The Home Stretch

Teaching in Texas, there was always this weird interim period between the end of standardized exams and final exams around this time of year. Usually this time spanned for about three weeks or so, during which disengagement was rampant. On top of this calendar quirk was the general end-of-school jitters, a mix of euphoria and senioritis. The end of the year was in sight and work slowed to a crawl. “Why are we cooped up in here when it’s so beautiful outside?,” I’d often hear.

The students were also ready for school to end.

So what do we do in this weird interim time? What do we do when there are two weeks of school left, there’s minimal accountability, and everyone’s got one eye on the clock and the other on the calendar? Many teachers, rightly or wrongly, feel this time period isn’t very conducive to student learning. After all, the kids know their grade is largely set in stone (with only small fidgeting based on the final exam). They also know that the standardized test that they just took is as or more important than the final exam. Often this time consists of a hodge podge of review packets and make-up work.

I found this time – The Home Stretch – to be quite liberating. To me, this was the time of the year I could try out new things, give tasks that weren’t necessarily aligned with my content standards, and perhaps undo some of the damage I had been doing all year via my fledgling teaching practice. I also felt the freedom to teach stuff that I was interested in, untethered to my Algebra, or Geometry, or Algebra 2 standards. One year this meant doing a book study on certain chapters of Freakonomics. Another year, it meant having students solve a series of bedeviling puzzles to earn their “super spy badge.”

I’m not suggesting anything I did during this time was any good, and that’s kind of the point. Maybe more than any other part of the school year, this is where I could try stuff without worrying too much about accountability – for me or for the students. I didn’t feel like a series of review packet days did much to result in long term student learning, so I decided to just… try stuff.

And this is where I’d implore you to try stuff these last few weeks. Consider doing this: head over here and click on a grade level above your class and click through until you find a task that you find particularly compelling. And facilitate that.

Some other ideas for The Home Stretch:

  • Give a task that you think is “too hard” for your students, perhaps stCfIdlImUUAA_VO1 (1)arting with a particular “Portfolio Problem
  • Head over to and find some good Level 3 questions and give those (in this case, I’d suggest you can sift through some grade levels below. Some of those Level 3 questions are challenging even years after the intended grade level.)
  • Do a number talk – yes, High School teachers & students: I’m looking at you here
  • Give a task that is richer, and more complex than anything you’ve given thus far, maybe a Low Floor/High Cieling task
  • Do Fawn’s Hotel Snap activity
  • Spend a day (or two) having student do logic puzzles or puzzles from the New York Times such as Set, or Battleships, right)
  • Have students do some free writing or metacognition on their learning from the past year

In other words, consider doing the stuff you like doing in math, but are always like “I would love to do that, but the standards!” Or stuff you wanted to do throughout the year (recall that first week of school where you were like “this is going to be the best math class ever”), but left it sitting on the backburner due to all the other urgent details of teaching.

If you need something with a little more explicit rigor (thanks, admin), consider just going with the first suggestion – a task that’s allegedly too hard – and align it with the CCSS Standards of Mathematical Practice and you’re good to go. The nice thing about that is that you’ll get those rich artifacts that I’m always yammering on about.

We’re in the Home Stretch now folks. Let’s finish strong and send kids into summer with some fun math, real math, while we’re at it.

Update 5/18/16: Jeez, just go look at what Nora‘s doing after her final exam. Seriously, go read about it. As someone who’s participated in an escape room experience, this excites me to no ends.


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A presentation format for deeper student questioning and universal engagement

(Editor’s note: this post is part of my not-necessarily math related posts. I spend a good portion of time in non-math classes these days. And thank goodness, because it exposes me to practices that I wouldn’t otherwise have experienced.)

I had the pleasure of sitting in on some student presentations on a recent site visit to a school. The presentations were for a English-Social Studies project in which students were giving a “Shark Tank” style pitch for a particular social issue. The format of the presentation was fantastic.

Before I describe the presentation format in question, let me describe how my (read: bad) presentation formats went.

