cheesemonkey wonders

cheesemonkey wonders

Sunday, December 15, 2013

Because it always pays to follow Sam around like a duckling #Made4Math

Last week of school before the break — and it's FINALS WEEK.

Because I'm so late to the game and don't really know what my kids have or haven't learned over the semester, all of the Algebra 2 teachers gave our classes two periods to work on the final. Last Thursday/Friday (our second block period of the week), they had the whole period to work on the exam. Then this week (our actual Finals Week), they'll have their whole block period to work on test corrections/finishing — using my markings as a guide.

So I've got about 150 finals to score preliminarily, which is why it made SO MUCH SENSE for me to use much of this precious Sunday to make my own version of Sam's amazing personalized planner.

Here is Sam's planner:

And here is my version:

I think this even counts as a #Made4Math entry, although it should probably be listed as a #MadeInsteadOfMath submission. :)

Sunday, December 8, 2013

A dented patchwork circle: new school, new impressions

This was my first week in my new school, which means I've been going through a few simultaneous transitions: (1) from middle schoolers to 11th and 12th graders, (2) from a 15-mile commute to a 1.5-mile commute, and (3) from a high-performing to a very diverse, high-need school.

I could not be more excited.

This first week was challenging because my partner-teacher and I were making a transition we could not inform them about fully until the end of the week. Also, he is beloved, which makes him a tough act to follow. But he is also my friend, so it was good, I think, for the kids to see that even math teachers have math teacher friends and that we are working hard to support them in a difficult transition. We did a restorative circle with Advisory so that everyone could be heard in the process of leave-taking, and we will do a round of circles with everybody tomorrow, Monday, to acknowledge the transition and to embody the process of support.

Our talking piece for circle practice is The Batman Ball — a small, inflated rubber ball with Batman on it that moved around the circle as each participant expressed his or her feelings about our shared situation.

What really struck me was their honesty and their authenticity. They honored the circle and each other. And they were willing to give me a chance. I know I will probably receive some of their displaced frustration and feelings of abandonment over the next few weeks, but they were making positive, honest effort that was moving to witness. For the guys in the class, it was especially hard. Most of them have at least one strong female authority figure in their lives, but for many of them, Mr. T was it — their one adult male role model: a young, whip-smart, kind, funny, warm, math-wizardy hipster with oversized glasses, a ready smile, and a heart the size of the ocean.

"Meetings end in departures," the Buddha said, but the fact that it's true doesn't make it any easier. They're still here, and now with me, but their hearts are going to be hurting for a little while. Plus we have finals coming up.

The other thing that made me happy to see was that they are incredibly capable math learners — more capable than they realize. Our department uses complex instruction pretty much exclusively, which was one of the reasons I really wanted to teach there. These gum-cracking wiseacres some of whom live in situations which are hard for most of us to imagine will sit their butts down in their table groups and do group work. I mean serious, collaborative mathematics.

The fact that they don't yet believe in themselves is a different problem. But that is a workable problem too.

My classroom is across the hall from the Special Ed department's special day class, and they are generous with their chilled filtered water and holiday cheer.

So tomorrow is another new beginning. I am trying to stay open and to notice and not to hesitate as I jump in. I am dressing warmly, drinking lots of water, and making effort to be present with an open heart. Looking forward to seeing what happens next.

Thursday, December 5, 2013

Thank you, Nelson Mandela

I first heard about Nelson Mandela and apartheid at Princeton, but it was not until I was a young lecturer at Stanford that he became real for me. 

A national political student movement had started during those years, aimed at educating students at elite private universities to demand that their universities' boards of trustees divest the universities of all their stock in companies that did any business in South Africa. 

When we young teachers and lecturers learned about apartheid and about the extent of Stanford's investments in companies that supported and did business with that brutal regime, it simply made sense to us to use our white privilege to speak up and speak out as part of the divestment movement. That was when all of my friends and our students simply held our classes outside the university president's office, in the quadrangle. 

Stanford was a pretty conservative university in those days, politically speaking (Condoleeza Rice was the provost!), and so it was only a matter of time before this caught the attention of the regional news media.

Several times a day, the university president had to pick his way through a sea of privileged white kids, sitting on the cobblestones, teaching and learning and protesting peacefully.  And these were not exactly the optics the administration was looking for.

This growing scene helped to create the pressure necessary to bring about Stanford's divestment from South Africa.

It would be almost another ten years before Nelson Mandela would be freed from prison, but it was inspiring to be a part of the divestment movement. At a time when a national motto seemed to be, "Greed is good," it gave me a belief in the power of ordinary good people speaking up peacefully.

Thursday, October 31, 2013

And *THIS* is why I love the MathTwitterBlogosphere, part 573 – Infinite Tangents interviews Lisa Henry, Part 1

There are some very good things about Halloween on a school day, but a mandatory classroom "party" in advisory last period of the day on a Thursday is not one of them.

When I had finished complaining to myself about the state of my classroom and cleaning up the last of what the 13-year-olds had left behind that really bothered me, I packed up my stuff and got into my car.

And I remembered that I had an Infinite Tangents podcast all cued up for myself — one I'd been saving for a moment when I needed it most. A moment like now.

So I got to spend the drive home with Ashli and Lisa Henry. Part 1. A glorious triumph of delayed gratification.

I feel lucky to have gotten to know Lisa first through Twitter and blogs and then in person at the first Twitter Math Camp in 2012. Lisa has a gift for teaching through community-building, and she has brought this gift to bear on Twitter Math Camp. I admire and appreciate her respectful and inclusive community-building, and it inspires me in my own classroom and in my life.

Now, like most teachers, I come from a family of storytellers, so it's probably no surprise that I love hearing other people tell the stories of events I participated in. I love the prismatic contrasts of perspective and memory – the way something that struck you as essential to an event gets bumped down or deflected sideways in another person's memory due to proximity or overtaking or whatever. So I love hearing Ashli and her guests telling stories of events I remember because that process invokes the same pleasure twice – the memory of the event itself and the joy in the retelling.

My mood lifted considerably as I gained distance from school and lost myself in the conversation and the memories they were weaving on my car stereo.

It was fun to hear about and remember the great Facebook "befriending" moment in 2011 or so—that moment over Christmas Break when a bunch of individuals who'd been nothing more than virtual colleagues on Twitter (but who were still basically strangers) decided to take the seemingly insane step of "friending" each other on Facebook.

It was a moment of enormous risk.

It's one thing to share teaching ideas or goof around on Twitter, but crossing that line between virtual and IRL felt profound. What if these people turned out to be crazy? unpleasant? dangerous? Or even worse — what if they turned out to have different political beliefs than I did?

The risk felt very real at the time, and sometimes it still does. I don't pretend to be something I am not. I am a liberal. I live in San Francisco. I am a practicing Buddhist and a Democrat. My representative in Congress is Nancy Pelosi. I believe in a lot of things I know that a lot of other people in other parts of the country do not.

But the one thing I know in my feet is that I am a teacher — and a learner.

And I knew that all of these other teachers all over the continent who had become my tentative friends and virtual colleagues on Twitter in exploring what it means to teach and learn math were every bit as committed to what that means as I am. So I guess I trusted it. I was willing to go with it, and to push myself beyond my comfort zone for the sake of connecting with a community of like-minded math teachers who want the same things for our kids and for our communities and for our countries — regardless of what we may believe at the grassroots personal level.

And with all of that as background, I have to admit — it was one of the best and most profound decisions I have ever made.

I was one of those crazy ten or fifteen people who was hellbent on attending Twitter Math Camp even if we had to hold it in a yurt outside a garbage dump. I knew that these were people I wanted to be connected to and spend time with and get to know, even if we seem like we'd be completely incompatible based on what you can see from examining our surfaces.

