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Creators/Authors contains: "Eckman, D."

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  1. ; ( , Research in Undergraduate Mathematics Education Conference)
    In this study, we report one group of students’ efforts to create a community meaning for set- builder notation collectively. Students’ ability to develop and interpret set-builder notation is essential to transition-to-proof courses. Conventionally, a colon is used in set-builder notation to (1) separate the universe of discourse from the set’s defining property and (2) indicate an ordering to these components, with the universe to the left and the property to the right of the colon. We describe one normative and non-normative interpretation of this notation and how the students’ individual attribution of conventional meanings for the colon to different inscriptions within the notation helped (or inhibited) them from interpreting these expressions. We report how communicative discourse between the students affected their meanings and discussions. 
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    Free, publicly-accessible full text available February 28, 2025
  2. ; ; ; ; ( , Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)
    Cook, S. ; Katz, B. ; Moore-Russo, D. (Ed.)
    This paper presents six categories of undergraduate student explanations and justifications regarding the question of whether a converse proof proves a conditional theorem. Two categories of explanation led students to judge that converse proofs cannot so prove, which is the normative interpretation. These judgments depended upon students spontaneously seeking uniform rules of proving across various theorems or assigning a direction to the theorems and proof. The other four categories of explanation led students to affirm that converse proofs prove. We emphasize the rationality of these non-normative explanations to suggest the need for further work to understand how we can help students understand the normative rules of logic. 
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    Accepted Manuscript Full Text Available
  3. ; ; ; ; ( , Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)
    Cook, S. ; Katz, B. ; Moore-Russo, D. (Ed.)
    Mathematicians often use set-builder notation and set diagrams to define and show relationships between sets in proof-related courses. This paper describes various meanings that students might attribute to these representations. Our data consist of students’ initial attempts to create and interpret these representations during the first day of a paired teaching experiment. Our analysis revealed that neither student imputed or attributed our desired theoretical meanings to their diagrams or notation. We summarize our findings in two vignettes, one describing students’ attributed meanings to instructor-provided set-builder notation and the other describing students’ imputed meanings to their personally-created set diagrams to relate pairs of sets. 
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    Accepted Manuscript Full Text Available
  4. ; ; ; ; ( , Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)
    Cook, S. ; Katz, B. ; Moore-Russo, D. (Ed.)
    This study explores how instructional interventions and teacher moves might support students’ learning of logic in mathematical contexts. We conducted an exploratory teaching experiment with a pair of undergraduate students to leverage set-based reasoning for proofs of conditional statements. The students initially displayed a lack of knowledge of contrapositive equivalence and converse independence in validating if a given proof-text proves a given theorem. However, they came to conceive of these logical principles as the teaching experiment progressed. We will discuss how our instructional interventions played a critical role in facilitating students’ joint reflection and modification of their reasoning about contrapositive equivalence and converse independence in reading proofs. 
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    Accepted Manuscript Full Text Available
  5. ; ; ; ; ( , Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)
    Cook, S. ; Katz, B. ; Moore-Russo D. (Ed.)
    Learning to interpret proofs is an important milepost in the maturity and development of students of higher mathematics. A key learning objective in proof-based courses is to discern whether a given proof is a valid justification of its underlying claim. In this study, we presented students with conditional statements and associated proofs and asked them to determine whether the proofs proved the statements and to explain their reasoning. Prior studies have found that inexperienced provers often accept the proof of a statement’s converse and reject proofs by contraposition, which are both erroneous determinations. Our study contributes to the literature by corroborating these findings and suggesting a connection between students’ reading comprehension and proof validation behaviors and their beliefs about mathematical proof and mathematical knowledge base. 
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    Accepted Manuscript Full Text Available