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Identifying Kinds of Reasoning inCollective ArgumentationAnnaMarie Connera, Laura M. Singletaryb, Ryan C. Smitha, PattyAnne Wagnera & Richard T. Franciscoa

a University of Georgiab Lee UniversityPublished online: 27 Jun 2014.

To cite this article: AnnaMarie Conner, Laura M. Singletary, Ryan C. Smith, Patty Anne Wagner &Richard T. Francisco (2014) Identifying Kinds of Reasoning in Collective Argumentation, MathematicalThinking and Learning, 16:3, 181-200, DOI: 10.1080/10986065.2014.921131

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Mathematical Thinking and Learning, 16: 181–200, 2014Copyright © Taylor & Francis Group, LLCISSN: 1098-6065 print / 1532-7833 onlineDOI: 10.1080/10986065.2014.921131

ARTICLES

Identifying Kinds of Reasoning in CollectiveArgumentation

AnnaMarie ConnerUniversity of Georgia

Laura M. SingletaryLee University

Ryan C. Smith, Patty Anne Wagner, and Richard T. FranciscoUniversity of Georgia

We combine Peirce’s rule, case, and result with Toulmin’s data, claim, and warrant to differentiatebetween deductive, inductive, abductive, and analogical reasoning within collective argumentation.In this theoretical article, we illustrate these kinds of reasoning in episodes of collective argu-mentation using examples from one teacher’s practice. Examining different kinds of reasoning incollective argumentation can inform how students engage in generating and examining hypothe-ses using inductive and abductive reasoning and move toward the deductive reasoning required forproof. Mathematics educators can build on their understanding of these kinds of reasoning to supportstudents in reasoning in productive ways.

It has long been accepted that reasoning is important in mathematics, both in learning mathemat-ics and in doing it. In fact, one of the distinguishing characteristics of mathematics as a disciplineis its reliance on proof as the mechanism for establishing results. Although a proof requiresa particular kind of reasoning (deductive reasoning), other kinds of reasoning (e.g., inductivereasoning) are necessary for the exploration involved in conjecturing or hypothesis generating.

Correspondence should be sent to AnnaMarie Conner, Department of Mathematics and Science Education, 105Aderhold Hall, University of Georgia, Athens, GA 30602. E-mail: aconner@uga.edu

Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/hmtl.

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182 CONNER ET AL.

Perhaps because of the nature of spoken language, distinguishing between the different kindsof reasoning in a classroom conversation is not straightforward. Researchers in mathemat-ics education have used the construct of collective argumentation to examine the particularlymathematical aspects of classroom conversations, often using Toulmin’s (1958/2003) model ofarguments (e.g., Krummheuer, 1995; Yackel, 2002). At least as it has been applied to mathemat-ics education, however, Toulmin’s model does not capture important differences in reasoning,and thus on its own is insufficient for tracking the nuanced evolution of reasoning in arguments.We propose that combining two perspectives, one addressing particular parts of an argument(Toulmin, 1958/2003) and the other addressing distinct elements of reasoning (Peirce, 1956),provides a useful way for mathematics educators to examine the various kinds of reasoning thatoccur in classroom conversations. Combining the perspectives may help mathematics educatorsanalyze the kinds of reasoning that occurs in mathematics classes, determine the impact of differ-ent patterns of reasoning on mathematics learning, and examine students’ opportunities to learnto construct arguments in mathematics, especially if teachers emphasize certain kinds of warrantsor privilege particular kinds of reasoning.

In this theoretical article, we give our definition of reasoning, briefly describe Toulmin’s(1958/2003) parts and structure of arguments, connect that structure to collective argumentationin mathematics classes, and review Peirce’s (1956) descriptions of different kinds of reasoningand the elements thereof. We demonstrate how combining the two perspectives into a single dia-gram provides insight into the kind of reasoning (deductive, inductive, abductive, or by analogy)present in an argument, and we provide examples from one teacher’s class to show how the com-bination of perspectives works in practice. We then give some examples of how this combinationof perspectives can be used in research and practice.

DEFINING REASONING

Mathematical reasoning is central to learning and doing mathematics. Despite the importanceplaced on reasoning by mathematics educators, however, there is no consensus on the defini-tion of reasoning (Yackel & Hanna, 2003). One difficulty in defining mathematical reasoningis that it encompasses a wide range of mathematical practice, as illustrated by calls for reason-ing in several of the standards for mathematical practice in the Common Core State Standardsfor Mathematics (National Governors Association Center for Best Practices & Council of ChiefState School Officers, 2010). An examination of the research literature in mathematics educationrelated to reasoning reveals emphases on quantitative reasoning (e.g., Moore & Carlson, 2012),algebraic reasoning (e.g., Blanton & Kaput, 2005), geometric reasoning (e.g., van Hiele, 1986),and reasoning and proof (e.g., Yackel & Hanna, 2003), to name a few. In addition, Stylianides andStylianides (2008) described a difficulty in discerning among deductive, inductive, and analogicalreasoning, “the boundaries among which are often faint” (p. 105).

Thompson (1996) defined reasoning as “purposeful inference, deduction, induction, and asso-ciation in the areas of quantity and structure” (p. 267), and the National Council of Teachersof Mathematics (2009) described it as “the process of drawing conclusions on the basis of evi-dence or stated assumptions” (p. 4). The common aspect of these two definitions is the referenceto inference or drawing conclusions. Moshman (2004) defined reasoning as “epistemically self-constrained thinking” (p. 223). He went on to explain that making inferences is a natural process

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IDENTIFYING KINDS OF REASONING 183

in which everyone, even young children, engages, but when people are aware of and “constraintheir inferences with the intent of conforming to what they deem to be appropriate inferentialnorms” (pp. 223–224), they engage in reasoning.

