Paying attention to students' ideas in the digital era

Sergei Abramovich

This paper demonstrates how recognition of a hidden potential of rather involved mathematical explorations in a student's unintentionally far-reaching response to an open-ended question about constructing a visual pattern allows for the development of the so-called TITE problem-solving activities that require concurrent use of computing technology and mathematical reasoning. The paper begins with the presentation of such a response by an elementary teacher candidate and it continues towards revealing the potential of the response as a springboard into the development of various TITE generalization activities with ever increasing conceptual and symbolic complexity. It is argued that whereas one of the goals of moving from particular to general is to assist in understanding special cases, the construction of workable computational algorithms for spreadsheet-supported problem solving and posing is not possible without experience in generalization. The mathematical content of the paper deals with polygonal numbers and their partial sums. Computer programs used are Wolfram Alpha (free interface) and Microsoft Excel spreadsheet.