# Basic Reasoning (Call-Response)

## Notes on LOGICMOO

Poetic reasoning uses call/response (C-R) mechanism for "evaluation" the same way poetry does.

Instead of S-R (stimulus/response) coding, *Reasoning *is done via C-R coding

C-R better explains how our neuro-biology works. The human brain contains a special class of cells, called mirror neurons,

C-R explains innate behavior and how memory works. https://crtandthebrain.com/the-neuroscience-of-call-and-response/

C-R explains/supports everything S-R does and explains things S-R never will

(see Malcolm Gladwell's "Why sometimes it is not a good idea to ask people what they think" (The coke/pepsi problem))*Because they will try to make up something that "sounds correct" to them (even if they didn't believe it before, now they will)*

Our innate C-R mechanism is possibly why music and poetry affects us..

Sometimes just the cadence of the C-R (inner mentalese) is good enough reason with.

We potentially have rules/heuristic that define how we "narrate to ourselves in order to think"

We vet Ideas on whether or not they "sound good"

S-R (stimulus response) is likely one day to be seen merely as a flat-world version for the round world C-R

Wittgenstein's language game would have worked had it been a C-R process.

This internal C-R litmus requirement becomes grammar recognition and cadence (except in cases of fox p 2 gene issues)

And likely why animals like to dance and drum

Antiphony is generally used for any call and response style of singing, such as the kirtan or the sea shanty and other work songs, and songs and worship in African and African-American culture. **Antiphonal music** is that performed by two choirs in interaction, often singing alternate musical phrases.^{[1]}

Sometimes just the cadence of the C-R (inner mentalese) is good enough reason with.

## C-R (poetic) structures

are the key to argumentation

can describe how we do math and logic* " p implies q, and sometimes [wh]y "*

merges procedural knowledge as well as non-procedural knowledge

In fact, it has even tricked a some people (like Doug Lenat) into believing humans think by using logic ( instead of merely poetically.)

**The poetics of argumentation: the relevance of conversational repetition for two theories of emergent mathematical reasoning**

Susan Staats - College of Education and Human Development, University of Minnesota, Minneapolis, MN, USA

- Notes on LOGICMOO
- C-R (poetic) structures
- ABSTRACT - Received 15 June 2016
- Introduction
- Identifying and representing poetic structures
- Poetic structures in social, cognitive and linguistic perspectives
- Two theories of emergent mathematical reasoning
- Collective construction of arguments
- Poetic structures in mathematical monologues: warrants, backings and qualifications
- Coactions through poetic structures in the geoboard conversation
- Competing arguments through poetic structures in the geoboard conversation
- Conclusion
- References

## ABSTRACT - Received 15 June 2016

Poetic structures emerge in spoken language when speakers repeat grammatical phrases that were spoken before. They create the Accepted 6 March 2017 potential to amend or comment on previous speech, and to KEYWORDS convey meaning through the structure of discourse. This paper Mathematical discourse considers the ways in which poetic structure analysis contributes argumentation; poetic to two perspectives on emergent mathematical reasoning: structures Toulmin’s model of argumentation and Martin, Towers, & Pirie’s theory of collaborative coactions in multi-speaker discourse. Poetic structures appear in varied argument types and at varied educational levels. They appear to facilitate speakers’ expression of warrants, backings, qualifications, and coactions.

Disclosure statement: No potential conflict of interest was reported by the author.

## Introduction

A common characteristic of conversation is that people repeat each other. Repeating the phrases of a prior speaker is documented widely. Repetition occurs in conversation and in single speakers’ discourse (Tannen, 1989), in folktales and storytelling (Hymes, 1981), in classroom talk (Staats, 2008; Wortham, 2006) and in teachers’ identity narratives (Oslund, 2012). In these anthropologically grounded examples, repetition is usually called a poetic structure, following Roman Jakobson’s treatment of oral poetry (1960). Aligning one’s phrases with others’ serves functions for both speakers and listeners. Repetition can validate another speaker’s contribution, it can confirm participation, and it can make interpretation and production of new sentences faster and easier (Tannen, 1989). A key aspect of repetition in conversation is that it is emergent—it arises in conversation without explicit agreement or awareness of speakers. Such a widespread, multifunctional behavior that is fundamental to the exchange of ideas is likely to be implicated in collaborative learning.

In current mathematics education research, two broad research areas address the emergent quality of collaborative classroom reasoning. First, Toulmin’s theory of practical argumentation has inspired models of collective, classroom construction of arguments and proofs (Knipping, 2008; Krummheuer, 1995, 2007; Toulmin, 1958). Second, the theory of improvisational coaction highlights the distributed nature of mathematical discovery through conversation (Martin & Towers, 2009; Martin, Towers, & Pirie, 2006).

