## 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.

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.

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?

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OK so let’s try this.

CCSS.MATH.CONTENT.HSF.BF.B.5
(+) 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?”)

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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.

## Thought experiment: combine Algebra 1 and Physical Education

(Part of the reason I started this blog is so I’d have a place to play around with ideas, no matter how non-field-tested they may be. Consider this one of my many half-baked ideas that I haven’t fully thought through.)

One of the hallmarks of a New Tech Network school – the network of schools to which I am happily attached contractually and emotionally, and spent part of my teaching career teaching at – are teaching using a Project Based Learning approach within combined courses: World Studies and English, Biology and Literature, and so forth. The first math class I ever really enjoyed taking was my combined Physics and Calculus class my HS senior year.

While I’m not suggesting that mathematics is impossible to combine with other courses, it is often fraught with peril. When we were starting out our journey as a New Tech school, the Science teacher and I splayed out our content standards on the table to see around which we could build projects around. We had a couple ideas for projects, but that would have left over half of our content standards either not combined in a project, or combined in contrived and unnatural ways. Often many of the math standards don’t play well with others.

Moreover, in a PBL classroom, it’s easy for math standards and skills to get dwarfed by the project’s product itself. That was part of my discomfort with PBL and began experimenting with what we now call Problem-Based Learning. It’s doubly easy for math standards to get dwarfed by the lab report, the prettiness of the art exhibit.

That said, I do think students learn the content better when it’s connected to other content. I got more out of my Calculus class by chucking things off the roof and bouncing tennis balls and seeing that the acceleration and the derivative of the speed magically matched. How do we reconcile the value in connecting math content to other physical, tangible subjects while maintaining fidelity to mathematical standards and quality pedagogy?

Here’s a class I’ve never seen implemented (at least, not implemented the way it exists in my head): combined Algebra 1 and Physical Education. That’s right the nerds and the jocks, hanging out together! The more I think about it, the more I like it – again, with the full disclosure that I’ve never seen it taught, never taught it myself, and haven’t even totally thought it through. I’m not sure I’d even consider this half-baked. This is a more 1/8th baked idea.

Still, here’s what I like about it:

The tasks themselves. The content can play pretty well together. I’ve created a couple of tasks just my little old self around physical fitness, and I’m not terribly fintessy. The tasks could either be directly about a student’s physical fitness or about sports and fitness at large. This allows for long term data tracking and regression. Even standards that don’t seem to play well with physical fitness still have physical fitness-like applications (like, say, quadratics … or… quadratics).

NBA.com has started making their SportsVU data public and it’s changing the way the game is played. Slow and fast people are running at the same time and it’s on video. Teams aren’t punting anymore. There’s fitness equipment to be constructed. There are NFL plays to be scripted.

For the PBL-practicing Physical Education teacher, this may hopefully push you beyond the “make a new sport” or “teach other kids sports” projects.

Seriously, why are we letting all this precious data from PE go to waste?

The way you could structure your weeks.

Another nice side-benefit of a combined course is that they are largely double blocked, giving you a full hour and a half or so a day. Seems to me a weekly schedule could look something like this.

Monday: do something physical that gives you data (and some math practice after cool-down, now that the brain has oxygen and blood and stuff)

Tuesday: do something mathematical with that data

Wednesday: do something physical that gives you more data (and some math practice after cool-down)

Thursday: do more mathy things with that data

Friday: spend 45 minute “maxing out” (or whatever) on that data-producing physical activity. Spend 45 minutes analyzing performance

Or just go halvsies the entire week and plot the progress of the students in whatever physical activity they’re doing.

Reduction of status issues in the math classroom

This might also be fraught. I mean, the only place that creates and supports status issues than a math classroom is a physical education classroom, right? On the flip side, it might allow students who are perceived to be low-level achievers in math to finally take the lead. You might get the athletes wanting the “smart kid” on their team, in their group.

