"Teaching Practices Associated with the Authoritative Proof Scheme
Author(s): Guershon Harel and Jeffrey M. Rabin
Source: Journal for Research in Mathematics Education , Jan., 2010, Vol. 41, No. 1
(Jan., 2010), pp. 14-19
Published by: National Council of Teachers of Mathematics
Stable URL: https://www.jstor.org/stable/40539362
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 A Brief Report
 Teaching Practices Associated With the
 Authoritative Proof Scheme
 Guershon Harel and Jeffrey M. Rabin
 University of California, San Diego
 This Brief Report summarizes case studies of the classroom teaching practices of
 2 algebra teachers. The data consist of videotaped classroom observations during
 1 academic year. The study identifies and characterizes specific teaching practices
 that establish the norm that the teacher is the sole arbiter of mathematical correctness
 in the classroom. The authors suggest that these practices are likely to promote the
 development of the authoritative proof scheme in students. The results can provide a
 basis for future research investigating the prevalence of these teaching practices and
 their impact on student learning, and they can be used as parameters to investigate
 teacher change.
 Key words'. Algebra; High School; Pedagogical Knowledge; Proof; Teacher Knowledge;
 Teaching (role, style, methods); Teaching Practice
 The notion of proof scheme (Harel & Sowder, 1998, 2007) is based on the notion
 of proving, defined as the mental act that a person (or community) employs to
 remove doubts about the truth of an assertion. In this subjective perspective, proof is defined as a particular argument that one produces to ascertain for oneself or to convince others that an assertion is true, and proof scheme is defined as a persistent
 characteristic of the proofs that one produces. This Brief Report concerns the
 proving discourse in the classroom: the nature of the persuasion-ascertainment
 processes involved in student-student and teacher-student interactions. Its focus
 is on teaching practices associated with the authoritative proof scheme (APS), in which proving depends on an authority such as a teacher or a book. Proving, here, is not limited to formal proof situations as in geometry but encompasses students enculturation into sociomathematical norms of how correctness is established in
 mathematics.
 Student difficulty with proof has been well established in the literature (Chazan,
 1993; Hart, 1994; Healy & Hoyles, 1998, 2000; Martin & Harel, 1989; Senk, 1985).
 Some of the findings in these studies were interpreted by Harel and Sowder (2007)
 as an indication of the APS, "in that in the eyes of these students proof must have
 a certain appearance ... as determined by the teacher . . ." (p. 822).
 This work was supported, in part, by a grant from the National Science Foundation (NSF)
 (REC 0310128, G. Harel PI). Opinions expressed herein are those of the authors and
 not necessarily those of the NSF.
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 Guershon Harel and Jeffrey M. Rabin 1 5
 McCrone and Martin (2004) found that what we refer  the schemes held by their subjects, who were ninth-gr found that more than one third of a sample of 106 eigh German Realschule possess the APS. Housman and P students who possessed the APS were unsuccessful in th ation, example usage, and reformulation learning strate One of the areas of research on proof that seems to  connection between teaching actions in the classroom an proof (Herbst, 2002; McCrone & Martin, 2004). Given t among students and its negative effects on different aspe the research previously cited suggests, there is a need t teaching actions that contribute to its formation. This stud addressing this need, in that it aims to answer the questio employed in mathematics classrooms manifest the phen the sole arbiter of mathematical correctness? Althoug teaching practices associated with this phenomenon, the its findings are empirical, not hypothetical, in that th observed in mathematics classrooms. As such, its findin future investigation of the impact of the observed teachin tion and institutionalization. Toward this end, we will p the potential impact of the observed teaching practices on tion of particular student behaviors that manifest the A METHOD
 The data for this study are drawn from classroom observations of two high sc mathematics teachers, Teacher A and Teacher B, who taught at the same sub high school with a predominantly middle-class population. These teachers w part of a larger cohort observed for a broader research project and were sel for the current study because they both taught algebra, had minimal student b ioral issues in their classrooms, and were observed regularly by us. Our data c of transcribed classroom videorecordings of each teacher at approximately 3 intervals during 1 academic year. Following principles of grounded theory (D 1999; Glaser & Strauss, 1967), we first conducted an open-ended analysis in w we attended to all teaching practices observed. As a result, several teaching p tices associated with the APS began to emerge. Through subsequent iter processes of refinement, we isolated specific practices in three catego (a) answering students' questions, (b) responding to students' ideas, a (c) lecturing. These categories might be viewed as part of what Mehan (1979) reproducible patterns of initiation, reply, and evaluation in interactions bet teachers and students. Some of these patterns include what is known ãsfunn (Sherin, 2002; Wood, 1997), in which teachers narrow students' attention to  particular issues. The emergence of the three categories was dictated by a jux sition of two sets of considerations: the nature of the classroom activities obs and the common manifestations among students of the three APS behThis content downloaded from