  1. Students present solutions
  2. I ask, “Any questions from the audience?”
  3. *crickets chirping*
  4. I say, “So uhh… how did you convert miles per hour into kilometers per minute?” Presenters respond.
  5. *every student except the three or four students presenting are staring at the clock and not paying attention in any way whatsoever*
  6. I say, “Any other questions?”
  7. *crickets have stopped chirping and have now also fallen asleep*
  8. I say, “Thanks gang! Who’s next?”
  9. Repeat with the next group

By about the third set of student presentations I was barely paying attention. Part of that was probably the oft-poor design of the problem or project at hand – it is said when multiple groups are presenting the same solution, they didn’t solve a problem, they followed a recipe – but it was also due to the nature of the presentation format. Even when solutions were different, or solution paths varied, I had the same problem of “any questions?” followed by silence.

Scripted initial questions, student panels, reconvening with deeper questions

The following presentation format was the brainchild of Kay and Chizzie. And it resulted in a level of student-to-student discourse more authentically than I ever achieved.

Here’s how the presentation format went according to this “Shark Tank” project.

  1. Students presented their work. They had about 30 seconds.student_prez1
  2. A few students served as a panel (if we’re sticking with “Shark Tank”, these are your Mark Cubans, your Mr. Wonderfuls, etc.). The teacher had prepared a few scripted questions, which the panel asked psuedo-randomly. The presenters knew these questions ahead of time and had to be prepared to answer them.student_prez2
  3. Students responded to the questions that were selected.student_prez3
  4. The panelists convened with their groupmates to discuss the presenters’ responses and to develop deeper, more probing questions. The presenters also had a couple minutes to regroup and confer.student_prez4
  5. After convening, the panelists return to their station and ask the questions that they and their group came up with. The presenters respond. From this point, it becomes semi-conversational as all the panelists are interested in getting their question answered.he presenters then answered those questions, which were generally more specific in nature and based on the initial responses of the presenters.


The two design features that separates this presentation format from my terribly ones are the following:

Design feature 1) Having a few “starter” questions that the students were aware of ahead of time, if only to get the conversation started.

Design feature 2) Letting the panelists confer about what they just heard with their group before proceeding to ask further questions.

There are so many things I like about this format over my non-format. Let me break down all the ways in which I like it:

  • Having a set of pre-written questions (designed by the teacher, presumably, but asked by the students). This got the conversation going. It normed the students into question-asking mode. Even though the questions were scripted, it was coming from the panelists and the presenters were speaking to the panelists. It became a conversation which allowed for more, in-depth explanation and gave the class more info to go on when designing their deeper, more specific questions.
  • The format involves the entire class during presentations. Even students who aren’t part of the panel or the presentation need to pay attention in order to design deeper questions upon convening.
  • Students have an opportunity to brainstorm and develop questions at a not-immediate pace, rather than coming up with questions on the spot. It’s hard enough for you and I as teachers to ask immediate off-the-cuff questions, in retrospect it was probably foolish of me to expect that of students in a consistent manner.
  • Conversations during the convening pretty much necessitated that kids pay attention to the content presented by the other groups.

Despite it being toward the end of the class period and several presentations already having been given, students were still engaged in the questioning process. And of the presentations I saw, student panelists asked excellent, varied questions after convening with their group mates. The group conversations during the convening were meaningful and related to the content presented. Everything I’d hope for in a formal presentation of learning to peers.

I’m sure there are also some meta-lessons to learn: giving students time and space to develop their questions, scaffolding the design of questions via some scripted prompts, and the like. While the premise of the project at hand was tied to the “Shark Tank” format, I see no reason why this or something like it couldn’t be a ubiquitous presentation format for a class throughout the entire year, or, better yet, an entire school.

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Portfolio Problems: Rebuilding Assessment with Rich Tasks

We have the technology. We can rebuild assessment. We can make it better than it was. Better, stronger, more accurate.

We all understand how assessment has served as a destructive force in our classrooms. And we’re all to blame. While the obvious perpetrators of destructive assessment are those foisted upon us by states and districts, let’s not forget to point the finger at our own shortsighted teacher-designed tests. And the latter culprit is one we have more control over.

I didn’t take a course in assessment in college. The only assessments I knew how to create were from a schema based on the problems at the end of the chapter or the standardized exams we all loathe.

