Rough-Draft Thinking & Bucky the Badger

Student work on the Bucky Badger problem.

Many thanks to Ben Spencer and his fifth-grade students at Beach Elementary for letting me learn with them on Friday.

In most of my classroom visits lately, I am trying to identify moments where the class and I are drafting our thinking, where we aren’t looking to reach an answer but to grow more sophisticated and more precise in our thinking. Your classmates are an asset rather than an impediment to you in those moments because the questions they ask you and the observations they make about your work can elevate your thinking into its next draft. (Amanda Jansen’s descriptions of Rough-Draft Thinking are extremely helpful here.)

From my limited experience, the preconditions for those moments are a) a productive set of teacher beliefs, b) a productive set of teacher moves, and c) a productive mathematical task – in that order of importance. For example, I’d rather give a dreary task to a teacher who believes one can never master mathematical understanding, only develop it, than give a richer task to a teacher who believes that a successful mathematical experience is one in which the number on the student’s paper matches the number in the answer key.

A productive task certainly helps though. So today, we worked with Bucky the Badger, a task I’d never taught with students before.

Bucky doing pushups.

We learned that Bucky the Badger has to do push-ups every time his football team scores. His push-ups are always the same as the number of points on the board after the score. That’s unfortunate because push-ups are the worst and we should hope to do fewer of them rather than more.

Maybe you have a strong understanding of the relationship between points and push-ups right now but the class and I needed to draft our own understanding of that relationship several times.

The scoreboard for the game. Wisconsin scored 83 points. Indiana scored 20 points.

I asked students to predict how many push-ups Bucky had to perform in total. Some students decided he performed 83, the total score of Bucky’s team at the end of the game. Several other students were mortified at that suggestion. It conflicted intensely with their own understanding of the situation.

I wanted to ask a question here that was interpretive rather than evaluative in order to help us draft our understanding. So I asked, “What would need to be true about Bucky’s world if he performed 83 push-ups in total?” The conversation that followed helped different students draft and redraft their understanding of the context.

They knew from the video that the final score was 83-20. I told them, “If you have everything you need to know about the situation, get to work, otherwise call me over and let me know what you need.”

Not every pair of students wondered these next two questions, but enough students wondered them that I brought them to the entire class’s attention as Very Important Thoughts We Should All Think About:

  • Does the kind of scores matter?
  • Does the order of those scores matter?

I told the students that if the answer to either question was “yes,” that I could definitely get them that information. But I am very lazy, I said, and would very much rather not. So I asked them to help me understand why they needed it.

Do not misunderstand what we’re up to here. The point of the Bucky Badger activity is not calculating the number of push-ups Bucky performed, rather it’s devising experiments to test our hypotheses for both of those two questions above, drafting and re-drafting our understanding of the relationship between points and push-ups. Those two questions both seemed to emerge by chance during the activity, but they contain the activity’s entire point and were planned for in advance.

To test whether or not the kind of scores mattered, we found the total push-ups for a score of 21 points made up of seven 3-point scores versus three 7-point scores. The push-ups were different, so the kind of scores mattered! I acted disappointed here and made a big show of rummaging through my backpack for that information. (For the sake of this lesson, I am still very lazy.) I told them Bucky’s 83 points were composed of 11 touchdowns and 2 field goals.

Again, I said, “If you have everything you need to know about the situation to figure out how many push-ups Bucky did in the game, get on it, otherwise call me over and let me know what you need.” The matter was still not settled for many students.

To test whether or not the order of the scores mattered, one student wanted to find out the number of push-ups for 2 field goals followed by 11 touchdowns and then for 11 touchdowns followed by 2 field goals. Amazing! “That will definitely help us understand if order matters,” I said. “But what is the one fact you know about me?” (Lazy.) “So is there a quicker experiment we could try?” We tried a field goal followed by a touchdown and then a touchdown followed by a field goal. The push-ups were different, so now we knew the order of the scores mattered.

