A program leader has been providing intensive professional development institutes for elementary teachers each summer for quite a few years. Depending on the scope of the institute, the duration varies from one week to three weeks. The following example is of a morning session in a one-week summer institute on whole numbers for a group of 25 teachers. There was a total of about 100 minutes of lecture, and teachers did an assigned activity, supervised by a staff of three. The program leader described the session as follows:
I began by asking the teachers if they knew why the standard algorithms were worth learning. None of the answers was given with a great deal of confidence, and none hit the mark. So I asked if they knew what the standard algorithms had in common. Again nothing came close. So I asked if they ever noticed that all standard algorithms boiled down to computations with a single digit. Slowly some nodded. So I briefly went through all four of them, pointing out how each time we would concentrate on just one digit. For example, the long division algorithm churns out the quotient, one digit at a time, regardless of how big the divisor or dividend may be. So I told them that the great virtue of the standard algorithms is that they reduce all whole number computations to single-digit computations.
As the program leader explained, this is not the only reason why they should learn the algorithms, or why their students should learn them. The real reason comes from the fact that this characteristic of the standard algorithms illustrates a fundamental principle in mathematics: Break up a task into a sequence of simpler tasks if at all possible.
Says the program leader, "Lots of important mathematics has been done using this principle. Therefore, if the teachers can explain to their students the principle behind these great algorithms, students would have a head start in learning real mathematics. Isn't this what mathematics education is all about?"
Elementary and middle school teachers in a three-week summer institute conducted independent inquiry investigations in science and read non-fictional accounts of scientists at work. The inquiry experience was carefully scaffolded for participants. Discussions focused on such topics as the nature of questions that scientists ask and the various approaches used by scientists to conduct empirical work. In addition, teachers were assisted in deciding how to determine what data need to be collected to address their questions, and in learning different analysis techniques. Further scaffolding was provided as teachers began to interpret their data and draw appropriate conclusions. Following the inquiries, teachers presented their work to fellow participants. The teachers were highly engaged in their own inquiries; written reflections in their science journals indicated that they gained a better understanding of how science has particular ways of developing new knowledge.
One observer explained that the focus in this institute was on inquiry as a way of knowing rather than the use of inquiry as an instructional strategy and noted how these two foci are sometimes inaccurately equated. She shared, "the idea of inquiry as a part of the science content...as a way of understanding of what makes science science -- how we know what we know in science -- has been too confused with constructivist learning theories; inquiry is used synonymously with instructional strategies that are aligned with constructivist learning theories. To teach inquiry is to teach science content (inquiry is the content)."