The language of engagement in two highly interactive undergraduate mathematics classrooms

Available online at www.sciencedirect.com

Linguistics and Education 21 (2010) 83–100

The language of engagement in two highly interactiveundergraduate mathematics classrooms

Vilma Mesa ∗, Peichin ChangUniversity of Michigan, 3111 SEB, 610 East University, Ann Arbor MI 48109-1259, USA

Abstract

We report an analysis of the language used by two instructors teaching two undergraduate mathematics classes that exhibitedhigh student participation yet differed in their level of dialogical engagement. We focus on the way instructors’ language contributesto opening or closing the opportunities for students’ engagement with mathematical dialog, in turn establishing parameters forstudents’ agency in the classroom discourse. We found ways in which instructors used language to determine different studentengagement. How instructors use their linguistic resources in engaging students may facilitate or forestall dialogic possibilities,which in turn can influence students’ performance.© 2010 Elsevier Inc. All rights reserved.

Keywords: Engagement system; Mathematics; College instruction

Calls for increasing student participation in mathematics classrooms in K-12 settings (e.g., National Council ofTeachers of Mathematics [NCTM], 2000) have also been promoted at the tertiary level (Blair, 2006), specifically formoving from a ‘teacher centered’ paradigm of instruction towards a ‘student centered’ one. In a setting in whichlecturing seems to be the dominant mode of interaction between students and instructors (Lutzer, Rodi, Kirkman, &Maxwell, 2007), increasing student participation seems a difficult task to accomplish. Moreover, as it is has been shownin the K-12 literature, it is not clear that having a high level of student participation is in itself enough to warrant studentengagement with the material nor is it necessarily conducive to authentic learning (Cazden, 1986; Hiebert & Wearne,1993; Voigt, 1985). In this paper we investigate how instructors teaching mathematics to college students invite orsuppress the dialog in the classroom and the implications of such moves on students’ agency. Attending to language isone of many ways that college instructors can change the dynamics of classroom conversations when such need exists.We asked, what characteristics of dialogic engagement are revealed by an analysis of the language used by instructorsteaching mathematics to college students? And, what can be said about the potential of such engagement for creatingopportunities for increasing students’ agency in the undergraduate mathematics classroom?

1. Theoretical and analytical orientation

We start by defining two central notions in our analysis: positioning and agency. Positioning refers to “how peopleuse action and speech to arrange social structures” (Harré & van Lagenhove, 1999 as cited by Wagner, 2008, p. 145).

∗ Corresponding author. Tel.: +1 734 647 0628; fax: +1 734 763 1368.E-mail address: [email protected] (V. Mesa).

0898-5898/$ – see front matter © 2010 Elsevier Inc. All rights reserved.doi:10.1016/j.linged.2010.01.002

dx.doi.org/10.1016/j.linged.2010.01.002

mailto:[email protected]

dx.doi.org/10.1016/j.linged.2010.01.002

84 V. Mesa, P. Chang / Linguistics and Education 21 (2010) 83–100

Positioning is an inevitable function of all utterances. However, “pedagogical activities, such as school mathematics,must logically construct a hierarchy of positions . . . dominant and subaltern positions are constructed via the distributionof practices” (Dowling, 1998, p. 140). Teachers, mainly unconsciously, use language forms to “structure a socialarrangement that resembles the physical arrangement [in] classrooms—students sitting apart from each other beneatha teacher who stands front and center” (Wagner & Herbel-Eisenmann, 2008, p. 145). Such organization is typical ofmathematics classrooms in undergraduate settings, and the analysis of the language used can reveal how utterances inthe dialog can construct particular positions for all participating in the dialog. In the broadest terms, human agency isdefined as the capacity for human beings to make choices and to impose those choices on the world. Although positioningis an inevitable function of all utterances, some discursive moves are more desirable than others in terms of definingwhat students can do (agency) and what mathematics is in undergraduate classrooms (e.g., ‘school mathematics’ or‘doing mathematics’, see Cobb, Wood, Yackel, & McNeal, 1992). With this analysis, we investigate the extent to whichlanguage used by instructors allows students to make choices, that is, the extent to which students are constructed asagents by the language. We look at how those language choices seek to align students to one particular voice (theinstructor’s) and how language limits the options that students might seek or investigate.

Our analysis is not focused on the genesis of mathematical objects in the classroom or on their cognition (e.g.,Bauersfeld, 1988; Rotman, 1988; Sfard, 2000; Steinbring, 1989); it does not trace mathematical meanings of particularwords or expressions used in classrooms (e.g., Moschkovich, 2008; Wagner & Herbel-Eisenmann, 2008), but is relatedto the genesis and sustainability of mathematical practices such as conjecturing and justifying (e.g., Cobb et al., 1992)and presents a linguistic description of mathematical language (e.g., Rowland, 1995) that allows us to make explicit howthe pedagogical discourse in mathematics realizes its dominant interpersonal dimension (Martin, 1993; O’Halloran,1998). Our analysis is embedded in the linguistic tradition and shares with the aforementioned analyses groundingin Bakhtin’s (1981) work, who points out that all utterances exist against a backdrop of multiple voices. At the sametime, we agree that even though the content of speech can both invite and suppress agency, the form of speech, ofrhetorical expression, is also powerful in establishing such agency, as the form is the medium through which contentis indexed (Dowling, 1998; White, 2003). The aphorism ‘our words are never neutral’ (Fiske, 1994) summarizes themain rationale for conducting this analysis and for concentrating our work in mathematics classrooms, which in collegecircles, tend to be perceived as objective and neutral learning and linguistic environments (see Ernest, 1991; Kline,1980, for a critique of this position).

Our work is informed by the tradition of Systemic Functional Linguistics (SFL), which views language as alwaysembedded in its social context and always meaning oriented. SFL systematically describes language as consisting offour strata: phonology, lexicogrammar, semantics, and context. Three metafunctions, ideational, interpersonal, andtextual, penetrate the four strata. Ideational meaning concerns the construction of experience through seeing, hearing,thinking, and so forth. Interpersonal meaning involves the construction of social relations, and textual meaning refers toorganizing the ideational and interpersonal meanings into discourse (Halliday, 1985; Halliday & Hasan, 1985; Sugeno,2008). An important premise of SFL is that language is a system of meaning potential; SFL-based analyses seek toexplore what linguistic choices fulfill what types of meaning.

It is within this framework that Martin and White’s (2005) appraisal system, the key methodological framework thatinforms this study, is situated. Martin and White’s appraisal system analyzes the semantic aspect of the interpersonalmeanings, exploring the attitudes projected by authors or speakers. The appraisal framework has three main subcom-ponents: engagement, attitude, and graduation (Martin & Plum, 1997; Martin & Rose, 2003; Martin & White, 2005).We focus on the engagement system to illustrate how instructors use language to engage the students by way of theinterplay between two major discursive voices, monogloss and heterogloss (including contraction and expansion).

Monogloss seeks less to engage than to give facts that ostensibly concede no room for the negotiation of meaning.For example, to attempt to align an audience to an author’s or speaker’s side, she or he can manipulate assertive devicesto elicit confidence in the statement she or he is making. On the other hand, heterogloss seeks to engage the audienceusing a variety of linguistic resources, to open up or close down options for dialog, each of these conveying a variedstrength of engagement (Martin & White, 2005).