We use these definitions to define mathematical reasoning as purposeful inference about math-ematical entities or relationships. We use mathematical entity in the sense of Zbiek and Conner(2006) as “any mathematical object from any area of curricular mathematics” (p. 92). In definingmathematical reasoning in this way, we emphasize the importance of purpose. Reasoning cannotoccur in the absence of a goal, although that goal might not be articulated clearly at the begin-ning of the activity, and the goal may be simply to make sense of a situation. For an activity tobe construed as reasoning, there must be some level of awareness of inference and constraintsthereon (Moshman, 2004). So, reasoning requires the presence of a goal (the endpoint or statedinference), premises from which the inference is reached, and some reasons that conform toappropriate constraints.

We view reasoning as being closely related to the activity of argumentation. Similarly,Toulmin, Rieke, and Janik (1984) equated arguments with “trains of reasoning” (p. 12). Toulmin(1958/2003) characterized arguments in any field as containing three main components: claim,data, and warrant, where a claim is a statement to be established, data include facts used tosupport the claim, and warrants show connections between data and claims. These componentsfrom Toulmin’s model correspond with our broad definition of reasoning. The stated inferenceis the claim, the premises from which the inference is made are the data, and the reason for theclaim is the warrant. When an individual is creating an argument, he or she is reasoning, andwhen an individual is reasoning, he or she is creating an argument; thus we contend that, whenconsidered as individual activities, argumentation and reasoning refer to the same process inmathematics.

When one begins to consider reasoning in classrooms, however, there are aspects of the socialsituation for which one must account. Ball and Bass (2003) used a definition of reasoning inclassrooms that is quite broad, asserting that “reasoning, as we use it, comprises a set of practicesand norms that are collective, not merely individual or idiosyncratic, and rooted in the discipline”(p. 29). Rather than changing our definition of reasoning to account for aspects of the socialsituation, we examine classroom interactions using the lens of collective argumentation. This isconsistent with Toulmin and colleagues’ (1984) description of reasoning as “a way of testingand sifting ideas critically . . . a collective and continuing human transaction” (p. 10). Becausereasoning is communicated during the process of collective argumentation, the construct of col-lective argumentation provides a useful part of our examination of the kinds of reasoning thatoccur in mathematics classrooms.

DEFINING COLLECTIVE ARGUMENTATION

In mathematics education, much of the research involving argumentation has used a model basedon the work of Toulmin (1958/2003) to investigate the structure of arguments constructed asteachers and students work together (e.g., Krummheuer, 1995, 2000, 2007) or as individuals con-struct proofs (e.g., Inglis, Mejia-Ramos, & Simpson, 2007). Following Krummheuer (1995) andothers (e.g., Forman, Larreamendy-Joerns, Stein, & Brown, 1998; Yackel, 2002), we characterize

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Data: Facts acting assupport for claim

Claim: Statementbeing established

Warrant: Link betweendata and claim

Qualifier: Describescertainty of claim

Rebuttal: Circumstancesunder which the warrant is notvalid

Backing: Usually implicit reasonthe warrant is valid in theparticular field

FIGURE 1 Generic Toulmin-style argument diagram with descriptionsof components (based on Toulmin, 1958/2003).

collective argumentation as multiple people working together to establish a claim.1 Toulmin dia-grammed the structure of an argument as shown in Figure 1. He asserted that the structure ofan argument could be examined separately from formal deductive logic and that this argumentstructure was similar in many fields, differing primarily in the content of the backings that wereacceptable in the fields.

Krummheuer (1995) introduced this model to mathematics education, choosing to use onlythe core of the argument (data, claim, and warrant) in his work. Many researchers (e.g., Hoyles& Kuchemann, 2002; Yackel, 2002) followed Krummheuer in using only the core. Inglis andcolleagues (2007) demonstrated, however, that Toulmin’s (1958/2003) full model is of value inanalyzing argumentation in mathematics. We use Toulmin’s full model in our diagrams of col-lective argumentation in general, but for the purposes of this article, we focus on argumentsconsisting only of the core components. We focus on the core because we are interested indescribing the kind of reasoning used in the argument, not its result or whether an outside observerwould consider it mathematically correct or appropriate. In the Analysis section, we argue that thecomponents beyond the core are not necessary in determining whether the reasoning is deductive,inductive, abductive, or by analogy.

To capture the collective nature of arguments in classrooms, we have modified Toulmin’s(1958/2003) model to reflect which participants in an argument contributed which components.We do not distinguish between particular student contributors, but assign color and line style indiagrams to denote whether a component was contributed by the teacher, only by a student orstudents, or whether the teacher and students worked together. (See Figure 2 for an illustrationof our modified diagram in a core argument from our data.) This argument occurred during aclass session in which the class was discussing various characteristics of triangles. The teacherdescribed a triangle with angles of 20, 72, and 88 degrees and asked students to identify thelongest side in the triangle. In the diagram, the student’s claim is indicated in blue with dashed

1We are not saying that every episode of collective argumentation requires multiple people to contribute orally to theargument. Rather, there are multiple people potentially involved in the argumentation.

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IDENTIFYING KINDS OF REASONING 185

Teacher: So, Tammy, what's the longest side? Thelargest side? The longest side?

Tammy: (inaudible)Teacher: c?Tammy: (inaudible)Teacher: Why is it the longest side?Tammy: I don't know.Teacher: Somebody wanna help her out? Okay,

Amanda?Amanda: Cause side c is corresponding to the largest

angle.Teacher: Right, cause it's across, side c is across from

the largest angle, angle 88 degrees. All right?That is the side angle inequality.

In triangle ABC, ifA = 20, B = 72, andC = 88, then side cis the largest side

Side c iscorresponding to(across from) the

largest angle

Side angleinequality

Data

Warrant

Claim

- Student- Teacher- Both

FIGURE 2 Modified diagram of an argument with transcript excerpt.

lines; the data, contributed jointly by teacher and student, is indicated by double purple lines; andthe warrant, contributed by the teacher, is bounded by a single red line.

