a teacher’s mediation of a thinking-aloud discussion in a 6th grade mathematics classroom

BETINA ZOLKOWER AND SAM SHREYAR

A TEACHER’S MEDIATION OF A THINKING-ALOUDDISCUSSION IN A 6TH GRADE MATHEMATICS CLASSROOM

ABSTRACT. This article presents a Vygotsky-inspired analysis of how a teacher mediated

a “thinking aloud” whole-group discussion in a 6th grade mathematics classroom. This

discussion centered on finding patterns in a triangular array of consecutive numbers as

a phase towards building recursive and direct algebraic formulas. By a “thinking aloud”

discussion we mean a conversation wherein students exchange and further develop ideas-

in-the-making with their peers under the teacher’s guidance. Drawing upon Halliday’s

systemic functional linguistics (SFL), we treated the selected discussion as a text. We then

analyzed how the teacher mediated the conjoined making of this text so that it served as an

interpersonal gateway for students to practice searching for patterns and signifying these

patterns in propositional form. This analysis was guided by the following questions: How

did the discussion as a text-in-the-making mean what it did? What was the role of the

teacher in the conjoined making of this text? Our study illustrates the power of SFL for

capturing the inner grammar of instructional conversations thus illuminating the complex-

ities and subtleties of the teacher’s role in mediating semiotic mediation in mathematics

classrooms.

KEY WORDS: interpersonal plane of learning, teacher mediation, text, verbal semiotic

mediation, mathematics discussion

1. INTRODUCTION

Over the last three decades, the study of interaction, communication, anddiscourse has become a central theme in the field of mathematics education.Within this overall theme, research on whole-class discussions occupies animportant place. While there is general agreement “that discussions are auseful medium for developing learning” views diverge as to “the role thatinterpersonal communication plays in the teaching and learning process”(Morgan, 2000, p. 94). O’Connor (2001) aptly describes this situation whenshe notes that “Despite the pervasiveness of the assumption that whole-group discussion in mathematics classrooms may promote mathematicallearning, we know little about the mechanisms that might underlie suchoutcomes” (p. 143). A better understanding of these mechanisms requiresthat we describe “the complex work of the teacher in conducting whole-class discussions” (ibid. p. 144).

Studies of “balance” and “conceptualization” discussions (BartoliniBussi, 1998a, 1998b), “position-driven” discussions (O’Connor, 2001), and

Educational Studies in Mathematics (2007) 65: 177–202

DOI: 10.1007/s10649-006-9046-0 C© Springer 2006

178 BETINA ZOLKOWER AND SAM SHREYAR

“collective argumentation” discussions (e.g. Forman, Larreamendy-Joerns,Stein and Brown, 1998; Martin, McCrone, Bower, and Dindyal, 2005), haveidentified how teachers may support students’ appropriation of the ways ofreasoning, speaking, and writing that constitute the practice of mathematics.Our aim is to contribute to this research literature an analysis of a “thinkingaloud” discussion in which a group of 6th grade students exchange mathe-matical ideas-in-the-making with their peers under the teacher’s guidance.We adopt a Vygotskyan perspective in that we view whole-class mathe-matics discussions as teacher-guided meaning-making experiences that canserve as interpersonal gateways for students to appropriate those meanings.

Despite much progress in understanding classroom interaction there re-mains “a growing need for theories, analytical tools, concepts, and researchmethodologies that would allow one to capture the subtleties and complex-ities” of this phenomenon (Steinbring, Bartolini Bussi, and Sierpinska,1998, p. 342). We believe that a Vygotskyan perspective on teachingand learning complemented by the systemic functional linguistics (SFL)developed by Halliday (1973, 1978, 1994), Halliday and Hasan (1989), andHasan (1996) addresses the above mentioned need by offering a frameworkwell suited for studying classroom interaction. In particular, SFL toolscan be used to capture the inner grammar of whole-class conversationsthus shedding light on how teachers can conduct these speech events inmanners that enlarge the mathematical meaning potential of their students.

We approached the selected whole-class discussion as a conjoined,teacher guided text-in-the-making. Utilizing SFL tools for studyinghow interpersonal meanings are made and exchanged, we analyzed theteacher’s mediation of this speech event. The questions guiding our analy-sis included the following: How did the discussion as a text-in-the-makingmean what it did? What was the role of the teacher in mediating the jointconstruction of this text?

2. THEORETICAL FRAMEWORK

2.1. Learning verbal thinking

Vygotsky (1986) suggests that “Schematically, we may imagine thoughtand speech as two intersecting circles. In their overlapping parts thoughtand speech coincide to produce what is called verbal thought” (p. 88).He argues further that “Verbal thought, however, does not by any meansinclude all forms of thought or all forms of speech” (ibid.). Although“the fusion of thought and speech, in adults as well as in children, is aphenomenon limited to a circumscribed area” it is nonetheless taken byVygotsky to be central to the higher psychological processes (p. 89).

TEACHER’S MEDIATION OF A THINKING-ALOUD DISCUSSION 179

In his study of tool and symbol in child development, Vygotsky (1978)notes that once the child can use speech to aid in the solution of practicalproblems she

. . .is able to ignore the direct line between actor and goal. Instead [s]he engages ina number or preliminary acts, using what we speak of as instrumental, or mediated(indirect) methods. In the process of solving a task the child is able to includestimuli that do not lie within the immediate visual field. Using words (one class ofsuch stimuli) to create a specific plan, the child achieves a much broader range ofactivity. . . (p. 27)

Similarly, Dewey (1938) argues that inquiry involves signifying experiencein descriptive and narrative propositions, inferences, hypothetical ideas(plans for action), and conclusions. The interconnection of thought andspeech makes possible the planning function of the latter, thereby the re-flective verbal-semiotic mediation of our experiences.

Vygotsky’s (1978) explanation of verbal thinking relies in part uponan examination of the evolving interrelation between speech and action inthe development of the child. In the early stages of development, “Speechfollows actions,” that is, it “is provoked and dominated by activity” (p. 28).At a later stage, “speech is moved to the starting point of an activity” (ibid).This shift in the location of speech within the structure of an experienceredefines the relationship between word and action. “Now speech guides,determines, and dominates the course of action; the planning function ofspeech comes into being in addition to the already existing function oflanguage to reflect the external world” (ibid).

