Prospective middle grade mathematics teachers’ knowledge of algebra for teaching

This study examined prospective middle grade mathematics teachers’ knowledge of algebra for teaching with a focus on knowledge for teaching the concept of function. 115 prospective teachers from an interdisciplinary program for mathematics and science middle teacher preparation at a large public university in the USA participated in a survey. It was found that the participants had relatively limited knowledge of algebra for teaching. They also revealed weakness in selecting appropriate perspectives of the concept of function and flexibly using representations of quadratic functions. They made numerous mistakes in solving quadratic or irrational equations and in algebraic manipulation and reasoning. The participants’ weakness in connecting algebraic and graphic representations resulted in their failure to solve quadratic inequalities and to judge the number of roots of quadratic functions. Follow-up interview further revealed the participants’ lack of knowledge in solving problems by integrating algebraic and graphic representations. The implications of these findings for mathematics teacher preparation are discussed.     

Errors associated with simple versus realistic models

This paper addresses the relative errors associated with simple versus realistic (or science-based) models. We take the perspective of trying to predict what the model will predict as we begin to build the model. Any model building process can get the model “wrong” to a greater or lesser extent by making a theoretical mistake in constructing the model. In addition, every model needs data of some sort, whether it be obtained by experiments, surveys or expert judgment, and the data collection process is filled with error sources. This paper suggests a hypothesis that