There was a (now-hilarious) period of several months when it seemed as though what my new friends most wanted us to do was to go on a cruise together and do Exeter or PCMI problem sets together. I remember that Julie looked into costs and group rates and I thought to myself, what in the name of everything sacred have I gotten myself into? I hate situations like cruises. I get seasick. I could imagine nothing worse than being trapped on the open ocean for days with people I don't know.

But there was something about the energy of the group that I innately trusted.

I kept my cruise-hating thoughts to myself, but I hung in there because I knew I did not want to miss out on what appeared to be happening. These were people I wanted to spend time with, and I supposed that if that meant I would HAVE to spend time on a cruise ship, I could probably get a prescription for some kind of anti-anxiety medication to have on hand in case I completely freaked out.

And I just hung in there.

Eventually, the cruise ship idea fell apart, thank God, and the math camp idea came together. And nothing has been the same in my life ever since. And it's been good. Very good.

By the time I went through the toll plaza at the city end of the Golden Gate Bridge, I was not only not crabby any more, I was actually happy.

I felt connected to something much larger than my own daily grumbles, and that was enough to wash all the grouchiness away.

By the time I had parked the car, walked the dog, and poured myself a beer, I realized I needed to blog about my drive as a way of remembering what was good and sane and life-affirming about this experience I am having of being part of a worldwide community of math teachers who see teaching as something much larger than what is happening just in our classrooms.

So this is my "One Good Thing" for the day. Thank you, MathTwitterBlogoSphere, for being there on the other end of the Twitter line whenever I need to feel connected.


Wednesday, October 30, 2013

Mathalicious New-Tritional Info Lesson – CC Math 6, units, calorie burn rates, celebrities, and oh, yeah, BURGERS!

DAY 1 - Tuesday

We started Mathalicious' New-Tritional Info today in my Math 6 class. We are not a fast-paced group, so it is going to take us more time to get through the lesson. We spent Tuesday working on Act 1 of the basic lesson.

I would echo everything my friend the inimitable Julie Reulbach had to say about the Mathalicious pacing and student worksheets. As Julie says,
I have to give a shout out to Mathalicious lessons right now.  I’m impressed with the way the student sheets are structured.  The directions are very clear and accessible to students so they can get right to work without tons of questions or further explanation from me.  This allows me to walk around and observe so I can see where my students are and help the ones that are struggling.  The questions also progress in the lesson so that students use their previous work to make discoveries.  This is really tough to do when creating lessons.  Kudos guys!
A couple of students were quicker than others to catch on, so I gave them the off-the-cuff extension idea to research online the calorie "burn rate" values of other activities such as singing (since Selena Gomez was the first example person given in the table). Once we found the burn rate, I divided those by the given poundage of the person performing the activity, and voilà – we had the burn rate per pound per minute. It was rather low – like around 0.0114 calories per pound for 1 minute worth of singing.

Then I had those students go back through the table and figure out how many calories Justin would burn by singing, how many Abby would burn, and how many LeBron would burn. The kids enjoyed imagining LeBron singing and figuring out his calorie-burning for that combination.

You could add on any variety of activities for this extension, such as kazoo-playing, knitting, thinking, etc. It gave the faster students a job to do (finding and converting) and it gave the slower students an incentive to hurry their little butts up on what they needed to focus on (can you spell "sixth graders"???).

A strange thing happened in my second class (5th period). An argument broke out between two factions: one who believed with all their hearts that Larry Bird had come along and stolen LeBron's lunch, and another faction that believed with equal vehemence that it was, in fact, former President Bill Clinton who had skipped out with the lunch at the end of the commercial.

Everyone in this class (except me) was born in 2002, so they probably aren't the most reliable eyewitnesses in the world. But I thought it was interesting how many of them insisted that the lunch-filcher was Bill Clinton.

Anyway, food for thought.

DAY 2 – Block Day

We worked on side 2 of the New-Tritional Info worksheet today. Students were more confident today with the various units, burn rates, unit rate conversions, and corresponding activities. For my classes, this was a good pace.

Tomorrow is our other Block Day, so I'll do the same lesson with my other 6th grade class.

Can't wait to integrate Julie's Desmos ideas and extensions next week!

Sunday, October 27, 2013

Noticing and Wondering as a practice with my 6th graders

When I'm using MARS tasks with my 6th graders, I have found no structure to be more effective in aligning their attentions and energies to the task than the Math Forum's Noticing and Wondering structure.

We kind of go into a "noticing and wondering" mode, in which we are choosing to limit our our monkey mind attention to just plain noticing and we let go of any other kind of attention that comes up during that cycle.

Noticing, in particular, is a quiet and nurturing structure for kids to simply be present with what they notice. We are not privileging noticings or knocking down noticings, we are simply welcoming them as valued and arriving guests.

6th graders love having a structure, so they loved the structure of noticing. Then once we'd heard from everybody,we did a round of noticing. It's powerful when space is allowed to sit with this first round of work.

Look at all these amazing insights they had:

We took a little time to admire this collection. It's a great list!

They even did great a great job of thinking about the value of doing noticing and wondering at the start. The first item from the link is from @fnoschese, whose wisdom even my middle schoolers can grasp.

Doing this task together and then talking about it made students reflect in deep ways about what kinds of growth processes were going on for them.
I just wanted to share this one implementation for anybody who is interested in ways you can use this.

Sunday, October 20, 2013

Using Exeter problems as an intro to algebra tiles

We are an Apple 1-1 school, so I am always happy to figure out lessons my students can use their laptops for.

I also like to use manipulatives in Algebra 1. It's not easy to get all students to accept the need to use multiple representations (such as an area model), but they help enormously to extend kids' conceptual understanding of the distributive property — plus they make a return appearance a lot when we get to the Festival of Factoring in the late winter.

So the National Library of Virtual Manipulatives seemed like a natural fit. But what problems to use to introduce them?

Enter the Exeter Math 1 problem sets.

I have been using the Exeter problems with my advanced 8th grade students taking Algebra 1 almost every week during our Problem-Solving Workshops on block days. Each page is a self-contained "problem set" that builds from simplest principles and often loops back on itself later in the page. This gives students a chance to give themselves a pat on the back for having discovered and developed an intuition for activating their own prior knowledge. I then have them write up one of the problems they solved as a problem of the week to give them practice in blending symbolic and graphical representations with verbal representations (don't forget the verbal representations!). So much Common Core math in such a small span of time!

I will write more about using the Exeter problems as a resource for long-form problem-wrestling with my students, but here I just want to talk about the specifics of introducing algebra tiles.

One of the features of the Exeter problems that we do not get to take much advantage of is the way they build page over page. They will introduce part of a concept or skill in a problem on page 5, say, then introduce the next part of the concept in a problem on, say, page 8. This makes so much sense if you are teaching using one page per day and working through all of them. But for those of us who dip in and nick out once a week, that isn't really possible.

But... by the time we got to around page 21, it dawned on me that I could collect the five problems they use that introduce algebra tiles and put them on a single sheet of paper.

Then I could give that to students during Problem-Solving Workshop, along with a quick intro to the NLVM, to let them teach themselves how to use algebra tiles!

So that is what I did this week.  :)

A couple of programming notes if you want to try this yourself with your students:
1. Start at the very beginning with NLVM and PREPARE FOR TECH HICCUPS
NLVM can be extremely persnickety. This is probably due to some perverse desire to help us cultivate CC standards of mathematical practice #1. Encourage yourself and others to persevere.

Your network may have restrictions on how students can use Java-enabled apps on school equipment. We had some hiccups getting NLVM to run on everybody's system at first. Firefox seems to be the most reliable browser for NLVM. Also, you need to have the most up-to-date version of Java on the student's system.

On our network and systems, students can only update Java by logging out and in again or by restarting their computers. No matter how many times I explained this, some kids still didn't quite figure it out. So much for being "digital natives." Plan to go around to each kid the first time to help them get their systems up and running.