Preparing the Brain for Cognitively Demanding Tasks. Physical activity makes for great pre-work for creative mathematical problem solving or a nice interruption from cognitively demanding tasks. Physical activity releases all sorts of good chemical stuff that make you more productive, more creative, more engaged, less stressed and presumable, more capable of taking on cognitively demanding work. It’s also a really nice way to break up an otherwise plodding work day. Here’s an example of a school that is trying to keep kids’ heart rates up for the sole purpose of preparing kids’ brains for learning (hat tip: @JimPa23). Even if the math task has no relation to the PE task, I’d rather have a bunch of kids who have just been exercising than kids who have just come back from the Taco Bell Express in the lunchroom.

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There are also lots of problems to address with such a mash-up. How much time would a class spend changing in and out of their gear? Would combining math and PE compound, rather than equalize status issues for some students? Do the facilitators have a similar vision for the class and what the kids should be getting out of it?

I also know that there are a ton of data in exercise to make good math tasks and physical might help prepare students’ brains for the cognitive ask that complex problems require.

What do you guys think? Is this feasible, or just one of those ideas that should stay in the “fun-to-think-about” realm?

## Equalizing practice and assessment

I’ve made it a habit to retweet this once a month or so from Jenn (@DataDiva) who I look up to as a leader in the field of teacher- and student-friendly assessment.

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

I retweet it because it’s a good reminder and, hey, it’s easy to miss in the never-ending scrawl of twitter. It’s so crucially important that it’s one of those things that should be shouted from the rooftops (on a regular basis, apparently). (PS: anyone know how to get rid of my dumb tweet? I already tried unchecking “remove parent tweet” but to no success.)

One of the side-benefits of transitioning to an inquiry-based, problem-based classroom is that you can slowly start to scrap those old entire-class-day-killing tests. Ideally, once you’re humming along, the Assessment Problems and Problems for Learning will be largely indistinguishable.

It took me a while to realize the power of this. It wasn’t until my final year of teaching that I have a single task to students for their final exam. Students worked on groups and developed a presentation on how to solve a particular complex task; it was assessed with a rubric, which was exactly how the class was structured throughout the year.

However, I’ll describe one thing I didn’t do that is crucial, but I need to get in to some rubric weeds.

There ought to be two sections for most assessment tools:

One thing I did not do throughout my classes that represents a huge gap in my practice was assessing against common standards of quality (“super-standards” is a term that I just made up that I need to sit with before I start using). I strictly assessed students against the particular content that was being taught at the time. “Demonstrated how this diagram proves Pythagorean’s Theorem? Great! PROFICIENT.” “Failed to simplify the quadratic into its simplest form? DEVELOPING.” What was missing was tracking growth in particular mathematical proficiencies over time. More generalized mathematical proficiencies such as “Developing a model”, “Using mathematical literary conventions”, “Representing scenarios in multple ways” that are ubiquitous across most worthwhile problems. Think Bryan’s Habits of a Mathematician. Shoot, think Common Core Standards of Mathematical Practice. By using indicators that lie outside the realm of the particular content addressed in a problem, students can demonstrate growth over time, and learn what it is to be a mathematician (and probably better articulate it).

Here’s an example of what I’m talking about: the top row is specific to this particular problem, the succeeding rows are to be assessed periodically throughout a course.

But this brings us back to equalizing the assessment and instruction. If these are the things you assess, then these are the things you have to teach. And it has to be ongoing.

Also be sure to check out Raymond’s analysis of Shepherd’s The Role of Assessment in a Learning Culture (2000). From which, I’m going to straight up crib his block quote:

## When to scaffold, if at all

It’s been a while since I’ve revisited the Taxonomy of Problems I threw together a while back, but I think it’ll be helpful to spend some time there when considering the following Most-Wanted question around Problem-Based Learning:

At what point after allowing the students to work on a problem do I scaffold the content knowledge?

It’s probably important to identify exactly what type of problem you’re implementing before deciding this.

One of the reasons I wanted to think about this as a potential framework is to address scaffolding (I’ve already addressed assessment). It might not be perfect or precise, but here’s what I basically envisioned.

