Tag Archives: questions

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.

What did you learn?

One thing Malke Rosenfeld and I agreed on over breakfast the other day is that the question, What did you learn? makes us uncomfortable. Weird, right? We are teachers and find both answering and asking this question makes us uncomfortable.

I have many reasons for not liking the question: that it implies the process has ended; that when I ask it of my students, they may be inclined to say what they think I want to hear; that it doesn’t invite further questions; on and on.

Being asked this question in Malke’s (fabulous) workshop* led me to something new, though.

New to me, anyway.

This coming school year, I will characterize learning—for myself and for students—in the following way.

Learning is having new questions to ask.

If I have learned something, it is because I can ask questions that I previously could not. Some examples…

example 1: Algebra II

Reading Nicholson Baker’s article on Algebra II in Harper’s [behind pay wall; also available at your local library. And seriously, a Harper's subscription is like $15 a year.] recently, I didn’t learn anything. Much of what he had to say about the course and the way students experience it is pretty familiar and the tone resonated with many of my feelings. But when I read Jose Vilson’s response to it, I had questions. Jose writes,

If someone said, “Let’s end compulsory higher-order math tomorrow,” and the fallout happens across racial, gender, class lines, then I could be convinced that this was a step towards reform.

I wondered whether I would view Algebra II differently if I were a man (or woman) of color. I wondered yet again about the place and effect of developmental math and College Algebra on the economically and culturally diverse population of my community college. I have new questions to ask, so I learned something from my colleague Mr. Vilson that I didn’t learn from Mr. Baker.

And you are reading Jose Vilson’s blog on a regular basis, right? If not, now would be a good time to start.

Example 2: Percussive Dance

At Malke’s workshop this week, I began asking about:

  • the relationship between variable and attribute,
  • the importance of decomposing things by their attributes and paying attention to one of these attributes at a time, and whether that is a fundamental characteristic of mathematical activity,
  • whether a characteristic of a novice is an inability to distinguish noise from pattern,
  • how children’s experiences with sameness in their non-mathematical lives informs and constrains their ability to work with sameness in mathematics,
  • whether I was taking seriously my responsibility and opportunity to use physical classroom space for student learning, and
  • what kinds of equivalence relations we could use in Malke’s percussive dance work, and whether we can form a group from the resulting elements, together with composition (my hunch is yes and that the resulting group is non-Abelian, but I haven’t worked out the details).

Now you should watch Malke in action. I’ll be surprised if this 3-minute video doesn’t give you some new questions to ask.


See, in math classes asking questions is usually a sign that you have not learned.

“Any questions?” is a signal to students to speak up if they don’t get what has just been explained.

We have it all backwards.

It shouldn’t be, “What questions do you have?” [I hope you have none so that I can tell myself you learned something.] 

It should be, “What new questions can you ask?” [I hope you have some because otherwise our work is having no effect on your mind.] 

*Asked by someone who is not Malke, for the record.

Best question of the semester

Quick break from prepping and grading final exams.

My future elementary teachers always struggle to name the denominator when they need to find \frac{1}{4} of 1\frac{1}{2}.

They draw the picture.


They know that the numerator needs to be 3. And then they argue about whether the denominator should be 12 or 16.

I struggle every year to get 8 on the table as an acceptable answer. I usually end up being a voice of authority for 8, and we discuss what the whole is if you use 12 or 16 as the denominator.

My students don’t like 8 because that means the answer is \frac{3}{8} of one square, but the pieces come from different squares.

This year, I had an insight that helped a lot. The question was this:

What are some situations in life when you get two same-sized parts of distinct wholes?

I opened the class session following our usual denominator debate with this question and it helped us to focus on the issue at hand.

After a few false starts (i.e. examples that didn’t really exemplify what we were after), we settled on this scenario.

When you buy a 75¢ pop from a vending machine, by inserting a dollar you get back a quarter. Do it again and now you have two quarters. Each quarter came from a different dollar, but they are still quarters. Each is one-fourth of a dollar and together they are half a dollar (even though collectively, they are one-fourth of the money you started with).

Back to the squares and we had a frame of reference for eighths.

I have been teaching this course for 8 years.

What is it like to be Christopher’s College Algebra student?

I happened across a review sheet for another instructor’s College Algebra exam today. I know not whose, nor do I wish to know. I just want to use it as an example of what my poor students have to go through.

There were eight tasks on the review sheet. I would like my students to have the skills represented in those tasks for sure. But I wouldn’t happy with just those skills.

So here are the tasks. Each is followed up by the sort of question I would ask my students on an exam. Pity them.

