Amusingly, I taught a college math class many years ago. A question on one exam was:
x^3 = 27
One student solved it in the following way:
1^3 = 1
2^3 = 8
3^3 = 27
ans = 3
There was a meeting for the teachers to discuss grading standards, and I brought up my student's answer. I got a number of responses:
"That's not algebra."
"The student obviously doesn't understand the problem."
And so forth. Being somewhat of a punk, I asked if any of them had, in their years of teaching, actually taught their students to recognize when an answer is "algebra" or how a right answer of 3 differs from a wrong answer of 3. It wasn't a pleasant discussion. This was a university with a prestigious math department, and I was just some guy off the street. I chatted with my students the next day. None of them had ever been told what it means to "show your work," or any of the other important trappings of school math. They either did it or they didn't.
Thinking about it more, I would have sidestepped the issue completely, by changing the problem to:
x^3 = 27
One student solved it in the following way:
1^3 = 1
2^3 = 8
3^3 = 27
ans = 3
There was a meeting for the teachers to discuss grading standards, and I brought up my student's answer. I got a number of responses:
"That's not algebra." "The student obviously doesn't understand the problem."
And so forth. Being somewhat of a punk, I asked if any of them had, in their years of teaching, actually taught their students to recognize when an answer is "algebra" or how a right answer of 3 differs from a wrong answer of 3. It wasn't a pleasant discussion. This was a university with a prestigious math department, and I was just some guy off the street. I chatted with my students the next day. None of them had ever been told what it means to "show your work," or any of the other important trappings of school math. They either did it or they didn't.
Thinking about it more, I would have sidestepped the issue completely, by changing the problem to:
x^3 = 26