Unintentionally, this kind of mirrors the ideal progression of both a PrBL Unit as well a classroom and high school experience.

So once you’ve figured out where you are on the taxonomy, where you are in the unit, you can think about your scaffolding.

What & When

I’ll toss out a couple broad-brush rules that oughtn’t be universally applied.

When you’re at the left end of the spectrum – the Content Learning Problems, I’d suggest the following.

If the need for the content is germane to the problem, intervene relatively quickly and with the entire class.

If the need us for an ancillary concept or “side-topic”, consider holding back and/or offering small, differentiated workshops.

For example, I threw Dan’s Taco Cart task into my unit on Linear Equations.

However, use of the Pythagorean Theroem is required to develop your linear equations to model. There will no doubt be a need for some – probably not all – students to revisit or relearn the Pythagorean Theorem. That is ancillary content knowledge: essential, but not the targeted content knowledge skill. Consider holding off on scaffolding that – another groupmate might be the better vessel to explain the concept. Or, if you deem yourself the ideal vessel, consider jigsawing that concept or holding a small pullout workshop with one groupmember per group (the groups’ “student-teacher liaison” as it were).

If the knowledge is germane and is the targeted content knowledge of the task, the scaffolding might need to be more prescriptive, more whole-group. You certainly could lecture (Grant Wiggins has an exceptional post on that), but you could also offer one of these scaffolding tasks. I’m a huge fan of manipulatives and students evaluating student work samples.

Ah, but when do you offer that scaffolding? How much productive struggle should we allow students before intervening? This is where teaching is more of an art than a science. Although if it is truly germane to the problem and it’s a Content Learning problem, I’d err on the side of quick-intervention. Twenty minutes after a problem is launched, perhaps? Thirty?

More important than a time demarcation for instruction is probably some classroom behavioral evidence. Here’s a short list of things to look for to initiate INSTRUCTION MODE:

• Over half the groups or students asking the same or similar thing
• Loss of cognitive demand in the attempted solutions
• Attempted solutions going totally off the rails

What have I missed? What are some indicators that it’s time for you to intervene with scaffolding? Or do you have a particular system or time-frame when considering when to cease the productive struggle time?

If

If your problem is more to the right on that arrow above – Exploratory or Conceptual Understanding problems – the question might not be “what and when” to scaffold but “if”. There is inherent value in an unscaffolded, nonroutine, “ill-structured” problem with a lugubrious associated standard. For these problems consider restricting yourself solely to small workshops devoted to ancillary content knowledge. Or perhaps follow up the problem with a standalone scaffolding task – perhaps, again, a manipulative or evaluation of work samples. Scaffolding for Assessment problems should focus on revision and peer-editing.

The tension between inquiry and instruction shifts from day-to-day, problem-to-problem, so I wouldn’t hold anyone to a hard-and-fast rule. I hope you’ve appreciated my self-indulgence as I continue to try to figure this out and establish a few basic tenets of solid PrBL practice. As always, feedback and commentary is appreciated.

## A Problem Based Learning Starter Kit

You’ve seen the tasks. You’ve read the research. You’re basically bought in. But how do you begin? More importantly, how do you introduce students to inquiry driven learning?

Or maybe you’re not convinced. Perhaps you maintain that the teacher is the primary knowledge constructor. Perhaps you’ve been burned in the past by inquiry driven instruction. You tried it and didn’t see kids learning much and you feel like you wasted some amount of class time when you could have been actually teaching. I can speak from experience: if I wasn’t part of a cohesive team (all subjects, as part of an entire school effort) I quite possibly would have tossed inquiry, Problem Based Learning, groupwork and everything else in the trash after my first miserable experience with it.

Or maybe your students are burned out and beaten down on math. They’ve been labeled “remedial” and by golly, they’re living up to that stamp that your district has placed on them. To them, math is an arbitrary bunch of rules to follow and steps to regurgitate. Their test scores stink and they have difficulty applying math in new and novel situations. Applying math in new and novel situations is probably an entirely foreign concept. Up until now they’ve had example problems or math instructional software to guide them through their problem packet.

