cheesemonkey wonders

cheesemonkey wonders

Thursday, December 1, 2016

The Festival of Reassessment (SBG)

December is when I am truly grateful for strong routines. They mean I can split the class up and still count on everything moving forward. I am running behind in my pacing, so I needed to set up a mass SBG reassessment opportunity for slope skills yesterday. I set up all the reassessors along the window side of the classroom and all of the non-reassessors on the hallway side of the room. The non-reassessors worked on linear systems skills and problem-based learning while the reassessors worked on demonstrating mastery of slope skills.

Whenever a reassessing student finished their work, they brought it up to be rechecked. I went through it right then and there while they watched. “Uh huh... uh huh... uh huh...” until “Oh noooooo! Why is this negative?!?!? It ruins everything! Go back and fix it!!!” And then the next student moved forward in the queue.

We went through this process of working and my checking and sending kids back for about 40 minutes. “Nooooo!!!” I would circle something and pretend to freak out, sending them back to fix stuff. It became a carnival. “AAACK!" I would say. "Go back and fix this!” Seeing this happen in real-time changed kids’ relationships with their misconceptions. It reminded me of piano practice as a child. Something would splatter in the midst of a phrase or figure and I would have to go back to the beginning, willing the figure to come out right through my fingers on the keyboard.

We kept going until all 12 students had triumphed over slope and point-slope form. What I loved about this process was how it transformed the social focus of the process of understanding to us (the whole class) against the skills. Kids were going wild and cheering when somebody finally triumphed over the "slope and point-slope” process, jumping up and down and hugging their newly non-reassessing classmates and friends.

The goal became one of getting everybody in the classroom over the finish line. From individual mastery to collective. Students stopped focusing on who had status in the classroom and who did not. They stopped thinking about their place in the social hierarchy and instead lost themselves in the flow of doing this f***ing slope and linear equation problem.

And then as each new classmate finally triumphed over the skill, the whole class felt truly victorious.

Everybody got the individualized attention they needed as they drove their skills forward, and we also bonded as a collective community, advancing our work as a group.

In these divisive times, I think this might be an important process to cultivate.

Tuesday, November 22, 2016

Blogs feed my teaching soul, but Twitter helps me feel less alone

As I've been working on my National Board certification, I have come to appreciate something that the #MTBoS gives to me every single hour of every single day.

My tweeps' teaching blogs feed my teaching soul. Seriously, any time I feel stuck, I can just fire off a search in the MTBoS search engine and BOOM, I get drenched in a downpour of borrowed genius.

But in the moment, when I am freaking out or feeling lost, Twitter makes me feel ever so much less alone.

So thank you, my tweeps, and keep going. I am thankful for you Every. Single. Day.

Sunday, October 23, 2016

Graphing Stories Meets Estimation 180: A Love Story

On Friday we started our Functions unit, and as always, my go-to is to start with Dan Meyer's Graphing Stories.

After they pick up a Graphing Stories worksheet, I give them a brief set-up, explain that the guy in the video is my friend Dr. Meyer (though now I have to explain that this was a long time ago, in a galaxy far, far away, when he was a young Jedi-in-training), and promise them that I will rewind the videos as many times as they want, to whatever point they want, for as long as they want, so we can figure out as closely as possible what their graphs ought to look like.
screen shot of Dan leaning over the railing of the footbridge
Portrait of the artist as a young Jedi warrior
Only this time, we encountered a spontaneous twist.

The first video went off without a hitch. Using QuickTime, I have an action-only version of the first 15-second video (Dan walks over a bridge in a Santa Cruz park), so that it doesn't accidentally reveal anything I'm not ready to reveal yet.

I explained that my friend Dr. Meyer is unusually tall (I gave his actual height, which can be found on his web site) and this gave students the idea to "measure" him onscreen so they could try to better estimate the rise of the bridge.

This gave me an idea.

The second video in the series is of Dan descending some exterior condo stairs, but after the first viewing, an argument broke out in the discussion as they tried to find some hook they could use to improve their estimations. "A car is, like 6 feet high!" "No, the stairs are about 5 feet in total!" Blah blah blah.

Enter Estimation 180 thinking.

My classroom is right next to the door to the stairs up to the small faculty parking lot. I pointed to two kids. "OK, you and you — take a yard stick, carefully cross the driveway, and go measure some average car heights in the teachers' parking lot."

The arguments continued so I pointed out another kid, one who was extremely interested in the stair height but who has never before piped up in class. "Pick a helper, grab a yard stick, and go out to the stairs and find an average for the stair height."

While our data gatherers performed their missions, the estimation arguments continued inside the classroom. "The car is only about 4 feet high!" "No, it's not!" "What about the lamp post—can we use that?"

Five minutes later, our intrepid explorers returned and we harvested our newfound information.

Then we continued working on the Graphings Stories task.

The funny thing was that our estimates led to wildly wrong answers, but that wasn't the most important thing. Instead, what was powerful was the level of engagement in discovery that electrified the room.

Instead of considering the need to quantify to be yet another tedious task that stood in the way of getting "the right answer," students started to lose themselves in the flow of the process of mathematizing their world.

And isn't that the whole point?

So thank you to Dan (@ddmeyer) and Andrew (@mr_stadel) for giving me the tools to slow my kids down and help them to find the wonder in everyday situations.

PS — I still haven't revealed the "answer" to Graphing Story #2 yet. But I'm curious to see what unfolds in class when we come back tomorrow. ;)

Wednesday, October 19, 2016

Scaffolding Proof to Cultivate Intellectual Need in Geometry

This year I'm teaching proof much more the way I have taught writing in previous years in English programs, and I have to say that the scaffolding and assessment techniques I learned as a part of a very high-performing ELA/Writing program are helping me (and benefiting my Geo students) a lot this year.

I should qualify that my school places an extremely high value on proof skills in our math sequence. Geometry is only the first place where our students are required to use the techniques of formal proof in our math courses. So I feel a strong duty to help my regular mortal Geometry students to leverage their strengths wherever possible in my classes. Since a huge number of my students are outstanding writers, it has made a world of difference to use techniques that they "get" about learning and growing as writers and apply them to learning and growing as mathematicians.

We are still in the very early stages of doing proofs, but the very first thing I have upped is the frequency of proof.  We now do at least one proof a day in my Geometry classes; however, because of the increased frequency, we are doing them in ways that are scaffolded to promote fluency.

Here's the thing: the hardest thing about teaching proof in Geometry, in my opinion, is to constantly make sure that it is the STUDENTS who are doing the proving of essential theorems.

Most of the textbooks I have seen tend to scaffold proof by giving students the sequence of "Statements" and asking them to provide the "Reasons."

While this seems necessary to me at times and for many students (especially during the early stages), it also seems dramatically insufficient because it removes the burden of sequencing and identifying logical dependencies and interdependencies between and among "Statements."

So my new daily scaffolding technique for October takes a page from Malcolm Swan and Guershon Harel (by way of Dan Meyer).

I give them the diagram, the Givens, the Prove statement, and a batch of unsorted, tiny Statement cards to cut out.

Every day they have to discuss and sequence the statements, and then justify each statement as a step in their proof.

This has led to some amazing discussions of argumentation and logical dependencies.

An example of what I give them (copied 2-UP to be chopped into two handouts, one per student) can be found here:

-Sample proof to be sequenced & justified

Now students are starting to understand why congruent triangles are so useful and how they enable us to make use of their corresponding parts! The conversations about intellectual need have been spectacular.

I am grateful to Dan Meyer for being so darned persistent and for pounding away on the notion of  developing intellectual need in his work!

Saturday, October 1, 2016

Algebra 1 inequalities unit - notes on a conceptual and problem-based approach

My Algebra 1 inequalities unit is now the unit with which I am most satisfied, including a solid conceptual launch and framework, some serious problem-based learning, a deep treatment of quantitative reasoning and logic, and an excellent amount of discourse-rich practice activities designed to achieve both procedural and conceptual fluency, so I want to document here how it works. If there is useful stuff in it for you, then great! But I'm not going to be posting a lot of user-ready materials here to print off and give to students, so I want to be totally up front with you.

