## 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.
 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 voilÃ¡! 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!