Learning verbal thinking is a process that entails the socialization of theindividual child’s mind. In this regard Vygotsky (1978) argues that

The greatest change in children’s capacity to use language as a problem-solvingtool takes place. . .when socialized speech (which has previously been used toaddress an adult) is turned inward. Instead of appealing to the adult, childrenappeal to themselves; language thus takes on an intrapersonal function in additionto its interpersonal use. . . .The history of the process of the internalization of socialspeech is also the history of the socialization of the child’s intellect. (p. 27)

Learning to think verbally in order to devise plans for solving problemsinvolves a transformation of an interpersonal communication with morecompetent others into an intrapersonal communication with oneself (Sfardand Kieran, 2001). The interpersonal plane of speech communication istherefore the gateway to the intra-individual plane of verbal thought.

Vygotsky’s (1978, 1986) zone of proximal development (zoped) un-derlines the social nature of learning. The upper reach of a child’s zopedis that space wherein she cannot handle a problem alone but can do sowith the collaborative assistance of a teacher or a more competent peer.This assistance may entail thinking aloud as s/he designs a plan for solvingthe problem at hand. When orchestrating a whole-class ensemble, if the


teacher is to mediate simultaneously the verbal-semiotic mediation of allher students, she must be in many zopeds at the same time (Carpay, 1999).At stake in such discussions is holding a meeting of multiple minds thatsupports the transformation of the interpersonal function of speech into theintrapersonal function of verbal thinking. Furthermore, since mediatingstudents’ verbal-semiotic mediation involves guiding their use of speechfor planning solutions, whole-class discussions serving this function wouldnecessarily precede the solution of the problem at hand.

Both spoken and written speech can be used as tools for verbal semi-otic mediation. The latter has certain advantages over the former. First,like all instances of spoken speech – either voiced or unvoiced – spokenverbal thinking is temporary and fleeting, and unless somehow capturedor recorded, leaves behind no permanent traces. Second, some (mathemat-ics) problems require rather lengthy and complex planning processes thatwould be greatly aided by capturing spoken ideas and plans in writing. Ver-bal semiotic mediation can thus be approached as a multi-phased practicethat involves back and forth movement between speaking what we havewritten and writing what we, as well as others, have spoken.

2.2. Systemic functional linguistics

Although Vygotsky explains that intellectual development is shaped byinteractions in social environments that are “influenced by the wider cul-ture, which varies according to the forms of organization of labor activitythat are practiced and the material and semiotic tools that are employed”(Wells, 1999, p. 37), he does not “explain how the discursive means thatare internalized to mediate instrumental functioning are themselves influ-enced by sociocultural factors” (ibid. p. 38). In Wells’ view, SFL accountsfor how those discursive means Vygotsky considered critical to explainingintellectual development are themselves shaped by social context.

Within SFL – which is also referred to as functional grammar – the unitof analysis is text conceived as language operational in a social context(Halliday and Hasan, 1989). As Halliday (1994) explains, “Text is some-thing that happens, in the form of talking or writing, listening or reading”(p. 311). He invites us to think of text “dynamically, as an ongoing pro-cess of meaning” (p. 311). SFL analyses aim at explaining how a giventext means as it does. Since meaning is viewed as function in the con-text of use, demonstrating how a text means amounts to explaining whatit means. Inasmuch as verbal semiotic mediation – mediation by means ofsign systems – is socially situated “participation in language use” (Hasan,1996, p. 181) that “takes the form of text in context” (ibid. p. 185), SFL isa well suited lens through which to examine this phenomenon.


According to Halliday (1994), language offers grammatical systems forrealizing ideational (experiential and logical), interpersonal, and textualmeanings. He refers to these as the three meta-functions of language. Lan-guage users exploit the ideational grammar to represent experiences andprocesses as well as to construct logical relations. The interpersonal gram-mar is used for realizing social relations and roles via exchanges. And,the textual grammar offers resources for organizing experiential, logical,and interpersonal meanings into coherent and cohesive texts. These threemeta-functions operate simultaneously, that is, “Speakers and writers si-multaneously present content, negotiate role relationships, and structuretexts through particular grammatical choices that make a text the kind oftext it is” (Schleppegrell, 2001, p. 432). Yet, certain situations call for textswith a relative predominance of one or another meta-function.

Functional grammar links texts to social contexts providing tools “todeduce context from text” and “predict language from context” (Eggins,1994, p. 7). The two main levels of social context in SFL are genre andregister. Genre accounts for the relationship between texts and culture.Martin (1984) defines genre as a “staged, goal-oriented, purposeful activ-ity in which speakers engage as members of a culture” (p. 25). Registerdescribes how texts are shaped by and, in turn, shape their immediate con-text of situation by accounting for the probabilistic relationship betweena given situation-type and the linguistic choices likely to be realized bythe participants therein (Halliday, 1978). In other words “Given a partic-ular context of situation – a ‘situation type’ – certain semantic featureshave a much higher probability of being selected in the construction of theassociated text” (Wells, 1994, p. 48).

Three aspects of any context of situation, or register variables, haveconsequences for language use, namely field, tenor, and mode (Halliday,1994). Field is the social activity within which the text is functioning aswell as the semantic domain (or subject matter) of the text. Tenor refers tothe relationship between the language users (status, power relations, anddegree of social distance and formality) as well as the degree to whichthe emphasis of the text falls on either realizing the interpersonal relationsat stake or the informational content of the message. Mode refers to thesymbolic organization of the text including the role of language (i.e. eitherconstitutive of or ancillary to the event) as well as the channel through whichthe text is produced (spoken, written, or both). Each aforementioned registervariable is associated with one meta-function: field with the ideationalmeta-function, tenor with the interpersonal meta-function, and mode withthe textual meta-function.

When making a text, language users must first interpret or recognizethe genre and register at hand. This recognition predisposes them to make


certain choices of grammar and vocabulary (lexico-grammatical choices)from their meaning potential. Meaning potential refers not to what parti-cipants know but to “what they can do linguistically” (Halliday, 1973,p. 44) that is, how they can use language to realize ideational, inter-personal, and textual meanings. If a text is to serve successfully its func-tions, participants need to make appropriate choices from their meaningpotential. Such choices depend on the degree to which participants are ableto accurately recognize the register and genre of the text at hand and arelimited by what they are able to mean. It is in this sense that any given textis an instantiation of a register-specific meaning potential.