The goal of this paper is to demonstrate that central discursive moves in argumentation, such as warranting, justifying, qualifying, and the collaborative adopting and amending of conjectures, are often conveyed through the structure of speech, through poetic structures, as much as through the lexical dimension of speech.

The paper opens with examples of poetic structures, recommendations for representing repetitions in transcriptions, and commentary on the treatment of poetic structures in several academic traditions. Following this, I review two theories of emergent mathematical reasoning, Toulmin’s model of argumentation, and perspectives on the development of collective mathematical understanding advanced by Martin, Towers and Pirie. I consider the contribution of poetic structures to argumentation in postgraduate students’ mathematical monologues in Inglis, Mejia-Ramos, and Simpson (2007). I also identify poetic structures in a highly collaborative mathematical conversation drawn from Martin and Towers’ paper on the co-construction of mathematical insight (2009), and indicate the ways in which coactions can be expressed through poetic structures.

## Identifying and representing poetic structures

Poetic structures can involve either syntactic or lexical elements, or both. In this paper, I focus on repetition in which some grammatical structure is repeated, because this usually involves a statement that is long enough to express a conjecture or a method. In the following excerpt, for example, a first-year university student explained her solution method for −4w + 4 = 3 − 2w to her classmates (Staats & Batteen, 2010, p. 50). The class had arrived at this equation while solving an equation composed of rational expressions. Shaniqua narrated the final steps as the instructor wrote her equations on the board.

**LOGICMOO defines/calls these poetic structures ****Sequegens**

### Selection 1

illustrates sentences that have grammatical repetition but not full lexical repetition.

Shaniqua’s explanation relies on four sentences that are similar to each other: the sentence over lines 1 and 2, at line 3, at line 4 and over lines 6 and 7. Each of these four sentences repeats a general grammatical structure:

Pronoun/modal auxiliary verb/main verb/secondary verb/equation as direct object (we; you)/(“gotta”; can)/(try; plus; minus; divide)/(get; gives)/(current equation)

The pronouns alternate between we and you. The modal auxiliary verbs are "gotta" and "can", and the main verbs are try, plus, minus and divide. Modal auxiliary verbs qualify their main verbs with a manner of knowing or level of certainty (Palmer, 1979). A secondary verb get or give introduces the equations that result from algebraic manipulation. In addition to this broad repetition, there is a secondary level of organization, in which each of the four combinatorial subject-verb arrangements of we, you, "gotta", and can also occur.

Though it’s impossible to know why Shaniqua spoke in this highly patterned way, it certainly lends a sense of cohesion to her solution statement. The scholarship of David Pimm and Tim Rowland allows us to consider more deeply the pragmatic force of this patterning. The pronoun we can represent coercive or authoritative qualities of a discourse community (Pimm, 1987) and the pronoun you can represent a sense of generalization (Rowland, 2000). Rowland suggests that modal verbs are important in mathematical discourse because they represent a “propositional attitude”, positioning the speaker “towards the factual content of what s/he says” (2000, p. 64). Shaniqua’s modalisation is based on gotta and can shifts between a sense of obligation and of option or agency. It is through her robust poetic structure that Shaniqua most clearly shows her confidence in algebraic procedure, but the repetition also allows her to offer quiet colourings of the work of algebra, a world in which one has agency to choose a problem-solving pathway even while participating within an academic community. It is well-known among discourse researchers that no form of transcription is objective (Herbel-Eisenmann, 2014; Ochs, 1979). Every layout represents the researchers’ theoretical choices, with some features of the spoken sound highlighted and others omitted. A poetic structure layout must present the repetitions that the researcher considers important, but it must also respond to the degree of repetition and the width of the printed page. In choosing a layout for selection 1, I tried to arrange the basic syntactic unit of pronoun / verb phrases / equation as direct object onto one line with subject/verb phrases in one column and the equations in another column. I identified the secondary level of repetition in the verbs and pronouns with several styles of underlining. Underlining highlights a stretch of discourse that the researcher thinks coheres together and that is repeated, even if there may be some variable words inside of it. The verb phrases with gotta … get received single underlining and the verb phrases with can … gives received double underlining. Indentation helps to place grammatically similar phrases into columns, and in cases like selections 2 and 3, to indicate a stanza of lines that shows very strong repetition.

## Poetic structures in social, cognitive and linguistic perspectives

The study of grammatical repetition has a fairly complex academic history. In a well known study from sociolinguistics, Dutch shopkeepers demonstrated conversational repetition (Levelt & Kelter, 1982, p. 89, original translation). When asked, “What time does your shop close?”, shopkeepers tended to answer, “five o’clock”. The question “At what time does your shop close?” prompted the answer “at five o’clock”, maintaining the prepositional phrase across question and answer. Tannen reports similar conversational turns, for example, when a woman teases her friend about sharing a soft drink called Tab (1989, p. 57, original underlining and emphasis, layout slightly modified).

