Creating a culture of (in)dependence.
In this study, the interactions in one middle-school mathematics classroom are examined for potential sharing of mathematical authority that takes place via a teacher's uses of authoritative discourse. The discourse of several classes is analyzed to investigate an alignment between the teacher's intentions and practices in the context of creating a sense of independence in students. Results indicate that by not allowing students much stake in judgments of mathematical validity, teachers may unwittingly create a classroom culture of students' dependence on "other" mathematical authorities.
This discussion begins with a brief narrative that situates the researcher in the mathematics and education community. The author's story is intended to provide readers with a notion of the potentially common struggles to align his pedagogical intentions with his practices. The author's practice began as a secondary mathematics teacher intent on bringing learning experiences to students in his classroom that were more beneficial than his own. After a short time in the classroom, the author's intentions to share mathematical authority and develop mathematical agency became moot for various reasons. He resorted to the controlling methods of teaching and knowing that he experienced as a student. He found it difficult to separate his authority over the events in the classroom and his authority as an "expert" in the field of high-school mathematics. Teaching from the mathematical pulpit became less demanding and more comfortable. On many occasions, he would encourage students to be creative with their solutions by modeling multiple ways to solve problems and encouraging multiple solution methods for a given problem. The author hoped by modeling multiple solution methods students would emulate his behavior when they were working individually or cooperatively. The author found, instead, he was trying to compel a way of thinking by forcing certain behaviors using his classroom authority to exert mathematical authority.
The author often made attempts to force, or allow, students to verify the mathematical validity of statements and solutions in class by responding to their oft asked question of "is this right?" with questions such as "what do you think?" These questions often frustrated students rather than inspired them to develop argumentative skills and share in the authority to justify and verify mathematical validity, as was intended. Students would often say that the teacher should know the answer, and, most importantly, tell them. With increased frequency, the answers were eventually revealed and judgments were made by the teacher. The author believes some reflection would have shown him that his practices of enabling students mathematically did not mirror his intentions of empowering them.
About midway through his first year of teaching, the author made the decision to leave the classroom and continue graduate studies as a fulltime doctoral student. His frustration with the inconsistency between his beliefs about the learning of mathematics and his teaching practices steered him toward the line of research in this investigation. The context of authoritative discourse and its effect on student's mathematical agency is one of many in which the author found his practices not reflecting his beliefs or intentions. As a researcher, these shortcomings as a well-intentioned teacher provide him with a critical lens by which to view teaching practices with the hope that other teachers can, through reflection, realign their practices and intentions.
Theory and Literature Review
The purpose of this study was to examine teacher's discursive practices in mathematics classrooms and offer a mechanism for teacher reflection. Specifically, the research question for this investigation was, "How do teachers' discursive practices align with their intentions of creating a sense of mathematical independence among students?"
Some reform efforts in mathematics education (e.g., National Council of Teachers of Mathematics, 2000) have suggested that an approach to classroom discourse that situates the teacher as an orchestrator of student interactions may give students a more active role in explaining and learning mathematics (Forman & Ansell, 2001). One of the most commonly used questioning frameworks is the Initiation, Response, and Evaluation ORE) sequence, where the teacher initiates the sequence with a question, a student or students respond(s), and then the teacher evaluates the response (Mehan, 1979). Edwards and Mercer (1987) found an alternative to the sequence where the teacher's evaluation of the student's response was replaced with or added to by an expansion, by the teacher, of the student's response. The differences between the types of interactional sequences are profound. The former reduces students' knowledge and understanding to being able to respond to their teacher's questions (Edwards & Mercer, 1987) and reduces student knowledge to bits of information. Conversely, the latter IRE sequence creates a learning community where knowledge "is constructed by and with those involved in the learning activities, by teachers and children alike" (Buzzelli, 1996, p. 525). The type of IRE sequence used by a teacher is therefore very important and should reflect the teacher's goals and beliefs about teaching and learning. The IRE sequence and its derivatives have been prominent in many studies on mathematics classroom discourse (e.g., Buzzelli, 1996; Nardi & Steward, 2003).
Classroom discourse tends to lose importance for students if the primary intention of teachers is to transmit knowledge of mathematics (Arlo & Skovsmose, 1998). For a teacher to establish classroom communication patterns whereby students share in mathematical authority, she should go beyond just asking questions (Carpenter & Lehrer, 1999). Rather, teachers' reactions to students' answers to questions go further to encourage or discourage future classroom communication than simply asking effective questions (Hamm & Perry, 2002). Hamm and Perry offer an example of an elementary school teacher that seemed to be developing the mathematical authority of students by actively engaging her students only to conclude the day's lesson by thanking them for being a "good audience" (p.136), a comment that overtly positioned the students as onlookers to the authoritative teacher.
In a comparison of the teaching of mathematics between elementary level classrooms in Japan and America, Stigler, Fernandez, and Yoshida (1996) note Japanese teachers' reluctance to solely base evaluation and justification of mathematics on their own of an outside authority, such as the text of the domain of mathematics. In Japanese classrooms, much of mathematical judgment was deferred to the authority of the "community" of students based on students' mathematical justifications with the teacher in a guiding role. In contrast, Stigler et al. found the American teachers they observed to frequently rely on various mathematical authorities including themselves and an ambiguous "they."