Inequalities are still one of the most procedurally taught of all Algebra 1 topics (see Holt Algebra 1, CPM, Exeter, or anything else you can find, if you have any doubts about this), and yet, it has seemed to me for a long time that they offer the opportunity for students to ground their new understandings in one of their most accessible, intuitive existing mathematical understandings — namely, their sense of when a quantity is "more than" or "less than" some other quantity.

As Nunes and Bryant clearly show, children have a very deep understanding of comparing quantities from a very young age — including both discrete and continuous quantities. For this reason, it has long seemed essential to me to hook Algebra 1 students' work with understanding with their own authentic and meaningful prior knowledge.

So in my unit,we start by activating prior knowledge with Talking Points and problem-based learning from the Exeter Math 1 sequence. In the initial part of the unit, we investigate whether quantities are going to be more than or less than other quantities. We go from comparing known, computable quantities (such as one sum or difference to another sum or difference) to more abstract quantities (such as, If x is greater than -1, then x + 2 is greater than 1), always pressing them to rely on their own understanding or constructed understanding.

The conversations that were sparked during the initial phase — both this year and last year (with Patrick Callahan in my room) — were fascinating to all.

My rule for this unit is this: whenever I give students a "rule" (or whenever we develop a "rule" together),  it has to be a guideline that will help them to ground their new ideas in their old understanding of something. This has led me to come up with much more sensible and conceptually-based guidelines that students both remember and rely on correctly. For example, one of our "rules" is that when you have some kind of inequality, it makes sense to put it into "Number Line Order" — in other words, always organizing statements or representations about smaller quantities to the left of statements or representations about larger quantities. This has had the benefit of getting students to realize that the real number line is a tool that they can use for their own benefit. It is not just another arbitrary math teacher whim.

Another "rule" we co-developed as a community was the idea of the subjectivity of x  namely, that in mathematical sentences and statements, we should organize our symbolic representations so that x is the subject of our sentences. Another way that students expressed this idea was that x is the hero or heroine of our story, so x should be the subject of our sentences.

This too has had a profound effect on students' understanding of inequalities. Most textbooks emphasize equivalence of the statements 2 > x and x < 2, but from the perspective of student understanding, privileging both statements equally is just stupid. What we are hoping for students is that they will see that as they are trying to represent values of x, that makes x the hero of our story. So our mathematical statements should be organized to reflect the subjective status of x — not the "correct" or "incorrect" equivalence of the statements. This gave students the idea that they have permission to change statements around to suit their own  learning, and that they do not need to distort themselves to please whatever mathematical authority decided to place given statements in a form that is backwards for them.

We have also developed a "rule" for distinguishing absolute value expressions from other groupings in equations and inequalities. The idea we have developed is that absolute value expressions need to be "isolated" and "unpacked" into all possible cases of ordinary equations or inequalities (i.e., NON-absolute value statements) before you can apply any solving or simplifying techniques. Our shorthand for this method is: 1 - ISOLATE, 2 - UNPACK, 3 - SOLVE. Then once you have solved, you should put everything into Number Line Order so you can INTERPRET and GRAPH your findings.

This idea of unpacking absolute values first into all possible cases is a strong way of getting students to reason quantitatively and abstractly rather than blindly applying rules. Since they have some guidelines that are grounded in their own logical understanding and personal experience, my students have been much more thoughtful and intentional when they approach absolute value inequalities and equations. It has also led to a deepening of students' understanding of when the distributive property can and should be applied to a grouping and when it should not. This has greatly deepened the discussions of the functioning of groupings among students to the better.

Because we have build a solid conceptual foundation for our work with absolute values, practice activities have become opportunities for investigation and application of our conceptual understanding, rather than blind shooting at a list of targets from 1 to 35 odd.  At every step of the way, we have been using Speed Dating as a discourse-rich activity in which students can apply their understanding to a wide range of different problems of varying levels of complexity. It also demands that they share their understanding and speak about it with their classmates, sometimes giving and sometimes receiving assistance.

In this way, practice activities have become a new form of conceptual investigation themselves, and are not mere mental exercises designed to enhance procedural efficiency. As I am discovering through my work on National Board Certification, our profession has a major problem with privileging "discovery" activities as the only ones that occasion and showcase conceptual understanding. As How People Learn clearly lays out, it is often the case that students do NOT have a 'Eureka' moment during the initial 'discovery' segment but rather, like the young learners they are, they have a lightbulb moment once they have experienced a problem in multiple different contexts. This leads me to one of my pet peeves about why teacher certification and National Board Certification overvalue only the initial discovery activity as a venue for showcasing student insight, but that is a complaint for a different blog post.

Another benefit of treating practice as an opportunity for applied investigation is that it avoids the contrived, dog bandana pseudo-context problems that strive to provide a real-world context, but only end up twisting themselves into senseless knots. The contexts should not be harder for students to make sense of than the absolute value problems themselves! The most accessible real-world absolute value problem we have worked with is the question, When is water NOT a liquid? This makes sense to my 9th graders. Convoluted problems about the conditions under which a value will land within a range of possible values are not helping here, people. I am finding that this is a case where treating absolute value problems as a system of equations or inequalities to be unpacked and investigated as number sense problems has been far more worthwhile — and far more rewarding for students. As Deborah Ball has said, Sometimes mathematics is its own best context.

Wednesday, September 7, 2016

Katamari and Speed Demons and Not-Knowing

I want to keep track of how I am learning from my speed demons and katamari in Algebra 1.

I want to be clear about two things. Not all speed demons are alike. And also not all katamari are alike.

Slowly I am finding my way to the most discouraged and shut-down learners. I group and regroup, based on formal and informal assessments. Also based on intuition. The great thing about using PBL and the adapted Exeter problem sets is that together they are giving me a lot better information and opportunities to identify, target, and support my students. Most of what we do in class is problem-solving in groups — interpreting, reasoning, and making sense using group whiteboards and whatever other tools we need.

I notice the need to provide much more intensive support to most discouraged katamari than I had expected. These are the most shut-down of my learners. These are the students who are being quiet to avoid being noticed. In math class, they live in a defensive psychological crouch. I know this posture well. But the only way I can understand it is to join the process and probe.  I ask questions: when you are AT such and such, how far away are you?

I am really shocked at how shut down and disassociated  these students are from their intuition and their natural intelligence. I wonder how long they have been cultivating this posture of hiding.

I draw a crude map on my paper and point to different positions. When you're here *pointing to the diagram*, are you further or closer to X? *moves pencil closer to destination* What about now? what about now?

By probing, I discovered something amazing — three tables of students didn't really understand what it meant to be AT the destination.

I tried different tactics until finally I asked, when you're IN your kitchen at home, how far away are you from your refrigerator? How many miles away are you? It took a while to convince them that they actually knew they were ZERO miles away.

Once they figured out what the meaning of being AT someplace or crossing the destination was/is, they could begin to build a table. One step after the other. Let's work backwards -- when you're one hour away, how many miles away are you?

This is a lack of basic trust in their own innate natural functioning. A lack of trust in their own intelligence and problem-solving skill. It takes skill to get this going. They have to be prompted/encouraged to start from where they're at — not to hide the fact that they are lost.

These students are good hiders. They know how to hide in plain sight. They are skilled spies. They know how to evade detection. I am more tenacious than most. I do not accept evasion. I probe, I question, I support, and if need be, I help. I help because they are too shut down emotionally to take the risk of humiliation. They are exhausted from not-knowing and from hiding their not-knowing. But not-knowing is the start of knowing. You can't begin to know until you know that you do NOT know.

Not-knowing is an empty space in which knowing can arise.

Katamari don't realize this yet, but they are closer to finding out. Speed demons have no clue about it yet, and they are so far away from discovering it there's not even any point in raising the issue yet. Katamari are so much closer to the truth. I don't mention any of this to any of them. I just keep probing and asking questions and asking what if and how much and how do you know.