COMBINING PERSPECTIVES TO EXAMINE KINDS OF REASONING INCOLLECTIVE ARGUMENTATION

Reid (2010) called for increased attention to kinds of reasoning and to the relationship betweenthem and what happens in mathematics classrooms. In particular, he called for attention to howstudents learn to prove and how that is related to the kinds of reasoning expressed in a classroom.Peirce (1956) distinguished different kinds of reasoning according to the ways each kind incor-porates the use of cases, rules, and results to arrive at a conclusion. When mathematics educationresearchers use Peirce’s perspective, they draw from his definitions of abduction, deduction, andinduction (e.g., Boero, Douek, Morselli, & Pedemonte, 2010; Pedemonte & Reid, 2011; Tall& Mejia-Ramos, 2010) and generally examine at most two kinds of reasoning (e.g., Ayalon &Even, 2008; English, 1998; Pedemonte, 2007; Pedemonte & Reid, 2011), at times in the contextof argumentation examined from Toulmin’s (1958/2003) perspective. Reid, in a reinterpretationof Peirce’s work for mathematics education, defined case, rule, and result and illustrated howthese three elements are related to each kind of reasoning, implying that different kinds of rea-soning have particular structures. The nuances of Peirce’s definitions with respect to case, rule,and result provide a useful analytic tool in exploring multiple kinds of reasoning.

According to Reid (2010), a case is “a specific observation that a condition holds” (p. 83).In this definition, a condition refers to a characteristic of something or an association betweentwo things. For example, the phrase “2 is a natural number” is a case in which “being a naturalnumber” is the condition. Reid’s definition of a rule is “a general proposition that states that if onecondition occurs then another one will also occur” (p. 83). “Natural numbers are integers” is a rulein which “being a natural number” and “being an integer” are linked conditions; if one occurs,the other will. Reid describes a result as similar to a case, but instead of simply being a specificobservation, a result is an observation that a condition holds, where the condition is related toanother condition by a rule. In other words, a result requires reference to a rule, whereas a casedoes not. In the example, the result “2 is an integer” depends on the link between natural numbersand integers provided by the rule.

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186 CONNER ET AL.

We offer the core of Toulmin’s (1958/2003) model as a helpful diagrammatic method to exam-ine the kinds of reasoning encountered in collective argumentation (see Figure 3 for elaboration).One of the strengths of the model is how it allows the structure of an argument to be examinedseparately from the contents of the components (e.g., Knipping, 2008). By characterizing thecontent of each component of an argument as a case, rule, or result, however, we can analyze theargument using both structure and content. In particular, we can examine the kinds of reasoningused in an argument by looking for the presence of rules, cases, and results and how they arerelated as claims, data, and warrants (see Figure 3).

Kinds of Reasoning

When engaging in deductive reasoning, one constructs conclusions as the logical consequenceof aforementioned assumptions or conditions. Deductive reasoning has been characterized as theonly kind of reasoning that allows one to arrive at a conclusion with certainty (e.g., Peirce, 1956).Using case, rule, and result, we can represent a simple deductive argument within the structureprovided by Toulmin’s (1958/2003) model. In deductive reasoning, a case is linked to a result bya rule. So, in a Toulmin-style diagram of a deductive argument, a case is data, a result is a claim,and a rule is a warrant (see Figure 3a).

When engaging in inductive reasoning, one draws abstractions or generalizations from individ-ual observations. Reid (2010) described inductive reasoning as moving from specific to general.

a. Deductive Argument b. Inductive Argument

c. Two Abductive Arguments d. Reasoning by Analogy

Case/Rule/Result

Analogy

Case/Rule/Result

Data

Warrant

Claim

Rule

Result

Case

Data

Warrant

Claim

Result

Rule

Case

Data

Warrant

Claim

Data

Warrant

Claim

Result

Case

RuleCase

Rule

Result

Data

Warrant

Claim

FIGURE 3 Toulmin-style diagrams of arguments reflecting differentkinds of reasoning.

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IDENTIFYING KINDS OF REASONING 187

In the mathematics classroom, inductive reasoning provides students with opportunities to noticepatterns among observations. The observed patterns provide the evidence needed to make a math-ematical prediction or to generate a conjecture. Although general statements can be made as aresult of inductive reasoning, these general statements are not certain in the mathematical sense.Inductive reasoning can be a bit more complicated to identify and diagram than deductive reason-ing, partly because there are often multiple cases and multiple results associated with an inductiveargument, but in the simplest case, in inductive reasoning, a result is linked to a rule by a case.(Often, inductive reasoning consists of many results linked to a rule by many cases, although notall the results or cases may be enumerated in an individual episode of argumentation.2) We assertthat a Toulmin-style diagram of an inductive argument includes a result (or multiple results) asdata, a rule as the claim, and a case or multiple cases as the warrant (see Figure 3b). Someresearchers, including Pedemonte and Reid (2011), have asserted that what we call the simplecase of an inductive argument (one case) is actually an abductive argument. When students areexplicitly establishing a rule as their claim, however, we see that as inductive; their intent is toestablish a rule even when it seems they are relying on only one case as the warrant.

Pedemonte (2007) defined abductive reasoning as making “an inference which allows theconstruction of a claim starting from an observed fact” (p. 29). Similarly, Reid (2010) charac-terized the structure of abductive reasoning as the reverse of deductive reasoning. Peirce (1956)originally called abductive reasoning hypothesis reasoning; he described it as a mechanism forexplaining a “very curious circumstance, which would be explained by the supposition that it wasa case of a certain general rule” (p. 135). Abduction can be seen in the mathematics classroomwhen students first come across a result and then have to guess or hypothesize which particularrule and case afforded (or might afford) such a result. According to Reid (2010), “in abductivereasoning a result and a rule lead to a case” (p. 83). The arguments we have found to exhibitabductive reasoning can be diagrammed with the case as claim and then either the result as dataand rule as warrant or the rule as data and result as warrant (see Figure 3c). As Krummheuer(1995) and others have pointed out, it is often difficult to determine precisely how students areusing particular information as either data or warrants.

