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.

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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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