At the end of class, one of my speed demons told me she would really like to collaborate (unlike the other speed demons at her table, who are just zooming along and only occasionally conversing). She asked if she could move to Table X, where some very discouraged Ss are. These are the students I was probing the most with. I was thrilled. She was eavesdropping on us when I was working with them. These are the questions she herself likes to ask and think about.

I don't have any conclusions to offer, just these noticings and wonderings.

We are still in the early community-building days, when discovery is young.

Wednesday, August 31, 2016

More on katamari and speed demons #MTBoSBlaugust

Gotta squeak this one in under the #MTBoSBlaugust wire. :)

In the world of meditation retreats, what I've come to think of as "Emotional Breakdown Day" of a five-to-seven-day sesshin is pretty much always on a Wednesday (or Day 3, if the retreat doesn't start on a Sunday). You can set your watch by this. There is something about getting psychologically and emotionally heated up that happens after you've been simmering for three days. I think this is true at all spiritual retreats around the world. At a three-day retreat like Twitter Math Camp, it comes at Day 1.5. All of a sudden, Twitter and blogs are flooded with snippets or full-on geysers of despair at how great everybody else seems to be doing and how totally crappy you [INSERT YOUR OWN NAME HERE] are as a teacher.

I have tried to learn not to take this seasonality personally, but it's hard not to. When you go deep, you get invested.

This cycle seems equally true for me during the school year. Week 3 is inevitably my emotional breakdown week. I can no longer keep up the pace (or the illusion of the pace) and the kids can no longer keep it up either. So things start to break. Students act out. Norms fall apart. I lose my shit.

This is just the nature of the cycle of practice. Like winter, or hurricane season, or the World Series, It. Happens. So the real test is how I am going to deal with it.

I changed the seating chart in 7th block and rolled out my best rethinking of katamari and speed demon problem-based learning practice (see "Lessons from 'Lessons from Bowen and Darryl'"). I put the speed demons with other speed demons so they would leave my katamari alone already. I want the katamari to learn how to trust their own minds, their own guts, their own hearts. They lack confidence. But put a bunch of them together, and they have no choice but to trust themselves and each other. Without the speed demons to carry them along, they have to think.

And I tell you, my friends, it was magical.

I revised the day's problem set to put the Important Stuff at the top (though I never label it as Important Stuff — that gives it too much weight for teenagers, plus too little weight for everything else), instructed them to get one table whiteboards, two markers, and a washcloth and I yelled, "GO!" I think the yelling is a particularly artful piece of instructional practice.

The room began to hum and glow in the late afternoon fog. This freed me to question and support the groups that felt particularly stuck — to help them get just unstuck enough to keep going.

They didn't even care when they worked beyond the time limit I had set for this work.

I especially loved the spontaneous alliances that formed across difference. After the first really juicy problem, two young men who hadn't said 'boo' to each other in the first two and a half weeks — a young black student and a Chinese-American student — gave each other a particularly complicated, multi-part handshake than made my heart smile. A table of girls cheered when they finished the same problem.

This is why I believe that getting students into a state of flow when they are doing mathematics is the most important thing. If you align yourself with everything we know is good and healthy and whole about doing math, then everything else will proceed smoothly.

Sunday, August 28, 2016

Week 3 — Talking Points — Desert Island Thinking #MTBoSBlaugust

When I arrived at Princeton, I had been placed with three other roommates in a Gothic dorm suite. I was proud of the things I had accomplished so far. I'd been tied for valedictorian at a huge and competitive public high school, I'd been a soloist at the All-State choral and orchestral concert, and I'd been president of and/or varsity lettered in all my extracurriculars.

So as I discovered that everybody I met had also been valedictorian, editor of the school newspaper, an All-State varsity athlete or musician, etc., I had quite an adjustment. I had to learn how to stay present with my own inner experience and on what was in front of me directly.

I've been thinking about that experience these past two weeks as I have been watched my incoming 9th graders at Lowell adjust to the shock of discovering what it means to arrive at the next level.

The classes are much, much more demanding than they are used to, even at the strongest middle schools. And in addition, as every visitor can see at the front entrance of our school, there is a board that celebrates accomplishments of many of our Lowell graduates since our founding in 1856. There are three Nobel laureates, Pulitzer Prize winners, Broadway and Hollywood stars, admirals and generals and and politicians and world leaders. There are sports legends and pop culture icons, civil rights heroes, the founder of The Gap, and friggin' Lemony Snicket, among others.

Every race, ethnic background, and gender seems to be well-represented.

No wonder my poor kids are freaking out.

Now I try to imagine what it must be like to be one of the very few African-American students in our school. Some of them appear to be doing just fine, but I imagine it is a very strange and disorienting experience to find yourself in what must seem like an endless ocean of whiteness.

We are trying to be intentional in how we are supporting these students and transforming our school culture. We are following best practices and reflecting critically on how we are doing and how we can support their experience. I wish that I could magically airlift in a larger number of faculty of color so that they felt more reflected in the adult community they see all around them. But that is not how public education works. And we have no time to indulge in magical thinking.

So this is the point at which I am introducing some Talking Points on what I like to call desert island thinking. It is the best way I have found to help students to cope with their own feelings of imposter syndrome and the need to be their own best supporters as they enter a completely new territory.

I call it desert island thinking because it is what helped me to cope when I felt overwhelmed and alone as a freshman at Princeton. I reminded myself over and over that, if I were stuck on a desert island, I would want to be with other smart and motivated and hopefully good-hearted people because that would give us our best chances to survive and thrive.

In my teaching life, I think of this as Otter Nation. Our motto is, Hold hands and stick together. When sea otters sleep, they hold hands so they don't drift apart from their tribe. The same is true of us math teachers. We hold hands through the #MTBoS and through #educolor , through Twitter and blogs, and through every social media-based method we can find.

We hold our students in our hearts and try to give them every possible support and advantage we can provide.

For me, a part of that involves helping them to become metacognitively aware and and self-reflective about what they are experiencing and how they can cope with it, how they are brave and well-equipped and the advantages of holding hands and sticking together.

Our Latin Club has hoodies with one of my favorite lines from Virgil's Aeneid emblazoned on the back: forsan et haec olim meminisse iuvabit, which I would loosely translate as, "Perhaps some day we will laugh about this."   This is pretty much where many of my students — and especially my students of color — find themselves at the start of Week 3 too. At this point in Aeneid I, Aeneas and his troops have been driven from their homeland in Troy and find themselves on storm-tossed seas, wondering how they are going to survive.

A growth mindset, and some desert island thinking, along with Talking Points about it, are the best support I can offer them.

Saturday, August 20, 2016

Week 1 - "very much like being shot out of a cannon" #MTBoSBlaugust

Week 1 is in the can and I wanted to blog one of my best ideas from my first week back.

I should start out by saying that Week 1 was very much like being shot out of a cannon — much more so than usual. My classes this year are huge — 36, 37, or 38 students per class — but my room is the smallest in the school. So it took a lot of clever angling and arranging to ensure that we could have enough desks in the room and that everybody could more or less see from their given position in the room. I always mark on the floor with a Sharpie so that it's easier to put the desk clusters back into their optimal positions. Once upon a time, I would have considered this a form of vandalism, but now...? Hey, that's just common sense.

SEATING CHART MANAGEMENTI had a major conceptual breakthrough with seating charts this August. I always use OmniGraffle to set up my basic seating chart/management chart template that I use on my clipboard to take attendance and make notes. This year, it occurred to me: instead of using those stupid little name card tags with names to make a wall chart (which takes up an unreasonable amount of time), why not just make a board with a sheet of vinyl across to hold blank copies of the week's seating charts?

So now, I  can just print off two copies of my updated charts — one for my clipboard and one for the wall pockets. And voil√°! Easy to change the seats around.

The stapled blob of paper charts makes it super-easy to make notes about collaboration or mathematical successes during group work. It also makes it easy to enter attendance on the computer each class because I just look for the 0s or 1s. My scribbled comments make it easy to enter "Professionalism" scores or comments, or to send e-mails to students or families. My working copy of charts gets stapled together and placed on my clipboard. When the week is over, I archive it in a big binder.