2.3. The interpersonal gateway

For Halliday (1993), teaching/learning is a semiotic process that consistsin enlarging students’ subject-matter specific meaning potential. He high-lights the centrality of “a generalized interpersonal gateway, whereby newmeanings are first construed in interpersonal contexts and only later trans-ferred to ideational, experiential and/or logical” (p. 103). This perspectiveresonates with Vygotsky’s (1978) assertion that “All the higher functionsoriginate as actual relations between human individuals” (p. 57) as well aswith his view of the zoped as a learning potential created in social interac-tion (Zacks and Graves, 2001, p. 232). The zoped can thus be conceived asa conjoined text with a predominance of interpersonal meanings that func-tions as a gateway to the appropriation of ideational and textual meanings.

The interpersonal meta-function provides linguistic resources for creat-ing and maintaining social roles and relationships via clauses that functionas exchanges. The natural unit of verbal exchanges is the turn or move,marked by a change of speaker. A move can consist of a single clause oran interconnected series of clauses. A clause is the smallest possible groupof words within a text that has meaning. There are two types of clauses:major and minor. A minor clause usually consists of one or two words andcontains little or no thematic content. Hereby “exchange” is used for refer-ring to the back and forth of moves made up of clauses whereby meaningsare being transacted, rather than to the combination of reciprocally relatedwithin the sequential organization of spoken discourse (Wells, 1999).

The linguistic resources afforded by the interpersonal grammar are or-ganized in the systems of speech roles and speech functions, mood, po-larity, and modality, which we briefly outline below. The general functionof speech interactions is to exchange either information (an intangible,purely verbal commodity) or goods and services (tangible commodities oractivities). The two speech roles available to participants in spoken ex-changes are giving and demanding. When goods or services are exchanged,language serves as a medium for bringing about the exchange. When


information is exchanged, language is both a medium for carrying outthe exchange and the commodity being exchanged. The semantic functionof the clause in exchanges of goods & services is a proposal, whereas ininformation exchanges, it is a proposition (Halliday, 1994).

There are five speech functions, four of which result from cross-tabulating speech roles and commodities exchanged. These are as follows:statement (giving information), offer (giving goods and services), question(demanding information), and command (demanding goods and services).As speakers select a speech function, thereby adopting a speech role, theyassign a complementary role to their interlocutors (e.g. the initiating func-tion of offer may be followed either by the expected response of an accep-tance or by the discretionary alternative of a rejection). The fifth speechfunction is the check, used by speakers in order to ensure that their ad-dressee(s) are with them so they can proceed with their words or actions.This speech function, which stands in between a command and a question,is usually realized as a minor clause at the end of a move (e.g. “okay?” and“right?”).

The mood system, another central component of the interpersonal meta-function, offers a set of choices for the grammatical realization of the abovelisted system of speech functions. Major clauses are either indicative orimperative in mood. Indicative clauses are, in turn, divided into interrog-atives and declaratives. Interrogatives are further differentiated into polar(or yes/no) and content or ‘wh’ (when, where, who, why, how, etc.) wherethe ‘wh’ specifies the element to be supplied in the response. While state-ments tend to be realized as declaratives, questions as interrogatives, andcommands as imperatives, the systems of speech function and mood donot always map neatly onto each other. The congruency – or lack thereof –between speech function and mood is particularly relevant with regard tothe use of commands. Commands may be realized either congruently inan imperative mood (e.g. “Let’s first focus on Anton’s idea”) or metaphor-ically, that is, non-congruently, in an interrogative or a declarative mood(e.g. “Can we first focus on Anton’s idea?” or “I would like us to first focuson Anton’s idea”).

Language includes also a system of polarity that allows speakers andwriters to choose between the positive/negative poles of is/isn’t and do/don’t(Halliday, 1994). The intermediate degrees between these two sets of polesare viewed as belonging to the system of modality which includes the sub-systems of modalization and modulation. Modalization is used in propo-sitions for realizing various degrees of probability (possibly, probably,certainly) as well as usuality (sometimes, usually, always), thus cover-ing a semantic space between is and isn’t. Modulation is used in proposalsfor realizing degrees of obligation and inclination between do and don’t.


In commands, modulation is marked by verbs such as have to/must/required to (high), should/will/supposed to (medium), and may/allowed to(low). Teachers often modulate their commands by using low modulationexpressions such as “I want you to. . .,” “Would you like to. . .?” or “Whydon’t we. . .?” rather than the congruent, high modulation “You must. . .”The former grammatical constructions appear to provide students withpossibilities by telling them not what they must do but what they could do.That is, this use of low modulation commands allows for the expression ofteacher authority in an oblique manner (Christie, 2000, p. 20).

As they participate in making a text, speakers exchange meanings byselecting from the lexico-grammatical systems that constitute the inter-personal meta-function. Since a discussion is an interactive text unfoldingin real time, speakers’ choices are also shaped by those already made byothers. To a significant extent the teacher is in charge of setting the tenor,the tone, and the agenda of whole-class discussions. The mood and speechfunction of her clauses will therefore delimit and guide the mood and speechfunction of her students’ contributions. Analyzing the teacher’s moves inthe conjoined making of a spoken text is thus central to explaining howthis text comes to mean what it does.

3. METHOD

3.1. Research site and data collection

The selected whole-class discussion was captured during a nine-monthstudy of a 6th grade classroom attended by African-American, Asian,Caucasian, and Latino students. The school is located in a large urban set-ting in the United States. This 6th grade class of 26 students was highly het-erogeneous in regard both to socio-economic class, racial, ethnic, cultural,and linguistic background and level of mathematical performance. At thetime of this study the teacher (Ms. L.) had 15 years of teaching experience.