I passed out the listing of the kinds of scores in order and students worked on the least interesting part of the problem: turning given numbers into another number.

I looked at the clock and realized we were quickly running out of time. We discussed final answers. I asked students what they had learned about mathematics today. That’s when a student volunteered this comment, which has etched itself permanently in my brain:

A problem can change while we’re figuring it out. Our ideas changed and they changed the question we were asking.

We had worked on the same problem for ninety minutes. Rather, we worked on three different drafts of the same problem for ninety minutes. As students’ ideas changed about the relationship between push-ups and points, the problem changed, gaining new life and becoming interesting all over again.

Many math problems don’t change while we’re figuring them out. The goal of their authors, though maybe not stated explicitly, is to prevent the problem from changing. The problem establishes all of its constraints, all of its given information, comprehensively and in advance. It tries to account for all possible interpretations, doing its best not to allow any room for any misinterpretation.

But that room for interpretation is exactly the room students need to ask each other questions, make conjectures, and generate hypotheses – actions that will help them create the next draft of their understanding about mathematics.

We need more tasks that include that room, more teacher moves that help students step into it, and more teacher beliefs that prepare us to learn from whatever students do there.

2018 May 23. Amanda Jansen contributes to the category of “productive teacher beliefs”:

Doing mathematics is more than answer-getting.

Everyone’s mathematical thinking can constantly evolve and shift. Continually. There is no end to this.

Everyone’s current mathematical thinking has value and can be built upon.

An important role of teachers is to interpret students’ thinking before evaluating it. Holding off on evaluating and instead engaging in negotiating meaning with students supports their learning. And teacher’s learning.

Everyone learns in the classroom. Teachers are learning about students’ thinking and their thinking about mathematics evolves as they make sense of kids’ thinking.

The list goes on, but I’m reflecting on some of the beliefs that are underlying the ideas in this post.

2018 May 26. Sarah Kingston is a math coach who was in the room for the lesson. She adds teacher moves as well.

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

15 Comments

  1. Reply

    Reminds me of something Christopher Emdin discusses beginning on pg. 145 of his book: For White Folks Who Teach in the Hood…and the Rest of Y’All Too. Reality Pedagogy and Urban Education, “When the students are fully engaged, their curiosity about the content is awakened, and they are constantly exploring the connections between the context and the content that the teacher identified and brought to the classroom. Once this happens they begin asking questions that go beyond the scope of the traditional lesson.”

  2. Reply

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    Regarding productive teacher beliefs, I’ve thought a lot about what they are, but I am less sure about how to help teachers develop them. Maybe they’re developed through just listening to kids and experiencing how much we learn from their thinking?

    But, yes, beliefs that support giving students opportunities to engage with doing mathematics (such as in ways you describe here) are likely to be those that you suggest in this post:

    Doing mathematics is more than answer-getting.

    Everyone’s mathematical thinking can constantly evolve and shift. Continually. There is no end to this.

    Everyone’s current mathematical thinking has value and can be built upon.

    An important role of teachers is to interpret students’ thinking before evaluating it. Holding off on evaluating and instead engaging in negotiating meaning with students supports their learning. And teacher’s learning.

    Everyone learns in the classroom. Teachers are learning about students’ thinking and their thinking about mathematics evolves as they make sense of kids’ thinking.

    The list goes on, but I’m reflecting on some of the beliefs that are underlying the ideas in this post.

    (I’d go so far as to say that misinterpretations are not a helpful way to think about students’ thinking (or anyone’s thinking), but instead I’d recommend “current” interpretations. [My advisor for my PhD was Jack Smith, who wrote this significant article that shapes my thinking on this matter.])

    In your post, I love love love how the everyone’s thinking about the problem itself evolved. That sounds like how mathematicians think and work!

    Also, this made me laugh: “That’s unfortunate because push-ups are the worst and we should hope to do fewer of them rather than more.”

