On designing tasks to elicit questions

Some interesting criticism of my most recent post on question mapping from Dan: the idea of considering questions you want your students to ask that will enable the teacher to more readily get into content.

There seems to be two strains of criticism, which I’ll attempt to distill here.

Criticism 1: By designing tasks to elicit specific questions, you are not allowing students to offer up their genuine questions and denying them a mathematical voice in the classroom.

Either I was unclear or it takes a pretty disingenuous reading of my post to land here, with me dismissing every question except the one I’m hoping to hear. In case it was the former, let me be clear: student ingenuity is great. There’s nothing better than when students ask a question I hadn’t thought of and we can explore it together. Students asking interesting questions is literally the best part of teaching. Full stop.

Perhaps the phrase “the right question” landed wrong and/or is ill-phrased (happy to take alternate phraseology in the comments!). But yes, I am looking to elicit (and hoping to promote and answer) content-oriented questions or questions I can address with content.

Which brings us to the second strain of criticism, the one I think Dan was getting at in his follow-up to a commenter,

Criticism 2: Lashing a prescribed question to a non-routine task is not realistic and it’s folly to rely upon a task to elicit particular questions. 

From (Harel, 2008):

For students to learn what we intend to teach them, they must have a need for it, where ‘need’ means intellectual need, not social or economic need.

My desire in all classrooms is to have students engage in problems that demand an intellectual need, preferably (but perhaps not necessarily always!) aligned to content I am to teach. That need often manifests itself in the form of students asking questions. In a response to a commenter, Dan says (emphasis mine):

I am interested in question-rich material that elicits lots of unstructured, informal mathematics that I can help students structure and formalize. But I never go into a classroom hoping that students will ask a certain question.

Well here is a point of real disagreement between me and Dan. I am hoping students ask certain questions: Who will win the race? When does the energy efficient light bulb pay for itself? How many sticky notes will cover the file cabinet? How many push-ups did Bucky the Badger do? These are questions I can synthesize into content. It’s more than hope though: with careful craftsmanship, I’d like to be able to predict what students will be curious about because I want to align it with my very real need to teach through my content standards in a meaningful way. Sometimes I’m able to, sometimes not. With practice I get better. These are the questions that evince intellectual need for the content I’m intended to teach.

I’ve never facilitate Bucky the Badger and not had “how many pushups did Bucky do?” be the overwhelming question in the room. I can safely predict (more than just hope) that this will be the primary question asked by students, and wouldn’t you know it? I have a “second act” ready to give you to aid you in your journey.

The point of Question Mapping is to consider how students might engage with the content in order to design a better, more clear task to hopefully alleviate the first two of four artifacts that Fuller et al describe as “problem free environments”:

Four categories of problem-free activity emerged from our analysis and reflection:

1. The situation or immediate goal is not understood by students.

2. The goal of the activity as a whole is unclear.

Problem-Based Learning contends that students learn best when there is an intellectual need for a concept. To me, student questions are the best evidence of that need. So as I teach content, yes, I am (hopefully!) designing tasks that gets students asking questions relating to that content while they are immersed in that scenario.

Anyhoo, comments, clarifications and pushback are welcome in the comments!


Harel, G. (2008b). DNR Perspective on Mathematics Curriculum and Instruction, Part II. Zentralblatt fuer Didaktik der Mathematik 40, 893-907.

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Question Mapping

I’m really good at enjoying the cleverness of a scenario and grafting (sometimes seamlessly, sometimes less so) it onto a mathematical standard (or two or three). I’m less good at starting with a standard (or two) and designing a scenario that appropriately and precisely maps onto it. Sometimes that results in a problem that doesn’t – in a targeted way – address the standard I’m hoping students will take away from it. Sometimes I wind up developing four problems that require students to develop a polynomial expression using the same idea without really introducing anything new or extending it. We do a lot of standards mapping and curriculum mapping, but rarely do we do question mapping.

For example, I’ve facilitated and messaged this problem followed by these problems. The scaffolding and teaching (I hope!) will address different standards. But the problems themselves don’t necessarily necessitate different methods or manipulation of polynomials or quadratics.

