem in her own way. His choice to work collaboratively with her to stay engaged in figuring out her solution reflects the value of collective work that is also prevalent in both the values system and the practices of the school. Consistent with constructionist theory, her initiative to think about mathematical ideas that were sophisticated for her was directly connected to her project of making a rainbow. Logo acted as a tool for her use to construct her rainbow and create mental constructions that grew out of making her rainbow. The ownership of her strategies was closely connected to her engagement with the ideas.
Logo Writer acted as both a palette upon which Shamia's idea to create a rainbow emerged and a drawing tool that required her to think about several mathematical ideas in order to create her rainbow. Her ideas about the relationships between the numbers for the turns and repetitions was elicited by her need to direct the turtle to draw with repeat statements. Logo acted as a tool for her knowledge construction because it was the context enabling her to generate her ideas, visualize her problems, and investigate solutions. The ownership of her programming project became a good backdrop for the interactions she had with Brother Joe so that they could discuss changes to make, why to make them, and ways to test them. It seemed as though doing this project on the computer helped her to assert her confidence in the way she wanted to think about the problem and find solutions.
This story is a glimpse into the life of a child whose developing ideas are meaningful to her and who has taken ownership of the process of understanding and applying complex mathematical ideas. Shamia worked on figuring out a way to understand the relationship between the numbers for the turning angle and the numbers for repetition because this understanding was going to help her make her rainbow. The ways that she asserted herself in calculating the statements for the arcs in her rainbow seemed similar to the kind of ownership a child can take in drawing a picture on paper. Her actions were like those of a young child taking