# 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

• 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.

### Conclusion

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.

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.

## Where do questions come from?

I have had the tremendous good fortune to meet and get to know Karim Ani and Dan Meyer in the last couple of years. The three of us have gone back and forth on blogs, Twitter and email. I have learned an awful lot from the conversation.

Last winter I watched the two of them in action at a conference organized by Keith Devlin. I found it really interesting. But I also got frustrated by the lack of critical questioning. The audience wasn’t asking any hard questions and they weren’t asking any hard questions of each other. These minds are way too sharp not to push back on each other a bit.

And if there’s one skill I have developed in my work in higher ed, it’s asking critical questions.

So I hounded these gentlemen and got them to sit down for a conversation at the NCTM meeting in Philadelphia. I compensated them with beer and cupcakes. As I get time, I’ll transcribe the recording of our conversation and post excerpts here.

Up first is a question Dan and Karim have debated a few times. Where do questions come from in math class?

Today we’ll hear from Dan. I’ll get Karim’s response up soon. The questioning is pretty softball here too. Later installments will be different.

NOTE: The people have spoken. They demanded editing so that conversational speech is more easily read as prose. I have done that with a light hand. The original, less edited version is available as a pdf.

Christopher: Traditionally, questions in math classrooms come pretty much exclusively from two places. They come from the teacher in which the teacher asks students questions with a known answer and students are expected to, either in unison or individually when called upon, provide an answer.

The other place questions come from is from students. When the teacher stops and says, “any questions”? Which is what I think you were playing with with your “anyqs” hashtag that turned into 101 questions, Dan.

But I think…those are sort of stereotypical…opportunities for questions to arise in math classrooms. I think we would each chafe against those as being particularly productive. And if that’s what math classrooms continue to be, I don’t think that produces a particularly productive math classroom and each of us has a vision of what questions should look like in math classrooms. So if you could just say a couple of words about-in your mind-where do, or should, questions in math classrooms come from?

Karim: The stork brings ‘em, huh?

Dan: I guess I would take that large question-it’s a good question-and start carving things off from it. Something I’ve dealt with a lot that is just tough to deal with is the idea that we should take a concept and that students should come up with their own questions for it. [This] has been a persistent critique that always bums me out. It bums me out because anyone who makes the point that students should have more control over their learning instantly occupies the moral high ground.

There’s…you asked at the very start about what compromises we’re willing to make for implementation’s sake. Having every student working on a question of their own device. Logistically, I might on my best day be willing to manage that. It’s not something I would construct for a national policy on teaching, or state or local or whatever.

So carving that off…the business of today’s class will not necessarily be on whatever question you the student just kind of came up with. So if you guys want to take that one on, I would love to… That’s been a tough one for me for a while now.

What I would say is this-that all things being equal-if the day has an objective; if there’s something on the agenda today that came from a standards sheet or the natural progression of the mathematics from previous days-all things being equal I would love for that to emerge from a question that the student came up with, or alternatively feels a lot of investment in. And you can gin up investment different ways.

But ideally I know that if I’m asking a question that students don’t care about, I can get them to work on it, and even answer it. But it’s gonna be at the expense of administrative managerial capital. It means I am putting my currency on the line insanely. You guys need to do this; it’s your grade, or you like me or whatever.

All things being equal, I would rather not have to spend that capital. That’d be a start, I suppose, on questions.

I remember reacting, thinking about trying to understand what the person who brought the task to the class, like…what were they hoping to teach a collection of graduate students about problem posing or teaching mathematics or research or cognition, like I didn’t get at all what…

I saw that the lesson would be completely unmanageable if I were teaching in high school, so I couldn’t imagine that it was modeling Here’s what we should be doing. And I struggled to understand what the point of the lesson for graduate students was.

Dan: Yeah. That was an interesting day.

I would say I’m referring to an even more Montessori, constructivist sense. Those questions came from a prompt; a very specific direct prompt about the sheep who cuts ahead in line.

I’m thinking more of like, why are you suggesting sheep? What if the student doesn’t want to deal with sheep? You know, let the student pose their own problem.

So there’s a spectrum here of student agency. And I’m saying I get that at some point the teacher’s going to have to say, This is what we’re going to do today. And I would love for that moment to be as closely aligned to what the student would like to do today as possible, acknowledging that isn’t ever going to be the case in a world that includes Call of Duty.

## More than you bargained for

This semester has been out of control since January. One of many consequences is that I haven’t had time to write about my teaching. There’s a serious backlog of stuff to document; no time to type it up.

Today’s post is a bit of a mix.

Nat Banting wrote this week about a lovely moment when a student looked at a task from a new perspective.

The look on my face must have been priceless, because she started to laugh. The scene went on for quite a while. Slowly but surely, every student had approached the desk to see what was up. The student beamed as she explained…

It got me thinking about similar episodes in my classroom; those moments when I take the time to ask instead of tell, and when my students’ ideas blow me away. The moments when my students teach me some mathematics.

Calculus 2 has had many of these moments this semester. We were studying approximate integration (about which, much more in future posts). We had a motto, “When you cannot integrate, you must approximate“. We had approximated with rectangles and trapezoids. We had built up the formulas for these methods based on students’ intuitions. And then it was time to deal with Simpson’s Rule.

If it’s been a while since you studied such things (or indeed if you never have), the basic idea is in the picture below:

We’re trying to find the area enclosed between the function above, the x-axis below, x=-1 on the left and x=1 on the right. Those rectangles give a pretty good approximation of that area. Each rectangle has a bit of extra area (above the function) and leaves out a bit of area; those roughly compensate for each other and the result is a good estimate.

The reason it’s not a perfect measure is that we are using straight lines (apologies to Chris Lusto) to approximate a curvy function.

So I asked my students what a reasonable solution to this problem would be. What curvy functions should we use to approximate f(x)? We all agreed that it would be desirable for these functions to have nice calculus properties, since that’s why we’re approximating in the first place (that original function doesn’t submit to the standard set of techniques for finding this area exactly).

We were building towards Simpson’s Rule, which is based on parabolas. Use parabolas as tops on those rectangles instead of horizontal line segments and you can get a really nice fit. Plus, parabolas are easy to integrate. Plus if you do a ton of complicated algebra, you can find a really nice formula so that you don’t even have to bother integrating (the pedagogical benefits of this are debatable; the calculational benefits are massive and undeniable).

So I asked. Not just rhetorically. I asked and I listened to their answers. They wanted to use sine (or cosine). And they wanted to use exponentials.

Of course. These are the things with the simplest antiderivatives. Polynomials get more complicated when we integrate. Sine, cosine and $e^{x}$ pretty much stay the same so they’re easy to evaluate. It’s brilliant, right? The algebra won’t work out nicely, and you won’t end up with a clean and tidy rule. But who cares? We’re trying to learn some Calculus here; some ways of thinking mathematically about relationships among functions.

I made the evaluation of this integral by a sine-based version of Simpson’s Rule into an A assignment. Several students are working on that right now.

So much more to report. It’s been a productive semester. I’ll get on that in just a few more weeks.