It’s always tough to be the first. In many cases, you might be the first teacher to actually ask students to solve complex math problems without pre-instruction. Students might look at you cross-eyed the first time you ask them to work in groups collaboratively on a problem that may not look like the stuff they see in their textbook. There isn’t an example problem for them to look at. Yes, you are the first line on the shores of Normandy.

Not all problems are created equally and some may be more easily acquired and delved-into by students. If you’re not careful with your first exposure of kids into a new way of mathematical task-posing, you and the students could easily frustrated with the process (if you even have one yet). As Dan states perfectly in one of my favorite posts this year on first-steps toward inquiry, “The Unengagables“, “you’ll be hearing from their attorney.” Dan poses three quick methods of introducing kids to mathematical inquisitiveness, be sure to check those out, and follow the comments. I’ll follow with a few tasks here that I think make for good first-foray’s into Problem Based Learning (PrBL).

I like these tasks as first-forays for a few reasons, pointing two directions.

For the teacher:

• The problems kind of “implement themselves.” That is, there isn’t a whole lot to do to massage the task to make it implementable. While I don’t necessarily advocate a plug-n-play curriculum, it’s ready to toss in the oven.
• It doesn’t take too long. Maybe a day, maybe two at most. I’m not sure any first-foray into PrBL should last more than a couple days.
• The task includes facilitation notes and/or other supporting resources.
• The task naturally fosters student and peer-to-peer dialogue. Obviously any good task should do just that, but these tasks especially do that with minimal teacher-prompting.

For the student:

• It’s naturally engaging or intuitively interesting. Real-world is nice, mathematically perplexing is better.
• The problem allows for multiple ways of being mathematically smart. Hopefully some of these tasks will spur the conversation about being smart in math in multiple ways. Habits of a Mathematician type stuff.
• The task at hand is clear. And gets to the point.

Here are a few problems that I’d consider starting with. Or, if you’ve been burned or you’re skeptical, problems to try and experiment with.

Why it’s a good starter problem:

It ties together a visual and number sense. There are several ways to prove or demonstrate a solution. It gets to the point.

Why it’s a good starter problem:

The task allows for guess and check. The task is intuitive and understandable. The scaffolding task involves analysis of samples of student work, a non-threatening way of fostering dialogue.

Why it’s a good starter problem:

The scaffolding involves manipulatives. The math naturally folds into multiple representations and modeling.

Why it’s a good starter problem:

The task prompts students to ask the question. There is an “either-or” possibility for initial guessing and estimating. The task allows for easy differentiated instruction (don’t know how to find the diagonal of a right triangle? how ’bout a workshop on Pythagorean’s Theorem?).

Why these are good starter problems:

You probably have a file cabinet in your room.You probably have a door through which students enter your room. Students have seen and interacted with post-it notes. Students have seen and interacted with styrofoam cups. And with a phone, you could recreate this exact Act 1 video. The task may incorporate multiple ways toward mathematical smartness. Kinesthetic learners might engage via experimentation with post-it-ing the file cabinet themselves.

Dialogue is an inevitability with Always/Sometimes/Never. It can be tailored to your specific classroom. The notion of finding counter-examples is one of the most mathematical ways of thinking I can come up with, and one that kids intuitively understand (it’s a shame we rarely bridge that). If you incorporate some Geometry-type Always/Sometimes/Never cards (like these), kids will be begging you for scratch paper.

Why it’s a good starter problem:

This task, like all of Yummymath’s, include well thought out worksheets with questions that allow for deep conceptual understanding. If you’re not comfortable with driving the car, let the questions that Bryan provides steer for a while.

Why it’s a good starter problem:

For teachers, I think this models nicely how to modify a textbook problem to something more interesting. For students, they have specific math-like things to do. It gives them exposure to modeling from an authentic scenario.

• Any of the stuff freely available from Mathalicious, specifically:

Why they’re good starter problems:

It’s got the presentation – usually with video – ready to roll. The Mathalicious team is adept at both humor and conceptual understanding. Like Yummymath, they can steer the ship for a while until you’re more comfortable with less lesson plan structure and organizing groupwork. The lessons are all aligned to CCSS.

