[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:

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[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.



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[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.


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[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.

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[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.

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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

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Students writing their own problems: a walkthrough

I imagine this is pretty high on whatever hierarchy of question you ascribe to, but it’s one that sure speaks to me. Malcolm Swan references Creating Problems (p.28) as a way of students demonstrating mastery. I’ve had mixed result with having students do just that.

Below is an attempt at streamlining the process, using a sort of “walkthrough” template. It begins by asking the students to describe the math we’ve been doing recently, followed by copying a recent prompt. Along the way are some (hopefully) helpful hints and key words that may be useful.

Feel free to use & modify as you see fit. Better yet, tell me how it goes.

Note: scribd doesn’t play well with some formatting, so here’s the original doc and PDF.

Problem Creation Walkthrough (PDF)

Problem Creation Walkthrough (doc)



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Thought experiment: combine Algebra 1 and Physical Education

(Part of the reason I started this blog is so I’d have a place to play around with ideas, no matter how non-field-tested they may be. Consider this one of my many half-baked ideas that I haven’t fully thought through.) 

One of the hallmarks of a New Tech Network school – the network of schools to which I am happily attached contractually and emotionally, and spent part of my teaching career teaching at – are teaching using a Project Based Learning approach within combined courses: World Studies and English, Biology and Literature, and so forth. The first math class I ever really enjoyed taking was my combined Physics and Calculus class my HS senior year.

While I’m not suggesting that mathematics is impossible to combine with other courses, it is often fraught with peril. When we were starting out our journey as a New Tech school, the Science teacher and I splayed out our content standards on the table to see around which we could build projects around. We had a couple ideas for projects, but that would have left over half of our content standards either not combined in a project, or combined in contrived and unnatural ways. Often many of the math standards don’t play well with others.

Moreover, in a PBL classroom, it’s easy for math standards and skills to get dwarfed by the project’s product itself. That was part of my discomfort with PBL and began experimenting with what we now call Problem-Based Learning. It’s doubly easy for math standards to get dwarfed by the lab report, the prettiness of the art exhibit.

That said, I do think students learn the content better when it’s connected to other content. I got more out of my Calculus class by chucking things off the roof and bouncing tennis balls and seeing that the acceleration and the derivative of the speed magically matched. How do we reconcile the value in connecting math content to other physical, tangible subjects while maintaining fidelity to mathematical standards and quality pedagogy?

Here’s a class I’ve never seen implemented (at least, not implemented the way it exists in my head): combined Algebra 1 and Physical Education. That’s right the nerds and the jocks, hanging out together! The more I think about it, the more I like it – again, with the full disclosure that I’ve never seen it taught, never taught it myself, and haven’t even totally thought it through. I’m not sure I’d even consider this half-baked. This is a more 1/8th baked idea.

Still, here’s what I like about it:

The tasks themselves. The content can play pretty well together. I’ve created a couple of tasks just my little old self around physical fitness, and I’m not terribly fintessy. The tasks could either be directly about a student’s physical fitness or about sports and fitness at large. This allows for long term data tracking and regression. Even standards that don’t seem to play well with physical fitness still have physical fitness-like applications (like, say, quadratics … or… quadratics).


NBA.com has started making their SportsVU data public and it’s changing the way the game is played. Slow and fast people are running at the same time and it’s on video. Teams aren’t punting anymore. There’s fitness equipment to be constructed. There are NFL plays to be scripted.

For the PBL-practicing Physical Education teacher, this may hopefully push you beyond the “make a new sport” or “teach other kids sports” projects.

Seriously, why are we letting all this precious data from PE go to waste?

The way you could structure your weeks.

Another nice side-benefit of a combined course is that they are largely double blocked, giving you a full hour and a half or so a day. Seems to me a weekly schedule could look something like this.

Monday: do something physical that gives you data (and some math practice after cool-down, now that the brain has oxygen and blood and stuff)

Tuesday: do something mathematical with that data

Wednesday: do something physical that gives you more data (and some math practice after cool-down)

Thursday: do more mathy things with that data

Friday: spend 45 minute “maxing out” (or whatever) on that data-producing physical activity. Spend 45 minutes analyzing performance

Or just go halvsies the entire week and plot the progress of the students in whatever physical activity they’re doing.

Reduction of status issues in the math classroom

This might also be fraught. I mean, the only place that creates and supports status issues than a math classroom is a physical education classroom, right? On the flip side, it might allow students who are perceived to be low-level achievers in math to finally take the lead. You might get the athletes wanting the “smart kid” on their team, in their group.

Preparing the Brain for Cognitively Demanding Tasks. Physical activity makes for great pre-work for creative mathematical problem solving or a nice interruption from cognitively demanding tasks. Physical activity releases all sorts of good chemical stuff that make you more productive, more creative, more engaged, less stressed and presumable, more capable of taking on cognitively demanding work. It’s also a really nice way to break up an otherwise plodding work day. Here’s an example of a school that is trying to keep kids’ heart rates up for the sole purpose of preparing kids’ brains for learning (hat tip: @JimPa23). Even if the math task has no relation to the PE task, I’d rather have a bunch of kids who have just been exercising than kids who have just come back from the Taco Bell Express in the lunchroom.