- Marge: Do you want to split a Tab?
- Kate: Do you want to split MY Tab? (laughter)

In Marge and Kate’s exchange, repetition influences the emergent quality of meaning in conversational interaction. When Marge uses the indefinite article in a Tab, she presupposes that the soft drink is a novel object in the conversation. This allows Marge to claim a position of generosity by offering to share the drink. However, a new meaning emerges when Kate repeats her entire sentence with a single shift to the possessive pronoun in my Tab. With a single word, Kate suggests that the drink, in fact, had a prior presence in their shared world, and that Kate is the legitimate authority over its dispensation. Neither sentence alone captures the full meaning of the exchange. The discursive structure of the repetition is the smallest unit that allows an adequate analysis of the scenario.

Sentence-level repetitions inspired intensive experimental work in psycholinguistics as a means to understand the mental representation of language (Bock, 1986; Branigan & Pickering, 2004). Experiments typically measure the tendency to repeat a grammatical construction after being exposed to or “primed” with one of two possible options. For example, the double object prime sentence: The police handed the judge the cake tends to create increased preference to say The pilot sold the teacher the book rather than the direct object alternative: The pilot sold the book to the teacher. This experimental work suggests that at least some forms of grammatical repetition are explained by cognitive processes; other cases of repetition, like the soft drink example, may be explained additionally or alternatively by social functions of communication.

A second academic tradition in the study of repetition was greatly influenced by the work of linguist Jakobson (1960). For Jakobson, whenever an addresser conveys information to an addressee, several communicative functions are active. The referential function specifies relevant elements of the context, the phatic function involves cues about when the speaker holds and relinquishes a turn at speech, and the emotive function represents the sentiments of the speaker. Jakobson describes the poetic function of communication in rather mathematical terms: “The poetic function projects the principle of equivalence from the axis of selection into the axis of combination” (Jakobson, 1960, p. 358, original italics). The axis of combination refers to the ordering of words over time, in a spoken comment or in a sentence, so that there are particular positions to be filled. The principle of equivalence refers to the act of selecting from among words that are similar in some way—in terms of meaning, grammatical role, rhythm, potential for alliteration, and so on, and inserting them from this axis of selection into an appropriate location in the axis of combination. In Tannen’s soft drink example above (1989, p. 57), when Marge asks, Do you want to split a Tab?, she establishes an axis of combination. Kate uses the principle of equivalence to notice that a Tab is similar to my Tab, which she projects into her poetic play on Marge’s words. For Jakobson, the key role of the poetic function is to draw attention to the form of a message; this self-reference is the central feature of spoken and written language that makes it poetic. The poetic structures that are described below involve similar small shifts in one of the components of a grammatically ordered comment about mathematics.

## Two theories of emergent mathematical reasoning

**Toulmin’s model of argumentation**

Toulmin’s model of argumentation offered a theory of reasoning that sought to be widely relevant. This model intends to explain informal ways in which a social group comes to accept a conclusion, without using formal logic, through six components: datum, warrant, backing, modal qualifier, rebuttal and conclusion (based on 1958, p. 104). The core of an argument is the connection between the foundational information, the “datum” of a question, to a conclusion. Toulmin’s model became an influential alternative to formal logic through its attention to several qualities of reasoning—warrants, backings and qualifications—that facilitate this connection and make it convincing to others. In the following commentary, we will consider the role of poetic structures in presenting warrants, backings and qualifications (Figure 1).

A warrant is a statement that connects a datum to a conclusion, which validates reaching the conclusion given that one accepts the datum. Warrants are often implicit and unstated, and they often hold force only in restricted conditions, for example, subject to qualification. In contrast, the backing is a “field dependent” statement that refers most directly to a warrant, and that asserts the validity of using the warrant in a particular social, professional or disciplinary context. Krummheuer, for example, considered finger counting as a backing for a child’s warrant on counting by grouping (Krummheuer, 1995, 244). In an adult domain, enumeration would be handled differently, for example, with survey data serving as a backing for warrants. Distinguishing backings and warrants can be difficult, but it is an important consideration for mathematics educators because a student’s argument may rely on mathematical methods that would not necessarily be recognised as persuasive or intelligible in other contexts.

Two components of Toulmin’s model that are sometimes neglected in analysis of informal argumentation are modal qualifiers and rebuttals (Inglis et al., 2007). Linguistically, modal qualifiers express a speaker’s epistemic orientation to an assertion, a level of confidence in an assertion. This can be accomplished through several linguistic means, including adverbs like possibly or probably, and modal auxiliary verbs like should, can, or must (Palmer, 1979). Inglis et al. (2007) argue that modal qualification is an important sign of mathematical sensitivity, particularly while students are in the process of developing a proof or an opinion about conjectures.

