Understanding the role of transactive reasoning in classroom discourse as students learn to construct proofs

Journal of Mathematical Behavior 34 (2014) 76–98

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The Journal of Mathematical Behavior

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Understanding the role of transactive reasoning in classroomdiscourse as students learn to construct proofs�

Maria L. Blantona,∗, Despina A. Stylianoub,1

a TERC, 2067 Massachusetts Avenue, Cambridge, MA 02140, USAb City College, The City University of New York, 160 Convent Avenue, New York, NY 10031, USA

a r t i c l e i n f o

Article history:Received 20 October 2012Received in revised form 22 January 2014Accepted 3 February 2014

Keywords:Sociocultural perspectiveUndergraduate educationTransactive reasoningMathematical proof

a b s t r a c t

This study uses a sociocultural perspective to examine the role of transactive reasoning in,whole-class discourse as undergraduate students learn to construct mathematical proofs.The research, setting is an undergraduate mathematics course with 30 participants. Dataare whole-class transcripts, of lessons focused on developing mathematical proofs andstudents’ written assessments on proofs. Transcript data are analyzed for (1) shifts in stu-dents’ knowledge about proofs; (2) the nature of, transactive reasoning (Berkowitz, Gibbs,& Broughton, 1980) in whole class discourse, including how it occurred and, indicationsthat students appropriated transactive reasoning as a practice of discourse; and (3) how,transactive reasoning supported students’ active constructions of proofs and understandingof proof. Results indicate that classroom discourse that helps students appropriate trans-active reasoning as a habit of interaction supports their capacity to build arguments aboutincreasingly complex, mathematical ideas and, as such, has positive implications for theirlearning of proof.

© 2014 Elsevier Inc. All rights reserved.

1. Introduction

1.1. Language as a mediator of learning

No utterance is neutral. As Bakhtin noted, “utterances are not indifferent to one another, and are not self-sufficient; theyare aware of and mutually reflect one another. . ..[and] every utterance must be regarded primarily as a response to precedingutterances” (1986, p. 91). If we accept this claim as an axiom of discourse, it not only points us generally to the reciprocalinfluence conversation has on its participants, it also raises the particular issue of how teacher utterances in classroom
conversation influence student learning. Cazden, noting that classroom discourse happens among the students and teacherwhile the heart of education is to effect learning within each student, raises the singular question, “How do the words spokenin classrooms affect this learning” (2001, p. 60)?
� The research reported here was supported in part by the National Science Foundation under Grant # REC-0337703. Any opinions, findings, andconclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National ScienceFoundation.

∗ Corresponding author. Tel.: +1 617 873 9640.E-mail addresses: Maria [email protected] (M.L. Blanton), [email protected] (D.A. Stylianou).

1 Tel.: +1 212 650 7000.

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M.L. Blanton, D.A. Stylianou / Journal of Mathematical Behavior 34 (2014) 76–98 77

The study of classroom discourse has its roots in language as a mediator of learning. In recent decades, Vygotsky’s1962/1934) sociocultural perspective has generated much interest in the study of how psychological tools—particularly,anguage—serve to guide human behavior. He argued that “higher voluntary forms of human behavior have their roots inocial interaction, in the individual’s participation in social behaviors that are mediated by speech” [italics added] (Minick,996, p. 33) and that language serves as a “manifestation of the transition between social speech on the inter-psychologicallane (between individuals) and inner speech on the intra-psychological plane (within the individual)” (Wertsch, 1988, p. 86).alliday further reasoned that the “distinctive characteristic of human learning is that it is a process of making meaning—a

emiotic process; and the prototypical form of human semiotic is language” (1993, p. 93). This dialectic of speech—that is,hat language is both tool and result of psychological functioning (Holzman, 1996)—captures the theoretical premise of thetudy reported here: Learning is shaped by the social context in which it occurs, and the speech occupying that social contextncodes a story of development that can be deconstructed through an analysis of discourse.

Prompted by Vygotsky’s groundbreaking work, the convergence of thinking embodied in international perspectives onathematics education (e.g., Wirszup & Streit, 1987) and the rise of social, situated learning perspectives (e.g., Gardner, 1985;reeno, 1989) has re-cast learning through participation metaphors (e.g., Sfard, 2002). It has also led to calls for educational

esearch that investigates “the relationship between discourse and knowing as it occurs in particular, situated activities”Wells, 1999, p. 102). All of this has brought what the teacher does—and says—in the mathematics classroom into relief. Were interested here in this dynamic as students learn to construct mathematical proofs in undergraduate classrooms.2

.2. Proof as an essential topic in learning mathematics

The ideas of proving and justifying have received increased attention as a necessary part of students’ mathematical devel-pment (e.g., Boero, 2007; Committee on the Undergraduate Program in Mathematics, 2004; National Council of Teachersf Mathematics [NCTM], 2000; RAND Mathematics Study Panel, 2002; Stylianou, Blanton & Knuth, 2009). Most recently,he Common Core State Standards Initiative [CCSSI] (2010) identified constructing viable arguments as one of its eight core

athematical Practices. Indeed, claims that the “essence of mathematics lies in proofs” (Ross, 1998, p. 2) and that proof isthe soul of mathematics” (Schoenfeld, 2009, p. 12) reinforce the centrality of proof in mathematical thinking.

In spite of this, students’ difficulty in learning how to understand and construct proofs is well documented (e.g., Senk,985; Stylianou, Blanton, & Rotou, 2014; Usiskin, 1987). Studies have found, for example, that students struggle with whatonstitutes a proof and with understanding the power of a generalized argument as covering all possible cases (e.g., Fischbein

Kedem, 1982; Healy & Hoyles, 2000). Moreover, they have difficulty with the logic and methods of proof (e.g., Duval, 1991)nd the problem solving skills necessary to construct arguments (Schoenfeld, 1985).

While studies such as these give us critical information in understanding how one learns to read and construct proofs,hey focus on individual cognitive processes. Yet, in light of contemporary ways of knowing, studies are also needed thatxamine the social aspects of teaching and learning proof (Alibert & Thomas, 1991). In response to this, the research reportedere is an effort to understand the role of whole-class discourse, specifically, the form of teacher and student utterances, inhe development of undergraduate students’ learning of proof.3

. Constructs constituting our frameworks for analyzing students’ learning of proof

In this section, we discuss the constructs constituting our frameworks for analyzing students’ learning of proof and usehese constructs to pose the research questions that frame this study.

.1. Using transactive reasoning to analyze classroom discourse

We found transactive reasoning, defined by Berkowitz, Gibbs, and Broughton (1980, presented in Kruger, 1993) as criti-isms, explanations, justifications, clarifications, and elaborations of one’s own or another’s ideas, to be a useful constructor analyzing whole-class discourse. Applying transactive reasoning to a study that sought to characterize the mechanism ofhange in peer discussions on socio-moral dilemmas, Kruger (1993) found that children’s transactive reasoning about solu-ions they challenged and eventually rejected was closely associated with positive posttest performance. Kruger argued thats children critiqued each other’s thoughts in the transactive discussion of rejected solutions, they co-constructed under-tanding. As such, dissent and conflict became vehicles for change in thinking. Elsewhere, Goos, Galbraith, and Renshaw2002) focused on how students used transactive reasoning in mathematical problem solving in small group peer dis-ussions. Similar to Kruger, they found that transactive discussions were a significant source of productive metacognitive
ctivity because these discussions led to the public scrutiny of ideas among peers. Further, Goos et al. (2002) claim thathis metacognitive activity due to transactive reasoning led to successful problem solving; students not only engaged inesirable behaviors (such as checking their work due to criticisms of others), but also increased their performance.
2 We take “mathematical proof” here to refer to a logical, deductive argument constructed within the accepted standards of the mathematics community.3 We take the term “discourse” here to refer to verbal utterances.

78 M.L. Blanton, D.A. Stylianou / Journal of Mathematical Behavior 34 (2014) 76–98

These studies suggest that when students are reasoning transactively, they are actively constructing an understandingof a particular concept or process. Although this research was conducted in laboratory or small group settings and is basedon analyses of conversations among peers, it is reasonable to consider whether and how these findings extend to whole-class settings as well, where the conversational dynamic includes the teacher as a more knowing other. As such, we usedtransactive reasoning here to analyze the form of teacher and student utterances in classroom discourse. Our initial purposewas to develop a framework to identify the forms of utterances based on how they supported or constrained students’transactive reasoning. Within this context, we wanted to understand how these particular forms of discourse occurredduring whole-class discussions that focused on constructing proofs, whether the transactive forms were appropriated bystudents as a practice of discourse, and if so, how students’ transactive reasoning supported students’ constructions ofproofs in-the-moment. We note here that, although our focus was on whole-class discussions, the day-to-day work of theclassroom included small-group discussions and group constructions of proofs. However, we are particularly interested herein whole-class discussions because the teacher was also a participant.

2.2. Using proof schemes and strategic knowledge to analyze shifts in students’ knowledge

In order to triangulate our findings on the role of transactive reasoning in discourse, we used the constructs of proofschemes (Harel & Sowder, 1998) and strategic knowledge (Weber, 2001) as a framework for examining shifts in students’mathematical knowledge about proof and to establish that the mathematical ideas being discussed developed into moresophisticated forms. As van Oers (2002) noted, it is not sufficient that students simply develop social norms of participation,but that their participation reflect the development of a mathematical discourse. Elsewhere, Williams and Baxter (1996)distinguish between “analytic and social scaffolding” (p. 24), where analytic scaffolding is specific to mathematical ideas andsocial scaffolding relates to norms and social behavior regarding discourse that are not specific to mathematics. Using theconstructs of analytic and social scaffolding, Nathan and Knuth (2003) found that high levels of interactions among studentsdo not necessarily advance mathematical knowledge unless the teacher functions as a mathematical authority, monitoringthe quality of students’ engagement with mathematical ideas through the use of analytic scaffolding. In what follows, webriefly describe the constructs of proof schemes and strategic knowledge.

2.2.1. Proof schemesHarel and Sowder (1998) propose three levels of student proof schemes: (1) external conviction proof schemes; (2)

empirical proof schemes; and (3) analytical proof schemes. External conviction proof schemes are schemes in which students’arguments are based on sources external to the student without reference to the symbols’ meaning. Such sources includethe form or appearance of the argument, the word of an authority, or a rote symbolic manipulation. Empirical proof schemescan be either inductive or perceptual. When a student uses examples or specific cases as the basis for an argument, he orshe is considered to have an inductive proof scheme. In a perceptual proof scheme, a conjecture is validated via rudimentarymental images, that is, “images that consist of perceptions and a coordination of perceptions but lack the ability to transformor to anticipate the results of a transformation” (Harel & Sowder, 1998, p. 255). One’s proof scheme is characterized asanalytical when the argument is based on the use of logical deduction. Analytical proof schemes can be either transformationalor axiomatic. A transformational proof scheme involves goal-oriented operations on objects. The student operates with adeductive process in which she considers generality aspects, applies goal-oriented and anticipatory mental operations, andtransforms images. An axiomatic proof scheme goes beyond a transformational one in that the student also recognizes thatmathematical systems rest on (possibly arbitrary) statements that are accepted without proof.