    Thanks so much for sharing the work you did with fifth graders in this post!

  3. Reply

    What would be the teacher moves? I feel that I have the most control and weaknesses in this area and could use some tools.

    • I’ve listed a few I found productive up above. Asking interpretive questions rather than evaluative questions, for one. I’m going to tag in Amanda Jansen here as well.

    • Ryan, I’m working on an entire book right now (Rough Draft Math, through Stenhouse) that includes productive teacher moves for this kind of engagement with mathematics through rough draft thinking! So you could imagine I have a long list… But Dan’s two posts about rough draft thinking have described some useful ones:

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      – Explicitly let students know that when they are sharing their thinking, you welcome them to share DRAFTS — thinking doesn’t have to be full worked out to be worth sharing. (So talking about talking… tagging talk as rough drafts.) — Dan talked about doing this in an earlier post. Try it. Ask students to “just share your draft ideas / your rough drafts.” It’s kind of magical. It frees students to talk more openly.

      – Yes, I agree with Dan that interpretive questions over evaluative questions matter. Also, having an interpretive stance rather than an evaluative one when listening to students is HUGE.

      – I noticed that at the end of the lesson, Dan had students reflect on what they had learned. Giving students opportunity to become aware of shifts in their thinking and giving them agency to name and label those shifts seems like a powerful teacher move, too.

  4. Chester Draws

    May 24, 2018 - 10:22 am -
    Reply

    It’s an interesting task. Well taught it would be an engaging task. What I am not convinced is that it is Maths.

    Maths includes problem solving, but not all problem solving is mathematical.

    The only important part of this task is to discover the variation in how the scores accumulate. That’s about football. The arithmetic part is trivial and hugely repetitive.

    At the end of the task, what mathematical concepts have they deepened? My bet is most learned more about football.

    Some would argue that “problem solving” is itself a useful task. I have two counters.

    Firstly, it’s probably not true. Research shows that generic problem solving skills are not transferable. Problem solving is hugely context dependent.

    Secondly, there are problems which involve a useful mathematical concept, which practice both problem solving and some technique or concept. We should be using those ones.

    So while you may have learned a lot about how students think, I believe the students would have been better off doing a genuine mathematical task. One that links to other mathematical skills and concepts, rather than being a stand alone task.

    • I quote myself:

      The point of the Bucky Badger activity is not calculating the number of push-ups Bucky performed, rather it’s devising experiments to test our hypotheses for both of those two questions above, drafting and re-drafting our understanding of the relationship between points and push-ups.

  5. Chester Draws

    May 25, 2018 - 5:53 am -
    Reply

    Sure Dan. But that’s not Maths.

    It would fall closer to Science. Hypothesis, test, conclusion.

    Maths is using appropriate Mathematical techniques to solve a mathematical problem. I see neither here. I would expect a Maths problem to require a hypothesis of which technique might work, which this does not. I would expect a Maths problem to yield a link to future Maths problems, which this does not.

    That it is interesting, engaging and problem solving doesn’t make it Mathematical.

    Iny view it is extended guess and check, which is literally the lowest form of mathematical reasoning.

    • William Carey

      May 25, 2018 - 10:43 am -

      Some lovely future problem that this points to:

      1. Are there series whose sum is preserved under reordering? What sorts?

      2. Inevitability. If I tell you the total number of push-ups, can you develop the score of the game? Can you construct inequalities that relate the total number of push-ups to a particular score sequence? What do those look like?

      3. This is a lovely segue to introduce the difference between summation notation and multiplication. Under what circumstances can we abbreviate our addition?

      4. Factoring and primes. Can you come up with a (nontrivial) score where I wouldn’t have to tell you the number or order of scores to know the number of push-ups?

      This is a fertile activity. At least #4 would be accessible to fifth graders.