Our system throws up a modal "Security Warning" dialog that forces you to check "I accept the risk" and "Run" before NLVM will load in the browser window. Again, a minor pain in the butt, but you do need to make sure that every kid gets through the security gauntlet to use the system.

Refresh the browser window if need be and be patient which Java and the applet cooperate in loading.
2. Get everybody to the *FIRST* page of the Algebra Tiles site on NLVM
For reasons that pass my understanding, NLVM dumps you into the sixth page of the algebra tiles site (the activity panel on the right, which loads as "Multiplying Binomials - 1").

You need to have students click the leftward-ho button at the top of this right-hand panel SIX TIMES to get back to the first page, which is called "Distributive Law - 1."

This is stupid but necessary because on the first two pages of this site, you can do things you need at the beginning that will quickly drop away as students gain fluency.

For example, the Distributive Law pages are the only ones where you can easily represent both multiplication over addition AND an area addition model in the same window.

3. Familiarize yourself with the syntax of the NLVM Algebra Tiles pages
You'll need to tinker with this a bit, to get comfortable with the syntax of the applet, but there are two essential features of the Algebra Tiles distributive property pages:

4. Click to CREATE tiles in the workspace; drag to MOVE tiles
You can create instances of any of the area blocks that are possible by CLICKING them in the menu bar along the bottom of the workspace. When you click the "x" button, for example, NOTICE that NLVM creates a single instance of a 1-by-x rectangle in the workspace. You can create as many "instantiations" of any of these blocks as you need for any expression you want to represent.

NOTICE that you can drag these critters around in the workspace and add them up, like LEGOs. Or you can drag them into the x-axis tray or the y-axis tray to represent lengths and widths of various area blocks of multiplication.

ALSO NOTICE that you can mouse over the corner of a block in the workspace to rotate it into the position you need.

5. How to show multiplication over addition (i.e.,  how to show  x (y+2) :
The x- and y-axis are basically x- and y-axis "trays" that students can drag tiles into. Drag a 1-by-x tile into the y-axis tray and it creates an x coefficient. Drag a 1-b-y tile and two unit blocks into the x-axis tray and they become the quantity in parenthesis over which your x coefficient will drape itself in multiplication.

NOTICE that as blocks snap into place in the second axis tray you fill, a red area outline appears in the main workspace between the x-axis tray and the y-axis tray.

6. How students can confirm for themselves that area addition and multiplication over addition produce equivalent area values (i.e.,  how to show that  x (y+2) = xy + 2x :
In this window, students can create blocks to fill in this red outline and verify for themselves that the area they get using the distributive property is equivalent to the area they can get using the area addition postulate approach.

Have students click to create blocks and then drag them around to fill the red outlined area perfectly.

My bastardized worksheet of the five Exeter Math 1 problems that introduce algebra tiles and an area model can be found here on the Math Teacher's Wiki.

NLVM Distributive Property pages are here and 6 pages to the left.


Monday, October 7, 2013

Teaching Mathalicious' "Harmony of Numbers" lesson on ratios, part 1 (grade 6, CCSSM 6.RP anchor lesson)

I started teaching Mathalicious' Harmony of Numbers lesson in my 6th grade classes today, and I wanted to capture some of my thoughts before I pass out for the night.

The Good — Engagement & Inclusion
First of all, let's talk engagement. This made a fabulous anchor lesson for introducing ratios. The lesson opens with a highly unusual video of a musical number that every middle school student in America knows — One Direction's "What Makes You Beautiful."

You'll just have to watch the video for yourself to see how the surprise of this song gets revealed.

What I wish I could capture — but I can only describe — was the excitement in the room as my 6th graders realized what song was being played. It took about eight measures for the realization to kick in. Imagine a room full of South Park characters all clapping their hands to their cheeks and turning around with delight to see whether or not I really understood the religious experience I was sharing with them.

Every kid in the room was mesmerized. Even my most challenging, least engaged, most bored "I hate math" kids were riveted to the idea that music might be connected to math. It passed the Dan Pink Drive test because suddenly even the reluctant learners were choosing to be curious about something in math class. My assessment: A+

We started with a deliberately inclusive activity to kick things off — one whole-class round of Noticing and Wondering (h/t to the Math Forum). Sorry for the blurry photography of my white board notes. They noticed all kinds of really interesting things and everybody participated:

From noticing and wondering, we began to circle in on length of piano strings and pitch of notes. This was a very natural and easy transition, perhaps since so many of the students (and I) are also musicians of different sorts. Five guys, one piano, dozens of different sounds, what's not to like?

The Not Actually 'Bad,' But Somehow Slightly Less Good
One thing I noticed right away was that, while the scale of the drawings on the worksheet worked out very neatly, it was kinda small for 6th graders to work with. The range of fine motor skills in any classroom of 6th graders is incredibly wide. At one end of the spectrum, you have students who can draw the most elaborate dragons or mermaids, complete with highly refined textures and details of the scales on either creature. At the other end of the spectrum, you have the students I've come to think of as the "mashers," "stompers," and "pluckers." These are the kids who haven't yet connected with the fine motor skills and tend to mash, crush, or stomp on things accidentally. Some will pluck out the erasers from the pencils in frustration ("Damn you, pointy pencil tip!!!").

This made me want to rethink the tools and scales of the modeling. It might be good to have an actual manipulative with bigger units (still to scale). Cutting things out is a good way for students this age to experience the idea of units and compatible units. Simply measuring and mentally parceling out segments is a little tough for this age group. Ironically, within a year or so, this difficulty seems to disappear. I'm sure there are a lot of great suggestions for ways to make this process of connecting the measurements to the ratios through a more physically accessible manipulative or model. But then again, I'm just one teacher, so what do I really know? My assessment: B

The Not Ugly, But Still Challenging Truth
The most difficult thing about this lesson is that 6th graders go S L O W L Y. Really slowly. My students' fastest pace was still about three times longer than the initial plan.

I am fortunate that this pacing is OK for me and my students. They need to wallow in this stuff, so I will simply take more time to let them marinate. We'll try to invent some new manipulatives for this, and I'll blog about them in a follow-up.

But the reality is that this lesson is going to take us three full periods to get through. They will be three awesome, deeply engaging learning episodes filled with deep connections as well as begging to have me play the video again (Seriously? Three times is not enough for you people???).

Even though this is a much bigger time requirement, I still give this aspect of the lesson an A+. Getting reluctant learners to be curious about something they're very well defended against is a big victory.

I'm excited to see what happens tomorrow! Thanks, Mathalicious!

Sunday, October 6, 2013

Reading aloud in math class — it's a developmental thing

One of the things you would see if you were to observe my mathematics teaching is that we spend a lot of time reading aloud, decoding, and rereading all kinds of texts.

A LOT of time.

We read word problems and problem set-ups aloud, we read instructions aloud, and we read texts by mathematicians and scientists aloud. We read texts and scripts that I have written out loud and we read texts that students have written out loud. We read boring stuff and serious stuff and whatever silly stuff I can sneak into an investigation set-up.

And what this has taught me is that all students need to do a lot more reading aloud in math class.

Research shows that kids whose parents read to them early become fluent and confident lifelong readers. I used to think that only high-poverty kids don't get much opportunity to be read to, but now I can confidently tell you that kids in wealthy communities need this too. Even kids with many advantages whose parents did and/or do read to them early and diligently are often still weak readers by the eighth, ninth, or tenth grades.

And what I have learned is that even well-trained students need and benefit from regular and intensive practice in reading aloud in their math class.

They need more than just primitive "reading comprehension" skills. They need practice in what gets called "textual response and analysis" in the ELA standards, in spite of the fact that these are actually just fundamental literacy skills for anyone who hopes to gain access to the kinds of career opportunities that provide socio-economic mobility.

