Investigating teachers and prospective teachers’ pedagogical content knowledge in the field of proportional reasoning with a focus on their problem solving activities on semantic types

Proportional reasoning is regarded as the gate of functional thinking and is a potent means for developing algebraic thinking which necessitates students’ understanding of the functional relationship between measure spaces. The ability to recognize and identify the structural similarity and multiple comparisons in the proportional reasoning process is the base of algebra and advanced mathematics. The concept of proportion and the necessities of developing proportional thinking are cognitively complex and its teaching demands concept-oriented approaches. Studing the quality of teachers' perceptions draws the perspective of the method and development of conceptual structures among the students. The present study focused on determining the extent of recognizing non-proportional situations and also the kind of selected strategies to solve proportional verbal issues in the teaching activity. Considering the importance of the context of this problem, the study focused on four semantic types of the problems in this field. Considering the pedagogical thinking of teachers in solving proportional problems provides the discussion on the obstacles of using the proportional reasoning among different semantic types .

The study was done by descriptive survey method. The statistical population included 180 teachers of primary schools and mathematics teachers of the first level of the secondary schools, and prospective teachers who participated in the study voluntarily. The research instrument was a researecher-developed test containing 17 problems comprised of 3 nonproportional situations of additive problem types and 14 direct proportional problems, presented in the missing-value type which were either researcher-devised or selected from reliable research sources. The content validity of the test was confirmed by professors in the field of mathematics and testing and psychometrics. The collected data were analyzed using inferential and descriptive statistics.

The results of the first study revealed that the primary school teachers and the prospective teachers were faced with some difficulties in recognizing non-proportional statements. It seems that the superficial characteristics of verbal problem including having a structure similar to the proportional problems of the type of missing value and also the multiplicative nature of numerical structure are involved in determining the situation as a proportional structure. In studying the the strategies of solving the proportional problems in the teaching activity, the responses of the participants were analyzed using the descrtiptive method based on 9 types of problem-solving strategies. The results of the analyses showed that all of the first levele of the secondary school teachers and the prospective teachers of both of these levels, at least in one of their first two priorities in teaching these problems, applied algorithmic proportional strategies or algebraic equation formulation while being slightly influenced by semantic types. Teachers of the primary schools had little desire to use the algorithmic proportional strategies. On the contrary, as compared to other teachers, they had a higher preference for using functional and numerical proportional reasoning. However, they did not prefer to use proportional functional reasoning in their activities. On the other hand, the first two priorities of the primary school teachers were not included in any semantic types, utilization of more complex proportional reasonings, and scale factor.

The results emphasize the necessity of the development of pedagogical content knowledge in this field in order to develop the application of the strategies of functional proportional reasoning and appropriate representations by teachers which are aimed at providing more desirable conditions for students’ proportional reasoning development. Unexpected behaviors of prospective teachers in this study emphasize creating higher sensitivity to the consequences of delaying the emergence of students’ relative thinking in the instructional plans of teacher training courses.