  1. Original: Find the domain and range of f(x)=\sqrt{x+2}. Follow up: Give two more functions: one that has the same range as f but a different domain, and one that has the same domain as f but a different range.
  2. Original:  Is f(x)=x^{2}+x even or odd or neither. Follow up: Can a function be both?
  3. Original:  Solve the absolute value inequality |2-5x|<7 and graph the solutions. Follow up: How do these solutions relate to the function f(x)=2-5x?
  4. Original: Graph the function f(x)=x^{2}+4. Then find the intervals on which f is increasing and decreasing. Are there any local maxima or minima? If yes, where are they located? Follow up: Choose two points near a maximum or minimum value (if such a value exists). Find and comment on the average rate of change between these two points.
  5. Original: If g(x)=6x^{2}+5, find the net change and the average rate of change between x=-2 and x=1 Follow up: Why are these values different? BONUS: Give a new function for which these values are equal between the same two points.
  6. Original: If h(x)=\sqrt{x}, write the transformations that yield g(x)=\sqrt{x+2}+1. Also graph both h(x) and g(x) on the same coordinate axes. Follow up: How many solutions are there to this system of equations? \begin{cases} y=\sqrt{x} \\ y=\sqrt{x+2}+1 \end{cases}
  7. Original: If t(x)=2x-1 and s(x)=\sqrt{x+1}, then what are t(x)+s(x), t(x)-s(x), t(x)*s(x) and \frac{t(x)}{s(x)}? Also list the domain for each case. Follow up: Choose one of the four functions you wrote. Write its inverse (if such a thing exists).
  8. Original: Graph this piecewise function f(x)= \begin{cases} 1, x\le 0 \\ x, x>0 \end{cases} Follow up: There is a gap in the graph. Change the second piece of the function to eliminate this gap.

Where do questions come from, part 2

In part 1 of this discussion, Dan Meyer gave his take on where questions in math class come from, or should come from. Dan’s position could be summarized this way: Everybody in my class can’t be working on their own question; I can’t manage that. At the same time, I don’t want to force questions down students’ throats because I use up a lot of my authority doing so. So I want to create classroom situations in which the questions we are answering seem as natural as possible to students. 

Here in part 2, Karim responds to the same question, having heard Dan’s thoughts. He points to an important difference between Dan’s work and his own. In a Mathalicious lesson, the questions are written down on a handout. In a Dan Meyer 3 Act lesson, the questions arise from the class’s experience with a video or photograph; the questions are not fixed in writing.

But neither are the questions especially open. Recall that Dan said in part 1, “I get that at some point the teacher’s going to have to say, This is what we’re going to do today.”

So maybe an important difference between these two isn’t so much where the questions come from (although you’ll soon see that Karim thinks about this very differently from Dan), as it is where they appear to students to have come from. In both cases, questions come from the lesson’s author. Dan is very concerned with student investment in this question-with whether students see the question as the natural one to ask. Karim seems less concerned with this latter issue.

Karim: So if you look at a Mathalicious handout; for example, how many color combinations are there available on Nike ID? And at what point does that cause paralysis by analysis? Or the health insurance one. That’s not something that can at all be encapsulatable in a single piece of multimedia.

And so therefore, we have to lay this narrative. But when we do it, if you look at the handouts, there are very few periods. They’re almost all questions. Every question is a legitimate question.

[We ask] So what do you think…if the insurance company has to charge the same price, how much are they gonna charge? And what do you think is gonna happen next?

And, yeah, like there’s gonna be an answer to that. But the question is a sincere one.

And so then the question is where does that question come from? Because that clearly is a bit more kind of paternalistic than Here’s a piece of multimedia, let’s as a class, let’s talk about what the questions are and then for the sake of efficacy let’s decide on one, but that one kind of came from you guys, kind of the democracy of you.

Mathalicious lessons are quite different from that. So where do those questions come from? I don’t know. I don’t know how to answer this. You know our goal is to be Socratic, and so where did Socrates’s questions come from?

And I don’t know how to answer that without just coming back to this idea of art. You know? Where did the Ninth come from? I really believe that if you ask Beethoven where the Ninth came from, I think he would say, The Ninth existed; my job was to conduct the Ninth. And I mean conduct as in I am a conduit for the Ninth. 

Christopher: In a Platonistic sense.

Karim: Yeah. The Ninth was out there and it’s this sublime piece of music. The Ninth was out there and it was just waiting for somebody to hear it and write it down. And similarly I do think that there are just questions that are really interesting. These questions want to be asked. And so…who’s asking that? I have no idea.

But I can tell you that, as someone who has spent years writing these scripts, I can tell you when a question feels forced and I can tell you when a question feels like it’s flowing. And when it feels like it’s flowing, it does not feel like it’s flowing from me.