Prospective elementary teachers’ claiming in responses to false generalizations

Journal of Mathematical Behavior 39 (2015) 79–99

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Prospective elementary teachers’ claiming in responses to false generalizations David A. Yopp ∗ University of Idaho, Departments of Mathematics and Curriculum and Instruction, 313 Brink Hall, Moscow, ID 83844, United States

a r t i c l e

i n f o

Article history: Received 19 November 2014 Received in revised form 12 June 2015 Accepted 15 June 2015 Available online 1 July 2015 Keywords: Counterexample Refutation Claiming Generalizing Argument Argumentation Proof Technical handles

a b s t r a c t When faced with a false generalization and a counterexample, what types of claims do prospective K-8 teachers make, and what factors influence the type and prudence of their claims relative to the data, observations, and arguments reported? This article addresses that question. Responses to refutation tasks and cognitive interviews were used to explore claiming. It was found that prospective K-8 teachers’ claiming can be influenced by knowledge of argumentation; knowledge and use of the mathematical practice of exception barring; perceptions of the task; use of natural language; knowledge of, use of, and skill with the mathematics register; and abilities to technically handle data or conceptual insights. A distinction between technical handlings for developing claims and technical handlings for supporting claims was made. It was found that prudent claims can arise from arguerdeveloped representations that afford conceptual insights, even when searching for support for a different claim. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Teachers need to be able to refute students’ invalid claims to help students develop an understanding of the mathematical situation (Giannakoulias, Mastorides, Potari, & Zachariades, 2010). Studies demonstrate that students and teachers have difficulty generating appropriate refutation arguments (Balacheff, 1991; Potari, Zachariades, & Zaslavsky, 2009; Giannakoulias et al., 2010). While previous research has noted that students and teachers give problematic responses to false generalizations, the literature lacks careful attention to the claims presented and the influences on those claims. Logically, one counterexample establishes that a generalization is false. Some literature suggests that further exploration of false statements can present opportunities for rich mathematical investigations. The basic idea is that once a counterexample is found, a student might attempt to classify all counterexamples, find counterexamples that provide insight into why the generalization is false, or develop a true generalization by altering the original claim (Peled & Zaslavsky, 1997; Komatsu, 2010; Yopp, 2013). On the other hand, as will be shown in this article, attempts to go beyond the existence of a counterexample, including making claims about classes of counterexamples and claims about cases that conform to the original claim, can lead to problematic responses when a counterexample would have sufficed. Barring two notable exceptions, Balacheff (1991) and Galbraith (1981), the literature has not addressed problematic responses developed after a counterexample has been identified.

Abbreviations: CI, conceptual insight; PST, prospective elementary (K-8) teacher; TH, technical handle. ∗ Tel.: +1 208 885 6220; fax: +1 208 885 5843. E-mail address: [email protected] http://dx.doi.org/10.1016/j.jmathb.2015.06.003 0732-3123/© 2015 Elsevier Inc. All rights reserved.

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Fig. 1. Wuan’s response to the prompt, “Develop a viable argument for or against a claim that the sum of five consecutive numbers is divisible by 6.”.

In the United States, teachers’ difficulties with communicating appropriate responses to false generalizations could prove particularly problematic for mathematics students. Common Core State Standards for Mathematics (CCSSM) call for students to construct counterexamples (NGACBP and CCSSO, 2010), and assessment designers are proposing middle-grade assessment items that ask students to test propositions or conjectures with examples (SBACS, 2013). Similar items for earlier grades may soon follow. Stylianides and Ball (2008) provide evidence that conjecture exploration and counterexample production are within the conceptual reach of children as early as third grade. This paper explores the following research question: When faced with a false generalization and a counterexample, what types of claims do prospective K-8 teachers (PSTs) make, and what factors influence the type and prudence of their claims relative to the data, observations, and arguments reported? A “claim” is a mathematical statement that an arguer believes to be true. “Prudent claims” are claims that can be supported by the data, warrants, or conceptual insights that accompany a PST’s claim. Claims are different than conjectures, in my lexicon, because conjectures have no connotation of truth. In this study I find that after PSTS acknowledge a counterexample, they often make imprudent and problematic claims. Furthermore, PSTs’ ability to articulate prudent claims, with a goal of creating a viable argument, is influenced by their perception of the task; their natural language usage; their knowledge of, use of, and skill with the mathematics register; their knowledge of argumentation; their knowledge and use of the mathematical practice of exception barring; and their ability to handle data and conceptual insights appropriately and prudently. 2. The issue In order to communicate a viable argument once they are aware of a counterexample, PSTs must report the counterexample and demonstrate that the example is indeed a counterexample, and/or they must present an alternative claim and support for that claim. This is the critical issue. How PSTs report this information can influence the correctness or appropriateness of their responses, even when the counterexample presented is otherwise correct. Student responses shown in Figs. 1 and 2 illustrate problematic reporting. These responses appeared on final exams in my mathematics courses for prospective elementary school teachers, which served as the context for this study.

Fig. 2. Bobbi’s response to the prompt, “Develop a viable argument for or against a claim that the sum of five consecutive numbers is divisible by 6.”.

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Wuan (Fig. 1) reports an appropriate counterexample in the foundation and demonstrates that the candidate is indeed a counterexample in the warrant. Yet the claim “the sum of five consecutive numbers is not divisible by 6” is literally equivalent to a generalization. This false claim also occurs in the warrant (narrative link) of Bobbi’s response (Fig. 2), although the statement that Bobbi labels as the claim is equivalent to an existence statement. In the second line of her warrant (narrative link), appearing faintly, Bobbi inserts the word “always” between “not” and “divisible.” It is unclear what Bobbi believes this insertion changes about the domain of the claim. The variable expression in Bobbi’s foundation complicates an interpretation of her argument. Bobbi’s discussion of this expression appears to support a generalization. In contrast, the example in Wuan’s data and warrant appears to support an existence statement. While a reader might make inferences about what each of these PSTs intended to write, the point is that it is not completely clear what either is claiming. We are left to infer the intended meaning of these claims, and we do not have clear evidence that the PSTs have mastery of the mathematics register or accepted mathematical argumentation practices. Such responses can be problematic for instructors to assess—and even more problematic when presented to other PSTs during collective argumentation, as will be shown in Section 7 that follows. This article explores the quality and prudence of PSTs’ claiming, the processes through which their claims arise, and influences on their claiming.

3. Background literature Much of the literature addressing students’ and teachers’ responses to false generalizations describes the difficulties these arguers have in generating counterexamples and the challenges they face in noting the status of counterexamples as refutations. Some teachers undervalue counterexamples as refutation arguments, and some teachers express views that counterexamples are exceptions as opposed to complete refutations of a claim (Potari et al., 2009; Giannakoulias et al., 2010). Some teachers view refutations based on theory as providing stronger and more general conclusions than counterexample arguments, even when the theory referenced does not actually disprove the claim (Potari et al., 2009; Giannakoulias et al., 2010). The “theory” that Potari et al.’s teachers use in refutations includes: maintaining that none of the “known” theorems apply, referring to a nonexistent theorem, noting an inappropriate use of a theorem, and stating a general rule in opposition to the claim. Studies have also found that both students and teachers may believe that counterexamples and a proof of a claim can co-exist (Balacheff, 1991; Galbraith, 1981; Potari et al., 2009; Stylianides & Al-Murani, 2010), although Stylianides and Al-Murani report finding no evidence of this misconception when they interviewed participants who at first appeared to express this view. This research is important to my study because a PST’s understanding and beliefs about what constitutes refutation in mathematics are likely to influence the type of claims a PST develops. Of particular interest to this study are the actions that PSTs take once they note a counterexample. In the mathematical philosophy literature, Lakatos (1976) uses the history of mathematics literature and a fictional account to provide a description of actions that mathematicians might take when facing a refutation. This description has proven useful in studying student and teacher refutations (see Balacheff, 1991; Giannakoulias et al., 2010; Larsen & Zandieh, 2008). Lakatos’s description includes monster-barring, in which a definition for the mathematical object at hand is revised to bar objects that make the claim untrue; exception-barring, in which the class of mathematical objects considered is reduced to bar the objects that make the claim untrue; and proofs and refutation, in which the “proof” is analyzed to find a class of mathematical objects for which the argument is valid. Along these lines, Balacheff (1991) found that once they “witnessed” a counterexample, the 13- and 14-year-olds in his study either modified their original conjecture’s conditions or conclusion, modified the definition of the objects concerned, or “brushed aside” the counterexample as not sufficient to call the conjecture into question. Influences on these actions (which Balacheff describes as determinations of choices) included students’ background in logic, students’ background in argumentation, and students’ perception of the didactical contract. Galbraith (1981) reports similar findings and notes cases when students made conjectures based on insufficient evidence and failed to acknowledge or define the domain of a conjecture. Balacheff (1991) is important to my study because his students’ responses can be associated with claiming, even though it is not clear in his work whether the students stated an explicit claim when they modified the conditions and conclusion or brushed aside counterexamples. If the students did state explicit claims, one wonders how they worded those claims.

4. Theoretical framework for this study In this study I asked PSTs to present arguments that included an explicit claim and support for that claim. I also requested that they present only claims they believed they could support with a viable argument. Therefore, it is assumed that PSTs’ claiming in this study was influenced by the arguments they developed and the actions they took in developing these arguments.