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So, there are a few problems to experiment with.

“But they don’t address my particular standards.” you say (For the record, all of these tasks do address specific content standards, see?, but possibly not yours). To that I’d say don’t worry about “coverage” for a day or two. Teachers lose teaching days all the time due to pep-rally schedules, fire-drills, or whatever. And (this is a whole other post waiting to happen but) coverage is overrated. If you can get kids buying into math – possibly for the first time ever – that’ll go a lot farther than coverage.

What are some of your favorite “starter problems”? What other advice would you give teachers that are starting out? Or maybe, better yet: what was your first experience with any kind of inquiry-based instruction like? What was it like for your students? Feel free to share in the comments.

Update 11/22/2013: Andrew, as always, doing great work. I feel like weekly POPs might be another way to dip you and your students’ toes into non-routine problem solving.

## A non-linear approach to curriculum mapping

I often hear teachers and parents talk about how math skills build on each other in a way that other subjects do not: you have to know how to add before you can subtract, you have to know how to multiply before you use exponents. This is certainly true to an extent, however, I’m wondering if we’re reinforcing these modes by our overly linear curriculum maps (*ahem*). In an inquiry based approach of mathematics, we often preach about “multiple solutions or solution paths” or “multiple entry points.” If we believe what we’re selling, doesn’t that fly in the face of a laddered approach to curriculum mapping? Are we just paying lip service to the whole “multiple solution paths” bit because we know the real way to solve the problem?

I was talking with Kelly Renier (@krenier), director at Viking New Tech, and we began discussing the concept of “power standards” or “enduring understandings” or “What are the Five Things you want your students to know when they leave your class?” then build out from there. However, we didn’t discuss building those Five (or whatever number) Things out into linearly progressing units, but rather concentric circles.

Tasks and/or concepts may go in some ring of each of these concentric circles.

Think of it as an outward moving spiral.

However, standalone, this still operates somewhat linearly: you start with the middle stuff (which is allegedly easier or essential) and progress outward, just like you would at the start of a unit, progressing to more complex concepts. But we make an entire curriculum of concentric circles and rotate from concentric circles cluster to concentric circle cluster every few days, or even in a week, potentially moving outward from the center of each set of concentric circles along the way.

There are two Moving Parts here, which probably should be addressed individually, but I’ve mashed together, either like a fluid Girl Talk album or Frankenstein’s Monster, take your pick.

• Moving Part 1: Constructing units as concentric circles
• Moving Part 2: Rotating through and revisiting topics

That said, I’m not sure you could do Moving Part 2 without doing Moving Part 1. We probably need a name for this type of Scope and Sequence / Curriculum Map: Circular Curriculum Mapping? Iterative Curriculum MappingArchimedean Spiral Curriculum Mapping?

This is getting a little mad-scientisty, I realize. Still, this may have a few potential benefits.

1) Students get to revisit a general topic every few weeks, rather than a one-and-done shot at learning a concept.

2) Students have time to “forget” algorithms and processes and when they see a scenario they have to fight their way through it accessing prior or inventing new knowledge, rather than relying on teacher led examples. Yes, I consider this a benefit.

3) Teachers may formatively assess more adeptly.

4) Students may see math as a more connected experience, rather than a bunch of arbitrary recipes to follow.

5) It probably better reflects the learning process, which happens in fits and starts, and frankly, cannot be counted upon to be contained within a specified time frame.

Personally, I find this framework compelling to a point. I think it better exemplifies recent research and advocacy toward math education. It certainly is messier than a linear approach to curriculum mapping. Your syllabus could potentially look elegant and beautiful or ugly and convoluted. Your administrator might back you, she might not. I’m guessing if you were forced to follow a district scope and sequence, or your math department wanted to be teaching the same things at the same time this would be a non-starter.