In Toulmin’s original model, if an argument involves a modal qualifier, then this component may be linked to a rebuttal, which indicates conditions under which a warrant is not a valid means of reaching a conclusion to an argument. In Toulmin’s model, a rebuttal is not a statement of opposition to an argument. Instead, it supports an emerging argument and its movement from datum to conclusion. A rebuttal recognizes that there are some situations in which the movement from datum to conclusion is not justified, but if these conditions are not present, the conclusion can indeed be reached. It will be

Figure 1. Toulmin’s diagram of an argument. The components of the argument are D, the datum; C, the conclusion; W, the warrant; B, the backing; Q, the modal qualifier; and R, the rebuttal. useful later to distinguish between arguments in which conflicting conjectures are in play, and arguments in which rebuttals are used to reach a conclusion.

Several researchers have applied Toulmin’s model to arguments that arise within mathematics classrooms in naturalistic ways, as a type of ethnography (Knipping, 2008; Krummheuer, 1995, 2007). Krummheuer (1995) considers classroom argumentation as a collaborative, social construction that takes place through speakers’ adjustment of their positions. This paper highlights cases when the adjustments of argumentation are conducted through the shared phrasing of conversational repetition.

Researchers have identified several difficulties in applying Toulmin’s model of argumentation to mathematical conversation. Toulmin may not have intended the model to make sense of emergent arguments, but rather, only complete ones (Simpson, 2015). Alternative typologies of warrants have been proposed; the blurred distinction between warrant and backing and the potential for different readers to create varied argumentation diagrams have been exposed (Nardi, Biza, & Zachariades, 2012; Simpson, 2015). The current discussion will be most useful to readers who accept close analysis of emergent mathematical arguments.

**Collective construction of arguments**

Another prominent approach to understanding collaborative argumentation is the body of work emerging from the research group of Martin, Towers and Pirie (Martin & Towers, 2009, 2015; Martin et al., 2006; Towers & Martin, 2014). Although several of their insights resonate with poetic structure analysis, the central connection is through their theory of improvisational coaction (Martin & Towers, 2009; Martin et al., 2006). In this view, collective understanding is constantly renegotiated. Coaction occurs when students build upon other’s previous mathematical work, so that a collective understanding—correct or not—can be shared, then dismantled and then re-collected by the group. Improvisational coaction is

a process through which mathematical ideas and actions, initially stemming from an individual learner, become taken up, built upon, developed, reworked and elaborated by others, and thus emerge as shared understandings for and across the group (Martin & Towers, 2009, p. 4).

This research group draws upon scholarship in musical and verbal improvisation to develop a framework for describing mathematical improvisation in collaborative conversations (Martin & Towers, 2009; Martin et al., 2006). Their perspective on improvisation captures the sense of unpredictability, emergence, and reference to prior expression that is also fundamental to poetic structure analysis. They find that four features of improvisation are relevant to mathematical collaboration. First, significant insights are spread across the comments of multiple speakers, and may never be articulated fully by an individual. Another characteristic of improvisational coaction is “collectively building on the better idea” (Martin & Towers, 2009, p. 15). If the group recognises a new idea as potentially useful, they may adopt it into the ongoing investigation. This decision may be implicit rather than explicit, a third feature of improvisational coaction termed “listening to the group mind” (Martin & Towers, 2009, p. 15). The collective nature of improvisational coactions also involves “an interweaving of partial fragments of images” (Martin & Towers, 2009, p. 14). Individuals may contribute parts of an idea, but the collective discussion of the group weaves them together into a more coherent commentary on the mathematical task.

While all four of these dimensions of improvisational coaction may be present in particular poetic structures, the characteristic of combining image fragments is always relevant. Grammatical repetition always involves incorporation of a prior comment into a new one. In multi-speaker conversations, the other dimensions of improvisation are likely to be in play as well.

Like Krummheuer and Knipping’s uses of Toulmin, the improvisational coaction model derives from a model of individual thinking. The theory of coaction extends the Pirie–Kieren analysis of dynamic, non-linear growth of mathematical understanding (Pirie & Kieren, 1994). Martin and Towers also acknowledge the significance of distributed intelligence perspectives on learning (Cobb, 1998; Cobb & Yackel, 1996), but they wish to emphasise the dynamic nature of mathematical conversations, and their focus is on smaller time segments of conversations. Martin and Towers make central use of Pirie and Kieren’s concept of image making, image having, and property noticing, three of the early stages in the growth of mathematical understanding, and recast them as collective mathematical activities (Martin & Towers, 2015; Towers & Martin, 2014). In collective image making, several students contribute to concrete activities to develop initial concepts about a mathematical object. In collective image having, students use an image to investigate a problem more deeply without returning to their initial concept building activities. In collective property noticing, multiple students collaborate to express a more general attribute of the mathematical object.

