[NCTMNOLA Processing Session 5] Networks and Silos

This will be the fifth and final NCTMNOLA Processing Session. It’ll be short too, just a quick debrief.

I vacillate between the poles of “math is different” and “math is just like other subjects.” Sometimes I wonder if math teachers use its alleged differentness as an excuse to teach it in an overly linear way. On the other hand, it sure seems different, doesn’t it? I’m not convinced either pole is correct, at least not for more than 72 hours at a time. I will say that math does feel especially silo’ed. I mean, here we were at a conference full of math educators and pretty much only math educators. We have our own vocabulary, our own best practices, our own standards of practice, our own conference, our own software. Yet still, we struggle as a profession to do the basics: get students to talk mathematically. Are we too buffered from other disciplines? I have the incredible opportunity to spend significant time with non-math teachers and much of what I do is taken from them: the way I conduct my debriefs, a See/Think/Wonder routine for interpreting works of art, fishbowls, Critical Friends for peer editing and solution review. These are protocols and facilitation moves usually reserved for non-math disciplines, to math’s detriment. 

In general, I wonder about the long-term sustainability of effective math teaching if single teachers are the unit of change, instead of systems. Here is what I mean: Seattle and University of Washington have an amazing system in place to keep their practice of Complex Instruction rolling, even as teachers move on. Most communities don’t have that systemic approach. When they lose a teacher, they have to start from scratch, hiring, professionally developing, and inducting (or shielding) that teacher into (from) the school culture.

However, what we do have is a network of educators online, on blogs and twitter, all the time at our beck and call. Maybe this is our permanent system that will outlast those of us who gracefully exit the classroom. 

Thank you to all my online collaborators that I got to meet in person and all the online collaborators that I have yet to meet.


Previous Processing Sessions:

[NCTMNOLA Processing Session 4] I may have missed my calling

Those most mathematical fun I had at NCTMNOLA was in sessions focused on Elementary School level math. Either by accident or by impulse I found myself drifting into sessions that one would associate with K-5. Well, if that’s the case, why did I have so much fun?

David and Kathryn led us through a gallery walk of sorts of mathematical games for K-2 Common Core standards. Immediately, selfishly, I and Alyssa began trying to adapt them for our contexts. Maybe we should have been more in the moment, and maybe that’s our High School education brains kicking in, where everything is a nail to our math hammer, but manipulatives and games by and large don’t happen at the high school level. A few months ago I had some non-math teachers do a little card matching game with functions and she remarked that she had just assumed that these manipulatives activities were for the kids in Talented and Gifted. It’s understandable where one would get that impression. Anyway, David and Kathryn were great hosts and provided several activities in a relatively short amount of time.


Marilyn Burns is as exceptional in person as you’ve come to know from her books and websites. Her presentation focused largely on her wealth of experience with math talks, presenting video of various methods kids use to mentally solve 99+17. Every time a kid solved 99+17 in a different way my heart fluttered a little bit. This looked like so. much. fun. After watching a few of these videos I wish my entire job was to ask kids how to solve 99+17 all day. As soon as I arrived on my home doorstep I asked Mrs. Emergentmath and emergent kids #1 and #2 to solve 99+17. Why did I feel like this was revolutionary? Why was this so much fun? Was it because these kids were practicing invaluable Algebra skills they will need later in school? That’s probably part of it. It also hearkened back to the idea, first presented to me by Bryan Meyer, of Mathematical Play. Dang, there’s that word again: play.

And speaking of Mathematical Play, perhaps the most fun I had in any single session was Christopher’s Hierarchy of Hexagons. Double points for an 8am start time. Christopher began by having us mentally sorting four different visual polygons into two groups (we hadn’t gotten to the hexagons just yet). Some people chose to sort the regular polygons from the irregular, others chose to sort by reflective properties. Either way, it gave us a good bit of practice until the real fun began. ENTER THE HEXAGONS.



We chose a hexagon that, in Christopher’s words, “spoke to [us].” I chose the one that kinda looked like a fox, or the FiveThirtyEight logo. Brandon had no clue whatsoever.


