Ph.D Student | Palatnik Alik |
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Subject | Middle-School Students Learning through Long-Term Mathematical Research Projects |

Department | Department of Education in Science and Technology |

Supervisor | Professor Boris Koichu |

Full Thesis text |

This work contributes to the growing body of research on Project-Based Learning of Mathematics (PBLM). It addresses a broad need to explore PBLM in specific contexts and situations, and particularly the analytical challenge of project diversity.

The study had two goals. The first was to explore the variety of the students’ mathematical projects and to identify sources of possible similarities and differences among them. This goal stemmed from the need to know how to plan and orchestrate students’ projects. The second goal was to characterize the phenomena pertinent to PBLM, including the appearance of insight solutions to mathematical problems, sense-making and student inventions that are perceived by them as creative.

To achieve these goals, the Open-Ended Mathematical Problems (OEMP)
initiative was devised and implemented in one of the Israeli schools for five
consecutive years. The researcher was the instructor. OEMP was designed as a
long-term opportunity for developing 9^{th} grade students’ algebraic
reasoning in the context of exploring numerical sequences. While participating
in OEMP, students choose an initial mathematical task from several suggested
tasks, engage with it, formulate a follow-up task, engage with it too, and
finally report the findings of their two-month-long exploration.

A qualitative multiple-case study approach was used. The data on 23 students’ projects, in which 45 students took part, were collected from the students' drafts, audiotaped meetings of the students with the instructor and follow-up interviews.

The main findings are as follows:

• Similarities and differences among the projects were identified in relation to landmark concepts and obstacles.

• A typology of the projects was constructed based on the following characteristics: types of mathematical tasks in the project, extent of scaffolding, and foci of exploration. Four (idealized) types of projects were proposed: Chain of Problems, Delving into a Proof or Solution, Guided Funneling, and Web-Based Research. The actual student projects were characterized in relation to the suggested idealized types.

• A case of the appearance of an insight solution to a challenging problem and a case of the invention of a particularly creative mathematical product were documented. The observed phenomena were explained by formulating grounded suggestions about the students’ shifts and structures of attention.

• The relationship between problem solving and sense making was explored, and a four-component conceptualization of algebraic sense making was proposed. The algebraic object can "make sense" to students when they sense how to justify it, sense its generality, sense a mechanism behind it, and sense how it comes to cohere with additional objects.

The contribution of the study is two-fold. First, the study highlights the dynamic nature of project tasks and the reciprocal re-focusing and re-structuring of students’ and instructor’s attention to particular aspects of these tasks. Second, a set of design principles for the future implementation of PRLM is suggested. Namely, the principles of unity, predictability, dynamics, attention and teacher enrichment are formulated.