[visualize the dazzling photo of my wall chart that will be posted here on Monday]

There's a lot more to say about Week 1, but I'm still recuperating. More soon!

Hey, Megan, Here's my
handouts hanger in situ!

Friday, August 12, 2016

#MTBoSBlaugust 4 - Dear Pam (starting the year and Exeter Math 1)

Dear Pam,

The closer the start of school gets, the more I realize I need to be wise about how I choose to introduce Exeter problems. I want to set my students up for success. They are an extremely diverse group, coming from all over the city and all over the world. They are new to our huge school and to high school. And my classes are huge. There is still load-balancing to do, but right now my Algebra 1 rosters are topping out at 42. This is a special problem in my room because we can only fit 36 desk-chairs in the room. Literally. So I have to practice basic trust that this is all getting worked out through placement and class balancing and other dark arts of administration.

But all of this thinking has made me realize that I need to rely on my proven, first-two-weeks training plan to get them ready for my course and to help accomplish all of the transitions I am expecting them to manage. Bottom line: I need to start by training them with Talking Points to ensure equity and access for all.

Accordingly, I've created a set of Day 1 Talking Points about ratios, units, and rates (actually for us it's Day 2 because Day 1 is a loss), which are the topics of M1:1#1. I want to use this to train them on norms and practices in my class so that we can begin our problem-based learning together. For me, the first two weeks are critical. They are when I train my students intensively on my rules and norms, using Talking Points and other practices. With 36 students per class, this is a must. Everybody has to get bought in and skilled at the structures so we can proceed.

I also need to work this beginning-of-the-year routine into my four-stage How People Learn groove. Talking Points is my go-to beginning-of-the-year Stage 1 task structure (initial encounter with new concepts and activation of old). Our debrief and shift into doing notes/INB organization is Stage 2 (initial provision of a new expert model).  We'll do some notes on ratios, units, and rates to review and organize our thinking, and then we'll do a little combo-plate Stage 3 / Stage 4 work (deliberate practice with metacognitive / transfer task) in the form of some opening Exeter problems.

I've changed my mind about ordering page 1. I'm going to have them do 1#1 first (easy application of ratios, units, and rates) as a "deliberate practice with metacognitive awareness" problem and then 1#5, the journey of a thousand miles problem (J-1000). What I like about this one as a first transfer task is that it pushes them to use their bodies and to involve their whole selves in the investigation. Once somebody in the class asks, "Hey, Dr. S — can we have a yardstick?" I know I've got them. ;)

From here we can move on to 1#2, which is really the heart of the intro to problem-based learning in my opinion.

But realistically, we won't get to this until Day 3 (which is my true Day 2). So I'm having to accept that this is the reality of where I can get them and when. I can tweak the rest of everything to make it fit the first two weeks, but this is my reality.

So Day 3 needs a new set of Talking Points to begin and then a handout for problem 1#2, the problem about counting non-stop by ones to one billion.

Friday, August 5, 2016

#MTBoSBlaugust 3 - The Bumper Car Theory of Anti-Racist Training for Teachers and Staff

This one is challenging to write because I want to honor all appropriate boundaries while inquiring into my own personal experience of the process.

This next week, during our whole-school PD on Wednesday, we are embarking on our first year of a multi-year program of anti-racist training for teachers and staff. Earlier this summer, I was one of 25 teachers and staff from our school who attended the initial training, and naturally, nothing went as planned. Does it ever? Heavy Sigh. So this morning, we did our reset and met about our plan to do this training with our whole school.

The enterprise of confronting privilege to teach and learn about privilege is daunting, and it is unavoidable that many people who encounter this work will quickly get rubbed raw. In some ways, that is by design. You can't remain comfortable while digging into uncomfortable territory. But at the same time, conceiving the work merely as a project of "disruption" dishonors the good will and long-term focus of individuals who have come together on their own out of their own deep-rooted belief that we need to do better, both for our students and for ourselves.

So you can see how it's a complicated and messy process to get started.

Your face here
What I am coming to understand about it all is this: in order to have courageous conversations about race, we need to learn how to see our own personal invisible beliefs. These are hard to see because they are by nature invisible. They are blind spots. For example, as a high-status, highly educated white teacher, I tend to feel confident in sharing my views publicly; but at the same time, I struggle to keep my passion and confidence from appearing as arrogance to those with different patterns of privilege. And I'm just one individual teacher in a very large faculty. I'm sure that other teachers struggle to notice their own patterns of privilege. Plus the nature of the dominant culture in our school is unusual and complex. So all in all, learning to see the individual and collective belief systems and blind spots is going to be a real challenge. We are going to need to spend a lot of time in a space of collective and individual not-knowing, together. And I fully expect that process to be uncomfortable.

What strikes me most is that this whole process is like being in racial identity bumper cars. Like at a carnival. We need to expect to be disturbed and surprised and confused as we discover how other drivers in their own identity bumper cars interpret and experience life from their own points of view, because everybody is so certain that their own personal bumper car point of view is clear-seeing and constructive and intentional. But every time the ride starts up, whenever you try to steer your own bumper car, you cannot help but crash into other people's bumper cars. So the process of investigation is complicated because there is no way to step outside of the bumper car bumping arena while the inquiry is ongoing.

From the 30,000-foot perspective, I can see that the bumper car system is designed to thwart objectivity. In their own bumper car ride, nobody is 100% in control of their own bumper car. We all have our own projections and privilege and beliefs that we project onto every other driver who crashes into us. If you consider how the bumper cars are designed, you may understand logically that the bumping is unavoidable. But after you've been in the arena for a little while, trying to steer your own car for a bit, it becomes hard not to take things personally. It becomes impossible to avoid lapsing into the belief that other drivers are intentionally crashing into you to push you off course.

I think this model is especially true when you've got a large room full of public school educators — smart, highly educated, open-hearted people who do what they do out of dedication to learning and to contributing to the common good. The moment you start to prod individual teachers into seeing how they benefit from various networks of privilege, things get painful. People shut down or break down. And I've never yet seen it handled well. In our culture, teaching is already pre-constructed as a "Wretched of the Earth"-level of profession. Poorly paid, micro-managed, and bullied by corporate reformers and unelected politicians. What could possibly go wrong when you try to confront public school teachers about privilege?

So I think it is going to take a certain gentleness, determination, and persistence to help a whole faculty to see how we as individuals benefit from different forms and degrees of privilege, both in our school culture and in our society. It is also going to take chocolate and a whole lot of radically appropriate self-care. I am hopeful in the long term that we will make progress, but I suspect that in the near term, things could get messy. Still, I remain optimistic and curious to see how things unfold.

Wednesday, August 3, 2016

#MTBoSBlaugust Post #2: Using Exeter Math 1 — down to brass tacks

I am shamelessly using the #MTBoSBlaugust challenge as a prod to organize my thinking about how to use Exeter Math 1 this year in Algebra 1.

Last year's Algebra 1 implementation was a mess. Not even a hot mess, just a mess. The materials from our district were inadequate and even with my enrichments, I found that the whole was not coherent enough or challenging enough for my students. It also started waaaaaaayyyy too slow out of the gate. And it didn't provide nearly as much work on modeling and sense-making as I wanted.

Other than that, it was fine.

So this year, to start with, here's my game plan for reordering units for the fall semester.

Sequence of Units
  1. Mathematical Modeling and Problem-Solving (EQ: How can we use mathematics and logic to make sense of real-world situations?) 
  2. Equations, Proportions, and Number Lines (EQ: How can we use our existing algebra toolkit to solve equations involving proportions, variables, and absolute value?) 
  3. Lines and Linear Functions (EQ: How do lines and linear functions enable us to analyze and predict real-world phenomena?) 
  4. Functions and Functional Thinking (EQ: How can functions and functional thinking help us in our modeling work?) 
  5. Systems of Linear Equations (EQ: How can we use multiple equations to model real-world situations?) 
  6. Working With Exponents (EQ: What makes exponents such powerful tools?) 
Unit 1 is going to be my two-week mathematical modeling and problem-solving boot camp.