2.2.2. Strategic knowledgeStudents’ proving activity should also entail strategic knowledge (Weber, 2001). Weber argues that even those students

who might appreciate a deductive argument and be aware of and able to apply the facts that should lead to a proof often failto do so. He attributes this failure to lack of strategic knowledge, that is, the knowledge to use facts, theorems and techniquesin appropriate ways in a proof. Moreover, he identified four types of strategic knowledge that are essential in one’s capacityfor proving: (1) recall of definitions; (2) search for properties, theorems and facts that are likely to be useful; (3) knowledgeof when to use syntactic (procedure-based) and semantic (conceptual) knowledge in a proof; and (4) search for appropriateproving techniques.

2.3. Research questions guiding the study

Using these constructs as lenses for analyzing whole-class discourse on proofs in an undergraduate mathematics class-room and for identifying shifts in students’ mathematical knowledge about proofs, we considered the following questions:

(1) What is the mathematical evidence for shifts in students’ knowledge of constructing proofs?(2) How do the forms of teacher and student utterances in whole-class discourse occur and what is the evidence of whether

students appropriated transactive reasoning as a practice of discourse?(3) How does transactive reasoning in in-the-moment whole-class discourse mediate students’ constructions of proofs?


Table 1Lists of topics addressed and topics (indicated by *) that included a focus on proofs and proving.

Fall semester Spring semester

Fundamental principles of counting RelationsRule of sum, rule of product *Properties of relationsPermutations *Equivalence relations and partitionsCombinations and the binomial theorem

Fundamentals of logic Rings and modular arithmetic*Basic connectives and truth tables *The ring structure*Laws of logic *Ring properties and substructures*Use of quantifiers *The integers modulo n*Quantifiers, definitions, techniques of proof *Ring homomorphisms and isomorphisms

Set theory Group theory introduction*Sets and subsets *Definition, examples, and properties of groups*Set operations and laws of set theory *Homomorphisms, isomorphisms, and cyclic groupsCounting and Venn diagramsProbability

Properties of the integers: mathematical induction Graph theory introduction*The well-ordering principle: mathematical induction *Definitions and examples of graphs*Recursive definitions *Subgraphs, complements, and graph isomorphism*The division algorithm *Vertex degree: Euler trails and circuits*The GCD: Euclidean algorithm

Relations and functions Trees

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*Cartesian products and relations *Definitions, properties and examples of trees*Functions: plain and one-to-oneOnto functions

. Methods

.1. Setting

We conducted a 1-year classroom teaching experiment (Confrey & Lachance, 2000; Steffe & Thompson, 2000) in anndergraduate, two-semester, discrete mathematics course that included an emphasis on learning to construct mathematicalroofs.4 While there was a common set of required topics for the course, there was some flexibility to increase the focusn proof by eliminating less essential topics. Table 1, which lists the mathematical topics addressed in the class, indicateshat 80% of the topics included an emphasis on constructing proofs or the logic and techniques associated with proving. Theophomore-level class met for 75-min lessons twice per week and consisted of approximately 30 undergraduate mathematicsnd computer science majors, all of whom participated in the study. The course was taught by one of the researchers, andt least two other members of the project team observed the lessons. The project team met weekly to debrief regarding theessons taught, plan subsequent ones, and begin informal analysis.

Because of the teaching experiment methodology used here, the teacher’s role was central to the work. The teacherramed the norms of action and interaction early in the course to build a classroom culture that allowed students to shareheir views (Yackel, Rasmussen, & King, 2000). In particular, the teacher worked to establish the socio-mathematical normhat students justify, explain and share with their peers their thinking and solution processes. As noted earlier, day-to-ay lessons were organized around both whole-class and small-group discussions, although the research focus here is onhole-class discussions.

The researcher-as-instructor model used in this teaching experiment allowed us to be on the “inside” of this instructionalrocess and, thus, understand the intents and purposes of the teacher’s actions rather than simply observe them. Suchlassroom-based research methodology, in which the role of the researcher/observer enlarges to that of teacher (Ball, 1993;obb, Confrey, diSessa, Lehrer, & Schauble, 2003), offers a critical lens into learning.

.2. Data

Data analyzed for this study were transcriptions of video-taped, whole-class lessons and students’ written assessments
dministered in the course. In order to record classroom discourse, all whole-class instruction was videotaped throughouthe 1-year teaching experiment. To facilitate data reduction (Chi, 1997), only the portions of lessons where whole-classiscussions focused on teaching and learning proofs were identified by the authors for transcription and analysis. As indicated
4 As is often the case with teaching experiments, this particular section course was chosen because it was accessible to us. Some demographic analysisuggests that the participants were representative of the whole population that takes the course, but we make no claim that this was a randomly assignedection. The course was designed to allow for certain types of interactions—these did not occur spontaneously—aiming to address the goals of this study.


Table 2Task from episodes analyzed and their point of occurrence.

Lesson Task from episode analyzed

September Prove: the sum of an even number and an odd number is oddOctober Prove:

√2 is an irrational number

November Prove: A ∩ ∅ = ∅April Let G be a group and H = {a ∈ G | ag = ga, for all g ∈ G}. Prove H is a subgroup of G

Table 3Items given on the September and December assessments.

September assessment December assessment

Prove:1. The sum of an even and an odd number is always odd2. Primes 2 and 3 are “consecutive primes” because they are

Prove:1. The product of an even number and an odd number is always even2. For all irrational numbers x, 5x is irrational

consecutive integers that are also prime numbers. Prove that there isno other pair of consecutive primes3. For all irrational numbers x, x-8 is irrational

3. Primes 2 and 3 are “consecutive primes” because they areconsecutive integers that are also prime numbers. Prove that there isno other pair of consecutive primes

in Table 1, not all of the topics addressed involved the construction of proofs; lesson excerpts that did not directly addressaspects of proving in instruction were not included in our data set.

To further reduce the volume of data, we concentrated our analysis on the first semester of the 1-year experiment, withan additional data point (lesson) selected at the end of the year as a retrospective point of analysis. Four lessons were selectedbecause of their strategic location throughout the year as well as the content they addressed relative to constructing proofs. Inparticular, one lesson was selected from each full month of the first semester of the teaching experiment (September, October,and November) and near the end of the second semester (April). While the content of these lessons came from differentmathematical domains (i.e., number theory, set theory, and group theory), their central feature was that the mathematicalclaims being proved were increasingly complex.

The September lesson represented the starting point for whole-class discussions on the nature of proving and constructingmathematical proofs. Prior to this, students had not had formal instruction in developing mathematical proofs. While theOctober lesson continued with number theory as in the September lesson, it involved a different proof technique—proofby contradiction—than the September lesson. The November lesson was based on set theory and, thus, represented a shiftaway from conjectures for which students might reason intuitively about a conjecture using their experiences with number.For example, while students had had many experiences with adding even numbers and odd numbers prior to this class,they had had little experience with sets and no experience with the notion of element arguments on which proofs in settheory are often based. In this sense, the November lesson represented a shift toward applying proof techniques in a moreabstract mathematical domain. The April lesson was selected to provide a critical retrospective point for comparing students’understanding of proof and how they reasoned transactively in the construction of proofs both near the end of the 1-yearexperiment and in a mathematical context—group theory—that was sufficiently different from those considered in the firstsemester. Our aim was to explore the robustness of students’ understanding of proof by examining how they applied theirknowledge to a novel and more complex situation that was chronologically distant from the lessons of the fall semester.

Lessons selected for analysis were first organized as a set of episodes (Lemke, 1990; Truxaw & DeFranco, 2006). Onetype of episode—whole-class discussions around the construction of a proof for a given conjecture—was the subject of ouranalysis here. Each such episode was approximately 45–60 min in length. Table 2 lists the particular proof tasks on whichthese episodes were based.

Because we focus here on the first semester of the teaching experiment, with the April episode included as a retrospectivepoint of analysis, our data also include students’ responses to individual written assessments given prior to any proofinstruction (September) and at the end of the first semester (December). Although our primary interest is on students’collective discussions, we include the analysis of students’ individual work to triangulate our findings about their collectivework. The assessments each included three conjectures for students to prove (see Table 3). The tasks selected were numbertheory tasks that required knowledge within the range of general knowledge of the participants and would, hence, allowparticipants to focus on the proof component. To avoid potential test/retest issues associated with giving identical items inclose range, items on the two assessments were designed to be mathematically similar, yet not identical.

3.3. Analysis: coding schemes and procedures

3.3.1. Developing a transactive coding scheme for teacher and student utterancesIn the episodes selected for analysis in each of the four lessons, a transactive coding scheme was developed to code

teacher and student utterances.5 Episodes were analyzed independently by the authors to develop the transactive coding

5 Utterances were defined within a speaker’s turn by shifts in conversational purpose.

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cheme. Codes were then mutually negotiated in order to refine this scheme and establish internal reliability in the analysis.ranscripts were coded again according to the negotiated scheme. Moreover, because transcribing produces a new texthere the change from speech to writing yields different perceptions of language (Lemke, 1998), subsequent iterations of

oding were completed while also viewing recorded episodes so as to ensure greater reliability in coding.Once data were coded, descriptive statistics were used to assign frequencies to each of the codes in order to analyze pat-

erns in how the forms of utterances occurred in instruction (Chi, 1997). We based the transactive coding scheme developedere on our earlier work that used a smaller scope of data (Blanton, Stylianou, & David, 2009). The more extensive data used

n the current study made it necessary to expand and modify the previous codes to account for more complexity in the typesf utterances. In what follows, we describe the modified coding scheme developed in the analysis of the current study.