Anyway, once we chose our hexagon, Christopher asked some participants to describe the hexagon they chose, why, and then led us through some authentic definition-making, that makes the hexagon belong (or not belong) to a category. Participants’ definitive categories included “waffle cones”, “reflectors”, “utah’s”, and more. From there, we created a flow chart/Venn diagrammy thing that showed which hexagons belonged to which categories and which were mutually exclusive. We’ve all done this with quadrilaterals (“a square is a rhombus, but a rhombus isn’t necessarily a square”). But hexagons are (as it turns out) an Undiscovered Country of polygons. It was fun and rewarding to explore those uncharted waters for a while. Here are half of those hexagons.


Explore. Play. I remember playing. I remember learning new things by conjecture, trial and error, sketches, etc. Why was I having so much fun in these sessions that weren’t in my self-defined “wheelhouse?” Did I miss my calling? Should I have been an elementary school teacher? Probably not, but it was fun to occupy that space for a while.

After volunteering in my daughter’s 3rd grade class for a year, and after these sessions, I’m more convinced than ever that we need to blur the lines between elementary and secondary math education. There’s no reason I couldn’t have done the Hierarchy of Hexagons in my Geometry class. There’s no reason I couldn’t have started off with a number talk in Algebra 2 once a week. If only for the fun of it.



[NCTMNOLA Processing Session 3] Summer School is Dead, Long Live Summer School

In between the instant they opened the door to Jo Boaler’s talk on “Promoting Equity Through Teaching for a Growth Mindset” and when she began speaking, I, Eleanor, and a few others in my row banded together as Jo Boaler groupies. We were those people at rock festivals who get to the stage several acts prior to your favorite band to ensure that we are front row, center. And to be honest, it kind of felt like that: like we were waiting for a performance for a favorite musician.

What’s Math Got To Do With It is the first book I recommend to math teachers and parents vaguely interested in math education. Its combination of research-based practices, accessibility, and price make it – in my opinion – the premier text on math instruction. And that was before the Boaler-Dweck tag-team was unleashed in full.

I’ve been thinking a lot about “death knells” recently, things that basically signal there’s virtually no going back. Remediation is a “death knell” for many students. Very few students who are labeled as needing remediation ever get caught up with their peers, fewer still ever exceed their peers. Some of that may be the remedial label itself, much of that is the methods in which these remediation classes are taught.

“These kids are so far behind, we need to do more traditional math to get them caught up!”

That is a sentence uttered by someone who doesn’t understand irony, and yet is the pervasive “methodology” (if one can call it that) for reaching students who are “behind.” The same math that got these kids behind in the first place … but more of it. More packets, more computerized instruction, more “I do/you do.”

I’ve heard enough “but these kids lack the basic skills to do complex math” to last a lifetime. First off, if you ever said “these kids” around Kelly Camak, you would probably never be heard from again. Second, the experience of doing challenging, fun, creative math is exactly what “these kids” need.

Jo Boaler shared a video of “these kids” in one of her Summer sessions. The students in the video persisted on a pattern problem for, according to her, 70 minutes. We saw about 5-7 minutes of three students working on a pattern, doing complex algebra, sharing ideas, and being 100% fully engaged in math. An individual problem packet would have not fostered that level of mathematical engagement.

More striking than the video and the numerical total was this:

Kids that have been told they are remedial know that they’re probably getting rudimentary math. Even if you call the course something cheery like “Gateway to Algebra!”, they know. They’ve been told, possibly by their math teachers, possibly by their peers, that they don’t have what it takes to be an exceptional math student.

Here’s a quote from Ilana Horn’s phenomenal Strength in Numbers I shared in my presentation.


So what of “these kids”? What about the kids that fail (and presumably have or will fail) math? They’ve lost credit in math for the year and must spend their entire summer sequestered in Summer School.