Informed by Bakhtin’s/Voloshinov’s notions of “dialogism and heteroglossia,” the engagement system regards allutterances as dialogic, suggesting that what is said is invariably implicated in a web of references (Martin & White,2005, p. 93). Bakhtin (1981) states that all utterances exist “. . . against a backdrop of other concrete utterances on thesame theme, a background made up of contradictory opinions, points of view and value judgments . . . pregnant withresponses and objections” (p. 281).

V. Mesa, P. Chang / Linguistics and Education 21 (2010) 83–100 85

Fig. 1. Examples of monoglossic and heteroglossic discursive voices.

Based on this notion, engagement analysis investigates “the degree to which speakers/writers acknowledge thisprior speaking . . . whether they present themselves as standing with, as standing against, as undecided, or as neutralwith respect to these other speakers and their value positions” (Martin & White, 2005, p. 93). Martin and White’sengagement framework aims to provide a “systematic account of how such positionings are achieved linguistically”(p. 93, emphasis added).

Generally, monoglossic text often sounds descriptive, report-like, and impersonal. Monogloss is akin to “bareassertions” where no “dialogistic alternatives” need to be recognized (p. 99). It designates an inherent value of taken-for-grantedness and presupposition that allows little room for advancing a counterpoint. As a result, monogloss construespropositions that do not need to be brought into active rhetorical play and are therefore construed as self-evidentlyright and just.

By comparison, heterogloss, as inherently “dialogistic locution,” overtly grounds a proposition “in the contingent,individual subjectivity of the speaker/writer” and thereby recognizes that the proposition is but one among a numberof propositions available (p. 100).

We provide examples from our corpus to illustrate how each discursive voice is realized linguistically (see Fig. 1).Key linguistic items are underlined to highlight the discursive effects each voice seeks to accomplish.

The ten examples in Fig. 1 show differences between monoglossic and heteroglossic rhetoric. In the four examplesof monogloss shown above, no dialogistic alternatives are given. They construe propositions that are self-evidentlyright (“that accounts for . . .,” “So the input here is . . .”). The speakers do not seek to engage the listeners but rather toshow a way of seeing things that is to be followed (“ask yourself . . .,” “what we have to do is . . .”).

By contrast, heterogloss is marked by authorial interpolation and engages the readers at different levels. In Example5 in Fig. 1, in interpolating the authorial subjectivity “I think,” the speaker, instead of construing the proposition asself-evidently right, foregrounds his opinion as confined in his very subjectivity, which can therefore be subjected toreexamination. By grounding the proposition in the personal subjectivity, the speaker acknowledges that the propositionis but one among a number of propositions available. In Example 6, by the countering “but,” the speaker restricts thescope of dialogic possibility in foregrounding his claim. “But” here counters a previous utterance to highlight the


Fig. 2. Engagement system: heteroglossia (adapted from Martin & White, 2005, p. 134).

current statement as more appropriate or reliable. As opposed to the contractive “but,” in Example 4, the conditional“if” renders the statement tentative until that condition is met. It leaves room for further discussions that may lead tomultiple interpretations. In Example 9, the speaker, in uttering “I would be very interested,” expresses his inclination inmodal terms. Contrary to “I am interested” which is a bare assertion, “I would be interested” expresses an inclinationthat would stand true if some other conditions are met.

Heterogloss is composed of complex values. We can choose to heteroglossically contract or expand an argument.The difference lies in “the degree to which an utterance . . . actively makes allowances for dialogically alternativepositions and voices (dialogic expansion), or alternatively, acts to challenge, fend off or restrict the scope of such(dialogic contraction)” (Martin & White, 2005, p. 102; see Fig. 2).

Contraction is directed toward confronting and defeating potential contrary positions in asserting or insisting andso seeks to align readers to the author’s point of view (Koutsantoni, 2004, p. 164). Its two main features are disclaimand proclaim. Disclaim, is used mainly to reject prior utterances or alternative perspectives by denying and countering.Proclaim, on the other hand, is used to overtly announce agreement with the projected dialogic partner by concurring,pronouncing, and endorsing. In proclaiming, the author simultaneously designates other interpretations or perspectivesas less valid, thus contracting the argument to align the readers to his or her side (Martin & White, 2005, pp. 117–127).

Expansion, by contrast, is an authorial voice set to entertain alternatives and possibilities as claims still opento question. Its two main features are “entertain,” and “attribute.” Entertain generally softens an otherwise sub-jective statement by a variety of linguistic resources. Attribute is used to open up discursive or dialogic space byreferencing an external source either in acknowledging or distancing the source. With acknowledge, the speakerstays neutral in introducing multiple perspectives. With distance, the speaker stands farther back withholdingjudgment.

The examples in Fig. 3 illustrate the differences between expansion and contraction and of some of the possibilitiesdescribed. The key linguistic items that set the two apart are underlined.

Blunt denial, “can’t” and the contrasting connective, “but” (Examples 1 and 2), confront or defeat potential contrarypositions. These two forms enable the speaker to designate other interpretations or perspectives as less valid and soseek to align the listener to alternative viewpoints the speaker himself or herself endorses. In proclaiming “I see whatyou are saying” and “you are on the way, yeah” (Examples 3 and 4), the speaker overtly aligns himself or herself with adialogic partner by announcing agreement with him or her. Both disclaiming and proclaiming work toward contractingdialogic possibilities as the speaker makes his or her position clear. By contrast, expansion options enable the speaker


Fig. 3. Examples of heterogloss-contraction and heterogloss-expansion.

to entertain alternatives and introduce divergent viewpoints. In the examples above, “it looks like,” “maybe,” “can,”etc., allow room for further discussions. The conditional “if” in Example 8 and the circumstance “when” in Example9 frame the speaker’s point as contingent thus allowing other dialogic possibilities.

As these examples show, this level of detail of the analysis of the language used helps reveal what the speaker’sstance is regarding the dialog: is the stance one in which the speaker states facts as they are without leaving roomfor negotiation or is it one by which the speaker invites others to engage with his or her position? Does the speakerdisclaim or entertain possibilities? Analysis of how particular wording is used to accomplish those moves is useful inuncovering differences that otherwise might not be recognized among college instructors who seem to be successfulat maintaining high classroom participation.

2. Methods

We analyzed transcripts of two undergraduate mathematics lessons that were collected as part of two differentprojects. The first project sought to establish how faculty teaching college introductory mathematics used their textbooksin teaching. Fifteen faculty from nine different institutions (including two community colleges, one private four-year colleges, two masters’ comprehensive institutions, and four research universities) were interviewed and theirteaching observed; their students answered a short survey regarding their use of textbooks. In this study the classesobserved ranged from remedial mathematics to graduate complex analysis (Mesa & Griffiths, 2007). The second projectwas a three-year evaluation of a program designed to increase minority students’ enrolments in science, technology,engineering, and mathematics majors at an elite university. The evaluation sought to determine the impact that theprogram had on students’ performance (course grades, number of mathematics courses taken, major chosen, degreecompletion). It also included pre- and post-tests of knowledge and attitudes, pre- and post-interviews with facultyteaching the program and with support personnel, classroom observations, and focus groups with the students; inthis, classes all the observations were from the same course (Mesa & Megginson, submitted for publication; Mesa,


Table 1Characteristics of the lessons in the corpus.