I’d like to propose a model of assessment based on resources that we have largely at the ready. Now that we’ve got all these great tasks floating around out there, let’s put them to use. Let’s do this (Martin-Kniep, G. & Picone-Zocchia, J. 2009):

Let’s untether ourselves to the 50 question, multiple choice/short answer test and start assessing in ways that are less destructive: by posing rich (even engaging! yikes!) tasks throughout the school year, assessing in a way that honors the student thought process and allows for demonstration of growth, while at the same time compiling artifacts that would serve well as a thesis-style defense.

Portfolio Problems

(n) Rich problems that serve as potential assessments worthy of a student to demonstrate proficiency in a group of standards. At the culmination of a school year, the artifacts created through the solving of such problems could constitute a demonstration of learning, either along side or in lieu of a comprehensive exam.

These problems may take a two or three days of class time when you take in to account the posing of the problem, the facilitation of the problem, workshops that need be taught or retaught, and revision of student work to proficiency.

One could pose, facilitate, and assess such a problem, say, once a “chapter” (since we secondary teachers seem to be quite beholden to chapters), perhaps eight a year. Think of it: by the end of the year, you’d have eight rich artifacts that demonstrate student learning per student. These artifacts could tell quite a story.

Perhaps the format of the artifacts are well-organized solutions on poster paper. Or perhaps they’re formal texts with figures and tables embedded. Or even a recorded presentation.

A two day Portfolio Problem may look like this.

Portfolio Problems Agenda.001

A Portfolio Problem should:

  • Be accessible at some level to all students in the room
  • Allow for multiple solutions or multiple solution paths
  • Require a deep demonstration of content knowledge and adept application of content knowledge
  • Align to standards you have taught, potentially synthesizing multiple content standards

Let me provide an example.

This task is from Illustrative Mathematics (simply, the go to spot for CCSS-aligned tasks).

A Linear and Quadratic System

Screen Shot 2016-03-21 at 9.50.15 PM

I love this task. There’s so much going on in this problem. A student must understand

  • how to find the equation of a linear function based on coordinates,
  • how to find the roots of a quadratic,
  • the actual meaning of a solution of a system of equations.

If a student can demonstrate competency on this task, there’s no need for me to give a test.

The Assessment: Rubrics with external standards

Assessing such problems will require a bit more than a “8/10” slapped on the top of a paper. It’s going to require a more holistic approach to assessment, namely a rubric. Moreover, since we’re assessing eight-ish portfolio problems in the course of a year, it makes sense to have a generalized rubric set to external standards. That is, one could assess two different problems using the same rubric; the rubric is not task-specific (or at least, in addition to task-specific indicators, you include general ones as well).

For example, here are the rubrics we at New Tech Network co-designed with the Stanford Center for Assessment and Learning Equity (SCALE). Here is the 12th grade math rubric. Note the generalized indicators for college-ready work.

But these particular rubrics aren’t the thing; any generalizable agreed-upon set of criteria for quality student work would do. Bryan has an excellent post from four years ago where he describes a portfolio system assessed on his agreed-upon Habits of a Mathematician. Be sure to check that post out ASAP both for the criteria he came up with and the system of portfolio/performance assessment he advocates.

The portfolio and teacher learning

I’ll leave the choice of platform for portfolio to the #edtech folks, but here are some nice recommendations from Tom Vander Ark, along with further motivation for a portfolio system. Manila folders also work well.

But now we have all these artifacts that demonstrate student learning and – provided the problems are rich enough – can expose gaps in student learning. Now we actually have work worthy of a session of learning from student work, which I’ve talked about before. The data provided by this work can be much more instructive than a spreadsheet telling you which students have answered 6-out-of-8 questions correct on HSA.APR.D.7.

You’re making it sound pretty rosy, Geoff

It’s true. I am. It’s a huge commitment. You have to take up, perhaps, twice as many days for assessment in your class than you currently offer. Perhaps three times if you want to give a more traditional assessment alongside a Portfolio Problem. It takes time to assess a Portfolio Problem against a rubric that you wouldn’t need in a scantron format. And if you’re going to use these artifacts as teacher learning opportunities (via LASW), that’s quite a lift for a math department there as well.

But, like I said, thankfully, many kindly bloggers and forward-thinking task designers have done some of the work for us: namely making quality tasks.

Which brings me to the “now what” portion of the post.

