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1.
Chamberlain, Darryl J, Jr.
Investigating the Development of *Proof* *Comprehension*: The Case of *Proof* by Contradiction.

Degree: PhD, Mathematics and Statistics, 2017, Georgia State University

URL: https://scholarworks.gsu.edu/math_diss/46

This dissertation reports on an investigation of transition-to-proof students' understanding of proof by contradiction. A plethora of research on the construction aspect of proof by contradiction is available and suggests that the method is one of the most difficult for students to construct and comprehend. However, there is little research on the students' comprehension of proofs and, in particular, proofs by contradiction. This study aims to fill this gap in the literature. Applying the cognitive lens of Action-Process-Object-Schema (APOS) Theory to proof by contradiction, this study proposes a preliminary genetic decomposition for how a student might construct the concept `proof by contradiction' and a series of five teaching interventions based on this preliminary genetic decomposition. Data was analyzed in two ways: (1) group analysis of the first two teaching interventions to consider students' initial conceptions of the proof method and (2) case study analysis of two individuals to consider how students' understanding developed over time. The genetic decomposition and teaching interventions were then revised based on the results of the data analysis. This study concludes with implications for teaching the concept of proof by contradiction and suggestions for further research on the topic.
*Advisors/Committee Members: Draga Vidakovic, Valerie Miller, Alexandra Smirnova, Yongwei Yao.*

Subjects/Keywords: Proof by contradiction; Proof comprehension; APOS Theory; ACE teaching cycle; Indirect proof

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Chamberlain, Darryl J, J. (2017). Investigating the Development of Proof Comprehension: The Case of Proof by Contradiction. (Doctoral Dissertation). Georgia State University. Retrieved from https://scholarworks.gsu.edu/math_diss/46

Chicago Manual of Style (16^{th} Edition):

Chamberlain, Darryl J, Jr. “Investigating the Development of Proof Comprehension: The Case of Proof by Contradiction.” 2017. Doctoral Dissertation, Georgia State University. Accessed December 11, 2019. https://scholarworks.gsu.edu/math_diss/46.

MLA Handbook (7^{th} Edition):

Chamberlain, Darryl J, Jr. “Investigating the Development of Proof Comprehension: The Case of Proof by Contradiction.” 2017. Web. 11 Dec 2019.

Vancouver:

Chamberlain, Darryl J J. Investigating the Development of Proof Comprehension: The Case of Proof by Contradiction. [Internet] [Doctoral dissertation]. Georgia State University; 2017. [cited 2019 Dec 11]. Available from: https://scholarworks.gsu.edu/math_diss/46.

Council of Science Editors:

Chamberlain, Darryl J J. Investigating the Development of Proof Comprehension: The Case of Proof by Contradiction. [Doctoral Dissertation]. Georgia State University; 2017. Available from: https://scholarworks.gsu.edu/math_diss/46

Texas State University – San Marcos

2.
Herrera, Christine A.
An Investigation of the Relationships between Conceptualization of Limits and *Proof* * Comprehension*.

Degree: PhD, Mathematics Education, 2016, Texas State University – San Marcos

URL: https://digital.library.txstate.edu/handle/10877/6051

This study investigates eighteen Real Analysis students’ informal and formal understanding of the mathematical concept of limits and the relationship on their comprehension of limit proofs. This study utilizes Tall and Vinner’s (1981) notion of concept image and concept definition. The framework examined students’ conceptualization of limit by eliciting their mental images, associated properties, processes, and example space (Watson & Mason, 2005). The study analyzed surveys on the conceptualizations of limits of sequences and limits of functions, class observations, and task-based interviews to explore the different varieties of conceptual understanding held by Real Analysis students. From the data emerged two cognitive categorization of participants’ thinking. Those whose concept image held serious potential conflict factors with the formal definitions of limits and those who had resolved their serious potential conflict factors by the end of the semester, called the cognitive conflict and cognitive resolution groups, respectively. The students were given an end-of-semester proof comprehension assessment that was designed based on Mejia-Ramos, Fuller, Weber, Rhoads, and Samkoff’s (2012) model for assessments for advanced mathematics’ proof comprehension. The analysis showed that Real Analysis students with cognitive resolution had a better local understanding of limit proofs. However, both cognitive groups had difficulty generating examples that illustrated the main ideas of the limit proofs.
*Advisors/Committee Members: Sorto, M. Alejandra (advisor), White, Alexander (committee member), Strickland, Sharon (committee member), Cory, Beth (committee member).*

Subjects/Keywords: Limits; Limits of sequences; Limits of functions; Real analysis; Undergraduate mathematics; Proof; Proof comprehension; Concept image; Concept definition; Example space; Mathematical analysis; Mathematics – Study and teaching

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Herrera, C. A. (2016). An Investigation of the Relationships between Conceptualization of Limits and Proof Comprehension. (Doctoral Dissertation). Texas State University – San Marcos. Retrieved from https://digital.library.txstate.edu/handle/10877/6051

Chicago Manual of Style (16^{th} Edition):

Herrera, Christine A. “An Investigation of the Relationships between Conceptualization of Limits and Proof Comprehension.” 2016. Doctoral Dissertation, Texas State University – San Marcos. Accessed December 11, 2019. https://digital.library.txstate.edu/handle/10877/6051.

MLA Handbook (7^{th} Edition):

Herrera, Christine A. “An Investigation of the Relationships between Conceptualization of Limits and Proof Comprehension.” 2016. Web. 11 Dec 2019.

Vancouver:

Herrera CA. An Investigation of the Relationships between Conceptualization of Limits and Proof Comprehension. [Internet] [Doctoral dissertation]. Texas State University – San Marcos; 2016. [cited 2019 Dec 11]. Available from: https://digital.library.txstate.edu/handle/10877/6051.

Council of Science Editors:

Herrera CA. An Investigation of the Relationships between Conceptualization of Limits and Proof Comprehension. [Doctoral Dissertation]. Texas State University – San Marcos; 2016. Available from: https://digital.library.txstate.edu/handle/10877/6051