Teacher utterances that were intended to promote transactive reasoning in whole class discussions were coded as trans-ctive prompts. Accordingly, transactive prompts were defined as teacher requests for critiques, explanations, justifications,larifications, elaborations, or strategies and were in the form of questions that asked students for immediate responsesequiring transactive reasoning.6 (See protocol lines 25, 27 and 31 for examples of transactive prompts.)7

Teacher utterances were coded as facilitative if they rephrased students’ ideas or served to structure classroomonversation.8 Through repeating or rephrasing students’ utterances, the teacher tacitly supported the direction of thetudent’s thinking. Structuring involved summarizing a discussion, pacing a conversation, or re-directing an utterance toocus students’ ideas or arguments. As such, facilitative utterances indirectly supported students’ transactive reasoning,ut did not directly prompt them to actively critique, justify, explain, or elaborate their thinking. Furthermore, facilitativetterances were akin to what Staples (2007) refers to as “establishing and monitoring a common ground”. As the teacherephrased student ideas or structured the conversation, the goal was, partly, to focus the collective attention of the classnd give access to ideas that were created toward the goal of constructing a proof. The following exchange illustrates aacilitative teacher utterance in that the teacher rephrases a student’s idea:

Student: There is an element there.

Teacher: There is some element that lives here.

Didactive utterances were defined to be teacher utterances on the nature of (mathematical) knowledge. Students mightot be obvious participants in the conversation, and the teacher’s utterances were not ideas to be negotiated. Through didac-ive utterances, the teacher introduced ideas such as axioms and principles, techniques of proof, or historically-developedotations that students were not expected to re-invent. The following is an example of a didactive utterance:

Teacher: You may come across a bi-conditional proof that you can do simultaneously, both parts at the same time.This is very rare [in my experience]. So normally we break it down into two separate implications, and think aboutthe proofs for each of those.

Directive utterances, on the other hand, were defined as teacher utterances that gave either immediate corrective feedbackr information toward solving a problem. In these, the teacher chose to follow her own thinking rather than question students.nlike didactive utterances (where it was presumed that students would not or could not re-invent the ideas at hand), withirective utterances students might develop the particular notion under discussion through transactive prompts, but theeacher made the choice to tell directly rather than elicit information through questioning. While the teacher might haveifferent intents with the use of directive utterances (e.g., to accelerate solving a problem when it is apparent that studentsave the necessary knowledge to answer relevant questions), these utterances primarily include the common “teacher aseller” instructional paradigm. The following utterance is an example of a directive utterance:

Teacher: So you are doing that.9 Then you want to show that S intersect T is empty. That is what is says up there.

Finally, teacher questions or responses that involved recalling prior knowledge or clarifying communication that did notontribute directly to the proof were coded simply as non-transactive. It should be noted that, although these utterancesid not require higher order thinking they still played an important role in facilitating conversation.10 For example, theon-transactive teacher utterance, “Which problem? Your problem or the one we’re doing?” is an example of clarifyingommunication (in this case, the teacher is clarifying the problem to which the student is referring).

Student utterances were coded at two levels. First, they were coded as either transactive or non-transactive. In particular,sing the definition of transactive reasoning given by Berkowitz et al. (1980), students’ utterances were defined as trans-ctive questions if they were requests for criticism, explanation, justification, clarification or elaboration of ideas. Similarly,

6 We include ‘request for strategies’ in our notion of transactive prompt because of its capacity to facilitate transactive discussion.7 Numbers refer to lines in the protocols. Line numbers are used with transcripts where utterances are referenced in discussions within the narrative.

ine numbers are not used with brief excerpts used to illustrate a definition or concept.8 We distinguish the use of rephrase and revoice here from that used in other literature (e.g., O’Connor & Michaels, 1993). Here, it was expected that the

eacher could rephrase to the whole class rather than only the student who made the initial utterance.9 “That” refers to a direct proof.

10 Designating an utterance as not involving higher order thinking can be subjective. As such, it was important in our analysis to code utterancesndependently and negotiate any disagreements.


Table 4Students’ proof schemes on the pre-assessment (September) and post-assessment (December) items.

Item Proof scheme

No response (%) External (%) Empirical (%) Analytical (%)

September pre-assessmentItem 1 0 30 58 12Item 2 48 24 20 8Item 3 48 28 20 4

December post-assessment
Item 1 0 0 0 100Item 2 19 0 0 81Item 3 11 0 4 85
students’ transactive responses were those utterances intended to critique, explain, justify, clarify, or elaborate one’s ownor another’s ideas. Moreover, in accordance with Goos et al. (2002), we also defined the proposal of or contribution toa new idea, plan or strategy to be a transactive utterance. (See protocol lines 35, 37, and 39 for examples of transactivestudent utterances.) As with teacher utterances, non-transactive student utterances were questions or responses involvingthe exchange of information that were based on either what might reasonably be inferred as recall of prior knowledge orclarifying communication that did not contribute directly to the proof. For example, in response to the teacher utterance“Which problem? Your problem or the one we’re doing?”, Tom responded, “The one we’re doing.”11 Both of these utteranceswere coded as non-transactive. Meanwhile, Tom’s utterance “We need to show that it is a subgroup, which means thatmultiplication has to be closed and the multiplicative inverse has to be there. . .. I think.” was coded as non-transactivebecause he is recalling a procedure, established previously in class, for showing a group is a subgroup.

Finally, student (or teacher) utterances that involved setting up the task or that made no substantive contribution to thetask at hand (e.g., off-topic discussions) were not coded.

In our second layer of analysis of student utterances, we employed a more fine-grained coding scheme in which we identi-fied each type of student transactive utterance according to whether it reflected a student’s elaboration, clarification, critique,explanation, justification, or contribution of a new idea or strategy. This more detailed look at the nature of students’ transac-tive utterances allowed us to analyze trends in the types of students’ transactive utterances to combine with our first level ofanalysis regarding how (i.e., frequency and placement) transactive utterances occurred throughout the data analyzed here.

3.3.2. Analyzing shifts in students’ mathematical knowledgeTo identify shifts in students’ proof knowledge, transcripts of the four episodes were analyzed to identify the proof

schemes (Harel & Sowder, 1998) used in the proofs constructed by the class. Students’ proving activity was also coded forinstances that reflected how students used strategic knowledge (Weber, 2001). In particular, we used the four categoriesidentified by Weber as codes to describe students’ proof constructions in each of the four episodes. We also identifiedinstances where types of strategic knowledge were invoked or failed to be invoked when they actually were necessary tofurther the development of the proof. Finally, we identified the proof schemes students used in their written responses toindividual assessments given prior to instruction on proof (September) and at the end of the first semester (December).

4. Results and discussion

In this section, we first discuss the results of our analysis of shifts in students’ understanding of proof as seen in theirproof schemes (Harel & Sowder, 1998) and strategic knowledge for proving (Weber, 2001). We then discuss results of ouranalysis of discourse using the transactive coding scheme and evidence that students appropriated transactive reasoningas a practice of discourse. Finally, we examine how students’ transactive reasoning mediated their understanding of proofconstructions during in-the-moment conversations.

4.1. Mathematical evidence of shifts in students’ understanding of proof

4.1.1. Students’ proof schemesWe gave students an individual pre-assessment (September) to identify their initial schemes for constructing proofs (see

Table 3 for items). All students responded to the first question, while, on average, only 52% of students responded to the
remaining two questions. Table 4 shows the percentage of students (out of 25 total) using a particular proof scheme.
Overall, students who attempted proofs at all responded primarily with only primitive (external or empirical) arguments.In particular, 88% of responses for item 1 (with 44% and 48% for items 2 and 3, respectively) were either external or empirical.Moreover, almost half (48%) of students made no attempt at proofs for items 2 and 3. Across the items, between 28% and 30% of

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tudents responded using a combination of two types of external proof schemes—authoritarian proof schemes and symbolicroof schemes—in which their arguments were based, for example, on ritual symbolic manipulation without meaning. Forxample, for item 1, one student wrote “if a is even and b is odd, then a + b is odd because its also not divisible by 2.” For item 3,

student tried to express an irrational number as “x.abcdef. . .” and show that the number remains irrational after subtracting from it. Similarly, between 20% and 58% of students responded with an empirical proof scheme in which they used a set ofxamples to validate the given conjectures. On average, only 8% of students attempted to use an analytical scheme in whichhey made an effort to construct deductive proofs. Of these, none of the proofs were completely accurate. Students facedifficulties in symbolizing their arguments or exhibited minor flaws in the flow of their logic. In summary, students had, atest, very rudimentary proof schemes and did not have the tools whereby they could build formal deductive proofs.

Data for the September episode, which occurred during the first lesson on proof, further suggest that students operatedargely with empirical proof schemes, using arguments based on examining specific numerical cases. As mentioned ear-ier, during this episode students were asked to develop a proof for the conjecture “The sum of an even number and oddumber is odd.” After students were given time individually to develop their own arguments, the teacher led a whole-classiscussion asking students to share proof strategies. The class then collectively developed a mathematical proof (that is, aormal symbolic deductive argument). By the collective development, we mean that no one person provided the proof, butontributions were made by different students and coordinated by the teacher through whole-class discussion.

Our coding of students’ individual attempts toward the proof, shared through whole-class discussion, indicated theredominant use of empirical arguments. Students generally seemed convinced, through several instances of numericalomputations, that the conjecture would hold in general. As one student noted,

“I started with some basic things like two or three, like a real easy number and then I used like a more, like a numberthat often disproves things, like zero or one. And then I used ridiculously large numbers and going all over the spectrumto all sorts of different things.”

It is worth noting that, while some students used a random selection of numbers, this student was trying to use specificumbers strategically as representatives of classes of numbers (e.g., “ridiculously large numbers”) and, by this strategy, wasttempting to move toward a more general argument. In a similar move toward a more general argument, another studentrew a diagram as the basis for his proof. However, only a few students (approximately 10%) attempted a general argumenthat would indicate an analytic proof scheme.

As the semester progressed, students began to exhibit more sophisticated proof schemes. In October, as the classttempted to prove that the square root of two is irrational (see Table 2 for problem statement), students suggested thathey needed to first define irrational numbers using a symbolic representation and then use this definition to constructhe proof. One student, Steve, attempted to apply this more general approach, but ultimately resorted to the use of a fewumerical examples: “So I, then I gave some possible values, like two over root two where root ten over root five wouldork.” (protocol lines 41–51 provide a complete transcript). One of his classmates, Mark, objected (see also line 45):

“I think it might work but you haven’t necessarily proven it because you just did it with some examples. . .. I thinkyou’d have to do something with the equation you have there to show that you can’t get two of those things withoutintegers (referring to the equation

(KR

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)= 2 that Steve had written on the board).”

e take Mark’s comment as evidence of a shift in thinking away from the predominant view, exhibited initially in theeptember episode and on students’ responses to the individual written assessment administered in September, thatmpirical arguments were acceptable forms of proof.

