Paul's goal was to have Joel see that he must first remove the last number, in this case the 12, before taking away the others. Joel was not able to make this transition initially, and Paul realized that the boy needed more practice in the strategy that he understood. Understanding Students' Thinking From his reading of research, Paul knew that direct modelers need experience in making symbolic representations in order to develop the generalizations necessary to progress from using counting strategies to using derived facts. A counter might continue to use fingers or objects even though he or she realizes that the first set of numbers doesn't need to be modeled. This is an important flexibility in thinking that direct modelers don't have since they must model each number and act out the sequence of the problem.
Flexibility is crucial for learning facts with understanding.
Facts should be learned through problem-solving experiences rather than by rote. Children will solve problems at their own level of understanding. As teachers, we are responsible for posing questions that challenge our students' thinking and expose them to new ideas or solution strategies. Teachers' understanding of students' thinking is the key to mathematics teaching and should be the driving force in choices regarding curriculum and pedagogy. Paul made strong use of research, student interviews, and observations to build his understanding of Jackie and Joel.