Blog Posts

2017 New Tech Network PBL Chopped Recap

I had the honor of co-designing and MC’ing the first ever PBL Chopped competition at the New Tech Annual Conference in July. While this is typically a blog about math instruction, this experience welcomed all content areas and all grade levels, teachers, principals and instructional coaches. It was an absolute blast and the teams were incredible. It’s so fun to be an observer to this cross-curricular design sprint. Below is a recap, followed by a link to additional resources and commentary – geoff

ST. LOUIS, MO — The only thing more intense than the current of the Mighy Mississip was the sweltering PBL Kitchen. Eight teams entered the Steelcase room on an overcast Saturday afternoon hoping to design the “tastiest” of projects based on three mystery “ingredients.” These ingredients came in the form of three randomly selected standards from across the curriculum.

Each team had 20 minutes to design the outline of a PBL unit based on the following standards which were drawn at the time of the competition.

  • Natural selection leads to adaptation, that is, to a population dominated by organisms that are anatomically, behaviorally, and physiologically well suited to survive and reproduce in a specific environment. (HS-LS4-3),(HS-LS4-4)
  • Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. (CCSSM.A-CED.A.4)
  • Integrate visual information (e.g., in charts, graphs, photographs, videos, or maps) with other information in print and digital texts. (CCSS.ELA-LITERACY.RH.6-8.7)

As soon as the standards were drawn, the first round of the competition began in a flurry of ideation and activity. Teams scribbled on whiteboards provided by Steelcase. Participants. The end of the 20 minutes was punctuated by a “3…. 2….1… Markers down! Cooking time is over, chefs!” Exasperated from the intensity, participants – teachers and principals alike – let out a cry of triumph. While the design of a PBL Unit – with carefully selected and uniform standards – can take hours, competitors had come together to create an engaging, meaningful project in 20 minutes.


Sadly, only three teams could advance to the final round. This proved to be the most difficult part of the competition – for the judges. After some deliberation, looking through the eight project ideas, they identified the three projects that allowed their creators to advance to the final round. At this point, standards were drawn again:

  • ELA: Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks, attending to special cases or exceptions defined in the text.
  • SS-Geo: Use geospatial and related technologies to create maps to display and explain the spatial patterns of cultural and environmental characteristics.
  • Sci: The complex patterns of the changes and the movement of water in the atmosphere, determined by winds, landforms, and ocean temperatures and currents, are major determinants of local weather patterns.

As before, 20 minutes flew by in a flash and the three teams developed PBL Units that would be uniquely engaging and meaningful to their students. They presented to the judges who asked them critical questions to get a sense of the scope of the project and the team’s understanding of the project.


After two rounds of high pressure, minimal time PBL design one team was left standing. The judges – after much deliberation and pained conversations – came to the unanimous conclusion that Scottsburg New Tech‘s final project was the one to take them over the finish line. They were announced as the winners at the next morning’s plenary session.

But while Scottsburg got to take home the lovely 2017 NTN PBL Chopped Trophy (which we can only assume is currently being prominently displayed in a proper trophy case), everyone who competed won something. Some teams walked away with a greater understanding and empathy for teachers of other disciplines. Some teams walked away with a greater camaraderie with their peers. Other teams walked away with an actual project that they promised to refine and implement in a cross-curricular experience.

So congrats are in order to the winner, Scottsburg New Tech, but given the creative explosion in the Steelcase room on that Saturday afternoon, congrats are also in order to the thousands of students who will benefit too, from PBL Chopped 2017.

(Note: in addition to the recap here, you can find additional information, commentary, and details on the New Tech Network blog. [link coming soon!])



Active Caring

I’m in awe of my son’s 3rd grade team.

Last year, when he was in 3rd grade, he had two teachers, a counselor, a GT specialist, a principal, and three specials teachers that cared for him. Not in a passive way, like a “my door is always open” kind of way. But in an active, give-him-hugs, come-to-multiple-parent-teacher conferences, “let’s figure this out” kind of way. One kid, eight adults just pouring love onto him. And he’s a kid that needs outward expressions of love.

As he enters fourth grade, he’ll have a lot of the same adults in his life next year as he did this past year, thankfully. I’m confident in fifth grade he’ll get that same level of care as well. Consider this blog post a partial paen to elementary schools that get that it truly does take a village.