In December, students were given a written post-assessment (see Table 3). The results, shown in Table 4, indicate substan-ial shifts in students’ individual proof schemes. For example, while in the pre-assessment only 12% of students attemptedo use analytical proof schemes for item 1, all students in the post-assessment used an analytical scheme on a comparabletem. Although not all proof attempts were correct on the post-assessment (about 70% of the students produced acceptableroofs), with the exception of one student who used an empirical argument on item 3, all students who attempted a proofttempted deductive arguments using logical deduction. Moreover, whereas 48% of students made no attempt at any proofor items 2 or 3 in the pre-assessment, responses on similar items in the post-assessment indicate that the majority oftudents (81% for item 2 and 85% for item 3) now attempted advanced (analytical) proof schemes for these items.

We see clear shifts during the Fall semester in students’ understanding of proof, as evidenced by the proof schemes theysed. As a measure of the robustness of those shifts, we compared these results with students’ understanding of proof athe end of the teaching experiment, in an episode that was chronologically far from the fall episodes and for which the

athematical claim being proved involved significantly more difficult concepts. For this April episode, the class was dividednto two large sections (about 15 students each), and each section was assigned a particular claim to prove. We examine here
he discussion by the students who were asked to prove that the center of a group is itself a group (see Table 2 for the task).
By the April episode, students seemed to collectively operate with (advanced) analytical proof schemes.12 In contrast tohe predominantly empirical arguments students made in the September episode (and which was supported by the high

12 Here, we mean those students who participated in the large group discussion. As is typical of classroom conversations, some students remained silent.hile this, too, is a point of interest, it is beyond the scope of this study.


proportion of empirical arguments students made individually in the written assessment administered in September (seeTable 4)), students in the April episode collectively exhibited deductive reasoning in their proof construction. Again, weintentionally use the term ‘collectively’ here. That is, because the April episode was based on students collectively building amathematical proof, we cannot make the claim that individual students used an advanced analytical proof scheme to provethe particular conjecture posed here. The following is excerpted from students’ conversation during the April episode:

1 Brad: Yeah, what we are trying to do is figure out. . .well. . .H. . .so. . .(inaudible) try to prove that H is a subgroup of G. But what H is, like,group H is really abstract (inaudible).

2 Tom: H is. . .is defined as ‘a times g equals g times a, where g is part of. . . (reading his notes)3 Mark: But it is the element a. . .a4 Brad: (inaudible). . .a is in G and g is in group G.5 Tom: g is an element of big G.6 Mike: H is just a set of elements that. . .that. . .ag equals ga. There might be other elements in G. . .that aren’t in H.7 Mark: So a is in H. Is g in H, too?8 Mike: Yeah, but only the g’s where ag equals ga.9 Sarah: The only things that are in H are a’s. . .. The element g’s are in group G.10 Mark: Yeah, but look at it. You could just switch the letters couldn’t you?11 Tom: That’s true. Everything in H is in G, too.12 Mark: Um, so what do we need to show?13 Tom: We need to show that it is a subgroup, which means that multiplication has to be closed and multiplicative inverse has to be there. . .. I

think.. . .. . ..14 Brad: . . .The group doesn’t have to be commutative. That would be a special case. That would be abelian. If it was commutative. . .(inaudible).

But then again, if it was commutative (inaudible) wouldn’t that make it closed?. . ..15 Brad: Well H is a group with all . . .with those crazy conditions. So we are going to have to prove two things: That the inverse is in H and that

multiplication is closed. Since we kind of know that multiplication would have to be closed because of the commutative. . .16 Mac: Yeah, how do we know H is closed?17 Mark: Yeah, how does that make it closed?18 Anthony: I think there is something there, but I don’t think just that statement does it.19 Mark: Right. Let’s say that the two elements were 3 and 2. 3 times 2 equals 2 times 3, but that doesn’t mean that 6 is in there.. . ..20 Mark: What we want to do is we want to take two arbitrary elements in H.

The above conversation is an excerpt from early in the April discussion. We would classify these students as collectivelyusing an analytical proof scheme, in particular, a transformational scheme. According to Harel and Sowder (1998), studentsoperating with a transformational proof scheme use a deductive process in which they consider generality aspects, applygoal-oriented and anticipatory mental operations, and transforms images. Indeed, here the students that appear in theepisode are searching for clear goals (“we need to show that it is a subgroup”—line 13) and for conditions that will lead themto this state (“multiplication has to be closed and the multiplicative inverse has to be there”—line 13). At the same time, theydisregard other related concepts that might not necessarily be helpful (e.g., “The group doesn’t have to be commutative.That would be a special case”—line 14), an equally important aspect of one’s ability to construct proofs. That is, consistentwith an analytical proof scheme, these students begin their proof construction by stating the definition of what it means tobe a subgroup and the properties that are necessary to show that H is a subgroup. Students do not attempt to explore thetask by using an example of a group, although at least one student (Mark) had the necessary knowledge to do so (see line 19,when Mark used an empirical example as a means to produce a counterargument—an acceptable practice in mathematics).

4.1.2. Students’ use of strategic knowledgeStudents’ whole-class discussions were also analyzed with respect to strategic knowledge, particularly their use of defi-

nitions. As mentioned above, in the September episode few students attempted general arguments. While students made aneffort in this episode to invoke definitions related to proving the conjecture at hand (such as definitions of even or odd num-bers), they were generally unable to coordinate aspects of definitions or use symbolic language to express proof statements.As two students described,

“Well basically what it broke down to was that the, umm, odd number could be separated into an even number andone and so two even numbers would add up to an even number and you would still have that one left over. But I hadno idea how to put that into an equation.”

“An even number is always divisible by two. So if you’re taking two numbers that are divisible by two evenly then anyodd number divisible by two is always going to be a fraction over two.”

In the first excerpt, the student invoked an appropriate definition of odd and even numbers. However, not only was heunable to coordinate these aspects of the definition, but as indicated by his claim, “I had no idea how to put that into an
equation,” we maintain that he was unable to invoke the symbolic language necessary to represent these numbers in gener-alized forms. The transformation of ideas from natural language into symbolic notation reflects an ability to (further) abstractmathematical ideas from the particulars of phrases (e.g., “any even numbers”) into generalized symbolic representations(e.g., ‘a’) that are independent of the contexts associated with these phrases. Moreover, coordinating aspects of a definition


Table 5Growth of strategic knowledge and proof schemes across the first three episodes.

Strategic knowledge September episode October episode November episode

Recall of definitions Students had a strong understanding of theconcepts involved, such as even numbersand odd numbers, but expressed theirthinking in narrative, non-symbolic terms.With substantial prompting from theteacher, students produced a more formaldefinition

Students initiated the attempt to defineirrational numbers symbolically, andthrough discussion, were able to defineirrational numbers. They also began to lookcritically at definitions based on examples

Students invoked formaldefinitions of empty set andintersection

Search for properties,theorems and factsthat are likely to beuseful

No evidence Students began to search for appropriatefacts and theorems (e.g., the Pythagoreantheorem). However, they possessed fewcriteria for deciding which of these factsmight be useful

Students actively searchedwithin their repertoire ofrelevant theorems to use. Theynarrowed their search withinrelevant set-theory theorems

Knowledge of whento use syntacticand semanticknowledge

While students had some of the semanticknowledge necessary for this proof (e.g., anunderstanding of even and odd numbers),they had difficulty using the syntacticaspects of this knowledge

Students began to operate on definitions inan anticipatory manner. They recognizedthe aspects of the proving process thatrequired syntactic strategies (e.g., symbolmanipulation)

Same as October

Search forappropriateproving techniques

No evidence. While this claim onlyrequired a direct proof, the notion of adifferent type of proof was not mentioned.Students did not seem to consider thatanother type of proof might be useful

Students discussed the type of proof to use.They now had different proof techniques,and were eager to use them. Various prooftechniques were tried in a trial-and-errorfashion. They could not argue why aparticular proof technique might be more

A number of students readilyproposed an indirect prooftechnique, without attempting adirect proof. They seemed torecognize that the former mightbe more appropriate to prove the

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y using the representational system that would allow for appropriate use of syntactic aspects of proof is one facet of thetrategic knowledge that is needed in producing proofs. As Weber (2001) suggested, even in the case where the studentight aim toward the development of a deductive proof and is able to recall a useful definition, the proof production still

ails when the student is unable to invoke this necessary component of strategic knowledge.In the second excerpt, the student was unable to further unpack his definition for even numbers (“divisible by two

venly”) to use in constructing the proof. For example, if he had been able to formulate his definition of even numbers as2a|a ∈ Z}, this might have facilitated his representation of a generalized sum for an arbitrary even number and an arbitrarydd number, a typical approach for a formal proof of the conjecture. However, his inability to represent these concepts inbstracted forms hindered his construction of a formal, symbolic proof.

Important shifts in the use of strategic knowledge were evident in the October episode. In this episode, students ques-ioned whether other theorems proven in class or elsewhere, such as the Pythagorean Theorem or generalizations aboutven numbers and odd numbers, might be useful in proving this conjecture. Students also attempted to use operations onefinitions in an anticipatory manner. For example, Lara shared her thoughts on how to proceed with the proof:

“P squared over Q squared equals two, and then move the Q squared over to the other side. So, because P squared isa multiple of two, it must be even and then P must also be even. You divide by two on both sides and then you get Psquared over 2 equals Q squared and Q squared can’t be both even and odd. So, it can’t be an integer and then it can’tbe, it can’t fit that definition of a rational number”.

While attempting to advance the proof being constructed by the class, Lara kept an eye on the definition of a rationalumber to look for a contradiction. Her thinking reflects an intentionality that we claim is a departure from earlier attemptso simply manipulate symbols ritualistically, that is, without a particular purpose or goal in mind. As this example illustratesnd as is further summarized in Table 5, the October episode suggests that definitions and previously proven theorems wereecoming objects to be shared, negotiated, and searched for possible connections and patterns as pieces of a broader puzzleather than a set of disconnected facts.

As Table 5 suggests, the development of strategic knowledge became more obvious in November (see Table 2 for the taskescription). As the following excerpt illustrates, students were now actively searching within their repertoire of previouslyroven theorems, while at the same time exploring definitions of the mathematical concepts involved in this conjecture:

1 Steve: Of the. . . I mean the empty set is a subset of A. . .2 Teacher: Is the empty set a subset of A? So how is that going to help you here?3 Steve: I am just thinking that the intersection of A would have to be the. . .the empty set. Like if you have A union B equal A, then A

intersection B would have to be B.4 Anthony: Just the basic definition of intersection. It is just the basic definition of intersection. If A is a subset of B and if A intersects B then the

intersection is equal to A.