Importantly, Martin and Towers note that collectively developed images and properties are not just visual or pictorial ones, but may be “any ideas the learner may have about the topic, any mental representations” (2009, p. 2). In selection 1, for example, Shaniqua’s poetic structure expresses the sense that every algebraic operation in solving this linear equation is similar to every other operation—she is in an image having stage for solving linear functions. This paper will assert that a phrase that is repeated and modified by a second speaker is a non-visual image that plays an important role in making and having mathematical ideas.

### Perspective on selection and interpretation of discourse samples

Most of the samples discussed below are extensions of analyses published by other researchers. The most important selection criterion was to identify discourse samples with relatively little teacher or researcher presence, to avoid an explicit relationship of guidance and authority that might influence discursive repetition. In extending the analysis of argumentation to include the relevance of poetic structures, I tried to acknowledge and as much as possible adopt the position of the original researchers. I took this as an ethical requirement of re-analyzing others’ data—I assumed that their analysis and interpretive intuition is stronger than those of researchers who were not involved in the primary research.

Applying Toulmin’s model to mathematical discourse can be hindered when a student has not spoken a particular component of the argument. Qualifiers, backings, even warrants may have been left unspoken. I prefer to instantiate components of Toulmin’s model with particular spoken phrases. Some components of Toulmin’s model may simply not be present or constructable. I prefer this partial application of Toulmin to an approach that relies on researcher-developed statements that allow the model be fully realized.

## Poetic structures in mathematical monologues: warrants, backings and qualifications

Several of our first examples are drawn from a study on qualification in mathematical argumentation (Inglis et al., 2007). The authors of this study recorded postgraduate students as they evaluated number theory conjectures to be true or false, surrounding the definitions: given an integer n, if the sum of divisors is 2n, the integer is said to be perfect; if the sum of divisors is greater than 2n, the integer is abundant; if the sum of divisors is less than 2n, the integer is deficient.

### Selection 2

Chris responded to the conjecture: If p1 and p2 are primes, then p1p2 is not abundant (Inglis et al., 2007, p. 7). Just before line 1 in the transcript rendering below, Chris decided that this conjecture is “probably” true, and he checked it for pairs 2 and 3; and 5 and 97. The authors develop a complete Toulmin diagram that includes all six components, but that highlights the student’s use of a modal qualifier, probably. The authors include the following phase of Chris’ statement as the warrant of the argument (Inglis et al., 2007, p. 8, layout modified, line numbers added). Here, the poetic structure occurs across lines 2, 3 and 4: I know (feel) / this statement / is (should be) true / for (large, small, middle) p1, p2.

Inglis et al. (2007) consider this as an inductive warrant type or argument because the student has performed calculations for small values and for large values, and the student uses this to persuade (lines 2–4). Warrants and backings are the most important part of an argument, because if they are unspoken, there is little else that can count as a justification of the argument. It is notable, then, that the most insightful part of the student’s commentary—the warrant—was conveyed through a poetic structure. Here, the repeated sentence structure expresses the sense that the conjecture holds for a breadth of cases, and legitimizes the plan to prove the conjecture. In this way, the warranting character of lines 2 through 4 is not merely conveyed by the content of the isolable words, but by the poetic structure itself.

In addition to expressing qualification, poetic structures seem to be very well-suited for expressing the warrants for inductive arguments, focusing on specific cases and alluding to a broader conclusion. Importantly, though, repetition does not mean that the argument is correct or complete. Analysis of poetic structures in mathematical discourse should be taken as a means of highlighting the authentic nature of student reasoning and insight, which may represent various degrees of completeness and correctness.

Chris did not use the poetic structure rigidly – there are small shifts, for example, from is to should be, and from the adjectives large and small to the prepositional phrase in the middle, and the shift from knowing to feeling. The shifts have some importance in tracking the quality of Chris’ argument—the shift from knowing to feeling may help him position his evaluation of the conjecture as tentative. The shift from large/small to in the middle could be explained in several ways, for example, avoiding a grammatically awkward alternative middle p1, p2. Another possibility is that ending the sentence with in the middle focuses attention on a range of numbers that still carries some uncertainty for the student. As important as modal qualifiers like “probably” are, students signal their level of certainty extensively through larger units of discourse. Each of these poetic shifts work together to convey a qualification.

### Selection 3

In the next selection, another postgraduate mathematics student responded to a number theory conjecture: If n and m are abundant, then n + m is abundant (Inglis et al., 2007, p. 12, layout modified, line numbers added). The student’s initial, and correct, argument is that the conjecture is false. As the final comments in this selection suggest, he decided to try to prove the conjecture, but eventually identified counterexamples. In this selection, there are several distinct types of repetition, and so I use indentation and several types of underlining to draw attention to them.