Let’s try revamping the Summer School experience. Boaler shared a small clip of a summer math experience that allowed kids to experience math in a rich way, possibly for the first time (though, to be fair, some of the kids in the class hadn’t failed and were there for some other reason, I’m not quite clear). My recommendation would be not to attempt to re-cover 8th grade math in a more rote way, but rather consider this an intervention, a lifeline, for math. Throw out the packets and books and spend a few weeks combing through and giving kids the following:

Summer School is one of those “death knells” for students. Students that get sacked into Summer School often are doomed to repeat it. At that point, cramming a year’s worth of mathematical content isn’t going to bring them back from the brink. It requires an entire rebuilding. I’m so thankful Boaler has offered teachers and students so many lifelines, via a low-cost book, a no-cost MOOC, and of course, a beautifully designed website for introductory resources and videos.

Instead of saying “these kids can’t do X,Y,Z”, let’s try this instead.

Thanks to Fawn and Jo and Ilana and Kelly, and my fellow #joboliebers.


[NCTMNOLA Processing Session 2] What math teachers applaud

I realize now I’m recapping NCTM in order of sessions that force me to process things. Dan’s presentation definitely forces me to do just that.

Dan’s talk focused on the lessons he learned while playing countless hours of Angry Birds, Portal, Flight Control, and Stickman Golf. I loved how he pointed out that the lessons of Angry Birds’ go beyond parabolic motion. Quadratics are a fun application of Angry Birds, and has resulted in great work, but there are other things at play here. With Angry Birds and these other games, Dan posits these six lessons learned.


Through the eyes of his in-laws, who are now among my favorite people in the world, he demonstrates these lessons adeptly.It was incredible how quickly the hour flew by. I hardly had time to tweet or think before he was on to the next great narrative.

But like I said, I need to process a bit. Please indulge.

Beginning from the end, Lesson 6 makes a beautiful argument for standards-based grading, or lessons on assessment in general. I love the idea of an open middle.

“Lesson #2: The Real World Is Overrated” drew – by far – the loudest applause. Audible hoots and hollers arose from audience the likes of which were not heard at any other point in the conference, let alone the talk itself.

I don’t have a problem with the point itself. I fully concur that the “real-world”, however one defines it, can be overrated. Before you pillory me in the comments, the following is not a criticism of the lesson: it’s an admittance of fear of what teachers take away.

I’ve blogged before about the value of the real-world, be it significant or insignificant, as a way to – among other things – begin thinking about redesigning your curriculum. A non-sugar coated version might read: a way to start giving a crap about your students and their interests. Considering your students’ interests may be the first of many necessary reparations in their introduction to mathematics. And students’ interests are generally not in abstract math, for good or ill.

To hear so many math educators cheering this lesson above all others was dismaying. I suppose it’s not surprising: teachers have witnessed or viewed enough artificial applications of math in their day; they may have even been forced to design some. But I suddenly felt like I was in a room full of teachers who were a bit too excited to have an excuse to stop making their math curriculum relevant and important to kids. I don’t know if there are analogous conferences in other disciplines, but I can’t imagine Social Studies or Science teachers cheering the news that they should’t worry so much about making their curriculum tangible to students the way that the Math teachers did. It was uncomfortable and convicting as a fellow math teacher. What does this say about math educators?

I’ve no problem with the lesson. I have no problem with abstract tasks. Shoot, I awarded Mr. Honner’s equilateral-er triangle problem as my highly coveted Problem of the Year (first and only ever winner!). Dan has provided great abstract tasks that are engaging, interesting, complex: all things I certainly want in my PrBL curriculum.

However, teachers – by and large – don’t and aren’t able to create nifty Adobe animations to stoke student curiosity. We’re not choosing between this ….

[Real World] Super-boring.
… and this:

[Decidedly not Real World] Kinda cool.
We’re choosing between this …

[Real World] A Tuvalabs investigation on amusement park attendance.
… and this:

[Decidedly not Real World] An easily find-able worksheet on box plots.
If this juxtoposition feels a bit straw-mannish, I’d implore you to google “box and whisker plot exercises.”

131112_1Using an example of a snowboarder artificially grafted on to a problem regarding growth in popularity of the sport sort of destroyed my opening slide (but I still stand by it). No, artificially plastering a picture of a snowboarder won’t suddenly get students interested in growth models, but it might give them a contextualized understanding of parabolic motion, the way a basketball shot, and yes, angry birds might.