Lesson ID No. of student turns No. of teacher turns Type of course Lesson length (min) No. of students’turns per minute

U1 1 2 Capstone 21 0.05U2 12 13 First-year honors 46 0.26U3 12 13 First year 43 0.28U4 13 14 First year 44 0.30U5 18 19 First-year honors 29 0.62U6 27 30 Capstone 59 0.46U7 32 34 First year 65 0.49U8 180 170 First year 56 3.21U9 221 114 First year 100 2.21

Boyle-Heimann, Mosher, Rhea, & Megginson, 2007). The interview protocols were slightly different for each project,but the observation protocols were the same. For this particular analysis, we focused on courses that corresponded toa four-year college program, as the population of students that these courses serve appeared to be more homogeneous:students in remedial courses had a wide range of degree interests, ages, and difficulties with mathematics; the graduatestudents although interested in mathematics had a wider range of countries of origin. These variations can play a rolein how agency is manifested in the classroom that might be outside the instructor’s control. Thus for this preliminaryanalysis, those classes were not considered. Instructors were interviewed before observing their classes about theirgoals for the course, their academic and teaching experience, and the activities they engaged when using their textbook.While observing classes we assumed the role of non-participant observers; we sat in the back of the room and took notesabout all events, avoiding interaction with the students. We audio recorded the lessons, with a small digital recorderconnected to a microphone that captured instructor’s speech and that of students within four feet of the microphone.Instructors alerted the students about our presence; and besides explaining the purpose of our visit, we had minimalinteraction with students or instructors during the lesson. In the first project, we used the last 15 min of every class toadminister the written survey to the students (ways in which they used their textbook for preparing for class, doinghomework, studying for tests, or other activities). After the observations we discussed with the instructor events in theclass and gathered their thoughts about how unusual the class was. These conversations were recorded as well, butwere very informal. In both projects, we strived to obtain a realistic account of events (Creswell, 2005). The purposeof the data collection was determined by the research questions in each project. For this analysis, we are focusing ontwo particular lessons, and performing an analysis that would shed light on teachers’ use of language and its relationwith students’ agency.

In Table 1 we present characteristics of the corpus from where the lessons were selected, sorted by the numberof students’ turns per minute (see last column in Table 1). This rough measure of student participation showed thatthere were two outliers, lessons U8 and U9, in which the students (and the teachers) were talking more per minute,relative to the other lessons in the sample. Common in the other lessons were segments in which the teachers wouldtalk without pause for several minutes (2–10 min), with students asking a question or offering a comment that theinstructor would address without elaborating on or without asking for further student elaboration. Because we wereinterested in investigating engagement in mathematical dialog during instruction, we focused on studying the portionsin lessons U8 and U9 in which students participated in the dialog more actively. This corresponded to the first 24 minin lesson U8; we contrasted the analysis of this segment to a comparable segment in lesson U9.

As indicated, we also had access to interviews with the faculty, regarding the goals they had for the classes, theiracademic background and teaching experience, and how they used textbooks for teaching. The instructor in lesson U8(hereafter Instructor A) was teaching at a master’s comprehensive university; the lesson was one of the three collectedfrom this instructor teaching other courses; the instructor in lesson U9 (hereafter Instructor B) was teaching at a researchuniversity, and the lesson was one of the four collected from this instructor teaching different sections of the samecourse. Both instructors were male, junior faculty with between five and seven years of teaching experience at the timeof the data collection. Additionally, both manifested an interest in creating classes in which students could participatemore, ask questions and offer answers, and rely on others to find solutions to problems. They also indicated that theselessons were representative of their teaching during the semester.


2.1. Context of the segments

Lesson U8 (hereafter Class A) was taken from a general education requirement course that covers topics suchas linear equations, linear programming, linear regression, probability, and statistics for non-math/science majors.Students with different interests take the class, as this is a course that would fulfill the quantitative requirement in theirprograms. Although it is a first-year course, a few sophomores and juniors enroll in it. End of semester evaluations forthis instructor in this class were good within the standards of the department. The course used a textbook created byfaculty in the department; the students were assessed throughout the semester via quizzes, projects, and exams. Theinstructor dedicated the first part of the class to letting students work in groups on a worksheet with problems of ahigher complexity than those in their textbooks. The observation was done in Winter 2007, about six weeks into the14-week course, when it was thought that most norms for classroom interaction had been established. The purposeof this particular lesson was to apply strategies to convert nominal to real dollars (and vice versa) in wages duringdifferent time periods. For the first 24 min, when most of the interaction occurred in this 80-min class, students workedin small groups solving six problems, two of which are shown in Fig. 4a, with the instructor going from group to group,answering questions as requested.

Lesson U9 (hereafter Class B) was taken from an elective course that seeks to engage first-year non-honors minoritystudents interested in math or science in learning to solve complex calculus mathematics problems. Students apply toenter into the program when they are applying to the university, and they are invited to join if they are admitted andplace into the first calculus course offered at the university. All the students are first year, but they exhibit differencesin academic preparation and ethnicity. The course did not have a textbook and, besides attendance, students were onlyrequired to participate in science and mathematics lectures offered in campus that would be of interest to them; thestudents received credit for the course. For each session, the instructor created a problem worksheet and after assigningthe students to small groups of three or four, let them work on their own, listening and intervening as needed. Infocus groups at the end of the semester, the students recognized his qualities as instructor, his caring approach, and hisknowledge. The lesson was recorded in October 2006 (about five weeks after the beginning of the term). For the first24 min of this 2-hr class students worked on three of the 10 assigned problems, two of which are presented in Fig. 4b.In both classes the problems could be solved using more than one approach and were challenging to the students.

2.2. Analysis

The focus of our analysis was the instructors’ utterances, but we also used students’ responses and the surroundingtext to make coding decisions. We decided to focus on instructors’ utterances because we agree with Brousseau (1997)that the teacher has the major responsibility for determining the didactical contract that manifests in the classroom.For the didactical contract to be sustained, the relationship between instructors and students needs to be asymmetrical,and thus it is the instructor’s responsibility to control and manage the instructional situation; in addition, there is anexpectation from the system (including students, curriculum, school, parents, etc.) that this be so. This assumption doesnot imply that students are not agents or that they do not play a role in how agency is developed. But methodologicallyit requires that we focus on the teachers’ utterances to understand how those possibilities for agency are constructed;therefore we use students’ utterances only to assist in the coding of instructor’s utterances. We parsed all instructors’turns into clauses1 and coded each clause using the categories of engagement defined previously. The second authorparsed and coded both transcripts in consultation with the first author. During these consultations we refined thecategorization, especially regarding entertain values. To test for consistency and reliability of the parsing and thecoding, two random segments of about the same length (5 min) were selected for recoding nine weeks after they hadbeen originally coded. Both authors then proceeded to parse and code the segments; the second author contrasted herlatest parsing with the initial one and with the parsing done by the first author. Two types of discrepancies in parsingclauses occurred: an utterance could contain a clause embedded in another clause (e.g., in “the first thing we do isconvert $3.35 into 2006 dollars” the embedded clause ends at “do”) or an expression would be split incorrectly, usuallyignoring a projected clause (e.g., the clause “I heard you say something earlier, Leslie” would be split incorrectly after

1 A clause, from the perspective of functional grammar, is “the best basic unit of grammatical analysis of text” (Schleppegrell, 2008, p. 551), asit is a unit of both spoken and written language; each clause presents a message that can be analyzed in terms of its stance and dialogic potential.


