Rough-Draft Talk in Front of Hundreds of Math Teachers

This was new. I was on a raised platform with seven middle school students to my left and six to my right and several hundred math teachers surrounding us on all sides.

This wasn’t a dream. The MidSchoolMath conference organizers had proposed the idea months ago. “Why don’t you do some actual teaching instead of just talking about teaching?” basically. They’d find the kids. I was game.

But what kind of math should we do together? I needed math with two properties:

  • The math should involve the real world in some way, by request of the organizers.
  • The math should ask students to think at different levels of formality, in concrete and abstract ways. Because these students would be working in front of hundreds of math teachers, I wanted to increase the likelihood they’d all find a comfortable access point somewhere in the math.

So we worked through a Graphing Stories vignette. We watched Adam Poetzel climb a playground structure and slide down it.

I asked the students to tell each other, and then me, some quantities in the video that were changing and some that were unchanging. I asked them to describe in words Adam’s height above the ground over time. Then I asked them to trace that relationship with their finger in the air. Only then did I ask them to graph it.

I asked the students to “take a couple of minutes and create a first draft.” The rest of this post is about that teaching move.

I want to report that asking students for a “first draft” had a number of really positive effects on me, and I think on us.

First, for me, I became less evaluative. I wasn’t looking for a correct graph. That isn’t the point of a rough draft. I was trying to interpret the sense students were making of the situation at an early stage.

Second, I wasn’t worried about finding a really precise graph so we (meaning the class, the audience, and I) could feel successful. I wanted to find a really interesting graph so we could enjoy a conversation about mathematics. I could feel a lot of my usual preoccupations melt away.

After a few minutes, I asked a pair of students if I could share their graph with everybody. I’m hesitant to speculate about students I don’t know, but my guess is that they were more willing to share their work because we had explicitly labeled it “a first draft.”

I asked other students to tell that pair “three aspects of their graph that you appreciate” and later to offer them “three questions or three pieces of advice for their next draft.”

  • I like how they show he took longer to go up than come down.
  • I like how they show he reached the bottom of the slide a little before the video ended.
  • I think they should show that he sped up on the slide.
  • Etc.

If you’ve ever participated in a writing workshop, you know that workshopping one author’s rough draft benefits everyone’s rough draft. We offered advice to two students, but every student had the opportunity to make use of that advice as well.

And then I gave everybody time for a second and final draft. Our pair of students produced this:

Notice here that correctness is a continuous variable, not a discrete one. It wasn’t as though some students had correct graphs and others had incorrect ones. (A discrete variable.) Rather, our goal was to become more correct, which is to say more observant and more precise through our drafting. (A continuous variable.)

And then the question hit me pretty hard:

Why should I limit “rough-draft talk” (as Amanda Jansen calls it — paywalled article; free video) to experiences where students are learning in front of hundreds of math teachers?

My students were likely anxious doing math in front of that audience. Naming their work a first draft, and then a second draft, seemed to ease that anxiety. But students feel anxious in math class all the time! That’s reason enough to find ways to explicitly name student work a rough draft.

That question now cascades onto my curriculum and my instruction.

How should I transform my instruction to see the benefits of “rough-draft talk”?

If I ask for a first draft but don’t make time for a second draft, students will know I really wanted a final draft.

If I ask for a first draft, I need to make sure I’m looking for work that is interesting, that will advance all of our work, rather than work that is formally correct.

How should I transform my curriculum to see the benefits of “rough-draft talk”?

“Create a first draft!” isn’t some kind of spell I can cast over just any kind of mathematical work and see student anxiety diminish and find students workshopping their thinking in productive ways.

Summative exams? Exercises? Problems with a single, correct numerical answer? I don’t think so.

What kind of mathematical work lends itself to creating and sharing rough drafts? My reflex answer is, “Well, it’s gotta be rich, low-floor-high-ceiling tasks,” the sprawling kind of experience you have time for only once every few weeks. However I suspect it’s possible to convert much more concise classroom experiences into opportunities for rough-draft talk.

To fully wrestle my question to the ground, how would you convert each of these questions to an opportunity for rough-draft talk, to a situation where you could plausibly say, “take a couple of minutes for a first draft,” then center a conversation on one of those drafts, then use that conversation to advance all of our drafts.

I think the questions each have to change.




[photo by Devin Rossiter]

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I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. How was this event different than the Chalkbeat event that you posted a strong disagreement with? There are many elements of the two events that seem identical to me.

    • (1) I wasn’t competing with another teacher.
      (2) My classroom was a group of students on the stage, rather than hundreds of teachers in the audience.
      (3) I had time to make a lot of the invisible aspects of the experience visible to the audience.

      It seems like the Chalkbeat event mostly arrived at those elements (adult students, tho?) but they weren’t evident in the first contest announcement.

    • 1. The Teach-off was never a competition, it just sounded like that from the description and for some reason you (and others) seem to refuse to see it any other way.
      2. We had students on stage, and at no point did we ever consider the entire audience part of the class. I don’t know where you got that idea from actually.
      3. Of the 95 or so minutes that the event ran, about 30 were devoted in various ways to making the teaching visible for the audience. In fact, making the thinking of the teachers visible during the Teach Off was the entire point of the exercise.

      The only differences I can see between this event and ours is that ours was organized by journalists, your event was organized by the conference, and that we had two different teams of teachers teach, you only had one. Other than that if a reporter was asked to describe what happened at each event, they would give virtually the same description.