Reading aloud, like counting circles, is a bedrock practice for students. It is often a great leveler and differentiator in the math classroom. When I announce that I need six readers for the overview we are going to dissect to begin a class, I routinely get three times as many volunteers as that. Even my most discouraged math learners will volunteer to read aloud. Reading is a window into another world. And it's an activity in which every learner can be actively included. It's an equal opportunity invitation into the concert of intention.

And as I often tell my students, much of adult life consists of responding to badly written instructions for ill-defined tasks. Doing your taxes, interpreting and responding to job posting, following your boss' instructions, interpreting conflicting instructions in e-mail sent by customers or clients with competing or downright troubling motivations.

So this is my plea to everybody to consider adding reading aloud as one more practice in your quest to introduce low-barrier-to-entry, high-ceiling problem solving in your classroom.

Thursday, September 26, 2013

"Wow, what a great question!" or Repeat After Me: A Unit Test is Still a Learning Opportunity"

I have met teachers who refuse to answer questions of any kind at all during tests, and I admit that this puzzles me because from where I sit, those are some of the highest-leverage teaching and learning opportunities I will ever have.

I just hate to waste them.

For they are the moments when I have my students' complete and undivided attention.

And that means they are my best hope for encouraging and guiding students in the process of productive struggle.

During a test, I will happily entertain anybody's question about anything. Really. Ask me anything. I will gladly offer encouragement and encourage their courage because for many students, THAT is the moment at which they are most deeply engaged and present in the process of struggling with their learning.

But I am apparently the most frustrating person in the world because the best answer I will ever give students is to say, "THAT is a GREAT QUESTION!"

I smile and nod and encourage them and urge them to keep going. And at first, they really think they hate me for it.

During today's Math 8 test, kids kept asking questions and I kept answering, "That is a GREAT question! What a terrific insight!" and leaving to move on to the next questioner. "But is this RIGHT?" They would ask, sounding wounded. And I would say, "You are asking a FANTASTIC question! Keep going!" and move right along.

Finally somebody thought to ask me in a tiny and supplicating voice, "Dr. S, is THIS a good question to ask?" And I peered over and looked at their paper and exclaimed, "Yes! That is a super-fantastic question!"

I was certainly the most annoying person in the room, but they are starting to catch on to this whole productive struggle business. Eventually it became a humorous trope. "Oh, yeah — don't bother asking. I'm sure that's a really GREAT question."

To which I would chime in, "Yes — it really IS a super-great question!"

Don't get me wrong — this is NOT an easy thing to do. It takes strength and practice and intestinal fortitude. It will never be featured as "great classroom action." But it is the most precious and valuable thing I know how to offer my students.

I can say this because I have also been on the receiving end of this kind of teaching. It is a teaching about the value of struggle. It is an incredibly precious gift, but nobody can ever explain it to you. You actually have to experience it in order to understand what a profound act of respect it is for the primacy and centrality of your own personal experience to your own personal learning.

I spent plenty of years complaining bitterly about meditation teachers who practiced this kind of bounded containment. But I sat with it. I stayed with it. I learned first to accept it, and then to embrace it. After a lot of struggle, it humbled me. It changed me in ways big and small. It opened my heart and empowered me to discover the value of struggling within my own life's journey, as well as in subjects like mathematics. That kind of preciousness and wholeheartedness is all too rare in America, but I hope that all human beings will at least taste it at some point in their lives.

So even though it was in some ways a crappy day and a frustrating day and an exasperating day, it was also a kind of gift day too. I want to remember that.

Sunday, August 25, 2013

Ode on a six-ootsie Tootsie — a love poem to @Trianglemancsd

What an amazing but crazy first week! OK, sure, it was only two days long, but still...

Sick of reading and rereading the various classroom and school rules, I wanted to dive into some mathematics lest I drive myself and my kids totally nuts. So I decided to use Christopher Danielson's Tootsie Roll-sharing problem from his #TMC13 presentation, along with some of his "fun facts" background, as a first math day diagnostic activity and as a platform for introducing my new norms and rubric for collaboration.

Even though I felt like I was just throwing together a slide show launch during my first-period prep, it worked incredibly well!

My Keynote slides and a PDF version of the slide deck are on the Math Teacher Wiki.

Here it is in a nutshell: 

•    Visualize a 6-ootsie Tootsie
•    Now imagine 4 kids who want to share it equally and completely.
•    Can your group come up with AT LEAST TWO WAYS to accomplish this?

Both the 8th graders and the 6th graders had a lot of pretty deep conversations about whether you were thinking of sharing it in terms of number of ootsies or in terms of parts of a Tootsie. Wholes and parts, plus funny-sounding words and the chance to introduce the word synecdoche.

One of the nicest things about doing this was that it helped a lot of kids see that "complicated" and "deeper" are not necessarily the same thing. 

What's not to like?

Sunday, August 18, 2013

Collaboration Literacy Part 2 — DRAFT Rubric: essential skills for mathematical learning groups

I have said this before: middle schoolers are extremely concrete thinkers. This is why I find it so helpful to have a clear and concrete rubric I can use to help them to understand assessment of their work as specifically as possible. I'm reasonably happy with the rubric I've revised over the years for problem-solving, as it seems to help students diagnose and understand what went wrong in their individual work and where they need to head. But I've realized I also needed a new rubric — one for what I've been calling "collaboration literacy" in this blog. My students need help naming and understanding the various component skills that make up being a healthy and valuable collaborator.

My draft of this rubric for collaboration, which is grounded in restorative practices, can be found on the MS Math Teacher's wiki. I would very much value your input and feedback on this tool and its ideas.

I don't want to spend a lot of time talking about how and why Complex Instruction does not work for me. Suffice it to say that the rigid assignment of individual roles is a deal breaker. If CI works for you, please accept that I am happy that you have something that works well for you in your teaching practice.

This rubric incorporates a lot of great ideas from a lot of sources I admire deeply, including the restorative practices people everywhere, Dr. Fred Joseph Orr, Max Ray and The Math Forum, Malcolm Swan, Judy Kysh/CPM, Brian R. Lawler, Dan Pink's book Drive, Sam J. Shah, Kate Nowak, Jason Buell, Megan Hayes-Golding, Ashli Black, Grace A. Chen, Breedeen Murray, Avery Pickford, "Sophie Germain," and yes, also the Complex Instruction folks. I hope it is worthy of all that they have taught me.

Wednesday, July 31, 2013

How Stats Bootcamp Saved My Life — #TMC13

I am trying to make sure I do ten- to fifteen-minute daily writing practices on my session- and topic write-ups from Twitter Math Camp '13 because otherwise I won't end up doing them and I made a promise to myself to try and capture the essence of what was important to me.

Hedge's Statistics Bootcamp session was particularly life-changing for me, primarily in that it saved my life. The CCSS Math 8 standards now require a juicy unit on statistics and probability, which are not my personal strong suit. The truth is, I spent so many years in high-tech marketing I no longer believed in the power of statistics to do anything more than be distorted for various nefarious purposes.

So it was good to go all the way back to the ground and to return to the way of beginner's mind. The term "beginner's mind" caught on due to one of the foundational texts of North American Zen practice, which is Shunryu Suzuki's Zen Mind, Beginner's Mind. If you have not read this, it will teach you everything you need to know about what I find valuable in classroom management (SPOILER ALERT: you will be reading about how "the way to control people is to encourage them to be mischievous. To give your sheep or cows a wide field to wander in is the best way to control them. Then they will be in control in a wider sense"). The key to Zen practice — and to math teaching practice — is always to keep your beginner's mind. As Suzuki Roshi says, "In the beginner's mind there are many possibilities, but in the expert's mind, there are few."

