Adapting the hypothetical learning trajectory notion to secondary preservice teacher training

We adapt the idea of hypothetical learning trajectory (Simon, 1995; Simon & Tzur, 2004) to preservice teacher training. Our approach is based on the notion of capacity, which is used to characterize a concrete teacher´s learning goal. Using links between capacities, we propose some tools that can be used by a teacher to analyze and select tasks and to produce hypotheses about studentsÕ learning processes. We describe possible uses of these tools by future teachers, considering that they will use standard available resources in preservice teacher training: meanings of a concept in school mathematics and students difficulties when facing the tasks. We exemplify this process considering a particular learning goal in a lesson on the quadratic function