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4.1. Arguments and activities associated with argument production 4.1.1. Arguments and viable arguments Toulmin (1958/2003) defines an argument as having a claim and support for that claim. Toulmin develops an argument analysis scheme that has been used extensively in mathematics education literature to construct plausible argument layouts (e.g., Conner, Singletary, Smith, Wagner, & Francisco, 2014; Krummheuer, 1995; Krummheuer, 2007). In particular, Giannakoulias et al. (2010) employ the scheme to describe teachers’ responses to false generalizations. Toulmin’s scheme consists of constructing a layout for an argument by identifying and labeling argument features as follows: a statement being established as true, labeled as claim; facts and other information supporting the claim, labeled as data; a link between the data and claim, labeled as warrant; reasons the warrant should be taken as valid in the particular field, labeled as backing; descriptions about the certainty of the claim, labeled as qualifiers; and circumstances under which a warrant is not valid, labeled as rebuttals. Krummheuer (2007) chooses to focus on the core of the argument – claim, data, and warrant – but other researchers adopt the full model or some modification of it (e.g., Conner et al., 2014). To understand PSTs’ support for their claims, I adopt Toulmin’s scheme to construct a layout of the PSTs’ arguments. I adopt the full model but focus primarily on the core. I use this core to define what I mean by viable argument. Viable arguments have a claim, data, and a warrant and meet the following criteria: (1) express a clear, explicit, unambiguous, prudent, and appropriately worded claim; (2) express support for that claim that involves acceptable data (or foundations); (3) express acceptable warrants (or narrative links) that link the data to the claim; and (4) identify the mathematics (definitions and prior results) on which the argument relies. Acceptable data/foundations include examples, diagrams, prior results, definitions, narrative descriptions, stories, etc., provided that the representation of the data/foundations can be appealed to appropriately in the warrant. For instance, examples are acceptable when crafting arguments for generalizations when they are generic examples (Balacheff, 1988; Sandefur, Mason, Stylianides, & Watson, 2013; Yopp & Ely, 2015) and are used as a referent to illustrate objects or relationships that support the claim. Viable example-based arguments become possible when students are able to express features common to all possible examples and appeal only to those features generically in support of the claim (Yopp & Ely, 2015). Acceptable warrants express how the data/foundations are used to support the claim. If the claim is a generalization, then the warrant must express how the data are used to represent all cases. In other words, acceptable warrants express how the conditions of the claim are used to support the conclusion. In generic-example arguments, warrants must communicate how the examples are appealed to in representing all cases. They also must communicate how the properties demonstrated in the examples are common to all examples in the domain of the claim. Criterion 4, identify the mathematics on which the argument relies, also needs elaboration. Mathematical claims are based on the meaning of the objects and operations involved in the claim and, in more colloquial terms, on previously established or accepted results. These “meanings” are determined by definitions, axioms, and theorems. Viable arguments must express these meanings, at least semantically. Example-based arguments for existence claims, including counterexample arguments, fit into my viable argument framework as follows: One or more candidate examples are presented as the data/foundations, and a demonstration that a candidate has the desired properties is presented as the warrant. For counterexample arguments, demonstrating that the candidate has the desired properties means showing that the candidate is in the domain of the generalization but does not satisfy the conclusion. At times, “that the candidate is in the domain of the generalization” is presented without support and taken as self-evident. 4.1.2. Manipulate, get-a-sense-of articulate Because I am studying both the types and quality of PSTs’ claiming relative to the types of supports provided or available, I needed a framework for analyzing how the PSTs report coming to those claims, as well as how they report developing support for a claim. In other words, what types of activities occurred and are associated with the claim? Mathematics education researchers such as Sandefur et al. (2013) and Mason (1981) have found Bruner’s (1966) description of activity during mathematical thinking useful in analyzing participants’ actions. I found the description useful as well. Sandefur et al. (2013) restate Bruner’s description as follows: Manipulation (manipulate) means using familiar mathematical objects for a specific purpose. This purpose is “to get-a-sense-of some underlying structure, pattern, or relationship by experiencing the effects of various actions and forming conjectures” (Sandefur et al., 2013, p. 327). As the first two phases become more coherent, students may be encouraged to articulate. In my study, PSTs are asked to articulate an argument, which includes a claim, so their overarching purpose is the construction of at least one viable argument in response to a false generalization. For my study, “forming conjectures” can be described as get-a-sense-of what can be claimed and supported. Often this stage is implied by the connections between the manipulate stage and the articulate stage. Online posts make up much of the data in my study; thus, PSTs are inherently in the articulate stage. I found that the PSTs typically report their activities in coming to a claim. In this way their manipulate and get-a-sense-of stages are identified through the activities they report. 4.1.3. Technical handles and conceptual insights When analyzing PSTs’ responses, I use a framework for understanding how they assembled their activities into an argument. This analysis needs to consider both the claim and the support for the claim, which are assumed to be related. This is

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important because PSTs might report using one type of activity to develop a claim and report using another type of activity to develop support for that claim. In terms of developing support for a claim, Raman, Sandefur, Birky, Campbell, and Somers (2009) suggest movements, not always found in order, in creating a proof. The first is getting a key idea (a reinterpretation of that proposed in Raman, 2003) that gives a sense of belief and understanding that may or may not provide clues about how to write a proof. The second is a technical handle that gives a sense of “now I can prove it” and a way to communicate the ideas behind the proof. The third is a culmination of the argument into a standard form (relative to the audience). Sandefur et al. (2013) recast the Raman et al. framework and report “two important components” (p. 328) to creating a proof: “(1) finding a conceptual insight (CI), i.e., a sense of a structural relationship pertinent to the phenomenon of interest that indicates why the statement is likely to be true, and (2) finding some technical handles (TH), i.e., ways of manipulating or making use of the structural relations that support the conversion of the CI into acceptable proofs” (p. 328). I found a need to reinterpret the Sandefur et al. (2013) and Raman et al. (2009) frameworks for the context of my study. These two frameworks seem to use the term “handles” as a metaphor for getting a grasp on how to develop a proof. In my study, the claim is not necessarily in place, so the term handles needs to be broader. I use the term handles to describe how an arguer makes use of any insights, data, or observations in developing a viable argument, which includes developing a claim. In this sense, I analyze how the student assembles (culminates) a variety of data, insights, knowledge, and perspectives to develop an argument to present to others. A similar modification is needed for the term conceptual insight in order for claiming to be an explicit component of developing insight (e.g., a sense of what can be claimed). An additional modification to Sandefur et al.’s (2013) term conceptual insight is to use the phrase “relational structure between a generalization’s conditions and conclusions” instead of “structural relationship pertinent to . . ..” I found it difficult to use the term pertinent structure in my analysis because “pertinent” is in the eyes of the beholder and seems to presuppose a claim. I prefer the phrase relational structure between a generalization’s conditions and conclusions in part because the PSTs encountered false generalizations. Counterexamples express a relational structure between conditions, which they meet, and conclusions, which they do not meet. I distinguish between relational structure that is limited to a few cases and relational structure that encompasses a class of objects, which offers opportunities for making prudent generalizations. This broader term allows me to examine relational structure as it is expressed in each PST’s data and insights. I then use the PST’s expressed relational structure to assess the prudence of the associated claim. For my study, the term conceptual insights (CI) can refer to either of the following: developing a sense or belief based in relational structure between a generalization’s conditions and conclusions about what might be claimed (including whether a conjecture is true or false); or developing a sense or belief based in relational structure between a generalization’s conditions and conclusions about why a claim is true or false or what causes the claim to be true or false. Because the PSTs had an explicit goal of articulating a viable argument, which includes making claims, I needed two distinct but related types of handlings of data and CIs that describe PSTs’ efforts to achieve that goal. Technical handles of type 1 (TH1) describe ways of making use of data and CIs in developing and articulating claims. Technical handles of type 2 (TH2) describe ways of manipulating and making use of data and CIs in developing and articulating support for claims. TH2s align well with the way technical handles are described in Sandefur et al. (2013). I describe technical handles first without any connotation of whether they are constructive. I apply adjectives (e.g., prudent, imprudent, problematic, adequate, inadequate) to note a technical handle’s subtype and potential for viable argumentation. The adjectives prudent and imprudent need some attention. Galbraith (1981) asserts that “avoidance of conjecture on insufficient evidence” (p. 26) is an essential component of successful proving. An example of imprudent claiming is when a PST puts forth a generalization with infinite domain when only a finite number of examples are presented and no relational structure shared by all cases is presented. I will exemplify this framework further in Section 7. I should also point out that in my earlier work (Yopp, 2014) I noted three technical handles in articulating an argument. In that work, expressing the data or CIs adequately relative to the claim and CI is a technical handle of type 2. Developing an adequate link between the data and claim is a technical handle of type 3. I still see these as related but distinct skills, particularly in the articulate stage when crafting an argument to share with others. However, for the purposes of this study, which has an emphasis on developing and articulating claims, collapsing these two handles into one type of technical handle made the framework easier to employ. The CI often becomes the warrant or link in elementary arguments such as those discussed in this paper. Thus, the technical handles associated with appropriately expressing data and appropriately wording a link between data and claims are assumed to be interwoven. 4.1.4. Reasoning To further describe the arguments that PSTs present, I look for reasoning types associated with making and supporting. Reasoning is a very broad term, and there is no consensus on its definition (Conner et al., 2014). For the purposes of this study, I define reasoning as ways in which PSTs come to a claim or support a claim. Inductive reasoning occurs when a PST makes a for-all claim from examining a proper subset of cases. Deductive reasoning occurs when a PST links data to claims using one or more inference rules in a chain of logical necessity. For this study, the key element here is the logical necessity, not whether the rule applied is explicitly stated. Deductive reasoning can be used either to support or to arrive at a claim, but inductive reasoning can only be used to arrive at a claim. When a PST uses the empirical data from an inductive activity to support a generalization, the reasoning is empirical. Empirical reasoning refers to supporting a claim by exhausting only a proper subset of cases to which the claim applies.

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It is a robust finding that there are students at all levels who are convinced of mathematical validity by empirical evidence. Stylianides and Stylianides (2009), however, found that many of the pre-service elementary teachers who developed empirical arguments were also aware that their arguments were not proof. Thus, my framework includes a distinction between PSTs who present empirical reasoning and those who express naïve empiricism by asserting that the empirical evidence is sufficient for viable argumentation. Another way PSTs may come to a claim or support a claim is by using an analogy. Analogical reasoning is observed when a PST develops or explores a claim, or support for a claim, by noting similarities between corresponding mathematical situations or by noting correspondences between structures among mathematical situations. (See Conner et al., 2014, for a discussion of how analogical reasoning has been used in other mathematics education literature.) Finally, when a PST searches for a deductive argument for a claim, the reasoning can be described as abductive. Pedemonte (2007) describes abductive reasoning as “an inference which allows the construction of a claim [about inference rules and cases] starting from an observed fact [sic]” (p. 29). Conner et al. (2014) elaborate on this idea, writing that “abduction can be seen in the mathematics classroom when students first come across a result and then have to guess or hypothesize which particular rule or case afforded (or might afford) such a result” (p. 187). Placing these perspectives into the language of my study, PSTs demonstrate abductive reasoning when they present a claim and then search for data and inference rules (warrants) to construct a deductive argument. 4.2. Influences on argument production 4.2.1. Knowledge of viable argument and argument practices My framework already has a notion of viable argument (defined above) that I communicated to the PSTs. However, PSTs come to any study with prior experiences in mathematics that may include arguing, justifying, and proving. PSTs may have learned argument standards and argument practices in previous courses that would influence their responses to my refutation tasks. Thus, I looked to the literature for argument knowledge frameworks useful in developing my theoretical framework for this study. Stylianides and Ball (2008) develop a framework for knowledge about proof for teaching that draws distinctions between the knowledge about the logico-linguistic structure of proof and the knowledge of situations for proving. The former refers to knowledge for communicating in accordance with a community’s norms for stating and supporting claims, particularly the norms canonical to the mathematical community. The latter refers to knowledge for mobilizing proving opportunities for students and is related to understanding of a task’s purposes. I will address this last construct in a different section. For this study, knowledge of argumentation will refer to PSTs’ understanding of what constitutes a viable argument. I applied a code of sufficient or insufficient understanding of a particular argument knowledge type (e.g., that testing a subset of cases does not prove a generalization) based on a PST’s claim and support of the claim—or a PST’s activity associated with developing the claim or its support. As mentioned in Section 3, Lakatos (1976) provides a theoretical framework, based on historical events, for actions that mathematicians might take when facing a refutation. This framework has proven useful in studying student and teacher refutations (e.g., Balacheff, 1991; Giannakoulias et al., 2010; Larsen & Zandieh, 2008). In this study, I adopted Lakatos’s framework to look for occasions when PSTs use the practices of mathematicians or practices akin to those of mathematicians. In such cases, I applied an influence code of sufficient or insufficient understanding or use of [the practice]. 4.2.2. Academic, natural, and semantic language and the mathematics register The way in which words are used in natural language can influence how those same words are used in mathematics. Semantic meanings of words influence how students engage with and assess proofs (Mejia-Ramos & Inglis, 2011). The mathematics register uses the precise technical meanings of conjunctives such as if, when, and therefore in ways that are different from their everyday uses (Schleppegrell, 2007). For instance, Galbraith (1981) notes that many students do not give proper attention to words like “every.” The appropriate use of the mathematics register is important for learning (Schleppegrell, 2007), and such use is viewed by some as intertwined with the practice of mathematics itself: Language is crucial for mathematical reasoning and communication with others about mathematical ideas, claims, and explanations, and proofs. Mathematical language is not simply an inert canon, inherited and learned from a distant past. It is also the medium in which mathematics is enacted, used, and created. (Ball and Bass, 2000, p. 205) Robotti (2012) argues that students’ natural language usage can be seen as a research tool for studying the evolution of students’ mathematical processes, especially during proving episodes. In my study, I assumed that PSTs’ claims are influenced by their use of language, including the mathematics register. I applied a code of sufficient or insufficient knowledge of, use of, or skill with the mathematics register to determine plausible influences on the claims that PSTs developed. 4.2.3. Perceptions of the task Participants in Balacheff’s (1991) study were asked to “give a way of calculating the number of diagonals of a polygon once the number of its vertices is known” (p. 93). Their assigned goal was to develop a generalization for all polygons and defend that generalization (Balacheff’s analysis was performed with this goal in mind), so it is unlikely that his participants would

