| This study investigated how urban, seventh-grade students from a low-income, urban community build ideas about linear functions and engaged in early algebra mathematical reasoning in the context of an after-school program while working individually or in pairs on Guess My Rule problems based on Robert B. Davis' work on early algebra, 'Discovery in Mathematics' (1967). The study examined the kinds of representations these students used to build and communicate their ideas and how they made use of a particular technological tool (Casio ClassPad 300 computer simulation) in building these ideas (Gjone and Anderson, 2003).;A particular interest was to examine how urban middle school students built ideas with meaning relating to linear functions and to follow the representations they used to build and communicate these ideas. The study is a qualitative case study of three of the participants and focuses on video data and students' inscriptions from an after-school program, Informal Mathematics Learning (IML) in Plainfield, New Jersey. The algebraic ideas investigated by the students included variable, truth values for statements, open sentences, truth sets for open sentences, negative numbers, and functions. The video data were analyzed by following the data treatment procedures and the analytical model proposed by Powell, Francisco, and Maher (2003).;The study reveals that the three focus students built powerful ideas related to variables, noticing patterns, recognizing isomorphic relationship between problems, finding finite differences, plotting points and making sense of the functional linear relationship between variables that enabled them to successfully solve the three assessment problems given at the end of the intervention. One of the students expressed solutions in the form of multiple rules that were meaningful to him. All the three focus students made conjectures and used evidence to test these conjectures. In the quest of building ideas with meaning and modifying the conjectures, one of the students attended to and built advanced mathematical ideas relating to piecewise functions and inverse functions. Another student made a judicious use of the Casio ClassPad that enhanced his learning and sense making capabilities. The findings of the study suggest that middle school students are capable of building early algebra ideas with meaning. Making sense was important to these students and the pursuit of sense making might help explain their success in building solutions that were relevant and meaningful to them.;Further, the three focus students used a variety of representations, including those available on the Casio ClassPad, to build and communicate their ideas. They moved back and forth among representations in building their ideas.;The work done by the students reinforces the effectiveness of Davis' approach of giving the opportunity to young students to build ideas with meaning with the help of open-ended problem solving activities. |
