Blog Posts

Stop Thief!, The Fugitive and introducing equations of circles

When I was a kid, we had this super high-tech board game called Stop Thief!. The gist was this: someone committed a crime somewhere on the game board, which was rife with jewelry displays, unattended cash registers and safes. Your job as the detective was to identify where the thief was. The location of the thief was tracked by a phone looking device that calls to mind those old Radio Shack commercials with car phones. After each turn, the invisible thief would move some number of spaces away from the crime scene. The phone made these noises indicating where he could be – opening a door, climbing through a window, breaking glass. Based on these clues and the number of turns that elapse, you’d try to identify where he was.


Fast-forward a few years. We all remember this scene from The Fugitive:

These are the artifacts that were going through my head as I designed this lesson, linking the pythagorean theorem and equations of circles. In it, students must overlay a circle to establish a “perimeter” (side note: shouldn’t Tommy Lee Jones have used the term “circumference?”).

While this task only starts from the origin, you could quickly modify it to have other starting points, which would allow students to explore what the equation of a circle looks like when you center it wound non-origin points. I’d expect that to occur the next day or later in the lesson as part of the debrief.

Feel free to tweak it to make it better. The desmos graph is linked below, along with a couple word handouts.


(Note: a version of this task will appear in my forthcoming book from Stenhouse Publishers, Necessary Conditions.)

The set-up: a crime has been committed and it’s up to the students to establish a perimeter based on how much time has elapsed. After using the pythagorean theorem a few times to identify buildings the thief could be hiding in.

PrBL - RunningFromTheLaw-01

Given the time that’s passed and typical footspeed, the criminal could be anywhere up to 5 kilometers from the crime scene.

Which of the buildings above could he be in?

[Desmos Graph]


Additional resources:

Running From the Law

Running from the Law Student Handout

Necessary conditions: understanding groupwork with a three-legged pedagogical framework

At some point this year (2018), I’ll have a book for you to read from Stenhouse that proposes a framework for effective math classrooms. These are the three broad ingredients that create a successful math classroom as well as how a student experiences math. They are:

  • Academic Safety – the social/emotional state of a student and her self-regard as a mathematician
  • Quality Tasks – the thing that students are doing, working on, and/or creating
  • Effective Facilitation – the short- and long-term moves that promote mathematical thinking and sensemaking

Every successful math classroom I’ve been in has each of these three hallmarks in spades. In fact no successful classroom I’ve been in hasn’t had each of these hallmarks working for it. They’re our necessary conditions for great classrooms.

They work independently and in concert and can be the lens through which we can better understand classroom issues. Let’s take a common issue of unproductive or inequitable groupwork. Effective strategies will tackle one, two or all three of these elements. Let’s use this framework to better understand the issue, before we jump into the solution.

Fig 1-2

Is the issue one of Academic Safety?

Students may not be engaged in groupwork if they self-identify as a “non-math person.” It’s possible they’re only living up to the social academic status they’ve been given. How do students see themselves as mathematicians? Do they see themselves as mathematicians? Do their peers see one another as mathematicians? If so, how so? Are they publicly acknowledging the mathematical smartness of their peers?

Is the issue one of Quality Tasks?

I’ve been in classrooms where the issue around groupwork began with the fact that students weren’t being assigned groupworthy tasks. If you’re going to require an assignment occur in a group setting, the task ought to require (or at least be enhanced by) groupwork. Tasks are often developed for individuals but assigned to groups.

Is the issue one of Effective Facilitation?

How was the groupwork time introduced? Did you just assign the task and say “go” or did you have a structure in place? Do students have specific and understandable roles or is it the onus of the student to figure out where they fit into the groupwork dynamic? What norms are present in your classroom (and no, not the norms that are on the wall, but the ones that are actually present)?

Once we start to answer some of these questions, we might be able to better identify potential solutions. Maybe the classroom needs defined group roles. Maybe a norm of “same problem, same time” needs to be enacted. Maybe tasks need to be developed to push students deeper into the math content. Or maybe it’ll generate additional questions or additional need for understanding the issue at hand.