The crux of Problem-Based Learning is to elicit the right question from students that you, the teacher, are equipped to answer. This requires the teacher posing just the right problem to elicit just the right question that points to the right standard.


The Experience of a Student in a PrBL Classroom.

In order to achieve this dance, there might be subtle differences in the way a problem is posed. Consider this an attempt to get better at that backwards design approach and to ensure that we’re eliciting the right question.


A design path of a task in a PrBL Classroom.

1. Start with the standard. Hey, here are some standards!

2a. What is the question that you want students to ask that points to the standard?

2b. What might be the language and vocabulary in which students ask it? Because students probably won’t ask “how do we find the roots of a polynomial?”, but they might ask “how do we find where the curve crosses the x-axis?”.

3. What is a possible scenario or task that will elicit that question?

[Optional?] Check your work: Are there other standards that this scenario might address? Are there other ways to solve it that skate around the standard you’re aiming at? Maybe consider giving it a trial run by posing it to a colleague and see if they get close to your intended question?


OK so let’s try this.

  1. I’ll pick a standard. How about this one.
(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

2a. What is the question that you want students to ask that points to the standard?

How do I find the inverse of this here equation that has an exponent (or logarithm) in it?

Perhaps something along the lines of y=ab×.

2b. What might be the language and vocabulary in which students ask it?

How do I find the solution of this here equation that has an exponent in it?

3. What is a possible scenario that will elicit that question?

Me thinking: Well there are lots of applications of things with exponential growth and decay. Populations, investments, radiation and half-life. Perhaps a solicitation letter asking students to analyze bacterial growth of a certain strain?

Or maybe we go abstract and posit something like:

What is the intersection of these two functions? (Or “what do you notice and wonder?”)



I’d also suggest that the practice of Question Mapping might actually help in facilitation as well? Namely that you have a question in the back of your pocket that you know you need to get the students to ask. And if they’re not asking it you need to pull it out of them with leading questions or other bread crumbs. For the problem draft above, I’m not moving on until we establish the questions in 2a and 2b as the impetus for the lesson.

It might be fun (and enlightening) to have a curriculum map of questions along with your standards. And shoot, you’d have your semester review already written months in advance.

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Global Math Department 12/15/15 – Designing Systems of Teacher Learning around Student Work

I’m chatting with the Global Math Department on 12/15/15 about using student work as the driver of teacher learning. Consider this post a repository for pertinent links, my slide deck, and a comment section for further conversation.



Link to working google doc for the session:

[Global Math Department 12/15]

LISTEN: Podcast in which we with Belleville New Tech facilitators about using Looking at Student Work (LASW) to drive teacher learning: [School Innovations Podcast Episode 305: Looking at Student Work, Driving Instruction]

WATCH: Video of a staff using a protocol to analyze student work

More on Looking at Student Work (NTN Website)

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Shuttling vs. Driving to the Airport

A quick Would You Rather today.

I was inspired by a teacher asking students to model airport parking as a means to get at linear equations. I thought I’d frame it similarly, but using the WYR format, because I like the WYR format.

I live a good distance away from a major airport, and I travel quite a bit for a living so I think about this all the time. I have to balance a flat, $80 fee (for both ways) vs. paying for parking (per day, the big variable here), gas, and tolls.



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Complaining about what students don’t know vs. learning how to teach that stuff yourself

It’s a lot easier to complain that students don’t know, say, their multiplication tables than to actually teach multiplication.

Setting aside the oft problematic mindset of a teacher complaining about what “these kids” don’t know for the time being, consider actually teaching to the gaps you feel are present.

Let’s be clear: not a single one of us have entered the school year 100% satisfied with where 100% of our students are at math-wise. There’s always something that was allegedly “covered” in previous years and for whatever reason was not retained by the students. A couple years ago I was at a relatively well-off suburban school where 99% of their students graduate and go on to college and even those teachers were complaining about what their students did and didn’t know.