A while back I critiqued myself for not writing about Social and Emotional Safety as I do Tasks and Facilitation. If I’ve evened out the ratios of those three elements, it’s only because I’ve written less on the blog in aggregate in the past year.

Nevertheless, I have been writing. And much of that has been about caring.

You see, we secondary teachers think we’re caring. I told kids they are welcome to hang out and talk after school. I left an open invitation for kids to come before school to get work done. I invited all students to participate. I said “good morning” at the beginning of the day. Shoot, I even greeted every kid with a handshake at the beginning of every class period. And, with a few students, I truly did have that special relationship such that I made an impact on their lives.

I showed passive caring. I opened the door and beckoned kids to come through.

Elementary teachers – at least the ones my son has – they show active caring. They open the door, beckon kids to come through, and when they don’t they’ll leave their room, grab them by the arm and bring them in. They don’t just invite kids to participate, they demand  it and make it a norm in their classroom. They don’t just say “good morning” at the beginning of the day, they hug my son, ask him how his dance class was, give him a specific word of encouragement, and then give him another hug.


I worry about the level of care it’s realistically possible to show a kid as he or she progresses through middle and high school. While it Takes a Village, kids graduate into an assembly line. They get 50 minutes with a Social Studies teacher, then they move down the assembly line to obtain their Science Parts. How many days in a row can a kid go without being shown care? I’ve talked to students who feel “invisible.” I’ve shadowed students that don’t get called on for an entire day.

Fig 1-2

I write about this in my book that will eventually be written, if I’d ever stop using the damn passive voice (see: this sentence). I was talking with a 10th Grade teacher who suggested, “maybe you should just call that chapter ‘Give a Crap.'” I laughed. And she’s right! While I don’t think I can get away with that title for a section (let alone a chapter), that’s probably the biggest differentiator between effective classrooms and ineffective ones.

Becca may have put it best:

I know it’s incredibly challenging for secondary teachers, who have limited time and expansive content to get through and, like, hundreds of students. But I’ve seen secondary teachers do it. And it’s absolutely a beautiful scene to behold, if only because it’s relatively rare in the hustle-and-bustle of the secondary learning ecosystem. I encourage you to show active caring, rather than just passive caring this year. Because that third grader who needs a word of genuine encouragement, still might need one in 10th grade.


Post-It Problem: Grades 2-3

If imitation is the purest form of flattery, then Graham should be pretty darned flattered. I imitated (read: stole) his The Big Pad problem for slightly younger grades. Graham’s task necessitates fractions, which was a bit further down the line for my intended audience, roughly grades two or three. In this task, the giant Post-It is 15 inches x 15 inches and the small Post-Its are 3 in x 3 in. Enjoy!

Screen Shot 2017-04-21 at 2.16.34 PM.png

(Coming soon: a 8 in x 6 in Post-It Problem for grades 4-5, with additional commentary)

Act 1


  • Watch the video. What do you notice? What do you wonder?
  • How many small post-its will it take to fill up the big post-it?
  • What do you know? What do you need to know in order to solve the problem?

Act 2

Post-It Problems 15x15 Act 2.png

Post it Problems Act 2 Post its.png

Act 3

Systems of Linear Inequalities: Paleontological Dig

(Editor’s note: the original post and activity mistook Paleontology for Archaeology. Archaeology is the study of human made fossils; paleontology is the study of dinosaur remains. The terminology has since been corrected and updated. Thanks to the commenters for the newfound knowledge.) 

Here’s an activity on systems of inequalities that teaches or reinforces the following concepts:

  • Systems of Linear Equations
  • Linear Inequalities
  • Systems of Linear Inequalities
  • Properties of Parallel and Perpendicular Slopes (depending on the equations chosen)

In this task students are asked to design four equations that would “box in” skeletons, as in a paleontological dig.