Fig. 4. Problems from the worksheet students were working on in Class A (a) and Class B (b).


“you”). Agreement in parsing clauses ranged from 84% to 96% and thus it was deemed reliable for the purposes ofthis analysis. The parsing of the two segments into clauses was revised prior to coding for engagement. The segment inLesson A had 80 clauses (26% of the total coded clauses), whereas the segment in Lesson B had 30 clauses (41% of thetotal coded clauses). The main source of disagreement was in the first author’s classification of monoglossic clauses,which prompted for clarification of their definition. The level of agreement and consistency ranged from 70% to 96%.Therefore, the coding was deemed reliable and we used the initially coded transcripts for the analysis. In order to checkthe appropriateness of the analysis and the interpretations, both instructors received copies of the analysis and theinterpretations and were invited to comment and voice their agreement or disagreement. Their comments indicated thatthe analysis was capturing important aspects of their practice, that they could see how they were prompting differentengagement levels, and that the findings were very useful for thinking about teaching. Their comments are included inthe discussion section.

As researchers, we are aware that we are not free agents, we are constrained by our choices of language and action(Taylor, 2001, p. 10). Yet, we see our researcher identities as a “position to be acknowledged” (Taylor, 2001, p. 17)rather than as a bias. The first author has multiple years of experience teaching college mathematics with numerousstudents who prefer not to speak even when they are invited, and is sensitized to the ways in which her use of languageinfluences what happens in the classroom. The second author has been involved in research on second language writing,and has encountered many linguistic difficulties and limitation of resources that second language writers have to engagereaders. Learning to create an effective argument has parallels to how instructor mobilizes language to engage students,yet the needed knowledge remains implicit. As non-native speakers of English, both of us are aware of the ways inwhich language interacts with the complexities of classroom culture and students’ and instructor’s identities, and theirinteraction with the societal norms at large. The expression of these complexities through language in the mathematicsclassrooms intrigues us, and the major role that instructors play, given our own experiences as instructors, sensitizesus to the possibilities and constraints for language use. Our positioning as researchers in education (rather than inmathematics) and as non-white foreign females (in a male dominated field) provide us with sensitivities and limits toour interpretations. As ‘outsiders’ (in terms of gender, ethnicity, and professions) we are allowed to question practicesthat we observe, and as “experts” we can propose interpretations that can be refuted and validated by the participants.These positions strengthen our connections with our participants; as they see us as outsiders interested in learning theirways, they can assume a role of ‘experts’ who can explain how things are to us. We see this as assets for ensuring bothdescriptive and interpretive validity in out study (Maxwell, 1992).

3. Results

As indicated before in these two classes students and instructors did a lot of talking compared to the other classesin the corpus. Compared to Class B, Class A appeared more lively and interactive, students took copious notes duringsegments in which the instructor was explaining or presenting material on the board, and appeared congenial; theexchanges were courteous and usually focused on the content; the instructor positioned himself (physically) in front ofthe class to present content, and moved around all the groups, making eye contact or reading what students had writtenon their notebooks and answering questions; the class had about 23 students and he visited all the pairs and individualswho were working on the problems; students appeared very comfortable asking questions and they did throughout thelesson, sometimes without an invitation to do so. In the initial interview, the instructor indicated that he had recently“started to read about mathematics education” and was excited about trying new things in his class that would increasehis students’ learning opportunities. In contrast Class B, with 12 students, appeared very quiet. The instructor rarelystood up; instead he would pull a chair to sit down and listen to the students’ conversations; there were long lapses inwhich the instructor would not say much and there were occasions in which he would not say anything to the students.Students would work on their papers, exchange ideas with each other, go to the board to put up solutions, or go toother groups to see what they were doing. They would raise their hand or call to the instructor by his first name andask for help. While listening to them, the instructor looked at their papers, made eye contact with all the students inthe group, or pointed at work on the board or on their papers. In the occasions in which the instructor stood up, hepositioned himself on the side or the back of the room, or close to the board, working with one or two students. In theinitial interview the instructor said that he didn’t “want to be seen as a fountain of knowledge,” instead he wanted thestudents to learn to use what they knew and learn to generate knowledge. He had not read literature in mathematicseducation, but had taught a similar class in a different university as a graduate student.


Table 2Descriptive information on the two segmentsa.

Class A Class B

Number of instructor turns 76 27Number of clauses 309 73Number and percent of monogloss clauses 157 (51%) 25 (34%)

Number and percent of heterogloss clauses 147 (48%) 46 (63%)Contracting 60 (19%) 12 (16%)Expanding 87 (28%) 34 (47%)

Number of student turns 76 43Student turn rate (no. of students’ turns/min)a 3.11 1.76

aThe segments were 24.4 min long.

In Table 2 we present descriptive information on the analyzed segments. Students in Class A took about three turnsper minute, whereas students in Class B took about two turns per minute. The difference in turns between both segmentssuggests a different level of student participation and also differences in the instructors’ talk, with the Class B instructorspeaking less than the Class A instructor. The segments differed in the number of clauses, with 309 in Class A and 73in Class B. In Class A, 157 (51%) of the clauses were classified as monoglossic, whereas in class B only 25 (34%)were classified as such. In both classes there were more clauses classified as expanding than contracting. However, theproportions differed in each class; about one third of the clauses in Class A were expanding (87) whereas in Class Babout half of the clauses (34) were classified as such.

We present now excerpts from each class in which we illustrate the coding, the characteristics of engagement thatthese two instructors exhibited, and also how differences were manifested in the segments. The segments were chosento be representative of the interaction in each class and to illustrate different engagement strategies. Each segment isfollowed by an explanation of our analysis. The coding of instructors’ speech is noted in parenthesis. Underlined textin the clause highlights terms and expressions that assisted in the coding.2 Bold text in the coding is used for showingthe abbreviations used later on (see Fig. 2).

In the following two excerpts we show how monoglossic and heteroglosic clauses were used to limit the dialogicspace.

3.1. Example 1, Class A

1. I: So we do want to compare old to new, right? (Heterogloss Contract-proclaim-pronounce)2. But the old that’s like really old. (HC-disclaim-counter)3. We can only compare old to new (H Expand-proclaim: pronounce)4. if they’re in the same units. (HE-en: conditional)5. This is in the units of 1989 dollars I think, yeah. (Monogloss)6. S: And this is in?7. I: That’s in units of 2006 dollars, (M)8. so we can’t divide them just like we can’t, (HC-disclaim-deny)9. we can’t add like meters to miles, (HC-dis-deny)10. we have to have everything in the same units. (HE-pro: pronounce)11. So what we have to do is compare 2006 dollars to 2006 dollars. (M)12. So 5 45 is a 2006 dollar (M)13. and what else is a 2006 dollar in this problem? (pause-5 seconds) (HE-en: Question)14. S: It would just be that.15. I: OK, but we want to compare the current federal minimum wage,

which is by definition 200 . . .

(HC-dis-counter)

16. S: So it’s going to be $5.15 then?17. I: Yeah. (HC-pro-concur)

2 We give the frequencies and percentages of all the clauses coded according to the engagement framework in the Appendix A.