Reasoning by analogy requires developing a claim based on noticing similarities betweencorresponding cases (Reid, 2010). English (1997) defined reasoning by analogy as noting thecorrespondence between the structures of one system and that of another system, depicting a kindof mapping that takes place from one system onto the other. Reasoning by analogy encompassesmuch of the reasoning that one might observe in classrooms. For example, students might makeclaims about how to solve a problem based on structural (or superficial) similarities to problemsthey have already solved. We focus primarily on the reasoning by analogy in which “one thing islike another” is given as the rationale for a claim. Although we acknowledge that reasoning byanalogy occurs frequently in other forms, our use of Toulmin’s model has facilitated identificationof reasoning by analogy only in this form. Other kinds of reasoning by analogy may be subsumedwithin deductive, inductive, or abductive reasoning; when that happens, our diagrams reflect thedeductive, inductive, or abductive reasoning rather than its analogical nature. For instance, Reid

2As a simple example, consider a student who used a dynamic geometry program to examine the sum of the measuresof the interior angles of a triangle. He or she might claim that the sum of the interior angles of a triangle is 180 degrees,showing only one example on the screen. The reasoning is inductive, with multiple results linked to the rule by multiplecases. The student, however, may say that the rule is established and cite only one case and one result.

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188 CONNER ET AL.

(2010) suggested that deductive reasoning could be used to verify a claim made by analogicalreasoning, or an analogy could be embedded within deductive reasoning, or analogues could existbetween separate deductions. Regardless of the relationships between the two kinds of reasoning,we can use Toulmin’s model to identify reasoning by analogy when an analogy is used as awarrant (see Figure 3d).

USING THE COMBINED PERSPECTIVE TO EXAMINE PRACTICE

Class Setting

The following examples are drawn from one class period in a ninth-grade integrated mathemat-ics course. We collected video and supporting data (such as worksheets) for one geometry unit inwhich students investigated, proved, and applied mathematical results related to triangles, quadri-laterals, and other polygons; this focus class period occurred on the second day of the unit. On thefirst day of the unit, the students began a task to explore the relationships between the measures ofinterior and exterior angles of various polygons. During this second lesson, the students becamefamiliar with the idea of an exterior angle, reviewed how to use a protractor to find angle mea-sure, and measured the exterior angles of various polygons while completing the task illustratedin Figure 4. We chose our examples from the focus class period because we wanted to reducethe amount of previous content the students might draw upon. The first class period had beenshortened, so there were not as many arguments on that day, and the students were involved ingeneralizing and conjecturing on the second day.

During the focus class period, a student teacher, Ms. Bell (a pseudonym), and her studentscontinued the task shown in Figure 4, measuring the exterior angles of different polygons to

Robot Gallery Guards Learning TaskThe Asimov Museum has contracted with a company that provides Robotic Security Squads to guard the exhibits during the hours the museum is closed. The robots are designed to patrol the hallways around the exhibits and are equipped with cameras and sensors that detect motion.

Each robot is assigned to patrol the area around a specific exhibit. They are designed to maintain a consistent distance from the wall of the exhibits. Since the shape of the exhibits change over time, the museum staff must program the robots to turn the corners of the exhibit. Below, you will find a map of the museum's current exhibits and the path to be followed by the robot. One robot is assigned to patrol each exhibit.

1. When a robot reaches a corner, it will stop, turn through a programmed angle, and then continue its patrol. Your job is to determine the angles that R1, R2, R3, and R4 will need to turn as they patrol their area. Keep in mind the direction in which the robot is traveling and make sure it always faces forward as it moves around the exhibits.

2. What do you notice about the sum of the angles (for R1, R2, R3, and R4)? Do you think this will always be true? Why?

3. Determine the sum of the measures of the interior angles of the paths the robots travel (try doing this without a protractor). Use this information to help you write a function that gives the sum of the interior angles of any n-sided polygon (n-gon). Justify your answer.

4. Looking at your results from #2 and #3, can you find a way to prove your conjecture about the sum of the exterior angles?

CR

R2R3

R4

R1

Exibit Z

Exibit BExibit C

Exibit A

Exibit D

Entrance

FIGURE 4 Task used in focus class period (adapted by the teacher fromGeorgia Department of Education, 2008).

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IDENTIFYING KINDS OF REASONING 189

explore the characteristics of both interior and exterior angle sums. At the beginning of the lesson,each of the student groups recorded their measures for the exterior angles of one of the polygonson the board. Using the measures recorded on the board, the students examined the sums of themeasures of the exterior angles of each polygon, found the measure of each interior angle, foundthe sum of the measures of the interior angles, and looked for patterns across their observationsto make and verify conjectures about interior and exterior angle sums. Later in the period, Ms.Bell and her students defined a regular polygon and investigated how to find the measure of anyinterior angle in a regular n-sided polygon. In the episodes we report, Ms. Bell’s students engagedin all four kinds of reasoning.

Analysis

In our analyses, our research team examined the data from the perspective of the students asmuch as possible. We view an argument as a reconstruction of the participants’ argumentation(see Krummheuer, 1995, pp. 232–234) that attempts to remain true to their intentions. What weput forth as the claim in an argument is whatever the students were attempting to establish or“get to.” It was their answer to a question they had posed or to their interpretation of a ques-tion they had been asked. That is, a claim might be a case, a rule, or a result. This interpretationis consistent with Toulmin and colleagues’ (1984) description of a claim as defining “both thestarting point and the destination of our procedures” (p. 29). We acknowledge that a differentgroup of students or a more mathematically knowledgeable person might have used the sameinformation in a different way. Thus, we are not claiming particular ontological status for anyparticular utterance used as claim, data, warrant, case, rule, or result. That same utterance ina different situation could be interpreted as a different part of an argument or used as a dif-ferent element of reasoning, based on how the students were operating at the time. We wereinterested in what these bits of information meant to the students in the ways they interpretedand used them; that is, we focused on the purpose of their inferences (i.e., their reasoning).We also are not claiming insight into the private thought processes of individual students; weare focused on the reasoning exhibited in the words and actions of individuals that were madepublic.

