### Understanding Pre-service Teachers Mathematical Concepts Representation for Problem Posing

#### Abstract

The purpose of the study is to apply the structure of problem posing as an instructional scaffold in posing problems. The conditions and explorations used by the pre-service teachers during the process of posing reflects how they manage previously learned concepts in mathematics. Results showed how participants posed textbook types of problems, utilized implicit assumptions, compare unrelated concepts, and insufficient information about the problem goal or the sentence structure. The implementation of the structured scheme using Pythagorean Theorem revealed that the instructional scaffold only serves as a “map” rather than a reflective guide so that posing of problems can be systematic and meaningful. Compartmentalization of concepts was a dominant conditions and actions used during the exploration and mainly focused on processing within the same area of representation. Integration of different concepts used during the manipulations of the Pythagorean Theorem reflects a lack of interrelatedness among the elements within the larger structure. Thus, pre-service teachers concerns mostly on how to complete the scheme. The algebraic and geometric representations of Pythagorean Theorem were treated independently rather than treating how the elements interrelates to enable them to function together. Hence, challenging a new concept to replace the existing concept was not successful. Thus, the structured scheme of problem posing suggests an important pedagogical role in understanding the mathematical knowledge of the pre-service teachers.

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