Mathematics & Dance Proofs
Written by: Dr. Christine von Renesse
In my Math and Dancing Class this fall we found all positions for two dancers that exhibit both reflectional and 180 degree rotational symmetry. After the students discovered their conjecture I asked them to prove that their conjecture is correct. This was our first activity of the semester and the students were new to the cycle of exploration - definitions - conjectures - proof.
Proof 1: The first student work (showing a first draft of a proof) shows how a student correctly states her conjecture but then uses only examples to show that she is correct. In fact, there is no sentence at all along the lines "I know that I am right, because...". Also, there are some sentences that are unclear, for instance "We knew that if we had our arms crossed it was different for each symmetry". What is "it"? And what does it mean for "it" to be "different"? It is not clear to the reader that the argument makes sense. Maybe it does make sense to the student, but we don't have evidence for that. This is typical for first drafts of proofs, because the idea of fully understanding why something is true — of sense making — is usually new to our students.
Proof 2: The second student work (showing a second draft of a proof) shows a proof by a different student who has accomplished the task of making sense of the reasons behind her conjecture and who also successfully explains her thinking to the reader.
I usually let my students do several drafts of their proofs to help them get closer to a complete argument. Notice that there is no formal language or symbolism used, the proof just consists of a few sentences that make sense and explain. Pictures help the reader to understand the main idea of the conjecture and the proof.
Why is proving relevant in a general education mathematics class? We want our students to be critical thinkers, who instinctively doubt unproven arguments and are eager to find truths for themselves. Math is the only science in which we can fully prove statements given axioms and definitions. This makes mathematics the perfect place for our students to practice feeling and expressing doubt, diving into exploration, having disagreements based on facts — not opinions, practice precision of their statements and arguments and make sense of a complicated situation.