18. S: Oh, so old is going to be $5.15.19. I: Well we have to . . . (HE-pro: pronounce)20. S: Right?21. I: We have two 2006 dollar values, $5.15 and $5.45, $5.45 comes

from 1989(M)

22. but it’s updated. (HC-dis-counter)23. So we’re asking (M)24. how many percent higher or lower is the current federal minimum

wage than this updated value?(M)

25. Anything that comes after that is usually our old or our basis forcomparison.

(M)

In this example, the instructor first emphasized the inclination of doing something in “We do want to . . .” byproclamation, which brought the students to focus on the problem space he set forth here, seeking concurrence withthe tag “right?” (line 1). Then he went on to highlight what “old” meant by countering “But” (line 2). He continued toset up a condition, “if they are in the same units,” for the proposition he set forth, and supplied with a “fact,” “in theunits of 1989 dollars” for the condition (lines 4–5). By responding to a short request from the student, the instructorwas able to explain what it meant to be in the same units by using a monoglossic statement, “That’s in” to justify thedisclaiming followed by negatives, “we can’t” and modalized obligation “we have to” to direct the course of action tobe taken (lines 7–11). He went on to offer a clue, “So 5 45 is a 2006 dollar” (line 12). Then he posed a question tothe students. With a short and apparently unclear response from the student, the instructor continued to explain firstby countering “But” to specify the problem (line 15). When the student asked for confirmation about the amount, “Soit’s going to be $5.15 then?” he responded with only a “Yeah.” When the student replaces the pronoun “it” with thespecific noun, “old” (line 18), the instructor opens up the options (line 19) to refine his next question (lines 21–24).The question functioned as a monoglossic statement and did not invite answers from the students (lines 24–25).

As seen in this segment, Instructor A used a relatively high proportion of monoglossic clauses to state “facts,”giving information to align students with the appropriate way to read and solve the problem. His responses to studentscontract options. The discursive strategies used, either monogloss or heterolgoss-contracting, resulted in interactionswhere dialogic space was limited. The first question he posed (line 13) did not seem to invite extended discussionwith the students, but instead led to monoglossic statements, modalized obligations, “have to” and countering “but,” tooffer more information to the student. A “how” question resided in another embedded clause, “So we’re asking,” (line23) which again fulfilled the instructor’s rhetorical purpose to offer more information that would sustain the expectedsolution to the problem rather than allow for potential new solutions however misaligned they could be.

3.2. Example 2, Class A

1. S: I, OK, I did this percent increase. I did it the way you always tell us to do itand I got the wrong answer.

2. I: OK. (not coded)3. S: I did it . . .

4. I: How do you know the answer’s wrong? (HE-en: Q)5. S: Well because um, I got 10% and that’s not right because I plugged into the

equation and I got a wrong answer.6. I: OK. Well . . .

7. S: It’s supposed to be new = 1 + r times old.8. I: That’s if it’s an increase. (M)9. Is it actually an increase? (HE-en: Q)10. You’re comparing here . . . (M)11. S: Well the minimum wage was $3.35 and it was $5.45.12. I: OK. But you’re trying to compare two things that are in different units. (HC-dis-counter)13. This is in 1989 units, (M)14. this is in 2006 units. (M)15. You can’t compare things that are in two different units. (HC-dis-deny)16. You have to compare 2006 dollars to 2006 dollars. (M)17. Ok. Yes? (HE-en: Q)18. S: [how do we] find the minimum wage for 1989 if it’s not on the table?


19. I: Um . . .

20. S: Go to 1990?21. I: No. (HC-dis-deny)22. It’d be nice (HE-en: suggest)23. if it worked that way, (HE-en: conditional)24. but you just take whatever the previous law was. So in 19 . . . (HC-dis-counter)25. S: 81.26. I: 81 they passed a law that said it’s $3.35. (M)27. S: So you’re assuming it’s staying the same?28. I: Yep. (HC-pro-concur)29. That’s what it does. (M)

In Example 2, the instructor responded to the student’s inquiry with a how-question seeking clarification about whythe student deemed his answer wrong (line 4). After responses from the student that indicated a wrong answer, theinstructor offered the explanation “That is if . . .” (line 8) and engaged the student “Is it . . .?” (line 9). More informationwas given next as a statement (though interrupted by the student) and then by a countering “But,” the instructor pointsout the problem the student was encountering, “But you’re trying to . . .” (lines 10–12). The explanation continuedusing monogloss clauses and a denying “can’t” (lines 13–15). In responses to the students’ next inquiries, the instructoragain used countering and denying processes to offer his explanation (lines 21–24). The “but” after the conditional“It’d be nice if” contracted the space for negotiation (of the alternative proposed by the student that the entry for 1990could be used instead) and instead underscored the statement coming after “but” as a more advisable approach offeredhere (lines 24–26). The last question posed by the student here, “So you are assuming . . .” was responded by definitivestatements, first by a pronouncement “Yep” to affirm the instructors’ explanation also reinforced by “That’s what itdoes” (lines 28–29).

In this episode, there were more interactions in the form of questions and responses between the students and theinstructor compared to the first example. But these interactions were usually initiated by short questions and respondedby either monoglossic or contracting options such as statements or countering/denying values to give information tothe students. In such situations, students got definitive answers instead of questions or hints. Students were thereforeexpected to be informed by the instructor’s explanations rather than invited to use the questions to pursue solutions ontheir own.

In the following excerpts we see how the instructor avoids making definitive statements or giving factual information,which results in opportunities for negotiation and exploration on the students’ part.

3.3. Example 3, Class B

1. I: So then let me ask you, (M)2. what is the purpose of setting this equal to zero in the first place? (HE-en: Q)3. S: To define the zeros so you can find numbers on the different intervals (inaudible)4. I: Oh I see5. So what are the points where dy/dx = 0 called? (HE-en: Q)6. S: (inaudible)7. I: OK. Right, right8. Probably [it is] better to say critical points (HE-en: suggest)9. because there can be places where y’ is 0 (HE-en: suggest)10. but it’s neither a min nor a max, right? (HC-dis-counter)11. So that’s one approach (M)12. but it looks like you’re seeing (HC-dis-counter)13. that it’s kind of intractable here, right? (HC-pro-pro)14. So then you might ask (HE-en: suggest)15. can I tell where this quantity’s positive and negative without knowing where its zeros are? (HE-en: Q)16. S: That’s what I was trying to do. I was trying to like think about it like without the math,

like, but I don’t know.

In Example 3, the instructor aligned with the student (lines 4 and 7) and then used a combination of questions,modalized suggestions, and possibilities to prompt the students to consider different options. The first question here


prompted for clarification of the purpose of a student action (line 2). It was followed by another question related to thenaming of “the points where dy/dx = 0” (line 6). After the questions and responses, the instructor offered suggestionsby using “probably better to” with an explanation, “because there can be . . .” (lines 9–10). The suggestion is offeredas an option when in both clauses the instructor uses the possibility words, “probably” and modal auxiliary, “can be”(showing possibility similarly) reinforcing this as a point for negotiation. The instructor continued with a countering“but” as a way to make the case that what was suggested was but “one approach” (lines 11–12). And the second “but”directed the discussion to the student’s idea framed in tentative terms, “looks like” and “kind of,” as “intractable” (lines13–14). The tentative terms show his interpretation as a probable way to take that interpretation, acknowledging at thesame time that such interpretation might not do justice to the student’s thinking. With this, he then asked the studentto consider the options by a question that he framed in tentative terms using “might” and “can” (lines 15–16). By notmaking definitive interpretations or giving factual statements, the instructor gave the students the option of consideringthe possibilities by him or herself.