Often under the guise of unspecific complaints about students “not knowing their basic math facts” or “numeracy”, teachers sometimes pass blame upon students, The Calculator, or their prior schooling. What I often don’t see happen is addressing those gaps in knowledge. Sometimes a cursory remediation worksheet is handed out, and after-school tutoring is offered, but many times I don’t see teachers actually teach to those gaps. Y’know: teaching kids these lugubrious “basic math facts.” Even more specific complaints about how students “don’t know how to do fractions” (whatever “do” means) are ripe for teaching opportunities, rather than tsk-tsk-ing.

Which is unfortunate because there’s never been a more robust cache of resources to remediate in a healthy, fun way. It reminds me of that bit from Arrested Development where Lucille Bluth brushes off her poor raising of Buster because “kids don’t come with a handbook.”

[Ron Howard voice] In fact, there are countless books that address the very learning gap you’re complaining about. NCTM has so many publications that would probably be perfect. Or go here and click on the grade lower than you. Shoot, just go to amazon and type in what you feel your students are struggling with.


If you feel your students lagging in a particular area of their learning, I’d suggest rather than complaining and sending them to a worksheet or instructional video, consider doing some learning yourself and find a book, blog, text, paper, resource, or teacher to teach you how to teach to this area. I’ve learned so much from my non-grade level colleagues about teaching number sense, rounding, fractions, ratios / proportions, and even an alleged area of expertise of mine: algebra (thanks Andrew!).

I mean, if you need a specific recommendation, I’d consider learning how to facilitate some number talks via Elham and Allison’s Intentional Talk and go from there. Or follow ’em on twitter and get their awesome advice for free! (But seriously, get their book.)

Also, we’re not talking about shutting everything else down classroom-wise, lest you’re worried about losing precious class time. While coverage is overrated, let’s put that aside for now, shall we? We’re talking 10-20 minute activities and discussion here, maybe a couple times a week. Stop complaining and start learning how to teach this stuff. If we want students to learn, we probably ought to do some learning ourselves, no?

Besides, teaching these skills and concepts is fun. This has probably been my biggest takeaway of the year so far: leading number talks is so fun, I’d do it even if I wasn’t addressing learning gaps.

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Lectures, various types

“I lecture, but I do it in a dynamic, interactive way!”

Teachers are sometimes justifiably defensive about their lectures. In many circles, lectures are a four-letter word. This flies against not only hundreds of years of pedagogical practice, but cuts against the teacher-as-expert model of instruction.

Of course, there are good reasons for that too. Lectures can be too long and a not terribly good medium for conveying information. I do think it is a good medium to generate excitement and momentum about a topic, as well as brief explanations of the pitfalls in a particular concept.

This isn’t a pro- vs. anti-lecture argument. This is just a brief atlas of four different lecture models, with animated .gifs of course.

Lecture Model 1: Teacher talks


This is our lecture hall model. The lecturer talks aloud, leading listeners through a topic.

It’s not as bad as it sounds though, potentially. This is basically our TED Talk model and people love TED Talks. Provided the material and/or the presenter are interesting enough, this is a perfectly fine model of information and interest conveyance. If you can pull it off and retain interest, more power to you.

Lecture Model 2: Teacher talks –> Teacher questions a student –> Student responds –> Teacher talks


This is actually the lecture model that inspired this post. When teachers claim their lectures are “interactive”, a lot of times this is what they mean. I’m not sure this is terribly interactive. It can be! But it also can not be.

This is probably my least favorite type of lecture, and probably the most prevalent. Under the guise of a “dynamic, interactive” lecture, it often becomes a “gotcha” lecture and “can you recite what I just said?” lecture. It’s basically the tactic I use when my kids start to tune me out.

Certainly there are benefits to such a lecture, but unless you’re being incredibly precise about it, it’s quite possible for a student to go the entire time without engaging with the topic at hand. I suppose the option is usually on the table: “students are free to ask questions in the middle of my instruction!” But in practice, unless that’s truly a norm the lecturer often goes unquestioned.

