# Questions as evidence of learning

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 $y=2$ counts as a function, and whether it counts as a one-to-one function.

(2) How do you solve $x=|y|$ for $y$?

This was asked by a student as were considering the relationships among functionsinverses 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 $x^{2}+y^{2}=1$. Swap $x$ and $y$ 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.

### 5 responses to “Questions as evidence of learning”

1. 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.”

2. 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.