    • The contest announcement required a lengthy explainer blog post and significant revisions. Maybe loads of teachers (myself included) just couldn’t figure out what was totally obvious or maybe you folks pulled the idea out of the oven a few minutes before it was ready.

      Happy to take this up via email or in the other teach off thread, but this particular thread is not about the teach off.

  2. Hi Dan,
    I wonder how I might use this kind of public teaching for some faculty PD…thanks for giving me a starting point.

    Some quick thoughts about
    Q1: What name could we create to describe this shape? Follow up: What characteristics did you consider?

    Q2: What numbers would be helpful for estimating 7483 – 209?

    Q3: Before solving, share how you think the quantities 5, 2 and 10 might be related.

    • Thanks for the follow-up, Norma. Q2 feels like it’s in the ballpark of a number talk. Give students some time to think about how they would calculate the value, then pause them before lots of folks will have finished and ask for rough draft strategies.

      I wonder about your approach to Q1. I’m not sure it converts correctness from a discrete variable to a continuous variable. Especially if I’m going to eventually introduce the term “isosceles trapezoid.” I need to think more about this.

  3. For the geometry question, what about:

    (1) Using a ruler, draw as many “different” 4-sided shapes as you can in 3 minutes. Draw one shape on each sticky note / index card.
    (2) With a partner, sort your combined shapes into groups. Put shapes with similar characteristics in the same group.
    (3) What characteristics did you consider as you sorted?
    (4) Did any shapes seem to belong in multiple groups? Why or why not?
    (5) To make your sorting decisions, were there any characteristics that you decided were more important than others? Any that you decided to ignore?
    (6) Make labels that describe the characteristics of each group of shapes.
    (7) Go look at another classmate’s sorting. What are two things you appreciate? What are two questions you have or changes you would make?

    As a teacher, I might go add a few index cards into each group if I noticed that they didn’t have any of the more common quadrilaterals (trapezoids, parallelograms, etc.).

    • A) I love this idea.

      B) I’m trying to figure out how to have a worthwhile rough draft talk here — one where a discussion of one person’s rough draft benefits everyone’s rough draft. Since we have different quadrilaterals, I wonder if I’ll connect your thinking about your quadrilaterals to my thinking about mine.

      What do you think would change about this activity if we all started from the same quadrilateral cards — essentially fixing one degree of freedom?

    • I think that’s a great idea and would definitely ensure that the discussion benefits everyone. I was envisioning that some students would draw shapes that are not quadrilaterals (perhaps the 4 “sides” would not create a closed plane figure, perhaps some would ignore the instructions to use a ruler) and hoping they would include both concave and convex quadrilaterals.

      If the quadrilateral cards only included convex quadrilaterals, it could be interesting to follow up an initial sorting by asking students to create their own card with a “four-sided shape.” Perhaps this could extend the discussion to concave shapes, as well as non-polygonal plane figures.

      I still feel like I would use this as an introductory instructional activity and feel compelled to follow it with an exam question similar to the geometry one that you’ve posted. When a student learning outcome is something like “classify shapes based on their characteristics,” I feel like I need to move past discussion and rough drafts to a final summative question.

      If you want correctness to be a continuous variable, then I think the curriculum itself needs to change, not just the instructional activities. While I enjoy the freedom to make curricular decisions in my current position, I know that many teachers do not. I’m not sure how to get past that.

    • For Q2, I wonder if an algorithm might better lend itself to asking for a rough draft if we ask students for potential ways to represent the solution or multiple ways to solve the problem?

  4. Summative exams? Exercises? Problems with a single, correct numerical answer? I don’t think so.

    I think so, sort of! It’s just that the sequence matters. As you’re starting to learn a topic, we want anxiety-reducing moves. As the sequence of activities goes on, my experience is that those anxiety-reducing moves start becoming frustration-inducing. I like your question in this post, but the language of “transform” implies to me replacement, in a way that I think you probably don’t mean.

    For the geometry question, I like “What else is guaranteed to be true?”

    For the arithmetic/algebra situations, you’re going to need to change things a bit to open it up to drafts, I think, since the “objects” you’ve provided aren’t diagrams; kids will see those as questions with “final” answers. You’ll need to change the object to be something closer to a “diagram,” like the one you provided in geometry, one that kids don’t associate with any particular question.

    Interesting stuff!

    • I like your question in this post, but the language of “transform” implies to me replacement, in a way that I think you probably don’t mean.

      Yeah, helpful check there. I’m not arguing for replacing those questions, rather fortifying them with opportunities for rough draft thinking.

      For the arithmetic/algebra situations, you’re going to need to change things a bit to open it up to drafts, I think, since the “objects” you’ve provided aren’t diagrams; kids will see those as questions with “final” answers. You’ll need to change the object to be something closer to a “diagram,” like the one you provided in geometry, one that kids don’t associate with any particular question.

      Everything about this fascinates me. Maybe diagrams are a special kind of mathematical representation that lends itself to rough draft thinking. Maybe kids not associating the representation with any particular question is an essential feature of rough draft thinking. I’m not sure I buy that there’s anything special about diagrams, though. And I know that my class’s rough draft thinking experience above was initiated by a particular question.

