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1. An elaboration on master narratives in mathematics and how undergraduates relate to counternarratives

   Melhuish, Kathleen; Guajardo, Lino; Lew, Kristen; Dawkins, Paul C; Roh, Kyeong_Hah (November 2024, Kent State University)

   Kosko, K W; Caniglia, J; Courtney, S A; Zolfaghari, M; Morris, G A (Ed.)

   Because the master narratives about mathematics in the US often play an exclusionary role in students’ educational experiences, educators have sought to integrate counternarratives into instruction that might disrupt these effects. As part of a larger project to develop a research-informed curriculum for undergraduate introduction to proof courses, we gathered author stories from a diverse set of mathematicians for students to read and reflect upon. To study student responses to these author stories, we synthesized a framework of the master narrative of mathematics in the US and identified how the author stories countered elements of this narrative. We then analyzed 80 student reflections from one introduction to proof course to identify whether and how students either endorsed or countered the elements of the master narrative. Our findings point to a positive, yet modest capacity for these stories as counternarratives. 

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2. Differences in Students’ Beliefs and Knowledge Regarding Mathematical Proof: Comparing Novice and Experienced Provers

   Ruiz, S.; Roh, K. H.; Dawkins, P. C.; Eckman, D.; Tucci, A. (August 2023, 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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3. Opposing dimensions in mathematicians' counter narratives written for undergraduate students

   Melhuish, Kate; Guajardo, Lino; Contreras, Norman; Dawkins, Paul C; Diaz-Lopez, Alexander; Garcia, Rebecca; Lew, Kristen; Harris, Pamela; Roh, Kyeong_Hah; Walker, Shanise; et al (November 2024, Special Interest Group of the Mathematics Association of America on Research in Undergraduate Mathematics Education)

   Cook, S; Katz, B; Moore-Russo, D (Ed.)

   In mathematics, counter narratives can be used to fight the dominant narrative of who is good at mathematics and who can succeed in mathematics. Eight mathematicians were recruited to co-author a larger NSF project (RAMP). In part, they were asked to create author stories for an undergraduate audience. In this article, we use narrative analysis to present five polarities identified in the author stories. We present various quotations from the mathematicians’ author stories to highlight their experiences with home and school life, view of what mathematics is, experiences in growth in mathematics, with collaboration, and their feelings of community in mathematics. The telling of these experiences contributes towards rehumanizing mathematics and rewriting the narrative of who is good at and who can succeed in mathematics. 

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4. Teaching children to be problem posers and puzzle-creators in mathematics

   Goldenberg, E.Paul (August 2018, Proceedings of Constructionism 2018)

   Seymour Papert’s 1972 paper “Teaching Children to be Mathematicians Versus Teaching About Mathematics” started with the summary statement “The important difference between the work of a child in an elementary mathematics class and that of a mathematician is not in the subject matter…but in the fact that the mathematician is creatively engaged….” Along with “creative,” a key term Papert kept using is project rather than the common notion of problem. A project is not simply a very large problem. It centrally includes a focus on sustained and active engagement. The projects in his illustrations were essentially research projects, not just multi-step, fullyprescribed, build-a-thing tasks, no matter how nice the end product might be. A mathematical playground with enough attractive destinations in it draws children naturally to pose their own tasks and projects—as they universally do in their other personal and group playgrounds—and to learn to act and think like mathematicians. They even acquire conventionally taught content through that play. Physical construction was always available, and appealed to such thinkers as Dewey, but for Papert computer programming, newly available to school, suggested a more flexible medium and a model for an ideal playground. A fact about playgrounds is that children choose challenge. In working and playing with children I’ve seen that puzzles tap some of the same personally chosen challenge that a programming centric playground offers. Children are naturally drawn to intellectual challenges of riddles (ones they learn and ones they invent) and puzzles; and adults are so lured by puzzles that even supermarkets sell books of them. So what’s the difference between real puzzles and school problems? What’s useful about creating a puzzle or posing a problem? How might puzzles and problem posing support mathematical learning? And what’s constructionist about this? This plenary will try to respond to these questions, invite some of your own responses, let you solve and create some puzzles, and explore how problem posing in programming and puzzling can support mathematics even in an age of rigid content constraints. 

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5. Instructional interventions and teacher moves to support student learning of logical principles in mathematical contexts

   Roh, K.H.; Dawkins, P.C.; Eckman, D.; Tucci, A.; Ruiz, S. (August 2023, 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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