Hedge teaches statistics using what I might call the case method on steroids. She opens with the story of a notorious serial murder trial — a trial that hinged on statistical analysis and interpretation of mortality rates and patterns of nurses' shifts in the VA hospital where the defendant worked. I would describe her Essential Question for the unit as, "How certain is certain enough in matters of life and death decisions?" Revealing one critical piece of statistical evidence after another, she guided our investigation and conversations through the key pieces of evidence in the trial.

One of the things I love best about blogs and Twitter Math Camp is that they give me a chance to experience the art of "the reveal" — more specifically, how other teachers handle "reveals" in their classrooms. Over the years, I have stolen adopted many of Sam's techniques for building reveals into his artful guided student investigation worksheets. Hedge handles reveals by demanding self-restraint. In this lesson, she provided each group with  three large manila envelopes labeled A, B, and C, and she starts class with her norms and consequences — namely, that you don't open anything until HEDGE SAYS it's time to do so. Failure to comply with this norm will result in a massive public test of physical fitness, such as giving her 20 pushups on the spot (I used to think this was a violation of the Geneva Conventions, but now I know better). Layer by layer, she has students uncover each next piece of evidence, which serves as a landing on a giant staircase of learning.

I want to be clear: this method is not "sage on the stage" so much as it is "trustworthy tour guide on a once-in-a-lifetime tour. She makes sure that her charges don't miss anything important along the way. I believe this is  our job — to set up experiences for our students so that they can uncover and develop what they need at each step. It's about helping kids to develop their capacity to slow down and attend to the world around them one thing at a time.

This is not something students can get from a MOOC or a videogame or other prepackaged online learning technique. It's a transfer of human praxis that happens mind to mind and heart to heart. This is why we say that the best teachers inspire.

Piece by piece, Hedge had us work through the statistical evidence and probabilities in the trial. She showed us how many parts of the trial hinged on competing interpretations of the statistical evidence by the lawyers, but the statistician expert witnesses each side employed, and ultimately, by the jury. I

I will definitely be adapting this unit for Math 8.

Saturday, July 27, 2013

Twitter Math Camp 2013 — reflections on a sustainable model of hope

At Twitter Math Camp 2013 (#TMC13) this morning, I was both amused and inspired to read these two tweets — one by one of my math ed inspirations and another by a colleague I could not respect any more than I do and whom I can also call a friend:
Like my spiritual and general life role model, Wile E. Coyote, I am invariably hopeful in a small sense that this will FINALLY be the moment — that perfect moment when all my best-laid "plans" will do the trick and I will, at long last, have the solid, effortlessly nourishing, and unshakable ground beneath my feet that I crave (and that I believe I so richly deserve).

But years of experience have taught me that that is the "hope" of an Indulging Baby — a person who looks like an adult on the outside, but who really walks around believing that my every problem, need, and desire in life should be solved by benevolent and invisible external forces. This is in harmony with my frequent conviction that my life really ought to operate like one of those behavioral experiments in which, each time I press the correct lever, the Universe promptly and consistently rewards me with a food pellet.

So I'm sure you can imagine my annoyance with the reality that life — and teaching — refuse to cooperate with my first-draft of things.

For the second year in a row, I have blown away by what I receive at Twitter Math Camp. The best, the most creative, the most resourceful, and the deepest-thinking math teacher I know in the English-speaking world show up and share with me their 'A' game. This is not so much a blessing to me as what I would describe as a complete fucking miracle. In sharing, in presenting, in participating, and in attending, every single person at this conference gives me a richer PD experience than many teachers ever get in an entire lifetime.

And in a sense, that is the point.

For me, this conference is about refilling the well at The Great Oasis of The Impeccable Warriors. There pretty much are no Indulging Babies here at TMC. If you want somebody to take care of you and make you feel better and wipe your butt, well, this is not going to be the place for you. Everybody here is truly impeccable. To me, that means that everybody does the very best they can in whatever situation they are in. It's a stone soup mindset. If everybody has crap, then we will be eating crap soup that night. But if everybody brings one small, precious ingredient to the soup, then we will be eating like royalty — or at least, like Silicon Valley-based organizations that are overfunded by the Bill and Melinda Gates Foundation (use your imagination, or consult @fnoschese's Twitter feed and/or blog).

That is not to say that everything is perfect. People are still people, which means we can all sometimes be thoughtless, stupid, impulsive, stubborn, rude, and a whole host of other things.

But what makes this work, I think, is that everybody here owns their own "stuff" and is willing to be accountable for what they put into the communal mystic cookpot.

The truth behind the truth is, I brought my 'A' game too. I worked for three months on my sessions, planning, preparing, reflecting. You guys are my tweeps. My tribe. Even though I had an almost totally crappy year, I did not want to let you down. And I have learned that I will get back in proportion to what I put in (cf. CCSSM 8.F.1 and 8.F.3, and passim).

So my challenge to everybody who is attending Twitter Math Camp for the first year this year is to reflect on this question:
Now that you have fifty percent as much experience with TMC as even the most experienced Twitter Math Campers among us, how are YOU going to help make Twitter Math Camp just as amazing next year?
I strongly believe that the people who show up for something are exactly the right people. So, hey — welcome to the club of Impeccable Math Camp Warriors! You certainly have something important to contribute, or you would not be here reading this.

You don't have to answer this question right now. But if you want this to be here next year — both for yourself and for others — it is important to hold this question in your heart as you process the experiences you've had these past several days.

I believe that hope is a process, not a destination, and I believe that what Steve Leinwand was responding to was the awesome force field of being in the presence of 125 impeccable warriors all being impeccable together — 125 math teachers who don't simply complain about what a mess things are, but rather who each grab a mop and say, oh, I see— I'll do it.

Thursday, July 11, 2013

Collaboration Literacy — essential skills in mathematical learning groups (i.e., shared mathematical thinking)

It has long been clear to me that there are important skills of interdependence that I value in mathematical group work, but I have always been dissatisfied with the assumptions inherent in the existing models. My dissatisfaction always stems from conflicts with some of my deepest-held values about collaboration and about learning about collaboration.

Sam's post on Participation Quizzes was the first model I ever heard about that felt harmonious with what I know about healthy collaboration. I love the idea of using formative assessment of a group's collaborative interactions as a lens for viewing the mathematical learning that was going on. But I also know myself well enough as a teacher to know that I need a clearer, more explicit framework in my head so I can be both clear and intentional about the skills I am cultivating and encouraging.

Fawn's epic and brilliant deconstruction of successful in-class activities the other day referenced a touchstone work that I too really value: Malcolm Swan's Improving Learning in Mathematics manifesto. This time through, though, I was struck by four skill areas for group work that Swan touches on but does not develop.

These, I have realized, form the basis of a set of group work skills I could envision developing into a rubric for participation quizzes as well as a set of foundational "collaboration literacy" skills I could wrap my mind and heart around.

Here are the four skill areas I am thinking about for the collaboration skills rubric, along with my early commentary and thoughts:

  • SHARING SKILLS — in other words, developing a sense of inclusiveness as a member of a mathematical learning group. These skills include: demonstrating patience when others have difficulty putting their ideas into words; allowing others adequate time to express their own ideas; not moving on until everyone understands; and actively making sure that everyone understands why or how a piece of shared thinking/reasoning is so
  • PARTICIPATING SKILLS — developing your own agency as a math learner (i.e., making your own personal contributions to the group's shared thinking). Skills include: overcoming shyness to share your thinking with the group; managing your desire to take the microphone more than your share of the time; genuinely "showing up" with your own unique insights and gifts as a thinker; encouraging and supporting others as they speak their ideas, confusion, or questions.
  • LISTENING SKILLS — developing your own openness as a collaborator. Skills here include: listening actively and deeply; not simply waiting for your turn to talk; making eye contact with those who are speaking; asking clarifying questions; disagreeing respectfully; and agreeing and extending others' thinking.
  • EXPLORATORY TALK SKILLS — developing your voice as a math learner and as a member of a learning group. The phrase "exploratory talk" comes from Swan's discussion (page 37), and I think it encapsulates the qualities we are looking for when we ask students to collaborate to develop a shared understanding. These are areas where I believe The Math Forum's work really shines. Skills here include: noticing and wondering; extracting information and a question; paraphrasing or rephrasing; acting out a problem; plus making explicit transitions from one topic to the next and preventing transitions from occurring until the whole group is ready to move on. 
I envision self-assessment and peer assessment being vital parts of the process in addition to teacher assessments of individuals and groups.