Begin At The Beginning: Unit 1
This is going to work because M1:1 – 8 cover all of the things I want to deal with as review topics anyway, but in a novel and challenging way: rates and units; "micro-" modeling (going from words to mathematics) and functional thinking; using the number line; modeling with non-standard rates, fractions, ratios, and proportions; distributive property and order of operations; "like" terms; and modeling with area and volume.

Meanwhile, the most important thing about the first two weeks is to really drive home my routines and norms and structures. That creates space to get to know students and do some formative assessment so I can make intentional groupings and establish the tone for the course.

This year, the material for the first 9 days will be the first eight pages of lightly adapted Exeter problems, i.e., replacing "Exeter" with my school's name and replacing all the kid names with character names of my own choosing (I tend to favor Batman and the character names from Sesame Street and Harry Potter).

Day 1 is a complete loss because (a) it's a ridiculously short period and (b) we are required to go over our syllabus (don't ask). But I think I will put M1:1#5, which I refer to as J-1000 (the "journey of a thousand miles" problem) on the opening slide as the intro task for students to do once they've found their seats. It's a little bit of math, but it sets the tone I want, which is that we get down to mathematical business in my classroom.

Day 2 is my true Day 1, so that will be my M1:1 day.

Background: Organizing Principles
For any day that I am using a page of Exeter problems, I want to organize the problems according to Bowen and Darryl's PCMI-based framework of Important Stuff / Interesting Stuff / Tough Stuff.

That way, I can also use their twin strategies of (a) intentional groupings (keep the speed demons away from the katamari) and (b) deliberately featuring katamari solutions and insights during whole-class discussion segments. For background on all of this, including PCMI, Bowen, and Darryl's group work strategies, first read Ben Blum-Smith's Lessons from Bowen and Darryl and then my post on Lessons from Lessons from Bowen and Darryl.

Any remaining problems left over from the day's classwork can be done as homework problems that night, and those can then be discussed during the next day's Home Enjoyment/Burning Questions segment. A beautiful thing.

Day 1: M1:1#2
Problem 1:#2 is a perfect rich task for starting off an Algebra 1 course. You can demand that students let go of their habits of learned helplessness and use whatever they know. You can encourage them with problem-solving process hints and without robbing them of the opportunity to do the thinking for themselves.

Everything else on page 1 is just review and activation of prior knowledge.

OK, that's all I've got for page 1.

Monday, August 1, 2016

Scan in your best worked examples #MTBoSBlaugust

This is going to be a short post because I am trying to take my own best advice:

Scan in your best worked examples.
Let me repeat that: Scan in your best worked examples. Scan in your best worked examples.

And if you are just joining us... today's advice is: scan in your best worked examples.

I spent about 2 months working through every problem on every page of Exeter Math 1. All 91 pages of problem sets. And finally, today is ScanFest 2016.

I first learned this advice from Sam Shah (of course), and it bears repeating. Your frazzled, middle-of-the-year self will thank you for it.

And now, back to scanning.

Thursday, June 23, 2016

Exeter Math 1 Reflection 3: A Course in Advanced Proportional Reasoning

This is the third in a who-knows-how-many-part-series I am doing on my experience and practice of doing and using Exeter Math 1 in my Algebra 1 classes. The three labels I am using for this series of posts are: Exeter Math 1, Algebra 1, and metacognition.

As I see it, there are two core developmental strands in Exeter Math 1 that are woven together throughout the course. One strand concerns advanced proportional reasoning. The other involves what I would characterize as Exeter's method of "micro-modeling"—an ongoing spiral of frequent, small, subtle modeling tasks that provide extensive both variety and depth of practice in modeling. Many variations are explored so that students get a lot of practice in making sense of similar and differing contexts.

What I love about this blend of proportional reasoning and micro-modeling is that it occurs at the intersection of advanced textual interpretation and advanced proportional reasoning. This means it is an immersive experience in relentless sense-making and meaning-making as students explore modeling. In this course, mathematical modeling is a full-contact sport. Having worked all the way through the entire course, I can see how it is going to develop great fluency and confidence in modeling for Algebra 1 students, regardless of where they are starting (assuming, of course, that they have the basic prerequisites for Algebra 1 success).

The opening problem sets are deceptively simple, although page 1 problem 2 (from here on out, I'm going to use the Exeter-style notation of 1#2 to mean "page 1 problem #2), would be a fantastic Day 1 in-class rich task that drops students right into a hard micro-modeling problem with whatever tools they have.

But other than 1#2, most of the problems in the first 7 pages are deceptively simple. They're clearly written to review prior knowledge and to establish individual and group norms of work, with the major themes being work on rates, distributive property, order of operations, functional thinking, notation, number line, negatives and opposites, #unitchat, fractions, reciprocals, and rational numbers. Major concept development focuses on distance = rate - time, distributive property, working with various kinds of graphs and graphical representations, and micro-modeling.

And then you arrive at 8#1, and BLAMMO.

This is what I'm thinking of when I talk about a truly rich task blast-off.

I'm not going to give away the punch line here, but this "box within a box" problem is an excellent example of what I mean when I say the course focuses on advanced proportional reasoning. The problem requires a very advanced analysis of many distinct moving parts, along with an ability to track back and explain your thinking. By my count, this problem requires the learner to navigate and articulate issues of area, volume, footprint, a difference of footprints, layering, and negative space. Perhaps you can see other ideas here as well.

So if you're just getting started with Math 1 and wondering what the heck all the fuss is about, I encourage you to hang in there. 

I imagine that we will get to page 8 around the middle to end of the second week of school. And when we do, students should know that the fun is just beginning.  :)

Wednesday, June 22, 2016

Exeter Math 1 Reflection 2: Growing Up as a Mathematical Thinker

This is the second in a who-knows-how-many-part-series I am doing on my experience and practice of doing and using Exeter Math 1 in my Algebra 1 classes. The three labels I am using for this series of posts are: Exeter Math 1, Algebra 1, and metacognition.

First and foremost, Exeter Math 1 is a course in growing up as a mathematical thinker. It is about leveling the playing field between and among rising 9th grade students. 

Here's how I would would frame this journey for students: This is a course in developing your own mathematical self-reliance and resourcefulness as a learner. Your Essential Question is always: How can I exhaust everything I already know before I ask the teacher for help?

You already know an enormous amount of mathematics. In this class, you will need to step forward with that and be willing to attack problems with the best thinking you already have. You may not know everything, but you always know something, and since that something is the best thing you know, you show up and start there and give it everything you've got.

Then, when you have struggled as much as you can and as hard as you can—both by yourself and with your table group—and when you can no longer do anything more with what you've got, that is the appropriate point at which you can ask the teacher for help.

That is the best use of the teacher.

If you are passive in this work or mess around, you are going to suffer.

This course works at two levels. At the content level, we are going to do all of the usual content work in an Algebra 1 class. But the more important work we will do always takes place at a metacognitive level. It is designed to help you learn how you learn advanced mathematics.

OK, back to the teacher perspective.

I have a Post-It on the inside-front cover of my binder on which I wrote this:
Exeter discovery is about guided sequential flailing.
 I think this is true. The Exeter Math 1 path definitely involves guided, well-sequenced flailing. It also integrates continual spiraling designed to activate prior knowledge. The purpose is always to discover how much math you already know and can put into service with the problems that are directly in front of you.

There is mathematical content and metacognitive content on each page.

This leads to the issue of practice. In Exeter Math 1, there is a very specific theory of action in the practice problems that are given and in how they are used. There are none of the usual taking-up-time, too-easy practice problems. If students need extra practice on certain specific procedures, then you have to source them yourself from someplace else, such as (for us) the Holt Algebra 1 textbook.

But that is OK because at this point in my career, I can do that in my sleep.

The Exeter Math 1 approach to practice problems is to provide juicy, meaningful, gimmick-free practice problems that are (a) always of medium difficulty or above and (b) integrated with metacognitive reflection and discussion. For this reason, I would be inclined to use these inflection points in the curriculum as opportunities to use Talking Points to solidify conceptual understanding and to get students exploring and articulating the subtle misconceptions and potential pitfalls inherent in practice problems of a medium level of difficulty or above.