3.4. Example 4, Class B

1. I: Then how does the volume change, (HE-en: Q)2. if you add an inch of radius then? (HE-en: conditional)3. S: The volume gets larger4. I: The volume what? (HE-en: Q)5. S: Gets larger6. I: Right, (HC-pro-concur)7. by how much? (HE-en: Q)8. S: Oh. It depends on what your radius is9. I: Right. (HC-pro-concur)10. But what’s the quantity? (HC-dis-counter)11. S: 4πr2. You’re asking me for the big picture?12. I: Yes. (HC-pro-concur)13. And so what I’m asking you is (M)14. does the volume change by exactly 4π2, 4πr2? (HE-en: Q)15. S: Yes. Approximately 4πr2?16. I: Approximately or exactly? (HE-en: Q)17. S: Approximately18. I: OK19. And why is that? (HE-en: Q)20. S: Because it’s just an approximation with tangent lines21. I: Right. Right22. And for this particular function it’s not a linear function bar, (HC-dis-deny)23. so it’s not equal to this linear approximation given by the derivative.. (HC-dis-deny)24. So then I’ll come back to you (HE-en: condition)25. and ask you how that, for the discussion we just had, (M)26. how it relates to this table that you’re doing, (HE-en: Q)27. OK? (HE-en: Q)

In this example the instructor first asked a how-question framed in a condition (lines 1–2), “if . . . then.” He raisedfurther questions by asking the student to specify “by how much” the volume changes (line 7). When the studentresponded with a more general observation, he rephrased the same question as “But what’s the quantity?” to get thestudent to focus on the “big picture” (lines 10–12). “But” in this question was emphatic in drawing the student’sattention to stay on the question. In other words, it closed down other options and highlighted the question of concernfor the student. After a few rounds of inquiring and responding, the instructor next framed the question more specificallyby building on the answers the student had given earlier, “does the volume change by exactly 4π2, 4πr2?” (line 14).When the student responded by “approximately” instead of repeating the instructor’s term, “exactly” in the question,he prompted for an explanation with a why-question (line 19). After the student responded, the instructor used twonegatives, “not” in two clauses connected by “so,” which narrow the possibilities for interpretation. This served thepurpose of aligning the student to the view the instructor held and allowed the instructor to formulate the how-questionrelated to “this table,” that the student was working on (lines 22–27). He thus helped the student to consider a furtherrelationship that she or he might have not anticipated. In this excerpt, we see how the instructor carefully scaffolds


student’s thinking with his language, either by framing a more rigorous way of thinking or by recasting student’sthinking in more appropriate terms.

A series of questions prompted opportunities for considering options; the instructor used alignment moves to focusone question, but still offered rhetorical moves that opened the discussion and suggested exploration on the student’spart. The student was forced to take a position, which while not necessarily apparent, had to be the one the instructor wasafter. The relative low incidence of monoglossic statements suggests that the instructor’s position was not necessarilyevident.

Compared to students in Class A, students in Class B were asked more questions and given more opportunities to shiftand choose positions and experienced fewer contracting or fact statements that would define a particular interpretationof the problem. Rather than being definite, the statements served the purpose of making the students take stock of whatthey had and what was asked or given in order to proceed. Thus, they had more opportunities to exert their agencyin making those decisions. In contrast, students in Class A received more definitive explanations or information inresponse to their questions, which seemed to prompt them to agree with the presumably right way to read and approachthe problems; heteroglossic forms in this segment were geared to highlighting the need for distinguishing between areduced set of options.

4. Discussion

The purpose of this study was to investigate the ways in which classroom discourse shapes students’ agency byanalyzing how instructors engage them in classroom dialog. The instructors whose classes we analyzed were committedto facilitating participation. They had created, what appeared to an observer to be very interactive classrooms, withstudents asking questions and offering suggestions more frequently than in other classes observed.

The analysis of the language instructors used in two segments of their classes revealed important differences inhow the use of certain linguistic forms opened or suppressed dialogic interaction between instructors and students. Ifcreating a setting in which students are dialogically engaged is important, then it is crucial to pay closer attention tohow the linguistic devices we use can act against this purpose.

First, our analysis reveals that monoglossic speech was relatively frequent in both segments analyzed (51% and 34%in Class A and Class B, respectively) compared to the frequency of heteroglossic discourse observed (48% versus 63%in Class A and Class B, respectively). This suggests that in these particular segments—devoted to work on problemson a worksheet—there was a need to state facts, clarify, and fend off argumentation or discussion: certain things aregiven, certain propositions are fixed, and presumably need to be learned and used as such. Because the two segmentswere recorded during a class period in which the students were working actively on predefined problems, the observedfrequency suggests that statements of fact were required for the problem solving session to be sustained, and alsoshows that the instructors were the source of a substantial number of those facts.

Second, our analysis revealed that within heteroglossic discourse, entertaining and contracting forms were used togive information, assess situations, and also to seek explanations and information from the students. We found thatthe strategy of entertaining options mitigated the authoritarian voice in the setting. An authoritarian voice has beenrecognized as a feature of mathematics, ostensibly present in textbooks (Love & Pimm, 1996; Pimm, 1987), even inthose assumed to be designed to include the students in the mathematics presented (Herbel-Eisenmann, 2007). Indirectevidence of this authoritarian voice in undergraduate mathematics classrooms comes from this corpus, as only two ofthe nine undergraduate mathematics lessons had a substantial level of student participation according to our analysis.

Third, we found striking differences in how the two instructors positioned themselves with respect to their students.The two settings differed in the extent to which the instructor maintained a monoglossic voice (over half of clausesanalyzed from Instructor A versus over one third of clauses analyzed from Instructor B), suggesting that Instructor Atended to maintain a more authoritarian position than Instructor B—therefore limiting student agency opportunitiesin his class. Furthermore, within the heteroglossic voices, the two instructors expressed almost diametrically oppositeattitudes in engaging students in the dialog. The instructor in Class B, in his choice of language, entertained dialogicallyalternative voices more frequently than the instructor in Class A. A predominant device was entertaining questions,with which the Class B instructor not only sought information or explanations from the students, but also managed toprovide suggestions or hints that would allow them to consider alternative options and give them the opportunity tomake a choice on their own. The Class A instructor preferred suggestions, but this device was used mostly for givinginformation that sought to limit students’ alternatives with a single interpretation of the problem.


When considering dialogic contraction which acts to challenge, fend off, or restrict dialog, the two instructorsalso differ. The Class B instructor used the authoritarian pronouncement to assess students’ progress and stress theimportance of particular information that was being considered. On the other hand, the Class A instructor preferred tocounter prior utterances or alternative positions, by denying a position previously recognized, with the main purposeof providing information and noting the importance of the points he made.

The analysis of the discourse of the two settings illustrates a subset of linguistic devices that actively include orexclude students from participating in mathematical dialog. This analysis reveals important elements of authority asexpressed by the language used by instructors that may limit or open possibilities for authentic dialogic engagementin classrooms in which students are expected to participate fully.