Ms. L. usually organizes instruction around multi-phased mathemati-cal inquiries that involve framing, studying, and solving open-ended andnon-routine problems. In this teacher’s classroom, activities are typicallystructured as series of interconnected whole-class discussions that alternatewith individual and small-group work. These whole-class discussions oc-cur not only at the beginning and the closing phases of a given mathematicalinquiry, but also when students are still thinking through a particular aspectof the inquiry underway. The teacher frequently interrupts students’ workand brings the class together for a discussion wherein ideas-in-the-makingare exchanged and further developed under her guidance.

Data were collected from 10 non-consecutive periods of mathematicsinstruction. Lessons were observed by one of the researchers and detailed


field-notes were taken to capture both verbal and non-verbal features of theinteraction, the latter including facial and body gestures as well as the use ofdiagrams, tables, graphs, and other artifacts. Lessons were also audio-tapedand transcribed verbatim. See the following Transcription Conventions.

Transcription Conventions

Layout Moves or turns are numbered consecutively; clauses within moves are

also numbered. Speakers are indicated by name (pseudonyms are used)

Italic Action, gesture, (facial) expression

Underlining Overlapping words spoken by more than one speaker at a time

CAPS Words spoken with emphasis

?! Utterances judged to have an interrogative or exclamatory intent

, Breathing space

. . . Short pause within a clause or a move

. . . . . . Long pause within a clause or a move

3.2. Coding and data analysis

In preparation for analysis, each transcribed lesson was sub-divided intoepisodes. Boundaries between episodes correspond to transitions from oneactivity structure to another one (e.g. from whole-class discussion to groupwork). Next, lesson episodes that consisted of whole-class discussions wereselected for fine-grained analysis. Each of these discussions was then fur-ther divided into fragments corresponding to the specific ideas exchangedduring the event. In coding each discussion, the moves made by teacher andstudents were numbered consecutively and the clauses within each movewere also numbered consecutively.

Our analysis of the selected whole-class discussion qua text centeredon the interpersonal grammar, that is, the grammar of exchanges. Fromamong the linguistic resources afforded by the interpersonal grammar, wechose to code for mood, speech function, and modulation. By coding theteacher’s choices with respect to these systems of choices, we aimed touncover the interpersonal relationships realized in this text. We coded bothteacher moves and student moves in the conversation, yet our analysisfocused on the former given our interest in describing how the teachermediated this speech event. Major clauses were coded twice, first for moodand then for speech function. We coded for mood as follows: declarative(clauses with subject-verb order), imperative (clauses with no subject), andinterrogative (clauses in which the auxiliary verb comes before the subject).The speech function of each major clause was coded as: offer, question,command, statement, and check. Minor clauses are considered mood-lessso we coded them only for speech function using the categories above.


In coding major and minor clauses for speech function, we used situ-ational information. For example, we coded an interrogative clause as acommand to do when (a) it contained the modals “can,” “could,” “will,”“would,” “going to”; (b) the subject of the clause was also the addressee;and (c) the predicate described an action that was physically possible at thetime of the utterance (Coulthard, 1985, p. 131).

The use of minor clauses is ubiquitous in teacher talk. Typical examplesare “okay” and “(al)right” when appearing as continuatives in the open-ing or closing of an inquiry phase. Teachers often use these minor clausesas commands in order to signal the beginning of a new activity structure(textual meta-function) as well as to demand that students attend to whatwill follow (inter-personal meta-function). In coding these minor clausesfor speech function, we considered situational information as well as into-nation. All minor clauses appearing at the end of a clause or a move andrealized with rising intonation, e.g. “okay?” and “(al)right?” were coded aschecks. Notwithstanding the rising tone, this minor clause does not func-tion as a question because no verbal information is required of the studentsother than an affirmative response. Whenever “okay” or “(al)right” ap-peared at the start of a move, uttered with strong stress and a high fallingintonation and followed by a short pause, these minor clauses were codedas commands. Nominations that function to give a specific student the rightto speak were coded as offers. The following is our coding key.

Coding Key

Mood Speech function

Declarative: Dec Statement: St

Interrogative: Question: Qu

Wh: Int(wh) Command: Co

Polar or yes/no: Int(p) Offer: Of

Imperative: Imp Check: Ch

4. ANALYSIS

4.1. The context of the situation

The mathematical problem at hand was located within an algebra unit inwhich students were engaged in the search for patterns and regularities invarious numerical and geometrical situations and the subsequent general-ization of those patterns via recursive and direct formulas. Below is thenumber array which was the object of the whole-class discussion under


analysis.

12 3 4

5 6 7 8 910 11 12 13 14 15 16

17 18 19 20 21 22 23 24 2526 27 28 29 30 31 32 33 34 35 36

... ... ... ... ... ... ... ... ... ... ... ... ...

The elements of this array are consecutive numbers. The triangular con-figuration of these numbers is an integral aspect of the array. This numberarray invites mathematizing, more specifically, algebraizing (Freudenthal,1991), in that it engages students in noticing, symbolizing, and generaliz-ing multiple patterns, that is, structuring this phenomenon with algebraictools. In regards to a similar arrangement of numbers, Zazkis and Liljedahl(2002) point out that, while it is generally agreed that patterns are centralto mathematics, “unlike solving equations or manipulating integers, explo-ration of patterns does not always stand on its own as a curricular topic oractivity” (p. 379). In the whole-class episode selected for analysis, teacherand students were engaged in the kind of pattern-searching activity calledfor by the above researchers.

The teacher had presented the number array in a previous lesson bysaying: “Imagine that we continue on and on writing down numbers onthis diagram. What formula can we use to predict what number would bein the middle of the 50th row?” This question may be answered by squaringthe row number and then subtracting from it 1 less than the row number[502 − (50 − 1) = 2,451]. Noticing that the utmost right number in eachrow is the square of the row number is central to arriving at this expression.Finding the middle number of any row given the row number calls for a the-oretical generalization in that it demands identifying essential invariants inthe given array (Dorfler, 1991). At stake in this lesson was not only noticingpatterns but also perceiving which among those patterns noticed were alge-braically most useful in light of the generalizing task at hand (Lee, 1996).