DOC version: (paleo-dig)

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  • Give students the entry event and instructions. Have one student read through it aloud while others follow along.
  • Consider getting started on the first one (Unicorn) as a class. Should our goal be to make a really large enclosed area or a smaller one?
  • Students may wish to start by sketching the equations first, others may chose to identify crucial points. Answers will vary.
  • If you have access to technology, you may wish to have students work on this is Desmos. Personally, I prefer pencil and paper. Here’s the blank graph in Desmos:
  • For students struggling with various aspects of the problem , consider hosting a workshop on the following:
    • Creating an equation given a line on a graph
    • Finding a solution to a system of equations
  • Sensemaking:
    • Did students use parallel and perpendicular lines? If so, consider bubbling that up to discuss slopes.
    • Who thinks they have the smallest area enclosed? What makes them think that? Is there any way we can find out?
    • Let’s say we wanted to represent the enclosed area. We would use a system of linear inequalities. Function notation might be helpful here:
      • f(x) < y < g(x) and h(x) < y < j(x) (special thanks to Dan for helping me figure this notation out in Desmos!)


=== Paleontological Dig ===

Congratulations! You’ve been assigned to an paleontological dig to dig up three ancient skeletons. Thanks to our fancy paleontology dig equipment, we’ve been able to map out where the skeletons are.

Your Task: For each skeleton, sketch and write four linear functions that would surround the skeleton, so we may then excavate it.

Check with your peers: Once you have it, compare your functions to your neighbors. Their answers will probably be different. What do you like about their answers?

Optional: For the technologically inclined, you may wish to use Desmos. (

Challenge: What’s the smallest area you can make with the four functions that still surround each skeleton.


Is bad context worse than no context?

In elementary classes we consider it a good thing to be able to move from the abstract to the concrete. We ask students to count and perform arithmetic on objects, even contrived ones. We ask students to group socks, slice pizzas, and describe snowballs. A critical person might suggest these are all examples of pseudo-context, and they’d be right! These are more-or-less contrived scenarios that don’t really require the context to get at the math involved. Why do we provide such seemingly inessential context? I’m venturing a little far away from my area of expertise here, but I’d guess it’s because it helps kids understand the math concept to have a concrete model of that concept in their heads.

My question is this: in secondary classrooms, is there inherent value in linking an abstract concept to an actualized context? Even if the context is contrived?

I mean, yeah that’s bad. Comically and tragically bad. It doesn’t do anything to enhance understanding. I’d say the context actually hinders understanding. The thickness of an ice sculpture dragon’s wing? That’s about three bridges too far.

But what about a slightly less convoluted, but also-contrived, example. Say:


This problem is still most certainly contrived: dimensions of tanks aren’t often given in terms of x. I’m not even suggesting this problem will engender immediate, massive engagement, but it might help students create a mental model of what’s going on with a third degree polynomial. Or at least the context allows students to affix understanding of the x- and y-axes once they create the graph of the volume of this tank.


We provide similarly pseudo-contextual in elementary classrooms in order to enhance understanding of arithmetic and geometry.

From Burns’ “About Teaching Mathematics”

Of course, we also compliment such problems with manipulatives, games, play, and discourse, which secondary math classrooms often lack. In the best elementary classrooms we don’t just provide students with that single task. We provide others, in addition to the pure abstract tasks such as puzzles or number talks.

Perhaps the true sin of pseudo-context is that it can be the prevailing task model, rather than one tool in a teacher’s task toolbox. In secondary math classrooms pseudo-contextual problems are offered as the motivation for the math, instead of exercises to create models and nothing more.

(See also: Michael’s blog post on Context and Modeling)

What does it mean to be problem based? An attempt to unwind “PrBL.”

Despite an increased awareness of this thing called “Problem-Based Learning,” (PBL/PrBL) there’s some nebulousness in what that word “based” means. Does it mean that students learn content within a problem? Does it mean students are honing their problem solving skills?

1If one were to ask me “what makes a lesson problem-based?” I honestly don’t have a great, specific definition at hand. To me, I think of a problem based lesson a thing where students are given a complex problem and they have to solve it. In the middle though, all kinds of wacky things happen: new learning is acquired, old learning is readdressed, information is researched, attempts are made at a solution.

That wacky middle is difficult to capture and package in a PD session, a conference talk, or even a modeled lesson study. Consider this an attempt to unwind a loaded term.

There are three ways in which one can deliver a “problem based” lesson. At least as I’d define it.