Again, this is not a criticism of the Lesson #2 (although, one wonders what lesson can be drawn with the successful Grand Theft Auto, Call of Duty, Madden, and other franchises steeped in realism. Or the highly acclaimed and hyper-realistic Gone Home, Polygon’s 2013 Game of the Year winner.) Dan’s not arguing against engaging real-world tasks, I’m not arguing for boring real-world tasks. No one is arguing for crappy tasks, even as our schools are currently festering in them. This is a fear about what teachers, specifically math teachers, tend to internalize and take home to their students. If I were to judge purely by applause, many internalized Lesson #2 so much so that I’m curious if the other five lessons had any room to find purchase.

In Gone Home, the protagonist explores her own house in search of her/your past.

So here I am, criticizing folks who I feel were over-focused on Lesson #2 by over-focusing on people who were over-focused on Lesson #2. Maybe I need to take my own advice. This is why I need to write to process. More of that coming, rest assured.

So thank you, Dan, for (yet another) engaging and challenging talk. I prefer presentations that force me to think, argue, and justify rather than just tickle my ear. I can think of few other educators who has advanced by thinking on math instruction. This presentation only further solidified that.

[NCTMNOLA Processing: Session 1] Classroom technology (and everything else): Start with the why

About that NCTM tech panel…

I’ll be writing a fuller, NCTM recap (hopefully) sometime this week but I wanted to get some thoughts out there for my own sanity (yes, I write to process). There was a panel during NCTM entitled Teachers Leveraging Technology in the Classroom. Here was the description:

“How can technology, from apps to blogs, help teachers create effective & innovative instruction? How can teachers use technology for their own professional development? This panel features the perspectives of five educational leaders: Karim Ani, Ashli Black, Chris Hunter, Dan Meyer & Kate Nowak who have incorporated technologies into their work.”

This was a weird one. Let’s start by noting that the title and description are disconnected: one refers to “technology in the classroom”, the description refers to “technology for … professional development.” Also, none of the panelists were teachers (correction: Chris Hunter is still in the classroom).  Still, the panel is filled with six of the most thoughtful educators I know (in addition to the five listed above, Raymond Johnson was on the panel). I wouldn’t consider any of these panelists technology cheerleaders. Quite the contrary. I feel like anyone that thinks these folks are going to be cheerleading classroom technology use doesn’t follow these folks very closely. Somehow the conversation turned quickly to NCTM and NTCM membership, which was tangentially related to virtual collaboration (which actually IS in the wheelhouse of these folks). But back to technology:

The panel all agreed that technology can often be solution without a problem. The audience participants wanted technology recommendations. Again: these are not the people to ask, because they’ll ask you right back (remember: the teacher thing): WHY? Show your work, district tech directors. Justify your reasoning.

At the risk of self-plagiarizing, I’ll refer to my own session in which I plagiarized Simon Sinek’s mantra “Start with the Why.” What is the goal of tech in the classroom? In fact, that doesn’t even feel like the right place to start: what is the goal of the math classroom?

Karim offered a nice test to whether a piece of technology is useful or not: if it increases the communication between students and teachers, it’s a good piece of technology. I’d add one more marker: if it increases communication between the student and self, i.e. technology that allows for reflection and individual sense-making. I’ll also toss “creating stuff” under that umbrella, but I understand if you’d make that its own category. These are my “why”s. Instructional software generally doesn’t achieve any of these communicative why’s, nor do I believe they even transfer that much content knowledge.

I feel bad for the participants that showed up to get more ideas for “whats”. I understand that grants run out and the funds are use-‘em-or-lose-‘em, so it’s incumbent on admins, tech directors, and teachers to spend money on tech quickly in a way that’s palatable to the grantors. So here’s some tech that I used, use, or have seen used relatively effectively (and sometimes ineffectively!) generally in 1:1 classrooms to achieve one or several of these three communicative goals. Note that some of these communication paths overlap – particularly the student-to-teacher and student-to-student routes.