Inglis et al. (2007) identify this argument as one centered on a structural-intuitive warrant, characterized by “using observations about, or experiments with, some kind of mental structure” (p. 12). In their analysis, the structural intuitive warrant is that the student asserts that divisors of a sum are not related to divisors of integers, a statement which occurs in lines 7–8. Divisors have certain properties—a mental structure—and the experiment with 3, 5, and 8 calls the conjecture into question. This interplay between mental representation and experimentation makes this a structural-intuitive argument. A different graduate student made a similar comment with very concise poetic phrasing: “because the factors of n + m don’t really have anything to do with the factors of n or m” (Inglis et al., 2007, p. 13, underlining added).

There are a couple of plausible interpretations of the role poetic structures play in this structural-intuitive argument. The strongest repetition is in the stanza of lines 3–5, starting with, is to do, with its divisors. This stanza is preparatory thinking to the warrant in lines 7–8. The warrant itself is repeated as a poetic structure in lines 9–10 as the student experiments with numerical examples. The poetic lines thus explore the nature of the mathematical structure—divisors of sums and products, so that they become the experimental or observational component of the structural-intuitive argument.

In the repetition of lines 7–8 and 9–10: when you add (two numbers) … I don’t know, the experimentation or exemplification is tied to the hedging quality of I don’t know. As in selection 2, poetic structures contribute to qualification of the argument. But in lines 11–13, the student shifts out of the exemplification strategy, and in this transition, he uses fewer poetic structures. One way to interpret this shift away from repetition is that the student has decided to reverse the direction of his initial argument—that the conjecture is false—and so the existing string of poetic structures is less useful for the future proof attempt. The repeating themes of multiples, divisors and examples did not uncover structures or relationships that could lead to a proof of the conjecture’s truth, and they assist in warranting the conjecture’s falseness. Continued repetition does not support the new direction of reasoning.

So far, we’ve seen that poetic structures lend themselves to expressing patterns, trials, and examples in both inductive and structural-intuitive arguments. In the following selection, a postgraduate mathematics student responded to the conjecture that if n is perfect, then kn is abundant for any integer k (Inglis et al., 2007, p. 15, layout modified, line numbers added). The conjecture is true for integers larger than 1.

### Selection 4

In this portion of the commentary, the student was close to achieving a proof for the conjecture for integers n larger than 1. He only needed to consider further that 1 is also a divisor of kn, so that the sum of divisors of kn is at least 2kn + 1. Inglis et al. (2007, p. 16) consider this as a deductive warrant argument type, because “the conclusion follows necessarily from the data”. The deductive character of this argument is conveyed primarily through two poetic structure transformations.

Line 1 summarizes the argument with an If … then structure; the phrase if n is perfect is the datum and then kn is abundant is the conclusion. This structure was repeated in lines 2 to 3 to accommodate stating the definition of a perfect integer, using the expanded, transformed structure If … take … divide … then … sum … is. This new form of the poetic structure was repeated again in lines 6 to 11 with small modifications, as take … divide … sum … get. This new repetition mostly conserves the verbal structure of the previous one, and in so doing, conserves relationships among the mathematical concepts. Maintaining the verbs (mostly) allows the noun positions to shift as the coefficient k is verbally distributed: pi becomes kpi, n becomes kn and 2n shifts to 2kn. This repetition culminates in a final poetic structure that provides the outline of a proof. As in previous selections, the support for the argument, warrants or backings according to one’s interpretive preference, emerge through conversational repetition.

As in selection 2, the poetic structure is prominent, but is not expressed rigidly. In line 7, we take kn instead of taking all kpi, which would have more perfectly replicated line 3. From line 3 to line 7, pi and kn occupy parallel discursive positions but they seem to represent different levels of analysis mathematically. How can we interpret this fluidity of a topic at a mathematically important moment? Generally, we cannot expect natural discourse to roll out in a consistently mechanical manner. Still, the most important mathematical relationships seem to be the divide … sum relationships, that is, to establish a parallel comparison between the pis that divide n and sum to 2n, and to establish a parallel comparison with kpis that divide kn and sum to 2kn. The introductory we take kn (line 7) initiates the topic that the conjecture must address, and then preserves the poetic structure as a way to apply the definition of abundant, decomposing the new integer kn into divisors kpi.

Overall, then, the student started with the conjecture, then repeated this if/then structure in order to introduce a definition, and then repeated it again to outline a proof. By transforming the datum into a definition and then into a proto-proof, the speaker uses poetic structures to preserve relationships among mathematical ideas. Maintaining a close verbal structure across the conjecture, definition and warranting commentary may be a means of reducing doubt. Discourse structure, beyond the mere words, facilitated the student’s first version of a deductive argument.