The research reported here stems from a doctoral dissertation which was a case study of a middle grades mathematics teacher, Ms. M, involved in the Middle School Mathematics Project at Texas A&M University . The class was a Pre-AP eighth-grade mathematics class in a rural, highly diverse, low socioeconomic status school district. The class was composed of 17 female and 11 male students. There were 14 White, 11 Latino, and 2 Black students in the class. The teacher had 14 years experience and was in her third year at this school. The teacher was chosen for the study because of her highly interactive style of teaching as witnessed from extant video data and previous classroom observations and discussions. Additionally, the teacher was a non-traditional, reform-oriented teacher using a variety of teaching methods including instructional technology and collaborative group work.
Curricular materials used during the time of observation by the teacher, interviews with the teacher, observation field notes, and the results of a critical discourse analysis of classroom interactions comprised the data collected in this investigation. The teacher was interviewed based on emergent themes from the classroom observations. The field notes and selected video recordings were used as "stimulated recall" to reconstruct classroom procedures and investigate issues related to the research question (Morine-Dershimer & Tenenberg, 198 I): How do teachers' discursive practices align with their intentions of creating a sense of mathematical independence among students? To help to re-create the contexts of the classroom and serve to inform the overall analysis, the interviews were unstructured, using questions such as "what were you trying to accomplish here?" or "what did you mean by this?" with accompanying documentation of the classroom referents. To gain an understanding of the daily interactions and the construction of the sociomathematical norms in the teacher's classroom, 20 daily lessons were observed involving one class of students over two months resulting in the videotaping of 10 classes over several weeks. The six most appropriate of the classes were transcribed and analyzed according to several discursive strategies shown to be authoritative in the literature as well as several others that emerged, including the use of sarcasm, questions that appear to transfer mathematical authority, and various non-literal uses of personal pronouns, e.g., we, us, and our. The research reported here was part of a larger study looking at classroom discourse practices with respect to mathematical authority and agency. For the purposes of this study, the primary concentration was on the teacher's statements of mathematical validation or judgment. For the sake of confidentiality, all names in the following examples are pseudonyms.
One of the most prevalent themes to emerge from the analysis of Ms. M's discursive practices was the use of statements that verified the validity of students' mathematical efforts, either in content or process. This theme is closely tied to Ms. M's stated intentions of developing students' sense of responsibility in and ownership of mathematics. In all but a few occurrences, Ms. M was the one that students came to for validation of their mathematical methods and results. In what follows, representative interactions ate provided to demonstrate the ways that students relied on Ms. M for validation and the ways that Ms. M encouraged this reliance through her discursive choices. Of all of the interactions Ms. M had with either individual students or with students in whole-class discussions specifically involving mathematical answers or processes, 50 percent of them were statements of mathematical validation or judgment. Each of these statements either verified the student's answer or mathematical process without requiring students to justify their answer or solution method.
This first example of an interaction where a student asks for verification and Ms. M obliges while students have just begun to work in groups. As was typical of the classes observed, when students worked in groups, Ms. M spent much of her time going from group to group verifying the procedures and answers of each group. In this example,
Ms. M is looking over homework papers when a student asks her a question, from across the room.
S: Isn't one out of four, twenty-five hundredths.
Ms. M: Say it again. Is it what?
S: Isn't one over four, twenty-five hundredths.
Ms. M: mm hmm.
Rather than having students confirm the answer to this question for themselves, Ms. M validates the answer perpetuating their reliance on her as the one with the answers. Ms. M continues this exchange by asking questions whose answers would lead the student to the answer of the original problem, but, by then, the student already had her answer validated. Alternative responses to the student's question, and ones that could have served to transfer mathematical authority to the student by turning the question back on the student, are "What do you think?," "Why would you think that?," or "How could you check that for yourself?" The next interaction is another example of the ways Ms. M exerts mathematical authority in her classroom.
S: Can we round? Like I have 24 and one hundredths. Is it supposed to equal 24?
Ms. M: Yes.
Ms. M: I would do an approximation. I would do it the, um. What did you, what is it? Is it, is that it? Oh, alright yeah. Okay.
In this example, notice that validation came in the form of granting permission and a suggestion. Ms. M confirmed that rounding was "okay" in this case. In other specific interactions involving rounding, Ms. M made it clear to students that they had to "get permission" to round, as well. These interactions regarding rounding show how Ms. M withholds mathematical authority from students. Of all the interactions where Ms. M validated or confirmed students' answers or processes, almost half, 49 percent involved some form of revoicing. In these evaluations, Ms. M revoiced students' answers by either repeating the student's response verbatim or rewording the student's answer. In each of these interactions, an evaluation was either implicit or explicit in Ms. M's response. In instances where judgments of validity were implicit, Ms. M either continued her line of questioning or asked her question again showing the value of her judgment. The last example shown below is a series of interactions characteristic of Ms. M's judgments by way of revoicing.