      More likely to me again is that correctness on the variable equation is perceived to be a discrete variable (“I got it or I don’t. My classmates get it or they don’t.”) rather than a continuous one (“I need to get this better than I currently do”). Rather than disassociating the representation from any particular question, it seems more likely to me that we need to disassociate it from any particular answer.

      That still leaves me pretty far from a design heuristic though.

    • I don’t think there’s anything special about diagrams — it’s just that (a) you can think about them and (b) kids don’t associate them with particular questions, so you have room to maneuver with your prompt. What’s tricky about arithmetic is it’s hard to make sure that kids are thinking about some particular subtraction without asking them to subtract.

      Maybe something like this: “Jenna subtracted two numbers and got 315. She got the answer correct, but found the subtraction challenging. What could her numbers be?”

      For algebra: “5 -2x. What could it be?”

    • In both of those last examples, you’re creating one more degree of freedom than existed previously. That’s a heuristic I’m going to tuck away and think more about.

  5. William Carey

    March 9, 2018 - 11:24 am -

    It seems like you’re gradually re-re-discovering Classical Education, which is great!

    Classical educators broadly divide the work we ask students to do into grammar, logic, and rhetoric (which broadly correspond to vocabulary, analysis, and synthesis). What you did with the students in front of the audience presupposed some grammar –the meaning of a cartesian plane–and asked the students to perform logical and rhetorical tasks on a given text–the video of Adam going down the slide.

    The geometry question you pose engages primarily grammatical knowledge. You have to be able to read the geometrical symbols and then recall that that particular combination is a trapezoid. The arithmetic question is also grammatical at heart: there’s an algorithm and you’ve got to apply it by rote. Many (good!) students would see the algebra question as grammar as well, leaning on a rote sequence to solve it.

    By the time they’re in 8th grade or so, students begin to crave logical and rhetorical questions and answers, and the pedagogy we use has to reflect that. The idea of drafts is an important part of that!

    • Super interesting. Thanks for the metaphor, William.

      I’m trying to be more cautious lately superimposing discrete categories onto areas of mathematics that we should actually measure continuously.

      Like, “That area of math is easy and that area of math is hard.”

      Or, “That area of math is concrete and that area of math is abstract.”

      Those are relative measurements.

      So is it possible that the same question can be perceived by three different students at three different developmental stages as about all three categories: grammar, logic, and rhetoric?

  6. From the author of the term “Rough-Draft Talk.”

    Yessssss…. I love this post, Dan. Obviously, I’m in full agreement that all students benefit from spaces to explore mathematical thinking (theirs and others) in a non-evaluative space. What intrigued me the most about your post was highlighting how workshopping someone else’s idea helps everyone refine their own thoughts. I completely agree with this on so many levels, and I love how you supported the students with engaging in this workshopping activity with your prompts — “three aspects of their graph that you appreciate” and “three questions or three pieces of advice for their next draft.” — Students benefit from support with learning how to engage with each others’ work. I love these prompts.

    Once rough drafts are explicitly invited, students start to tag the thinking themselves. They say things like, “This is just my rough draft, but…”

    For more on rough draft talk, there is the article you cited in MTMS (thanks!) and here is another one:

    Thanheiser, E., & Jansen, A. (2016). Inviting prospective teachers to share rough draft mathematical thinking. Mathematics Teacher Educator, 4(2), 145-163.

    Back to your original wonderment about modifying the tasks…

    I agree with Michael Pershan, above, r.e. summative assessments — “As you’re starting to learn a topic, we want anxiety-reducing moves.” Rough draft talk tends to make the most sense to use when exploring new ideas.

    For the arithmetic problem, rough draft talk is about using talk to work on understanding something, like an idea or concept. With multi-digit subtraction, you might want to understand how subtraction behaves (as a take away operation, as a comparison situation, etc), so then the task could be shifted to write two different types of story problems to represent the number sentence. But this totally changes the goal of the task away from finding the solution.

    Maybe then if the focus needs to be on finding the solution, the rough draft talk could be about representing place value relationships, so the task could be about creating representations or diagrams that represent how to find the solution. Some people might use base ten blocks, tally marks, number lines, and then the discussion could be about how to see the solution in these different representations.

    Same with algebra… writing story problems to represent the equation… but creating representations to illustrate the equation feels weirder to me in this one,.

    For geometry, again, I’m tempted to change the goal and have the concept be properties of various quadrilaterals and create a shape sorting task to compare and contrast attributes.

    For me, for rough draft talk, you need a CONCEPTUAL goal, something to make sense of, not a low level cognitive demand task.

    But, as I said on Twitter, you could make the move that I’ve heard Annie Fetter make — instead of asking, “What answer did you get for that problem?” Instead, you could ask, “How are you thinking about that problem?”

    Some disjointed thoughts on a Friday afternoon…

    I’m working on a book called Rough Draft Math (through Stenhouse Publishers), and Chapter 3 is all about the role of the task! So this post is super exciting for me…

    • William Carey

      March 9, 2018 - 6:42 pm -

      > So is it possible that the same question can be perceived by three different students at three different developmental stages as about all three categories: grammar, logic, and rhetoric?

      All three ways of thinking are definitely present in every question to varying degrees, and there’s some overlap between them. Broadly speaking, grammar involves the interpretation of symbols, logic involves analyzing the relationship between objects and rules, and rhetoric involves the synthesis of those two to produce a new and meaningful result. The categories aren’t really ways of discretely categorizing work, but of recognizing the different mental operations that contribute to a particular piece of work.