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view a counterexample argument as a complete response to the task. Perceived didactic contracts can influence responses, and theoretically there is always a didactic contract in place, either explicitly or implicitly given. This perceived contract could come from the researcher, the teacher, or some previous experiences in an argument context. For my study, I adopted a broader term, perceptions of the task, to account for these influences. The Stylianides and Ball (2008) conception of knowledge of situations for proving has some relationship to my conception of perceptions of the task. Within the Stylianides and Ball construct are two types of knowledge related to tasks: knowledge of different kinds of proving tasks and knowledge of the relationship between proving tasks and proving activities. Knowledge of proving tasks refers to an understanding of proving tasks and the classification of tasks. For example, does the task involve refuting or verification? Or does the task involve finitely many or infinitely many? The latter refers to knowledge of the critical aspects of the proving activity that can be provoked by a task and its purposes. (For instance, refuting a claim might provoke a counterexample search, while a claim about finitely many objects might provoke systematic enumeration.) These constructs are not an exact match for my study because PSTs enrolled in a content course are often given a task and asked to produce an argument in response, as opposed to being asked to develop the task itself. Also, the word knowledge might not tell the entire story because PSTs likely combine existing knowledge with perceptions of teacher/researcher expectations (i.e., the didactic contract). Finally, I do not view viable argument and proof as synonymous. I preserve the word proof for formal, logical arguments that explicitly appeal to prior results and make all the logical progressions explicit. Aspects of the Stylianides and Ball knowledge-of-proving-tasks conception, however, do resonate with the knowledge and perceptions likely to be found among PSTs when they encounter an argument task. PSTs must make judgments about the purposes of the task and what mathematical practices are expected from them. When PSTs encounter a false claim, they must ask themselves whether they are expected to produce a counterexample and declare the original claim false, or whether they are expected to alter the claim based on their counterexample(s) and develop a new generalization—or both. Thus, I found it necessary to recast Stylianides and Ball’s framework for this context. Perceptions of the task refers to the types of responses PSTs view as appropriate, required, or desired (e.g., present a counterexample argument or develop an alternative generalization). This construct differs from the earlier construct knowledge of argumentation in the following way. A student who asserts that a counterexample is all that is needed for a complete response to a task with a false generalization expresses a perception of the task. A student who asserts that a refutation task is not complete until an alternative generalization that classifies all counterexamples or conforming cases is developed also expresses a perception of the task. A student who asserts that a counterexample is sufficient for showing a generalization is false expresses knowledge of argumentation. A student who asserts that general support, not specific cases, is required to create a viable argument for a generalization also expresses knowledge of argumentation. 5. Methods I conducted the study reported in this article within a larger research project investigating the effects of the teaching sequence for improving PSTs’ ability to construct and critique arguments. This larger project involved a course on mathematics for elementary (K-8) teachers with an emphasis on number and operation. I was the teacher/researcher doing research akin to that described by Cobb and Steffe (1983). The larger study was comprised of several tasks that involved making claims and developing arguments for those claims. This section briefly describes the participants, the teaching they experienced, and the research strategy I chose to address the research question. 5.1. Participants and data collection Data were collected during a 10-week teaching experiment (Cobb & Steffe, 1983) designed to improve PSTs’ ability to construct viable arguments. Twenty-one PSTs who were enrolled in an undergraduate mathematics content course for elementary school teachers in the Western United States participated in the study. Data were collected from five sources: (1) PSTs’ weekly posts in an online environment (147 posts in total); (2) my observations during in-class work (30 lessons that had at least some type of argumentation practice); (3) PSTs’ written responses to in-class tasks; (4) PSTs’ responses on paper-and-pencil assessments (two assessment items on the final exam); and (5) four task-based clinical interviews with PSTs (as described in Goldin, 2000), which were audio-recorded and transcribed. I performed clinical interviews in groups of four, based on PSTs’ online discussion groups. 5.2. Teaching and tasks In the course I gave PSTs the framework for viable argument (described in Section 4.1.1) as a standard for constructing arguments. PSTs received instruction on creating viable arguments and critiquing the arguments of others (see NGACBP and CCSSO, 2010, for a description of this phrase). This instruction reflected the constructivist practices described in Cobb and Steffe (1983). I demonstrated the argument practices canonical to mathematicians and the viable argument framework through both explicit instruction and interactions with the PSTs during collective argumentation activities. PSTs frequently worked on argument tasks alone and in groups in a manner resembling what Cobb and Steffe might call “working at a distance” from the teacher. I reflected on PSTs’ responses/arguments and built models of their thinking and practices. I reflected on

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the distance between (1) their responses and current practices and (2) the knowledge and practices I hoped they would learn and adopt. In this way, I viewed the PSTs as constructing their own thinking and practices, perhaps choosing what to learn based on prior experiences; and I attempted to bridge the gap between the intended learning and what was actually learned or adopted by the PSTs. Argument tasks included developing and exploring conjectures, mostly in number theory, and responding with arguments. I gave explicit instruction about the differences between generalizations and existence statements, as well as the standards for viable argument for each type. PSTs received instruction on arguing with counterexamples; with the method of exhaustion; with prior results to show that a conclusion follows necessarily from conditions; and with referents such as diagrams, variable expressions, and generic examples to demonstrate that the conclusion of a claim follows necessarily for all cases in the domain of a claim. PSTs also received instruction on some of the refutation practices described in Lakatos (1976). After a counterexample argument was constructed, I emphasized the practice of developing and arguing for generalizations about conforming cases (finite and infinite domains) and generalizations about classes of counterexamples (finite and infinite domains). I made explicit that one counterexample is sufficient to refute a generalization, but that often mathematicians look for alternative generalizations after a counterexample argument is complete. Accordingly, PSTs practiced writing a new argument separate from the counterexample argument. During such episodes, I emphasized exception barring, in which I modeled the practice of restricting a claim to only the cases we had actually considered. The PSTs completed a variety of argumentation tasks throughout the semester. Here, for brevity and clarity, I present only two of the tasks. I presented these tasks to the PSTs toward the end of the semester, after PSTs had received the instruction described above. The PSTs’ responses to the tasks and to the follow-up interview questions are assumed to reflect the knowledge and practices the PSTs constructed from the experiences in the course (and perhaps prior experiences). Task 1: You are teaching a sixth-grade class. You ask the class to investigate the sums of consecutive [counting] numbers and develop some rules about the types of numbers that are sums of 2, 3, 4, and 5 numbers. After some set time, three students offer rules. Sally says that the sum of two consecutive numbers is odd. Sophia says that the sum of three consecutive numbers is divisible by 3. Isabella says, “I think that the sum of four consecutive numbers is divisible by 4.” Write exemplary responses that include viable arguments. Task 2: Examine the conjecture and respond with a viable argument. Conjecture: For all natural numbers n, n2 + n + 41 is a prime number. 5.3. Data analysis I addressed the research question about PSTs’ claiming, stated in Section 1, using a generative study (Clement, 2000) and interpretive analysis to modify and revise existing frameworks. This process can include developing new categories as well as modifying existing ones, in order to analyze and interpret the study’s data (e.g., build a model for the data). Teaching experiments are well-suited for generative studies that build models for students’ thinking and practices because participants are studied as they engage over time in a particular learning experience. This article reports on the thinking and practices that emerged within the teaching experiment—not on the impact of the experiment as a model for developing argumentation skills. Analysis methods resembled those described by Miles and Huberman (1994), in which the analyst begins with a theoretical coding framework that is constantly compared to the data until a model that fits the data emerges. I employed the entire scheme described in Section 4. In practice, the literature discussed in Section 4 was an a priori lens for the data analysis. The final theoretical framework emerged from constantly comparing the existing literature to my data and adapting themes until a fit emerged. As new or modified conceptual themes emerged, I verified the themes through triangulation. For example, as I coded PSTs’ posts, I confirmed emerging themes with task-based interview data, in-class written work, and assessment data. In Section 7, I illustrate the findings with cases to give empirical support for the existence of practices and influences. Other than noting whether or not the occurrence was isolated to one or two participants, I make no assertions about the frequency of occurrences based on this sample. (Therefore, I do not include statistics such as “7 out of 21.”) 5.4. Summary of framework as an analysis scheme Tables in Appendices A and B summarize the final theoretical framework as it was used as an analysis scheme. For clarity and brevity, I have included only aspects of the framework that I actually observed in my data. For example, I found no instances of monster barring, perhaps due to the context, so this practice is not included in the table. 6. Illustrating the analysis framework The example below, though developed from my data, does not represent any one PST’s work. Instead it is an illustration of the framework based on an accumulation of responses found in my data. (No single PST in my study expressed the framework so coherently.) The purpose of this example is to provide the reader with a clear understanding of how the framework is

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applied. Using a fictionalized version of data and events to develop a coherent illustrative model for argumentation has precedents in the mathematics philosophy literature (e.g., Lakatos, 1976) and mathematics education literature (e.g., Leron & Zaslavsky, 2013). 6.1. Discovering that the claim Is false Kate encounters the prompt in Task 1 and is asked to develop a response. She begins by generating some examples in a manipulate stage. She writes: 1 + 2 + 3 + 4 = 10;

