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Mistakes occur frequently in mathematics. Reframing mistakes into positive moments can be psychologically important in a student's educational journey. We investigated two tertiary math classes that explicitly valued mistakes through a pedagogical requirement called "productive failure". For a percentage of their grade, students demonstrated how they made mistakes in their problem solving and, most importantly, how they overcame those mistakes. Through interviews, video-stimulated recalls, and evaluations of the course all from students, we initially looked for affectual responses to the pedagogical allowance and student-led demonstration. Many of the responses, both benefits and drawbacks of the productive failure, were interpreted by the research group to resemble the psychology literature on peer-led support groups. Descriptions of both productive failure and support groups, as well as quotes from the students, aim to shed light on psychological benefits of valuing mistakes. Finally, we believe that productive failures benefitted many students because it made the human aspect of mathematics more explicit.

It is no easy feat to engage young people with abstract material as well as push them to greater depths of understanding. Add in the extra pressures of curriculum expectations and standards and the problem is exacerbated. Projects designed around standards and having multiple entry points clearly offer students the best opportunity to engage with otherwise daunting material and construct rich understanding. A project the author has used in his classes addresses these issues. In this project, which is presented in this article, students explore how high a person can jump. They work in groups to design experiments and collect jump data with which they develop assumptions related to projectile motion. The project encourages students to explore topics in algebra, although suggestions for extensions in a calculus course are presented too. Students actively engage in the study of quadratic functions through this hands-on approach that emphasizes the engineering design process. Using this design process in the classroom has promoted curiosity, improved self-confidence, and encouraged resilience in students. (A bibliography is included.)