Tag Archives: nctm

A short talk I gave this spring

The constraints are these: Five minutes, 20 slides. They advance every 15 seconds whether you are ready or not.

Here is my first stab at the genre, from this spring’s NCTM/NCSM conference in New Orleans. The others who presented that day are all worth watching. You can get the complete list, links and a bit more context from The Math Forum, which hosted the talks.


The hexagons are here! [#nctmnola]

Forgive the delay. Here are pdf files of the hexagons we built for use in my hierarchy of hexagons lessons. You should be able to open and edit them in Adobe Illustrator. Consider them CC-BY-SA.

Set 1 (pdf)

Set 2 (pdf)

Shout out to former students Jen Carlson, Nadaa Hassan and Brenna Magnuson for collaborating on these.
Creative Commons License
Hexagons by Christopher Danielson is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Update: Below is the current complete set, with added hexagons from former students Ruth Pieper, Brandon Schwab and Mona Yusuf.


What WAS that? or, What was THAT? [#NCTMDenver]

We watched this video during the closing session in Denver.

I love this video. I find it amusing and clever. The moment where Vi Hart folded the guacamole into the interior of the hexaflexamexagon was marked by an audible gasp of delight in the room.


Somebody needs to explain to me what this is. Is it a lesson? Is it a tasty bite-sized morsel of entertainment? Is it an inspiring call to mathematical action?


Because Vi Hart works for Khan Academy.

Khan Academy is a school (at least metaphorically, but we have reason to believe that Khan and Gates see it more literally than that). So is this an assembly in the auditorium?


It turns out that Hart thinks it’s a lesson. Lessons have objectives. Can you guess hers? She began a sentence this way, “The main educational purpose [of this video] is…”

Watch the video again if you need to. Then you may scroll down for the answer, which will be in the comments.

We also watched Hart’s “i” video.

She said, “Technically, it was a bad video because I lost subscribers [on YouTube]. But numbers don’t matter.”

She cringed at her own words and observed that saying numbers don’t matter in a ballroom full of math teachers is probably a bad idea. I think we all understood that she meant to say that popularity is different from quality, and is a direct indicator of neither quality nor effectiveness. It is in this spirit that her numbers don’t matter quip is strange.

Has anyone from Khan Academy ever given a talk which did not use the number of views, or hits, or followers, or lessons served?

She and her father (George Hart) had already exchanged the number of views of each of their first viral videos. Number of views as a measure was discussed on at least five separate occasions during the hour including the introduction by Outgoing Past President Michael Shaughnessy (whose title I am absolutely not making up, and which is strangely not redundant).

Which brings me back to my original question. What was that? What was that video, exactly? What was that talk? Anyone?

The goods [#NCTMDenver]

Good turn out for my session Saturday morning (EIGHT O’CLOCK!).

Thanks to Ashli Black (@Mythagon) for the shot of title screen.

I’ll get some more details up here sometime soon. In the meantime, here’s the handout (.pdf). And here’s the slide deck (.zip, and which—to be honest—was just a photo album on the iPad; the simplicity of this was liberating).

Here are Alison Krasnow’s notes from the session.

road.to.calculusOne last thing…this is the absolute best form of session feedback, as far as I am concerned—getting to read someone else’s notes on the session speaks volumes about what participants experienced (in contrast sometimes to what I think we did).

The slides:

UPDATE: This talk has been adapted to a paper submitted to Mathematics Teaching in the Middle School. I’ll keep you posted on its progress.

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.

Christopher: So you made me think of your Eric the sheep post. You had a graduate course, or someone who came and spoke in your graduate program who came and had you work on the Eric the sheep problem and I remember reading that post and you were writing about the really quite wide variety of questions that people came up with in response to this really open task that had been posed.

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.

Moments from #nctm12

Freshly back from Philadelphia, prepping for the last week of classes. Moments, quotes and ideas from sessions that left a lasting impression on me…


Constance Kamii is amazing. I love a person who takes a strong stand and lives true to it. Kamii is that person. Among my favorite Kamii quotes from her session on Friday:

“You can tell kids all kinds of things…and they will obey you, and that’s sometimes called learning.”

“If I have not published on the uselessness of base-10 blocks, it’s because Teaching Children Mathematics has rejected my manuscripts. They are ignorant and in power, so…”

“Algorithms unteach place value”

“I think math educators are too hung up on writing. The important things in math is thinking. If they can think, they can write and it doesn’t really matter what they write.”

“Numbers are not seeable. Three is three and ten is ten, no matter how you arrange them.”

“These errors are usually considered careless errors. They are not careless errors; they come from an inability to think.”

Like I said…a woman with principles.


Karen Fuson’s session, on the other hand, left me puzzled. I could not tell whether she was doing her duty in reporting out details of the Common Core Progressions for 5—8 rational number and proportion, or whether she actually believed that nonsense. Two examples…

Regarding the proportion \frac{6}{10} = \frac{x}{5}, she asked Is that a fraction? The answer, surprisingly was No.

Consider the ratio A:B. She claimed that the C in y=Cx was \frac{B}{A} and emphatically not \frac{A}{B} because of the proportion \frac{y}{x}=\frac{B}{A}. But this presumes that we always write ratios so that the independent variable is first. But this is nonsense. In most mathematical relationships, which variable is dependent and which independent is arbitrary, and it is certainly arbitrary which we write first in a ratio.

I left feeling like no progress had been made in the last eight months since I hashed out some of these issues with Bill McCallum.


Jack Smith and Funda Gonulates spoke about their research work on the measurement strand in the Common Core. Their talk was everything the Progressions are not. Thoughtful, research based and useful. If you are at all interested in the development of measurement in elementary and middle school curriculum, you should read their paper.

My favorite quotes from the session were both Jack’s:

“The cognitive action is at the corners.”

“Length is the workhorse of learning measurement.”


Breedeen Murray put together the kind of teacher session I love. Active and practical with a huge helping of ideas. Lots to think about.

“In the puzzle world, the rules are like your axioms. And if you change the rules, you get a different system.”

“It is productive to build the formal structure of proof outside of working on the formal content…is students are juggling both proof and content, the cognitive load may be too high.”


Once I finished chuckling immaturely over how much Ed Berger was talking about my “doodle”, (and his own, to be fair), I captured the following:

“Where do questions come from? Out of thin air.”

This was in the context of a mnemonic (which I have mostly forgotten-I found it to be rather forgettable) for his 5 keys to creative thinking (or something like that). And as such it’s not really clear whether he meant that.

I’ll be coming back to this topic later in the week, when I’ll report on moments outside of sessions.