Here, Steve and Anthony (lines 21, 23 and 24) mentally searched for known facts, theorems and definitions, a practiceoticeably different than their earlier approaches to proof conveyed in the empirical arguments constructed in their written


assessments and observed in classroom discourse. From our analysis of the three episodes, we found an overall posi-tive growth in students’ strategic knowledge. Table 5, which characterizes students’ strategic knowledge across the threeepisodes, indicates qualitative shifts in the types of strategic knowledge students used during the episodes. It also identifieswhat we describe as a “collective” proof scheme for each episode and indicates a gradual move toward a collective analyti-cal proof scheme—that is, the ability to reason together as a group, with guidance from the teacher, while maintaining thegeneral characteristics of the analytic proof scheme.13

The clear shifts we observed in students’ use of strategic knowledge during the Fall semester continued during the Aprilepisode. For example, in the April excerpt presented earlier (lines 1–20), students fluently refer to aspects of definitions intheir symbolic forms (e.g., lines 2–6). In the September episode, students initially had great difficulty in even expressingtheir ideas in symbolic notation. Hence, they were unable to operate on and transform these concepts in ways that wouldadvance the proof. As Table 5 indicates, by October students were beginning to take a more active role in developing andinvoking definitions symbolically. In November, they were able to do this without the support of the teacher. By the Aprilepisode, students were able to reason collectively with these definitions in their symbolic forms (lines 2–6). Fluency inoperating on symbols is indeed one of the characteristics of advanced proof production ability. As Weber and Alcock (2009)argue, the use of symbolic notation is central to deductive proof production. Working within this particular representationsystem (symbolic representations of definitions) allows one to “move from a collection of initial configurations to a desiredconfiguration” (p. 326), which lies at the heart of a transformational proof scheme.

Furthermore, in contrast to the fall episodes, students seemed to more successfully navigate definitional aspects of theconstructs used in the conjecture. We see evidence of this in lines 1–20. In particular, in April students spent a considerableamount of time “unwrapping definitions” (Weber, 2001, p. 114) as well as negotiating the aspects of syntactic knowledgethat they would use toward completing the proof. There are several properties one might recall about a group (e.g., itsdefinitional properties, whether it is abelian or not). In lines 1–20, students were grappling with which group conceptsmight be useful in this proof, recalling properties about groups that could be relevant to this task (e.g., closed under theoperation and having an inverse—line 13) as well as properties that might not be as useful (e.g., commutativity and abeliangroups—line 14). In essence, students in this episode were able to recall (or look for) definitions of concepts and ideas thatwere central to the proof and express and reason with them in symbolic terms.

We take this as further evidence of their emerging strategic knowledge in proof construction. As Weber notes, in proofconstruction

“there are typically a large number of actions that one can perform, but only a small subset of these actions will beuseful in completing the task. . .. There are many inferences one can derive, but most of these inferences will not berelevant” (2001, p. 111).

Hence, one has to use strategic knowledge to decide which of these actions and inferences are relevant to the task at hand.Even though the complexity of mathematical concepts had increased significantly in the April episode, students were ableto strategically select an appropriate choice of actions, based on definitional aspects of a group, that would be salient toproving the given conjecture.

4.2. Patterns in forms of teacher and student utterances

In the previous section, we provided evidence of shifts in students’ mathematical knowledge as it pertained to under-standing and constructing proofs. We provided evidence from students’ written assessments that they shifted individuallyaway from the use of empirical proof schemes toward an ability to use analytical proof schemes to build deductive argu-ments. We also provided evidence that they exhibited positive shifts collectively—that is, through whole-class negotiationsof a proof—in their use of strategic knowledge and the proof schemes they used for constructing proofs. That is not to say thatall students exhibited the kinds of thinking that occurred in the April episode. Learning is rarely, if ever, so synchronized.However, we do claim that the data—both individual written assessments and collective whole class discussions—provideevidence of growth in students’ understanding of how to construct proofs.

Knowing that shifts in proof knowledge occurred raises the question of how, or in what manner. In this section, wefocus on Research Question 2, particularly, the forms of teacher and student utterances in whole-class discourse, how theyoccurred across the four episodes, and whether this supports that students appropriated transactive reasoning as a practice
of discourse. In particular, we analyze frequencies of and patterns in the occurrence of these utterances across the fourepisodes. We first discuss frequency of utterances in the three fall semester episodes, then compare and contrast theseresults with those in the April episode. Finally, we examine patterns in the types of transactive utterances as they occurredacross these four episodes.
13 We intentionally use the term “collective” here. That is, because the classroom episodes were based on students collectively building a mathematicalproof and, in that process, negotiating its constituent parts, we cannot make the claim that individual students used a particular proof scheme to prove thegiven conjecture. A critical indicator of students’ learning is their ability to operate with analytical proof schemes without guidance from the teacher. Aswe will see in our analysis of classroom discourse, the teacher necessarily participated in a scaffolding role throughout the first semester, but was able towithdraw her support for the fourth lesson.


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As we detail in what follows, our findings suggest a key point. In particular, the form of utterances that occurred indicatehat a main goal for the teacher was to encourage the development of students’ transactive reasoning by initially engaginghem through her use of transactive prompts. Over time, the significant increase in students’ transactive utterances andecrease in the teacher’s transactive utterances and prompts suggests that students did begin to internalize a discourseractice characterized by transactive utterances and were increasingly able to participate, with decreasing support from theeacher, in discussions focused on constructing proofs.

We emphasize that the scope of our analysis here is on those whole-class episodes around which proofs were beingonstructed, not all whole-class discussions. The primary objective was to tighten our focus and reduce the volume of data.s described earlier, the episodes analyzed were about 45–60 min in length and occurred within a 75 min lesson. We cannotay whether other aspects of a lesson would have the same structure of teacher and student utterances. For example, when

proof technique (such as the Principle of Mathematical Induction) is being introduced, it might be reasonable to expecteacher utterances to be more didactive in nature because students would not be expected to ‘discover’ such a technique.owever, those types of episodes are not the focus of this study.

Data for the September episode (see Fig. 1) indicate that utterances that were transactive in form (that is, either teacherransactive prompts or student transactive utterances) comprised almost half of the utterances in classroom discourse (47%f all utterances). By comparison, only 26% of all utterances were non-transactive, 20% were facilitative, and 7% were directive.

Similarly, data for the October and November episodes (see Figs. 2 and 3) further support that teacher and studenttterances that were transactive in form comprised the majority of utterances in classroom discourse, while the frequenciesf other types of utterances (i.e., non-transactive, facilitative, directive, and didactive) were considerably less and similar toheir frequency of occurrence in the September episode. Table 6 provides a summary of the analysis of all three episodes.

On average, about 51% of utterances across the three episodes were transactive in the form of transactive prompts fromhe teacher or transactive questions and responses from students. In contrast, on average only about 6% of utterances acrosshese episodes were directive utterances by the teacher. We take this as evidence that the teacher’s attempts to cultivate
tudents’ transactive reasoning was a routine part of her practice, not an isolated occurrence. Moreover, it suggests that theeacher’s instructional purpose was to support students’ transactive reasoning about proofs rather than follow her own linef thinking through directive utterances.
Fig. 2. Comparison of total utterances (teacher and student combined) in October episode.


Fig. 3. Comparison of total utterances (teacher and student combined) in November episode.

Table 6Summary of types of teacher utterances across the first three episodes.

September (%) October (%) November (%)

Transactive (total) 47 58 49• Teacher 64 68 57• Students 65 77 63

Non-trans 26 20 30
Facilitative 20 16 11Directive 7 4 6Didactive 4 2
Focusing only on the teacher’s utterances across the three episodes, Fig. 4 indicates that transactive prompts constituteda primary form of the teacher’s total utterances. In particular, on average 34% of the teacher’s utterances were transactiveprompts. We further maintain that most of the teacher utterances intended to support transactive reasoning directly throughtransactive prompts or indirectly through facilitative utterances that tacitly served to affirm students’ ideas. In particular, anaverage of 63% of utterances across the three episodes were either transactive or facilitative in form. In contrast, on averageonly 10% of the teacher’s total utterances were directive, further supporting our claim that the teacher’s purpose was tominimize giving information directly.

Finally, if we focus only on student utterances, Fig. 5 shows that the majority of student utterances were transactiveacross the first 3 episodes, with an average 68% of student utterances classified as transactive.

There are several points to underscore about these data. First, as we noted earlier, there is a consistent trend in the formof discourse across these episodes that suggests that the teacher’s purpose was to support students’ transactive reasoningthrough the use of transactive prompts and, to a lesser extent, facilitative utterances that prioritized students’ thinking. Atthe same time, the data also indicate the teacher’s intent was to minimize the use of directive and didactive utterances.

In addition to this, if we examine teacher and student total utterances (Figs. 1–3), we see a slight decrease in teachertransactive prompts over time in conjunction with an overall increase in student utterances. We would hope to see thisdownward trend in the teacher’s transactive prompts, coupled with an increase in students’ transactive participation in the

Fig. 4. Comparison of teacher utterances in the September, October, and November episodes.


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iscourse, because it suggests that students are beginning to internalize transactive reasoning as a practice of discourse. Onhe other hand, it could be argued that the decrease in transactive prompts and increase in student utterances across thesehree episodes is so slight that it might be attributed to normal variation in discourse. Given the predominance of empiricalroof schemes and limited strategic knowledge in the September episode, it is reasonable to expect that students had notet developed a deep habit of transactive reasoning about proofs on their own, that is, without significant scaffolding by theeacher.

Moreover, while it is promising that a majority of student utterances were transactive (on average, 68% of studenttterances out of the total student utterances were transactive—see Fig. 5), we interpret this cautiously. For example, whilehe frequency of student transactive utterances increases substantially from September to October in comparison to theotal teacher and student utterances, it decreases in November (see Figs. 1–3). Although this, too, might be due to normalariation in discourse, we suggest that it also might be explained by the fact that the November content was substantiallyifferent than that addressed in the previous two episodes. In particular, in the November episode students were learning setheory, and the types of arguments (i.e., element arguments) and concepts associated with set theory were new to studentsnd less rooted in their prior mathematical experiences than those encountered in the previous number theory arguments.s students encounter new ideas in instruction, it is reasonable that this might affect the structure of discourse.

