At the close of last year, I was working with my fifth grade students on equivalence. I gave them the following question after seeing something similar in an NCTM monthly article:
My hope with this question was to understand how my students interpreted the equal sign. As I had guessed, they had two understandings equality and “the answer.” In 5th grade, I expected these two responses and we worked together to clarify the meaning.
As this school year began, I was curious how my 12th grade Calculus students would answer the same question. I gave my seniors the same prompt and I got one of two responses below.
It was extremely interesting to me that both 5th graders and 12th graders had the same misunderstanding. It made me think about how many other mathematical misunderstandings students acquire through grade school without being corrected and as teachers what are we doing to help prevent these subtle, but critical misunderstandings.