WHY: To increase communication between student and teacher

  • The computer’s built-in webcam. I saw a teacher have students create “video shorts” and they’re fantastic. Students have about 60 seconds to describe their solution to a problem into their webcam, which the teacher can then assess for understanding quickly. It has the nice side benefit of getting kids comfortable with using mathematical vocabulary without the stigma of fumbling in front of the class. Students can rerecord if they like.vs
  • Geogebra and Desmos. Free, intuitive, sharable. You don’t need a step-by-step do-as-I-do walkthrough to use them. You can just get in and play around. Also, the Geogebra and Desmos user-communities are vibrant and responsive.

WHY: To increase communication between student and student

  • Google apps. We’ve all used google docs, forms, spreadsheets, etc. at this point. You can use it to collect need-to-knows about a topic or problem, reflect and journal,
  • Some sort of flowcharting software. There are some free ones in Google Apps.


  • Modular furniture. Look, I’m not saying Steelcase is cheap, but their furniture is great for enhancing collaboration. With all this money you’re saving on free tech tools, maybe you can use some of the extra dough on workspaces. Hey, you asked (you didn’t ask).
  • That whiteboarding paint stuff. Also not terribly cheap, this stuff is still great. You can use it to scribble all over the wall. It does need to be wiped down and reapplied pretty regularly though. It’s also probably not easy to write this stuff into a cool tech grant, but still.

WHY: To increase communication between student and self

  • Word processing. Yup, good old Microsoft Word, perhaps with an equation editor (free or paid) tossed in for good measure. Students should spend time doing disciplinary writing and get comfortable writing mathematically. You can do this with pencil and paper, but saving your work, revising and improving your work, and embedding images, graphs, and data tables is difficult or impossible to do with paper.
  • tdg
    Student work sample courtesy of Mr. Eberly.
  • “Free creating-stuff software suite”. Inkscape. Gimp. Google Sketch-up. I use Inkscape for pretty much every diagram I make, even over existing images. Google Sketch-up is good for geometry and dimension. right triangle
  • Jo Boaler’s “How to Learn Math” (student edition, coming soon) and maybe some other MOOCs. I can’t say I know much about the student edition of this course, being released soon, but the teacher-facing one was illuminating and I only assume the student-facing one will be too. You could possibly have a “math lunch group” or after school thing based around this course.
  • Data research websites. Sites such as Tuvalabs, Gapminder, NASA, and probably lots of others should be open for students doing data-driven projects.

The tech that enables all this to happen

  • Wifi. Clean, stable wifi. I don’t want to wade into the “block or unblock certain websites” debate here; I just want a stable internet connection that allows students to email, upload and collaborate on their work. You can’t do that without an internet connection, preferably one that doesn’t tether students to fixed locations in the classroom. You’ll also need a space for students to upload their work. You can do that with email, but something like google sites or some sort of LMS might be easier to manage.
  • A device that contains all this stuff standard. My preference would be chromebooks or laptops, if only because I find it cumbersome to type, create, and share fluently without a keyboard and mouse. Also iPads keep their software allowances under pretty harsh lock and key. Inkscape, for instance, can’t be installed on chromebooks, iPads or Macs.
  • A place to store all this tech when it’s not appropriate for the day or week’s topic. Get it outta the way. Moreover, consider teaching and practicing norms for retrieving and replacing the tech.

If I were to summarize, this tech is the same tech I use for my work. You’ll also notice that much of the software is free.

For the record, I generally hate the “50 TOP TECH TOOLS FOR EDUCATION” blog posts. In part, it’s because they don’t start with (or even consider) the “why.” Relatedly, it’s also because they’re basically just click-bait and don’t think through the actual classroom issues they are intended to address.

Tech may be the “what.” It may not be the “what.” Make sure you identify your “why.” As in, “why math?”

I have many more disparate thoughts on this panel (like, I wonder what it would have looked like had there been a tech-warrior on the panel?) and NCTM at large, but I thought I’d throw in my two cents, even though, I too, am no longer in the classroom full time.

My NCTM Slides and Resources: Designing Your Problem-Based Classroom

Here is the powerpoint and additional resources for my NCTM 2014 presentation:

Setting the Scene: Designing Your Problem-Based Classroom

NCTM – Setting the Scene [PPT]
NCTM_Problem Based Learning one pager_Krall [PDF]

The Source Texts

The Tasks

The Resources