Poetic structures in collaborative discussions: coactions and competing arguments

In Martin and Towers’ analysis of collaborative coactions, future primary school teachers use a geoboard to classify triangles as scalene, isosceles and equilateral (2009, p. 7, layout modified). Their discussion compares the side lengths of a right triangle. On the geoboard, the hypotenuse consists of two line segments from a vertex to an interior vertex to another vertex, just like each leg of the triangle (see Figure 2, based on Martin & Towers, 2009, p. 7). Throughout much of the conversation, Mary and Shauna believe that the triangle

Figure 2. Geoboard representation of a triangle.

might be equilateral, because each side is “two pegs away.” Hilary believes that the hypotenuse has a different length compared to the legs.

### Selection 5

As the conversation continues, Hilary recalls the Pythagorean Theorem (line 67), and with this information, the group is able to resolve the question at line 85, concluding that An equilateral can’t have a right angle in it (p. 12). In the following discussion, we will consider the association between Martin and Towers’ coactions and conversational repetition. We will also consider the role repetitions could play if one analyses the conversation using Toulmin’s model.

**Coactions through poetic structures in the geoboard conversation**

Current work on collective coactions does not identify the types of evidence that indicate a coaction occurs, but rather, accepts that readers will notice the referential connections between students’ spoken ideas. Attention to conversational poetic structures can make coaction analysis more concrete. In lines 17 to 25 of this collaborative discussion, there is substantial repetition—both speaker-internal repetition and repetition across speakers —especially between Mary and Hilary. In Mary’s first comment, she establishes a phrase that distance should be the same as that. This phrase sets into motion a series of repetitions that allow the students to express their fundamental disagreements about their mathematical images. The repetitions rely on small changes such as that to this (lines 17 and 18); should be to would be (lines 18 and 19) to this is one peg away (line 22) and this is two (line 25). Though Hilary seems persuasive at line 25, Shauna later suggests that the triangle is really equilateral, by saying This one looks equal … Because it’s two and two and two. (lines 27–28). Emphasizing repetition and de-emphasising speaker, we could portray the conversation as:

This rendering of the conversation highlights the transformation of that should be into this is into it ’s. The repetitions suggest that the students are intently trying to bring their mathematical models of the geoboard triangle into alignment with each other. As the repeated phrases become shorter, the students seem to agree on the features that they need to look at in order to resolve the question, but not on how to visualize or interpret these features. The emergent, collective argument, focusing on several proposed images of distance, has a prominent poetic structure.

Just as repetition does not signal correctness, nor does it signal agreement. Poetic structure coactions in selection 4 demonstrate that shared phrasing does not necessarily mean that the mathematical image is shared fully. Instead, students have agreed upon a verbal sign, and they have agreed to work towards a shared interpretation of it. Repetition is a signal that coaction in emerging, collective mathematical thinking, is happening.

**Competing arguments through poetic structures in the geoboard conversation**

We can also analyze the collective nature of the students’ mathematical reasoning in terms of Toulmin’s model. Despite Hilary’s suggestion in line 25 that the hypotenuse and the sides have different lengths, an argument emerges that the triangle is equilateral. We can identify components of this incorrect argument with the datum, warrant/backing and conclusion at lines 17–18; lines 22 and 28; and line 27 respectively.

Mary expressed the datum for this argument, that the horizontal and vertical sides of the figure have equal length, in line 17: That distance should be the same as that, and Hilary repeated it in line 18, this should be the same as this. Hilary and Mary used similar phrases to express a shared datum. From this datum, Mary and Shauna argue for their false conclusion, which Shauna states directly at line 27, saying This one looks equal. Hilary argues that the side lengths are not all equal, which she states in line 20, … but they’re not. Hilary’s comment is not a rebuttal in the sense that Toulmin uses the term, because a rebuttal supports the current argument. Instead, Hilary’s comment at line 20 is better understood as a statement of a conclusion for an alternative argument. The components of Toulmin’s model for these two arguments, the Mary/Shauna argument and the competing Hilary argument, are shown in Table 1.

Just as in earlier selections, poetic structures play a prominent role in the emerging supportive statements for the arguments—the warrants or backings. In this multi-speaker

conversation, though, poetic structures link the competing warrants/backings of the competing arguments. All speakers use poetic structures to establish agreement on what the conflict is, and to create contrasting support for the two arguments.