During subsequent lessons, Ms. L’s class continued working on thisproblem. Using rn for the row number and m the middle number, studentseventually built the direct formula m = r2

n − (rn −1), its equivalent m =r2

n − rn + 1 and, through a different route, m = 1/2{[(rn − 1)2 + 1] +r2n )}

which describes the averaging of the first and the last number in the givenrow. In order to build these algebraic formulas, students had to be able torealize in propositional form the patterns noticed on the array. In particular,arriving at m = r2

n − (rn − 1), requires a thinking (aloud) text that includes


the following propositions: “The utmost right number of every row is thesquare of the row number;” “In any row, the distance between the utmostright number and the middle number is 1 less than the row number;” and“Therefore the middle number of any row may be found by subtracting 1less than the row number from the square of that row number.”

4.2. Lesson episodes

Below we analyze two episodes of a whole-class discussion that occurred inone of the lessons recorded during our study. In between these two episodesof spoken interaction, a brief episode occurred during which students wrotesilently in their notebooks.

4.2.1. Task orientation


The first episode is a task orientation phase wherein Ms. L. as-serts what she expects to happen during the lesson. In this prelude, theteacher demands her students’ attention with a boundary-marking mi-nor clause and declares that time constraints will allow them to workon only one problem (1.1–2). Next, she introduces the task using al-most all declarative statements (1.3, 1.4, 1.7–9). Yet despite the use ofdeclaratives, her opening is rather implicit. Until move 7 (7.21–22), shedoes not asks questions nor does she tell her students what they are todo.

Ms. L. declares that the activity is a revision of one recently doneand that some students had used the same number arrangement in de-signing board games (1.3–4). Here the teacher uses two declarativesthat function to build continuity across lessons and as a reminder thatrevisiting problems is central to the joint activities that occur in theirclassroom. The teacher then chooses a polar interrogative by which shecommands the class to recall that past event, and immediately afterwardspresents the number arrangement written on a large chart paper sheet(1.5–6).

The teacher uses three declaratives to justify revisiting the number array(1.7–9). Her emphasis on the pronoun “all” and the numerative “a lot”indicate, respectively, that she expects every student to work on the taskat hand and that crucial patterns are yet to be found. Thus, although shepresents the search for patterns as an open-ended activity, Ms. L.’s use of“missing” indicates that she has in mind a specific set of patterns studentsare expected to find, as if the text under co-construction had one or moremissing pages. While making room for Dianne’s (2.10) and Victor’s moves(3.11) whereby they claim that the original array, as they remembered it, wasslightly different than the present one, the teacher quickly regains controlof the discussion with a statement realized in the mood of a rhetoricalinterrogative as well as with a boundary-marking minor clause wherebyshe expresses her eagerness to resume providing instructions to the class(7.17–8).

Ms. L. specifies that the task involves copying down the array and writ-ing “four things that you notice” (7.21–22). This move may be read asevidence that the jointly produced text is to be simultaneously realizedthrough written and spoken channels. Worth noticing is that the teachermakes no explicit reference to the ultimate purpose of the search for pat-terns, namely to build a direct algebraic formula for the middle number onthe 50th row. She will not pose the key mathematical question until the lastmove in the episode.


4.2.2. Victor’s ideaThe fragment below took place immediately after the students spent fiveminutes writing down in their notebooks patterns noticed in the numberarray.

Ms. L.’s minor clause “okay” marks the transition from the individ-ual written work of searching for patterns in the number array to thewhole-class conversation in which students are to exchange their find-ings (10.25). Next are two high modulation commands realized in declar-ative mood, both containing the modal “have,” by means of which theteacher instructs the students that they are responsible for listening andcopying down the ideas they like (10.27–28). Ms. L. does not assumethat her students would be inclined spontaneously to take selective notesduring the discussion. She reminds them of this important obligation.Here we find further evidence of the back and forth movement be-tween writing and speaking in the conjoined making of this discussion(text).

In the exchange between Victor and the teacher (11–13), the latter usesa declarative to inform him, and everyone else in the class, that no stu-dent is allowed to contribute more than one idea. Victor’s use of the modal


adjunct “actually” (13.36) and Ms. L.’s implicit acknowledgement of hiscontribution by laughing indicates a relaxed tone. This student feels enti-tled to briefly negotiate with her the very meaning of telling “only one”pattern.

Ms. L. does not react to Victor’s contribution with a declarative eval-uation (e.g. what an interesting idea!), nor does she follow up with aninterrogative request inviting him to recount how he had figured thatout. Instead, she laughs and uses a polar interrogative that functions asa command by which she indirectly requests that someone repeat Victor’scontribution (14.37). The teacher’s request is not met by a verbatim ver-sion of Victor’s statement (15.38–39). Instead, Ivan rewords the latter’snon-relational clause as a relational one, thus translating it into a form thatis typical of the mathematical register (Halliday and Martin, 1993; Veel,1999).

4.2.3. Leslie’s and Alex’s ideas

Ms. L. uses a minor clause with the speech function of an offer in order tonominate a student to speak (16.40). In response to Leslie’s contribution(17.41–42), the teacher chooses to use a polar interrogative as an explicitcommand for the student to further elaborate her idea (18.43). After Leslie


does so using a more precise wording (“columns” replaces “corners”)(19.44–45), Ms. L. enhances her contribution with a non-verbal move inwhich she inserts symbols for “odd” and “even” on the array (20.46). Thisclarifies Leslie’s statement: the number 26 is located in a column withan odd number of numbers. Right afterwards, when Alex’s idea comesup unsolicited, the teacher reacts non-verbally by translating “slanted go-ing up” into arrow language and inscribing this observation into the array(22.48).

4.2.4. Mimi’s, Donald’s, and Dianne’s ideas

Mimi’s contribution (23.49) has the kernel for answering the still un-formulated question: What is the middle number on the 50th row? YetMs. L. does not explicitly evaluate, dwell, or pounce on this contri-bution. Instead, she requests collective agreement by addressing a po-lar interrogative question to the whole class (24.50). This move high-lights the centrality of the idea without ‘funneling’ the discussion (Voigt,1985). Although Dianne’s pattern (27.53–56) is not as mathematicallycentral as Mimi’s, Ms. L. responds to her in a similar fashion as shehad to Mimi. The teacher’s use of a tagged declarative serves thedual function of rewording Dianne’s contribution by replacing “it” with“each row” and, once again, requesting agreement from the entire class(28.57).