I am particularly excited about ways to integrate thinking from restorative practices into this framework for mathematical learning groups because I believe they hold a lot of promise for improving the quality of student interaction.

I hope there will be a vigorous discussion in the comments!

Thursday, June 27, 2013

Quotes hanging over and around my desk — or, what the MTBoS means to me

I got a new job today, plus I had a birthday earlier this week, plus Topper had a hot date in Doodle Land, so I've been noticeably in the bleachers of the latest debates about the mathtwitterblogosphere, what it is, what it means, what's good about it, what's not so good about it, and so on... and so on... and so on.

It's probably just as well. I don't find myself to be particularly constructive when I'm tired and strung out and stressed.

I have one of the TMC12 photos from last summer hanging over my desk, along with a number of quotes all around that remind me, among other things, what I value about the MTBoS. Today they feel wiser about the whole thing than I do:

"I was a late bloomer. But anyone who blooms at all, ever, is very lucky."
    — Sharon Olds

"Success is going from failure to failure without loss of enthusiasm."
    — Winston Churchill

"Ancora imparo." ["I am still learning."]
     — Michelangelo at age 87

"If you bring forth what is within you, what you bring forth will save you."
     — The Gospel of Thomas

"Rock bottom became the solid foundation on which I rebuilt my life."
     — J.K. Rowling

Welcome difficulty.
Learn the alchemy
True human beings know.
The moment you accept what troubles you've been given,
The door opens.
    — Rumi

My learning to swim drowns no one.

     — Dr. Fred Joseph Orr

"There are years that ask the questions and years that answer."
     — Zora Neale Hurston

"Continue under all circumstances.
  Don't be tossed away.
  Make positive effort for the good."
    — Dainin Katagiri Roshi

"We're lost. But we are making good time."
     — Yogi Berra

"Tell me, what is it you plan to do
 with your one wild and precious life?"
     — Mary Oliver

Saturday, May 25, 2013

Oreos, Barbies, and Essential Questions: framing projects for differentiated learning

The Oreos lesson/unit has been going swimmingly. Students loved the project and the poster, though I'm feeling a little bored with my project ideas right now. Not every project culminates in making a poster. I have a bunch of other ideas I want to write about in another blog post, but that is not where I am headed in this post.

What I want to ponder is, I wonder if we have not been making our Essential Questions (EQs) in mathematics too small. Too narrow. Ever since my Global Math presentation (the one where I had the epic #micfail that left me playing Harpo to Daniel and Tina's Groucho and Chico), I have been thinking about Understanding by Design more and more, and that has led me to ask myself if I don't need to make my Essential Questions in math lessons a whole lot bigger and deeper. There are so many ways to bring the real world into my math classroom, and one of those ways is to frame our work using questions that adolescents are obsessed with thinking about in their everyday lives — questions such as, How dangerous is too dangerous? How do we define what is fair? truthful?

These EQs can form a frame around the activities we do to connect the mathematics to the real world around us. They help provide a situational motivation for learning — and for wallowing in — the mathematics that starts from a place where all students are naturally. And they also make the work we do more, well, essential.

Some of my "major" Oreo lesson EQs blossomed into, Are Nabisco's claims about their Double Stuf Oreo products fair? Are they truthful? just? And my "minor" EQs started revolving around, How can systems of linear equations in two variables help us to model and assess the validity of this claim in the real world?

I am finding more and more that when I frame our work in this way, I hear less and less of the question, "When am I ever going to use this?" And frankly, that's less wear and tear on my soul as a teacher.

With Barbie Bungee, in addition to creating an occasion for more practice in reading aloud and practicing decoding and interpretation skills, I used my situation set-up to raise the EQ, how dangerous is too dangerous? This is a question every adolescent has had to wrestle with since the dawn of time (or at least, since the dawn of puberty as a social construct). In their effort to keep him safe, Siddhartha's parents built him a golden cage of pleasure palaces and theme parks so he would marry and have a life there and never want to leave home. And I can only imagine what Moses' parents must have gone through ("Put down that rod! You're going to put someone's eye out with that thing!").

And the things that matter about that question are (a) the assertion you come up with and (b) the way you marshall concrete evidence and interpretive scaffolding and support to persuade someone else (such as your parents) of the validity and rightness of your assertion.

Coming from a writerly and an entrepreneurial background, I often find that the math of a thing — the essential mathematics of a thing — comes down to what I can persuade someone else of. 

For example, How dangerous is too dangerous? 

Well, it turns out that 28 rubber bands can be empirically demonstrated to be one rubber band too many. With 27 rubber bands, Barbie can have a thrilling — but still safe enough — ride, but at 28, she cracks her head open on the sidewalk, the lawsuits begin, and her parents return to her graveside frequently to tell her "I told you so!" throughout eternity.

And isn't that something we ALL dread?

With the Oreos experiment, the EQs were, Am I being cheated? Are Double Stufs, in fact, double? Is this fair? Is this a good deal? and of course, also, "Does this seem universally and predictably true?"

These are questions every adolescent wallows in every day of their lives. How many times a day do YOU hear, "But that's not FAIR!" or "Mr. C decided such and such. Do you think that is FAIR?"

Fairness is about our own personal beliefs and interpretations of the evidence in light of our own experiences in our world. And only you can answer a question like that for yourself.

Sunday, May 19, 2013

Oreos on Trial — Expert Witness Edition (or The Curious Case of the "Double Stuf" Oreos)

This is the week when everyone in our eighth grade class needs to give his or her formal presentation as the final part of their already super-huge culminating assessment project. This public speaking trial by fire will take place this week in English classrooms full of other eighth grade students, eighth-graders' parents and grandparents, school board members, our superintendent, principal, and assistant principal, seventh graders being introduced to the expectations they will have to meet next year, and any other interested parties, politicians, or luminaries from the community who wish to stop by.

The net effect of this situation is that the eighth graders are all nervous wrecks this week, the seventh graders are all bundles of hyperkinetic rubber-band energy, and our mornings will be filled from start to finish with formal presentations and — for me, as their teacher — high-stakes tech support for PowerPoint and Keynote slide shows that have been tinkered with more than is recommended.

For all these reasons, I have realized that this is the PERFECT week for us to do the great Double Stuf Oreo investigation project in my mixed 7th and 8th grade Algebra 1 classes during the afternoons.

I have created a minimally scaffolded version of this lesson that has an embedded literacy component in the situation set-up (a whole-group reading aloud activity) because (a) I have finally started to understand this year how much even the strongest adolescent readers can benefit from practicing their reading and decoding skills in a low-stakes, whole-class setting, (b) this is in keeping with my understanding of Common Core's cross-curricular demand for literacy activities in every subject area, with significant practice in reading, writing, speaking, and thinking along the way, and (c) I have also realized that writing these situation set-ups is fun for me and that reading them aloud is fun for the kids and a welcome change of pace from what they are used to. They love being "written into a situation," and this experience gives them a healthy receptivity to practicing the reading and decoding skills they will still be developing well into their high school and college years.

So here is a link to the PDF version of my Oreo investigation on the Math Teacher Wiki. And here's a link to the Comprehensive Oreology reference site curated by the inestimable Christopher Danielson.

