ing motion, and the students recognized a new context in which the idea applies.

Mr. Hull queried, “It sounds like vectors might be useful for representing force? How would you use them to represent forces?” A student responded, “Well, a longer arrow would represent a bigger force, and the direction of the arrow would represent the direction of the force.”

Mr. Hull waited while the students talked about this representation for a while. He then placed a book on the demonstration table in the front of the room and asked students to use vector arrows to represent the forces on the book, while it remained at rest on the table. He also asked students to pay attention to both the length and direction aspects of the vector representation and to add a label to each force arrow stating what exerts it. While each student drew and labeled his or her own representation of the situation, Mr. Hull walked around the room observing to get some idea of which students were suggesting what forces.

Although there were several variations in the students’ representations, there was one main difference between the representations that he knew would occur. Some students had drawn and labeled an upward force by the table and others had not. From his experience in the workshops run by the local university, he had learned that this difference is evidence that the students have very different conceptions of force. After the students had finished their representations, Mr. Hull drew two books on the front blackboard. On one he drew only a downward arrow. On the other he drew both an upward arrow and a downward arrow. Between the two diagrams he drew a large question mark.

“I noticed one big difference in the diagrams,” he said. “About half the class had an upward force by the table and half did not. That suggests a difference in the ways you are conceptualizing force. Since we are just beginning a unit on force, we’d better resolve this difference. So, why do some of you think we need to include an upward force by the table? And, why do others of you think we should not include an upward force by the table?”

Some students shared their ideas, suggesting that if the table did not exert a force on the book, it would fall. Others said there only needed to be a downward force in order to hold the book to the table. Still others argued that the table could not push or pull anything because it was not alive; it did not have any energy. Mr. Hull recognized that many of the students were thinking that force can be exerted only by active agents, so that passive agents, like tables, cannot exert force.

Mr. Hull asked the students to each pick up a book and hold it in an outstretched hand. He then asked the

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