## Problems: then a miracle occurs

This is a post in the ongoing Emergent Math mini-series: Routines, Lessons, Problems, and Projects.

Ah problems. I have to reveal my bias here: I love problems. Problematic problems. Problems are where I honestly cut my teeth as an educator. If you’re reading this blog, might have stumbled across my Problem-Based Learning (more on that specifically in a second) curriculum maps. I’ve blogged about Problem-Based Learning (PrBL) a bit. I’ve learned so much from teachers and math ed bloggers about what makes a good problem, how to facilitate a problem, what kinds of problems are out there. Some of that I’ll share here. Let’s just start with Problems.

The questions on voluminous review packets? Not problems. My first resource on problems, problem-based learning, and problem solving is NCTM’s research brief on problem-solving, Why is teaching with problem solving important to student learning? (2010). In it, it hints at the “what really is a true problem” question:

Story or word problems often come to mind in a discussion about problem solving. However, this conception of problem solving is limited. Some “story problems” are not problematic enough for students and hence should only be considered exercises for students to perform.

This brings us to my personal, current definition of Problems: Problems are complex tasks, not immediately solvable without further knowhow, research, or decoding of the prompt. Problems can take anywhere from one class period to three or four class periods.

So when I say “problems” I mean problems that are genuinely challenging to the problem solver. Even the difficult, toward-the-end-of-the-section questions may not be problematic enough for some students. Also, a problem ought not to be so obtuse or convoluted as to not be accessible for all students. Just because something is real hard doesn’t necessarily mean it’s a problem. If someone were to challenge me to make the U.S. gymnastics team, I wouldn’t consider that a problem; I’d consider it futility.

I like to think of good math problems like this: a good problem is accessible enough so students a couple grades lower can attempt it, yet challenging enough so students a couple grades above have to think about it. I actually think this of all mathematical tasks, but it’s particularly apropos of problems.

Here’s a good problem (from Illustrative Mathematics):

I like this problem for many reasons. One, it combines two not-often mathematical things: lines and quadratics. In most curricula you have your unit on linear functions and your unit on quadratics. Why aren’t these two things combined more often? I have no idea. Most textbooks presents lines in one unit and quadratics in another, as if they’re in a different universe. It’s like we’re reading a geography textbook about pre-Columbian South America and Europe. But back to our discussion of problems, it’s the confluence of these concepts that makes this such an interesting, challenging, and worthwhile problem.

There’s straight up problems – just give students a prompt and facilitate as you see fit.

There are countless other modes of problems, here are a few.

• Would You Rather? problems

I’m not sure of John Stevens is the first “would you rather” problem designer, but he certainly codified it with his stellar website. A Would You Rather (WYR) provides students two possible choices and students must decide which one makes more sense to choose: which one is cheaper? which one is better? what deal gives the greatest value? etc.

There are several things that make this format incredibly appealing: 1) Providing students an initial choice naturally facilitates guesses and estimates at the beginning of the problem. 2) Making it a choice makes CCSS.MP3, making arguments and critiquing the reasoning of others, a necessary part of the task. 3) In many cases, either answer may be correct, depending on how it’s interpreted, the desired outcome, or the input variables (in the WYR above, the answer may depend on how far away one is from the airport, how much airport parking is, etc.). And 4) there’s something delightful about the “would you rather” framing. Maybe because it reminds me of the “what’s worse?” scene from So I Married an Axe Murderer.

Dan Meyer gave us this format years ago and countless of math teachers have built upon it sense. Following the narrative structure of movie, in act 1 the “conflict” is established and we’re drawn into the plot of the movie/problem. In act 2, our hero / students go questing for the solution. In act 3, we come to a resolution.

Most often these act 1’s kick off with a video or picture to pique the interest. What do you notice/wonder? What do you think will happen? In act 2, students will work through the scenario presented in act 1, sometimes provided with additional information or knowhow that might be useful to solve the problem. In act 3, students make their final answer and we come to some sort of resolution (often by playing the last part of the video).

Dan has the most comprehensive list of 3-Acts, but others have followed suit with their own libraries.

I’m sure I’m missing others. Please let me know in the comments who I’ve missed.

Like WYR, there’s something inherently appealing about a narrative structure that we’re already used to. We’ve all seen movies, plays, TV shows, and read books. If you can provide a successful hook, we’ll want to see how the movie ends.

• Just straight up puzzles

While sometimes challenging to align directly to required content, give students mathematical puzzles. NRICH has a great library of puzzle-like maths, or perhaps maths-like puzzles.

And I don’t know if the authors (or you) would consider these puzzles, but I quite enjoy the tasks from Open Middle as puzzle-esque math.

Problem-Based Learning

Let’s take a slight birdwalk into the practice of Problem-Based Learning or PrBL. It uses problems as a means to teach new concepts or knowhow. The problem creates a need (and in the best cases, a desire) that requires the intended content knowledge, additional information, or mathematical dispositions.

I suppose in some ways this may not differ much from just giving students the problem and teaching as-needed, as you go. In PrBL there’s an intentionality (and even predictability) with how the problem is posed and how the learning is facilitated (for instance, you prepare the lesson beforehand, rather than just winging it).

Facilitating a problem

One of the biggest mistakes teachers make when using Problems for the first time is that they think that by posing a clever enough problem, students will intrinsically work their way through it dilligently, testing out different methods along the way. And to be sure, it’s understandable to think that when you watch a presentation on problem solving in math or participate in a conference session and the participants or audience dilligently work their way through a problem. But here’s the dirty little secret about conference sessions: the audience is entirely composed of adults who are excited about math and presenters are showcasing their absolute best problems. It’s easy to present engaging problems as a panacea when the audience is entirely bought in and the presenter gets to cherry pick which problem or lesson he or she gets to present. So it’s easy to walk away from these experiences thinking that – just like in that session – I’ll present this super-cool problem to my students and they’ll collaborate, problem-solve, and stick to it just like at that conference.

