I have argued that learning is having new questions to ask.
Here are a few questions that have surfaced in the early weeks of the semester. These are all student questions in College Algebra.
(1) Can it still be a variable if it only has one value?
This was asked by a student as we were sorting out whether 
(2) How do you solve 
This was asked by a student as were considering the relationships among functions, inverses and inverse functions.
(3) Is the inverse of a circle an inside-out circle?
See, we were using a set of equations, considering x as the domain and y as the range. We were asking whether each equation—so viewed—is a function and whether it is one-to-one.
Then we were switching domain and range (i.e. swapping x and y) and asking the same questions about this new equation. Bonus question was to solve each of the new equations for y.
One of our equations was 
But my purpose here is to check in on the progress I am making in fostering and noticing student questions as evidence of learning.
Overthinking my teaching 
These are great questions! Nothing makes me happier than a student asking an insightful question because I see it as an indication of two things. The first being a a deep enough understanding of a topic to ask an insightful question and the second being the desire to know the answer.
I deal WAY too often with “I don’t know and I don’t care that I can’t know.”
Christopher
The beginning of my unfinished dissertation (long, sad story) is a fantastic quote from Voltaire “Judge others by their questions rather than by their answers” This quote came rushing back into my brain as I read this post. All of these are fantastic questions and I think it speaks to your classroom environment that they are engaged enough to think of them and safe enough to ask them. I especially like the variables question here. This is a tough distinction in calculus when we are discussing unknown constants rather than variables.
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