This is a very deep teaching idea to me — to keep practice problems at or above a medium level of difficulty and to have students explore and give voice to these subtleties as rich opportunities to make meaning in their work.

More thoughts coming soon.

Tuesday, June 21, 2016

First thoughts on completing Exeter Math 1

I just finished doing the 2010 edition of Math 1 (91pages) today.  Now begins the synthesizing and summarizing, which I will put into blog posts.

Math 1 is an Algebra 1 course that includes an incredibly deep coverage of proportional reasoning, in addition to the usual linear, quadratic, and exponential function topics.

I did Math 1 because most of our incoming students are incredibly bright and hard-working but they were not the math monsters in their middle schools. They have many of the typical middle school gaps, but they are much more sophisticated than most 9th grade Algebra 1 students. So the fact that Math 1 is a REALLY TOUGH course that dives very deep into Algebra 1 material is a great thing because it will give my students the deep rich course they deserve, even though they are placed into Algebra 1 based on their current skill level.

My Algebra 1 learners find themselves stuck in a ZPD no-man's-land: their ZPD as math learners is nowhere near their ZPD as readers. 

This presents a huge problem in the classroom. The math in CPM Algebra 1, for example, is rich and interesting, but the text is written for reluctant readers, discouraged readers, and English Language Learners, which is a huge turn-off for the vast majority of my enthusiastic and highly capable readers.

They feel insulted by it, and they are not shy about expressing these feelings. So my student population tends to dismiss it and resist it, even if they really do need to learn the content. This raises the question of how best to serve a population of learners who need to be challenged with greater nuance in textual interpretation and presentation in an introductory high school math class.

For all of these reasons, Math 1 is going to form a terrific problem-based “spine” for my Algebra 1 classes. The problem sequences are rich and interesting and engaging with sophisticated contexts, though they start from first principles. They develop to a point where even a mathematically sophisticated adult will find them very challenging.

To get started, I printed all pages of the problem sets, answer keys, and commentaries and created a binder with the following sections:

1 - Problem Sets plus glossary at the end

2 - Commentaries

3 - My Worked Solutions (for each page of problems, I have one stapled cluster of my worked solution pages)

4 - Answer Keys

I did all of my work on three-hole binder paper, with each new page from the problem set being its own stapled packet (or "blob") in the Worked Solutions section. 

Whatever problem set I was working on I would take out of the binder along with the relevant answer key page. That way I could work on binder paper without having to carry the whole damn binder around all the time. Much of this work was done on a lap desk with my iPhone/Desmos for graphing, my TI-83-plus (sorry, Eli) for computation, and my monkey pencil case including my mechanical pencil, my ProRadian protractor, and my colored pencils.

A lot of people have asked me why I started at the end and worked from the end forwards, about 10 pages at a time. The answer has two parts: (1) whenever I started from the beginning, I bogged down or got sidetracked; and (2)  it enabled me to see where we were going and where students would end up. By seeing where they would land at the end of the course, I could better understand how things worked from the beginning.

More thoughts coming soon, but I wanted to capture these ideas right away. If you have specific questions you'd like to discuss, please put them into the comments section below.

Saturday, June 18, 2016


In addition to our school vegetable garden and internal CSA (Community Supported Agriculture program), we have a coop with 8 laying hens. The AP Environmental Science classes work in the garden every Monday and harvest the produce that will be provided to faculty and staff subscribers during the school year. In this way, the garden program is self-funding, and students learn from experience exactly how difficult it is to grow food to meet expectations.

The chickens in the garden are our program's local celebrities. The kids love the chickens and the fresh, local chicken eggs are a very prized item for CSA subscribers.

The chickens, however, are a little bitchy. Bitchy and not too bright.

The reason we know this is that the faculty have to take care of the chickens once the school year is over and the kids are out of school.

Taking care of the garden is one thing. It's very meditative to come in to the garden once a week and pull weeds in the sunshine (or more likely, in the fog). Taking care of the chickens, on the other hand, can be a pain. As I said, they are bitchy and often peck at their caregivers as well as at each other. They knock over their feeding bins, poop in their communal water trough, and hide their eggs away from sight.

For this reason, it's always a good idea to tend our chickens on the buddy system.

So during our weeklong training session at school this week, I often brought my lunch down to the garden with my friend and colleague Cathy Christensen to hang out in the rare sunshine and assist with the chickens. One of the other AP Environmental Sciences teachers, Kathy Melvin, was already there between the rows of kale and chard, pulling weeds and listening to Bob Dylan.

Cathy and I ate our lunch, bravely fought off the chickens as we righted the water bins, filled up their food stocks, and gathered eggs.

When we were done, I asked Kathy M whether I should put my compostable fork into the compost pile with my plate and napkin. As a new composter, I am sensitive to the limits of low-tech composting systems, as opposed to our city's industrial strength composting system. I was trying to be mindful.

She said, "It won't decompose in there, but it will spark a good conversation."

That answer told me everything I need to remember about why I love our school and my colleagues there. As Michelangelo wrote to himself in his 80s (on a note found in his studio after his death), "Ancora imparo — I am still learning."

Friday, May 20, 2016

Because Anne

And because finals.  I never do these things, but Anne and Julie, you are both awesome.  So here goes...

A- Age: I'm better with age.

B- Biggest fear: that Donald Trump will be elected President

C- Current time:  6:43 a.m.

D- Drink you last had: Death Wish ™coffee

E- Every day starts with: Death Wish ™coffee

F- Favorite song:  Tweet Me Maybe

G- Ghosts, are they real? Definitely

H- Hometown: Cherry Hill, NJ

I- In love with: M.C. Escher

J- Jealous of:  Fawn

K- killed someone?:  No

L- Last time you cried?:  the other day from laughing so hard in the math office

M- Middle name: N/A

N- Number of siblings:   1

O- One wish:  truth and reconciliation

P- Person you last called: my mom

Q- Question you’re always asked:  Why "cheesemonkey"?

R- Reason to smile: Summer

S- Song last sang:  Let the Mystery Be

T- Time you woke up: 5 a.m.

U- Underwear color:  beige

V- Vacation destination:  anywhere the people I love are

W- Worst habit:  self-neglect

Y- Your favorite food:  pasta

X- X-Rays you’ve had: Teeth

Z- Zodiac sign: Cancer

Wednesday, April 27, 2016

"De-tracking" Versus Mastery: Is This Our Dirtiest Little Secret...?

There has been so much heat and noise (and not very much light) on all sides of the so-called "de-tracking" debate, it has made me want to raise a question I have been thinking a lot about:
What is the difference between "tracking" (i.e., ability grouping, as in "high-," medium," or "low") and an SBG-style measure of mastery?
I ask because as someone who thinks about classroom instruction in a deeply Vygotskian way, I value the Zone of Proximal Development (ZPD) above almost all else in figuring out how to ensure that all my students receive meaningfully differentiated instruction.

But if there IS no reasonably common ZPD, there's no way I can see to differentiate — apart from simply allowing everybody to work at their own pace... in which case, what good am I in the room?

How do you make sense of this distinction?

Wednesday, April 6, 2016

Volume of a Pyramid: Proof by Play-Doh

This is the best idea I never had.

My colleague, Tom Chan, asked me in the Math Office this morning, "Where are you guys at?"

I told him, "We're on volume of a pyramid."

"Me too!" He's usually a pretty cool cucumber, so this caught me by surprise. He said, "We're doing proof of the volume formula by Play-Doh. Wait here a minute."

He dashed out and came back within a minute with a fist-sized cube made of three different colors of Play-Doh.

"Each table gets three little tubs (so three colors) and they have to make three identical pyramids that fit together into a cube. Then they can move on and do the next piece."

I was dumbfounded. The best I'd been able to do for today was to produce tiny, helpful diagram handouts to fit into our INBs.

But I'm bookmarking this for myself for next year by blogging it, and by giving full credit.