And as I mentioned, it’s possible (probable) that issues will bleed between our three pedagogical elements. Certain tasks can reinforce messages about mathematical self-regard. Unstructured groupwork can reinforce issues of academic status. It’s messy work, this teaching. Hopefully this framework will help you better understand the complex dynamic of a classroom ecosystem.




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.

Why we teach the “other stuff”


“I don’t know what to say.”

“I don’t know how to talk to him.”

I’m sitting in a coffee shop with my back facing a mentor and her mentee, a college student who is apparently struggling through her semester. I can hear them clearly, even though I’m trying not to eavesdrop. The mentor is pleading with her mentee to email one of her professors to get help with an assignment, or even figure out when office hours are. “I don’t even know what to say!” the student response. The mentor patiently describes what questions to ask and how to start off the email, to no avail. “I don’t know how to talk to him!” she keeps responding. Pretty soon the student starts doing that thing that teenagers do where they start laughing when they’re really uncomfortable . It’s an unintentional defense mechanism employed by nearly all adolescents. She’s embarrassed by her inability to perform a seemingly simple task, so she starts blushing and laughing.

“I don’t want to talk to someone else.”

“That sounds like so many words.”

Later on, the mentor is trying to get the mentee to call the registrar’s office to find some information about something or other. Again, the student giggles that she doesn’t know how to talk over the phone. The mentor clearly cares deeply for her mentee. The mentee is clearly embarrassed at how nervous she is communicating to professionals.


Teachers sometimes scoff when we implore them to teach those other things. Things like communication skills, groupwork skills, self-reflection, these are “soft skills” that don’t appear in our scope and sequence. The state test doesn’t address them, and we have too much content already to cover to worry about these phony-baloney skills. I’m a math teacher, I teach math, that’s what I do. That’s my responsibility.

I can’t speak to the mentee’s instruction, but I’ve seen it frequently enough. Listening to her, she was paralyzed when it came to communicating with other adults, or at all. This is something she did not learn, let alone practice, while in High School. And now it’s preventing her from succeeding at the post-secondary level. Her paralysis wasn’t that she didn’t know her content well enough, it was that she didn’t have the ability to find out how to further her content knowledge. Whether that was technically the purview of her secondary math teacher or not, the responsibility to prepare students for post-secondary life falls on each of her teachers and the system in which they teach.

Does that mean that it is your responsibility – as, say, a math teacher – to teach students how to make phone calls? Or email professionals? Or create a study group? Or manage their time? Or teach all of those non-math skills?

Yes. Yes, it is.

The reason I taught math was because I love math. The reason I taught math the way I did was because I wanted students to grow as communicators and problem-solvers. Thankfully, I taught in a school in which we were unified that these were indeed skills we wanted our students to have when they graduated. I now coach in a model that aspires to make that happen for all students.

In our classrooms, we have students call and set up meetings with mentors in the business or academic communities. We have students stand and defend their work. We assess students on eye contact as well as content knowledge. And we teach them how to do it, not just tell them to and let them flounder. We do these, not in isolation, but as an entire school. In my class, this included these activities, as well as complex math problems that required collaboration, presentation of new and novel ideas, and practice and structures to handle it when these proved exceptionally difficult.


I’m heartened by these mentor-mentee relationships. Whether they were established via a specific program or they grew organically, there’s genuine care there. As exasperated as the mentor was, she showed active, authentic caring. I’m glad the mentee has that support system in place now that she’s in college. She’ll need the skills her mentor offers, as she clearly didn’t receive them in High School.

Vivienne Malone-Mayes and Waco, Texas

Vivienne Malone-Mayes grew up in Waco, TX, a highly segregated community in a highly segregated state. She attended a highly segregated high school where she graduated two years early at the age of 16 so she could pursue Mathematics at Fisk University, where she graduated in four years with a bachelor’s’ degree and another two years with a masters’. She worked as a professor at Paul Quinn College and Bishop College.