In our initial analysis, we diagrammed the episodes of argumentation according to the compo-nents of Toulmin’s (1958/2003) model (for details of our analysis, please see Conner, Singletary,Smith, Wagner, & Francisco, 2014). Our secondary analysis involved a purposeful examination ofpreviously constructed core arguments using Reid’s (2010) explanation of Peirce’s (1956) case,rule, and result. Our choice to use the core was linked to our interpretation of the kinds of rea-soning present in the class. Specifically, we do not believe that qualifiers, rebuttals, and backingschange the kind of reasoning used by the students. For example, a qualifier describes the certaintywith which a person makes a claim. This degree of certainty is relevant in assessing the degreeto which the participants believe the claim to be established, but it does not change the kind ofreasoning in which they are engaged. For instance, a person could say, within an inductive argu-ment, certainly the rule must be this. That does not change the inductive nature of the argument.Similarly, within a deductive argument, a person could say I think this is the result (perhaps heor she is not certain of a particular mathematical calculation). This does not change the deductivenature of the argument.

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190 CONNER ET AL.

Likewise, a rebuttal may be offered in the context of an argument, and the rebuttal may includea counterexample or other circumstances in which the rest of the argument would need to bemodified. This kind of challenge would not, however, change the underlying kind of reasoning inwhich the participants were engaged. For instance, suppose a student claimed that the length of aside of the given triangle was five because of the Pythagorean Theorem “unless the PythagoreanTheorem didn’t apply.” The student has made a deductive argument with a rebuttal. This rebuttal(or any other that might contain a counterexample, even one contributed by a different student orthe teacher) does not change the deductive nature of the argument, even if the original student’sconclusion was mathematically incorrect. We recognize that an argument is a possibly interme-diate step within a larger chain of reasoning. A rebuttal does not change the kind of reasoningin which the participants were engaging prior to its contribution, but it may change the directionof the larger chain of reasoning. We remain open to the possibility that analyzing larger chainsof reasoning, including arguments with rebuttals, will yield more complex characterizations ofreasoning within collective argumentation.

Finally, because the backing is related to the warrant of a given argument, and because it mustusually be inferred from the given warrant, we used the warrant rather than the backing in ourexamination of kinds of reasoning. Because the backing of an argument is usually implicit, it isan individual element even within collective argumentation and would provide insight into anindividual’s epistemically constrained thinking while not changing the kind of reasoning that iscontained within the collective argument.

We examined the core components in relation to each other to determine whether each com-ponent was a case, rule, or result. In particular, we used Reid’s (2010) definitions of case, rule,and result to classify each claim, data, and warrant. During this analysis, we often found it use-ful to first determine which component should be classified as a rule, because that classificationinformed our identification of the other components as case and result.

Four Arguments Exemplifying Different Kinds of Reasoning

For each example, we examined the mathematical content and the intent of the speakers of eachpart of the argument to identify it as a case, rule, or result. The position of each of the elementswithin the Toulmin (1958/2003) diagram distinguished the kind of reasoning that the class useddeveloping these arguments.

A Deductive Argument

Our first example of argumentation illustrates deductive reasoning (Figure 5) in Ms. Bell’sclassroom. This episode occurred during a small group discussion about ways to calculate themeasure of an interior angle of a polygon given an exterior angle. A member of the group hadoriginally proposed a rather complicated method of calculation. A different student suggesteda simpler method. He claimed that they could calculate the measure of the interior angle bysubtracting the measure of the exterior angle from 180 degrees. This student’s proposed methodwas the claim, as interpreted both by the student making the suggestion and by Ms. Bell, whorequested support for it. Ms. Bell asked the student, “Why do you think that?” He responded byproviding the data for his claim. He stated, “You’re making a half a circle if you have one exterior[angle] and one interior [angle].” Ms. Bell asked him to define what he meant by a half circle.

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CASE: One exterior and oneinterior angle make a half

circle

RESULT: The interior angle is180 degrees minus the exterior

angle

RULE: A half circle has180 degrees

Data

Warrant

Claim

- Student- Ms. Bell- Both

FIGURE 5 Example of deductive argument from Ms. Bell’s class.

His answer, that a half circle is 180 degrees, completed the warrant because it related his originalclaim that you could subtract the exterior angle from 180 degrees to his statement that an exteriorangle and interior angle form a half circle.

In this episode, the data is a case because the data is a specific observation that the interiorand exterior angles of a polygon form a half circle. The condition in this case is an associationbetween the interior and exterior angles; that is, they form a half circle. Similarly, in this episode,the warrant is a rule that links the concepts of a half circle and 180 degrees together to say that ahalf circle is 180 degrees. The student’s claim, in this instance, is the result because it is a specificobservation made through the application of the rule to the case. The condition is the 180 degreesminus the exterior angle, where the rule that a half circle is 180 degrees is used to relate theoriginal condition of the case to the final condition.

An Inductive Argument

The second episode of argumentation is an example of inductive reasoning (Figure 6) in thesame geometry lesson. When interacting with a different small group of students, Ms. Bell askeda student to consider what possible relationships exist between the number of sides of a givenpolygon and the sum of the measures of its interior angles. The student considered a possiblerelationship using the sum of the measures of the interior angles of a quadrilateral and of a pen-tagon. The student noticed that the sum of the measures of the interior angles of a four-sidedpolygon was 360 degrees and the sum of the measures of the interior angles of a five-sided poly-gon was 540 degrees. The student attempted to establish the relationship between the number ofsides of a polygon and the sum of the interior angles with the claim, “We’re 40 away [from thesum].” By this claim, he meant that the sum of the measures of the interior angles of a polygon is

RESULT: 360 and 540 are40 away from 400 and 500

RULE: Pattern: If you multiply thenumber of sides by 100 you will be 40

away from the sum of the interiorangles

CASE: 4-sided is 360degrees; 5-sided is 540

degrees

Data

Warrant

Claim

- Student- Ms. Bell- Both

FIGURE 6 Example of inductive argument from Ms. Bell’s class.

