טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentBordo Nadia
SubjectThe Power of Intuition Traps in Teaching and Learning
of Probabilistic Thinking
DepartmentDepartment of Education in Science and Technology
Supervisors Professor Emeritus Uri Leron
Professor Orit Hazzan
Full Thesis text - in Hebrew Full thesis text - Hebrew Version


Abstract


Many studies in cognitive psychology have described the existence of two separate modes - intuitive and analytic - in human thinking, and documented "intuition traps" in which intuition leads respondents to nonnormative solutions in mathematics, logic, and statistics. Researchers in mathematical education, who consider intuition to be a powerful and useful resource, have begun to design ways to bridge the gap between the two modes of thinking and to establish "peaceful coexistence" between them. In this study we design and evaluate such bridging means in the field of conditional ("Bayesian") probability, by integrating recent developments in psychology, mathematics and science education as a theoretical basis. The guiding principle underlying our bridging methods is to help the students achieve a conceptual change by “debugging” their intuition.

The methodology used in this study is design research, in which we have designed bridging tasks and corresponding pedagogical interventions. The research tools for evaluation and exploration were questionnaires, semi-structured one-on-one interviews (with the teacher-researcher interviewing students), student exercises and a researcher's journal. The research population consisted of undergraduate students that completed a basic probability course .

Pedagogical intervention included: (a) written tasks with bridging potential. The tasks used were mathematically equivalent to the original task, but presumably psychologically easier; (b) teacher’s assistance. The teacher (a role fulfilled by the researcher) did not teach how to solve the problems, but assisted students by providing minimal scaffolding activities.

The main findings of the research are:

         “Reduction of randomness”: students experienced difficulty constructing a two-process mental model for probability, and over-simplified by treating only one process as a random process.

         In this study, different answers were observed, with different frequencies, compared to previous studies using identical problems. Furthermore, we offer alternative explanations for some common errors.

         Effect of frequency format: contrary to common claims, the frequency format does not always make Bayesian problems easier to solve, especially when it does not highlight the nested subset relations.

         Extended framing effect: students often treated differently two mathematically equivalent problems, when presented in single event probability format or in frequency format.

         Equivalence insensitivity: students often did not recognize the mathematical equivalence between two consecutive problems presented in different formats.

         In response to the problems indicated above, we propose an approach to the design of probability instruction that focuses on the development of coherent basic principles of randomness and probability, based on the stochastic conception of probability. The study showed that short-term targeted intervention can be effective in developing a conceptual understanding of key concepts in probabilistic thinking and can help students solve Bayesian problems in a way that uses their intuition as a positive resource.

The contribution of the study to mathematical education is the design and development of a theoretical and practical framework for bridging the gap between intuitive and analytical thinking in the context of Bayesian reasoning. Further, the findings that arose in the design process are expected to contribute to an empirically grounded instruction theory for probability and statistics education.