While these data suggest that the teacher was cultivating a habit of transactive reasoning in students’ discourse, it woulde important to examine data that was still within the teaching experiment yet chronologically and conceptually removedrom those episodes in the fall semester. Habits of discourse are developed over periods of time and might be fragile in theace of increasingly complex mathematical ideas. To this extent, we compared the results of the fall episodes to the Aprilpisode that occurred at the end of the teaching experiment. Although the teacher’s effort to cultivate transactive reasoningad continued during the intervening period, the mathematical claims being proved—now in group theory—were muchore complex than those encountered during the fall episodes.Fig. 6, which compares the total teacher and student utterances in the April episode, shows clear shifts in how teacher and

tudents participated in constructing a proof. Out of all teacher and student utterances, the teacher’s utterances decreased
rom an average of 52% in the fall episodes to 14% in April, indicating that students were able to more fully participate inhe discourse. But perhaps more importantly, the frequency of students’ transactive utterances increased from an averagef 33% in the fall episodes to 64% of total utterances in April, while the frequency of transactive prompts by the teacher
Fig. 6. Comparison of teacher and student utterances in the April episode.


Fig. 7. Discourse thread for the September episode.

Fig. 8. Discourse thread for the April episode.

decreased from an average of 18–4%. The higher frequency of teacher transactive prompts in the fall episodes suggests thatpart of the teacher’s instructional purpose was to establish a norm of mathematical participation that called on students toexplain, critique, justify, and so on—that is, to reason transactively. But simultaneous shifts in the form of both teacher andstudent utterances in the April episode imply that the teacher was able to withdraw the support of transactive prompts inresponse to the developing competence of students to engage in transactive discussions. In other words, we maintain thatstudents seemed to be developing a practice of reasoning transactively about a proof on their own, without the teacher’sdirect intervention.

Further support for this position is found in Figs. 7 and 8, which provide an alternative representation of classroomdiscourse that we refer to as discourse threads. By capturing the chronology of utterances as they occurred in the episodes,
discourse threads help visually preserve the patterns of classroom interaction and how transactive discussions occurredin these interactions. Because of the similarities of the discourse structure for the Fall episodes, we compare here only theSeptember and April episodes.
Table 7Percentage of types of student transactive utterances across the four episodes.

Transactive utterances September (%) October (%) November (%) April (%)

ResponsesElaborations 9 37 36 17Clarifications 13 13 17 36Critique 13 7 6 –Explanation 52 13 12 18Justifications 4 14 1 6New Ideas 4 4 14 1

Requests for/questionsElaborations – 1 1 1Clarifications 5 4 8 7Critique – 1 –Explanations – 5 3 8Justifications – – 6

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In particular, Fig. 7 (September episode) indicates that the teacher’s transactive prompts (graph line 5) and facilitativetterances (graph line 3) are interspersed with students’ transactive utterances (graph line 6) in a relatively equal manner.his scattering distribution suggests that, initially, the teacher needed to continually “prop up” transactive reasoning byrompting students to explain, justify, clarify and so forth (transactive prompts) and by affirming or rephrasing students’

deas (facilitative utterances). The following excerpt from this episode, in which a discussion occurs about how to representn arbitrary even number and an arbitrary odd number, illustrates how these forms were interspersed as the teacher workedo prop up transactive reasoning. A student (Tom) has suggested that an arbitrary even number and odd number could beepresented, respectively, as 2k and 2k + 1, for an arbitrary integer k.

5 Teacher: What do you think? (transactive prompt)6 Sarah: But with that you get stuck with, you can only use twenty and twenty-one as part of a proof. You can’t use, say, like twenty and

fifty-seven. You can’t have fifty-seven be your odd number with that model. (transactive)4 Teacher: What do you think Tom about what Sarah just said? (transactive prompt)8 Tom: That makes sense but I don’t think it matters in this problem. (transactive)7 Teacher: Which problem, your problem or the one we’re doing? (non-transactive)0 Tom: The one we’re doing. (non-transactive)1 Teacher (to the class): Does it matter in the problem we are doing? (transactive prompt)2 Jemma: Well, yes. If you use k for both and you prove it over there it is only going to be true for two that are next to each other. (transactive)3 Teacher: So it matters. Okay. (facilitative)

As the interaction suggests, the teacher consistently prompts students (lines 25, 27, and 31) to think about the nature ofhe representation posed by Tom—it is not a conversation that students facilitate on their own. At this point (September),e maintain that students had internalized neither a transactive practice of interaction about mathematically sophisticated

deas such as developing rigorous mathematical proofs, nor the mathematical knowledge of how to build sophisticatedrguments. In other words, while they might have had the capacity to explain, justify, and so forth, in the normal course ofonversation, these practices of communication had not yet become a mathematical one (van Oers, 2002), in the context ofroving, that students could sustain without the support of the teacher.

In contrast, the April episode (see Fig. 8) shows thick clusters of students’ transactive utterances (graph line 6)bout complex mathematical ideas that occurred in conjunction with a significant decrease in the teacher’s transac-ive prompts (graph line 5) and facilitative utterances (graph line 3). In other words, we maintain that the teacheras able to withdraw her support because students had become legitimate participants (Lave & Wenger, 1991) in theiscourse.

Moreover, we would like to underscore that understanding the form of teacher utterances that supported the devel-pment of students’ transactive reasoning is as much about what did not occur as what did occur. In particular, Figs. 1–3nd 6 indicate that teacher utterances did not dominate instruction and that conversation was shared among teacher andtudents (on average, 58% of utterances were made by students across all 4 episodes). But perhaps more importantly, thesepisodes were not only characterized by the presence of transactive utterances, but also the (near) absence of directiveeacher utterances, that is, those utterances most closely aligned with what we might view as direct instruction. In partic-lar, Figs. 1–3 and 6 indicate that directive utterances decreased from an average of 6% in the Fall episodes to 1% in April,uggesting that direct instruction did not feature prominently in classroom discourse. The discourse thread of the Aprilpisode supports the transcripts we provided earlier. That is, in lines 1–20 of the episodes we provided, the teacher wasearly absent from the discourse, while students ably debated definitions and proof construction. Consistent with these data,igs. 7 and 8 indicate that classroom discourse might be viewed as “discussion orchestration” (O’Connor & Michaels, 1993),n which teacher and students are co-participants in conversation. Taken together, these results suggest that instructionhat scaffolded students’ transactive reasoning did not, in this study, resemble traditional direct instruction paradigms (e.g.,teacher as teller”).

Finally, we note that the total percentage of teacher and student non-transactive utterances was relatively stablecross all episodes (see Figs. 1–3 and 6). Moreover, Figs. 7 and 8 suggest that non-transactive utterances were distributedn a relatively equal pattern across interactions. We see non-transactive utterances as a normal—and essential—partf conversation and as such, we interpret the random spread in which non-transactive utterances occurred across thepisodes as reasonable. That is, we would not necessarily expect a significant fluctuation in the total amount of non-ransactive utterances because their purpose generally serves to move conversation along through exchange of rotenformation.

While the preceding discussion gives us a sense of how transactive utterances occurred in terms of their frequencynd placement in the discourse, we can obtain a more complete picture of the role of transactive reasoning in students’onstruction of proofs through a more fine-grained analysis of the particular types of transactive student utterances thatccurred in these episodes. Table 7 summarizes the percentage of occurrence of such utterances across the four episodesnalyzed here.

While one might expect an increase in transactive utterances over time whose intent was to justify or explain mathe-
atical claims because of their importance to proving, the data show little increase in these dimensions. In fact, we observe
he opposite effect, that is, a decline in the number of justification and explanation utterances over time. At the same time,here is a notable increase in the number of elaborations and clarifications. Further, we see a strong increase in the numberf transactive questions (requests) students ask over time.


Thus, while, on the surface this may look like a decrease in activity that is central to proving (e.g., acts of justifying), wesuggest an alternative explanation. In particular, the increase in requests for elaboration and clarification of each other’sstatements, as well as actual follow-up elaborations and clarifications, suggests further development of strategic knowledge.That is, students seemed increasingly reluctant to accept new ideas without questioning. Rather, they spent time analyzingand acting on definitions and ideas that were being proposed. It is these elaborations and clarifications that helped studentsbring to the surface those aspects of the definition that might prove fruitful in the construction of the proof. For example,in September there was little questioning of the concept of “evenness”. In November, however, students questioned eachother and gradually unpacked the idea of an “element” of a set. Thus, while elaborations and clarifications increased, quickexplanations and justifications decreased.

4.3. The role of transactive reasoning in constructing proofs in-the-moment

We have provided evidence that the teacher sustained her effort to develop students’ transactive reasoning throughinstruction in which transactive prompts were a primary form of teacher utterance, that teacher utterances were marked bythe absence of direct instruction, and that, in this context, students seemed to internalize a transactive habit of interactionin which they could reason about complex mathematical ideas without scaffolding by the teacher. Moreover, we providedevidence that students were able to engage in increasingly sophisticated mathematical practices in which they transitionedfrom the use of predominantly empirical proof schemes with little evidence of strategic knowledge, to the use of advanced(analytical) proof schemes and strategic knowledge in constructing proofs (see Tables 4–6).

Our goal here is to examine how the particular form of transactive communication in whole class discourse helpedmediate students’ understanding of proof over time as well as their in-the-moment proof construction. We look first at howtransactive reasoning mediated students’ understanding of proof by focusing our analysis on shifts in the proof schemesstudents used across episodes.

We have already established from the pre-assessment and the September episode that, initially, students’ proof schemeswere predominantly empirical. Thus, a goal of instruction was to help students understand the limitations of empiricalarguments and to develop more mathematically-appropriate proof schemes (that is, analytical proof schemes). The fol-lowing excerpt from the September episode is part of a lengthier transactive-based discussion where the teacher’s intentwas to challenge students’ use of empirical arguments to prove that the sum of an even number and an odd number isodd:34 Teacher: Are you sure, absolutely sure that this will hold for all the numbers? . . ..Do you think somebody could come along and. . .say “Aha,

here is an example where it does not work”? (transactive prompt)35 Steve: No because they would have to prove (inaudible). . .. (transactive)36 Teacher: Ok. (facilitative)37 Steve: An integer is any whole number, increase it by one, so how can you. (voice trails off)? (transactive)38 Teacher: So you’re saying nobody could. . .find a case where that is not true? (transactive prompt)39 Steve: Unless they redefine what an integer or infinity is. (transactive)40 Teacher: Does anybody think you can come up with an example of where that is not true? (transactive prompt)

At this point, students seem fairly certain that the claim is true: It fits with their experiences with adding evennumbers and odd numbers, and they could not produce a case where it fails to be true. However, neither could theyproduce the kind of general argument that guaranteed that no such counter-example existed. Moreover, while most ofthe class shared their own arguments based on empirical proof schemes, no one acknowledged the limitations of thesearguments.

