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.


Posted in algebra, linear functions, linear inequalities, problem based learning, systems of equations, Uncategorized | 6 Comments

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)

Posted in commentary, problem based learning, Uncategorized | 7 Comments

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.


Posted in problem based learning, Uncategorized | 6 Comments

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?


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Using August inservice to plan for May

In case you hadn’t noticed, school is starting soon for many teachers and students. Some have already started! Much of teachers’ inservice time is gobbled up by sometimes-helpful, sometimes-not professional development, new school procedures, supply gathering and those other necessities that come along with having a captive staff for perhaps the only time all year. Some of that time is devoted to planning. At that point teachers often scuttle off into their rooms and begin writing lesson plans or sifting through resources on their first unit.

My recommendation this year has been to not (necessarily) just think about that first unit, but rather to think about the year as a whole: What 10-12 problems do you want students to wrestle with at various points in the year? Let’s spend our inservice time finding those 10-12 problems that encapsulate the near entirety of our course, accompanied potentially with a rough indication of when in the school year you anticipate these problems to be deployed. Portfolio Problems, as it were.

zoom calendar.001

Having these portfolio problems identified might just help keep yourself accountable to implementing rich task throughout the year, rather than getting “behind” and feeling like you have to scramble to catch up.

The last few days of summer and inservice can be the last few days a teacher has to think deeply about the structure of their year. I know that I often didn’t take advantage of that fact and focused instead of the first few days and lessons. That approach may have made my first week more planned out in my head, but I’m not sure it did much to make my classroom in February, March and April any better.

So – once your mandatory meetings on hall pass policies have concluded – think about a Top 10 Problems for My Geometry Class and make a little note somewhere about when you’ll deploy them. Maybe you can put it on your Google Calendar. Maybe you can write in in your planner. Maybe you could even put it on your syllabus. Once those are planned it’ll be easier to move outward from there. And your May self from the future will thank you.

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Just how high was the Big Thompson Flood? And how often will a flood like that occur?

Recently, the family and I were taking in an afternoon in Boulder, CO. After taking in a lunch at the lovely Dushanbe Tea Room we took a stroll along Boulder Creek. Right by a retaining wall stands this object.


This monument demarcates how high the waters rise for a flood of various magnitude. Zooming in a bit to the demarcations we see the following (from bottom to top):

Version 2

Version 2Version 3

The thing that makes flood levels so interesting mathematically is that in addition to height, they’re measured in probabilistic time. That is, every 100 years we can expect one flood to reach as high as the demarcation of the “100 Year Level”. Every 500 years we can expect one flood to reach as high as the “500 Year Level” and so on.

So… what does that suggest for the marker near the tippy top of this monument, marking the height of the Big Thompson flood of 1976?

Version 4


Suggested Facilitation

Provide the following (enhanced) picture.


Follow up with your favorite problem kicking off protocol. I’d suggest either a Notice/Wonder or a Know/Need-to-Know.

Potential Questions

  • How high did the water level get during the Big Thompson flood?
  • How often does an event like that happen?
  • How high are these markers off the ground?

For this last one, you’ll probably need some sort of base level unit to measure the heights, for perspective’s sake. Allow me to provide one additional picture.


(Hey, if it’s good enough for Stadel, it’s good enough for me.)

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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.


Posted in culture | 6 Comments