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always 40 degrees away from 100 times the number of sides. His data were his observations ofthe two interior angle sums; he observed that 360 is 40 away from 400 and 540 is 40 away from500. His warrant, connecting his data to his claim, was that he had established that the sum ofthe measures of the interior angles of a 4-sided polygon is 360 degrees and of a 5-sided polygonis 540 degrees.

The observations of the polygon interior angle sums are two cases, where the conditions arethe sums of the measures of the interior angles. These cases served as the student’s warrant,linking his data to his claim. The observations used as data, that 360 and 540 are 40 away from400 and 500 respectively, are his result, where the conditions of the sums of the measures of theinterior angles are linked to the conditions of being close to a multiple of 100 by the rule of being“40 away.” In this episode, the student was making a general claim based on observations ofparticular results. This episode provides a picture of the structure of inductive reasoning in whichthe combination of a result and the student’s observations of particular cases implied a rule (whichin this instance was not correct but reasonable based on a limited number of observations).

An Abductive Argument

The third episode of argumentation illustrates one possible structure of abductive reasoning(Figure 7). This episode occurred during a small group discussion in which the students offeredobservations about the patterns among the sums of the measures of the interior angles of differentpolygons. Ms. Bell asked the group, “What did you figure out?” One student, Mike, responded,“They [the sum of the measures of the interior angles] all end up being a multiple of 180.” Ms.Bell read out loud the list of interior angle sums recorded on the board: 360, 540, 720, and 897.Immediately after she said 897, Mike claimed, “Although, I think it’s supposed to be 900.” Ms.Bell questioned, “Why do you think that?” Mike provided his reasoning by stating, “Because900 is a multiple of 180.” The observation that the sum of the measures of the interior angles of apolygon is a multiple of 180 degrees served as the data for the argument. Mike’s claim was thatthe sum 897 degrees should be 900 degrees, and his warrant was that 900 is a multiple of 180.This example of abductive reasoning allowed the students to continue reasoning about the patternthey had observed.

In this argument, the claim, the value 897 should be 900, served as the case, in which thecondition is being the sum of the measures of the interior angles of a 7-sided polygon. The datafor this argument was the previously observed rule concerning the sums of interior angles being

RULE: Interior anglesums are multiples of 180

degrees

CASE: The value 897should be 900

RESULT: 900 is amultiple of 180

Data

Warrant

Claim

- Student- Ms. Bell- Both

FIGURE 7 Example of abductive argument from Ms. Bell’s class.

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multiples of 180 degrees. The student’s observation that 900 is a multiple of 180 was interpretedas the result, in which the condition of being the sum of the measures of the interior angles of a7-sided polygon is linked to the condition of being a multiple of 180. In this episode of collectiveargumentation, the student’s claim is a case, a specific observation. To support this case, thestudent provided the rule and result. Some might argue that this argument could be diagrammedwith the result (900 is a multiple of 180) as data and the rule (interior angle sums are multiplesof 180) as the warrant. The statement about interior angle sums, however, was stated as a fact onwhich Mike based his argument, and the result that 900 is a multiple of 180 was stated as a linkbetween the data and claim. Thus we infer he meant it as a warrant. Even if someone characterizedthe result as data and the rule as warrant, we would still classify this as an abductive argument.

An Argument by Analogy

The final episode we consider is an example of reasoning by analogy. In this episode, whichoccurred in a large group discussion after the students had presented their thoughts about the sumsof the measures of exterior and interior angles in polygons, Ms. Bell used an analogy to help thestudents construct a definition of a regular polygon (Figure 8). She initially asked her studentsto provide a definition (asking them to make a claim), and when they were unable to do so, shedescribed a regular polygon as a special kind of polygon. She continued by saying “a regulartriangle is an equilateral triangle,” characterizing regular polygons as like equilateral triangles(introducing the analogy as the warrant). Making this comparison, she asked her students whatwas special about an equilateral triangle, to which they responded that equilateral triangles haveequal sides and equal angles, thus providing the data. They then applied this characteristic ofequilateral triangles to describe a regular polygon, where this description is the claim. Becausethe warrant was an analogy, this is an example of reasoning by analogy. As we discussed in ourintroduction of reasoning by analogy, analogies may occur alongside or within other kinds ofreasoning. We believe Ms. Bell’s intent was to reason by analogy here. However, it is possiblethat students may have interpreted the analogy as an example and thus seen this as inductivereasoning.

As illustrated by the examples, each of the kinds of reasoning was helpful in moving stu-dents toward a more robust understanding of the mathematics being considered. Within one classperiod, Ms. Bell and her students used each of the four kinds of reasoning to develop and sup-port hypotheses and write definitions. In addition, in the next class period, Ms. Bell asked herstudents to use deductive reasoning to prove their previous claim about the sum of the measures

Equilateral triangles haveequal sides and angles

Special (regular) polygons haveequal sides and angles

ANALOGY: Regularpolygons are like equilateral

triangles

Data

Warrant

Claim

- Student- Ms. Bell- Both

FIGURE 8 Example of reasoning by analogy from Ms. Bell’s class.

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194 CONNER ET AL.

of the interior angles of a polygon. Even though she encouraged and supported all four kinds ofreasoning, she privileged deductive reasoning in the establishment of mathematical certainty. Theanalysis of this larger episode is beyond the scope of this article.

Observations From Using the Combined Perspective

The examples that we have presented provide only a small sample of the arguments that weobserved during Ms. Bell’s geometry unit. These four examples and the classroom interactionssurrounding them, however, illustrate trends that we observed in Ms. Bell’s classroom practice.She encouraged and supported her students’ engagement in multiple kinds of reasoning. Whethertheir claim was a case, rule, or result, she expected them to provide relevant data and warrants.At times, as in the example of a deductive argument, Ms. Bell assisted her students in contributingrelevant components to the argument, such as when she assisted in contributing the rule that a halfcircle has 180 degrees as the warrant in that argument. Even when the students’ reasoning resultedin an incorrect claim (as in the example of inductive reasoning), Ms. Bell asked her students toexplicate their reasoning and requested that they attempt to apply their rule to other polygons,allowing them to see the limitation of their conclusion.