So this is just a sort of framework I’ve been playing around with, mostly in my head and I thought I’d throw it out there. I haven’t really developed anything useful. I’d be interested to hear your thoughts. How would you feel about a framework such as this? Do you think it adheres to best practices around mathematics instruction? Would this just work to create more confusion within students? Just how impossible would this be to develop in a public school? Maybe some math departments or curricula are already doing this or something like this? And if it does adhere to best practices and it isn’t implementable due to external constraints, then there may be additional implications for a teacher, school and district. For now, we’re just trying things on. And possibly tearing things apart and starting from scratch. Again.

## Kicking things off: How do I start the facilitation of a problem?

So you’ve decided to undertake inquiry-based learning. That’s great. I’m really glad you see the inherent value in having students swim through a challenging problem on their own a bit before the teacher jumps in with instruction. I’m also glad you’ve been creative at creating new mathematical tasks with cool entry videos, perplexing pictures, and solid scenarios. Looks like you’ve got your curriculum mapped out, all ready to go for the 2013-2014 school year. Really, you’ve done incredible work this summer as you’ve restructured your curriculum with the help of awesome, engaging tasks from the MathTwitterBlogosphere. It’s fantastic. You’ve come a long way. You’ve shown tremendous agency.

Now what? We’ve got all these nifty tasks tied to standards, but what do we actually, you know, do with them? Sadly, even though we’re all rowing the same direction with regards to inquiry based learning and complex mathematical task driven learning, your students are (probably) not at the place where you can just say “GO!” and they’ll spring in to action. Facilitation needs to happen. And while it’s great to have a protocol like the Know/Need-to-Know process (below) handy, if you’re doing 3-5 tasks per unit, any single protocol, no matter how effective, can get pretty boring after a couple rounds of it. While I do believe in giving students the power in common language, it needn’t be that common.

Here are a few ways facilitate the transition from the entry event (the artifact or problem scenario that launches the task) to the student work time.

1. The Know/Need-to-Know Process (NTK).

I’ve blogged a bit about the NTK process before. It’s certainly my go-to protocol. It works well when deconstructing longer (or wordy) problem scenarios. It’s got its problems though. If you’re not adept at facilitating the protocol or just leave it to the students to fill in some blanks, you’ll get some pretty crappy Need-to-Knows, heavy on logistics (when is it due?) or worthless Next-Steps (teach us how to do the math in this here problem). The point of the NTK process isn’t to establish how many words are in a written task, it’s to aggregate prior knowledge and begin brainstorming solution strategies.

2. #anyqs

The good old “Do you have any questions?” “protocol”. Certainly one of the more fun ones. I’d suggest having students jot down their questions before aggregating them as a class. Dan Meyer does a nice job of this by adding “+1’s” when there’s a repeat question. Ideally you’ll have an overwhelming majority of students asking the same question.

2 1/2. Related: Jeff  de Varona (@devaron3) does a nice bit about “what do you think I’m going to ask you?” after producing the problem scenario. I’ve never done that but it seems on point to me. Here’s an entirely stolen-and-published-without-permission of one of Jeff’s worksheets that has that little nugget in there.

3. Visible Thinking Routine: See, Think, Wonder (STW)

Also similar to #anyqs, but slightly more structured, STW was developed as a way of interpreting and discussing works of art, which, if you’ll allow me to opine, ought not to be so different from math problems. It also has the added bonus of adding a layer of evidence-based things the students notice about the picture or video that #anyqs sometimes lacks. Students observe an artifact and discuss what they see, what they think about what they see, and what it makes them wonder. This protocol also works well when having students peer-evaluate each other’s work.

4. Estimations

I think many of us know the power of having students put some estimations up before launching in to the problem. See, they know how to do it over here.

5. Have students develop a concept map before they begin working on a solution.

This is a nice way of having students recall previous lessons and mathematical knowledge. It also helps to bin those logistical need-to-knows that often muck of the NTK process (above).

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So there are 5 (and a half) quick ways to move from that awesome, engaging entry event of yours into actual mathematical work. What are some additional protocols or structures you have in your classroom to elicit mathematical strategery?