A problem in which students need to identify or find additional information in order to solve the problem.



Consider Graham’s “Downsizing Ketchup” 3-Act lesson, and most 3-Act’ers for that matter. The problem is posed via Act 1 and the setup of the scenario (or “conflict” if we’re being true to the 3-Act narrative terminology). A student or teacher may ask about and will need to know the information contained in Act 2. Act 2 yield the information that students need. Ostensibly (and again I should caveat: generally) that should be enough to complete the problem, with possibly side workshops as needed.

A problem in which students need to learn new knowhow in order to solve the problem.


This is the model of lesson under which I tried to teach most often. Like in the previous problem, students are given a problem to solve via an initial event: a video, a letter, an image, or even a straightforward word problem. After some initial brainstorming and pulling apart of the problem, students begin working toward a solution. At some point throughout the student-working portion a need for new knowhow will emerge.

Consider a problem in which the need to solve a system of equations arises. Energy efficiency electronics and appliances work quite well. How about light bulbs? Upon developing a model for both the cheaper, but energy guzzling light bulb and the more expensive, but energy consuming bulb. Upon graphing these, the need arises to solve for this system of equations. When I facilitated this lesson in class, students had not yet learned how to solve a system of equations, graphically or otherwise. We would deconstruct the problem, create a couple models of energy usage and graph them. At this point in the problem-solving process, I’d deliver a quick class lesson on how solve a system of equations. Once I felt like students had the hang of it, I’d turn them back to their light bulb problem and allow them to apply that new knowhow.screen-shot-2016-11-10-at-1-46-02-pm

The thinking is that students learn better when there’s an authentic need to understand, which is what the problem context can provide. I found this to be both highly effective and incredibly difficult. How do you design a problem that necessitates the knowhow? At what point do you take that problem “timeout” to deliver the lesson? I’ve written a bit on that before. But it’s certainly more of an art than a science.

A problem in which students have everything they need and must demonstrate mathematical thinking in order to solve it.


Of course, there are excellent problems that may be given when students generally may not need additional info or new knowhow. Perhaps there are multiple pathways or methods that yield a solution. Consider a “puzzle” type problem, such as Youcubed’s Four Fours or Leo the Rabbit task. These are interesting, rigorous problems that don’t require new methods per se, but rely on a more general notion of mathematical thinking, such as Bryan’s Habits of a Mathematician.screen-shot-2016-11-10-at-1-48-03-pm

I’d also put Fawn’s Hotel Snap in this category. There isn’t any information students need or instruction from the facilitator in order to achieve a solution. But it does require creativity, persistence, and organization, all mathematical skills.


Each of these types of (*extremely academic professor voice*) PROBLEM BASED LEARNING have their time and place, depending on the objective, the standards involved, the students, the problem itself, and teacher comfort level with Problem Based Learning. And even providing these three models perhaps draws unnecessary boundaries between Problem Based Learning and just generally good math teaching and even between each particular model mentioned here. Still, I hope it’s somewhat clarifying, if only to generate additional future conversations.


Math and the Message

“This isn’t right,” she says. “This can’t be right. All my friends got Math 7.”

My soon-to-be 6th Grade daughter is near trembling as she held her the schedule for the upcoming school year. She compares her paper with friends who were both part of her peer group as well has having the last name L through S. This is the day incoming 6th Grade students pick up their daily schedules from the gymnasium. She is at first dismayed that she isn’t in the same class as her friends. This is a bummer to kids, to be sure. But one that we all deal with and are able to handle. However, eventually it dawns on her that she was placed in a different math course from her peer group altogether: Math 6. Plain old, Math 6.

Last year, as a fifth grader, she received the message that she’d be placed in an accelerated math program. Students who were identified as Gifted and Talented were all part of the same cohort and participated in pull-out math throughout the year. There they received enrichment opportunities. She – and presumably her peers that were not part of these pull-out options – knew full well what this opportunity meant: she was one of the smart kids.

It’s in those gifted and talented pull-outs that she made her closest friends. Because why wouldn’t you? These were the well behaved kids. This was the fun class where kids get to play math games. These were the kids who were told that they were uniquely gifted and talented at mathematics.They were X-Men. They were invited to attend Hogwarts School of Wizardry. They all had that in common.