4.2.5. Erick’s idea


This fragment centers on a pattern found by Erick. Ms. L. rejoins hisstatement with a declarative functioning as a high modulation command(via the modal “have to”) for him to work further on his idea by writ-ing a general rule (31.65). When Erick regains the floor and contin-ues explaining his pattern, he is met by a chorus-like complaint (33.68–69). As he tries again, this time complementing his speech and writingwith an ostensive gesture, Dianne interrupts him claiming to have un-derstood. In response to the latter’s unsuccessful attempt at explainingErick’s idea, the teacher ‘re-voices’ (O’Connor and Michaels, 1993) thisstudent’s contribution with two declarative statements followed by an in-terrogative question aimed at checking with him the accuracy of her re-formulation (36.76–8). Erick’s use of the modal adjunct “really” maybe read as further evidence of a relaxed tone, an atmosphere in whichit is acceptable for a student to question the accuracy of the teacher’srevoicing.

Having failed to accurately re-voice Erick’s idea, as noted by his use ofthe modal “really” (37.79), Ms. L. uses an imperative command to authorizethe student to speak for himself (38.82). After Erick makes a new attemptat presenting his idea, the teacher once again reacts with a high modulationcommand – realized in declarative mood – asking him to improve the writ-ing of his idea (40.88). When Ricky offers to rephrase Erick’s contribution,Ms. L. at first ignores him, pursuing instead her exchange with Erick bytelling the latter, via both imperative commands and declarative statements,to continue working on his idea (42.90–2). After Ricky takes the floor to tryexplaining Erick’s finding, the teacher responds non-verbally by insertingarrows in the array (44.97).


4.2.6. Donald’s and Ivan’s ideas

In this brief fragment the teacher makes only one brief move using an im-perative command in response to a student’s request to come to the board(48.103). In this fragment there is an instance of an explicit evaluation oc-curring within a student-to-student exchange (45.98 and 46.99–100). Theensuing contribution by Ivan (49.105–6) extends Mimi’s recognition ofthe square number as the last numeral in every row suggesting that eachrow could be replaced (or summarized) by the square of the row num-ber. Once again, Ms. L. does not explicitly evaluate or highlight this idea,or ask for students to link it to Mimi’s contribution. She also does notmake this link herself. As will be clear in both the next and the final frag-ments of this episode, the evaluation of ideas is to happen later, as studentsinteract with each other’s contributions in writing, studying them and de-ciding which may prove the most valuable tool for algebraizing the array athand.

4.2.7. Jason’s idea


The exchanges in this fragment center on the ideas contributed by Jasonand by Mimi. The teacher again speaks only once. When Jason suggeststhat they could modify the array by making it into two triangles with “a linein the middle” (50.107–111), Erick dismisses this contribution (51.112–4)while Leslie embraces it by further considering the geometric shape ofthe transformed array (52.115). Ms. L. intervenes by evaluating Jason’ssuggestion via a tagged declarative statement and a declarative with thespeech function of a high modulation command asking students to take anote of Jason’s idea for further study (53.116–7). She praises this student’sidea by declaring that it belongs to the text jointly being constructed.

4.2.8. Closure

Ms. L. closes the discussion – and the lesson – with a brief monologue thatincludes a declarative functioning as an evaluative statement (56.121–5).After expressing her satisfaction with the number of patterns found by theclass, she continues with two declaratives functioning as commands, onewith a low modulation interpersonal metaphor (“I want you to. . .”) and theother one with high modulation (“You need to. . .”) (56.121–3). A declar-ative statement follows by means of which the teacher makes explicit forthe first time in the discussion the question that motivated the search forpatterns (56.124). Yet, she does so rather tentatively, with a clause thatbegins with “maybe” and a suggestion that the task at hand goes further


than finding the middle number on the 50th row towards building a generalformula for the middle number in any given row (“or the 100th row”).At the end of her last move, Ms. L. chooses an imperative mood for com-manding her students to continue investigating patterns in the number array(56.125).

5. DISCUSSION

The selected whole-class discussion may be viewed as paradigmatic ofteaching and learning algebraizing via guided reinvention (Freudenthal,1991). Within algebraizing, searching for patterns is conceived of as anopen-ended activity. Organizing a phenomenon via algebraic means en-tails first collecting as many patterns as possible and then studying (i.e.comparing, selecting, combining) these patterns with an eye to buildingthe sought formula. Key to this pattern-finding stage of algebraizing is thesuspension of belief (Dewey, 1910) that prevents us from following the firstpattern found into a formula thereby missing all the others and perhaps thebest out of the bunch.

The field of this discussion text was framed by two mathematical ques-tions: What patterns can you find in this diagram? And what formula couldbe used for finding the middle number in the 50th row, the 100th row, or anygiven row? Between these there was another, albeit an implicit question:Which among the patterns found could be most helpful for constructing ageneral rule for the middle number on any given row? In her prelude tothe activity, the teacher did not ask or tell her students to solve the mathe-matical problem at hand. Instead, she instructed them to study the array ofnumbers with an eye to finding as many patterns as possible. At the end ofthe lesson, she requested the class to continue studying the patterns sharedduring the discussion.

Students in this 6th grade class were not merely searching for patterns inthe sense of seeing or finding as such, but were also reconstructing the givensituation via verbal thinking. Dewey (1938) argues that central to reflectivethinking within an inquiry is signifying observations in propositional form,a process that Halliday (1994) describes as realizing experiences by meansof resources afforded by the ideational grammar. These propositions aresemiotic tools for reflective thinking that are super-added into the situa-tion at hand. During the discussion, students were asked to find, signify,and collect as many patterns as possible and then study these further. Oneor several of these patterns were to be eventually reconstructed into an-other semiotic tool (i.e. an algebraic formula) for solving the problem athand.