Also, you should know that Nabisco is having a HUGE sale on Oreos this week (at least in Northern California), and between my Safeway club card and the coupons I got from the Nabisco rep at Safeway this afternoon, I feel that the company has gone a long way toward repairing the damage done by some blockhead in customer service who clearly didn't get the importance of Christopher Danielson's and Chris Lusto's attempts to receive an answer that was worthy of the mighty Oreo itself.

More news as it breaks!

Saturday, May 18, 2013

An act of wisdom

"One thing I know for sure is that when you are hungry, it is an act of wisdom each time you turn down a spoonful if you know that the food is poisoned."                            
— Anne Lamott, Operating Instructions
There are some truths you have to live, even when that path is hard. For me, this is one of those times. I have this quote hanging over my desk, which is helpful because I have really had to live it this school year. Every morning I need to remind myself of the wisdom and sanity of this perspective.

For me, this truth is bedrock. 

I resigned from my current school in March to remove myself from a toxic situation that is still unfolding. My conscience told me I could not be a part of the direction that is being pursued.

I had to turn down the spoonful to save my soul because I knew in my bones that the food being offered had been poisoned.

Hence my current job search.

I may have resigned that position, but there is no way on earth I am going to leave this profession.

I am a very effective and highly qualified teacher of mathematics, which is an area of desperate need and critical shortage around here. But we are living through an extraordinary period of economic uncertainty and complete political insanity — a time in which our leaders oscillate between one extreme of grandiose talk about "reforming" public education and its opposite of all-out panic at the crisis-level reality of our schools' current situation. 

Our leaders are lost, and our children are bearing the brunt.

The Serenity Prayer instructs me to accept the things I cannot change, the courage to reach out and change the things I can affect, and the wisdom to discern the difference between these two very different kinds of things. 

So as I apply for new jobs and do interviews and give demo lessons, I am also choosing moment by moment to renew my focus on growing and improving my practice as a teacher of mathematics.

And as I do this — even as I fret or worry about finding a new position — a curious thing keeps happening: I keep falling in love with math teaching all over again.

I've created a really great project-based learning (PBL) version of the Barbie Bungee activity (see here and here and here and of course, here), and I'm doing the same thing for the Double Stuf Oreo measurement extravaganza I plan to guide my students through this week. I am learning a ton about differentiation through teaching problem-solving from the online course I am taking from Max Ray at The Math Forum, even though I feel like I can never do enough of the coursework. And Kate Nowak (now of Mathalicious!) and I are having a blast brainstorming our 'PCMI Problem-Solving, TMC-style' problem-solving session for Twitter Math Camp '13 in late-night Google doc chat sessions.

I am hoping that all of this work will be of benefit to me in the fall, but the reality is, of course, that there are no guarantees.

I remind myself daily of the three great teachings my own teacher Natalie Goldberg passed on to me from her root teacher Katagiri Roshi. These are:
  •  Continue under all circumstances
  •  Don't be tossed away
  •  Make positive effort for the good
I am working on writing up and sharing all these lesson ideas and learnings that I'm figuring out, but to be honest, I am struggling to find the time right now. So I am taking good notes to help me write up these blog posts over the summer.

I also remind myself of my amazing good fortune to have my tribe of math teacher-bloggers in the math twitterblogosphere. You support and inspire me every day, and my gratitude for you is bottomless.

Wednesday, May 15, 2013

Substitution with stars

This one is for Max, who asked about it on Twitter, and for Ashli, who interviewed me for her Infinite Tangents podcasts.

As Ashli and I were talking about some of the struggles we see as young adolescents make the transition from concrete thinking to abstraction, I mentioned substitution.

For many learners, there comes a point in their journey when abstraction shows up as a very polite ladder to be scaled. But for others (and I count myself among this number), abstraction showed up as the edge of a cliff looking out over a giant canyon chasm. A chasm without a bridge.

This chasm appears whenever students need to apply the substitution property of equality — namely, the principle that if one algebraic expression is equivalent to another, then that equivalence will be durable enough to withstand the seismic shift that might occur if one were asked to make it in order to solve a system of equations.

Here is how I have tinkered with the concept and procedures.

Most kids understand the idea that a dollar is worth one hundred cents and that one hundred cents is equivalent to the value of one dollar. I would characterize this as a robust conceptual understanding of the ideas of substitution and of equivalence.

One dime is equivalent to ten cents. Seventy-five pennies are equivalent to three quarters. You get the idea.

We play a game. "I have in my hand a dollar bill. Here are the rules. When George's face is up, it's worth one dollar. When George is face down, it's worth one hundred cents. Now, here's my question."

I pause.

"Do you care which side is facing up when I hand it to you?"

No one has yet told me they care.

"OK. So now, let's say that I take this little green paper star I have here on the document camera. Everybody take a little paper star in whatever color you like."

Autonomy and choice are important. I have a student pass around a bowl of brightly colored little paper stars I made using a Martha Stewart shape punch I got at Michael's.

Everybody chooses a star and wonders what kind of crazy thing I am going to have them do next.

We consider a system of equations which I have them write down in their INB (on a right-hand-side page):

We use some noticing and wondering on this little gem, and eventually we identify that y is, in fact, equivalent to 11x – 16.

On one side of our little paper star, we write "y" while on the other side, we write "11x-16":

I think this becomes a tangible metaphor for the process we are considering. The important thing seems to be, we are all taking a step out over the edge of the cliff together.

We flip our little stars over on our desks several times. This seems to give everybody a chance to get comfortable with things. One side up displays "y." The other side up displays "11x–16." Over and over and over. The more students handle their tools, the more comfortable they get with the concepts and ideas they represent.

Then we rewrite equation #1 on our INB page a little bigger and with a properly labeled blank where the "y" lived just a few short moments ago:

"Hey, look!" somebody usually says. "It looks like a Mad Lib!"

Exactly. It looks like a Mad Lib. Gauss probably starts spinning in his grave.

"Can we play Mad Libs?" "I love Mad Libs!" "We did Mad Libs in fifth grade!" "We have a lot of Mad Libs at my house!" "I'll bring in my Mad Libs books!" "No, mine!"

It usually takes a few minutes to calm the people down. This is middle school.

I now ask students to place their star y-side-up in the blank staring back at us.

Then it's time to ask everybody to buckle up. "Are you ready?"

When everybody can assure me that they are ready, we flip the star. Flip it! For good measure, we tape it down with Scotch tape. Very satisfying.

A little distributive property action, a little combining of like terms, and our usual fancy footwork to finish solving for x.

Some students stick with substitution stars for every single problem they encounter for a week. Maybe two. I let them use the stars for as long as they want. I consider them a form of algebraic training wheels, like all good manipulatives. But eventually, everybody gets comfortable making the shift to abstraction and the Ziploc bag of little stars goes back into my rolling backpack for another year.

I'd like to thank the Academy and Martha Stewart for my fabulous star puncher, without which, this idea would never have arisen.

I wore out my first star puncher, so I've added a link above for my new paper punch that works much better for making substitution stars. Only eight bucks at Amazon. What's not to like? :)

Thursday, April 18, 2013

Sometimes I teach, and sometimes I just try to get out of the way...

We are in the midst of our giant 8th grade culminating assessment extravaganza — a multi-part project that includes a research paper, a creative/expressive project, a presentation with slides, and several other components I'm spacing out on at the moment.

I have to admit something here: I used to be an unbeliever when it comes to projects.

I used to think they lacked rigor and intellectual heft.

But I was wrong.

Two years of this process has made me a believer in the power of project-based learning.

Sometimes the creative projects are merely terrific, but every year, there are a few that are incredible. This year, this has already happened twice... and only two projects have been turned in so far (they are due on Monday, 22-Apr-13).