It’s also inefficient. Consider a 30-minute lecture in which only one person is talking at a time (the student or the teacher). That’s 30 minutes of “discourse.” Even in Lecture format one (outward), you’d have 15 minutes of outward lecture and then 15 extra minutes to do something to get other kids talking. This lecture could easily have been replicated via Zaption or other interactive videoing program.

Lecture Model 3: Teacher talks –> Teacher questions a student –> Student responds –> Student responds


This is what a “dynamic, interactive” lecture ought to look like. This is what I think teachers are going for when they go down the path of Lecture model 2; it’s more difficult to pull off, takes more time, and allows enough grey area that some teachers aren’t willing to cede.

I’m talking about the teaching move where a student responds to a question about something and instead of the teacher confirming, denying, or expanding on the response, a fellow student responds. The teacher sets the conversation in motion but the students become the primary questioners and conversationalists. We see this modeled well in literary circles, but not often in math class.

To encourage this dynamic, a teacher may prompt “Jane, what do you think about Jack’s idea?”, “What words in Jim’s statement resonated with you?”, or, as Kate adeptly incants, “Hey, so-and-so, would you explain your understanding of Bianca’s solution?” Some classrooms have gotten so good at this that the prompt is no longer needed. In Brette’s classroom it became a norm that students would routinely ask things like “can you prove that to me?” and “how are you certain about that?” Music to any lecturers ears.

Lecture Model 4: Teacher talks –> Prompts students to talk –> Students talk


Not unlike Lecture Model 3, in this case a teacher prompts additional thinking. The teacher may talk for about 30% of the lecture time. There is a lot of “turn and talk…”, and “tell your elbow-partner…”, and “in your groups, discuss…” and so on. At this point, a teacher becomes untethered to the front of the room and can join in the small conversations, prompting additional questions or deeper thinking.


Next time you’re lecturing, I encourage you to think about the animated .gifs and consider diagramming the conversation, replete with arrows and dots. Researchers do this regularly. Consider becoming a researcher in your own class by tracking the conversation. Invite a peer in to do it for you or video yourself and watch it back. If you’re going to lean on the lecture as a tried and true means of conveying critical content, at least make sure you spend some time diagnosing the lecture model itself.

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Starting with the What


The Golden Circle

It’s become almost de rigueur to “Start With The Why” or at least to share that title track from the Simon Sinek TED Talk playlist. I often use it to structure my talks or professional developments. I don’t necessarily disagree with the mantra. However, in practice, I seem to often get more out of teachers only once they implement the “What.”

The start of the school year has seen a lot of the “What” thus far. The “What” in the teachers I’ve had the pleasure working with, has fed the “why”, not the other way around. That is, when a teacher facilitates a 3-Act Task, or a Would You Rather?, or a Which One Doesn’t Belong, or a Desmos Activity, or PBL or PrBL Unit, their practice changes as a result of the succeeding reflection. Jo Boaler’s Low Floor / High Ceiling tasks actually start to find and bubble up the cracks in student learning (or in teacher practice). Starting with the “why” feels almost like getting teachers to reflect on something that hasn’t happened yet, or at least the way I clumsily package it. Perhaps even more often, a teacher may be falsely reassured that he/she is attending to the Why if they don’t ever offer a compelling What. A teacher that assigns solely rote problems may never expose the gaps in his/her Why: I want kids to be good problem solvers, and just look at all these problems they’re solving!

I wonder if that’s in part due to the fact that most of us are kind of aligned with the “Why.” We all want students who are persistent problem solvers and like math. That’s true of basically 100% of teachers. You don’t need to build buy-in around that “Why.” The part where we differ and where teachers have had disparate training experiences is in the “How” and “What.”