2 + 3 + 4 + 5 = 14;

3 + 4 + 5 + 6 = 18

In a get-a-sense-of stage, Kate notices that none of her examples has a sum that is divisible by 4. Her conceptual insight is that there are examples of four consecutive numbers that satisfy the conditions of the claim but not the conclusion. She suspects that no sum of four consecutive numbers is divisible by 4, but she does not have an insight about why this is so. She is aware that the relational structure between the conditions and not the conclusion does not address all cases, but only the three examples considered. In the articulate stage, Kate writes the following argument: Claim Isabella’s claim is false. Data: 1 + 2 + 3 + 4 = 10. Warrant: 1, 2, 3, and 4 are four consecutive counting numbers. These numbers sum to 10. 10/4 = 2.5 is not an integer. Therefore, the sum is not divisible by 4. 6.1.1. Analysis Kate expresses a prudent TH1 by articulating a claim that does not go beyond what she can support with her data and CI. Kate does not claim that no sum of four consecutive numbers is divisible by 4. Such a claim would not be prudent relative to the data and CI expressed. This is because the CI, up to this point, does not attend to any relational structure shared by all possible examples. The relational structure observed depends on the specific examples tested. Kate’s foundation expresses an adequate TH2 because she has assembled her data and CI into a viable argument by meeting the standards for arguing for an existence claim (there exists a counterexample). Her data exhibit a candidate in the domain of the original claim, and her warrant demonstrates that the candidate has the desired properties (not the conclusion). 6.2. Kate’s movements toward a generalization Based on her data, Kate wonders whether there exists a sum of four consecutive numbers that is divisible by 4. She suspects there is not one. She wishes to get-a-sense-of the relational structure between the conditions of the claim (the sum of four consecutives) and not the conclusion (not divisible by 4) to understand why there are counterexamples. She reenters the manipulate stage to look for this structure. She recalls seeing consecutive numbers written as n, n+ 1, . . . and writes the conditions of the claim as n + (n + 1) + (n + 2) + (n + 3). She manipulates this sum to form the expression 4n + 6. Reentering the get-a-sense-of stage, Kate notices that dividing 4n + 6 by 4 results in a remainder of 2. This observation becomes a CI about the relational structure between the conditions of the claim and not the conclusion that applies to all possible examples (cases) in the domain of the claim. She returns to the manipulate stage to express her CI in a manner that can be appealed to in a viable argument. She writes 4n + 6 as 4(n + 1) + 2. In an articulate stage, Kate writes the following argument: Claim For all sums of four consecutive counting numbers, none are divisible by 4. Data: Let n represent any counting number. n + (n + 1) + (n + 2) + (n + 3) = 4n + 6 =4(n + 1) + 2 Warrant: My foundation shows that any collection of four consecutive numbers has a sum equal to a multiple of 4 plus 2. A multiple of 4 plus 2 has a remainder of 2 when divided by 4 and is therefore not divisible by 4. 6.2.1. Analysis Kate expresses a prudent TH1 of her data and CI by developing a generalization that is appropriate relative to her data and CI. This is because her CI expresses the relational structure shared by all possible examples in the domain of her claim. Kate expresses an adequate TH2 by assembling her data and CI as a viable argument. The handlings occur in various stages of argumentation. One technical handling occurs when she represents the conditions of the claim generically as n + (n + 1) + (n + 2) + (n + 3). Another handling occurs relative to her CI when she expresses 4n + 6 as 4(n + 1) + 2. This handling

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occurs in a movement from a get-a-sense-of stage back to a manipulate stage with a goal of developing an adequate representation of the CI for viable argumentation. This last “movement” is unique from the framework illustrated in Fig. 1 in Sandefur et al. (2013, p. 339). Sandefur et al. place technical handles (TH2) in the get-a-sense-of stage. In my data they occur in both pre-articulate stages. 7. Results In the subsections that follow, I apply the analysis framework to report findings that emerged in my data to address the research question about the types of claims PSTs produced and plausible influences. 7.1. Problematic and imprudent claims and influences PSTs often presented problematic claims in response to false generalizations in which the literal meaning of their claims was unclear or the literal meaning was different from what they intended to express. These problematic claims were influenced by PSTs’ use of natural language and PSTs’ limited knowledge of, use of, and skill with the mathematics register. PSTs often presented imprudent claims about conforming cases after noting an infinite class of counterexamples. These imprudent claims were influenced by insufficient understandings of the mathematical practice of exception barring, perceptions that responses to refutation tasks should be general, insufficient understanding of the limitations of empirical evidence, insufficient knowledge that general claims require data about the relational structure for all cases, and insufficient understanding of and skill with the mathematics register. 7.1.1. Problematic claims influenced by language use and knowledge of, use of, and skill with the mathematics register In the exchange between Jerri and Gabbi in Appendix C, Jerri’s first post contains a problematic claim. Jerri claims that “for all Natural numbers N, N2 + N + 41 is SOMETIMES a prime number.” To understand the types of claims Jerri believed he was presenting as well as his thoughts about the domain of his “sometimes” claim, I asked Jerri about the post. Below are excerpts from the interview: Yopp: So the original conjecture . . . what kind of statement is that? Jerri: That’s a for-all. Yopp: . . . What does that say to you mathematically as a for-all or a there-exists? [Yopp points to Jerri’s first post.] Jerri: There exists a counterexample. Yopp: Mathematically, have you done what you are to do? Jerri: As far as writing a viable argument or proving it false? Yopp: What would your claim be? In the transcript Jerri then restates his “sometimes” claim found in his first post and mentions that all multiples of 41 are counterexamples. Jerri continues, “So I guess that would be more of a there-exists statement because it’s sometimes. It’s not for all numbers. But I wasn’t thinking that way.” Jerri is using “sometimes” as neither an existence qualifier nor a for-all qualifier; yet he wishes to express his observation about an infinite and unbounded set of counterexamples, as well as express observations about the existence of conforming cases. Jerri wishes to do all of this without making a claim about all the real numbers. Jerri’s reported activities (in the sense of Bruner, 1966) and conceptual insights are important for understanding his claiming practices. In his first post, Jerri reports a manipulate stage during which he discovers a variety of conforming cases and counterexamples. Jerri also reports a get-a-sense-of stage in which he found two CIs that can be associated with claiming. One CI expresses a relational structure between the conditions and the conclusion for the inputs N = 3, 4, and 41 (i.e., particular conforming cases and a particular counterexample). The other CI expresses a relational structure between the conditions and not the conclusion for an infinite class of counterexamples (i.e., multiples of 41). The combination of Jerri’s posts and interview responses offers insights into the handling of his data and CIs in developing claims. Jerri’s “sometimes” claim expresses a problematic TH1 because the literal meaning of his claim is unclear relative to other information he gives. In the first post, Jerri appears to make claims about both the existence of counterexamples and conforming cases, but the interview data suggest otherwise (“I guess that would be more of a there-exists statement . . .. But I wasn’t thinking that way.”). Yet his response to Gabbi in his second post (“We are trying to show that this formula works almost every time or SOMETIMES”) suggests that Jerri is using “sometimes” to express existence notions. Next, toward the end of his exchange with Gabbi, Jerri is attempting to state that the claim is false and then classify all conforming cases and all counterexamples (see the claim at the end of the second post – “if not for 41 and its multiples, it will work ALL the time” – and similar claims in his third post). What is clear in his posts is that Jerri has either limited knowledge of or limited skill with the mathematics register for expressing his observations. In particular, he wants to express a claim about an infinite class of counterexamples and, at least in the first post, a finite number of known conforming cases. However, at the end of his first post, Jerri reports that he is not sure how to make this claim. In sum, Jerri makes rich observations about the

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mathematical situation in Task 2 and wishes to express them, but he lacks sufficient mastery of the mathematics register to express his observations correctly or in a manner that communicates them clearly to his peer and instructor. Other problematic handles of data and CIs occur when the literal meaning of the PST’s articulated claim is different from what the PST intended to claim. An example of this occurs in the posts from another student who worked in a different online group from Jerri and Gabbi: Jean: [Reports on the first five conforming examples.] But what happens if we use 41 as n? n = 41[;] 412 = 1681[;] 1681 + 41 + 41 = 1763[;] 41 goes into 1681 41 times, as 1683 is 41 squared. However, 41 goes into 41 + 41 twice as well, so I am noticing something really big: 1763 is NOT PRIME . . . Check it out: 1763/41 = 43. Boom. So this claim is FALSE, as n = 41 is a counterexample. Jean: So, new claim: n2 + n + 41 does not produce a prime number for all natural numbers, n . . .. Jean’s reported manipulate stage expresses a purposeful testing of the claim in Task 2 against examples. In the get-a-senseof stage, Jean notes conforming cases and a counterexample. He expresses the structural relation between the conditions (a natural number) and not the conclusion (41 is a factor of 1683), and thus, he identifies a CI. Here the relational structure is limited to a finite set of particular cases. In the articulate stage, Jean reports his earlier stages and writes a claim that the original claim is false. Jean then returns to the conversation, responding to his own post, and presents a “new claim” that incorporates the phrase “for all.” This new claim is literally a generalization. Unfortunately, Jean’s new claim states that no outputs are prime, and this interpretation is inconsistent with what Jean likely intended to express. Jean reports both conforming cases and counterexamples, so it is unlikely that he intended to write an impossibility claim for all inputs. Jean’s issues are with natural language usage, not a misconception about what can be claimed from the data and insights. Jean may not see a difference between stating, “n2 + n + 41 is not prime for all natural numbers” and stating, “n2 + n + 41 is not prime for at least some natural numbers.” Moreover, Jean may not be aware that among mathematicians, the use of the phrase “for all” invokes perceptions that the claim’s conclusion applies to all cases in the claim’s domain. The position expressed in this article is that Jean’s technical handlings of his data and insights in forming a claim are problematic because the claim is not worded appropriately with respect to the data and insights expressed. I view this type of formal claiming as a technical skill, a skill in which Jean may not have received formal training in his schooling outside of this course. 7.1.2. Imprudent claims influenced by insufficient understanding or use of exception barring In Jerri’s exchange with Gabbi, shown in Appendix C, and in his follow-up interview, an imprudent claim is present. Jerri claims that all natural numbers other than multiples of 41 are conforming cases. Jerri has offered no conceptual insight about the conforming cases other than demonstrating the existence of a handful of conforming cases. Jerri expresses skepticism about empirical evidence when he writes in his third post, “As far as I know there are no other [counterexamples].” Jerri is also skeptical in his first post when he explores the original claim in Task 2 by examining cases past the first few conforming cases until he finds a counterexample. Moreover, Jerri expresses an understanding that generalizations require general support when he presents the foundation N(N + 1) + 41 in support of his claim about an infinite class of counterexamples. Consequently, Jerri is not a naive empiricist. Yet, in his third post, Jerri claims that 41 is the least counterexample, a claim that is false (40 is also a counterexample). In fact, when I review Jerri’s reported manipulate and get-a-sense-of stages, I find no evidence that Jerri is looking for a structural insight that accommodates a claim about all non-multiples of 41. Thus, Jerri expresses an imprudent TH1 when he assembles his data and CIs into alternative claims about the least counterexample and all conforming cases. The structural relationships he acknowledges do not apply or attend to all cases in the domain of these claims. In my data, the type of overgeneralizations that Jerri makes seems to be associated with inadequate representations— those that do not offer structural links between the conditions (e.g., not a multiple of 41) and the claim (e.g., prime output). Such overgeneralizations are also associated with a limited understanding of or use of the practice of exception barring. Throughout Jerri’s exchange with Gabbi, Jerri continues to put forth claims about conforming cases (e.g., all natural numbers except 41 and its multiples work), but Jerri never focuses on developing restricted claims about what he knows for certain to be true. This is despite having a compelling CI about multiples of 41 and having adequately handled this CI by writing N(N + 1) + 41, which would serve as an adequate TH2 for a claim restricted to multiples of 41. Jerri seems to feel that claims restricted to proper subsets of the original claims domain are inadequate responses to the task (I will address this in detail in the Perceptions section). This is evidenced in his first response to Gabbi: “We are trying to show this formula works almost every time or sometimes.” In Jerri and Gabbi’s discourse, neither PST attempts to develop a claim restricted to “known” conforming cases. Absent throughout my data, except for one case described below, is any use of exception barring to restrict a generalization to the finite number of cases tested. In the one exception, a PST offers a claim restricted to the first 40 cases; but a peer PST dismisses the claim as insufficient for the task and asks, “I wonder what the point of proving 1–40 is . . . what patter[n] does that build?” This suggests perceptions of the task, but it also relates to exception barring. PSTs seem quick to use exception barring by writing a generalization that “throws out” known counterexamples, but the practice of restricting claims to known conforming cases appears in my data only once.