In 1961, she applied to take additional courses to begin her PhD work at Baylor University – my alma mater – but was rejected explicitly for her race. Required by federal law, the University of Texas admitted her (Baylor is a private university). Though she was admitted, she was not welcomed. “My mathematical isolation was complete,” she noted as she described her experience being the only female and the only African-American in many of her courses*.

Despite these challenges, Malone-Mayes obtained her PhD in 1966 with her thesis A structure problem in asymptotic analysis. Throughout her education she took part in civil rights demonstrations.

After she graduated with her PhD, she became the first full-time African American professor at Baylor University, the institution which had rejected her from taking courses just a few years prior. There she was voted the most outstanding faculty member by the student congress in 1971.


This is an excerpt profiling mathematicians from my forthcoming book. It’ll be in the appendix along with other famous mathematicians and should-be-famous mathematicians.

I say “should-be-famous” because I’d never heard of Malone-Mayes. And I graduated from Baylor University, where she was rejected from taking classes because of her race and later a high profile university professor. I lived in Waco, TX for four years.

Part of the reason I’d never known Malone-Mayes was because of my own stupidity and probably-desired ignorance. But it’s kind of also on the University, isn’t it? I would have liked to have known of the achievements of this incredible woman while I was there. My classmates would have as well.

Why isn’t a fucking building named after Vivienne Malone-Mayes? Again, maybe things have changed since 2008. I’m sure there is a plaque somewhere on campus while I was attending, but I feel such shame for not knowing about Vivienne Malone-Mayes then and up until just a few days ago. I encourage you to look for what mathematical heroes may have been buried in your community. Please share them in the comments so we may unearth this invaluable, generally unspoken history and these amazing men and women.

For more on Malone-Mayes, I recommend this article from the Waco History Project, where the photo is taken from. See also, Complexities: Women in Mathematics.

*Case, Bettye Anne; Leggett, Anne M. (31 May 2016). Complexities: Women in mathematics.

You are the Name Rememberer


I’m just not good with names: It takes me a long time to remember them and even then I sometimes forget. 

Not anymore. For you are the Name Rememberer. The One Who Remembers Names.

It’s difficult with so many students. These first few months of school it’s hard to get names straight. I’m not a names person. 

It matters not. You will remember their names, Name Rememberer.

I’ve never been good with names. I can remember a face, but names are hard.

Maybe names are hard, maybe they are not. It doesn’t make any difference to you, the Name Rememberer.

I’ll get their names by the end of the month. It’s October, there’s still plenty of school year left.

You will get their names by tomorrow morning 8am. If this means you need to print out flash cards that is what you will do. If it means you need to find their address, drive to their house, wake their house up, and have a 30-minute conversation to remember their names, that is what you will do. That’s what the Name Rememberer does.

But I don’t know their addresses and I don’t have a car.

For every student that is in your class tomorrow whose name you do not know, you will give them free 100’s on assignments for a month. You will give each of them $100 a day until you remember their name. You will do 50 pushups for each student whose name you can’t immediately recall. And you will NOT under any circumstances make that face where you’re trying to think of their name.

But I have these kids that sit next to each other and they really look alike. What if I —

QUIET! You will learn every student’s name and you will learn them all now. Tomorrow morning you will stand by your door, greet each student BY NAME, and welcome them in to your classroom BY NAME.

But – 

SILENCE. You are wasting valuable time, Name Rememberer. Time that could be spent on remembering students’ names. The school day starts in 10 hours and you have names to remember. Or, as the Name Rememberer, do you have them all memorized? You have the proper pronunciations down seamlessly? First and last? There will be no beat skipping when you want to call on a student with his or her hand raised? 

Good. For tomorrow, you – in addition to being the Student Name Rememberer – will also become the What Students are Passionate About Knower. 