      Featured Comment

      Math education has historically favored an almost purely analytical logical approach (i.e. unthinkingly apply this rule to this ill-understood object) without much real grammatical understanding or rhetorical synthesis.

      There are totally different ways to teach the different types of reasoning.

      The ultimate goal is to get kids to think about doing math the way mathematicians think about doing math. And what to mathematicians produce? Papers. (And not just one draft!)

    • Mandy, you took the words out of my mouth when you talked about the need for a conceptual learning goal. I paused on all three tasks needing transforming/fortifying because my initial thought was–how can I change or write a task if I don’t know what the Learning Goal is? Then I thought if the Learning Goal is procedural there often isn’t a rich opportunity for RDT.

      Dan, I really appreciated the article, particularly making explicit a goal of teachers being to reduce anxiety for students (I know you’ve talked about this very idea a bit before), but I think offering specific teacher moves to help reduce anxiety for our students is something that is worth writing about. Thank you.

  7. Your algebra problem reminds me of one of my favorite algebra problems.

    Oo watch this transformation, y’all.

    “I have two whole numbers a and b, and 2a^2 = b^2. What are a and b?”

    This problem doesn’t offer much oppportunity for a rough draft: you either have a solution or you don’t. (In fact, you definitely don’t, so this problem doesn’t even offer an opportunity for a final draft.)

    Let’s reword it a little.

    “Look for whole numbers a and b with 2a^2 = b^2.”

    Rather than asking you to have something, this version of the problem asks you to look for something–and it suggests a way of judging how close you are. Let’s try guessing (3, 7) for (a, b). Then 2a^2 = 18, and b^2 = 49, which are 31 apart–pretty far off. If we guess (4, 7), we end up off by 17–a little better than last time. If we guess (5, 7)–holy smokes! 2a^2 = 50 and b^2 = 49, so we’re only off by 1!

    This kind of near miss–just one step from the goal!–happens tantalizingly often. If you start small and work your way outward, here are the first few you’ll find:

    (0, 1)
    (1, 1)
    (2, 3)
    (5, 7)
    (12, 17)
    (29, 41)

    There are some weird relationships between the a and b columns of this list. If you spot two of them, you might be able to guess that the next a value is 70 (or 29 + 41), and the next b value is 99 (or 29 + 70). You can use the same pattern to extend the list as far as you want. Since the problem doesn’t actually have a solution, that list of near misses is as close as you can get to a final draft.

    (If you ignore the rule about making a and b whole numbers, you can solve the problem just by typing 2x^2 = y^2 into Desmos. And, come to think of it, Desmos has that handy grid showing points with whole-number coordinates…)

    • Thanks for the example, Aaron. I’m trying to narrow in on a couple of design heuristics for enabling more rough draft talk. It feels like you’ve added a degree of freedom to the problem.

      Instead of implying there is only one solution to the equation, you’re allowing the possibility that there are many.

    • While I agree that this rewording might have the side effect of adding freedom to the problem, I think its most important effect is to lower the floor of the problem in a certain way. You could say it enables rough draft talk by turning a step into a ramp.

      For me, the original wording creates a barrier I have to get over to get any reward from the problem. If I don’t have a solution, it feels like I’ve failed to achieve anything. I’m not going to hold up my failed attempts as rough drafts; I’m just going to throw them out.

      The new wording invites me to just try something! I don’t have to find a solution; I just have to look for one. I can plug in random numbers, see that I didn’t find a solution, and still feel like I’m doing what the problem asked me to do. That’s a pretty low floor.

      But this low floor slopes upward. When I’m plugging in numbers, I might notice that some guesses feel closer than others. If my first draft is a random guess, my second draft could be a better guess, my third draft could be a proposed rule for identifying good guesses before I plug them in, my fourth draft could be a refinement of that rule, and so on.

      To illustrate the difference I see between adding freedom and “turning a step into a ramp,” consider this problem.

      “Find the positive real number x for which x(x-1) = 1.”

      I can add freedom by allowing more solutions.

      “Find a real number x for which x(x-1) = 1.”

      However, I think this leaves the problem just as inaccessible to rough draft talk.

      I can “turn a step into a ramp” by asking for approximate solutions.

      “By adjusting the real number x, make x(x-1) as close as you can to 1.”

      Now every guess is a rough draft, and revising that draft might lead me to explore methods for approximating the solutions of polynomial equations.

      I’d be interested to see examples where adding freedom is enough, by itself, to enable more rough draft talk. I also wonder if the heuristic would work the opposite way: if a rewording doesn’t add any freedom to the problem, is it unlikely to enable more rough draft talk?

    • Helpful caveat to my suggestion above, Aaron, and usefully illustrated.

      Another approach I’ve tried in similar situations is to invite students to choose any three numbers before they know the problem we’ll be considering.

      “Choose three numbers. Okay now which of them makes x(x+1) closest to 1. How will your results inform your next guess.”

