|Ph.D Student||Klineshtern Michal|
|Subject||Problem Posing and Choosing Problems by High School and|
University Mathematics Teachers
|Department||Department of Education in Science and Technology||Supervisors||Professor Emeritus Abraham Berman|
|Professor Boris Koichu|
|Full Thesis text - in Hebrew|
Mathematical problems play a central role in teaching and learning mathematics. Mathematics teachers are engaged in selecting problems for their teaching and for students’ assessment. This study explored the process of choosing and posing mathematical problems by teachers for their teaching. The process of problem choosing was explored as a decision-making process, through the lenses of teachers’ resources, goals and orientations.
At first, a pilot study was conducted in order to spot high-school teachers who claim that they pose their own problems. The data were collected from on-line surveys conducted on 154 teachers and from personal interviews with eight selected teachers. The results show that more than 50% of the survey participants see themselves as “problem posers”, and that they interpret “problem posing” in diverse ways. Generally speaking, “problem posing” means for the teachers: constructing a problem that somehow differs from problems that appear in their available resources, has not been approached by their students, and can be used in order to fulfill teaching needs that otherwise could hardly be fulfilled.
The goal of the main study was to investigate the working patterns of and conciderations for choosing and posing mathematical problems for teaching. Data analysis leaned mainly on semi-structured interviews with nine teachers in three educational settings: high school, pre-academic study and academic study. Additional data were collected from lesson observations accompanied by pre- and post- interviews, narratives initiated by the participants and the researcher’s diary. The data were recorded, transcribed and inductively analyzed.
Four clusters of the teachers’ working patterns for selecting problems were found: Adhering Textbooks; Searching in Variety of written Resources; Posing or Recalling; Human Assistance. Seven considerations for selecting problems were inferred: students’ engagement, conceptual understanding, procedural understanding, connections between and within mathematical issues, a sequence of problems, using technological tools and a challenge for the teacher. In many cases, more than one consideration was involved in problems’ selection.
Similarities were found in the teachers’ assertions about their pedagogical goals and orientations. All the teachers declared that they wanted to evoke discourse and bring students to conceptual understanding. Dissimilarities were found in the teachers’ practices. These dissimilarities can be attributed to the following: differences in educational settings, teachers’ views on students' learning, lesson ownership and the use of time. Particularly, high-school teachers, in many of the cases, stated that they lack the time for lesson preparation, while the university teachers stated that the time limitation during the lessons is driving them to invest time in careful lesson preparation.
The suggested categorization of the teachers’ considerations and working patterns helped us to understand their motives in enriching the textbook “problem space”. It also provided a glimpse into their enacted resources goals and oreintations. The theoretical anchoring of the data, by systematic use of a theoretical framework for decision-making, provides an elaborated picture of how and why mathematics teachers choose problems for their teaching.