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Fig. 3. Drew continues to express a “density” notion—here on her final exam.

PSTs made overgeneralizations not only in the form of imprudent exception barring, but also in the form of overgeneralizations about the density of conforming cases. The online post and follow-up work from Drew illustrate this finding: Drew: You made a good point, Jean, that the claim has a counterexample and is therefore false. I also wonder why, though, that adding 41 most of the time works in this case. My thoughts are that it might have to do with the fact that 41 is a prime number itself. If you add a prime number to a prime number, then the end result will no [sic] be prime unless you add another prime number to it, at least most of the time. Drew’s peers had already posted the counterexample 41 and several conforming cases before Drew entered the conversation. Prior to Drew’s post, Jean posted a claim that n2 + n + k is composite when k = n. Jean supported this claim with the general representation k(k + 2). In the post above, Drew reports being in a state of abduction in which she is trying to get-a-sense-of why the expression n2 + n + k outputs a prime “most of the time,” which is apparently her claim. Drew offers no information about a manipulate stage and may only be referencing data from previous posts. We know little about the details of Drew’s get-a-sense-of stage, although her claim that “most of the time the output is prime” appears to be based in a false insight about adding primes and not in a representation of the conditions or conclusion of the original claim in Task 2. Drew’s “most of the time” claim is imprudent based on the data and CI expressed in previous posts and the false insight about adding primes. Drew presents no relational structure about adding primes to support the subclaim about primes. She exhibits an inappropriate TH1 when she develops a “most of the time” claim based on the limited amount of data and the limited scope of the CI. Comparing her practice to the practice of exception barring, Drew does not limit her claiming to those cases that have been sufficiently explored. Some readers might find Drew’s use of terms like “most of the time” to be sensible based on empirical observations. I don’t disagree. Some readers might even note that for countable subsets of countable sets, terms like “usually” and “most of time” are not necessarily meaningful (e.g., it does not make sense to say there are more rational numbers than integers). I acknowledge that reasoning is a messy business and that pursuing the “why” based on an observation about a set of finite data is a noteworthy mathematical practice. But finding ways to validate Drew’s claim based on an analyst’s knowledge of mathematics is not the purpose of this study. The purpose is to explore the claims developed in the articulate stage relative to the data or conceptual insights presented and plausible influences. The data and insights Drew presents are insufficient for supporting even a claim that the conforming cases are infinite. Even if we give Drew the benefit of the doubt and assume that she means that up to some finite value, the number of conforming cases exceeds the counterexample cases, we would still be overlooking the fact that the claim is not appropriate relative to the data or CI presented. Let’s not forget that primes become less frequent over intervals of set length as the interval bounds grow larger. So intuitions about primes and “frequency” are problematic at best. Drew is likely unaware of this and unaware of the mathematical standard of not making claims for which you have insufficient support (Galbraith, 1981). The concern is that Drew is expressing a type of empiricism about all counting numbers based on a relatively small, specialized sample of values. In fact, Drew seems to have a habit of overgeneralizing about density based on finite data. Fig. 3 shows one of her final exam responses that also makes a density (“usually”) assertion without support. The practice of exception barring can be viewed not only as “throwing out” known counterexamples, but also as a practice of restricting one’s claims to cases of which we are certain. In this sense, Drew expresses a limited understanding or use of this practice. While the cases above illustrate situations in which the domain of the claim is infinite, examples of insufficient understanding of exception barring can also occur when claiming about a finite number of cases. In my data, only one case emerged in which the arguer suggested restricting the domain to a finite number of conforming cases:

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Fig. 4. Corey’s problematic claim and unsupported, imprudent assertion in the warrant.

Corey: . . . So, maybe 41 is the only number that doesn’t work and then we could change our claim to all natural numbers under 41. I think this definitely needs some further looking into by our group members. I would say that any multiple of 41 would make this conjecture false. Jean: Yes, I would imagine that any multiple of 41 would make it false, as it would then be divisible by 41. But one counterexample is all we need to prove it wrong. I wonder what the point of proving 1–40 is, though? What patter[n] does that build? Corey is responding to others’ posts about Task 2 that included a few conforming cases and the counterexample 41. Corey does not report enough detail to understand her manipulate and get-a-sense-of stages; yet there is enough information in her post to get a picture of her TH1. Corey’s proposed claim, that inputs 1–40 have prime outputs, is false because 40 is also a counterexample. It appears that Corey has not explored any cases other than those presented by peers: “So, maybe 41 is the only number that doesn’t work.” Recall that Galbraith (1981) asserts that avoiding making conjectures on insufficient evidence is an essential component of successful proving. Corey appears to have a habit of including information about cases not sufficiently supported, even when that information is not necessary to support her claim. In Fig. 4, Corey offers a problematic claim that is equivalent to a generalization, and she supports it in part with an imprudent assertion, “I’m sure there are some . . . that work.” 7.1.3. Imprudent claims influenced by insufficient understanding of the limitations of empirical evidence Some PSTs made imprudent claims and expressed no understanding of the limitations of empirical evidence. One example is Gabbi’s first response to Jerri (Appendix C). In her first post, Gabbi moves Jerri’s “sometimes” qualifier to a different position in the claim. In her second post, Gabbi asserts that she is adding a non-multiple of 41 to the infinite list of counterexamples. I was interested in Gabbi’s use of the word sometimes in her claim, her understanding of counterexample arguments, and her goals for developing an argument (e.g., why is she searching for one more counterexample?). In the follow-up interview, I read to Gabbi her first response to Jerri and asked her about her thinking. Below is an excerpt: Gabbi: I guess I was just thinking that maybe since he had two [conforming examples] right here that maybe he just could have had, maybe, more [counterexamples] over here, just to, kinda like, balance it out. I just thought maybe it would just have made it stronger, in my opinion. Yopp: So he had two cases where it was a prime number . . . Gabbi: Yeah, and then he only had one [where n2 + n + 41 isn’t prime]. Yopp: So to prove to someone or to argue that something is false, how many counterexamples do you need? Gabbi: I was just thinking more than one, but that’s just me. Yopp: I do want to say, mathematically, you only need one counterexample to show it’s false . . .. Gabbi: So that’s all you needed? Gabbi’s use of the word sometimes is confusing and expresses a limited understanding of the mathematics register with respect to how quantifiers are used in mathematics to make assertions clear. We know she wishes to create a generalization (“I think it . . . needs to be generalized”), and she may be asserting that the number of counterexamples is the same as the number of conforming cases. Most interesting are Gabbi’s assertions about “balancing it out” as a way to improve the support