Ms. L.’s students had so little time to find and signify patterns on theirown that it is reasonable to assume some of them continued to do so duringthe conversation. The transition from writing individually to talking to-gether, a switch in the mode of the text, did not necessarily end the noticing-signifying. Evidence of this continuity may be found in some students’ useof the present tense in comments made regarding their noticing of patterns.Some of the students who contributed to the discussion were quite possi-bly still thinking, and were therefore thinking aloud verbally on a socialplane of learning. Furthermore, some appeared to have been led to find newpatterns as a consequence of those offered by their peers. By not allowingsufficient time for students to finish their thinking and, instead, bringingthem together in an ensemble, the teacher set the stage for verbal semioticmediation to occur on a social plane under her guidance.

Ms. L. often commanded her students to signify noticed patterns inpropositional form, thereby directing them to think verbally in regardto the given array for the purpose of mediating their mathematical ac-tivity. She adopted the speech role of demanding verbal mathematicalthinking from her students and assigned to them the complementary roleof giving, that is, signifying and exchanging patterns. The teacher fre-quently contributed statements – mostly realized congruently in a declar-ative mood – which did not add new mathematical content to the dis-cussion. It was up to the students to signify and exchange the ideas andto thereby offer the mathematical propositions called for by the task athand. Ms. L.’s lexico-grammatical choices guided the students in rec-ognizing the genre and register of the text in the making. This accuraterecognition led them to make the appropriate lexico-grammatical choicesin their contributions to this text. In doing so, the teacher structured thewhole-class discussion as an interpersonal gateway for expanding her stu-dents’ potential to mediate their emergent algebraic activity via verbalthinking.

Ms. L. guided her students to interpret the task at hand as an open-endedsearch for patterns, in part, by refraining from explicitly evaluating theircontributions. She never let on that the best fitted pattern had been found,or even that it existed at all. Her interpersonal lexico-grammatical choiceswere consistently proposals commanding the students to signify or brieflyexplain on the social plane a pattern they had noticed. The teacher usedcommands – realized either as imperatives or as modulated declaratives – orstatements no matter what the algebraic significance of the pattern offered.This sheds light on an important lesson to be learned by the students inthis class, namely, how to conduct an open-ended search and verbalizationof patterns so that it successfully serves its appointed mediating functionwithin the larger context of an algebra inquiry.


While in the opening episode students were asked each to find four pat-terns on the array, no student was allowed to contribute more than one to thediscussion. Given the open-endedness of the task, many different studentswould therefore have the opportunity to add to the conjoined text-in-the-making. The aimed at text of signified patterns would thus be made on thesocial plane of a whole-class discussion. The open-endedness of the taskand the constraint of one contribution (pattern) per student facilitated thearticulation of multiple zopeds. The teacher’s lexico-grammatical choiceswere effective in opening the social plane of verbal thinking to many ofher students. She was able to move the text along quickly from one stu-dent’s pattern to the next, thereby making room for as many as possiblewithin the brief period of time made available for the discussion. Elevenstudents out of a total of 26 made spoken contributions, with the students asa group contributing 60% of the total number of clauses exchanged duringthe discussion.

After the initial instructions, the students engaged in a brief episode ofindividual work in which they wrote down in their notebooks those pat-terns they noticed in the number array. These written texts were then usedto begin the conjoined construction of the spoken verbal text of patternson the social plane. Throughout the discussion, Ms. L. consistently com-manded her students to take notes. This whole class discussion was in fact amultimodal text which entailed, in part, writing down what was being said.We view the teacher’s commands to take notes and write down ideas asdirections to abstract the text of propositionally formulated patterns fromthe text of learning how to make such a text. This amounts to editing outthe teacher’s proposals keeping only the conjoined collection of signifiedpatterns. She was, in fact, commanding them to edit out her commands. Indemanding that her students continue studying their notes, Ms. L. directedthem not only to study the patterns as such, but also to treat their classnotes as a mediating semiotic tool. Moreover, this text could function as asocially constructed paradigmatic instance of written verbal mathematicalthinking that students might appropriate and use for framing and solvingsimilar problems in the future.

6. CONCLUDING REMARKS

In this paper we analyzed how a teacher engaged her 6th grade students inmaking a conjoined text of patterns found in a number array. The teacher’srole in the discussion was to guide the co-production of an open-endedtext of observed-signified patterns that functioned as a social plane for stu-dents to learn how to use verbal thinking to mediate mathematical inquiry.


Viewed through the lens of functional grammar, Ms. L.’s role in the con-versation was to command her students in regard to their role. That is, herlexico-grammatical choices were geared toward commanding the studentsas to their choices. Given the context of this lesson, with this group ofstudents, at that given point in the year working together, we can say thatshe was successful in accomplishing this goal, to the degree that she guidedher students’ participation in a thinking-aloud conversation in which theycreated a text of signified patterns that could be used as a semiotic-verbaltool for mediating algebraic activity.

The above should not be read as a sweeping assertion that teachercommands are instrumental to the success of all whole-class discussionsthat aim at teaching mathematical thinking. In fact, we would argue justthe opposite, that is, that no one kind of lexico-grammatical choice onthe teachers’ part within whole-class discussions will be suited to guideverbal semiotic mediation. The teacher’s lexico-grammatical choices de-pend upon what is appropriate for the genre and register of the con-joined text, the nature of the problem at hand, the phase within the in-quiry process, and the level of appropriation of the targeted tools anddispositions.

We intended this analysis to illustrate the power of SFL for studyingthe inner grammar of classroom interactions so as to illuminate the com-plexities and subtleties in the teacher’s mediating role. At the surface level,the authoritative manner in which the teacher mediated the whole-classdiscussion may have seemed un-conducive to engaging her students ingenuine mathematical exchanges. Our analysis of Ms. L.’s moves in theconversation allowed us to demonstrate that the contrary was the case,highlighting how through her very lexico-grammatical choices, the teacherguided these sixth graders in thinking aloud together about algebraicallysignificant patterns in a number array.

REFERENCES

Bartolini Bussi, M.G.: 1998a, ‘Verbal interaction in the mathematics classroom,’ in H.

Steinbring, M. Bartolini Bussi and A. Sierpinska (eds.), Language and Communicationin the Mathematics Classroom, NCTM, Reston, VA, 65–84.

Bartolini Bussi, M.G.: 1998b, ‘Joint activity in the mathematics classroom: A Vygotskian

analysis’, in F. Seeger, J. Voigt and U. Waschescio (eds.), The Culture of the Math-ematics Classroom: Analyses and Changes, Cambridge University Press, Cambridge,

pp. 13–49.

