M.Sc Student | Elkine Yulia |
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Subject | Learning Mathematical Principles through Problem Solving by High Achieving Middle School Students and Pre-Service Mathematics Teachers |

Department | Department of Education in Science and Technology |

Supervisor | Professor Boris Koichu |

Full Thesis text - in Hebrew |

Problem solving is considered by many a major constituent in the process of mathematics learning. Learning through problem solving has been broadly debated by the mathematics education research community over a long period of time. Nevertheless, consensus on the learning processes taking place when solving mathematical problems remains beyond grasp.

This study is based on an in-depth analysis of two classroom activities, in which two mathematical principles, the Dirichlet's principle and the principle of symmetry, were taught through problem solving. The sample of the study consisted of 15 high achieving middle school students and 17 prospective mathematics teachers. The participants worked in small groups using two sets of worksheets of similar structure. The data gathered throughout the study and employed for analysis included videotapes, their transcriptions and the authentic worksheets as filled out by the participants.

The research questions were (1) How do the two groups solve the given problems before and after they have been acquainted with the novel mathematical principle? (2) What are the learning scenarios of the chosen mathematical principles in the two groups? (3) What are the inter-group similarities and the differences of the learning processes of the chosen mathematical principles?

The findings indicate that in cases of a successful learning of the selected principles, the participants went through the following stages of learning: formation of the intellectual need to acquire the new mathematical knowledge, gradual integration of the new ways of understanding in the process of solving challenging mathematical problems, using the acquired knowledge for solving the problems that could not have been solved by the initial knowledge and skills. These findings are generally in line with the theory of learning that underlies the research.

Furthermore, three learning scenarios, in terms of the aforementioned stages, were found. The scenarios are associated with different levels of success: an intended change in ways of understanding, a blend of intended and initial ways of understanding and unintended change in ways of understanding. This subtle classification (as opposed to success/failure dichotomy) sheds new light on the construction processes of new problem-solving schemes based on the ones already possessed by the learner and allows to refine the learning theory that underlies the research. From practical perspective, better understanding of the processes of learning through problem solving allows availing a teacher of recommendations that could potentially improve the effectiveness of teaching.