One student had constructed (what I thought to be) a very interesting theory. His theory was (and I'm paraphrasing with terminology that he didn't use, exactly) that two 2-dimensional figures were congruent if there was a composition of translations and rotations that would lead to an exact mapping. But, the two figures would not be congruent if a reflection was required to achieve that mapping. According to his theory/thinking, a reflection requires you to (in essence) "pick up" the shape and "turn it over," which moves the 2-dimensional figure into three dimensional space. This, he believed, should not be allowed when talking about 2-dimensional figures.

To clarify, these two figures would be congruent....

"...in mathematics teaching, a focus on conocimiento (Spanish for "knowing with") offers a way to acknowledge students' ways of making meaning in mathematics, regardless of whether those meanings are 'forbidden knowledges' or not socially sanctioned as mathematical."

"when students offer a different view, they are seen as having deficient, underdeveloped, or misconstrued understandings of mathematics. Let me be clear, I am not advocating for an 'anything goes' kind of mathematics teaching. Rather, I am suggesting that when teachers can recognize a student's unique perspective along side of but equally important to a mathematician's or math educator's view, there is greater potential for connection between the teacher, student, and new possible forms of mathematics."