Carpay, J.: 1999, ‘The interdependence between forms of mutuality and the development

of theoretical interest in the classroom’, Mind, Culture, and Activity 6(4), 314–324.

Christie, F.: 2000, Classroom Discourse Analysis: A Functional Perspective, Continuum,

London and New York.


Coulthard, M.: 1985, An Introduction to Discourse Analysis, 2nd edition, Longman, London

and New York.

Dewey, J.: 1910, How We Think, D.C. Heath, Boston.

Dewey, J.: 1938, Logic: The Theory of Inquiry, Henry Holt, New York.

Dorfler, W.: 1991, ‘Forms and means of generalization in mathematics’, in A.J. Bishop (ed.),

Mathematical Knowledge: Its Growth through Teaching, Kluwer Academic Publishers,

Dordrecht, pp. 63–85.

Eggins, S.: 1994, Introduction to Systemic Functional Linguistics, Continuum, London.

Forman, E.A., Larreamendy-Joerns, J., Stein, M.K. and Brown, C.: 1998, “You’re going

to want to find out which and prove it’: Collective argumentation in a mathematics

classroom’, Learning and Instruction 8(6), 527–548.

Freudenthal, H.: 1991, Revisiting Mathematics Education: China Lectures, Kluwer, Dor-

drecht.

Halliday, M.A.K.: 1973, Explorations in the Functions of Language, Elsevier, New York,

Oxford, and Amsterdam.

Halliday, M.A.K.: 1978, Language as Social Semiotic: The Social Interpretation of Lan-guage and Meaning, Arnold, London.

Halliday, M.A.K.: 1993, ‘Towards a language-based theory of learning’, Linguistics andEducation 5, 93–116.

Halliday, M.A.K.: 1994, An Introduction to Functional Grammar, 2nd edition, Arnold,

London.

Halliday, M.A.K. and Hasan, R.: 1989, Language, Context, and Text: Aspects of Languageis a Social-Semiotic Perspective, Oxford University Press, Oxford.

Halliday, M.A.K. and Martin, J.: 1993, Writing Science: Literacy and Discursive Power,

Falmer Press, London.

Halliday, M.A.K. and Matthiessen, C.: 1999, Construing Experience through Meaning: ALanguage-Based Approach to Cognition, Cassell, London and New York.

Hasan, R.: 1996, Ways of Saying, Ways of Meaning, Cassell, London and New York.

Lee, L.: 1996, ‘An initiation into algebraic culture through generalization activities’,

in N. Bednarz, C. Kieran and L. Lee (eds.), Approaches to Algebra: Perspec-tives for Research and Teaching, Kluwer Academic Publishers, Dordrecht, pp. 65–

86.

Martin, J.R.: 1984, ‘Language, register, and genre’, in F. Christie (ed.), Children Writing:A Reader, Deakin University Press, Geelong.

Martin, J.R. and Veel, R. (eds.): 1998, Reading Science: Critical and Functional Perspec-tives on Discourses of Science, Routledge, London.

Martin, T.S., McCrone, S.S., Wallace Bower, M.L. and Dindyal, J.: 2005, ‘The interplay of

teacher and student actions in the teaching and learning of geometric proof’, EducationalStudies in Mathematics 60, 95–124.

Morgan, C.: 2000, ‘Language in use in mathematics classrooms: Developing approaches to

a research domain’, Educational Studies in Mathematics 41, 93–99.

O’Connor, M.C.: 2001, ‘Can any fraction be turned into a decimal?: A case study

of a mathematical group discussion’, Educational Studies in Mathematics 46, 143–

185.

O’Connor, M.C. and Michaels, S.: 1993, ‘Aligning academic task and participation sta-

tus through revoicing: Analysis of a classroom discourse strategy’, Anthropology andEducation Quarterly 24(4), 318–335.

Schleppegrell, M.: 2001, ‘Linguistic features of the language of schooling’, Linguistics andEducation 12(4), 431–459.


Sfard, A. and Kieran, C.: 2001, ‘Cognition and communication: Rethinking learning-by-

talking through multi-faceted analysis of students’ mathematical interactions’, Mind,Culture, and Activity 8(1), 42–76.

Steinbring, H., Bartolini Bussi, M. G. and Sierpinska, A. (eds.): 1998, Language and Com-munication in the Mathematics Classroom, National Council of Teachers of Mathematics,

Reston, VA.

Veel, R.: 1999, ‘Language, knowledge and authority in school mathematics’, in F. Christie

(ed.), Pedagogy and the Shaping of Consciousness, Linguistics and Social Processes,

Cassell, London and New York, pp. 185–216.

Voigt, J.: 1985, ‘Patterns and routines in classroom interaction’, Recherches en Didactiquedes Mathematiques 6(1), 69–118.

Vygotsky, L.S.: 1978, Mind in Society: The Development of Higher Psychological Func-tions, Harvard University Press, Cambridge, MA.

Vygotsky, L.S.: 1986, Thought and Language, MIT Press, Cambridge, MA.

Wells, G.: 1994, ‘The complementary contributions of Halliday and Vygotsky to a

“Language-based theory of learning’”, Linguistics and Education 6, 41–90.

Wells, G.: 1999, Dialogic Inquiry: Toward a Socio-cultural Practice and Theory of Educa-tion, Cambridge University Press, Cambridge, UK.

Zack, V. and Graves, B.: 2001, ‘Making mathematical meaning through dialogue: Once you

think of it, the z minus three seems pretty weird’, Educational Studies in Mathematics46, 229–271.

Zazkis, R. and Liljedahl, P.: 2002, ‘Generalization of patterns: The tension between algebraic

thinking and algebraic notation’, Educational Studies in Mathematics 49, 379–402.

BETINA ZOLKOWER

School of EducationBrooklyn CollegeCity University of New York2900 Bedford AvenueBrooklyn, NY, 11210Telephone: 1-718-951-3113Fax: 1-718-951-4816E-mail: [email protected]

SAM SHREYAR

Lehman CollegeCity University of New York

a teacher’s mediation of a thinking-aloud discussion in a 6th grade mathematics classroom

Documents