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 1 6 Authoritative Proof Scheme
 identified by Harel and Sowder (1998). Th and often ask to be told the procedure to  on problems without first making a seriou (c) students are reluctant to ask question presented and the reasoning behind their  OBSERVATIONS AND DISCUSSION
 Answering Students ' Questions
 Typically, the dialogues between Teacher A and his students are brief qu answer exchanges. The dialogues exemplify the teaching practice in whi teacher deals exclusively with procedures regarding how to perform tas students mostly ask for confirmation, and the teacher responds either by con or rejecting the student's suggestion. When the teacher does not directly  or reject a particular response by the student, he responds with a leading  hinting to the student what to do. The exchange that follows narrows the  ties for actions on the part of the student, leading him or her to the answer by the teacher. The teacher's expressed satisfaction indicates to the student task is completed or when its result is correct. In these practices, the teac not attempt to elicit evidence of the students' conceptions, help them reflect they might answer the question themselves, or elicit the thought processes up to their question.
 Teacher B, in contrast, emphasizes that correctness is determined by a p of checking, which is public, objective, and open to debate. When asked w an answer is correct, he often reminds students that he would not know unt performed the same check they could perform themselves. He does not  point out a student's mistake but asks the student to reflect on his or her re When the student corrects his or her error, the teacher still does not conf result but asks another student to do so.
 Responding to Students 'Ideas
 Teacher A provides little space for students to reason independently. He is the
 one who raises a question, and he is the one who answers it. When a student's answer
 is not the one he expects, he rejects it and responds in a way that narrows the
 student's choices. When a student makes the correct choice, he confirms it without
 investigating students' understanding of the answer. When disagreements occur
 among students, he serves as the arbiter, rather than a facilitator of debate among
 students.
 Teacher B responds differently to students' ideas. For example, he makes explicit
 use of the process of checking - not only to determine whether an answer is correct
 or incorrect but also to find the correct answer by analyzing a flawed solution
 attempt. Often he does not merely certify the correct answer but attempts to elicit
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 Guershon Harel and Jeffrey M. Rabin 1 7
 other students' positions and directs them to the meaning  He does, however, make assertions on his own authority w an alternative justification.
 In other cases, we observed Teacher B repeatedly asking a assertion, and the student responding by doing so in acco standing of a particular concept. From the student's stand adequate. Rather than attempting to elicit the student's unde tial consequences, the teacher responded with a justificat understanding. Although correct, this understanding was no The end result was that the student agreed with the teache socially, rather than intellectually, based.
 Lecturing
 We focused on the mode and organization of the discourse and the nature of
 justifications in the discourse. Typically, Teacher A delivers lessons with a
 PowerPoint presentation. Students seldom ask questions, and, when they do, their
 questions primarily seek clarification. Teacher A first states the general rules or
 principles that apply and then illustrates these with examples. Then, he assigns
 students tasks that correspond to the general rules and principles he introduced in
 his presentation. Justification is typically authoritative or based on examples. No
 purpose, intellectual need, or motivation is offered for new material. Instead, the
 teacher often cites relevant curricular standards when introducing a new topic. This
 amounts to an appeal to authority for the usefulness, interest, or importance of new
 ideas. Teacher B, on the other hand, tries to ensure that students understand the goal
 and purpose of the problem at hand, and he waits until students have seen that a
 concept or procedure is useful before introducing terminology for it.
