Creating opportunities for problem solving in mathematics classes as a teaching practice related to anticipating students' solutions

their own, is still not a common school practice. The aim of this dissertation is to propose a way for encouraging teachers to adopt the practice of integrating activities for students' APS in their mathematics classes - as part of teachers' professional development (PD) based on strengthening teachers' relevant knowledge for teaching problem solving – through:

• Conceptualization of the concept of readiness, meaning a tendency to adopt the practice based on relevant knowledge.
• Conceptualization of the concept of thick anticipating, meaning a detailed and extensive anticipation of student's expected solution to a given task, prior to the lesson. This is because the classification of a task as a problem is dependent on the solver, hence teachers' mathematical knowledge for teaching problem solving is focused on knowledge about students as problem solvers and in particular on strengthening the skill of anticipating students' solutions in advance.

 

Accordingly, the research questions are:

1. How can teachers' readiness to create opportunities for students' APS in math classes be evaluated?
2. What are the characteristics of teachers' thick anticipating for their students' expected solutions to given problems?
3. What are the anticipation patterns of teachers with different readiness levels regarding integrating APS in lessons?
4. How does teachers' readiness to create opportunities for students' APS in math classes, develop during a PD based on enriching teachers' knowledge for problem solving teaching with emphasis on anticipating students' solutions?

 

The research was spread over four research activities: two separate rounds of teacher interviews, annual PD, and a teacher survey.

The first activity was an interview about problem solving instruction, with six reputable teachers. The interview was structured according to the Hypothetical Learning Trajectory model, HLT, proposed by Simon (1995). Based on the concept that teachers' beliefs and practices involve and influence each other (Kagan, 1992), we choose from the data collected in this activity expressions about problem solving instruction, both beliefs and reported teaching practices. These expressions were converted into statements on a Likert scale, and from them a 20-item questionnaire was compiled to rate teachers' readiness to integrate students' APS in their lessons. The questionnaire was distributed to teachers, and Cronbach's alpha of 0.77 was found for all items (n=60). The tool was also validated as part of the PD, according participants' statements in the first meetings of the PD in parallel to their filling out of the questionnaire.

Another activity of the research was an annual PD in which 12 mathematics teachers with diverse background data participated. The PD outline was designed based on a fusion of two models, the model for challenging teachers' beliefs, values, and practices in a PD framework, proposed by Swan (2011); and the model for adopting new teaching practices by teachers, proposed by Barker et al. (2015) following TDI, Diffusion Theory of Innovation (Rogers, 2003). Along the PD, emphasis was placed on experiencing the anticipating of students' solutions in various activities. The experimentation stage of the PD was first carried out as a mental exercise and only after the thought experiment were the teachers required to try it in the classroom reality. Analyzing the explicit statements of the participants regarding the practice of integrating problem solving in the lessons, led to a refinement of the model proposed by Barker et al. (2015). The findings of the present study suggest five stages in the adopting path of a new teaching practice by teachers: opposition, skepticism, inclination, readiness, and intercession. Shedding the light on the important place occupied by readiness and its role as a link between the stages of awareness and experimentation and between the stage of routinization, may help the non-trivial transition between the PD and its application as a daily teaching activity. During the PD year, the changes in teachers' readiness to integrate problem solving in their classrooms were explored both with qualitative and quantitative tools. Among all PD participants a clear progress in their readiness level was found.

The last research activity was an interview about anticipating students' solutions. The interview protocol was designed as an extension to the use of the virtual monologue tool (Ejersbo, 2005). Ten teachers with diverse backgrounds voluntarily participated in the interview, during which the they anticipated solutions for three selected tasks with different characteristics. Inspired by the term thick description from the field of qualitative data analysis (Ponterotto, 2006), we coined the concept of thick anticipating, for a detailed and extensive prediction given by the teacher for a student's expected solution to a given task, prior to the lesson. Thick anticipating characteristics found for a single scenario are: reference to a specific student, a detailed description of a multi-step mathematical process, reference to "non-mathematical" moves in pursuit of a solution, such as organizational and communicational moves, reference to the voice of the solver, his thoughts and considerations, and first-person quotes of solver's statements or thoughts. In addition, the skill of thick anticipating is expressed in a wide variety of scenarios for the same task.

According to these criteria, the findings emerging from the analysis of all the collected anticipations indicate that teachers tend to make thicker anticipating for tasks in a situation they are familiar with, i.e. a task that belongs to materials they are used to teach; in comparison with their anticipating for tasks in an unfamiliar situation. Among teachers with a high level of readiness to integrate students' APS, more characteristics of thick anticipating were found in comparison to the teachers with a low readiness level.

 

The methodological contribution of the study is expressed in the two developed research tools, the questionnaire for rating readiness and the criteria for characterizing thick anticipating. These research tools may be used by teacher educators as well as by mathematics education researchers, who wish to follow the development of teachers and to design adapted PD activities. The main theoretical contribution of the research is expressed in shedding the light on the stage of readiness as a connecting link between teachers' PD frameworks and their current teaching practices. In addition, pointing out a possible connection between the two mathematics teaching practices, the practice of integrating problem solving and the practice of anticipating solutions, strengthens the hope that strengthening teachers' anticipating skill will increase their readiness to integrate problem solving in their lessons, which is, as mentioned, highly recommended mathematics teaching practice that is still challenging for teachers (Chapman, 2016).