This type of analysis provides an alternative way to look at classroom discourse by foregrounding the role oflinguistic devices in engaging speakers in the dialog to see how speakers’ agency is established. The instructors, whoparticipated and read the analysis and the interpretation, indicated two main implications of this work, the need toattend to language and the possible influence of context in the findings.

Instructors indicated that the detailed focus on language was important because it allowed them to understand thehidden messages that it conveys. They said that it raised their awareness that language is not neutral and that their usewas not “innocent” either. They also wondered how much of these discursive patterns “traveled” to their other lessons,that is to what extent their “teaching speech” was defining their own teacher identity.

They suggested that the different contexts in which the courses operated could in part explain the findings. Instructor Bsaid that because he did not have the constraint of assessing whether students had learned a predetermined curriculumwith his class, he felt he did not need to push the students to solve problems in certain ways; because he was notresponsible for their grading, he could let them wander (both physically and intellectually) without exerting too muchpressure on them to do things in “a” particular way. In addition, the course was a corequisite not a prerequisite, so hedid not feel the pressure to cover content that he would normally feel in other more lock-step courses. At the same time,however, the course was designed with the goal of assisting students excel in their calculus classes, so even though inthis particular class the students were not graded, both students and the instructor measured the effectiveness of thecourse by the students’ grades in the calculus exams, and likewise, even though the course had not a set established setof topics to ‘cover,’ the problems used knowledge that would be encountered in any calculus course in the sequence.These two contextual aspects, curriculum and assessment, were different for Instructor A. Although not a prerequisitefor other courses, the instructor had one of many sections of a course that had a set curriculum and the expectation tocover all of it. Students were graded in this class and he was responsible for the grading, and thus within each class,he felt the need to make sure the activities were scaffolded in a productive way. This same instructor, indicated that inhis Mathematics Modeling class in which students are prepared to take part of modeling competitions, he purposefullygives students more space for learning modeling and believed that his language would probably be very different in thatclass. Thus these differences in context, determined different didactical contracts for the particular courses analyzed,and the instructors felt they would play a role in interpreting the findings.

Such differences can explain why making changes to encourage students’ agency across undergraduate mathematicsclassrooms can be very difficult and suggest possible factors to consider (e.g., type of course, expectations, gradingrequirements). These instructors both had the will to create classrooms in which students participated, but both weresubject to different circumstances that simultaneously constrained and facilitated how they were enacting such partici-pation. These circumstances impacted how they used the language: Having more dialog served the purpose of showingthat student interaction was occurring, but using the dialog in ways that maintained control fulfilled accountabilityrequirements established by their programs (Herbst, 2003).

5. Conclusion

Our analysis is promising in helping us understand the mechanisms by which students are actually dialogicallyengaged with classroom discourse. We have seen that even in seemingly highly interactive settings, there may be littleroom for students to include their own perspectives or voices into the dialog. However, our analysis shows that it ispossible, to organize classroom discourse in a way that does. We conjecture that the varying discursive strategies usedby these two instructors shaped different levels of students’ agency: Students participated and responded differently ineach case. In both cases, such agency depends on the ways in which the instructors interacted with their students usingthe discursive options they deemed appropriate.


The findings from this study suggest further avenues for research. First, it would be of paramount importanceto determine the extent to which some of these discursive practices “travel” with the instructors as they teachcourses that are more or less constrained than the ones whose classes we analyzed here. If indeed there are strik-ing differences with more monoglossic language in courses in which there are demands for content coverage andstudent assessment, and more heteroglossic language in courses in which such demands do not exist, then we couldsuggest that curriculum and assessment—elements that shape the didactical contract in the classroom—could beimportant areas for effecting change in promoting students’ agency in the classroom. But if the discursive prac-tices persist, which is what we think might be the case—because language determines our identities in the roles weassume in life (Gee, 2000–2001)—then attention to language would be central for any strategy focused on alteringhow students’ agency in mathematics classrooms is promoted. Analyzing the dialogic engagement used by theseinstructors teaching other courses and observing other teachers teaching similar courses, could shed light on theseconjectures.

Another area that would be important to consider is the extent to which students are aware of the different effectsof the classroom language in influencing their own agency. Interviews with students after particular lessons or withspecific classroom excerpts could shed light on the process by which students develop their own linguistic devices asa response to teachers’ invitations to engage in the dialog.

While it is assumed that the relationship between instructors and students needs to be asymmetrical with the instruc-tors assuming more responsibility for controlling and managing the instructional situation to fulfill the expectationof the didactical contract in the classroom, a long term project could probe how students responding and interactingas active agents actually impact their learning experience. Such work would give support to research that proposes adifferent approach to teaching mathematics using more post-modern, chaos based, framing of reference (e.g., Davis,1996).

Within an authoritarian discourse such as mathematics, understanding what it means to dialogically engage stu-dents and under what circumstances is somewhat urgent, given suggestions that an authoritarian voice de factoexcludes some groups of students (female, low ability, minority) from participating in the mathematical dis-course (Dowling, 1998; O’Halloran, 1998), and in turn from careers in the sciences, mathematics, technology, orengineering.

Likewise, raising awareness of the role of language in sustaining dialogic engagement is an important area forprofessional and faculty development. Perhaps, if the goal is to create a truly engaged classroom, some of these devicescan be part of what instructors need to learn in order to teach undergraduates. An analysis of how their language affectsthe extent to which they invite students into the dialog seems to be an important area for consideration. While awarenessabout students’ misconceptions has been highlighted as important for preparing future faculty (Kung & Speer, 2007;Speer, Gutman, & Murphy, 2005), information about the impact of language in classroom work is also important for allinstructors. We use language to deliver information and to assess students’ progress. How we use it conveys powerfulmessages that might exclude the students that we need to engage in the dialog.

Acknowledgments

This research was funded in part by a grant from the Office of the Vice-President for Research at the University ofMichigan. We thank Mary Schleppegrell, Jim Martin, and the reviewers for their feedback on earlier versions of thispaper. Portions of this work have been presented at the 9th Conference on Research in Undergraduate MathematicsEducation (San Diego, 2008, February), the International Conference of the Psychology of Mathematics Education(Mexico, 2008, July), the Joint Meeting of the American Mathematical Society and the Mathematical Association ofAmerica (Washington, 2009, January), and the annual meeting of the American Educational Research Association(San Diego, 2009, April).

Appendix A.

See Table A1.


Table A1Frequencies and percentages of clauses coded in each segment.

Engagement Class A (N = 309) Class B (N = 73)

n % N %

Monogloss (EM) 157 51 25 34Heterogloss (EH) 147 48 46 63

Contracting 60 19 12 16pro-pro 12 4 5 7pro-con 11 4 3 4dis-coun 21 7 3 4dis-deny 16 5 1 1

Expanding 87 28 34 47en-imagine 0 0 3 4en-suggest 46 15 10 14en-condition 14 5 4 5en-question 27 9 17 23

Not Codeda 5 2 2 6

Note: Bolded entries correspond to major categories in the coding system; entries in italics correspond to totals of the two subcategories in heterogloss.Underlined entries correspond to the largest values within each subcategory.

a Corresponded to clauses that did not express a particular engagement position (e.g., OK, Uh?).