 SUMMARY AND IMPLICATIONS
 In this study, we identified nine teaching practices that we believe are linked the APS:
 TP1 The teacher's answers to students' questions mainly tell how to perform task and whether an action is correct or incorrect.
 TP2 When responding to student ideas, the teacher guides students toward t teacher's preferred solution approaches.
 TP3 The teacher rejects a student's approach to solving a problem if it does n coincide with the teacher's approach.
 TP4 The teacher is the sole arbiter of disagreement between students.
 TP5 The teacher lectures to students in a manner such that interaction amon students is nonexistent and interaction between teacher and students is
 minimal.
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 1 8 Authoritative Proof Scheme
 TP6 The teacher's lectures begin with a illustrate it, and end with tasks simil TP7 The justification of the need for c lectual.
 TP8 The teacher's justifications are mai authoritative are mainly empirical rat TP9 When interacting with students, th conceptual bases for their questions a Teaching practices likely to promote  included:
 TP 10 The teacher responds to students' questions by probing their reasoning or
 asking them to check correctness for themselves.
 TP1 1 The teacher allows students to debate and resolve disagreements.
 TP 12 The teacher offers deductive justifications.
 Using these case studies of Teachers A and B, we have identified teaching prac-
 tices TP1-TP9 that are likely to establish the teacher as the sole arbiter of correct-
 ness in the classroom and lead, as a result, to the APS behaviors outlined in Harel
 and Sowder (1998): (a) Students expect and often ask to be told the procedure to
 solve problems; (b) students ask for help on problems without first making a serious
 effort to solve them on their own; and (c) students are reluctant to ask questions
 about the motivation for the content presented and the reasoning behind their
 teacher's and classmates' assertions. Collectively, TP1-TP9 seem likely to lead to
 the formation or promotion of these APS behaviors. This research lays a foundation
 for future studies examining the prevalence of these practices, hypothesizing their
 effects on student learning, and testing these hypotheses. Further, to complement
 existing results on how teachers change their knowledge, beliefs, and overall class-
 room organization, researchers can look at how their teaching changes in a specific
 category: practices associated with the APS. Similar studies investigating change
 in other categories of teaching practices could help paint a more detailed picture
 of how teachers change what they do in the classroom.
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 Chazan, D. (1993). High school geometry students' justification for their views of empirical evide and mathematical proof. Educational Studies in Mathematics, 24, 359-387.
 Dey, I. (1999). Grounding grounded theory: Guidelines for qualitative inquiry. San Diego: Acad Press.
 Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualit research. Chicago: Aldine.
 Harel, G., & Sowder, L. (1998). Students' proof schemes: Results from exploratory studies. In  Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Issues in mathematics education: Vol. 7. Resea in collegiate mathematics education III (pp. 234-283). Providence, RI: American Mathemati Society.
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 Heinze, A. (2002). ". . . because a square is not a rectangle": Students' k cal concepts when starting to learn proof. In A. D. Cockburn & E. N 26th conference of the International Group for the Psychology of  pp. 81-88). Norwich, UK: University of East Anglia.
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 McCrone, S. M. S., & Martin, T. S. (2004). Assessing high school stu ric proof. Canadian Journal of Science, Mathematics, and Techno Mehan, H. (1979). Learning lessons: Social organization in the classr University Press.
 Senk, S. L. (1985). How well do students write geometry proofs? M Sherin, M. G. (2002). A balancing act: Developing a discourse comm room. Journal of Mathematics Teacher Education, 5, 205-233.
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 Authors
 Guershon Harel, University of California, San Diego, Department of Mathematics, 0112, 9500 Gil-
 man Drive, La Jolla, CA 92093-01 12; <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>
 Jeffrey M. Rabin, University of California, San Diego, Department of Mathematics, 0112, 9500 Gil-
 man Drive, La Jolla, CA 92093-01 12; <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>
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