Thursday, March 24, 2016

Algebra 1 quadratics — which method and why

When kids demonstrate that they don't yet have a solid-enough fluency to move on to deliberate practice with metacognitive reflection, it's time to go back to the drawing board.

That's what happened this morning with my second-block Algebra 1 class.

So during third block, I went to the library and started fishing in the MTBoS Search Engine. I wanted a card sort activity or an idea for one.

It didn't take long to find out that Dane Ehlert and Geoff Krall had already come to the same conclusion independently — and that they had even done some of the work for me!

Everybody's kids are at different levels when you slam into a new topic. So it's great to be able to find the structure of an activity that you can easily adapt to fit your own students' actual depth of knowledge.

This is why, even though I love a lot of the Shell Centre activities, I often find that MTBoS adaptations (or my own) are best for the reality of my classroom. They've given us some fantastic models to use in our actual teaching and learning.



Monday, March 21, 2016

What to do when strong students struggle

Jessica Lahey just posted this column on the New York Times web site and I think it may be the most important read I have seen for parents of the kinds of students I teach:

All students encounter struggle. Even strong students struggle. And when this happens, parents often ask me how they can make their child care more about doing better in math.

The only answer I know — the only answer I trust — is that you have to be willing to allow them to struggle.

Only then can they truly own their own success.

If they don't own their own failure, then they can't own their own success.

The eminent child psychologist Rudolf Dreikurs wrote about this more than 50 years ago, and it is as true now as it was then. You have to step back and let them own it. Dreikurs called this the practice of using natural and logical consequences. If the child doesn't own the problem, then s/he cannot own the solution.

I also love Dr. Charlotte Kasl's framing of this. She calls it the "Good luck with that!" response. I have seen this approach be very successful with students who have internalized a kind of passivity or learned helplessness that drives adults crazy. They have learned how to get adults to rescue them.

I think of this not as "tough love" or "grit" or a growth mindset. I think this is about the practice of maintaining — and helping adolescents learn how to maintain — strong, healthy boundaries.

Friday, March 18, 2016

Let the kids teach themselves how to do Talking Points

In thinking about the ways in which I try to push authority downward into student groups, I have been searching for ways to get my students to teach themselves about how to do Talking Points.

scene from the forthcoming Harry Potter and the Chapter on Inequalities
from the forthcoming mathematical
blockbuster, Harry Potter and the 
Chapter on Inequalities
So far, this has been by far my most successful method.

I have written "deleted scenes" from unmade (or yet-to-be-made) movies to introduce new concepts by having kids do a readers' theater activity instead of lecturing. So, I thought, why not do the same thing for Talking Points?

The results have been much better than I expected. Because all the voices and rules come to them through their own voices, they seem much more bought into the guidelines. They also act as their own enforcers of norms, rather than my having to circulate around the room constantly on the lookout for infractions.

So here is a link to my deleted scene for having kids teach themselves about Talking Points.

I will also be using this scene in my NCTM workshop on Talking Points next month.

Let me know what you think!

Saturday, February 6, 2016

Lessons from "Lessons from Bowen and Darryl"

I had the beginnings of a blinding insight this week that I wanted to write down and think about. It all started with Ben Blum-Smith’s blog post about what he learned from Bowen and Darryl’s master class at JMM earlier this year. He wrote up his takeaways and it sparked many great Twitter conversations.

I wanted to write about what *I* took away from his post and what I have learned from subsequent Twitter conversations with Bowen and Darryl.

My burning question this week that I posed for them after reading Ben’s post was, how do they manage the mixture of “speed demons” and “katamari” in their class work. This distinction between speed demons and katamari was really obvious once I read about it; yet, it remains the dirty little secret of math pedagogy — and the unspoken, problematic truth of group work strategies.

Both of these approaches—the “speed demon” approach and the “katamari” approach —  are highly personal, highly developed inner world views within mindsets. They are not labels or designations applied from without. They are basic “come-from” attitudes that arise from within.

I have my own truth of this distinction from my own experiences, and I believe that we all approach it with our own biases and theoretical frameworks. I freely admit that I speak as a katamari with a lifetime of bad experiences in mathematical group work, both with fellow students and with professional colleagues. In mathematics and in group work, I find that I still experience these hidden assumptions over and over again: what I perceive as the tyranny of the speed demons and my own resigned sense of hopelessness that my tortoise-like katamari learning style — including my tortoise-shell defense mechanisms against the feelings of rage and powerlessness and inner worthlessness as a math learner I experience whenever I am asked to do mathematical learning in a group with others.
[EDITORIAL NOTE: Please don’t worry that you need to rescue me. Or that I need you to rescue me. I don’t and you don’t. These are only thoughts flowing down the river of mind, and I have learned how to notice them and work with them through many years of self-noticing, meditation, inner development work, and therapy. I’m not held back by them, and I don’t need to be reassured about them. But the reality of inner development is that those deep-rooted holes and feelings don’t go away. We just learn how to notice them and work with them more skillfully and artfully so we can continue under all circumstances. Over time, they lose their power. They just become the voice of monkey mind yakking away in the background. And I have learned how to work with that noise in the background and to tune it out].
For math learners with a high degree of natural aptitude and curiosity, the speed demon world view is a very natural attitude to develop. You love math, you want more of it, and you are totally and completely voracious. I see this in many of my students. They want to devour math. They’re not just hungry, they are  completely driven by their appetite for math. More more more. Faster faster faster. Nom nom nom. They find math delicious, and they want to eat all they can as fast as they can. They want to climb every mountain they encounter. The higher, the better. More mountains, please. More math.

They’re joyful — they’re not intentionally being aggressive in the classroom. But they are children. They don’t have a lot of self-control or self-regulation skills, and they’re adolescents, so they don’t have a lot of awareness of how others are feeling in the moment. That is why they have to be managed and fed in the classroom learning process.

Meanwhile, the katamari in my classes are experiencing things quite differently. But first, we should mention that there is a HUGE range of ability, interest, and aptitude among the katamari. In fact, the population of katamari is where the greatest range of learners exists. But they are distinguished by their katamari worldview, which arises in binary opposition to the speed demon worldview. They work through things at their own pace and they alternate between individual think time moments and collaboration. And they build as patiently as they can on what they figure out.

Bowen and Darryl propose a revolutionary approach to managing the mixture of speed demons and katamari within the problem-based, group-work-centric math class. They divide classroom time into “doing” segments and whole-class “discussing” segments.

Here is my summary understanding of the three key parts of their deployment strategy for the problem-based learning experience.
1. Problem sets are designed to be a treasure map — but the essential treasure in any day’s work is located within the “Important Stuff” initial section
2. Groupings must facilitate individual discovery together — and they must eliminate all social and emotional obstacles to including *everybody* in the process of discovery See #1.
3. Whole-class discussion segments exist only to feature the discoveries that katamari have made by slowing down and really noticing the often subtle and deep mathematics that can be noticed through careful work
Some elaboration on all of this:


The daily problem set is designed as a treasure map, but the secret is that all essential mathematics in any day’s work is located in the first section, not at the end of the problem set.

Interesting Stuff and Tough Stuff exist to provide nourishment for anyone who is ready to explore it. But it does NOT contain the keys to the kingdom. If speed demons wish to zoom ahead and tackle the “tough stuff” that is there to satisfy their appetite for zooming into zoomy, zoomy heights, then they are welcome to do so. Meanwhile, the katamari can find reassurance in the fact that their worldview and their approach is explicitly being valued.


This point is key. Our students are still adolescents, and they are starting from a place of little impulse control when it comes to their own self-interest. Since the only way to develop intrinsic motivation is through autonomy, mastery, and purpose, we need to tap into that rather than try to manipulate everybody into clamping down on their natural orientations.

Since what motivates people is a combination of autonomy, mastery, and purpose (see Dan Pink, Drive), our grouping strategies in the classroom HAVE TO support student autonomy. In other words, if speed demons believe they’ve gotta speed, then our groupings need to support their desire/need for speed.