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for the “sometimes” claim and her critique in her first post: “Having one counterexample does not really prove your point.” Gabbi expresses an insufficient understanding of counterexample argumentation, but this follows a limited understanding of the role of empirical data and its potential for viable argumentation. Gabbi clearly wants to make a claim that is more general than the one Jerri presents, and she bases her claiming on examining an insufficient number of cases. Readers might view Gabbi’s claiming as presenting an imprudent claim that is false. I want to make clear, however, that my findings are not based in truth of the claims that PSTs present. Below is an example of a true claim that is still imprudent because the PST does not report a sufficient CI for making the claim. Charli: Data for Claim #3. My example of the claim: 1 + 2 + 3 + 4 = 10 or 4 + 5 + 6 + 7 = 2 [22] supports the claim that says: “The sum of four consecutive counting numbers is not divisible by 4.” Because shown here proves that these consecutive counting numbers are not divisible by 4. The claim Charli presents in quotes is equivalent to a generalization, although it is not completely clear in the post whether she intended to make a generalization. In the third sentence, Charli specifies “these consecutive counting numbers,” raising the possibility that she has restricted the claim to just the examples presented. Yet during a follow-up interview, Charli reports that she was making a generalization about all sums of four consecutives in her post. When asked to read her post and discuss whether her claim applies to all sums or just those included in her post, Charli responds, “I was doing data, so I just picked those right off the top of my head.” Follow-up questions reveal that Charli thinks of data as being examples or a list of numbers as in science experiments. Charli says that she intended to “get general about it in the warrant.” When asked whether her support was sufficient for a generalization, Charli responds, “So then did I need to include more [examples] . . . like, a different [example], like, instead of 4, 5, 6, 7, or? [sic].” At this point, a peer from Charli’s discussion group who participated in the interview writes n + n + 1 + n + 2 + n + 3 =4n + 6 and says, “This is what you are basically saying in more general form.” Charli responds, “So mine was just an example. So, like, what data could we have used to make it more [general]? Because all I did was 1, 2, 3, 4. I could have done, like, 21, 22, 23, 24, 25.” Charli’s manipulate and get-a-sense-of stages involve only examples, and she expresses an imprudent TH1 and an inadequate TH2. Even though Charli expresses some understanding of mathematical argumentation by stating that she wishes to “get general . . . in the warrant,” she does not express understanding that in order to get general about it in the warrant, the data must express the relational structure shared by all cases. Charli does not show an understanding that data should be handled in a manner so that the relational structure between the sum of four consecutives and not divisible by 4 is transparent. Finally, Charli does not express an understanding about limiting her claim to what she can support or an understanding that no amount of examples is sufficient support for a generalization with infinite domain. In Charli’s case, I note the plausibility of two influences. One, Charli expresses an insufficient understanding that arguing for generalizations requires attention to structural elements shared by all cases. Two, Charli does not express an understanding of the practice of exception barring that limits claims to the cases examined. Thus, her imprudent TH1 is influenced by insufficient understandings of both how generalizations are supported by referents (e.g., the structure expressed in examples) and what claims can be made based on data and CIs. 7.1.4. Imprudent claiming in there-exists contexts Gabbi’s data offer an example of another type of imprudent handling in developing claims. This imprudent handling relates to making existence claims. After her first post (Appendix C), Gabbi enters a manipulate stage in which she searches for another counterexample. After an apparent get-a-sense-of stage, Gabbi reports that 632 + 63 + 41 = 4073 is not prime. This stage is problematic because she does not express a structural relationship between the conditions and not the conclusion. This would involve demonstrating the pertinent structure for a counterexample: that 4073 has a factor other than 1. In fact, Jerri contests the counterexample, and Gabbi reports that she did not actually check. Incidentally, she responds with another candidate counterexample, 40, which is indeed a counterexample; but again, Gabbi offers no relational structure that connects 1681 to “not prime.” I coded Gabbi’s TH1 of her data as imprudent because she lacks an appropriate CI to warrant the existence claim. This example offers a chance to reiterate a key component in my theoretical framework: that to explore PSTs’ claiming appropriately, one needs to examine both the TH1 and TH2. This is because the term “argument” incorporates both the claim and the support for the claim. Prudent claiming is inherently associated with knowledge about the support for that claim, or at least a CI about how the claim can be viably supported. An appropriate CI for Gabbi’s counterexample claim can be based in a relational structure that connects 402 + 40 + 41 to not the conclusion “not prime.” Viable existence arguments not only offer a suitable candidate, but also demonstrate that the candidate has the desired properties. Thus, Gabbi imprudently makes her claim without appropriate support for her claim. This claiming is associated with incomplete pre-articulate activity stages. 7.1.5. More on exception barring and perceptions of the task Jean’s response to Corey in Section 7.1.2 offers insight into PSTs’ limited use of prudent exception barring and its influences on claiming. Corey’s proposed restricted claim (cases 1–40 for Task 2) offers a legitimate form of exception barring, even though Lakatos’s (1976) characters might call this playing it safe. Unfortunately, Corey’s approach to restricting the domain

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of her claim to a finite “exhaustible” set is dismissed by her peer for not finding a pattern. The group never returns to this claim or any other claim about the known conforming cases. While a reader might value Jean’s desire for a more general claim (and I do as well), let’s return to the scope of this article. My goal is to document types of claiming and influences on claiming that might be observed in PSTs in this type of course. Regardless of the reasons for Jean’s dismissal of Corey’s claim, its effect is that a viable mathematical practice – restricting claims to those cases the arguer feels certain of – is no longer part of the PSTs’ collective argumentation. Such practice has the potential to limit overgeneralizing, but Jean dismisses the practice as not meeting the task’s objectives. 7.2. Prudent claims and their influences Jerri’s second claim (in his first post, Appendix C) that all multiples of 41 are counterexamples expresses a prudent TH1 relative to the data and the CIs he presents. The expression N(N + 1) + 41 conveys a structural relationship for a class of objects that meet the conditions (multiples of 41) and not the conclusion (output is composite). In the interview, Jerri reports developing this representation because he felt that he needed to “figure out why that is” [that all numbers other than 41 seem to conform to the original claim]. He reports changing the constant term to other numbers, including composites, to try to meet this goal. In my data, these types of prudent claims are associated with CIs and with data that were sufficiently handled so that the structural relationships for all cases in the domain of the resulting claim were transparent. These claims also are associated with perceptions that complete responses to refutation tasks explain why there are counterexamples, conforming cases, or both. In a group different from those previously discussed, a PST named Sam develops a prudent alternative generalization from a CI developed while searching for an explanation for why there are counterexamples to Isabella’s claim (Task 1). Sam’s peer, Andy, makes clear that only one counterexample is needed to complete that part of the task. Sam returns to the conversation and offers a CI for why counterexamples exist. Sam ultimately develops an alternative generalization for Isabella’s claim based on her search for why counterexamples exist. Along the way, Sam makes only claims about counterexamples and does not overgeneralize relative to her data—except, perhaps, when she claims that Isabella’s claim “isn’t always true” (although it is not completely clear how she uses the quantifier “always”). Yet the prudent claims that Sam and Andy present appear to be influenced by their knowledge of argumentation and by Sam’s desire to explain why. The following excerpt from Sam and Andy’s online discussion about Task 1 demonstrates this point: Sam: Sophia’s claim makes sense as well. N + (N + 1) + (N + 2) = a number divisible by 3. I like writing it out like this because you can see the group of three that the N + 1 and N + 2 makes, therefore verifying that it will make a number divisible by 3. Isabella claims that N + (N + 1) + (N + 2) + (N + 3) = a number divisible by 4. For the same reason, we can see that this isn’t always true. Take the numbers 1, 2, 3, and 4 for instance. Added together their sum is 10, which is not divisible by 4. Andy: Isabella’s claim is not accurate, and you proved it by showing (1 + 2 + 3 + 4 = 10). All it needs is one counterexample, and then no further talk is necessary. Sam: I found a problem with using our same reasoning. Of course, proving it false by counterexample works, but I was trying to figure out a true claim to make and our basis seemed shaky. . . . Now I’ve done some thinking about number 3 [Isabella’s claim] and I think I came up with a solution. I’ll just type this one. N + (N + 1) + (N + 2) + (N + 3) = ? N + N + N + N + N + 1 + 2 + 3 = 4N + 6 [sic]. So the answer is always going to be some multiple of 4, plus 6 more. In [Sophia’s claim] it just so happened that having a remainder sum of 3 worked so that it was divisible by 3 because it was [sic] also 3N, or a multiple of 3, plus another 3. In this problem we see that a multiple of 4 plus 6 will not always be divisible by 4 or 6. It will however always be divisible by 2. We could change the claim to make it true by saying that N + (N + 1) + (N + 2) + (N + 3) = an even number[.] I wanted to know more about Sam’s understanding of counterexample arguments and arguments for generalizations. I also wanted to know more about her assertion that their “basis seemed shaky.” Yopp: [After the counterexample,] are you done at this point? Sam: For proving it false? Yes. Yopp: But you were trying to find a true statement about four consecutive numbers. What was the true statement you were after? Sam: Um, what I . . . I didn’t even start with trying to think of a claim. I started with writing it out as general and found that it’s always going to equal, and I made it [inaudible], but 4N and plus 6 is going to be this, going to be the sum of four consecutive numbers. Yopp: [Sam, you write,] “Proving it false by counterexample works, but I was trying to figure out a true claim . . . and our basis seemed shaky.” What do you mean there?

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Sam: Well, I think, um, actually I’m not even sure if our basis was shaky. The basis that I was trying to make for it was shaky in my head. I was saying that [N,] N + 1 and N + 2 makes a group of three, so obviously that’ll work [referring to her work on Sophia’s claim]. But what I wanted to say was that this makes 3n and then plus three. That’s why that works because it’s a multiple of 3 plus another multiple of 3. And so I was thinking that I hadn’t explained that clearly [for the sum of four consecutives]. I was just referring to the remainder [when divided by 4], which is why I went further on my next post to talk about what made the remainder of the multiples in the remainder. I then guided Sam and her group toward the generalization, for all sums of four consecutive numbers, none are divisible by 4, that appeared in a later post by one of their peers. This generalization had not been supported with a viable argument: Andy: I also did a bunch of examples but I didn’t do them on the post. I just was like, hmmm. Because once I found my counterexample I was, like, maybe there is . . . so I tried to pick some random numbers to see if that was going to work and I couldn’t find anything, so I didn’t comment. Yopp: On the floor, there’s a generalization that these two [Sam and Andy] have established. For all consecutive numbers, never divisible by 4. They’ve each done a bunch of examples. Are you all satisfied with their argument? Andy: I’m not even satisfied by that. Sam: I’d like to look into it, but I’m not satisfied with that. Yopp: Why are you not satisfied? Andy: Because it’s not general enough. . . . Because you have all those but there are, like, infinite possibilities. Sam: I’m satisfied in a different way . . . because the answer is always going to be a multiple of 4, plus 6. So it’ll be divisible by 4 with a remainder of 2.

7.2.1. Further analysis of Sam’s claiming and reasoning In Sam’s first post, she offers an argument for Sophia’s claim (Task 1) and an argument against Isabella’s claim (Task 1). In an apparent manipulate stage, Sam has symbolically expressed the conditions of Sophia’s claim. In a get-a-sense-of stage, Sam has found a structural reason for the truth of the claim (the 1 and the 2 make a group of three). There is a clear CI about the relational structure between the conditions (sum of three consecutives) and the conclusions (divisible by 3). This structure is expressed, in part, by N + N + 1 + N + 2. Sam develops a similar representation for Isabella’s claim in a manipulate stage, but in a get-a-sense-of stage, Sam produces a particular counterexample. The TH1 associated with her claim (“isn’t always true”) is difficult to label as prudent because she presents only one counterexample. Later posts indicate that perhaps she is using “isn’t always” to express general observations – possibly that there are multiple or even infinitely many counterexamples – yet this is unclear in this post. After Sam presents a counterexample argument and claim, she returns to the conversation and reports her search for a true claim. She also critiques her previous reasoning associated with her counterexample argument. Part of her reasoning in this second post can be described as analogical. This is because she refers to the approach she used to address Sophia’s claim in order to explain why there are counterexamples to Isabella’s claim. It is evident that Sam’s earlier get-a-sense-of stage is not complete because she returns to the conversation and reports another manipulate stage. In this stage, Sam adds like terms and to form 4N + 6, an expression that has potential for a variety of prudent claims about divisors. Here, Sam reports developing CIs about divisors. She enters the articulate stage when she claims that the sum of four consecutive numbers is not always divisible by 4 or 6, but the sum is always divisible by 2. Ignoring the problematic use of the word always and the fact that stronger claims can be made about 4 and 6, these claims are prudent relative to the data expressed. Thus, I labeled them as prudent TH1s. In the interview data, Sam affirms the stages described above: “I didn’t . . . start with trying to think of a claim. . . . I started writing it out as general and found what it’s always going to equal.” Sam also says that she wanted to explain why there are counterexamples and affirms her understanding that general arguments require general support. In Sandefur et al. (2013) the focus is on learner-generated examples developing into CIs and, ultimately, a proof. When an arguer searches for a proof of a claim, the arguer engages in abductive reasoning. When the arguer pieces together already identified cases and rules, sequencing them using logical necessity, the arguer engages in deduction. Sam develops a general representation of the relational structure prior to having her specific alternative claim in mind. In that moment, the claim arises from the structure expressed. Such acts situate better with TH1 codes than TH2 codes. The claiming arises from arguer-developed representations of data that afford general conceptual insights. This type of reasoning is unique from inductive reasoning, in which claims are developed from empirical observations or patterns observed in a subset of cases. What is interesting in the post is that Sam is in an abductive state when she explores the structure of 4N + 6 to “explain why” not all sums of four consecutives are divisible by 4. From this expression, Sam reports seeing a “true claim”: that the sum is divisible by 2. Interestingly, Sam never asserts that all sums of four consecutive numbers are not divisible by 4. She does not seem to realize that until the cognitive interview. When Sam learns of this claim during the interview, she says