There is no guarantee that multiple kinds of reasoning would be found in every classroom.In fact, it is possible that a particular kind of reasoning might be privileged in a classroom. Forexample, when a teacher is interested in pattern-finding, he or she may emphasize inductivereasoning. Or in classrooms in which the logical structure of mathematics is the focus, deductivereasoning may take priority. One of the interesting things about Ms. Bell’s classroom is that thetasks were seemingly chosen to promote student exploration and engagement, and within thatengagement, one could see multiple kinds of reasoning used by different groups of students.As Ms. Bell encountered different kinds of reasoning, she supported her students in their linesof reasoning rather than immediately asking them to reason differently. More research is neededto examine the implications of different kinds of reasoning on how knowledge is established inclassroom communities or how privileging one kind of reasoning affects student engagement andlearning.

Connections to Other Characterizations of Reasoning and Argumentation

In An Introduction to Reasoning, Toulmin and colleagues (1984) presented numerous additionalexamples of arguments, including an entire chapter devoted to the classification of arguments.Although the authors do not use the terms inductive, deductive, and abductive in their descrip-tions of different kinds of arguments, they do provide multiple examples of arguments that theydescribe as “reasoning from analogy” (p. 216), “reasoning from generalization” (p. 219), “reason-ing from sign” (p. 222), and “reasoning from cause” (p. 226). Their description of reasoning fromanalogy agrees with our definition of reasoning by analogy, but their reasoning from generaliza-tion, sign, and cause do not directly correspond to our other categories. Some of the argumentsunder these categories, however, do correspond to our description of deductive, abductive, andinductive reasoning, and it is important to note that Toulmin and colleagues were writing aboutreasoning in a host of fields, not just mathematics classrooms. For instance, their argument aboutwetlands (p. 229) as an example of reasoning from cause gives a claim that is a rule, data that are

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cases, and a warrant that is a result, corresponding to our diagram of an abductive argument inFigure 3c. Toulmin and colleagues include one more kind of argument, “reasoning from author-ity” (p. 229), which we have not specifically included in this article, but we acknowledge thatthis kind of arguments is not infrequent in some students’ mathematical experiences (which alsoparallels Harel and Sowder’s, 1998, authoritarian proof scheme). Toulmin and colleagues alsoacknowledged, as we do, other kinds of reasoning that are beyond the scope of their (and our)analysis.

Toulmin’s (1958/2003) model has been criticized as too structural (Sampson & Clark, 2008),perhaps because it (as generally used) lends itself to analyzing arguments that are primar-ily deductive, although clearly not only proofs. Toulmin spent much of the Uses of Argumentarguing that deductive arguments were not the only kind of arguments that are valid. He partic-ularly distinguished between warrant-using arguments (which might be deductive in nature) andwarrant-establishing arguments (which could be used in inductive reasoning). Although Toulminpointed out that it was possible to have arguments that were not deductive, the majority of theexamples in the chapter in which he describes “the layout of arguments” (p. 87) can be charac-terized as deductive. Toulmin’s model has been used most often for arguments that are or can beseen as somewhat deductive in nature or structure, and a deductive structure may be imposed bythe researcher who is reconstructing the argument (see Krummheuer, 1995, for a description ofhow arguments must be reconstructed from episodes of argumentation to be diagrammed).

Although many mathematics education researchers acknowledge that Toulmin’s (1958/2003)model was designed to include other than deductive arguments (e.g., Pedemonte & Reid, 2011),they sometimes make implicit assumptions of the deductive (or at least noninductive) nature ofarguments in descriptions of warrants as rules, formulas, principles, and definitions. For instance,Pedemonte (2007) suggested a deductive structure when she described warrants as rules: “Thewarrant, which can be expressed by a principle, or a rule, acts as a bridge between the data andthe claim” (p. 27). Forman and colleagues (1998) also suggested a generally deductive nature ofargument in their description of a warrant: “In discussions in mathematics classrooms, warrantsmay be formulas or algorithms that allow us to find the values of unknown variables from the val-ues of known variables” (p. 532). Inglis and associates (2007) described a warrant as “[justifying]the connection between data and conclusion by, for example, appealing to a rule, a definition orby making an analogy” (p. 4). In this description, Inglis and associates implied that most war-rants are rules, since a definition can be interpreted as a general rule of the form “This is that,”but they also allow for the possibility of reasoning by analogy. Inglis and colleagues also distin-guished between argumentation (and warrants) that “reduce uncertainty” and those that “removeuncertainty” (p. 9), suggesting implicitly that not all arguments are deductive.

In later work, Pedemonte and Reid (2011) specifically characterized some arguments asabductive and diagrammed them using a modification of Toulmin’s (1958/2003) model, usingquestion marks and multiple possibilities for backings as a way to distinguish abductivearguments from deductive arguments. Their purpose in the analysis was to demonstrate theprogression from abductive arguments to deductive proofs, and by diagramming the abductivearguments in this way, they preserved the same claims in the abductive and deductive argu-ments. Our alternative diagram of an abductive argument may not preserve the same claimwhen a deductive proof would be constructed from an initial abductive or inductive argument.However, we believe it illustrates the reasoning of the students by foregrounding the purpose oftheir inference—placing the case rather than the result as the claim.

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DISCUSSION

Combining Peirce’s (1956) and Toulmin’s (1958/2003) perspectives provides opportunities toextend research related to argumentation and reasoning and explore connections between thetwo. We emphasize that Toulmin’s model does not necessarily require a deductive structure fora mathematical argument as evidenced by our examples of the kinds of reasoning that occurredin Ms. Bell’s classroom. Researchers focused on argumentation in mathematics education areat risk of imposing a deductive structure on the Toulmin diagrams they create, perhaps becauseof their own mathematical biases. By attending to the kinds of reasoning that are being usedin arguments, researchers may ensure that these diagrams are more accurate representations ofclassroom interactions.