The teacher then asked students whether they thought the claim “The sum of two even numbers is odd” was true or false.Again, students expressed a degree of certainty about this claim. One student argued that he was “even more sure” that thisclaim is false because “it’s a lot easier to disprove something.” Some students thought that it was sufficient to produce onecounter-example, while others needed several counter-examples. The teacher summarized the discussion:

“If you’re trying to prove it’s true, even if you have five examples, ten examples, one hundred, a million examples,how certain are you, are you absolutely one hundred percent certain that the examples will prove that it’s true?. . ..Sowe’ve got to figure out how can we develop arguments that are general arguments that apply regardless of what theparticular number is that convinces you that [the claim] is true.” (didactive)

There are two points to make here with regards to the role of transactive reasoning. First, the teachers’ utterances werepredominantly transactive (lines 34, 38, 40). Moreover, by prompting students to explore any limitations in an empiricalargument, she positioned them in an active stance toward the goal at hand. Her questioning implicitly called into questionthe practice of using examples as the basis for an argument and invited students to publicize and discuss their ideas. Theintroduction of the false claim pushed one student, near the end of the discussion, to express degrees of certainty (he was“even more sure” that the second claim was false).

We maintain that as students thought about these two different types of claims—a process facilitated and made public bythe transactive prompts of the teacher—they were able to get to a point where a measure of uncertainty about the empiricalargument for the true claim, in comparison to their certainty about the false claim, could be expressed. We don’t know thatall students yet held this perspective, but we do think that the critical role of the transactive discussion—discussion that

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nvolved a substantial number of exchanges of transactive utterances—was to place students in a position to analyze theirmpirical arguments and whether these arguments were sufficiently rigorous.

The acceptability of empirical arguments was another focus of a transactive discussion in the October episode. As theollowing excerpt opens, Steve shares part of his argument that

√2 is irrational:

1 Steve: If you rewrite K squared over R squared equals two and it’s K over R times K over R. (Steve writes(

KR

KR

)= 2 on the board.). . .. And I

basically said there are no integers, therefore there are no integers or fractions that are represented by integers that produce an answer oftwo. To achieve a proper two you must use the irrational number as a fraction or an integer. So I, then I gave some possible values, like twoover root two where root ten over root five would work [emphasis added]. (transactive)

2 Teacher: In what sense would two over square root of two work? Work for what? You mean it could be one of the r’s? (transactive prompt)

3 Steve: Well, yeah. And then you can have like square root of two over one times square root of two over one( √

21 ×

√2

1

). But the thing is that

the definition of it is that they’re (referring to the numerator and denominator of an arbitrary rational number) in a set of integers and these(referring to

√2,

√5, and

√10) aren’t integers so even though these could work, they’re not integers so that wouldn’t make any sense. So I

basically just said that that would violate the definition of a rational number, therefore root two is a rational number. (transactive)4 Teacher (to the class): What do you think? (transactive prompt)5 Mark: I think it might work but you haven’t necessarily proven it because you just did it with some examples. . .. I think you’d have to do

something with the equation you have there to show that you can’t get two of those things without integers (referring to the equation(KR

KR

)= 2 that Steve had written on the board) (transactive)

6 Steve: Well, I’m just saying that in order—two would have to be a perfect square. If you square something—because k over r [times] k over r.This isn’t something different. These are the exact same things. So two would have to be a perfect square and it’s not. There’s no number thatyou could square that would be equal to 2 unless you use an irrational number. (transactive)

7 Mark: You know what I mean (referring to his previous comment (line 45) about the use of examples)? (transactive)8 Steve: What? (transactive)9 Mark: If you do a proof for that then that will work, too. (transactive)0 Steve: Yeah. Well, I don’t know. (non-transactive)1 Mark: Just something to show me. (transactive)

n this excerpt, Mark has taken up the idea, introduced in the September episode, that empirical arguments are not sufficientlyigorous (line 45). His transactive discussion with Steve (lines 45–51) helps mediate Steve’s understanding of the weaknessf his argument, after which Steve concludes, “So I don’t know if I proved anything, but I don’t know, I just—I don’t know.o that doesn’t prove anything.”

There are two points we would like to make here. First, the transactive exchange between Steve and Mark helped pro-ote a shift in Steve’s thinking (by the December written assessment, Steve was able to use an analytical proof scheme on

ome arguments, indicating that he had come to value these types of arguments over empirical ones). While this exchangeith Mark was not the only challenge to Steve’s thinking (e.g., lines 34–40 from the September episode represented another

hallenge to Steve’s thinking), it contributed to the shift in Steve’s thinking. Second, lines 41–51 are quite different thanines 34–40 in a fundamental way: Mark is moving toward fuller participation in the classroom community and is tak-ng on the role of arguing against empirical schemes, a role that the teacher necessarily held in September. In this, we

aintain that Mark and Steve were beginning to appropriate a practice of (transactive) discourse that would enable stu-ents to construct proofs without the support of (or with very little support from) the teacher. Moreover, this discourseas mediating student thinking (for example, Steve’s thinking) toward the use of more mathematically rigorous proof

chemes.By the November episode, when students were constructing element arguments in set theory, no one in the

hole-class discussion suggested empirical schemes as the basis for a proof. While we do not know what everyndividual thought about empirical proof schemes during this episode, that the class collectively was shifting awayrom empirical schemes is further supported by the results of the December assessment (see Table 4) and the Aprilpisode.

Our claim here is that by making students’ thinking visible and public, transactive reasoning seemed to help students,ver time, confront their misconceptions about proof (e.g., that empirical arguments were mathematically acceptable) ande-negotiate these ideas toward a more sophisticated understanding of proof. As Kruger (1993) suggests, the dissent andonflict enabled by transactive discussions became agents of change in students’ thinking. Moreover, over time, studentsegan to take on a more active role in the negotiation of ideas, while the teacher gradually withdrew support. We triangulatehe data provided in the previous excerpts about the type of proof schemes students negotiated in whole-class discussionsith results from the written assessments (see Tables 4 and 6), which show that individual students made significant shifts

way from the predominant use of empirical proof schemes and toward the use of analytical proof schemes. That is, whileur focus is on the collective analysis of proof construction, our analysis of individual work serves to support our claimsbout the collective work. We maintain that the kinds of transactive discussions that occurred during the construction ofroofs (as illustrated in lines 41–51 above) supported these shifts because they were the avenue by which ideas were madeublic and negotiated.

Finally, in order to understand how transactive reasoning supported students’ actual construction of a proof within
n episode, we take a more fine-grained look at transcripts from the April episode. We look specifically at this episodeecause its discourse structure shows the fullest level of participation by students. In other words, our aim is to look at howransactive reasoning supported the construction of a proof “in-the-moment”, when students were able to, independentlyf the teacher, engage in this type of reasoning. We consider excerpts from the April episode in which the task was to


prove that the center H of a group G is itself a group (see Table 2 for task description). In the following excerpts, studentsnegotiate what it means to show that set H is closed.14 Initially, Tom proposed an argument for why H is closed undermultiplication:

52 Tom: Wait, we already know that G is closed under multiplication. Closed under multiplication is easy because we know that everything in His in G and we know that everything in G is closed under multiplication, or the operation, right? (transactive)

53 Anthony: Dot a, dot b (Anthony re-expresses the operation more generally as ‘dot’ rather than ‘multiplication’.) (non-transactive)

Tom pauses to analyze his own explanation. (We infer this from line 54, where he indicates from his response—“Nevermind”—that his thinking has changed.)

54 Tom: Never mind. (transactive)55 Mark: Is that right (referring to Tom’s argument)? (transactive)56 Tom: No. (transactive)

As he verbalized his thinking to the group, Tom realized that the fact that G is closed, which follows from the groupproperties of G, does not imply that H is also closed. That is, for arbitrary elements a and b in H, ab is an element of G since His a subset of G, but ab is not necessarily an element of H. Tom’s argument generated different (inaudible) brief conversationswithin the group, then an exchange between Tom, Mark, and Anthony developed:

57 Tom: (inaudible) . . . everything in G has to be in H? (transactive)58 Mark (to Tom): We want to prove H is closed under multiplication . . . (inaudible) in H. (transactive)59 Tom: Oh yeah. (transactive)60 Anthony: H, not G. (transactive)

When Tom explained his argument, members of the group—including Tom—implicitly took an active stance toward itthrough requests for justification (line 55), statements of clarification (lines 58, 60), and evaluation of one’s own or other’sthinking (lines 54, 56). That is, Tom’s idea was not accepted at face value, but was sifted through the transactive reasoning ofthe group so that the mode of communication—transactive reasoning—itself became the avenue by which ideas were refined.As such, transactive reasoning served as the (public) channel through which discursive conflict arose and was resolved.

In the conversation that ensued, Tom was eventually able to describe that showing H is closed entailed establishing that(ab)g = g(ab), where a, b are in H and g is in G:

61 Teacher: Where are you trying to get to? (transactive prompt)62 Tom: We are trying to get to ‘a-b-g equals g-a-b’. (transactive)

However, he went on to explain that his argument was a repeated application of the Associative Property (deriving fromthe associativity of G under the operation of G). While the argument does involve the use of the Associative Property, itfundamentally requires using the commutativity of elements a and b as elements of H. His confusion about this is the subjectof the following exchange:

63 Mark: No. I think, you know. . .look you’ve got this thing (points to “(ab)g = g(ab)” written on the group’s paper). You’ve got to start with this(referring to (ab)g) and you have to end up with ‘g-a-b’. (transactive)

64 Tom: That’s all we need to do. Just use the associative property. We have it right above that. It’s just all messy and bad. (transactive)65 Brad: So turn to a-b-g and from here. . . (transactive)66 Tim: Where did we go? (transactive)67 Brad: I’m trying to remember. (non-transactive)68 Tom: We know that, um, since a is in that (referring to set H), so we can switch that around (pointing to the group’s paper), plus a is in H. So

a-b-g equals b-g-a. (transactive)69 Mark: As soon as you switch the order of them you’re talking commutative, you’re not talking about associative. (transactive)70 Mark: You can use commutative, can’t we? (transactive)71 Anthony: Well, if we are talking about elements of H, then yes. (transactive)72 Tom: We can do b times ga. (transactive)73 Mark: If we can use commutative, then we don’t need to (inaudible) any of the middle stuff (points to something on the paper). We can just

flip these, and you got it. . .. (transactive)74 Tom: We can’t do that because we don’t know that either of those are in H. But we know that a is in H and that b is in H. (transactive)75 Anthony: But you know independently that a and b are in H (inaudible). . . (transactive)76 Tom: Yeah, so just do b times ga. (transactive)77 Brad: Oh, b times g. . .oh. (transactive)78 Tom: And then um. . .. (no code)79 Brad: Wait a sec. Isn’t that backtracking? (transactive)80 Anthony: Just re-associate (inaudible). (transactive)81 Tom: No, then it says b is in H. ‘b’ is in H, so that can. . .that means that, uh, b times ga.. . .. (transactive)82 Tom (to Mark): It doesn’t matter. ‘bg’ is in G. That is all we needed to know. So a is in H, so. . .and g is in G. That’s all we needed to know. So this

is in G. So as long as this is in G, we can switch them. (transactive)83 Mark: Alright. (transactive)