Most of those peers of hers were placed in a 7th Grade math course for the upcoming school year. They were deemed to be far enough along according to a few different metrics such that they could skip 6th grade math in order to take higher math earlier. There are three different metrics the school uses and kids have to excel in two of the three exams, including an “Algebra Readiness Test.” My daughter only excelled in one of the three exams, and not the “Algebra Readiness Test,” which, according to the school counselor, is the one that really matters. She was placed into 6th grade math, which makes sense: she’s a 6th Grader. But that’s not the message she’s receiving today.


Flashback Charles Dickens-style 6 years ago when my daughter brought home the following artifact from her Kindergarten class.


Screen Shot 2016-08-22 at 11.14.33 AM

When I saw this artifact emerge from the trappings of her backpack I was stunned. Where had she gotten this message that she was a “late bloomer” at math? She wasn’t (and isn’t) a “late bloomer” at math in any sense of that loaded word-slaw. And she was in Kindergarten for crying out loud!

Regardless of where the message came from, the next six years were an attempt at combating the stereotype threat associated with being a young female mathematician. Roughly halfway through those six years, she took a test – the CogAT – in 2nd grade which ostensibly identifies Gifted and Talented individuals. She scored well enough to be identified as such in Math and English Language Arts. I was so proud of her, and that may have been a crucial mistake. My thinking at the time was that she had worked hard (she had!) and developed a sense of self-confidence (she had!) in math, as evidenced by the results of this test and teacher observations which placed her in this special cohort of special students.


Every night during dinner our little nuclear family of four have a conversation of questions. That is, one of us asks a question and then we go around the table and respond. This past Saturday, my question is “If you could go back in time to any year, what year would it be and why?” My wife says when she was 10 years old: that was the best age. My son says 1883 so he could warn everyone about the eruption of Krakatoa (what?). My daughter says she wishes she could go back in time to 5th grade and study for the “Algebra Readiness” test so she can be placed in the accelerated math class.

It’s probably worth noting that she didn’t have her Gifted and Talented identification revoked. I’m not even sure that’s possible or legal. Moreover, she is still on track to take Algebra in the 8th Grade, which – as we all know – places you on a track to take Calculus as a Senior in High School. Make no mistake: she’s still on a pathway of being in the “upper track” which feels gross to type on my computer screen. But she’s apparently not in the “upper upper track” which is deeply concerning to her; so much so that she brings it up at dinner as her point of time travel.


Let’s not let me off the hook here: I’d have been pleasantly pleased to have her be in a math class a year ahead, just as I was happy to see she was receiving special pull-out enrichment services in elementary school. I’m all too willing to take advantage of these opportunities from a place of privilege.  I’m not even in favor of the existence of such an accelerated math program, but I’d sure let her be in it given the chance.

This message is almost always unintentional. It is a bi-product of our dysfunctional understanding of the discipline of math that values correctness over effort, memorization over creativity, speed over thoughtfulness. It is also commonly used as tool with which we rank order and separate students. Sure, other subjects bi- and trisect the student body but none quite have the rusty edge that math does.


Math makes everyone feel stupid at some point. For many, it’s early on when you’re not fast enough during Math Minute. For others, it’s not until AlgebScreen Shot 2016-08-23 at 9.45.00 AMra. For others still, college Calculus is the first stomach punch. For many it’s all of the above. For my daughter, it was when she got that schedule and received the message that she was no longer one of the “smart kids.” Of course, the Original Sin was the message that there exist smart kids and not-quite-as-smart kids in math to begin with.

It took six years to communicate to my daughter that she was a brilliant mathematician. We do little Algebra exercises on the whiteboard. We worked through a SuScreen Shot 2016-08-22 at 11.37.52 AMmmer curriculum to keep her brain finely tuned during the summer slog. Together, we once made an instructional video on an iPad to help a co-worker’s friend’s kid. And it took one piece of paper to undo that messaging. Of course, when a structure falls apart that quickly, that’s an indication it was built on a flimsy foundation. I wonder how long, if ever, she’ll re-receive the message once again that she is a brilliant mathematician. Or will we just have to wait for the next round of test results and keep our fingers crossed?