There are also lots of problems to address with such a mash-up. How much time would a class spend changing in and out of their gear? Would combining math and PE compound, rather than equalize status issues for some students? Do the facilitators have a similar vision for the class and what the kids should be getting out of it?

I also know that there are a ton of data in exercise to make good math tasks and physical might help prepare students’ brains for the cognitive ask that complex problems require.

What do you guys think? Is this feasible, or just one of those ideas that should stay in the “fun-to-think-about” realm?

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Equalizing Practice and Assessment (Part 2): What You Value Should Be What You Assess

Have you told your students how much you value honest attempts at solutions to a problem? Even incorrect solutions? Then you have to assess this way.

You can’t tell students that you value their incorrect attempts at solutions when you take off points when they get an answer wrong. Worse, you can’t say you value the process over the solution when you assess using a multiple choice or short answer examination. Students are too smart and they will see right through that facade, as well they should.

“Mr. Krall, you say you want us to be persistent problem solvers and you value our mathematical thinking, but you still took off half-credit for my solution attempt.”

“And you said the highest I can make on my re-test is a 70.”


“One of those new age feel gooderies.”

I’m not suggesting an “everyone gets an A” or a “crocodile in Spelling” method of assessment, but just that one needs to put a grade where their mouth is. 

Similar to Dan’s “What Do You Worship?” question, I’d ask what do you, the facilitator, value? Value, both in an ethereal “boy I sure would like this!” way and a “yes, this is what you will be assessed on” way.


In the Why/How/What framework, “why” has been addressed all over the blogosphere (but here’s a thing), “how” was partially addressed in my previous assessment post. As for “what”:

You will be assessed on your growth. You will be assessed on your persistence. You will be assessed on your various methods of solutions. You will be assessed on your communication.

You will not be assessed on the correctness of your answer. You will not be assessed based on the boxed number on the right side of the page. You will not be assessed using rote tasks that are easily solvable using a formula chart. 

Also, this goes beyond “I allow retakes.” Retakes is a way of saying that students have one more chance to get it right (usually accompanied by a significant numerical penalty). It’s not penalizing a student for a wrong answer whatsoever. Or at least honoring the solution attempt that isn’t actually a penalty in disguise (i.e. “partial credit”). This is a huge assessment shift, and requires a more sophisticated assessment tool than an answer key can provide (such as these).

It’s really difficult to switch gears like this in the middle of the year though. There’s a certain foundational work that needs to happen first. And frankly, it’ll probably take a few rounds of assessment before students even believe you. You’re probably not the first person to say they value honest – if incorrect – solution attempts, only to turn around and dock students in the name of “well, the SAT doesn’t allow redo’s”.

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Getting Better: I can improve anything for students, but I can’t improve that

I can get better at almost everything. You can get better at your practice, regardless of your teaching style. I know I often come across as dogmatic with regards to


Figure 1

Problem-Based Learning (see Fig. 1), but really, it’s all about steady improvement, irregardless of your teaching style. My personal preference is inquiry and complex task oriented groupwork 100% of the time (even if I fall short), but yours might be different. You can get better at it. You can improve it. 

Like to do inquiry learning? You can improve that.

Like to utilize real-world tasks? You can get better at it.

Like to do Project-Based Learning? 3 Acts? AnyQs? You can improve at that. And you can improve the stuff provided to students: better projects, more compelling videos and pictures,

Like to use a textbook? There are ways of improving its use.

Shoot, like to do worksheets? I know that’s allegedly a bad word but man, some of the worksheets – YES, WORKSHEETS – that Sam (@samjshah) and Jeff (@devaron3) have put together put most PrBL lessons to shame (or are included in PrBL-ish lessons!).

Whatever you find compelling, you can get better at. And, you can mix-and-match, depending on the day/week/content area.


Except math instructional software.

I’ve always had a problem with instructional software and I think I’ve found the root cause: you can’t make it better or adjust it to your students’ interests or curiosities. Sure, you can adjust it according to their needs, most, like ALEKS, Cognitive Tutor, and Khan Academy can be adaptive to do that for you, but not according to students’ interest or curiosities. You can’t change on the fly. And districts spend so much on this software, or invest in so much PD in this software, that you, the teacher, kind of have to use it. Maybe this isn’t news to you but to me – who has had the pleasure of working with teachers who use such software expertly – it was an “aha” moment.

I’m not suggesting these tools have no use. But that their use is quite limited by nature. ALEKS can determine and teach a lot of things about and to a student, but it can’t determine what the student finds compelling about math. And that’s kind of the whole ballgame.

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