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Fig. 5. A model for coming to a prudent claim when a counterexample is noted.

she is not satisfied with the empirical evidence and develops a CI about why all sums are not divisible by 4 (4n + 6/4 has a remainder of 2). Thus, Sam expresses a strong understanding of viable mathematics argumentation. In summary, Sam’s reasoning in developing the “divisible by 2” claim can be described as deductive reasoning, but in a different order than the way the term is often used. Sam uses an expression as a starting case for a sequence of rules used to come to (or observe) a claim. One implicit rule found in Sam’s reasoning is if two terms are divisible by 2, then the sum is divisible by 2. The chain of reasoning is as follows: sum of four consecutives implies the sum is of the form 4N + 6, which implies divisible by 2. Figs. 5 and 6 are diagrams developed from my data that capture Sam’s progression to prudent claims. Fig. 5 offers a model for the process that Sam reports in coming to a counterexample claim (idealized to remove the issues with use of the word “always”). Fig. 6 offers a model for the process that Sam reports in coming to a prudent alternative generation.

Fig. 6. A model for coming to a prudent claim when a CI about a relational structure emerges.

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Table 1 TH1s when the relational structure expressed is limited to particular cases, or the expression of that structure is vague. Example

Conceptual Insight

Claim Assessment

Observed Influences

Task 2 response: Andy acknowledges another PST’s counterexample 41 and claims that no other work is needed to show the generalization is false

Limited to the relational structure expressed in a particular case (meets conditions and not the conclusion)

Prudent relative to the data referenced

Task 2 response: Corey notes that 41 is a counterexample (demonstrated by a peer in a previous post) and writes, “Maybe we could change the claim to all natural numbers under 41 [produce prime numbers]” Task 1 response: Charli presents two examples of four consecutive numbers that are not divisible by 4 and claims that the sum of four consecutive numbers is not divisible by 4

Unclear, but the arguer does claim that any multiple of 41 would make this conjecture false

Imprudent relative to the data presented

Knowledge that one counterexample is all that is required to refute a claim Perception that providing one counterexample is a sufficient response to a false generalization Insufficient understanding or use of the mathematical practice of exception barring and restricting claims to only those cases considered

Limited to the relational structure expressed in particular cases (meets conditions and not the conclusion) Limited to the relational structure expressed in particular cases (meets conditions and not the conclusion) Relational structure expressed for particular cases only, both conforming and counterexample Arguer does claim that any multiple of 41 would make this conjecture false

Imprudent relative to the data presented in the argument

Insufficient knowledge that general claims with infinite domains require data about the relational structure for all cases

Problematic. The literal interpretation is different from the arguer’s intent.

Insufficient understanding of and skill with the mathematics register

Imprudent and problematic relative to the data presented in the argument

Insufficient understanding of and skill with the mathematics register Insufficient knowledge of the limitations of empirical evidence as support for generalizations

Task 2 response: Jean gives both conforming cases and counterexamples for Task 2 and claims that n2 + n + 41 does not produce a prime number for all natural numbers Task 2 response: Gabbi acknowledges or presents conforming cases and a counterexample and claims that sometimes for all natural numbers n2 + n + 41 is prime. Gabbi views this claim as a generalization and claims there are infinitely many cases of each type: conforming and counterexamples

7.3. Summary of influences on claiming Tables 1 and 2 reorganize the findings presented in the previous sections. These summaries are based on the domain of the data and CIs presented or acknowledged by the PST relative to the claim the PST articulates. Table 1 summarizes the influences on claiming when the relational structure presented or acknowledged is either limited to particular cases or is too vague to viably support a generalization. Table 2 summarizes claiming when the relational structure presented or acknowledged encompasses an infinite class of objects. This organization was motivated by my hypothesis that prudent claiming is associated with the domain of the data and CIs presented and with awareness of that domain. 8. Discussion PSTs’ arguments in response to false generalizations, after a counterexample(s) was acknowledged, were influenced by their abilities to develop appropriate claims relative to their data or CIs (TH1 skills). At times PSTs’ claiming was also influenced by their abilities to adequately represent, use, and appeal to their data or CIs (TH2 skills). PSTs might develop any one of the THs without the other present, demonstrating that viable argument features are not necessarily constructed in a particular order. I found that adequate representations of data or CIs for viably arguing about an infinite class of counterexamples can exist prior to the explicit development of the generalization. Claiming that arose from arguer-developed representations of data that afford CIs was unique from a type of abductive reasoning, in which representations and rules are sought to support an existing claim. This claiming also was unique from inductive reasoning, in which claims are developed from empirical observations or patterns observed in a subset of cases. In one instance, an arguer was in an abductive state searching for an explanation for why counterexamples existed. The arguer reported developing an alternative generalization from a representation created for the aforementioned purposes. TH1s were profoundly influenced by a PST’s knowledge of argumentation. Overgeneralizing about the number of counterexamples or conforming examples was common. Problematic claims that made assertions about the density or frequency of the counterexamples were present in two PSTs’ responses. I also observed confusing quantifiers (e.g., sometimes). In their discussion (Appendix C), Gabbi’s and Jerri’s “sometimes” claims illustrate why a lack of standardized language for this community is problematic. Gabbi perceives her sometimes qualifier as creating a generalization, but it is not clear that Jerri always uses “sometimes” in this way. Gabbi appears to interpret the sometimes qualifier as asserting that there are an infinite number of counterexamples and an infinite number of conforming cases—and perhaps the same number of each. At minimum, their use of this term creates dissidence in their conversation. Their discussion might have been more productive if they had a shared mathematical vocabulary for communicating their ideas and observations.

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Table 2 TH1s when the relational structure between conditions and conclusions is explicit and encompasses an infinite class. Example

Conceptual insight

Claim assessment

Plausible or observed influences

Task 1 response: Sam presents a counterexample and then returns to the conversation looking for a true generalization. Sam develops the expression 4N + 6 for the sum of four consecutives and claims that the sum is not always divisible by 4 or 6

An explicit CI was not expressed but is likely present (e.g., that only one of the terms in 4N + 6 is divisible by 4 or 6)

Prudent relative to the structural elements expressed

Task 2 response: Jerri observes that 41 and its multiples are counterexamples and asserts that if it were not for 41 and its multiples, n2 + n + 41 is prime Task 2 response: Jerri observes that 41 and its multiples are counterexamples and claims that “there are infinite that will not be prime using this formula” [n2 + n + 41] Task 2 response: Jean observes that n2 + n + p, where p is a prime and n = p, produces a composite Task 2 response: Corey notes that any multiple of 41 is a counterexample and suggests claiming that n2 + n + 41 is prime for all n except 41 and its multiples Task 2 response: Drew acknowledges another PST’s counterexample 41 and asserts that the original claim works most of the time

N2 + N + 41 can be written as N(N + 1) + 41, so multiples of 41 are counterexamples

Imprudent overgeneralization relative to the CI and structure expressed Prudent relative to the CI and structure expressed

Knowledge of the limitations of empirical evidence as mathematical support for generalizations Knowledge that general claims require general support Desire to develop a true generalization and explain why there are counterexamples Insufficient understanding or use of the mathematical practice of exception barring

N2 + N + 41 can be written as N(N + 1) + 41, so multiples of 41 are counterexamples n2 + n + k can be expressed as k(k + 2) when n = k Multiples of 41 are counterexamples because you can create groups of 41 Adding a prime to a prime is not prime, unless you add another prime, at least most of the time (this is false)

Prudent relative to the CI and structure expressed Imprudent relative to the CI expressed

Imprudent relative to the CI expressed

Perception that responses to refutation tasks should explain why Perception that refutation tasks involve finding a pattern Insufficient understanding or use of the mathematical practice of exception barring Insufficient understanding or use of the mathematical practice of exception barring

Many of the PSTs lacked an adequate understanding of what claims are appropriate relative to their data and CIs. Many also lacked adequate knowledge of how to use the mathematics register, particularly with respect to the precision of quantifiers. We should not overlook that PSTs who say “most of the time” or “usually” based on testing a proper subset of examples are expressing empiricism. In his conversation with Gabbi, Jerri illustrates how PSTs who are otherwise skeptical about empirical evidence may fail to apply this skepticism appropriately to their alternative claims in response to false generalizations. After developing a variable representation for demonstrating why all multiples of 41 are counterexamples, Jerri overgeneralizes about conforming cases. While Jerri qualifies his claim about conforming cases in one of his online posts (“As far as I know there are no other [counterexamples].”), in other posts and in the interview, Jerri abandons this skepticism. Of course, we want PSTs to be free to share thoughts and intuitions. That does not change the analysis in this article, however. Jerri is not conjecturing; he is offering a proposition that he believes is true, and he reiterates that during the interview. Recall that Jerri also asserts that 41 is the least counterexample, which is false. The point of this article is not to discourage PSTs from offering a variety of conjectures for which the truth is uncertain. Rather, the point is to document claims that an arguer puts forth as true and to note whether the claims are prudent, imprudent, or problematic relative to the data and CIs expressed. Jerri presents no viable reason to believe his claim about conforming cases other than empirical evidence from examining relatively few cases. Furthermore, he fails to acknowledge Gabbi’s correct counterexample of 40 (Gabbi’s last post). Previous literature has established that PSTs who offer empirical arguments for generalization may express awareness that the arguments are not proofs, if they are given the opportunity to critique their arguments (Stylianides & Stylianides, 2009). The results of this study suggest that training in exception barring might also give PSTs opportunities to express awareness of the limitations of empirical evidence. The results demonstrate that the PSTs were not skilled in exception barring or did not use the practice at all. One PST dismissed the practice of restricting an alternative generalization to a finite domain because it did not build a pattern for a large class or all cases. Natural language usage also influenced PSTs’ ability to express appropriate claims. While it was possible for some of the PSTs to use natural language to communicate an appropriate claim, several PSTs’ claims, when restated with proper quantifiers, were false or imprudent relative to their data or CIs. The types of claims that PSTs pursued were also dependent on their perceptions of the task—in particular, whether they felt that demonstrating that the original conjecture is false was sufficient as a response, or whether an alternative generalization was desired. In one case, a PST perceived that the task was to show that the formula in the original claim “works” sometimes or almost every time. Not all of the PSTs expressed an understanding that arguments for generalization require data or CIs in a format that could be appealed to generically, even when they acknowledged that the warrant should be general. This knowledge issue can be associated with overgeneralizing with respect to the data or CIs present.

