In our classrooms we create an environment where students are active participants in authentic mathematical experiences in which proof plays a natural and essential role. In this setting students begin to see proof as a key component of sense-making, as an extension of narrative and rhetorical structure that is fundamental to human communication, and as a complement to how we come to know and understand things in other areas of study.

We describe this approach in a full-length paper, have blog posts describing our use of this approach, have examples of student proofs, and classroom videos all listed below.

## Discovering the Art of Mathematics – Proof as Sense-Making

**by Julian Fleron, Christine von Renesse & Volker Ecke**

We describe a model for Mathematics for Liberals Arts courses where students are active participants in an authentic mathematical experience in which proof plays a natural and essential role. In this setting students begin to see proof as a key component of sense-making, as an extension of narrative and rhetorical structure that is fundamental to human communication, and as a complement to how we come to know and understand things in other areas of study. Having reframed proof in this broader context, we discuss benefits of this view for Mathematics majors as well.

Do your students experience the messiness of doing mathematics, searching for patterns, trying and failing in explaining an idea before they come to a complete argument?

## Mathematics & Dance Proofs

**by Christine von Renesse**

We found all positions for two dancers that exhibit both reflectional and 180 degree rotational symmetry. After the students discovered their conjectures, I asked them to prove that their conjectures were correct. This was our first activity of the semester and the students were new to the cycle of exploration - definitions - conjectures - proof. In the full blog post, we provide examples of student work and a classroom video.

## $3a+5b$ Proofs

**by Julian Fleron**

Asked to determine all possible values generated by the Diophantine equation $3a+5b$ when $a,b ≥ 0$, students discovered their first proofs involving the infinite. The diversity of *entirely* different proofs was both a challenge to the teacher and a great affirmation of the importance of inquiry-based learning. In the the full blog post, we provide examples of student work and several classroom videos.

## Pennies & Paperclip Proofs

**by Julian Fleron**

Pennies and Paperclips is a beautiful game whose winning strategies students determine inductively and then work to prove. Proof of the winning strategy for Penny is remarkable for its clarity and simplicity. Proof of the winning strategy for Clip seems to be as straightforward but really offers important lessons in what constitutes proof.

## Pythagorean Theorem Proofs

**by Julian Fleron**

Initial instructions to create 4 congruent right triangles of their choice set students on a path to re-discover the beautiful "proof without words" attributed to Bhaskara. Despite their success, when writing up their proofs many sensed that pictures weren't sufficient and modified their approach to something that involved algebra. In the the full blog post, we provide examples of student work and a classroom video.