Find, Adapt, Create: A Path Towards More Agreeable Task Design Time

In the past I consistently struggled with making the turn from the excitement toward problem-based learning (PrBL) to the actual design of complex, engaging problems. Typically I would spend the morning building the buy-in (the “why”), another part of the morning conducting some sort of problem simulation to showcase PrBL (the “how”). Then my instructions were along the lines of “OK gang, after lunch you’ll start designing your own tasks!” If you’re like me, you find it difficult to be creative on demand*. (I mean, if you’ve been keeping up with the infrequency of my blog posts in the past year you probably know that already).

Don’t get me wrong, I have little patience for math teachers who say “they’re not the creative type.” And I do think creativity is an under heralded attribute teachers need to have. It’s difficult to be creative at gunpoint.

I’ve started codifying what I believe is a more agreeable framework. Many of the successful implementation of inquiry-based, complex tasks has followed this progression (often over the course of multiple coaching sessions):

First: Find

Then: Adapt

Finally: Create

We start by finding (and often trying out) a task; then, at a later date, we try adapting a task (which we then implement); finally – and this is a tall ask – we try out creating a task more-or-less from scratch. This final step is probably more of a slow-walk from adapt than a full on design sprint.



There are countless websites with open accessible tasks of ever-increasing quality and navigability. You know ’em, you love ’em. You can find a bunch on the “Math-like Blogs” list on the right side of this page. You can also find well organized tasks at IllustrativeMathematics, Shell Centre,, and NCTM’s Illuminations.

An afternoon of PD spend simply clicking through your favorite, say, three of these resources is an afternoon well spent. That’s how the ol’ curriculum maps came to be.

Find something compelling and pretty soon you’ll find a ton of stuff you find compelling.


Once you’ve found some good stuff, try to see if you can take something that’s pretty good and make it better. I’ve presented about that before: [NCTM] Adaptation.

This requires a bit more discussion and contemplation. You start to turn from “I like this task” to “What do you like about it?” Once we start adapting we are developing an implicit or explicit criteria for what makes a quality problem.

Maybe you adapt a problem by removing the sub-steps. That would suggest you like problems that allow for a lot of “open middleness.” Maybe your colleague adapts a problem to a hands on activity a la Fawn. That speaks to how much you value tactile experiences and students actually doing stuff.

Now – and only now – ought we turn to the ever challenging work of creation.


Most of the time, creation of a task comes from either inspiration and/or sheer luck. I’ll see an advertisement or watching a movie and see something that’s kinda mathematical. Like I said, really tough to do on-demand, and also really tough to do in any kind of standards-aligned way.

But it’s also absolutely crucial! Not only does it work out your creative muscles, it generates tasks for the rest of us to find! It’s a give-a-penny / take-a-penny situation. Even if you’re not teaching, say, geometric constructions in your Algebra 2 class, maybe you get struck by a divine lightning bolt of inspiration that the rest of us can draw on. In that respect the Find –> Adapt –> Create framework could be seen as a cycle.

Find –> Adapt –> Create –> Other people find your creation

But yeah, it’s difficult to do on a good day, it’s much more difficult to achieve when I’m hovering over individuals harping on them: “got anything yet?

And this isn’t just true for tasks. Consider other instructional tools.

  Find Adapt Create
Rubrics NTN Learning Outcome Rubrics (Math) Pull from a few of the rubric indicators Design your own, based on your grade level, school context and content area
Lesson Plan Template Problem Planning Form Modify based on your class time Design a lesson plan template that works for an entire department
Math attitudes survey Here’s one I developed Steal a bit from it, but identify a few of the specific things you’re trying to deduce On your next iteration, make it totally your own!

I do believe that the best instructional experiences students have are by-and-large teacher-designed. Getting to that point is challenging so start with the stuff we have and slow-walk yourself into creation mode.

Y’know, unless inspiration strikes you like a lightning bolt while you’re sitting on the couch. In that case, disregard everything I said and go nuts, Creator.

* – Note: to contradict myself, this was not true of PBL Chopped! That was absolutely a fantastic experience of solely creation with incredible project ideas.