    • Chester Draws

      May 25, 2018 - 11:05 pm -

      Well, you can sort-of force standard mathematical contexts into Badger, but they are not good fits. You can’t really use Summation Notation for such a sequence. Any student doing Sequences and Series would be past finding Badger engaging for very long anyway.

      Why not do an exercise which *naturally* fits into the curriculum we are meant to be teaching?

      In a ninety minute time frame I could have *finished* teaching simple arithmetic series. Then they would have a technique that they could use problem solve.

    • Sorry, but I kindly disagree. Perhaps it’s not applicable to the Math that you teach, but I teach Statistics and my first thought is that this would be a perfect activity for my Seniors just before introducing Hypothesis Testing.

    • If Chester is dependable for anything, it’s to stop by and try to referee what should and shouldn’t be considered mathematics. I don’t care to argue the point with him because by the measures that matter to me (eg. do I think it’s mathematics, do people whose opinion I care about think it’s mathematics, do the governing bodies that decide what is mathematics where I live think it’s mathematics) what we did was mathematics. For Chester it isn’t, and he shouldn’t spend class time on the problem. (To say nothing of the fact that American football probably won’t have much purchase in New Zealand.)

      What I do think is interesting is that our personal and generally unspoken definitions of mathematics are very predictive of the experiences we allow students to have in our math classes. That’s wild to me.

  6. Sarah Kingston

    May 25, 2018 - 2:39 pm -
    Reply

    I am a Math Coach at Beach Elementary and had the good fortune of sharing in this session with Dan at the helm. Although I have a fair bit of experience with and comfort in facilitating 3 Act Tasks K-5, watching Dan in action, and reading his blog and exchange with Amanda have pushed my thinking. My recent focus has shifted to harnessing the learning potential using intentional teacher moves. As Dan referenced, he knew going into the task that the relationships between points and push ups leant themselves to testing hypotheses. In the words of a student, “toying and tinkering with, and testing the math” became the work of the day. Dan deliberately, but artfully guided the process in this direction by using a series of teacher moves.

    These moves aren’t always elaborated upon in these online resources, but I noticed they impact the rigor of the experience significantly. Although Dan and Amanda have highlighted a few, here is a list that another teacher and I came up with in preparation for teaching the Bucky the Badger task ourselves.

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    –Anticipating: anticipate the student interpretations of the point/push-up relationship and be prepared to engage in the students’ thinking

    –Soliciting Estimates: solicit estimates (too high, too low, just right) These become a quick formative assessment and create some productive tension and curiosity within the learning community.

    –Questioning: ask an interpretive question about the relationships in the problem’s context to move students’ understanding forward.

    –Withholding Act 2 Information: encourage students to make sense of the scenario so they request necessary information. Dan said, “If you have everything you need, get to work, otherwise, call me over and let me know what you need.”

    –Conferring: check in with students who may be struggling unproductively. Ask them some relational questions or have them do a quick gallery walk to jump-start their thinking without over-scaffolding.

    –Sharing: students share and justify why the needed information will support their investigation before the teacher reveals it.

    –Identifying Critical Questions: identify critical questions in advance and bring them to the forefront when students begin to ask them.
    Does the kind of score matter?
    Does the order matter?

    –Highlighting Efficient Student Strategies: wonder aloud if there is a more efficient way….mathematicians can be lazy! Also, efficiency oftentimes leaves less room for error.

    –Reflecting: provide time for closure by asking students to privately write a response to, “What did you learn today as a mathematician?”. Then share out as teacher types them for the class to see. Was your learning objective apparent to your students?

    As teachers, we are constantly trying to reflect and refine these practices and there is nothing quite like a window into these teacher moves in action. Thank you, Dan!

  7. Reply

    I am impressed that students stayed on task with one problem for 90 minutes. It seems as though you could’ve spent more time on this one problem. I teach high school students and it is a struggle keep them focused on the same task for any amount of time. They tend to give up, lose focus and hence are off task. Interpretative questioning must be an excellent strategy to help students persevere.

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