Sometimes it is the quiet, timid kid who really blows my mind. Sometimes it is a kid who is kind of rowdy who reveals another, hidden side. But I never fail to be humbled at the potential inside each of these people, and I am honored to teach them.

So this is a reminder to myself that sometimes my job is simply to get out of their way.

Tuesday, April 9, 2013

Allegory, iambic pentameter, and 8th graders

In 8th grade English we have just started our poetry unit, which is probably my favorite literature unit, and today was probably my favorite lesson of my favorite literature unit.

I had to start by finishing up what I think of as the "poetry bootcamp" section. There are all the basic terms, the mandatory vocabulary, bleep, blorp, bleep, blorp, and a yada yada yada. BO-RING. That is no way to engage 8th graders.

So I took my opening when I got to allegory, which, as I explained to them, is what we call an "extended metaphor," or as I like to think of it, a "story-length metaphor."

Like the fable of The Ugly Duckling.

I am a believer in the power of storytelling and poetry to save lives. They've saved my life many, many times over, and I know many others who've been saved by them as well.

I told them a version of Clarissa Pinkola Estès' version of The Ugly Duckling. I wove the story from the perspective of the bewildered, misfit duckling who cannot belong but who tries so hard to belong until he JUST. CANNOT. EVEN. At which point, he gets driven out of the flock into the landscape of despair.

He wanders through the landscape of despair — through the forest of his fears — until he has reached the end of all that he knows.

Finally, exhausted and hungry, he paddles out on the lake in search of solace and food. As he is paddling around, lost and spent, a pair of magnificent swans paddle up alongside him and ask if they can swim with him.

He looks over his shoulder to see if there is somebody else behind to whom they must be talking. The water is empty.

After many backs and forths, he relents and allows himself to swim with them. And as the sun peeks through the thick cloud cover, the glassy surface of the water turns into a giant reflecting glass, into which he looks, expecting to see his familiar, unlovable image.

But instead, he sees quite another image looking back at him — the reflected image of a third, equally magnificent swan on the lake.

I told them, we all wander lost at some point in our lives, but if we hold on and remain clear about what we are searching for, we will all eventually find our flock, our tribe, our true pack. The people with whom we can be authentic and with whom we belong. Estès talks about "belonging as blessing" as a promise, and I have learned that this is true, even though I always find the needle on my gas gauge quivering around the "E" end of the spectrum by this point in my journey.

On my own path right now, I'm not "there" yet. I don't know where I'll be teaching this time next year, but I do know the shape of this journey, and I understand that now is the moment when I need to redouble my faith in the archetype — even though every fiber of my being is ready to just lie down and allow myself to be eaten by whatever hungry ghosts are passing my way.

I told my students that there are patterns to our experience, just as there are patterns in mathematics and the natural world and in human history. And I think that I told them what I needed to hear for myself, namely, that education and growing up is the process of discovering and learning to trust the patterns that are bigger and greater than our own, fidgety little monkey minds.

Tuesday, April 2, 2013

Intro to Quadratics — from "drab" to "fab" (or at least, to something less drab)

Recently, I created a new anchor lesson for my Algebra 1 quadratics unit. I found that, while I really liked the sequencing of activities and questioning in the NCTM Illuminations lesson on "Patterns and Functions," I found their situation and set-up simultaneously boring, contrived, and inane.

Actual photograph of San Francisco monkeys

hosting a tea party in the wild
As is so often the case, I find that a certain, judicious sprinkling of silliness and fun in the set-up can really liven up the lesson. A certain amount of contrivance is necessary in many activities, even those that are based on "real-world situations." So why not stretch the real world to make it conform to the needs of my algebra students?

The Made To Stick elements are all here: multiple access points are provided through manipulatives, storytelling, and humor.

My student investigation sheet, Table for Eighteen... Monkeys is available on A PDF of the Table Tiles master is available here on

Tiny plastic monkeys sold separately. :)

UPDATE: Worksheets now also on the Math Teacher's Wiki, at

Sunday, March 24, 2013

Thoughts On Making Math Tasks "Stickier"

Last year, the book that changed my teaching practice the most was definitely Dan Pink's Drive: The Surprising Truth About What Motivates Us. It helped me to think through how I wanted to structure classroom tasks in order to maximize intrinsic motivation and engagement.

This year, the book that is influencing my teaching practice the most would have to be Made To Stick: Why Some Ideas Survive and Others Die by Chip and Dan Heath. I bought it to read on my Kindle, and I kind of regret that now because it is one of those books (like Drive) that really needs to be waved around at meaningful PD events.

The Heath brothers' thesis is basically that any idea, task, or activity can be made "stickier" by applying six basic principles of stickiness. Their big six are:

  1. Simple
  2. Unexpected
  3. Concrete
  4. Credible
  5. Emotional
  6. Story
The writer in me is bothered by the failure of parallel structure in the last item on this list (Seriously? SERIOUSLY? Would it have killed you to have used a sixth adjective rather than five adjectives and one noun? OTOH, that does make the list a little stickier for me, because my visceral quality of my reaction only adds to the concreteness of my experience, so there is that). But that is a small price to pay for a very useful and compact rubric. It also fits in with nicely with a lot of the brain-based learning ideas that @mgolding and @jreulbach first turned me on to.

This framework can also help us to understand — and hopefully to improve —a lot of so-so ideas that start with a seed of stickiness but haven't yet achieved their optimal sticky potential.

I wanted to write out some of what I mean here.

For example, I have often waxed poetic about Dan Meyer's Graphing Stories, which are a little jewel of stickiness when introducing the practice of graphing situations, yet I find a lot of the other Three-Act Tasks to be curiously flat for me and non-engaging. Some of this has to do with the fact that I am not a particularly visual learner, but I also think there is some value in analyzing my own experience as a formerly discouraged math learner. I have learned that if I can't get myself to be curious and engaged about something, I can't really manage to engage anybody else either.

Made To Stick has given me a vocabulary for analyzing some of what goes wrong for me and what goes right with certain math tasks. The six principles framework are very valuable for me in this regard, both descriptively and prescriptively. For example, Dan's original Graphing Stories lesson meets all of the Heath brothers' criteria. It is simple, unexpected, concrete, credible, emotional, and narrative. The lesson anchors the learning in students' own experience, then opens an unexpected "curiosity gap" in students' knowledge by pointing out some specific bits of knowledge they do not have but could actually reach for if they were simply to reach for it a little bit.

But I would argue that the place where this lesson succeeds most strongly is in its concreteness, which is implemented through Dan's cleverly designed and integrated handout. At first glance, this looks like just another boring student worksheet. But actually, through its clever design and tie-in to the videos, it becomes a concrete, tangible tool that students use to expose and investigate their own curiosity gaps for themselves.

Students discover their own knowledge gap through two distinct, but related physical, sensory moments: the first, when they anchor their own experiences of walking in the forest, crossing over a bridge, and peering out over the railing as they pass over (sorry, bad Passover pun), and the second, when they glance down at the physical worksheet and pencil in their own hands and are asked to connect what they saw with what they must now do.

This connection in the present moment to the students' own physical, tangible experience must not be underestimated.

Watching the video — even watching a worldclass piece of cinematography — is a relatively passive sensory experience for most of us.

But opening a gap between what I see as a viewer and what I hold in my hands — or what I taste (Double-Stuf Oreos!), smell, feel, or hear — and I'm yours forever.

"My work here is done."
This way of thinking has given me a much deeper understanding of why my lessons that integrate two or three sensory modalities always seem to be stickier than my lessons that rely on just one modality. Even when the manipulatives I introduce might seem contrived or artificial, there is value in introducing a second or third sensory dimension to my tasks. In so doing, they both (a) add another access point for students I have not yet reached and (b) expose the gap in students' knowledge by bringing in their present-moment sensory experiences. And these two dimensions can make an enormous different in students' emotional engagement in a math task.