Bridging the gap between graphical arguments and verbal-symbolic proofs in a real analysis context

Zazkis, Dov; Weber, Keith; Mejía-Ramos, Juan Pablo

doi:10.1007/s10649-016-9698-3

Cited by 27 publications

(22 citation statements)

References 23 publications

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“…To overcome such difficulties, two types of actions can be taken. The first type is "rewarranting" (Zazkis et al, 2016). Rewarranting, as employed by Zazkis et al, is related to translation from graphical warrants to verbal-symbolic warrants; in our study, instead, it involves adding additional reasons in order to make sufficient warrants, as illustrated in Moe's case in Section 5.2.3.…”

Section: Discussionmentioning

confidence: 92%

“…Following Pedemonte (2007) and Zazkis et al (2016), we used the simplified version of Toulmin's (2003) framework to analyze students' attempts at proof validation and modification. This simplified framework consists of the claim (C) being argued, the data (D) used to demonstrate the claim, and the warrant (W) describing how the data support the claim.…”

Section: Framework and Data Analysismentioning

confidence: 99%

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Proof validation and modification in secondary school geometry

Komatsu

Jones

Ikeda³

et al. 2017

The Journal of Mathematical Behavior

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Proof validation is important in school mathematics because it can provide a basis upon which to critique mathematical arguments. While there has been some previous research on proof validation, the need for studies with school students is pressing. For this paper, we focus on proof validation and modification during secondary school geometry. For that purpose, we employ Lakatos' notion of local counterexample that rejects a specific step in a proof. By using Toulmin's framework to analyze data from a task-based questionnaire completed by 32 ninth-grade students in a class in Japan, we identify what attempts the students made in producing local counterexamples to their proofs and modifying their proofs to deal with local counterexamples. We found that student difficulties related to producing diagrams that satisfied the condition of the set proof problem and to generating acceptable warrants for claims. The classroom use of tasks that entail student discovery of local counterexamples may help to improve students' learning of proof and proving. first category encompasses studies in which different types of mathematical argument (visual, inductive, generic, or deductive) are presented to students and the students are asked to evaluate whether the arguments are personally convincing and whether the arguments can be considered to constitute proofs (e.g., Healy & Hoyles, 2000; Martin & Harel, 1989; Segal, 1999). A second category relates to students' understanding of given correct proofs. Based on models of proof comprehension (Mejia-Ramos, Fuller, Weber, Rhoads, & Samkoff, 2012; Yang & Lin, 2008), studies in this category seek to identify effective proof comprehension strategies and examine the effectiveness of specific training to improve students' proof comprehension (Hodds, Alcock, & Inglis, 2014; Samkoff & Weber, 2015). A third category involves proof validation; here, researchers show students purported deductive proofs and ask them to determine whether the proofs are valid or invalid (

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Section: Discussionmentioning

confidence: 92%

Section: Framework and Data Analysismentioning

confidence: 99%

Proof validation and modification in secondary school geometry

Komatsu

Jones

Ikeda³

et al. 2017

The Journal of Mathematical Behavior

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show abstract

“…The construction of arguments that meet the standard of proof is often the last step in a complex and multifaceted activity typically referred to as proving that includes also other processes such as the following: work with examples or exploration of particular cases, identification of patterns and generation or refinement of conjectures or other Students' proof constructions in mathematics 157 kind of mathematical claims, and attempts to develop informal arguments for these mathematical claims that may offer insight, or ultimately translate, into a proof (e.g. Mason, 1982;Boero et al, 1996;Weber & Alcock, 2004;Mariotti, 2006;Stylianides, 2007Stylianides, , 2008Alcock & Inglis, 2008;Zazkis et al, 2008Zazkis et al, , 2016. In this article, I focus on the presentation of mathematical arguments whose constructors perceive they meet the standard of proof and who might have engaged previously in some other processes within the broader activity of proving.…”

Section: Introductionmentioning

confidence: 99%

Secondary students’ proof constructions in mathematics: The role of written versus oral mode of argument representation

Stylianides

2018

Review of Education

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Prior research showed that many secondary students fail to construct arguments that meet the standard of proof in mathematics. However, this research tended to use survey methods and only consider students presenting their perceived proofs in written form. The limited use of observation methods and the lack of consideration of students presenting their perceived proofs orally—in tandem with their written proofs for the same claims—might have resulted in a skewed picture of the potential of students’ constructed proofs, and this raises concern about the validity of research findings. The research reported in this article substantiates this concern. Using data from a design experiment in two secondary mathematics classrooms (14–15‐year‐olds), I explored the role of the written versus the oral mode of argument representation in students’ proof constructions. Findings from the comparison between the written arguments (perceived proofs) that the students produced during small group work and the oral arguments that the students presented in front of the class for the same claims showed that the oral mode of representation is more likely than the written mode to be associated with the construction of arguments that meet the standard of proof. Thus if a study had analysed students’ written arguments only (as in survey research), it would have reported a less favourable picture of the potential of students’ constructed proofs than another study that would focus only on students’ oral arguments (as in observational research). Implications for methodology, research and practice are discussed in light of these findings.

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“…Research falling into the first category has highlighted the important role that semantic reasoning can play in proof generation. Various types of semantic reasoning have been shown to inform proof generation (Gibson 1998;Sandefur et al 2013;Zazkis et al 2016). For example, drawing diagrams (Gibson 1998), generating examples (Sandefur et al 2013), conjecture generation (Boero 1999;Pedemonte 2007), and even the generation of fully formed semantic arguments (Zazkis et al 2016), have all been shown to inform students' proof generation.…”

Section: Introductionmentioning

confidence: 99%

“…Various types of semantic reasoning have been shown to inform proof generation (Gibson 1998;Sandefur et al 2013;Zazkis et al 2016). For example, drawing diagrams (Gibson 1998), generating examples (Sandefur et al 2013), conjecture generation (Boero 1999;Pedemonte 2007), and even the generation of fully formed semantic arguments (Zazkis et al 2016), have all been shown to inform students' proof generation. Additionally, some researchers have argued that the generation of certain types of proofs is difficult without conceptual insights gained from semantic reasoning (Raman 2003;Weber and Alcock 2004).…”

Section: Introductionmentioning

confidence: 99%

Student Conceptions of What it Means to Base a Proof on an Informal Argument

Zazkis

Villanueva

2016

Int. J. Res. Undergrad. Math. Ed.

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In this paper we explore how students construe what it means for an informal argument to be the basis of a proof. We use students' attention as a proxy for their conceptions and document what they pay attention to when assessing whether a proof is based on an informal argument. The data illustrate that some undergraduate mathematics students pay attention to only a limited cross section of the connections that exist between informal arguments and proofs. This indicates that their conceptions of what it means to base a proof on an informal argument involve relatively superficial similarities. Students' attention to only a small cross section of what can be carried forward from an informal argument toward the goal of creating a proof can account for some student difficulties with, and avoidance of, using their own informal arguments as a basis for proof construction.

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Bridging the gap between graphical arguments and verbal-symbolic proofs in a real analysis context

Cited by 27 publications

References 23 publications

Proof validation and modification in secondary school geometry

Proof validation and modification in secondary school geometry

Secondary students’ proof constructions in mathematics: The role of written versus oral mode of argument representation

Student Conceptions of What it Means to Base a Proof on an Informal Argument

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