Ms. M: We're multiplying. So I need to figure out what do need to do to three. What do I need to multiply it by to get to four and a hall?. S: A half.
Ms. M: A half?, Is three times a half, four and a half?
Ms. M: No. What is half of three?
S: One and five tenths
Ms. M: One and five tenths. So I need to multiply but I need that one and five tenths but what do I also need? The three I started with, right? So I need to multiply by, I need the three I started with. What do I multiply my three by to get the three I started with? Three times what is three?
Ms. M: So I need to multiply it by one to keep the three and:: I need a half again more. Yes?
In this interaction, Ms. M implicitly evaluates the student's response, "a half," by echoing the response and then rewording the question. Ms. M's remaining three statements in this example are also implicit validation of students' answers. By revoicing the answers and moving on with the line of questioning, Ms. M implicitly validates the students' answers. In isolation, each of Ms. M's responses in the examples shown above could be justified and explained as an efficient way of delivering instruction to a classroom of students. However, these examples characterize a pattern found through an examination all of Ms. M's interactions with her students. Because students were not accustomed to having to validate their own work mathematically and appeared to have accepted the classroom norms, they were habituated to a reliance on the teacher for endorsement of most of their mathematical endeavors, whether during or after each process.
By directing a majority of students' mathematical actions and ways of thinking, Ms. M created a sense of dependence for mathematical validation. This dependence was strengthened through Ms. M's use of confirmatory statements. Because she established herself as primary mathematical authority, students looked to Ms. M as having "the answer." The result of a classroom where various forms of authoritative discourse was the primary means of communication for the teacher was the establishment and support of a classroom culture where students' mathematical actions and knowledge were directed by an authority besides themselves.
Authoritative discourse practices of teachers, such as directive and definitive statements, can inhibit students' independent thoughts and actions, unless these statements take a form such as "you must think for yourself," "you are going to argue that your own answers are mathematically valid," or "it is always the case that the explanations of your results are more important than the actual results." These potentially productive directive and definitive statements, which would have been aligned with Ms. M's stated intentions, did not appear in her discourse practices in the classroom.
Utilizing particular authoritative discourse practices can result in a development of students' dependency on a higher authority, which may or may not be the teacher, to define what is mathematically correct and acceptable. This is consistent with Prevost's (1996) and others' assertion that students should rely less on the authority of their teachers to tell them if they have the correct answer or if they are solving the problem in the correct way. This dependency can preclude students from sharing in mathematical authority and result in the underdevelopment of students' mathematical agency. Without learning habits of argumentation and proof to support their ideas, students may end up dependent on someone or something for mathematical validation. How do students know if they are right or if they are thinking correctly without first checking the back of the book or taking their ideas to their teacher? If independent mathematical thinking by students is a goal for mathematics teachers, then discursive choices that include a large quantity of authoritative discourse may be counterproductive.
For teachers who wish to share mathematical authority with students and to develop a culture of mathematical independence, taking a less pronounced authoritative role, mathematically, could better align their intentions and their practices. Richards (1996) agrees that teachers should realize a role as negotiator between students and the mathematical community or mathematical knowledge teaching students methods of mathematical argumentation and proof. With these skills, students can then begin to exercise mathematical independence. Certainly, teachers have an obligation to make the final decisions on the validity of students' answers, but if their intentions are to foster and support a classroom culture of independence where students share in mathematical authority, teachers may consider reflecting on how their interactions with students indicate the prominence of their mathematical authority.
The purpose of this study was to investigate the ways a teacher's communicative practices, more specifically her uses of authoritative discourse, and her willingness to share mathematical authority with her students and create a sense of mathematical independence in students. The findings showed that although teachers may be willing and intent upon sharing mathematical authority with students, a consistent use of discursive moves that serve to validate students' mathematical efforts can serve to withhold mathematical authority and create a culture where students are dependent on the teacher for validation, contrary to teachers' intentions.
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Prevost, F. J. (1996). A new way of teaching. Journal of Education, 178(1), 49-59.
Richards, J. (1996). Negotiating the negotiation of meaning: Comments on Voigt (1992) and Saxe and Bernudez (1992). In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 69-75). Mahwah, N J: Lawrence Erlbaum Associates, Inc.
Stigler, J. W., Fernandez, C, & Yoshida, M. (1996). Traditions of school mathematics in Japanese and American elementary classrooms. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 149-175). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
 This study was funded in part by the Improving Mathematics and Student Learning through Professional Development grant, Gerald Kulm, Principal Investigator, and the American Association for the Advancement of Science (REC-0129398). The author would like to graciously acknowledge the editorial contributions of Robert Capraro and Mary Margaret Capraro.
Adam P. Harbaugh, University of North Carolina, Charlotte
Harbauyh, Ph. D., is an Assistant Professor of Middle, Secondary, and K-12 Education.
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|Title Annotation:||mathematics education|
|Author:||Harbaugh, Adam P.|
|Publication:||Academic Exchange Quarterly|
|Date:||Dec 22, 2005|
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