  8. Although intimidating, I will share my perspective.
    Love the workshopping analogy. Students taking about student work to students empowers them to build math and social confidence as well as social grace.(choosing kind words)
    Disclaimer (I ❤️ Number Talks in High School)
    Q2: With a friend developed a story to give these numbers meaning? Share the story “first drafts”(love this language) on a padlet for peer review. SS could improve stories, but the real meat could be to take a story and Describe the meaning of the solution in context of the story. Giving meaning to ‘real’ student meaning to numbers gives Ss access to meaningful Mathematics. (In my humble opinion)
    Q2. I have used a similar prompt as a Number Talk with the question “what would be your first move in solving) my prompt had a couple more options which allowed for different entry ways to be explored and compared for mathematical equivalence. The Ss only had to start the solving journey and with NTs we value all ideas and tried out everyone’s what I will know call “first draft”
    Open to new ideas : and will continue reading posts. Thx Dan

    • Thanks for sharing, Jackie! My feeling is that a number talk is a really useful form of rough-draft talk for problems that are otherwise about the computation of an expression. Thanks for suggesting it.

  9. Long time reader, first time commenter.

    Featured Comment

    I’ve been following you for most of my five year teaching career. But some how I have never heard this sentence.

    “What kind of mathematical work lends itself to creating and sharing rough drafts? My reflex answer is, “Well, it’s gotta be rich, low-floor-high-ceiling tasks,” **the sprawling kind of experience you have time for only once every few weeks**””

    I’m sure you’ve said it before – but the frequency piece jumped out at me as I read today’s article. I’ve somehow had the idea that you were this super teacher with all of these deep critical thinking exercises embedded into every. single. lesson. every. single. day. And that was what all of your instruction looked like. And while on one hand I knew that wasn’t what wasn’t what was going on – at some level I believed that it was and I was a poorer teacher because I wasn’t attaining that.

    I couldn’t make that kind of thing work with my students. I have 6 preps a day with at-risk teenagers. I’m the only math teacher. I thought it was an all or nothing approach. I struggle getting my kids to engage in deep thinking problems – let alone doing them every day. We still do them, but I’ve not attained what I *knew* you (and others) were doing.

    So I guess I’m trying to say thank you. I’m sure I’ve read that sentence – or similar ones many times over the last five years – but I HEARD it today. Thank you for reminding me that there is a balance. That there is still a place for the concrete problems in math. And if it’s not 3-act tasks every day -that’s ok too.

    And thank you for being one of the teachers I can look to, learn from, and steal from. When I’m all alone in my classroom trying to create todays days lessons that span pre-alg through pre-calc it’s nice to know this is one of the places I can turn. And maybe I’m not so far off the mark after all.

    • Ditto Erin! I also appreciate the suggested frequency of every two weeks for the same reasons Erin described above. The first draft concept is something I have started using with some groups of students when they are preparing responses to modeling and reasoning questions. This post has provided much more to consider and use in planning and delivering those type of opportunities for my students.

    • Hey Erin, thanks for commenting! I hope you’ll drop by again.

      First, I hate the thought that you’ve been measuring yourself against an utterly unrealistic benchmark for the last five years, and that I’ve done something to push that benchmark higher. No one should feel like less of a teacher on my account, simply because I’m not a teacher. You’re still there, doing the work.

      Second, while I wouldn’t use lengthy tasks in the classroom every day, I do want to encourage myself (and you!) to have a theory of learning (what makes it effective; what makes it interesting) and apply it consistently. How do people learn best? What does that imply for days when your classes are learning something new? When they’re developing automaticity in what they’ve learned? When you’re assessing what they’ve learned?

      What I don’t want is a schizophrenic classroom — where once every two weeks we act and think like mathematicians, and every other day we act like windup toys.

      That’s why I’m interested in ideas beyond 3-Act Tasks, ideas like “developing the question” or “intellectual need” or Amanda Jansen’s rough-draft thinking. They align with my broader goals for learning but they don’t require unrealistic commitments of our time.

  10. Love the fact that you invited students to graph it in the air! I love this post. Thank you for sharing.

    I mean, life and math is part of guessing. Trying to tell where to go based on the information we have. Guess by our experience. Prove things step by step.

    I like what you said- trying to make it more correct. I see this process as creating the meaningful experience so our students can apply this experience next time when they encounter the same situation.

    Yes, it’s not that important to say first or second draft. Apparently, the latter draft will be better, but without the first draft or without the practice, the better will never come.

    I think of the recent post from MathyCathy. Not only my students, I am also the same, as a teacher. I am improving every day so let’s just forget about the accuracy and focus more on the thinking/reasoning- the fun of math. :)

    • This idea of frequency is interesting to me. I certainly understand and appreciate the relief granted by feeling like you have the freedom to work on the very hard practice of teaching the way we think about it (as curriculum designer, feedback-giver, idea-nurturer, discourse facilitator, summative assessor, etc etc) in a reasonable and manageable timeframe.

      However, I wonder about the potential for success of a “once in a while” non-traditional teaching attempt. My fear is that something outside of the norm will garner resistance from students in ways that may feel overpowering when they are able to define it as something ‘weird’ as opposed to just ‘what we do in here’ or even better, ‘what math actually is.’ And, I wonder about the inclination for students to avoid thinking hard or exploring if they recognize that they can just hold out and things will be back to normal in a day or so… and what they ought to think may be delivered to them with much less brain effort on their part.

      It’s also likely at that point that a teacher may identify the task or overall approach as not working for their kids or their personality or some other reason besides the potential culprit of the structural flaw of the infrequent nature.

      I don’t think I have an answer here, I am just really interested in this idea of how much “grey area” exists between teaching traditionally and non (presuming the dichotomy could even be defined that cleanly for the sake of argument). I suspect Dan may say that good teaching is a continuous random variable and not a discrete one, but I’m not so sure it isn’t two mutually exclusive sets.

