Investigating and Improving Undergraduate Proof Comprehension

Alcock, Lara; Hodds, Mark; Roy, Somali; Inglis, Matthew

doi:10.1090/noti1263

Cited by 16 publications

(4 citation statements)

References 14 publications

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“…Comments such as these and the data in figure 3 suggest that students may not have been challenged by following the step-by-step, lab manual-led experiment but found it difficult to link that procedure to forming a conceptual understanding of the underlying physics content. This might hint at the idea that while the lab may have been 'easy', it led to shallower engagement, and therefore poorer learning outcomes, in relating the experiment to concepts and theories studied in lectures [20]. Student responses from this cohort did not make reference to engagement, with only 1% of comments referring to their enjoyment.…”

Section: Correlation Between Student Engagement and Scientific Habits...mentioning

confidence: 93%

Exploring problem-based cooperative learning in undergraduate physics labs: student perspectives

2018

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This study examines the potential of problem-based cooperative learning (PBCL) in expanding undergraduate physics students’ understanding of, and engagement with, the scientific process. Two groups of first-year physics students (n = 180) completed a questionnaire which compared their perceptions of learning science with their engagement in physics labs. One cohort completed a lab based on a PBCL approach, whilst the other completed the same experiment, using a more traditional, manual-based lab. Utilising a participant research approach, the questionnaire was co-constructed by researchers and student advisers from each cohort in order to improve shared meaning between researchers and participants. Analysis of students’ responses suggests that students in the PBCL cohort engaged more in higher-order problem-solving skills and evidenced a deeper understanding of the scientific process than students in the more traditional, manual-based cohort. However, the latter cohort responses placed more emphasis on accuracy and measurement in lab science than the PBCL cohort. The students in the PBCL cohort were also more positively engaged with their learning than their counterparts in the manual led group.

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Section: Correlation Between Student Engagement and Scientific Habits...mentioning

confidence: 93%

Exploring problem-based cooperative learning in undergraduate physics labs: student perspectives

2018

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“…Students can, in fact, be trained to ask similar questions when they are reading mathematical text. When students are trained to do so, their comprehension of the text improves dramatically (Alcock et al, 2015;Hodds, Alcock, & Inglis, 2014).…”

Section: Task-based Interviews On How Mathematicians Develop An Under...mentioning

confidence: 99%

Mathematics Education Research on Mathematical Practice

Weber¹,

Inglis²

2021

Handbook of the History and Philosophy of Mathematical Practice

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“…24 There is some empirical support for such a claim. In (Alcock, Hodds, Roy, & Inglis, 2015) it is shown that even modest training in "self-explanation" when reading proofs improves subsequent proof comprehension. Students are encouraged, in their training, to apply "a series of techniques"; those techniques are described in the training material as follows: "After reading each line: Try to identify and elaborate the main ideas of the proof.…”

Section: General Structural Lemmamentioning

confidence: 99%

Natural Formalization: Deriving the Cantor-Bernstein Theorem in Zf

Sieg

Walsh

2019

The Review of Symbolic Logic

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Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame is a definitional extension of Zermelo–Fraenkel set theory and has a hierarchically organized structure of concepts and operations, and (3) the construction of formal proofs is deeply connected to the frame through rules for definitions and lemmas. To bring these general ideas to life, we examine, as a case study, proofs of the Cantor–Bernstein Theorem that do not appeal to the principle of choice. A thorough analysis of the multitude of “different” informal proofs seems to reduce them to exactly one. The natural formalization confirms that there is one proof, but that it comes in two variants due to Dedekind and Zermelo, respectively. In this way it enhances the conceptual understanding of the represented informal proofs. The formal, computational work is carried out with the proof search system AProS that serves as a proof assistant and implements the above inference mechanism; it can be fully inspected at http://www.phil.cmu.edu/legacy/Proof_Site/. We must—that is my conviction—take the concept of the specifically mathematical proof as an object of investigation. Hilbert 1918

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Investigating and Improving Undergraduate Proof Comprehension

Cited by 16 publications

References 14 publications

Exploring problem-based cooperative learning in undergraduate physics labs: student perspectives

Exploring problem-based cooperative learning in undergraduate physics labs: student perspectives

Mathematics Education Research on Mathematical Practice

Natural Formalization: Deriving the Cantor-Bernstein Theorem in Zf

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