References

Bakhtin, M. M. (1981). The dialogic imagination. Austin, TX: University of Texas Press.Bauersfeld, H. (1988). Interaction, construction, and knowledge: Alternative perspectives for mathematics education. In D. Grouws, T. Cooney, &

D. Jones (Eds.), Perspectives on research on effective mathematics teaching (pp. 27–46). Reston, VA: NCTM & Erlbaum.Blair, R. (Ed.). (2006). Beyond crossroads: Implementing mathematics standards in the first two years of college. Memphis, TN: American

Mathematical Association of Two Year Colleges.Brousseau, G. (1997). Theory of didactic situations in mathematics (N. Balacheff, Trans.). Dordrecht, Netherlands: Kluwer.Cazden, C. (1986). Classroom discourse. In M. C. Wittrock (Ed.), Handbook of research on teaching (pp. 432–463). New York: Macmillan.Cobb, P., Wood, T., Yackel, E., & McNeal, B. (1992). Characteristics of classroom mathematics traditions: An interactional analysis. American

Educational Research Journal, 29, 573–604.Creswell, J. W. (2005). Educational Research: Planning, conducting, and evaluating quantitative and qualitative research (3rd ed.). Upper Saddle

River, NJ: Pearson.Davis, B. A. (1996). Teaching mathematics: Toward a sound alternative. New York: Garland.Dowling, P. (1998). The sociology of mathematics education: Mathematical myths/pedagogic texts London: Falmer Press.Ernest, P. (1991). The philosophy of mathematics education. London: Falmer.Fiske, J. (1994). Media matters: Everyday culture and political change. Minneapolis: University of Minnesota Press.Gee, J. P. (2000–2001). Identity as an analytical lens for research in education. In W. G. Secada (Ed.), Review of Research in Education (pp. 99–125).

Washington, DC: American Educational Research Association.Halliday, M. A. K. (1985). An introduction to functional grammar. London: Edward Arnold.Halliday, M. A. K., & Hasan, R. (1985). Language, context, and text: Aspects of language in a social-semiotic perspective. Oxford: Oxford University

Press.Harré, R., & van Lagenhove, L. (1999). Positioning theory: Moral contexts of intentional action. Oxford: Blackwell.Herbel-Eisenmann, B. (2007). From intended curriculum to written curriculum: Examining the “voice” of a mathematics textbook. Journal for

Research in Mathematics Education, 38, 344–369.Herbst, P. (2003). Using novel tasks in teaching mathematics: Three tensions affecting the work of the teacher. American Educational Research

Journal, 40, 197–238.Hiebert, J., & Wearne, D. (1993). Instructional tasks, classroom discourse, and students’ learning in second-grade arithmetic. American Educational

Research Journal, 30, 393–425.Kline, M. (1980). Mathematics: The loss of certainty. Oxford: Oxford University Press.Koutsantoni, D. (2004). Attitude, certainty, and allusions to common knowledge in scientific research articles. Journal of English for Academic

Purposes, 3, 163–182.Kung, D. T., & Speer, N. (2007). Teaching assistants learning to teach: Recasting early teaching experiences as rich learning opportunities. Paper

presented at the tenth RUME conference.Love, E., & Pimm, D. (1996). ‘This is so’: A text on texts. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International

handbook of mathematics education (pp. 371–409). Dordrecht: Kluwer.


Lutzer, D. J., Rodi, S. B., Kirkman, E. E., & Maxwell, J. W. (2007). Statistical abstract of undergraduate programs in the mathematical sciences inthe United States: Fall 2005 CBMS Survey. Washington, DC: American Mathematical Society.

Martin, J. R. (1993). Technicality and abstraction: Language for the creation of specialised texts. In M. A. K. Halliday, & J. R. Martin (Eds.), Writingscience: Literacy and discursive power (pp. 203–220). London: Falmer.

Martin, J. R., & Plum, G. (1997). Construing experience: Some story genres. Journal of Narrative and Life History, 7, 1–4.Martin, J. R., & Rose, D. (2003). Working with discourse meaning beyond the clause. London: Continuum.Martin, J. R., & White, P. R. R. (2005). The language of evaluation: Appraisal in English. New York: Palgrave-Macmillan.Maxwell, J. A. (1992). Understanding and validity in qualitative research. Harvard Educational Review, 62, 279–300.Mesa, V., Boyle-Heimann, K., Mosher, B., Rhea, K., & Megginson, R. (2007). The Douglass Houghton Scholars Program: Lessons learned in the

first year. Paper presented at the Educating a STEM Workforce: New Strategies for U-M and the State of Michigan. Ann Arbor: University ofMichigan.

Mesa, V., & Griffiths, B. (2007). Collegiate mathematics instructors: Textbooks and teaching (unpublished manuscript). Ann Arbor: University ofMichigan.

Mesa, V., & Megginson, R. (submitted for publication). Equity and quality in a program for under-represented students at an elite university. AnnArbor: University of Michigan.

Moschkovich, J. (2008). “I went by twos, he went by one”: Multiple interpretations of inscriptions as resources for mathematical discussions.Journal of the Learning Sciences, 17, 551–587.

National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston, VA: Author.O’Halloran, K. L. (1998). Classroom discourse in mathematics: A multisemiotic analysis. Linguistics and Education, 10, 359–388.Pimm, D. (1987). Speaking mathematically. New York: Routledge & Kegan Paul.Rotman, B. (1988). Toward a semiotic of mathematics. Semiotica, 72, 1–35.Rowland, T. (1995). Hedges in mathematics talk: Linguistic pointers to uncertainty. Educational Studies in Mathematics, 29, 327–353.Schleppegrell, M. (2008). Grammar, the sentence, and traditions of linguistic analysis. In C. Bazerman (Ed.), Handbook of research on writing.

New York: Erlbaum.Sfard, A. (2000). Steering (Dis)course between metaphors and rigor: Using focal analysis to investigate an emergence of mathematical objects.

Journal for Research in Mathematics Education, 31, 269–327.Speer, N., Gutman, T., & Murphy, T. J. (2005). Mathematics teaching assistant preparation and development. College Teaching, 53, 75–80.Steinbring, H. (1989). Routine and meaning in the mathematics classroom. For the Learning of Mathematics, 9(1), 24–33.Sugeno, M. (2008). Towards elucidating language functions in the brain. In V. Torra, & Y. Narukawa (Eds.), Modeling decisions for artificial

intelligence (pp. 1–2). Berlin/Heidelberg: Springer-Verlag.Taylor, S. (2001). Locating and conducting discourse analytic research. In M. Wetherell, S. Taylor, & S. Yates (Eds.), Discourse as data: A guide

for analysis (pp. 5–48). Thousand Oaks, CA: Sage.Voigt, J. (1985). Patterns and routines in classroom interaction. Recherches en Didactique des Mathématiques, 6(1), 66–118.Wagner, D., & Herbel-Eisenmann, B. (2008). Just don’t”: The suppression and invitation of dialogue in the mathematics classroom. Educational

Studies in Mathematics, 67, 143–157.White, P. R. R. (2003). Beyond modality and hedging: A dialogic view of the language of intersubjective stance. Text, 23(2), 259–284.

Documents

The language of engagement in two highly interactive undergraduate mathematics classrooms