Nachlieli,Talli, Herbst, Patricio, and González, Gloriana. “Seeing a Colleague Encourage a Student to Make an Assumption While Proving: What Teachers Put in Play When Casting an Episode of Instruction.” Journal for Research in Mathematics Education 40 (July 2009): 427–459.
Taking our lead from the article’s title, we could say that teachers are like directors, their students the actors, and the classroom their stage. Lessons are the play, and the students and teacher work together to achieve a performance faithful to the script.
Mathematics classrooms can indeed be complex environments. The teacher and students are joint stakeholders in classroom results. Teachers and students alike have claims on mathematical knowledge, both its teaching and its learning. A teacher expects at some point to be able to claim that he or she has taught a particular skill; likewise, the student hopes to claim that skill at some point before the course is over. They do this by interacting, and with the help of tools, they engage in mathematical problem solving.
The major goal of the authors in this research article is to understand what is acceptable for a mathematics teacher to do to move the proving process along and what steps he or she should take to help students understand and solve the problem.
This article reports on how teachers of geometry perceived an episode of instruction presented to them as a case of engaging students in doing a proof.
Who Was Studied
A total of 26 geometry teachers from 19 high schools participated in five focus group sessions dedicated to the topic of engaging students in proving. Each session lasted between one and a half and three hours. A few of the teachers came to more than one session, but most of them participated in only one session. In each of the sessions, the participants watched the same 12-minute edited video that included clips from one geometry lesson, and then they discussed this video with their peers.
The teachers who participated in the investigation perceived the episode in various ways, resulting in 10 different “stories” of the same episode. The participants’ perspectives on the episode can be compared to the classic tale of the four blind men and the elephant. Different participants focused on different elements of the episode—how a proof should be conducted, the mathematical knowledge involved, or the students’ perceptions of their mathematical ability or status—and thus came to different conclusions. Yet, just as in the fable of the blind men, all were viewing the same thing.
The study was conducted under the direction of Patricio Herbst. Parts have been reported in an article by Nachlieli and Herbst with González in the July 2009 issue of the Journal for Research in Mathematics Education (see below for full citation). See also Herbst and Chazan (2003) and Weiss, Herbst, and Chen (2008).
The authors videotaped a high school geometry lesson that involved a deviation in the common practice of working a proof. They then explored the perceptions that the teachers who viewed the video shared in the discussion groups.
The videotaped episode was a lesson preparing students for a midyear exam. The teacher had distributed a sheet showing a parallelogram and asked the students what they could say about its angle bisectors. In response, some students immediately drew the diagonals of the parallelogram and stopped working; others endeavored to draw the angle bisectors and conjectured that they made a rectangle. The teacher asked them to prove that conjecture as a class and drew the T form that the students were accustomed to using to show statements and reasons.
One student volunteered to start, and listed what was given—that the figure was a parallelogram. He continued with the proof, stating that the angle bisectors of opposite angles in the parallelogram are parallel. But at this point the student was unable to justify that statement with a reason. By not providing a reason, the student departed from the norm of doing proofs.
The teacher then also stepped away from the normal proof process, allowing the student to make an assumption that the lines are parallel and continue with the proof, with the understanding that they would justify the assumption later in the proof.
The authors wanted to explore whether other teachers would regard this as a viable choice for a geometry teacher. What else could the teacher have done? What made that move possible for a teacher? What impact did that choice have on the meaning that the teacher could give to the work being done?
By confronting practitioners with episodes of teaching in which some norms had been breached, the authors hoped to understand the underlying reasons for the norms of customary teaching. This study contributes to understanding teachers’ perspectives on what is appropriate for a teacher to do while engaging students in proving.
The study showed that the focus group participants considered the breach of the customary norm of doing proofs by the teacher noteworthy, but they responded to it differently. By focusing on different aspects of the situation, they cast the episode in as many as 10 different stories.
The stories describe a range of resources that teachers of geometry might use to respond to a student’s deviation from the norm in solving. Although some stories emphasized the need to hold the line and thus supply a reason, other stories articulated mathematical or pedagogical reasons to allow the student to make an assumption for now with the understanding that he would return later and fill in the gap.
The different accounts of the episode identified resources for professional practice that teachers could use to negotiate norms of a situation in which they had made a tactical, but possibly problematic, move. The findings presented in this research article can be of use for teacher educators to stimulate teachers’ thinking about alternative courses of action in the face of impasses or dilemmas.
Even though there were 10 different stories, all stories in one way or another revealed that teachers of geometry take for granted that it is normative for each statement to be justified before proceeding to make the subsequent statement.
The authors acknowledge that their investigation elicited and examined stories by practitioners without attention to which individuals espoused which stories. A natural continuation of this research could look for individual differences among teachers—do individuals with different levels of teaching experience or mathematical knowledge find different stories more compelling or prefer different courses of action?
Herbst, P. “Establishing a custom of proving in American school geometry: Evolution of the two-column proof in the early twentieth century.” Educational Studies in Mathematics, 49 (2002b): 283–312.
Herbst, P., & Chazan, D. “Exploring the practical rationality of mathematics teaching through conversations about videotaped episodes: The case of engaging students in proving.” For the Learning of Mathematics, 23 (1) (2003): 2–14.
Weiss, M., Herbst, P., & Chen, C. “Teachers’ perspectives on mathematical proof and the two-column form.” Educational Studies in Mathematics, 70 (2009): 275–293.