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 counts as a function, and whether it counts as a one-to-one function.
(2) How do you solve for ?
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 . Swap and and get back the same thing. Thus, a circle (as a relation) is its own inverse. Which fact I had never considered.
But my purpose here is to check in on the progress I am making in fostering and noticing student questions as evidence of learning.