In line 25, for example, Hilary supports her conclusion (the lengths of the legs and the hypotenuse are different) through a poetic structure transformation of Mary’s line 22. Mary’s comment at 22 of … this is one peg away and this is one peg away warrants her conclusion that side lengths are all the same, and at line 25, the sentence becomes Hilary’s … This is two and this is two and this is two and this is two … But that is not two. Hilary transforms Mary’s warrant in order to warrant (or back) a different argument. At line 28, Shauna transforms Hilary’s line 25 with … Because it’s two and two and two, using Hilary’s words to reassert her commitment to her own argument. Each of the lines 22, 25 and 28 are attempts to support the speakers’ arguments, and the later ones are clearly poetic structure transformations of the earliest one. The earliest of these supportive statements, Mary’s line 22, is a poetic transformation of lines 17 and 18 about side length distances being the same, which was the shared datum for both arguments.

I’ve left the interpretation open as to whether these supportive statements are warrants or backings, to acknowledge the likelihood that different readers would interpret them differently. My own preference is to consider Mary and Shauna’s poetic structures as warrants, and Hilary’s as a backing. When Hilary’s comments … This is two and this is two and this is two and this is two … , she seems to use a field dependent, mathematical approach of introducing side lengths and a figure that are not present in the original question; this seems to be a mathematical problem-solving strategy. By identifying all side lengths as … this is two … and contrasting this with the hypotenuse length … But that is not two, Hilary seems to build what Toulmin called an analytical argument, in which the backing directly includes the conclusion as a special case (Toulmin, 1958). Overall, though, the supports for the arguments, whether one categorizes them as warrants or backings, are highly facilitated by poetic structures.

In this conversation, poetic structures convey almost all of the mathematical thinking. The speakers use closely related poetic structures to agree upon the datum, to pose competing arguments, and to offer conflicting support—warrants or backings—for the arguments. Just as in selections 2–4, repetition is deeply engaged in establishing the warrants, backings, qualifications and more generally, the support for moving from datum to conclusion.

Conversational repetition is important for describing emerging, collective arguments because it allows speakers to use comparatively few words to signal topics on which they agree or disagree. Nearly all of the coactions in this conversation are literally co-phrases.

## Conclusion

Students engaged in emergent mathematical reasoning can convey central components of their arguments, warrants, backings and coactions, through poetic structures. All of the warrant types described by Inglis et al. (2007)—structural-intuitive, inductive and deductive warrants—co-occurred with prominent poetic structures. During a multi-speaker mathematical task, poetic structures were the primary discursive means of presenting coactions (Martin & Towers, 2009). Speakers at many educational levels—from primary to postgraduate—appear to use poetic structures to express their mathematical thinking across all argument types (Staats, 2008; Inglis et al., 2007).

In part, poetic structures facilitate argumentation because they allow speakers to investigate the interplay of similarities and differences. A poetic structure retains a previous part of an image while another part shifts. In selections 1–4, identifying the similarity among things was important to the speakers. In selection 1, Shaniqua’s highly repetitive cadence expressed her sense of image having (Pirie & Kieren, 1994). In selections 2–4, students were engaged in image making, but used poetic structures to express a particular level of similarity and difference. In selection 3, for example, the poetic structure: [It] is to do with its divisors … things that divide it … with multiples allows the speaker to consider alternative concepts that represent nearly the same thing. In selection 2, the poetic structure stem of I know that this statement is true for … allows the speaker to suggest that seemingly different numbers in fact may behave in the same way, in the environment of this conjecture. In selection 4, the initial statement in the poetic structure was a definition. The student expanded the units of the definition slightly to outline the beginnings of an algebraically oriented proof. His uncertainty about the conjecture’s truth may have compelled him to use a poetic structure to stay close to the relationships within the definition. In selection 5, though, Hilary first established similarity This is two and this is two and this is two and this is two … so that she could then forcefully assert the contrast But that is not two. This ability to express gradations of similarity and difference allows poetic structures to explore the kinds of precision that mathematical thinking requires.

A broader accounting for the association of emergent argumentation and grammatical repetition may derive from Toulmin’s distinction between warranting and field dependent backings, ambiguous though this may be. Warrants are less field dependent than backings, so that warrants are components of reasoning that may be widely accepted across disciplines or social settings. As a tool of ordinary conversation for creating a sense of legitimation, coherence or distinction, poetic structures can function as an informal warrant. Poetic structures place concepts into relationships of similarity and difference, and through repetition, assert the impression that these relationships are inevitable.

Quotes of students’ mathematical speech are commonplace in research literature, and in most cases, interpretation of speech is conducted through word-level decoding. Rarely are any discourse features recorded other than the words. And certainly, vast research insights have been garnered through this attention to words and not discursive form. Still, mathematics is a discipline committed to the study of abstract form. Mathematics education is relatively committed to the idea that mathematical meaning can be expressed through multiple representations—not just the “declarative sentence” of an equation, but through graphs, diagrams, and for some, proofs-without-words, in which the meaning is expressed through a non-verbal image. Mathematics education researchers can consider poetic structure analysis as a new form of representation for understanding how mathematical discourse amplifies, qualifies, and advances the word-level message.

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