14 Because of the length of the discussion, we only include selected portions here.

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We see in the above excerpts a shift in Tom’s thinking from his (flawed) argument that H is closed because G is closed,o one in which he understands that it must be established that for elements a and b in H, element ab commutes with eachlement of G (although he initially uses the term ‘associative’, he later refers to ‘switching’ the terms, suggesting he meansommutative). Moreover, what seems to be his mistaken reference to the associative property (line 64) generates confusionor Mark, who then explains to Tom that “switching” connotes the commutative property (line 69). At this point, Mark startso question his own thinking as to whether or not the commutative property can be used (line 70), leading Anthony andom to justify its use (lines 71, 72) and Mark to further refine his own thinking and propose an alternative argument (line3). Tom rejects Mark’s reasoning (line 74) and elaborates on his own thinking. Anthony contributes to Tom’s idea (line 75),hich Tom validates and extends (line 76). In lines 78 and 81, Tom hesitates as he organizes his thinking about the next step

hen, ultimately, seems to convince Mark that his (Tom’s) argument is sound (line 82, 83).We take the excerpts given here as part of the evidence that students had learned to contribute progressively to the

evelopment of a proof by interpreting contributions “in terms of the information it introduces as well as the speaker’s stanceo that information”, [comparing] these contributions to his/her own understanding, and [formulating] a contribution thatwould] add to the discourse by extending, questioning or qualifying it (Wells, 1999, pp. 107–108).” Borrowing participation

etaphors (Lave & Wenger, 1991), we might characterize their actions between the September and April episodes as shiftingrom that of legitimate peripheral participants, to full(er) participants in a mathematical discourse. As evidence of this, recall

ark’s attempted proof in the September episode:

“Well basically what it broke down to was that the, umm, odd number could be separated into an even number andone and so two even numbers would add up to an even number and you would still have that one left over. But I hadno idea how to put that into an equation.”

At this point, Mark did not have the tools to engage in a mathematical discourse—he was unable to invoke a notationalymbolic system whereby he could express abstracted quantities, an understanding of the methods of proof, and the essentialtrategic knowledge to invoke ideas in ways that would lead to a proof. However, by the April episode, he had developed toolshat enabled him to participate more fully in a mathematical discourse. In particular, Mark, like his peers, employed moreophisticated forms of argumentation consonant with mathematical proof, engaged in habits of (transactive) argumentationithout teacher scaffolding, and spoke in terms of more abstract mathematical ideas.

What also strikes us, however, is that each of the utterances taken by itself—from any of the participants—was not aomplete or even fully accurate argument. Instead, there were subtleties of meaning, including lacks of clarity and fragmentsf ideas, in individual utterances that reflected what might be described as “exploratory talk” (see Truxaw & DeFranco, 2006),here one’s ideas are presented in draft form.15 In other words, no one person seemed to own the proof or its constituent

rguments. Rather, we see what emerged as a collective argument that had its origins in the social plane of public discourse.eflecting back on our earlier discussion on shifts in students’ mathematical knowledge, students showed in April that,

ndependently, they could invoke relevant and useful mathematical facts. However, it was through public discourse thattrategic knowledge—the unpacking and coordination of these ideas in ways that led to a mathematical argument—wasnvoked and refined. In our view, this privileges the role of the social in the development of students’ individual (private)

athematical understandings.Perhaps more importantly, we maintain that that collective argument was mediated through students’ transactive reason-

ng. For example, Mark (line 58) clarifies the task at hand, which helps refine Tom’s understanding (line 59); Tim requests clarification (line 66) which Brad is unable to give (line 67), then Tom offers a general explanation to the group (line 68);ark requests a clarification (line 70) to which Anthony and Tom offer a justification (lines 71–72); Mark then builds on his

efined thinking to propose a strategy (line 73) which Tom critiques (line 74); and Anthony clarifies Tom’s explanation (line5). In this, as we noted earlier in reference to lines 52–60, it seemed that transactive reasoning became the particular meansy which ideas were able to be taken up, critiqued, discarded, preserved or extended in the development of the argumenthat H is closed. As Sfard (2002) has argued, learning seemed to be inherent in—and a result of—this type of discursive conflictn that the communication, characterized here by transactive reasoning, forced the cognitive actions. As such, the transactiveeasoning that provoked discursive conflict became the very means by which students refined an “objectified discourse”Sfard, 2002) that allowed them to talk about mathematical concepts and processes (e.g., the center of a group, the meaningf closure, operations on arbitrary elements) “as if these were some real, self-sustained objects. . .existing independently ofhe discourse itself” (ibid, p. 45).

Moreover, it seemed that the process of verbalizing one’s ideas—not just the form or content of the utterance—wastself a catalyst for mediating thinking. This is seen, for example, in lines 52–56 when Tom presents a flawed argument,ut re-examines his own thinking as he verbalizes his idea. It also seems to occur when Mark asserts that the notion ofswitching” refers to the commutative property (not associative property). In particular, verbalizing this idea prompts him
o re-examine whether or not the commutative property could be applied (line 70) and ultimately make an alternativergument (line 73). This, again, points to the significance of public discourse as a means by which students could negotiateheir private knowledge.
15 This can be different than the way that Mercer (1995) defined “exploratory talk”, that is, talk in which students engage critically but constructivelyith each other’s ideas.


Finally, we would like to point to an implicit component of the preceding discussion. In particular, we maintain that theseexcerpts illustrate how students had come to view what it meant to engage in proving a mathematical claim by how theyparticipated in the construction of their argument. In particular, their discourse from these excerpts as well as the full lessonsanalyzed here, where the number of student transactive utterances increased from 27% to 64%, suggests that they had cometo view proving as a habit of mind that involved explaining, critiquing, justifying, and so forth, as the means to establishwhat conjecture was to be proved, to identify the assumptions from which deductions would ultimately be made, and tonegotiate meanings for the component parts of a conjecture (e.g., “center” of a group). In our view, the sociomathematicalnorms reflected in how they engaged in constructing an argument in the April episode further indicate students’ fullerparticipation in a mathematical discourse community.

5. Conclusion

This study suggests that classroom discourse that helps students’ appropriate transactive reasoning can support theirlearning of proof. In particular, we have explored here how an instructional practice that promoted transactive reasoningsupported students in developing a habit of interaction based on critiquing, clarifying, justifying, explaining, and elaboratingtheir mathematical ideas. More importantly, we suggest that transactive discourse became a means by which studentscould improve their “abilities to participate in mathematical practice, both the operational part (the symbolic technologyof mathematics) and the discursive part” (van Oers, 2002, p. 72). Indeed, the findings presented here, including that therewere shifts in the complexity of proof schemes and strategic knowledge students used in constructing proofs, that classroomdiscourse seemed to promote a transactive habit of reasoning, and that transactive reasoning allowed students to mediatetheir understanding of proof and proof constructions in particular ways, suggests that transactive reasoning functioned as acritical discursive tool in the development of students’ proof understanding. In this sense, we maintain that the excerpts givenhere—and the larger episodes in which they occurred—show an important reliance on transactive discourse when studentsare developing an understanding of proof and constructing proofs. We suggest, for example, that in the April episode theargument that H is closed was established as participants were able to mutually negotiate ideas that they might not haveestablished on their own and without transactive discourse.

At the same time, we caution that while this qualitative study points to transactive reasoning as a critical discursivetool in students learning to prove, we cannot claim that students’ transactive reasoning is the singular reason for shifts instudents’ proof schemes or strategic knowledge. It is likely one of a variety of reasons. However, we think it is promisingthat students’ exhibited positive shifts in their thinking in a pedagogical context in which they were encouraged to—andwere ultimately able to—fully participate in classroom discourse focused on constructing proofs. Our goal here has been toexplore how students’ transactive reasoning, as one mechanism for change, might support this learning.

While the results of this study are promising, more systematic study is needed to understand how classroom discourse islinked to the nature of mathematical knowledge that emerges. The current study suggests several avenues of further research:(1) the development of case studies of shifts in student learning as a function of whole-class and small group interactionscould provide further insights into the theoretical premise of this study that learning originates in the social plane and isconstructed through participation in a community of practice; (2) a schematic analysis of the flow of argumentation capturedin discourse threads might help us see how the forms of argumentation evolved (e.g., What happens at points of clustering andhow do clusters occur over time?); and (3) a more fine-grained analysis of transactive utterances, including how particulartypes of transactive utterances are related to specific forms of shifts in strategic knowledge and proof schemes, could furtherilluminate our understanding of how transactive reasoning can support learning proof.

We think our focus on the construct of transactive reasoning within a tradition of classroom pedagogy that has been dom-inated by direct instruction models of teaching is appropriate. Indeed, in our view, this study has significant implicationsfor classroom practice in undergraduate mathematics broadly, not just instruction focused on teaching and learning proof.In particular, because undergraduate instruction is often characterized by direct instruction, we wonder how opportunitiescan be created for students to engage in the types of (lengthy) transactive discussions that characterized this classroom. Wewonder also to what extent students’ difficulties with proof (or other subject areas) are rooted in these direct instructionalexperiences and what kind of evidence is necessary to shift deeply-held notions of classroom practice in undergraduateclassrooms to include transactive-based pedagogies. Undergraduate mathematics instructors typically belong to a popu-lation that has been highly successful in learning mathematics. How do those experiences prepare them as instructors ofstudents who might find undergraduate mathematics more challenging and who might, as we think, benefit greatly fromthe public negotiation of ideas through transactive discourse?

Moreover, what is the nature of student learning in classrooms where transactive-based pedagogies are not predominant.While we cannot claim that the kind of mediated thinking that occurred here would not occur in instructional contexts whereother discursive forms are predominant (e.g., those which might follow teacher-as-teller paradigms), we do postulate thatthese contexts (i.e., those where directive utterances are prevalent) might not give students the space for refining their own
ideas publicly, and as such, might not build in students’ thinking the critical, active, dialogic stance toward mathematicalideas that fosters a mathematical discourse. To this end, experimental studies that compare transactive-based pedagogieswith direct instruction might provide the kind of quantitative evidence of best practices that are needed to shift instructionalpractices in undergraduate classrooms.

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Documents

Understanding the role of transactive reasoning in classroom discourse as students learn to construct proofs