It’s important to separate the speed demons from the katamari, but there is a crucial misunderstanding as to why. Many people believe that this places an undue burden on the speed demons, but that’s actually backwards from the truth. The reality is, mixing the groups during the “doing” segments actually places an unfair burden on the katamari. It requires them to allow the speed demons to dominate the learning process, to be the center of attention at all times, and to cheat them out of their understanding.

So for this reason, it’s important to let the speed demons go off in zoomy groups and zoom away. This is not the place where they are going to learn the social and emotional interpersonal skills they need because it’s NOT the place where they are receptive to these lessons. Instead, this is the place where we, the adults, need to create safe space for katamari to work at their own pace and to develop the learnings they need in order to move forward.

*This* is meaningful differentiation.

So during the doing segments, I am now going to let the speed demons zoom. In fact, I am going to set up my speed demons so they can go off and do their zoomy zoomy zoom investigations with the “Interesting Stuff” and the “Tough Stuff” in the daily problem sets.


This strategy allows for a wonderfully cross-pollinating atmosphere to arise in whole-class discussion segments. Since everyone has received what they individually needed during the “doing” segments, they are now free to be more open and receptive to what others experienced and discovered while they were lost in their own autonomous worldview.

They can also pay attention to what the instructors really want everybody in the room to experience.

What I value about this strategy is how it turns whole-class discussion segments into resonant, experiential learning sessions for all participants — whatever their starting-point orientation.

Over and over, speed demons are exposed to — and required to notice — the kinds of majestic mathematical discoveries that are possible when you relinquish your foundational belief that only faster can be better.

And katamari experience that “slow and steady” is not an inferior way to approach mathematics but rather, a powerful orientation and set of talents that can reveal mathematical depth and structure that are hidden to the naked speed demon eye.

It also strikes me that much of this is completely at odds with the narrow-minded, and often obtuse insistence in Complex Instruction that everybody always stay together on everything, working on the “same problem” at the “same time.”

I find this obtuse because I have seen how the richness of our human experience comes from coming together and bringing our whole selves to our collaboration — not by holding ourselves back and playing small to avoid making anybody else feel less empowered.

This is what I am trying to get my learners to understand about the value of collaborating with others. Speed demons are rewarded by the mountains they climb and the spectacular landscapes this allows them to experience. Katamari are rewarded by the dazzling richness and microscopic hidden structures they discover. When we bring these experiences together and allow ourselves to share our most powerful insights, that is when we discover the full spectrum of what it means to be mathematical and to be human.

Saturday, January 30, 2016

Algebra 1 Systems of Inequalities - Dan Wekselgreene's Ohio Jones & the Templo de los Dulces treasure map

Some of Dan Wekselgreene's early puzzles, lessons, and projects are truly love poems for Algebra 1 students. And I have loved his Ohio Jones and the Templo de los Dulces systems of inequalities puzzle since the first time I read about it, did it, and used it.

So there was never any question that I would use it with my Algebra 1 students this year. The only question was, how would I make it accessible to my blind student?

Susan Osterhaus of the Texas School for the Blind and Visually Impaired has been generous beyond words with her ideas for teaching math to blind students. Her web site is filled with ideas, best practices, and links to resources for making mathematics accessible to blind students. I cannot recommend it strongly enough.

Here is how I adapted this activity:

I typed out each of the three clues as a quote by a statue — i.e., Statue 1 says...

Then I plugged each quote into an online Braille Translator (I like and downloaded the Braille text file as an image. I copied and pasted each image file onto an Omni Graffle document (though you could also use Word or Pages) next to the regular text quote. That way the student and the paraprofessional aide could easily collaborate and share information.

This took three pages, but it worked.

I traced the basic map at the bottom of the page but without all the grid lines. This became the "map" for this part of the puzzle. 

Then I copied these pages onto capsule paper and ran them through the PIAF (Pictures In A Flash) machine to create a tactile worksheet with Braille and a raised map. The PIAF machine (affectionately known around the math office as "the toaster") takes the capsule paper with all its delicious black carbon-heavy areas and raises them to create a tactile graphic that can be read by a Braille-literate blind person.

After solving the system and figuring out the target region, my student used Wikki Stix on the map to make a graph.


I traced the "big picture" outline of the map to remove as much visual noise and clutter as possible from the main image.  I added Braille labels to indicate the start and the hint at the end of the map:

I scanned this file, printed it on capsule paper, and ran it through the PIAF machine. Again, the student worked on Braille graph paper, then transferred her results to her tactile treasure map using Wikki Stix.

For each "sector" of the map, my student used Braille graph paper and Wikki Stix while her classmates used pencil and the grid on the worksheet.

It was such a joy to see her as just another team member at her table, doing mathematics and solving a puzzle. It was even more exciting to see how her table mates appreciated her mathematical skills.

All in all, a successful experience in creating an inclusive classroom!

My reduced version of the Teacher Packet (including the worksheet and instructions) plus the Braille-ready package are all on the Math Teacher's Wiki.

Wikki Stix are available in a big box on Amazon or any kids' art supply store.

Monday, January 18, 2016

Algebra 1 Inequalities – A minor 'How People Learn' unit

Here's a perfectly imperfect model unit of how I use the How People Learn stages and cycles in a typical unit in my Algebra 1 class. I'm documenting this for myself, so anybody else who finds this useful is just icing on the cake! :)

All of the files I use are in this downloadable zip file on the Math Teacher Wiki:

     Algebra 1 Inequalities unit

Here's a rough overview of how this works.

How are algebraic inequalities related to our basic number sense concepts of "more than" and "less than," and how can we use this understanding to build a more generalizeable algebraic understanding?

STAGE 1 - hands-on introductory task

(1) Deleted scene (readers' theater activity):  groups read the deleted scene about how to do Talking Points, which also contains a review of number sense concepts of more than and less than.

(2) Talking Points - set #1 — more than and less than: what does your group think?
Students follow the protocol they have just learned and do set #1 of Talking Points that active prior knowledge about which is more and which is less, given two sets of quantities.

STAGE 2 - initial provision of an expert model

Each day is different, but usually I spend about 10-15 minutes working with the whole class to "do some notes" (combination of mini-lecture and note-taking and modeling). This is often the last thing we do in the class period.

STAGE 3 - deliberate practice with metacognitive self-monitoring

Practice happens at both a macro- and a micro- level. On the micro- level, each day's homework (which is completely distinct from the day's classwork) gives a chance to practice and review. Then the first activity of the next day's class is comparing answers in your table groups and answering all questions that groups or students are able to answer for each other or for themselves. I take only Burning Questions (like only taking group questions during Complex Instruction problem-solving tasks).

At the macro-level, we are building toward two big days of Speed Dating, which is differentiated deliberate practice with metacognitive self-monitoring and peer tutoring or reteaching as needed.

We cycle back through all of these for several days, as you can see, with a new set of Talking Points each day that students have to work through and puzzle over. Each day's Talking Points build in a new piece of knowledge that is in students' Zone of Proximal Development so that they can encounter it, wrestle with it, and formalize their understanding of it. Then they get the nightly chance to practice some more.

This is a highly Vygotskian model of learning.

Eventually we need to introduce a new concept into our exploration of greater than and less than. I call this concept the concept of "betweenness." We do this through (5) another deleted scene.

Once again, we are investigating numbers and quantities, but we are extending our investigation to more abstract conceptions of quantities. We go back to Talking Points. We do some problem-solving. We struggle together. We organize our learning.

STAGE 4 - transfer task

I don't have a particularly great transfer task yet for this unit. That's why it's such a good one to use as an introduction or reintroduction to my Talking Points norms and practices.

If you have a super-terrific transfer task for Algebra 1-level linear inequalities (not yet at systems), I'm all ears. Please let us know about it in the comments section.

Let me know what you think if you try any of this!
A Postscript —
Today was the kind of day that makes all this work worth it. To review for tomorrow's unit test, we did group whiteboarding of some pretty hard problems. And even though many kids were still stumped by some of the harder problems, they felt excited. They were understanding it.

And doing it.

Doing math.

Now that is the kind of thing that makes it worth waking up at 5 in the morning five days a week for ten months of the year straight at near-poverty wages. :)