This book summarizes new directions in mathematics education research on proving at the university level, thereby providing contemporary extensions of the sub-fields of proof that Annie and John Selden introduced to the field.

Part I: Introductory chapters.- Chapter 1. The legacy of John and Annie Selden on proof and proving.- Chapter 2. My take: Proof research at the undergraduate level—how it evolved.- Part II: New directions for research on proof writing.- Chapter 3. Undergraduate students’ mathematical proof skills: Examining the impact of six cognitive resources.- Chapter 4. New directions for domain-specific aspects of proof: The case of combinatorics and graph theory.- Chapter 5. Researching collaborative proving in real university classrooms: The role and utility of theoretical framing.- Chapter 6. Categorizing undergraduates’ proving processes through the lenses of their “stuck points”.- Part III: New directions for research on proof reading.- Chapter 7. Student performance on proof comprehension tests in transition-to-proof course.- Chapter 8. The summary task and its potential for proof comprehension.- Chapter 9. Incorporating self-explanation in undergraduate proof-based courses: The role of expertise, textual coherence, and instructor modeling.- Chapter 10. Investigating strategies undergraduates and mathematicians use for enriched proof-reading experience.- Part IV: New directions for research on proof as a genre.- Chapter 11. Exploring how undergraduate students cope with learning the genre of proof: Their conception about linguistic convenctions.- Chapter 12. Microcosms in the classroom: An ethnographic study of equity in inquiry-based introduction to proof courses.- Chapter 13. Researching proof viewed as a genre of text.- Part V: New directions for research on affect and proving.- Chapter 14. In-the-moment affect and proving: Research on student emotions.- Chapter 15. The aesthetic challenge of doing visual proofs.- Part VI: Conclusion.- Chapter 16. Theoretical and methodological diversity in undergraduate mathematics education: The influence of John and Annie Selden on our field.

This book summarizes new directions in mathematics education research on proving at the university level, thereby providing contemporary extensions of the sub-fields of proof that Annie and John Selden introduced to the field. The chapters each describe an emerging new area of proof research, review the relevant findings in this area, present open research questions and the tools to address them. The book also discusses proof as a literary genre, and how students' feelings during the proof writing process can influence their behavior. The concluding chapter of the book reflects on new directions for research on proving. As such, this book provides mathematics educators, who have extensive experience researching proof, with an up-to-date review of the new methodologies and research questions with regard to proof, and young scholars, interested in proof, can use these chapters as primers on which they can build a research program.

Highlights exciting new areas of mathematics education research on proof by building on the work of Annie & John Selden Reviews new methodologies and research questions with regard to proof Provides open research questions and describes methodological tools for addressing these questions

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