  11. Sorry for the second comment:

    Here are my answers:

    For the listed Geometry questions, I would say:

    Nice move!

    Okay, I have a shape. I don’t know what it looks like, but…BUT… it has four sides. Two of the sides are parallel (not sure if I should use two of the sides or one pair). Two of the sides are congruent. (Once again, I feel like if I said one pair of the sides, then I will kill the curiosity.) Would you mind to help me- maybe make a guess – about what it is?

    Start with 9-3, what’s your answer?
    Then, 3-9, what’s your answer?
    Then, 13-9, what ‘s your answer?

    Now, I will make something up, if I have 7000 pieces of hair and 200 pieces of them are gray, then how many pieces of them are black? What if I have 7483 pieces of hair in total and 209 of them are gray, then how many of them are black? How do you depict this question? Make something up to describe the situation.

    (If Ss write it horizontally, how are they going to solve it? What is their algorithm?
    If Ss write it vertically, then why not 3-9 is 6 or -6? How does subtracting numbers vertically really work? )

    Algebra: Not sure yet. Good question though. Negative slope and negative “X” make it difficult to write a prompt for this question.

    • Thanks for the idea, Joanne.

      I think the idea of gradually revealing information is an interesting one. “What could the shape be?” seems like a question that can elicit rough-draft thinking more effectively than “What is the shape?”

    • Hi Dan,

      Thanks! I agree! A more general question would more effective for students to formulate the math concept before they finalize what the math concept means to them.

  12. Thanks so much for this post. My head is spinning with thoughts, mostly (as you’d guess) for how to implement the idea in elementary classes.

    Featured Comment

    For elementary teachers, it seems much in line with what they do for teaching writing — rough drafts, conferring, polishing, etc. In the same class in writing, however, students’ work is typically personal and, therefore, different from their classmates. Yet, while they are all writing about different things (though a genre may be the same), there are ways they can communicate and support one another. There are differences, of course, but there are things I think we can learn.

    About the arithmetic subtraction example you suggest to jog our thinking, the problem for me is that it makes getting an exact numerical answer the goal. And that runs the risk, to me, of defining the “work” in math of getting answers. And that too often leads to learning efficient procedures as the “real” goal of instruction. Plus the problem didn’t relate to the two goals you set for yourself: The math should involve the real world in some way, and The math should ask students to think at different levels of formality.

    So what do we do to embrace your ideas in the realm of the cornerstone of elementary math instruction? Here’s something that Phil Daro did with elementary teachers asking them to think about which was closer to 1―4/5 or 5/4. Actually, he gave them the answer, telling them that 4/5 is closer to 1 than is 5/4. Then the “work” he gave was for them to explain why this was so, using drawings, words, pictures, whatever, but in some way that they could present their explanation to the others. After a few minutes of working on what I think could be considered “rough drafts,” he had them share ideas, and then get back to their work to continue, revise, polish, whatever.

    Inspired by Phil, I presented fifth graders with a similar problem that I learned about at a talk given at NCSM by Susan Jo Russell. I blogged about it here: After students did some work, we had a gallery walk so they could see and comment on each other’s papers.

    So . . . now I’m thinking about problems that would work in this way.
    Why do you always get an even sum when you add two odd numbers?
    Why do you always get an odd product when you multiply two odd whole number factors?

    Or maybe arithmetic problems where multiple strategies.
    If notebooks cost $1.39, how many can I buy for $5.00.
    I have $1.00. Snacks cost $.39. Do I have enough money to buy three of them?

    Thanks for getting me thinking.

    • About the arithmetic subtraction example you suggest to jog our thinking, the problem for me is that it makes getting an exact numerical answer the goal.

      This seems like a helpful anti-pattern for rough-draft thinking. In the same way that there isn’t “an exact answer” to a rough-draft essay prompt, perhaps there can’t be a definite answer to a rough-draft math prompt. So I’m wondering if that condition excludes certain kinds of math entirely, or if it’s possible to rehabilitate that math.

      An idea that’s surfaced repeatedly in this thread is to take existing questions and add one or more degrees of freedom to it. Instead of “what is the shape in this diagram?” we could say, “I have a shape with two parallel sides and a right angle. What could it be?”

      You’re also asking questions with multiple strategies, questions that involve the precise communication of information or an argument.

      Is it possible that all of math lends itself to rough-draft thinking, provided we’re creative in the questions we ask?

    • “Is it possible that all of math lends itself to rough-draft thinking, provided we’re creative in the questions we ask?”

      I wonder if sometimes our creativity in shifting the question ends up shifting the goal of the task? I don’t necessarily have a problem with this, but I want to be aware of this happening, if it does. (I am personally more inclined to value the goals for learning that align with rough draft thinking, many times.)

    • I am personally more inclined to value the goals for learning that align with rough draft thinking, many times.

      A personal liability I am growing more and more aware of is my tendency to decide that the math that doesn’t lend itself well to my pedagogical preferences isn’t math that’s worth doing. Considering arbitrary mathematical objectives and asking “how well does this preference fit?” is how I try to protect against that liability.

    • Can you say more about this? I’m not quite following.

      I recognize that I personally PREFER more conceptual goals for students, but students also benefit from opportunities to develop procedural fluency so I should not deprive them from those opportunities just because of my preference. Is this what you mean?