Toulmin’s (1958/2003) model offers structure to the sometimes messy and often ill-definedconstruct of reasoning in school mathematics. Our modifications of the model allow researchersstudying reasoning to unravel the complexities of who is providing which elements of a col-lectively produced reasoning episode. For instance, researchers might examine which membersof the collective generally provide the rules, cases, or results. In addition, the structure thatToulmin’s model provides allows researchers to parse important aspects of reasoning in theclassroom in order to examine details related to the support of student reasoning. For example,using this structure, one can explore the types of teacher questioning that lead to particular kindsof reasoning in the classroom or identify how often the teacher, as opposed to the students, isresponsible for a deductive argument. The answers to such questions can shed light on the elusivequestion of how students learn to reason.

By combining the two perspectives, one can see more clearly the kinds of reasoning presentin collective argumentation in a classroom. It is useful to document the presence of argumentswithin a classroom conversation, and doing so has proved enlightening with regards to the estab-lishment of mathematical knowledge in a class (see, e.g., Rasmussen & Stephan, 2008). Withthis combination of perspectives, one goes a step farther in analyzing the kinds of reasoning usedin the argumentation. Using this combined perspective, researchers may be able to investigatehow different kinds of reasoning are involved in the establishment of mathematical knowledge inclassrooms.

In addition, we believe that this combined perspective allows researchers to place an appropri-ate focus on the warrant as related to the data and claim in characterizing the kind of reasoningin the argumentation. Even though examining the contents of warrants alone does not alwaysdistinguish the kind of reasoning (i.e., in both deductive and abductive reasoning the warrant maybe a rule), the content of the warrant gives much insight into how the students and teacher arereasoning. Our description illustrates Inglis and colleagues’ (2007) distinction between warrantsthat reduce and those that remove uncertainty. As used in mathematics education, warrants areoften seen to remove uncertainty because of the generally deductive assumption that is madeby researchers when they reconstruct an argument using Toulmin’s (1958/2003) model. Whenone considers arguments to be inductive or abductive, however, one allows for warrants to onlyreduce, rather than remove, uncertainty. For instance, when the case was used as a warrant in theexample of inductive reasoning, the students’ uncertainty was only reduced, not removed. Theteacher pointed out the limitation of that warrant by asking about their observations of the sumsof the interior angles of polygons with more or fewer sides; that is, pointing out the possibilityof different cases. On the other hand, when a known rule is used as a warrant in the example

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of deductive reasoning, the claimed result is not questioned and is treated as certain. Looking atwarrants within the structure of arguments can give teachers insight into the ways students reasonin collective argumentation. Future research should examine how teachers strategically supporttheir students in contributing appropriate warrants in different kinds of reasoning.

Combining Toulmin’s (1958/2003) model with Peirce’s (1956) perspective may allow one torecognize instances of reasoning that are consistent with what Garuti, Boero, Lemut and Mariotti(1996) and others call cognitive unity, which is based on the connection between reasoning duringthe production of a conjecture and the reasoning in the eventual proof. Garuti and colleaguesexplained that:

During the production of the conjecture, the student progressively works out his/her statementthrough an intensive argumentative activity functionally intermingled with the justification of theplausibility of his/her choices. During the subsequent statement-proving stage, the student links upwith this process in a coherent way, organizing some of the previously produced arguments accordingto a logical chain. (pp. 119–120)

The combination of Toulmin’s and Peirce’s ideas allows one to examine the kinds of reasoningused in the arguments created while conjecturing and the ways in which the argument and rea-soning shift as the arguments transition to a proof. More importantly, the combined perspectivemay allow researchers to investigate how teachers support their students in producing argumentsas they transition from conjecture to proof. Specifically, this combined perspective along withrecent work in supporting argumentation (Conner et al., 2014) can shed light on how teachershelp students to move from nondeductive arguments to deductive arguments.

Without the use of both perspectives, being able to determine the kinds of reasoning that occurin a particular argument may be difficult. We see our combined perspective as an alternativeto Pedemonte’s (2007) examination of abduction and deduction in the construction of proofs.Pedemonte’s diagrams privilege the structure of the deductive proofs that are the goal of theinstruction in the classrooms she studied and so are useful in examining the cognitive unity (orlack thereof) in the arguments that lead to proofs. However, there are many times when teachersand students engage in collective argumentation in which the immediate goal (or even the eventualgoal) is not a deductive proof. The avenue of research opened by this combined perspectivecould inform instructional practice used to build on students’ reasoning and support the students’development and understanding of proof. The application of this perspective to other teachers’practice and their students’ reasoning may enable mathematics educators to identify the processesinvolved in supporting their students to reason deductively after they have reasoned in other waysto reach a conjecture.

The mathematics education literature base on proof and proving is extensive, and much ofit refers to the importance of deductive reasoning in the construction of proofs (e.g., Ayalon &Even, 2008; Stylianides & Stylianides, 2008). Two of the most prevalent results in the literatureon proof and proving are that proving is difficult for students at all levels (e.g., Weber, 2001), andthat many school and beginning undergraduate students accept empirical evidence (which wouldoften be characterized as inductive reasoning) as proofs rather than expecting the deductive rea-soning that characterizes a proof (e.g., Chazan, 1993). Our combined perspective offers a way formathematics educators to examine the kinds of reasoning students are using in often-messy class-room interactions and begin to build on students’ existing ways of reasoning. If a mathematicseducator’s goal is to help students know how to establish mathematical results with appropriate

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certainty, helping them to understand the differences between different kinds of reasoning may bean important step. And if proving is difficult for students, it seems that understanding and build-ing on how they are already reasoning might help to bridge that gap into the deductive reasoningnecessary for proof.

ACKNOWLEDGMENT

The authors would like to thank Dr. Jeremy Kilpatrick for his assistance in critiquing andimproving this manuscript prior to publication.

FUNDING

This research was supported by the University of Georgia Research Foundation under GrantFRG772 and the National Science Foundation through the Center for Proficiency in TeachingMathematics under Grant 0227586. Opinions, findings, and conclusions in this paper are those ofthe authors and do not necessarily reflect the views of the funding agencies.

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