Suggested Portfolio Problems (the rest of the school year)

It’s late March (wait, really?) so we’re closer to the end of the year than the beginning. My imploration would be for teachers to undertake a year of Portfolio Problems for all the reasons mentioned above.

Still, the last few months of the school year might be a good time to get some baseline data for the 2015-16 school year with the facilitation of one Portfolio Problem. We wouldn’t be able to (or need to) get into the whole digital portfolio system now. But it would give you some reps in facilitating a rich problem, collecting the student work, and be able to compare next year to this one.

Here are some suggested Portfolio Problems that may match with late-in-the-year standards you may be tackling in class right now.

Grade 4

Grade 5

Grade 6

Grade 7

Grade 8

Algebra 1 / Grade 9

Geometry / Grade 10

Algebra 2 / Grade 11

These are problems that allow for significant demonstrations of student understanding of multiple content standards, and often the synthesis between multiple standards. With all the great stuff that our colleagues have created, we have the ability to do that UT’s Dana Center has been advocating for decades now.

Suggested Portfolio Problems (going forward)

I’ve added asterisks to (what I think are) exceptional potential Portfolio Problems in the ol’ PrBL-CCSS curriculum maps. I admit it’s only my intuition that’s marking them as such. In almost all sections of the curriculum maps I was able to identify at least one problem that I felt was rich enough to be a Portfolio Problem, or at least had a kernel of potential that could fully form it into a Portfolio Problem.


Wait, so why are we doing this again?

For the student, hopefully the implementation of 6-10 Portfolio Problems throughout a year will be a positive learning experience, even though it’s assessment. It could help blur the lines between assessment and instruction (see tweet above). Also, a student could be proud of and reflect on the work they’ve accumulated over the course of the year.

Screen Shot 2016-03-22 at 9.59.35 AMFor the teacher, hopefully the implementation of 6-10 Portfolio Problems throughout a year will provide a much more instructive data set to determine whether students have
truly mastered the content. It could also help with our messaging around math. We can authentically say “this is math,” rather than saying “math is fun y’all, except we have to do this really boring math to see if you’re doing it right.”

One last note: I was chatting with some teachers and principals recently and was informed that their pro-active assessment system (similar to the one I describe here) found purchase in the district. They were able to provide documented evidence of student learning using a performance assessment-style system and now they are awarded the agency from the district to not give the common standardized benchmark assessments other local schools have to give. I’m not going to suggest your school, district, or state would be amenable to such a system of demonstrated student learning in lieu of standardized tests, but it’s possible. And I’d submit that you might be surprised how open district officials might be to an alternative assessment system if presented with a convincing case.


Martin-Kniep, G. & Picone-Zocchia, J. (2009) Changing the Way You Teach: Improving the Way Students Learn

See also: Raymond’s review of Shepard’s The Role of Assessment in a Learning Culture(2000)

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Culture and the kitchen sink

I was finishing up a whirlwind site-visit at a school last week, trying to collect my thoughts. It had been a ping-pong kind of a day, bouncing to several different classrooms, meetings with teachers, tours with students, a quick look at a school. In the short time I had between exiting the school doors and catching my plane I was trying to piece together why the student-staff culture was so exceptional at the school.

Student and staff and student-staff culture and its tending-to has been drifting to the forefront of my mind lately as the supreme thing that a school ought to work on (to be fair though, other things often drift to the forefront from time to time, so take it with a grain of salt). And I was struck by the culture of the school. Not because there was a lot of “rah-rah” stuff going on. There wasn’t a pep rally or a celebration ceremony or anything. Shoot, I didn’t even host a common practice of assembling a student focus group to get their perspective. The culture – student to student, student to staff, staff to student and staff to staff – was just professional and kind.

Going through my notes from classroom observations and conversations I had with teachers, even old notes from phone calls I had with the principal and other staff, I was trying to put my finger on just what it was that led to that series pleasurable interactions I had and witnessed in the previous eight hours.

Maybe it was the student focus group the teachers had just hosted in which they were able to put in practice some of the things that came up. That could be it.

Or, perhaps it was their practice of including students on staff meetings: yes, afternoon staff meetings! That’s remarkable! Surely that’s what it was.