It’s never that smooth. Rather than – like Carrie Underwood – letting “Jesus Take the Wheel” – you need to keep your hands on the wheel and your foot on the pedal (and sometimes the brakes as well). Problems should be facilitated, not tossed in like a hand grenade. So how do we facilitate a problem?

Use routines. The biggest tip I can provide for facilitating problems is something we’ve already covered in this mini-series: provide routines. Routines to get started on the problem, routines to facilitate discussion in the middle of a problem, and routines when students are sharing their solutions.

Consider this sample Problem facilitation agenda:

• Introduce the problem
• Facilitate a Notice & Wonder routine
• Identify next steps and let students begin working
• 20 minutes later, take a quick problem time out and have groups do a gallery walk routine to see how  and what other groups are doing
• Give a problem “time in” and have students continue working toward a solution
• After finishing the problem, have students show appreciations to one another via a routine.

One problem, three routines. And who knows? If students are struggling, you may want to hold a small workshop lesson in there as well. We’re starting to see our Routines, Lessons, Problems, and Projects framework become a set of nesting dolls.

Provide consistent group roles. Assuming students are working in groups, provide consistent, well-understood group roles.

And – like the problem itself – don’t just provide the group roles and hope for the best, check in with them and how they’re operating. Mix them up. Talk with them.

• “I’d like the Recorder/Reporter from each group to meet with me at the front of the class for five minutes to discuss your progress.”
• “Harmonizers – at this point give one of your teammates a compliment.”
• “I’d like all the Facilitators to swap groups for the next ten minutes.”
• “Resource Monitors – come up with a question as I’m going to go to each group and you can ask me one question.”

Use these roles, don’t just assign them.

Make Problems the cornerstone of your class

Quality problems won’t be the most often employed mode of teaching in your class, but make them the essential thing that students do in your class. Rich problems make for excellent assessment artifacts. They help teachers find the nooks and crannies of what students can do and know and what gaps in understanding still remain. They foster mathematical habits in a way that lessons and routines often can’t.

To be transparent, part of the reason I began thinking about this mini-series is because I was wrestling with the question: what’s the “right” number of problems to facilitate in a school year? And what are those problems? That’s when I began to think of the music mixing knobs analogy from my intro post.

There are endless ways to facilitate problems – use routines early, often and throughout a problem. Use Problems often and throughout a class. They are the bedrock of your class, and the discipline of mathematics more broadly.

Also in this mini-series:

## Counting Idling Cars: An Elementary Math Project Based Learning Unit

I’m sitting in my car, waiting to pick up my son from school. It’s too cold to wait outside  this time of year so I keep the heat on, the engine running, and continue listening to the Dunc’d On Basketball Podcast, the nerdiest podcast about basketball out there. I’m also quite anti-social, so I prefer to sit in the car, rather than, like, talk to people.

The driver of the car in front of me is doing the same (presumably, minus the podcast listening), ditto for the car behind of me. Maybe they’re reading “The Pickup Line,” an e-mag specifically for parents who sit in the car, waiting to pick up their kid from school. It occurs to me: boy there are a lot of cars idling in front of the school right now. I’d guess about 40. But y’know, someone should really count these up.

I get typically get to the school about 10 minutes before the release bell rings and I’m sort of in the middle of the pack of idling cars. I’d guess it’s about the average of when most cars arrive, again, most of which are idling. While I don’t conduct this environment-destroying practice all year long – when the weather is nice I’ll get out and check my phone, rather than talk to other parents – I practice it for maybe half the school year. That’s about 80 days or so.

80 days x 10 minutes of idling per day. Boy, 800 minutes of idling seems like a lot doesn’t it? And if there are indeed 40 cars at my son’s school, averaging a similar amount of idling time, we’re looking at 32,000 minutes of car idling. That’s over eight days of just idling.

We have a train that goes through town and we have signs encouraging us to turn off our car, rather than sit there idling, while we wait for the train to pass through. And I sometimes follow that instruction! I should probably follow it more often and more aggressively. But what about idling in the school pickup line? Or along the side of the school for us anti-socialites?

• How much gas are we wasting?
• How much Carbon Monoxide are we putting in the air?
• How much gas waste / CO is the entire town/state/country contributing?
• Would it be better to just switch off the car and start it later?

Boy, oh boy, someone oughta do the research on this…

What about at your school? How much gas is wasted in a day, week, or school year? Could students do the research? Could they create an awareness campaign for reducing gas waste (and presumably promoting cleaner air at their school)? Seems like something a bunch of go-getter students could handle.

If this scenario interests you and these questions intrigue you, consider adapting it to fir your school.

Update 8/25/2019: Stay tuned on this post or subscribe to follow ups. I like this project idea so much I’m going to be developing it and adding resources and sample project calendars.

## 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)

This slideshow requires JavaScript.

===

Facilitation

• 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:  https://www.desmos.com/calculator/y1qkrfnsw2
• 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.

Optional: For the technologically inclined, you may wish to use Desmos. (https://www.desmos.com/calculator/y1qkrfnsw2)

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

https://www.desmos.com/calculator/6v0rb28bsl?embed

## 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.

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.

## 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?

If 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.

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.

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.

## 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.

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.

## On designing tasks to elicit questions

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

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

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

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

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

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

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

From (Harel, 2008):

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

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

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

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

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

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

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

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

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

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