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Consistent with previous studies, there was evidence in my study that some PSTs did not have a clear understanding of the role of counterexamples in refutation. They also did not have a clear understanding of what constitutes a sufficient argument against a generalization. On the positive side, several PSTs acknowledged that counterexample arguments are sufficient. They pursued explanations beyond presenting a counterexample for what appeared to be intellectual curiosity. Sam provided evidence that it is plausible for PSTs to value “explaining why” there are counterexamples while still communicating minimally sufficient counterexample arguments. I am not asserting that PSTs should be discouraged from presenting causal information in counterexample arguments. Instead I contend that PSTs should know that causal information and generalizable warrants and data are not necessary to show that a generalization is false. This knowledge might prevent PSTs from including unnecessary and incorrect information in arguments that would otherwise be viable. In particular, having a standard that a counterexample argument should attend only to a counterexample and nothing more might prevent PSTs from making unsupported or false alternative claims within the counterexample argument itself. I also assert that PSTs might benefit from instructors’ giving them explicit “license” to restrict their claims to finite domains when only that domain has been tested and no general CI is present. Emphasizing this type of exception barring in response to false generalizations might help PSTs avoid presenting empirical arguments by stating only claims that their data and CIs support. These findings suggest that PSTs might also benefit from explicit instruction on the mathematics register and canonical standards for argumentation—in particular, how to express claims that are appropriate relative to the data or CIs presented. Secondary mathematics education majors likely take coursework where the language of mathematics and standards for proof and proving are stressed. PSTs who do not take advanced mathematics coursework, however, may not receive sufficient training and practice with these mathematical practices. The use of the term viable argument in CCSSM instead of proof suggests plausible alternatives to formal, logical arguments. Yet argumentation instruction that does not explicitly stress “the rules” and “the register” of mathematics may not be adequate for argumentation in mathematics courses, even in early grades. The PSTs in this study had difficulty making appropriate claims. At times these difficulties led to significant communication issues, which often resulted in false claims (e.g., Jerri and Gabbi’s debate about claims and support for those claims). Even otherwise proficient PSTs such as Jean and Sam, who in their discussions appear to have appropriate perceptions of the task and strong argumentation skills, might benefit from explicit instruction on what types of claims can be stated prudently in response to data and CIs, as well as instruction on what language is appropriate (e.g., being clearer about the domain of an “isn’t-always-true” claim). These findings also suggest that PSTs might benefit from explicit instruction on the distinction between arguing that a conjecture is false and developing an alternative generalization. Constructing both types of arguments as separate and distinct responses might be viewed as the standard for a fully developed response to a false generalization. Such a standard might help teachers avoid some of the issues noted in previous literature (e.g., not valuing counterexample arguments as complete refutations and viewing refutations based on theory as stronger than counterexample arguments). Developing counterexample arguments and alternative generation arguments as distinct products offers teachers the opportunity to express correct counterexample knowledge while also embracing the mathematical practice of going beyond refutations to understand why counterexamples exist or to use exception barring to discuss known conforming cases without confounding and confusing these practices. Finally, direct instruction on THs might also prove beneficial for PSTs. It would be interesting to know whether practice with each of the THs would improve PSTs’ skills in handling their data and CIs to construct viable arguments. A PST might be successful in producing a prudent claim but might need scaffolding to develop appropriate representations of data or CIs, as well as adequate warrants when supporting a claim (or visa versa). Careful distinctions between these types of handles might help PSTs organize the products of their explorations for the articulate stage by attending to each handle separately.

9. Limitations While these findings address some of the affordances to communicating a viable argument, findings should not be viewed as a model for how PSTs develop viable argumentation skills. Teaching experiments use tasks, lessons, and interviews to induce learning so that researchers may study and build models for how PSTs develop practices. Teaching experiments also create environments in which PSTs’ practices and thinking become transparent. Ultimately this work uses empirical findings to build one plausible model for influences on PSTs’ ability to communicate a viable argument. In this study I introduced PSTs to a definition of viable argument and types of viable arguments to give them objectives, concepts, and language for argument production, and to improve argument communication. While my teaching actions likely influenced the types of responses, this is true in all case studies and teaching experiments. I have been careful to give the readers an overview of the argumentation practices that I emphasized in my instruction. My views about viable arguments in mathematics courses may not represent the views of others. Because the definition of viable argument was unique to this setting, findings may not be generalizable to all settings. The definition used here, however, is consistent with Toulmin’s (1958/2003) definition of argument and CCSSM’s description (NGACBP and CCSSO, 2010).

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Acknowledgment The author would like to thank the reviews and editors whose feedback greatly improved this article. References Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216–235). London, United Kingdom: Hodder & Stoughton. Balacheff, N. (1991). Treatment of refutations: Aspects of the complexity of a constructivist approach to mathematics learning. In E. von Glasersfeld (Ed.), Radical constructivism in mathematics education (pp. 89–110). Dordrecht, The Netherlands: Kluwer. Ball, D. L., & Bass, H. (2000). Making believe: The collective construction of public mathematical knowledge in the elementary classroom. In D. C. Phillips (Ed.), Yearbook of the National Society for the Study of Education: Constructivism in education (pp. 193–224). Chicago, IL: University of Chicago Press. Bruner, J. S. (1966). Toward a theory of instruction. Cambridge, MA: Harvard University Press. Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In A. E. Kelly, & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 547–590). Mahwah, NJ: Lawrence Erlbaum. Cobb, P., & Steffe, L. P. (1983). The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education, 14(2), 83–94. Conner, A., Singletary, L. M., Smith, R. C., Wagner, P. A., & Francisco, R. T. (2014). Identifying kinds of reasoning in collective argumentation. Mathematical Thinking and Learning, 16(3), 181–200. Galbraith, P. L. (1981). Aspects of proving: A clinical investigation of process. Educational Studies in Mathematics, 12(1), 1–28. Giannakoulias, E., Mastorides, E., Potari, D., & Zachariades, T. (2010). Studying teachers’ mathematical argumentation in the context of refuting students’ invalid claims. The Journal of Mathematical Behavior, 29(3), 160–168. Goldin, G. A. (2000). A scientific perspective on structured, task-based interviews in mathematics education research. In A. E. Kelly, & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 517–546). Mahwah, NJ: Lawrence Erlbaum. Komatsu, K. (2010). Counter-examples for refinement of conjectures and proofs in primary school mathematics. The Journal of Mathematical Behavior, 29(1), 1–10. Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb, & H. Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 229–269). Hillsdale, NJ: Lawrence Erlbaum. Krummheuer, G. (2007). Argumentation and participation in the primary mathematics classroom: Two episodes and related theoretical abductions. The Journal of Mathematical Behavior, 26(1), 60–82. Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge, United Kingdom: Cambridge University Press. Larsen, S., & Zandieh, M. (2008). Proofs and refutations in the undergraduate mathematics classroom. Educational Studies in Mathematics, 67(3), 205–216. Leron, U., & Zaslavsky, O. (2013). Generic proving: Reflections on scope and method. For the Learning of Mathematics, 33(3), 24–30. Mason, J. (1981). When is a symbol symbolic? For the Learning of Mathematics, 1(2), 8–11. Mejia-Ramos, J. P., & Inglis, M. (2011). Semantic contamination and mathematical proof: Can a non-proof prove? The Journal of Mathematical Behavior, 30(1), 19–29. Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded sourcebook. Thousand Oaks, CA: Sage Publications. National Governors Association Center for Better Practices & Council of Chief State School Officers (NGACBP & CCSSO). (2010). Common core state standards for mathematics. Washington, DC: National Governors Association Center for Better Practices and Council of Chief State School Officers. Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66(1), 23–41. Peled, I., & Zaslavsky, O. (1997). Counter-examples that (only) prove and counter-examples that (also) explain. Focus on Learning Problems in Mathematics, 19(3), 49–61. Potari, D., Zachariades, T., & Zaslavsky, O. (2009). Mathematics teachers’ reasoning for refuting students’ invalid claims. In Proceedings of the sixth congress of the European society for research in mathematics education France, (pp. 281–290). Raman, M. (2003). Key ideas: What are they and how can they help us understand how people view proof? Educational Studies in Mathematics, 52(3), 319–325. Raman, M., Sandefur, J., Birky, G., Campbell, C., & Somers, K. (2009). Is that a proof?”: Using video to teach and learn how to prove at the university level. In F.-L. Lin, F.-J. Hsieh, G. Hanna, & M. de Villiers (Eds.), Proceedings of the ICMI Study 19 conference: Proof and proving in mathematics education (vol. 2) (pp. 154–159). Taipei, Taiwan: National Taiwan Normal University. Robotti, E. (2012). Natural language as a tool for analyzing the proving process: The case of plane geometry proof. Educational Studies in Mathematics, 80(3), 433–450. Sandefur, J., Mason, J., Stylianides, G. J., & Watson, A. (2013). Generating and using examples in the proving process. Educational Studies in Mathematics, 83(3), 323–340. Schleppegrell, M. J. (2007). The linguistic challenges of mathematics teaching and learning: A research review. Reading & Writing Quarterly, 23(2), 139–159. Smarter Balanced Assessment Consortium Staff (SBACS). (2013). Content specifications for the summative assessment of the common core state standards for mathematics. Retrieved from http://www.smarterbalanced.org/wordpress/wp-content/uploads/2011/12/Math-Content-Specifications.pdf. Stylianides, A. J., & Al-Murani, T. (2010). Can a proof and a counterexample coexist? Students’ conceptions about the relationship between proof and refutation. Research in Mathematics Education, 12(1), 21–36. Stylianides, A. J., & Ball, D. L. (2008). Understanding and describing mathematical knowledge for teaching: Knowledge about proof for engaging students in the activity of proving. Journal of Mathematics Teacher Education, 11(4), 307–332. Stylianides, A. J., & Stylianides, G. J. (2009). Proof construction and evaluations. Educational Studies in Mathematics, 72(2), 237–253. Toulmin, S. (1958/2003). The uses of argument. Cambridge, United Kingdom: Cambridge University Press. Yopp, D. A. (2013). Counterexamples as starting points for reasoning and sense making. Mathematics Teacher, 106(9), 674–679. Yopp, D. A. (2014). Viable arguments, conceptual insights, and technical handles. In C. Nicol, S. Oesterle, P. Liljedahl, & D. Allan (Eds.), Proceedings of the 38th conference of the international group for the psychology of mathematics education and the 36th conference of the North American chapter of the psychology of mathematics education (vol. 5) (pp. 401–408). Vancouver, Canada: PME. Yopp, D. A., & Ely, R. (2015). When does an argument use a generic example? (Manuscript submitted for publication).