    • And really that’s why I want to be aware of the ways that shifting the task can shift the goal! If I still have a particular goal for students, and I shift the task to invite rough draft thinking in a way that drifts from a goal, I also want to see if I can get back to that goal, if I still have that goal in mind for the students…

    • I too thought of Phil Daro when reading Dan’s post here. I’ve heard him talk about the Japanese classroom, where a student’s math notebook is central. In the notebook, students craft their response to the prompt, with the aim of forming an argument to convince the class. If a student is stuck, they’re told to roam the class a bit to generate some ideas. Peers can also be useful for helping as sounding boards to help sharpen classmates’ ideas. Ultimately, students make their presentations to the class; the students in the class are the arbiters of correctness. Once the discussion wraps up, students return to their notebooks to revisit their work.

      Contrast that with a model I’m more familiar with, where the teacher acts as the arbiter while engaging one student at a time in verbal ping-pong (aka initiate-respond-evaluate, on repeat).

      In the former model, the notebooks are a place where ideas become arguments and are workshopped and refined throughout the lesson. In the latter, there’s a lot of improv, and it’s easier for students to avoid rigorous thinking.

      In other words, I think Dan’s on to something here! And it’s unquestionably true that the pedagogical shift requires a related curricular shift. Excited to see how these ideas develop.

    • Whoa! Zack, “verbal ping-pong” is exactly what most of my classroom interactions feel like to me, and it’s one of the things I dislike most about teaching in front of a classroom instead of talking with a small group. I never saw this discomfort clearly until you pointed it out, and I never knew there were classroom setups designed to avoid it.

  13. This is legitimately the blog post I have always dreamed of reading. I think there are a ton of opportunities to explore how rough draft thinking can be incorporated into the summative assessment side of a math class.

    I’ve imagined a math classroom stealing some elements of a project-based learning model + ideas from design thinking to create a process that replaces the word “exam” with “project” and treats it as such.

    What would it look like for students to be presented with a low floor, high ceiling task and the first thing they are asked to do is to turn in a first draft. Then they get feedback and critique from their pairs, turn in a second draft, get more feedback and critique, a final draft, then a reflection on what they learned and what they still want to learn? (I’ve seen good materials from Problem of the Month, explanation and status posters).

    My biggest wondering is what’s powerful peer critique look like in a math classroom? For some inspiration on critique, I always watch this video about Austin’s butterfly:

    • “They were more specific but they weren’t mean about it!”

      I love that video, Chris. Thanks for sharing.

  14. I love the idea of “rough draft” because it signals to the student that what they are learning/showing is in process. I switched my thinking a few years back from “what do I want my kids to know” to “what relationship do I want my kids to have with learning,” and it’s been transforming my classroom ever since. It’s nothing new in the math world (growth mindset), but it was new to my WAY of teaching. I look forward to incorporating rough drafts into my monthly explorations – another technique I use to expose students to the PROCESS of doing math. Thanks, Dan.

  15. Hey Dan,


    I started using the language of rough draft thinking this last year after discussions with our humanities department. Every other desciine in our school embraces the revision process as an excpected part of the learning process. What if we did the same thing in math? The change in my students willingness to participate has been dramatic. Since I’m only asking for an initial thought not a finished solution, most students are willing to put an idea on the table. That gives the class a starting point. Just like when writing an essay, the hardest part is getting started. Super excited about you talking about this idea with the wider math community!

    I even got a chance to talked with Amanda about is this past September. She’s spectacular!

    • Have you written this up on your blog anywhere, Andrew? In particular, how has an emphasis on “drafting” in mathematical thinking changed the work you assign your class and how you assign it? Is it as easy as taking the usual problem sets and asking for “an initial thought” or have you made other changes?

    • Dan,
      Surprisingly, there wasn’t a lot I needed to change other than the language we used in the classroom. I asked students to present their ideas as rough drafts. Over and over again we normed the idea that having the correct answer was not expected in your initial thoughts on a topic anywhere in school–not even in math. We just needed a 1st draft to start our revision process as a class. Together, as as class, we were going to revise the idea but we had to have a place to start. I used the same tasks as last year. However, this year the courage to make mistakes in front of the class started happening in August rather than in November. That’s huge. That’s the power of language.

      Here’s the blog post:

    • Amanda,
      That’s true. This was still huge though! Probably one of the top 3 pieces of advice I’d give to my younger self.

      Just in case you ask me, here’s my current top 3:
      1) Use tasks with more than one solution method.
      2) Pick which groups/students present initial attempts at a solution methods (and in what order) rather than asking them to volunteer (this works best in conjunction with the next one).
      3) Talk about the importance of rough draft thinking and the revision process in absolutely EVERY part of life–including math class. We are not expecting a finished piece of work, we are asking for your starting ideas on the problem.

  16. Judy Mendaglio

    March 11, 2018 - 7:14 pm -

    I love coincidences. I was just today reading Whitin and Whitin’s Math is Language Too (2000) that was quoted in the Ontario Ministry of Education document “A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 — Volume Two” from 2006.
    “It is this ‘rough-draft’ talk that allows peers and teachers a window into each other’s thinking. As we talk with freshly fashioned ideas in our minds, we all witness the birth of still further ideas.”
    (Whitin & Whitin, 2000, p. 2)

    • Thanks for this comment, Judy. Just ordered the book! I have read other Whitin books in the past and there was a lot there.