Screen Shot 2015-09-10 at 9.01.17 PMOne of the more successful coaching experiences I had last year was with a teacher I never had the chance to “pre-brief” the Why. She was hired early in the year, but after school had started. And she took the resources and started trying stuff out. It was through the reflection of those facilitation experiences that she grew as an educator. We never talked about the Why around inquiry or student engagement or anything like that. At the end of the year, she was rocking and rolling with Problem Based Learning, using many of the tasks that had been provided and developing some of her own. Early this school year I’ve had the pleasure of seeing teachers implementing a lot of “Whats” and the reflection and feedback has pointed to the Why, rather than the other way around. And when things go poorly we can talk about that as well. Oftentimes it’s when the tasks break that teachers reconsider the inner sections of the Golden Circle (the How and Why).

Similarly, the most influential PD activities I’ve participated in or facilitated are those that involve some sort of student work analysis. Once we have a What, we can start to pull apart how it points to (or doesn’t) our Why. Or only then do we start to develop our Why. I love Math Mistakes because the tasks and student work provided elicits conversation about the purpose of mathematics and how we teach it.


Does mindset change practice or does practice change mindset? Or does practice change practice?

Of course it’s possible that I’m using the lens of my own predisposition to break and try new things. I didn’t really start reconsidering my practice and how I did things until I tried something quite new and foreign to what I’d been doing before. Only after I made a huge mess did I start thinking more deeply about my craft, including the Why. While I’ll probably continue Starting With The Why ™ for my own planning and reflection, I’ll probably ease up on my emphasis about its importance.

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I am a tree / I am a nest: What I learned after Day One of Elementary PBL Training

I have had the pleasure of “facilitating” Elementary PBL sessions this week. I placed “facilitating” in quotes because I’m essentially window dressing while the venerable Stacey Lopaz, Jodi Posadas, and Megan Pacheco do the actual facilitation. My knowledge base of Elementary PBL is probably lower than the median participant. Even in just a day I’ve had several takeaways.

I am a tree.

Stacey led us through an activity called “I am a tree.” It goes like this:

  • The first person says “I am a tree” and then the person poses as a tree.
  • The second person thinks up something to add to the scene and poses like that. In our scene, Jodi posed as and said “I am a nest” and made a nice little nest handshape.
  • The next person adds on: “I am a leaf on the tree.”
  • The last person adds on: “I am the grass under the tree.”
  • The first three are dismissed and we start again with the last person – the grass – and create another scene: “I am a chicken eating the grass”, ”I am the hen house”, etc.

The idea is to practice building on one another’s ideas. Or using existing ideas and more fully develop them. Either way it was fun, and I feel like it’s what we do here on the ol’ Internet.

I am a tree: I threw out this potential entry event.

I am a nest: Dane made it more interesting and more fully fleshed out.

I’d like to tree/nest one the first non-tree activity we partook of.

I am a tree: Stacey and Jodi passed out unlabeled contour maps of Chicago by year.


Participants talked about what they thought it represented. Guesses were made (“hotels”, “ozone”, “population”, etc.). Analysis occurred. We did a Notice/Wonder protocol about the maps.

There was so much for a math teacher to take away. The curiosity was initiated via the unidentified maps. We got some guesses on the board before the answer was revealed.

I am a nest: I’m not the first to suggest that guessing is good. The unlegened map could be a great way to get students oriented to the idea of using their intuition and estimation abilities.

I began to think about how could I translate this experience to other math content. Such as,

Given the x-axis and the trend, what do you think the y-axis is?


Given the axes, what do you think the trend will look like?


(Note: this reminds me of that really cool NY Times activity from a couple months ago, predicting the trendline of family income and college attendance.)


What do you Notice and Wonder about the following data?

Screen Shot 2015-07-21 at 4.15.42 PM

(all data and graphs are from Tuvalabs)

Elementary teachers often have a sensibility that a lot of math teachers could learn from. In our session today, our facilitators channeled our energy in a way that allowed us to practice a norm and initiated curiosity using visual prompts. I’m curious if you’re able to take away things from sessions that don’t necessarily fit your exact context. Because I feel like that happens all the time when I participate in Elementary conference sessions.

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Putting the pieces together

Just a quick post to draw your attention to a new “selected posts” page. I’ve attempted to map my past writings on Problem Based Learning (PrBL) – at least the ones I don’t totally hate – onto a Problem Process.