But wait, maybe it was because they had their students sit in on the hiring of a new front desk attendent. That’s a pretty unique thing! Maybe that’s what I should tell the other schools I support to do to build school culture.

One teacher had a wall of celebrations of students by students. Sticky after sticky was placed around a periodic table and the words “We celebrate ~periodically!~ (this was a Chemistry class). Things like that tend to build a positive culture, no?


You know what though? Despite being a small school (about 400 high school students, 9-12) they had a “book nook” / library. When a couple of students were giving me a tour, they showed me the Book Nook. It is unattended four days out of the week, but operates as a library and study area for students that need it. It is unlocked and had a sign in/sign out and book checkout sheet. Students can come and go as they please. Man, that shows the kind of trust in students we want, so that’s probably it, right?

Maybe it was because the school didn’t have bells to signal the start and end of class. That tends to engender trust and professionalism.

Then it dawns on me: none of these things produce exceptional school culture. All of these things produce exceptional school culture. It’s all of these things that are producing the culture that I just experienced, and certainly more practices that I didn’t witness.

Attending to student and staff culture requires the kitchen-sink treatment. No one practice is going to be sufficient. Three consistent practices probably isn’t enough.

In fact, in isolation, many of these practices would probably be inert, if not deleterious, to student-staff culture. “Trust cards” is a great example of a fine practice that is useless if it’s the only means of establishing trust and mutual respect: “You can use the bathroom when you want but if you’re not in your seat by the time the tardy bell rings you get points taken away.”

I guess that’s what makes it so difficult to treat. If you – as I have often communicated! – host a student panel or focus group, that’s lovely, but it alone will not change the culture. “Trust cards” won’t change the culture. Including students on the hiring process won’t change the culture. Shadowing a student for a day won’t change the culture. But maybe all of those things together – and even more things – will. If you’re looking to improve the culture at a school, you can start with a few changed practices, but to expect significant results, consider a long-term, sustained kitchen-sink approach.


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On sequestering math

I struggle with how “special” we treat math in schools. It’s not uncommon for math teachers and departments to run professional development apart from all other subjects. Or use different classroom norms. Or instruct entirely differently. Or blog or tweet exclusively about math. Math teachers have their own software, their own language, different and separate from the rest of the school.

The entire design of New Tech Network schools, for whom I work as a Math and School Development Coach, purports that schools work best when staff practices, protocols, norms, and buy-in are all aligned. For example, entire staffs are committed to the same norms, the same school culture, and the same protocols. So I’m often biased toward thinking about how kids are experiencing school, rather than just math.

Still, even with that alleged common understanding, I’ve heard so many times from non-math teachers, “oh, that’s the math department, we just let them do their own thing.” Or, from math teachers, “oh, this is math, we do it differently in here.

It’s also quite rational behavior for teachers. If I have limited time for planning and reflection (if any), I’m not going to use it to explain what we’re going over to teachers who won’t give precise feedback. And If I’m employing instructional software as my teaching tool, my peers have literally nothing to offer me.

Is this OK? Is this best for a student’s schooling experience?

I often wonder how are students experiencing and witnessing this. Do they see the disconnect between math departments and the rest of the staff, like the way children intuit when parents aren’t getting along? Do they experience firsthand the pedagogical isolation of a math class compared to the (more often) aligned approaches of other subjects?

It makes me wonder if there’s an upper limit to how great a school – or even a math department – can be if they’re so often sequestered from the rest of the staff.

I worry sometimes that perhaps a great folly of all these rich math resources online is that they can allow math teachers to remove themselves from school norms and ways of being (editor’s note: I want to emphasize the word “can” here, not that it does, but that they can. Are we cool, now? Cool.) Great tasks and lessons are the technical solutions. A staff coming together to determine how to best support students is an adaptive one. (For more an the technical vs. adaptive terminology, head over here.)

So how can math teachers engage productively with the rest of the staff?

At the most successful schools I work with, there are a few common threads, which I wrote about at the beginning of the school year. I would like to highlight/reiterate the notion of, as a staff or grade-level department, examining samples of student work across subject areas. In addition, consider intentionally inviting peers into your classroom. Facilitating a really cool task sometime in the next couple weeks? Send an invitation and/or record yourself so others can see what you’re doing, and perhaps by proxy, get oriented to what fun math can actually be! Follow it up by inviting yourself in to a non-math classroom to observe and learn.