    • Thanks for the reference, Judy. I found a link to the chapter. (Over at NCTE, interestingly.)

      The whole article is great. Their distilled recommendations were particularly useful:

      Highlight the process.
      Recognize the thinking of others.
      Honor surprise.
      Invite reflection.

  17. David Robinson

    March 12, 2018 - 12:18 pm -

    For Q2 – write a rough draft of a story that this equation represents (could also draw one). This would work for Q3 as well. Love this whole post, especially that you made us grapple with it at the end.

  18. Nico Rowinsky

    March 15, 2018 - 10:53 am -

    Q1) Write a rough draft “list of things” you think you need to make a square, a square?
    It doesn’t have to be complete, and it may include extra attributes.
    Students can compare lists and share things they liked, and things they questioned about each other’s lists. Then you can ask a student to share their list with the class.
    They can then refine (second draft) their own list to cut out things they think are ‘extra’ and add things they think are necessary. Counter-examples might pop-up…controversies!
    Now try this with Isosceles Triangle, or Trapezoid, or Isosceles Trapezoid.

    I like that a LIST OF THINGS can be written as a draft, can be adjusted and refined easily (cut, add or change) for future drafts. Old drafts can be seen in new drafts. An editor’s work on a script.

    How does the LIST OF THINGS work for the other questions?

    Q3) Write a list of things you know about the answer to 5-2x=10.
    A student might write:
    x does not equal 2, I tried it.
    x does not equal 0.
    the answer won’t be positive.
    I think it’s negative.

    Future drafts may include:
    I think x is a decimal?
    x=-3 is close

    As always, a very thoughtful post and tremendously thoughtful comments.
    Thank you for sharing Dan.

    • NICE! I love that “list of things” requires more than one thing. I teach college, and my students are everywhere on the spectrum from “I already know what you’re teaching” to “I have never heard of what you’re teaching.” Even if students know “the” answer, making a list requires them to push their knowledge. And it also gives students a safe way to enter the conversation, even if they don’t know much (or anything!) about the topic.

      Another thing I like about “list of things” is that it’s transferrable. I could use it when teaching transcendental equations. I could use it in geometry. I could use it when connecting graphical and algebraic representations of equations. Really, it’s brilliant.

  19. sean genovese

    March 20, 2018 - 10:30 am -

    I’ve been waiting for someone to do a live presentation like this for years. It was just what was needed to illustrate technique.

    There are signs in my classroom that say, “It’s o.k. not to get it the first time” Rough draft and final draft of work is a common concept that students need to realize applies to math.

    I thought The presentation went great – especially I came up and bugged you right before you went on . . . sorry. ;-o

  20. Nice post! I am attracted to models of professional learning that are built around “actual teaching instead of talking about teaching” just as you described. Your experience reminds me in of Lesson Study (or Collaborative Lesson Research – Takahashi & McDougal 2016). You were curious about reducing anxiety (which, if one felt the need, could be framed as increasing: student agency, perseverance, confidence to communicate/critique…) while also teaching math content. You built on your assets and experiences – and reached a little deeper to try something different. You devised/revised some teaching moves (rough draft talk, multiple drafts, using student thinking to inform learning for all), designed a lesson, and taught it live in front of peers with real students. Then you reflected, wondered out loud, pushed your own practice and the practice of your peers. In many forms of Lesson Study, small teams make similar moves. In her post, Norma wondered about adapting this kind of public teaching for some faculty PD. I’m curious about this too. I am finding that using Live Lessons as a platform for professional learning is super energizing for teachers (both those designing/modeling/reflecting on the teaching and those observing, gathering data, reflecting on their own practice). I’m interested in adapting/inventing/testing ways to use public instruction that are cost-effective and time-efficient, and found this example to be inspiring on a number of levels.

    • I’m interested in adapting/inventing/testing ways to use public instruction that are cost-effective and time-efficient, and found this example to be inspiring on a number of levels.

      Same! I know you’ve been working on that project for a long time, Kevin. If you’ve published anything on your work at NTC or elsewhere, please let me know.

  21. I teach preservice math teachers (among other subjects) and my observation of a model lesson the other day (in a HS prealgebra class) led me to a similar conclusion around rough-draft talk/thinking.

    Love it.

    The preservice teacher asked the students “Okay, who has the answer?” after an independent practice problem. Afterwards I’d asked her what would have happened if she’s asked “Who has an answer?”

    It was a real ah-ha moment for the both of us. I’m also an ELA/writing teacher by training, so am all about workshop, revision, talk, and the like. I might suggest a good place to read more about this kind of talk are Peter Johnston’s books, Choice Words or Opening Minds. Both are short and incredibly powerful. He’s in ELA, but his writing about talk is useful for all disciplines in all kinds of classrooms.

    • Love that distinction between “the answer” and “an answer,” Ann. Thanks for sharing that, and for the reference to Peter Johnston as well.

  22. Chelsea Moorman

    April 27, 2018 - 5:40 pm -

    What if students treated daily learning as a rough draft? What if notes, problem sets, projects, and assessments were treated as their portfolio? Instead of treating math as something terminal, treat it like art, where it’s truly never finished. Students create portfolios through synthesizing their work with mathematical evidence. It’s an ongoing process that is part of your classroom norms.