Check it out! Let me know what you think.


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Necessary and Sufficient Conditions for School Improvement

These past couple school years have afforded me the opportunity to stretch a bit beyond my math-y world. I’ve had the opportunity to coach entire schools and facilitate whole-staff PDs. It’s definitely invigorating lest I become too myopic in my approach to instruction. I’ve also had the luxury of seeing many different types of school of varying grade levels, demographics, and physical spaces. Yet, what’s most interesting is just how similar these different-looking schools are.

We all play the “build your ideal school” game in our heads from time to time. This post isn’t exactly that. It’s more about the starting blocks that will allow for schools to be successful (understanding, of course, that the term “successful” is pretty loaded). Regardless of whether you’re an public urban high school in Detroit or a charter middle school in Colorado there are several things that schools can do to ensure the best outcomes for students.

In other words, this is a “call me when you’ve done these things” list.

Analysis of work the students are producing. You want a PD plan for the year? Start with looking at student work and go from there. In fact, if all a school is doing for their PD time is looking at student work and using some sort of student work analysis protocol, then they’re probably a high functioning staff. It takes great courage and truly high-flying teachers to look at artifacts students are producing. These artifacts suggest so much (everything?) about task design, instruction, student understanding, and expectations. If you’re giving low-level tasks that’ll become readily apparent. If students aren’t understanding the concepts with any particular depth, that too will become glaringly apparent. A staff that can and has the time to analyze student products in a critical but supportive manner will improve their practice.

At least one additional period of prep time for teachers to spend time coaching other teachers or developing PD. In other words, teachers-as-coaches. Find your highest capacity, best organized, most creative, strong teacher and give him or her the time and space to work on the professional development for the school. The larger a school, the more teacher release time you’ll need. During this time, a teacher may develop professional development, protocols, a year of PD schedules, classroom learning walk schedules, etc.

Regular student input panels: what’s working at the school / what’s not. Students have a much better idea of what’s really happening in the school. If you’re not tending to the student culture at a school, don’t be shocked when the culture deteriorates. Student culture isn’t something that just gets better with time, nor is it something that can be solved by a day of trust falls.

PD for teachers designed largely in house (that is, at the school, not the district). Your district doesn’t know your kids or your classroom and school context. They know their student IDs and probably some quantitative data about them, but not much beyond that.

Teachers spending time in one another’s classroom. There’s so many reasons to unsilo teachers in this way. The obvious is that you get to see neat little facilitation moves. More subtle ways this improves a school include developing consistent practices and structures and identifying certain students that you may struggle reaching but others are having success with. Also, it’s pretty funny when kids are like “hey Mr. Krall, what are YOU doing in here??!?!” It’s pretty conducive to modeling a professional culture for students. Even if you’re just a warm body: a colleague once told me that her staff spent their prep periods prepping just as usual, but in one another’s classroom. They just sat in the back and graded papers, wrote assignments and all that other good stuff, but in a peer’s class. Even by osmosis you’ll pick up stuff.

The time and financial investment by the school to make these things happen. I understand the monetary constraints districts are facing “these days.” It’s cumbersome for schools to find that time for an additional prep-period, that substitute for teacher release time to observe other classes, that allowance of a school to attend a school-focused PD, rather than a district-led PD. A school might not be able to hit the mark 100% of the time, but these are the hills I would die on as an administrator. This is where a director or principal ought to fight tooth and nail. And truth be told, these investments can pale in comparison to, say, a consultant brought in at the beginning of the year, or a smart board.

Really a lot of this is just your classic Data–>Analysis–>Strategy cycle, just with a focus on teachers and students.

Screen Shot 2015-06-06 at 11.07.38 AM

The two umbrellas that these fall under could be broadly generalized as teacher capacity building and student input: how can we as teachers continue to learn from one another and by what our students have to say and are producing? While I’m not going to go so far as to say these things are magic bullets that’ll solve everything, I will say that in the many schools I’ve been in, the ones that are doing these things consistently are getting better every month. And I have no counterexamples.

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