To be sure, the discipline of math is peculiar, to the point of being existential. There are certain teaching moves that are special to the discipline. There are attitudes within students unique to the subject developed over time that we must examine and treat. And to be fair, every subject has their own discipline-specific ideas and norms. Math classrooms just seem to take it to the extreme in terms of looking different.

This is the tension I find myself in: just how different is math as a subject? When is it appropriate and beneficial to think about math-specific teaching strategies vs. general strategies? To what degree does math’s “specialness” hinder or help the overall goal of a school where students and adults feel connected and successful?

I’m not sure I have any concrete answers to these questions, but I do know a couple things: that teaching math is enhanced by leaning upon fellow awesome math teachers and their lessons AND students experience school best when overall teaching practices and norms are aligned.

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Protocol me maybe (teaching edition)

Facilitating is really hard.

We miss things. Perhaps is little misconceptions that we hear but don’t make it to our brain because we’re too busy taking roll and pointing out where to pick up missing work for the umpteenth time. We (ok, I) need to figure out a way to slow things down, so we can better listen, think, and respond appropriately. That’s incredibly challenging to do ad hoc.

Some of us, either through hard work and reps or a divine gift bestowed by R’hllor, have a knack for being able to do just that on the fly. The rest of us have a ways to go.

Also, it’s really difficult to ensure that all kids are speaking. Even in classrooms where the teacher knows better than to call on the quick hand-raisers, we do it anyway, because it keeps things moving. The use of protocols in a classroom can be a way to facilitate better and more equitably.

For one, they can give kids equal voice. Too, the give us time to process and develop a better response than an on-the-spot, seat-of-your-pants teaching moment.

Here are a four protocols I like to utilize in classrooms.

  • The Know/Need-to-Know process. This was/is my go-to means of kicking off a problem. Students identify what they know about the problem and what they need to know (either content-instruction related or additional-info-needed related). I’ve blogged at length about this one, and others have made it even better.

Screen Shot 2016-01-31 at 10.03.35 PM

(Editor’s note: if you have students with IEP, consider giving them this along with the task the day before so they can come in with pre-ideated Knows and Need-to-Knows.)

  • Notice and Wonder. Max has written about this before on his blog and in his book (no, seriously, why haven’t you bought this yet?). This is great for data explorations and interesting visuals and diagrams. See also: See/Think/Wonder. Notice and wonder allows access for all students to describe what they’re seeing and generating authentic wonders.


  • Gallery walks. Once students have solved a problem, they post them around the room and students circumnavigate to each solution for a prescribed period of time (say, 5 minutes). While observing solutions, students are asked to make comments and ask additional questions via post-it note or some other asynchronous medium. Be sure to require at least one comment and at least one question per student per gallery walk “exhibit.” We want everyone’s voices here.
  • I like / I wonder / Next Steps. Another feedback protocol, the sentence starters are exceptionally helpful for students. Five minutes of “I like…”s, another five of “I wonder…”s, and five for “Next Steps” if there are things to potentially do after the feedback.

There are also a bunch of great protocols from, say, NSRF that can be used to facilitate discourse on non-content oriented stuff. I’ve used the final word protocol such that students can demystify, clarify, and expand upon a text. The block party protocol is great to do with students and adults when you want to get them talking about a text or selected parts of a text.

Just a couple quick tips upon using protocols:

  1. Stick to the protocol. You’re going to seem like an overbearing ogre at first, but among the chief value of protocols is giving equity to student voice. The moment the protocol is abandoned you are paving the way back in an inequitable discussion.
  2. Use a protocol iteratively. The power of protocols comes with repeated use. Once students have mastered the protocol itself, it’s incredible how rich the content-oriented discussions can become. I’d say use a particular protocol no less than three times a month.

A small while back a teacher described the use of protocols as “scaffolding for adults.” The context was in staff collaboration, but I think it works well for classroom instruction too. In an ideal world, kids would be quick to voice ideas and we’d be just as quick to answer them in a way that produces sense-making. Until then, we can use protocols to help us get there.

What are some of your favorite in-class protocols?

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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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