Metaphor as a possible pathway to more formal understanding of the definition of sequence convergence

Dawkins, Paul Christian

doi:10.1016/j.jmathb.2012.02.002

Cited by 15 publications

(8 citation statements)

References 20 publications

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“…In what ways does the White Tigers analogy do so? First, we observed that such analogies were very common in Dr. B's parlance throughout the semester (Dawkins 2009(Dawkins , 2012. She used them as a tool to address confusion or for surfacing student misconceptions that she anticipated based on her teaching experience.…”

Section: Comments On the White Tigers Analogymentioning

confidence: 99%

Promoting Metalinguistic and Metamathematical Reasoning in Proof-Oriented Mathematics Courses: a Method and a Framework

Dawkins

Roh

2016

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

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Students' apprenticeship into proof-oriented mathematical practice requires that they become aware of a range of conventions and assumptions about mathematical language, reference, and inference that rarely appear as explicit components of the content of undergraduate courses. Because these matters of interpretation are often preconscious for students and taken for granted by instructors, students may fail to apprehend the problems at hand and thus misunderstand the mathematical community's solutions. So, we argue that proof-oriented instruction must simultaneously help students understand problems of ambiguity and referencewhich requires them to engage in what we call metalinguistic and metamathematical reasoningas well as provide some informal language for discussing possible solutions to these problems. We provide three illustrative episodes from real analysis classrooms to highlight our common way of framing this class of learning challenge and the pedagogical difficulties they entail. In each case, we highlight the use of analogies as a method of helping students apprehend a problem of mathematical language, reference, or inference and discuss possible solutions. This paper's contributions are thus both practical and theoretical: the instructional method is intended to serve as one possible approach to training students in mathematical conventions and assumptions, and it helps demonstrate our theoretical perspective on this class of teaching and learning phenomena.

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Section: Comments On the White Tigers Analogymentioning

confidence: 99%

Promoting Metalinguistic and Metamathematical Reasoning in Proof-Oriented Mathematics Courses: a Method and a Framework

Dawkins

Roh

2016

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

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“…Instructors' metaphors have a strong influence on a student's understanding of mathematics (Dawkins, , ; Font et al., ). Knowledge of instructor's metaphorical reasoning could be used to shape teaching practices to better assist a student's opportunity to learn.…”

Section: Implications For Future Researchmentioning

confidence: 99%

“…This may be due to the multiple perspectives and shifting metaphor clusters available to solve items correctly. It would be interesting to study whether the direct modeling use of metaphorical reasoning by the instructor, a la Dawkins (, ), would promote better understanding of limits. Of course, future research should investigate whether metaphorical reasoning is appropriate for other problem areas of mathematics, for example derivatives, integrals, and series.…”

Section: Implications For Future Researchmentioning

confidence: 99%

Metaphor Clusters: Characterizing Instructor Metaphorical Reasoning on Limit Concepts in College Calculus

Patel

McCombs

Zollman

2014

School Sci & Mathematics

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Novice students have difficulty with the topic of limits in calculus. We believe this is in part because of the multiple perspectives and shifting metaphors available to solve items correctly. We investigated college calculus instructors' personal concepts of limits. Based upon previous research investigating introductory calculus student metaphorical reasoning, we examined 11 college instructors' metaphorical reasoning on limit concepts. This paper focused on previous research of metaphor clusters observed among students to answer the following: (a) Do college instructors use metaphorical reasoning to conceptualize the meaning of a limit? (b) Can we characterize instructor metaphorical reasoning similar to those observed among students? (c) Will an instructor's self‐identification of metaphor clusters be consistent with our metaphor coding? We found that college instructors' perspectives vary, either graphical or algebraic, in their explanations of limit items. All the instructors used metaphors, and instructor metaphorical reasoning aligned with student metaphor clusters. Instructors tended to change their metaphors with respect to the limit item. Instructors were not aware of their use of metaphors, nor were they aware of their inconsistency in their choice of metaphor. We believe that instructor awareness of their own distinct perspectives and metaphors would assist students' understanding of limit concepts.

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“…In addition, some students do not infer the arbitrariness of error bounds from the ε-N definition. They tend to test with only some ε > 0 for the condition Bfor all n >N, |a n -L| < ε^instead of considering every ε > 0 (Dawkins 2012;Oehrtman et al 2014;Roh 2010). Students do not often grasp that Bany ε^implies the value of ε decreases towards 0 so that smaller values can be continuously chosen for ε (Roh 2010).…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, an explicit algorithm to find the value of N is not suggested in the ε-N definition. As a consequence, it might not be easy for students to reason properly why ε and N are necessary in defining the convergence of a sequence, how N could be determined dependently on ε, and why ε must be independent of N (Cory and Garofalo 2011;Dawkins 2012;Mamona-Downs 2001;Roh and Lee 2011).…”

Section: Introductionmentioning

confidence: 99%

Designing Tasks of Introductory Real Analysis to Bridge a Gap Between Students’ Intuition and Mathematical Rigor: the Case of the Convergence of a Sequence

Roh

Lee

2016

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

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The purpose of this study is to explore how an introductory real analysis (IRA) course can be designed to bridge a gap between students' intuition and mathematical rigor. In particular, we focus on a task, called the ε-strip activity, designed for the convergence of a sequence. Data were collected from a larger study conducted as a classroom teaching experiment for a semester-long IRA course. Fischbein's notion of secondary intuition was employed to elucidate the development of student intuition throughout the ε-strip activity. We discuss how the ε-strip activity played a role in developing students' intuition and how it impacted students' learning of the subsequent topics in the course, including definitions and theorems about convergence.

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Metaphor as a possible pathway to more formal understanding of the definition of sequence convergence

Cited by 15 publications

References 20 publications

Promoting Metalinguistic and Metamathematical Reasoning in Proof-Oriented Mathematics Courses: a Method and a Framework

Promoting Metalinguistic and Metamathematical Reasoning in Proof-Oriented Mathematics Courses: a Method and a Framework

Metaphor Clusters: Characterizing Instructor Metaphorical Reasoning on Limit Concepts in College Calculus

Designing Tasks of Introductory Real Analysis to Bridge a Gap Between Students’ Intuition and Mathematical Rigor: the Case of the Convergence of a Sequence

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